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Estimation of dry matter and nitrogen removal by the logistic equation

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Title:
Estimation of dry matter and nitrogen removal by the logistic equation
Creator:
Wilson, Denise Marie, 1970-
Publication Date:
Language:
English
Physical Description:
xxiii, 269 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Analysis of variance ( jstor )
Corn ( jstor )
Logistics ( jstor )
Modeling ( jstor )
Nitrogen ( jstor )
Nutrients ( jstor )
Parametric models ( jstor )
Scatter plots ( jstor )
Soils ( jstor )
Statistical discrepancies ( jstor )
Agricultural and Biological Engineering thesis, Ph. D
Dissertations, Academic -- Agricultural and Biological Engineering -- UF
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1995.
Bibliography:
Includes bibliographical references (leaves 265-268).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Denise Marie Wilson.

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University of Florida
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34345131 ( OCLC )

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ESTIMATION OF DRY MATTER AND NITROGEN REMOVAL
BY THE LOGISTIC EQUATION















By


DENISE MARIE WILSON















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1995



i"vERSITy OF FLORIDA LIBePARlS

-� ,... - .� CT-A *!y, "-3 - -.* - *-***

























Copyright 1995

by

Denise M. Wilson

























I dedicate this dissertation to my parents, Robert and Marian Shelton, my second-

parents, Brooks and Carol Wilson, and especially to my husband, Russell.














ACKNOWLEDGMENTS

First, I would like to thank the many field plant-soil science researchers throughout

the world for having well-designed experiments resulting in great data. Developing and

maintaining a field plot experiment are no easy task and I wish to recognize and thank

those people.

I would like to thank my husband, Russell, for the many hours of brainstorming and

editing. He was a stabilizing influence during a swift three years of graduate work, and I thank

him for the encouragement he provided during my weak moments.

I would like to thank my parents, Robert and Marian Shelton, for the many sacrifices

that were done in order that I might be able to attend college in the first place. Furthermore, I

would like to thank them for instilling in me a sense of honor, a hard work ethic, and

determination that I could accomplish anything.

I would like to thank my major professor and advisor over the last five years, Dr. Allen

Overman. He instilled in me a sense of professionalism and always demanded high quality

work. I am thankful for the opportunity I was given the summer of 1991 to work with him,

since this work helped establish the foundation for my graduate work.

I would like to give my appreciation to the members of my committee, Dr. Larry

Bagnall, Dr. Stanley Wilkinson, Dr. Paul Chadik, and Dr. Frank Martin for giving their time. I

would like to give special thanks to Dr. Frank Martin for also serving as my committee chair on

my concurrent master's program in statistics. Your wisdom, guidance, and encouragement

were greatly appreciated.

I would also like to thank the National Science Foundation for selecting me as a fellow

for the period of August 1992 to August 1995. The stipend and cost of education allowance

iv










enabled me to concentrate solely on my studies. I considered it a high honor to have been

selected for this award out of the thousands of applicants in science and engineering. The

fellowship provided me with a freedom to plot my own course and research project.














































v














TABLE OF CONTENTS

page

ACKNOWLEDGMENTS ......................................................... ........................ iv

LIST OF TABLES ....... ......................... ............... vii

LIST O F FIG U R E S............................................................................... ............. xii

A B STR A C T .............................................................................................. xxii

CHAPTERS

1 IN TRO D U CTIO N ................... .................................... ........................... 1

2 LITERATURE REVIEW............................. ........................... 6

3 MATERIALS AND METHODS............................ ...................... 18

Analysis of D ata ........................ .............................. ................................ 18
Models to be Investigated................................................... 20
D ata Sets to be Investigated ............................................... .............. 21

4 RESULTS AND DISCUSSION........................... ....................... 44

Evaluation of the Simple Logistic Model ........................................ 44
Evaluation of the Extended Logistic Model .......................... ....... 46
Evaluation of the Extended Triple Logistic (NPK) Model........................... 78

5 SUMMARY AND CONCLUSIONS .................................................. 253

R E FE R EN C E S......................................................................................................... 265

BIOGRAPHICAL SKETCH ........................................................................ 269





vi














LIST OF TABLES

Table page
3-1 Common and Scientific Names of Grasses Studied .......................... ...... 26

3-2 Seasonal Dry Matter Yield of Coastal Bermudagrass and Pensacola Bahiagrass
Grown on Greenville Fine Sandy Loam at Thorsby, Alabama............................. 27

3-3 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Dallisgrass Grown at Baton Rouge, Louisiana ...................... ...... .......... 28

3-4 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bermudagrass Grown at Thorsby, Alabama over 3.0 and 4.5 Week Harvest
Intervals (196 1)............................................................................... ................ 29

3-5 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Dallisgrass Grown in Maryland.......................... ... ............................ 30

3-6 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bermudagrass Grown at Tifton, Georgia..................... ................... ................ 31

3-7 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Perennial Ryegrass Grown in England with a Different Number of Harvests
over the Season.. .......................................................... .................................... 32

3-8 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bermudagrass Grown at Fayetteville, Arkansas over Three Years, Irrigated and
N on-irrigated. ....................................................... ...................................... 33

3-9 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Tall Fescue Grown at Fayetteville, Arkansas over Three Seasons, Irrigated and
N on-irrigated. ...................................................... ..................................... 34

3-10 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bahiagrass and Bermudagrass Grown at Eagle Lake, Texas ............................... 35

3-11 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Grain and Total Plant of Corn Grown on Dothan and Goldsboro Soils at Clayton
and Kinston, North Carolina, Respectively. ..................................... ... ......... 36
vii








3-12 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Grain and Total Plant of Corn Grown on Portsmouth Soil at Plymouth, North
C arolina. ..................................................................... .............. 37

3-13 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bahiagrass Grown on Entisol and Spodosol Soils in Florida............................... 38

3-14 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Ryegrass Grown in England. ..................................................... .............. 39

3-15 Seasonal Dry Matter Yields and Nutrient Response to Applied Nutrients for
Gator Rye at Tifton, Georgia. ........................ ............... . 43

4-1 Analysis of Variance of Model Parameters Used to Describe Coastal
Bermudagrass Yield Response to Nitrogen at Thorsby, Alabama, 1957-
1959. .................... . ...... ................. 81

4-2 Analysis of Variance of Model Parameters Used to Describe Pensacola
Bahiagrass Yield Response to Nitrogen at Thorsby, Alabama, 1957-1959 .... 82

4-3 Analysis of Variance of Model Parameters Used to Describe Coastal
Bermudagrass and Pensacola Bahiagrass Yield Response to Nitrogen at
Thorsby, Alabama, 1957-1959. ............................................ ............. 83

4-4 Error Analysis for Model Parameters of Coastal Bermudagrass and
Pensacola Bahiagrass Grown at Thorsby, Alabama...................................... 84

4-5 Seasonal Dry Matter Yield for Coastal Bermudagrass and Pensacola
Bahiagrass Grown at Thorsby, Alabama Averaged over Years 1957-1959..... 85

4-6 Error Analysis for Model Parameters on Averaged Dry Matter Yield of
Coastal Bermudagrass and Pensacola Bahiagrass Grown at Thorsby,
A lab am a .................. . ..... ............. ......... .. ......... ...................................... 8 6

4-7 Analysis of Variance of Model Parameters Used to Describe Dallisgrass
Grown at Baton Rouge, LA. ........................................ ..................... 87

4-8 Error Analysis for Model Parameters of Dallisgrass Grown at Baton Rouge,
LA. ...... ........................................................ ...... 88

4-9 Analysis of Variance of Model Parameters Used to Describe Bermudagrass
Grown at Thorsby, AL with Two Clipping Intervals.................................. 89



viii








4-10 Error Analysis of Model Parameters of Bermudagrass Grown at Thorsby,
A L . ............................................ .................................. 9 0

4-11 Analysis of VarianCe of Model Parameters for Bermudagrass Grown at
M aryland and Cut at Five Harvest Intervals................................ ................... 91

4-12 Error Analysis for Model Parameters of Bermudagrass Grown at Maryland
and Cut at Five Harvest Intervals.......................... ........................ 92

4-13 Analysis of Variance on Model Parameters for Bermudagrass Grown at
Tifton, GA, over Two Years and Cut at Five Different Harvest Intervals...... 93

4-14 Error Analysis for Model Parameters of Bermudagrass Grown at Tifton, GA
over Two Years and Cut at Five Different Harvest Intervals....................... 94

4-15 Analysis of Variance of Model Parameters on Ryegrass Grown at England,
with Three Different Numbers of Cuttings over the Season for 1969 ............ 95

4-16 Error Analysis for Model Parameters of Ryegrass Grown at England, with
Three Different Numbers of Cuttings over the Season for 1969 ................... 96

4-17 Analysis of Variance for Bermudagrass over Three Seasons, Irrigated and
Non-irrigated, Grown at Fayetteville, AR. .................................. ...... ..... 97

4-18 Error Analysis for Model Parameters of Bermudagrass Grown at
Fayetteville, AR, over Three Years, with and without Irrigation ................... 98

4-19 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bermudagrass Grown at Fayetteville, AR. Averaged over Three Years......... 99

4-20 Analysis of Variance for Model Parameters for Bermudagrass Grown at
Fayetteville, AR, Averaged over Three Years, with and without Irrigation..... 100

4-21 Error Analysis for Model Parameters of Bermudagrass Grown at
Fayetteville, AR, Averaged over Three Years, with and without Irrigation..... 101

4-22 Analysis of Variance of Model Paramters for Tall Fescue Grown at
Fayetteville, AR, over Three Seasons, with and without Irrigation............... 102

4-23 Error Analysis for Model Parameters of Tall Fescue Grown at Fayetteville,
AR, over Three Seasons, with and without Irrigation ................................ 103

4-24 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Tall Fescue Grown at Fayetteville, AR, Averaged over Three Seasons.......... 104

ix








4-25 Analysis of Variance of Model Parameters for Tall Fescue Grown at
Fayetteville, AR, Averaged over Three Seasons, with and without Irrigation. 105

4-26 Error Analysis for Model Parameters of Tall Fescue Grown at Fayetteville,
AR, Averaged over Three Seasons, with and without Irrigation..................... 106

4-27 Analysis of Variance for Bermudagrass Grown at Eagle Lake, Texas, over
Tw o Y ears. ............................................. ..... .................................. 107

4-28 Analysis of Variance for Bahiagrass Grown at Eagle Lake, Texas, over Two
Years. ....... ........ . . . ................... .......... ................ 108

4-29 Analysis of Variance for Bermudagrass and Bahiagrass Grown at Eagle
Lake, Texas, over Tw o Y ears ...................................................................... 109

4-30 Averaged Seasonal Dry Matter Yield, Plant N Removal, and Plant N
Concentration for Bermudagrass and Bahiagrass Grown at Eagle Lake, TX.. 110

4-31 Analysis of Variance on Model Parameters for Bermudagrass and Bahiagrass
Grown at Eagle Lake, TX, Averaged to Estimate b and c Parameters ........... 111

4-32 Error Analysis for Model Parameters of Bermudagrass and Bahiagrass
Grown at Eagle Lake, TX Averaged over Years..................... .............. 112

4-33 Error Analysis for Model Parameters of Bermudagrass and Bahiagrass
Grown at Eagle Lake, TX over Two Years. .................................... 113

4-34 Analysis of Variance on Model Parameters for Corn Grown on Dothan
Sandy Loam at Clayton, NC, Both Grain and Total...................................... 114

4-35 Analysis of Variance on Model Parameters for Corn Grown on Goldsboro
Sandy Loam at Kinston, NC, Both Grain and Total................................... 115

4-36 Analysis of Variance on Model Parameters for Grain and Total Plant of Corn
Grown on Dothan and Goldsboro Sandy Loam at Clayton and Kinston, NC.. 116

4-37 Error Analysis of Model Parameters for Grain and Total Plant of Corn
Grown on Dothan and Goldsboro Sandy Loam at Clayton and Kinston, NC.. 117

4-38 Analysis of Variance on Model Parameters for Grain and Total Plant of Corn
Grown on Portsmouth Very Fine Sandy Loam at Plymouth, NC.................. 118

4-39 Error Analysis for Model Parameters for Grain and Total Plant of Corn
Grown on Portsmouth Very Fine Sandy Loam at Plymouth, NC................ 119

x








4-40 Analysis of Variance on Model Parameters for Bahiagrass Grown on Two
Soils: an Entisol and Spodosol at Williston and Gainesville, FL,
R espectively ............................................................................. .... . ...... .. 120

4-41 Error Analysis for Model Parameters for Bahiagrass Grown on Two Soils:
an Entisol and Spodosol at Williston and Gainesville, FL, Respectively ......... 121

4-42 Analysis of Variance on Model Parameters for Seasonal Dry Matter Yield
and Plant N Removal of Ryegrass Grown on 20 Different Sites in England.... 122

4-43 Error Analysis of Model Parameters for Seasonal Dry Matter Yield and
Plant N Removal of Ryegrass Grown on 20 Different Sites in England ......... 123

4-44 Summary of Model Parameters for Ryegrass in England................................ 126

4-45 Summary of Model Parameters, Standard Errors, and Relative Errors for the
Extended Triple Logistic (NPK) Model for Rye Grown at Tifton, GA........... 127

5-1 A Summary of the Ab Parameter for Various Studies .............................. 262

5-2 A Summary of c and N' Parameters from Various Studies.............................. 263


























xi














LIST OF FIGURES

Figure page
1-1 Response of dry matter yield, N removal, and N concentration for
bermudagrass as a function of applied N grown over two years at five
clipping intervals at Tifton, GA. ............................. ............................... 5

2-1 Response of linearized dry matter yield and plant N removal to applied N
for dallisgrass [Paspalum dilatalum Poir.] grown at Baton Rouge, LA. ....... 17

4-1 Response of seasonal dry matter yield for Coastal bermudagrass and
Pensacola bahiagrass to applied N at Thorsby, AL. ...................................... 128

4-2 Scatter plot of seasonal dry matter yield for Coastal bermudagrass and
Pensacola bahiagrass at Thorsby, AL. ................................................... 129

4-3 Residual plot of seasonal dry matter yield for Coastal bermudagrass and
Pensacola bahiagrass at Thorsby, AL. .................................................. 130

4-4 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for dallisgrass grown at Baton Rouge, LA.......... 131

4-5 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for dallisgrass grown at Baton Rouge, LA. ............................... 132

4-6 Dimensionless plot of dry matter and plant N concentration as a function
of plant N removal for dallisgrass grown at Baton Rouge, LA .................... 133

4-7 Scatter plot of dry matter yield for dallisgrass grown at Baton Rouge, LA... 134

4-8 Scatter plot of plant N removal for dallisgrass grown at Baton Rouge, LA... 135

4-9 Residual plot of dry matter yield for dallisgrass grown at Baton Rouge,
LA .................. ................................... .. ... . . ...... ..... 136

4-10 Residual plot of plant N removal for dallisgrass grown at Baton Rouge,
L A .......................... ..................... ......... ........... . 137



xii








4-11 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bermudagrass grown at Thorsby, AL and
cut at two harvest intervals. ... .......................................... 138

4-12 Seasonal dry matter yield and plant N concentration as a function of N
removal for bermudagrass grown at Thorsby, AL and cut at two harvest
intervals .......................................................... .................. .... ............ 139

4-13 Dimensionless plot of dry matter and plant N concentration as a function
of plant N removal for bermudagrass grown at Thorsby, AL and cut at
two harvest intervals........................................... .............. 140

4-14 Scatter plot of dry matter yield for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals.............................................................. 141

4-15 Scatter plot of plant N removal for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals........................................... ......... ......... 142

4-16 Residual plot of dry matter yield for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals........................................ ....... ......... 143

4-17 Residual plot of plant N removal for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals................................................. 144

4-18 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bermudagrass grown at Maryland and cut
at five harvest intervals. ...................................... ................ 145

4-19 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bermudagrass grown at Maryland and cut at five harvest
intervals ................... .......... ........................................... ..................... 146

4-20 Dimensionless plot of dry matter and plant N concentration as a function
of plant N removal for bermudagrass grown at Maryland and cut at five
harvest intervals .................. .............................................. ....................... 147

4-21 Scatter plot of dry matter yield for bermudagrass grown at Maryland and
cut at five harvest intervals......................................................................... 148

4-22 Scatter plot of plant N removal for bermudagrass grown at Maryland and
cut at five harvest intervals............................................... ............... . 149

4-23 Residual plot of dry matter yield for bermudagrass grown at Maryland and
cut at five harvest intervals..................................................... ................. . 150

xiii








4-24 Residual plot of plant N removal for bermudagrass grown at Maryland
and cut at five harvest intervals. .................. ..................... 151

4-25 Estimated maximum dry matter yield and estimated maximum plant N
removal as a function of harvest interval for bermudagrass in Maryland........ 152

4-26 Seasonal dry matter yield, plant N removal, and plant N concentration for
a two week clipping interval over two years for bermudagrass grown at
Tifton, GA .............. .............................. .............................. 153

4-27 Seasonal dry matter yield, plant N removal, and plant N concentration for
a three week clipping interval over two years for bermudagrass grown at
Tifton, G A ..................................... ............. ........ .......... ............. ... 154

4-28 Seasonal dry matter yield, plant N removal, and plant N concentration for
a four week clipping interval over two years for bermudagrass grown at
T ift o n, G A ..... ....... ....... ............... ..................................... ................ 155

4-29 Seasonal dry matter yield, plant N removal, and plant N concentration for
a six week clipping interval over two years for bermudagrass grown at
Tifton, G A .......................... ............................... ......... ................ 156

4-30 Seasonal dry matter yield, plant N removal, and plant N concentration for
a eight week clipping interval over two years for bermudagrass grown at
Tifton, G A ............................... ............ .... .... ................ 157

4-31 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a two week clipping interval over two years for
bermudagrass grown at Tifton, GA ......................................... ................ 158

4-32 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a three week clipping interval over two years for
bermudagrass grown at Tifton, GA. ................ ........................ .............. 159

4-33 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a' four week clipping interval over two years for
bermudagrass grown at Tifton, GA. ................ ........................ .............. 160

4-34 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a six week clipping interval over two years for
bermudagrass grown at Tifton, GA. ........................................................ 161

4-35 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a eight week clipping interval over two years for
bermudagrass grown at Tifton, GA. .................................................. 162
xiv








4-36 Dimensionless plot of seasonal dry matter yield and seasonal plant N
concentration versus seasonal plant N removal ........................................... 163

4-37 Estimated maximums of seasonal dry matter yield and plant N removal as
a function of harvest interval for two years of bermudagrass grown at
T ifton, G A .............................................................. ................. ........ 164

4-38 Scatter plot of seasonal dry matter yield for bermudagrass grown over
two years at Tifton, GA and cut at five different harvest intervals............... 165

4-39 Scatter plot of seasonal plant N removal for bermudagrass grown over
two years at Tifton, GA and cut at five different harvest intervals............... 166

4-40 Residual plot of seasonal dry matter yield for bermudagrass grown over
two years at Tifton, GA and cut at five different harvest intervals............... 167

4-41 Residual plot of seasonal plant N removal for bermudagrass grown over
two years at Tifton, GA and cut at five different harvest intervals............... 168

4-42 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for three different number of cuttings over the
season for ryegrass grown at England. ................................................... 169

4-43 Seasonal dry matter yield and plant N removal as a function of plant N
concentration for three different number of cuttings over the season for
ryegrass grow n at England.......................................................................... 170

4-44 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for three different number of cuttings
over the season for ryegrass grown at England......................... ................. 171

4-45 Scatter plot of seasonal dry matter yield for ryegrass grown at England
and cut different times over the season....................... ........... 172

4-46 Scatter plot of seasonal plant N removal for ryegrass grown at England
and cut different times over the season.................................... ............... 173

4-47 Residual plot of seasonal dry matter yield for ryegrass grown at England
and cut different times over the season ....................................................... 174

4-48 Residual plot of seasonal plant N removal for ryegrass grown at England
and cut different times over the season................................................ 175

4-49 Estimated maximums of seasonal dry matter yield and plant N removal as
a function of average harvest interval for ryegrass grown at England........... 176
xv








4-50 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bermudagrass grown over three years at
Fayetteville, AR, with and without irrigation .......................... ................ 177

4-51 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bermudagrass grown over three years at Fayetteville, AR,
with and without irrigation........................................................ ......... ..... 178

4-52 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for bermudagrass grown over three years
at Fayetteville, AR, with and without irrigation ........................................ 179

4-53 Scatter plot of seasonal dry matter yield for bermudagrass grown over
three years at Fayetteville, AR, with and without irrigation........................... 180

4-54 Scatter plot of seasonal plant N removal for bermudagrass grown over
three years at Fayetteville, AR, with and without irrigation........................... 181

4-55 Residual plot of seasonal dry matter yield for bermudagrass grown over
three years at Fayetteville, AR, with and without irrigation......................... 182

4-56 Residual plot of seasonal plant N removal for bermudagrass grown over
three years at Fayetteville, AR, with and without irrigation......................... 183

4-57 Response of seasonal dry matter yield, plant N removal and plant N
concentration for bermudagrass averaged over three years at Fayetteville,
AR, with and without irrigation ........... ............................. ................. 184

4-58 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bermudagrass averaged over three years at Fayetteville,
AR, with and without irrigation......................... ........................ 185

4-59 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for bermudagrass averaged over three
years at Fayetteville, AR, with and without irrigation............................... 186

4-60 Scatter plot of seasonal dry matter yield for bermudagrass averaged over
three years at Fayetteville, AR, with and without irrigation........................... 187

4-61 Scatter plot of seasonal plant N removal for bermudagrass averaged over
three years at Fayetteville, AR, with and without irrigation......................... 188

4-62 Residual plot of seasonal dry matter yield for bermudagrass averaged over
three years at Fayetteville, AR, with and without irrigation........................ 189

xvi








4-63 Residual plot of seasonal plant N removal for bermudagrass averaged over
three years at Fayetteville, AR, with and without irrigation........................... 190

4-64 Response of seasonal dry matter yield, plant N removal and plant N
concentration for tall fescue grown over three years at Fayetteville, AR,
with and without irrigation ........................................... ....................... 191

4-65 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for tall fescue grown over three years at Fayetteville, AR, with
and without irrigation. ....................................... ............. 192

4-66 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for tall fescue grown over three years at
Fayetteville, AR, with and without irrigation......................................... 193

4-67 Scatter plot of seasonal dry matter yield for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation. ................................ 194

4-68 Scatter plot of seasonal plant N removal for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation. .................................. 195

4-69 Residual plot of seasonal dry matter yield for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation. .................................... 196

4-70 Residual plot of seasonal plant N removal for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation ................................. 197

4-71 Response of seasonal dry matter yield, plant N removal and plant N
concentration for tall fescue averaged over three years at Fayetteville, AR,
with and without irrigation............................................ .................. 198

4-72 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for tall fescue averaged over three years at Fayetteville, AR,
w ith and w ithout irrigation......................................................................... 199

4-73 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for tall fescue averaged over three years
at Fayetteville, AR, with and without irrigation ........................................ 200

4-74 Scatter plot of seasonal dry matter yield for tall fescue averaged over three
years at Fayetteville, AR, with and without irrigation. ............................... 201

4-75 Scatter plot of seasonal plant N removal for tall fescue averaged over
three years at Fayetteville, AR, with and without irrigation......................... 202

xvii








4-76 Residual plot of seasonal dry matter yield for tall fescue averaged over
three years at Fayetteville, AR, with and without irrigation........................... 203

4-77 Residual plot of seasonal plant N removal for tall fescue averaged over
three years at Fayetteville, AR, with and without irrigation........................... 204

4-78 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bermudagrass grown over two years at
Eagle Lake, TX. . ....................... ............................. ............................ 205

4-79 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bahiagrass grown over two years at Eagle
L ake, T X .......................................................................... ................. . .. 206

4-80 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bermudagrass grown over two years at Eagle Lake, TX........ 207

4-81 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bahiagrass grown over two years at Eagle Lake, TX.............. 208

4-82 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for bahiagrass and bermudagrass grown
over two years at Eagle Lake, TX...................... ..... .......... .............. 209

4-83 Scatter plot of seasonal dry matter yield for bahiagrass and bermudagrass
grown over two years at Eagle Lake, TX.................... ................ 210

4-84 Scatter plot of seasonal plant N removal for bahiagrass and bermudagrass
grown over two years at Eagle Lake, TX.................... ................ 211

4-85 Residual plot of seasonal dry matter yield for bahiagrass and bermudagrass
grown over two years at Eagle Lake, TX......................................... 212

4-86 Residual plot of seasonal plant N removal for bahiagrass and
bermudagrass grown over two years at Eagle Lake, TX............................. 213

4-87 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for grain and total plant of corn grown at
Clayton and Kinston, NC on Dothan and Goldsboro soils, respectively. ........ 214

4-88 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for grain and total plant of corn grown at Clayton and Kinston,
NC on Dothan and Goldsboro soils, respectively ....................................... 215


xviii








4-89 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for grain and total plant of corn grown at
Clayton and Kinston, NC on Dothan and Goldsboro soils, respectively......... 216

4-90 Scatter plot of seasonal dry matter yield for grain and total plant of corn
grown at Clayton and Kinston, NC on Dothan and Goldsboro soils,
respectively.......................................................................... ... ...... 217

4-91 Scatter plot of seasonal plant N removal for grain and total plant of corn
grown at Clayton and Kinston, NC on Dothan and Goldsboro soils,
respectively ........................................... .................... ............... 2 18

4-92 Residual plot of seasonal dry matter yield for grain and total plant of corn
grown at Clayton and Kinston, NC on Dothan and Goldsboro soils,
respectively ............................. .................... 219

4-93 Residual plot of seasonal plant N removal for grain and total plant of corn
grown at Clayton and Kinston, NC on Dothan and Goldsboro soils,
respectively ......................... ......................... 220

4-94 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for grain and total plant of corn grown at
Plym outh, N C .......................................................................... .......... 221

4-95 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for grain and total plant of corn grown at Plymouth, NC............. 222

4-96 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for grain and total plant of corn grown at
Plymouth, NC. .......................................................... ............... 223

4-97 Scatter plot of seasonal dry matter yield for grain and total plant of corn
grown at Plymouth, NC ...................................................... ............... 224

4-98 Scatter plot of seasonal plant N removal for grain and total plant of corn
grown at Plymouth, NC ...................................................... ............... 225

4-99 Residual plot of seasonal dry matter yield for grain and total plant of corn
grown at Plymouth, NC ......................................................... ............... 226

4-100 Residual plot of seasonal plant N removal for grain and total plant of corn
grown at Plymouth, NC ...................................................... ............... 227

4-101 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bahiagrass grown on two soils in Florida....... 228
xix








4-102 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bahiagrass grown on two soils in Florida ............................. 229

4-103 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for bahiagrass grown on two soils in
Florida. ......... ................... .................. .............. . ....................... 230

4-104 Scatter plot of seasonal dry matter yield for bahiagrass grown on two soils
in F lo rida. .. . . ............................... .. ....... ...... ......... ................................. 23 1

4-105 Scatter plot of seasonal plant N removal for bahiagrass grown on two
soils in Florida. ........................................ ..................... .. .......... .. 232

4-106 Residual plot of seasonal dry matter yield for bahiagrass grown on two
soils in Florida. ................................................................................... 233

4-107 Residual plot of seasonal plant N removal for bahiagrass grown on two
soils in Florida. ................................................. ............................... 234

4-108 Plot of the mean and �2 standard errors of Ar/A and Ab for twenty sites in
E n g land . ................ . ........................... ............. ....... ....... .................... 2 3 5

4-109 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for twenty sites in England......................... 236

4-110 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for twenty sites in England...................... 237

4-111 Response of seasonal dry matter, plant N removal, and plant N
concentration to applied N for rye grown at Tifton, GA and fixed
application rates of 40 and 74 kg/ha of P and K, respectively ..................... 238

4-112 Response of seasonal dry matter, plant P removal, and plant P
concentration to applied P for rye grown at Tifton, GA and fixed
application rates of 135 and 74 kg/ha ofN and K, respectively..................... 239

4-113 Response of seasonal dry matter, plant K removal, and plant K
concentration to applied K for rye grown at Tifton, GA and fixed
application rates of 135 and 40 kg/ha of N and P, respectively ................... 240

4-114 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for rye grown at Tifton, GA............................. ................. 241



xx








4-115 Seasonal dry matter yield and plant P concentration as a function of plant
P removal for rye grown at Tifton, GA. ................................................ 242

4-116 Seasonal dry matter yield and plant K concentration as a function of plant
K removal for rye grown at Tifton, GA................................ .................. 243

4-117 Dimensionless plot of seasonal dry matter yield and plant nutrient
concentration as a function of plant nutrient removal for rye grown at
Tifton, GA ....................................................... ....................... ... 244

4-118 Scatter plot of dry matter yield for rye grown at Tifton, GA...................... 245

4-119 Scatter plot of plant N removal for rye grown at Tifton, GA...................... 246

4-120 Scatter plot of plant P removal for rye grown at Tifton, GA......................... 247

4-121 Scatter plot of plant K removal for rye grown at Tifton, GA ...................... 248

4-122 Residual plot of dry matter yield for rye grown at Tifton, GA.................... 249

4-123 Residual plot of plant N removal for rye grown at Tifton, GA...................... 250

4-124 Residual plot of plant P removal for rye grown at Tifton, GA....................... 251

4-125 Residual plot of plant K removal for rye grown at Tifton, GA.................... 252

5-1 Sensitivity of logistic to the c parameter ................. .............. ................ 264




















xxi













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


ESTIMATION OF DRY MATTER AND NITROGEN REMOVAL
BY THE LOGISTIC EQUATION

By

Denise Marie Wilson

December 1995

Chairman: Allen R. Overman, Ph.D.
Major Department: Agricultural and Biological Engineering Department


Environmental issues are shaping today's world. Land application of treated

wastes and effluent is often done to manage the excess nutrients. Forage grasses are

grown to accomplish two objectives: remove the nutrients from the waste or effluent and

to produce food for livestock. Engineers, regulators, and managers use nutrient budgets

in the design of systems to reduce the chance of pollution. This analysis was performed to

establish a form of a model that would describe forage grass response to applied nutrients

and provide reasonable estimates.

Forms of the logistic equation were used to relate dry matter yield and plant N

removal to applied nitrogen for seven different forage grasses. Many different factors,

such as water availability, harvest interval, and plant partitioning, were investigated by

examining thirteen studies in the literature.

The logistic equation has shown a high correlation between grass yields and

applied nitrogen. It is the purpose of this analysis to show that the form of the logistic

xxii









equation adequately describes dry matter yield and plant N removal response to applied

nitrogen. The parameters of the models are determined using nonlinear regression.

Analysis of variance is used to search for simplification in the form of common parameter

values.

The results of the analysis showed that the logistic model is well behaved and

relatively simple to use. Harvest interval, water availability and plant partitioning can be

accounted for in the linear parameter. Dimensionless plots are a valuable tool in

evaluating the form of a model. The logistic equation exhibits symmetry suggesting

conservation of something. For this use of the equation, the total capacity of the system is

conserved. The assumptions of the extended model suggest a hyperbolic relationship

between dry matter yield and plant N removal. This relationship was observed throughout

the analysis. Although parameter values cannot be determined without prior

experimentation, estimates for the parameters can be assumed. If parameter values are

needed before any investigation, they should be underestimated in order to overdesign the

system. Furthermore, for large error in the b, b', and c parameters, the seasonal estimate is

not affected greatly (<15% error).



















xxiii













CHAPTER 1
INTRODUCTION


In today's society, there is an increasing focus on environmental issues. Concerns

are being raised about pollution in soil, water and air. One of the major pollution concerns

in Florida is nitrate and phosphate contamination in the aquifer and lakes, respectively.

Engineers are using nutrient budgets to ensure that excess nutrients are not applied and to

help control and eliminate contamination. Often nutrients are applied to forage grasses as

treated wastes or effluent. Many municipalities are using water reuse systems as a way to

remove high levels of nutrients from treated reclaimed water (Allhands et al., 1995). This

process is beneficial to two main parties, those who wish to clean the water and return it

to the aquifer and those who benefit from the addition of the nutrients to their system.

Since a large amount of the required nutrients is applied in the water, less fertilizer is

needed. The effluent is often applied to forage grasses, golf courses and lawns. A simple

procedure is needed to assist engineers in estimating and predicting for various forage

grasses the amount of nutrient removed and dry matter produced given a specific nutrient

application rate.

This research project primarily deals with various forage grasses and their nitrogen

response curves. Consider the response data presented in Figure 1-1. The data were

taken from a study by Prine and Burton (1956). Dry matter yield, plant N removal, and

plant N concentration was recorded for bermudagrass [Cynodon dactylon] grown at

Tifton, GA, for two years at five harvest intervals. As expected, there is a relationship

between yield, N removal, and N concentration to applied N, but what exactly is the

relationship? Linear, hyperbolic, quadratic, sigmoid? If one form of an equation would








2

adequately describe all the behavior, the task of identifying the specific relationship would

be greatly simplified. In theory, the form of the model should represent the physical

system beyond the range of data. The challenge is to identify patterns in the data (such as

Figure 1-1) and identify relationships to describe such data.

The objective of this project is to establish a model that provides reasonable

estimates of dry matter yield and nutrient removal given a nutrient application rate.

Studies from the literature will be used to document the fit of the model to numerous data

sets with varying factors. These estimates could be used by engineers, managers, and

regulators. The form of the equation should work regardless of forage or site. Water

availability and harvest interval should also be quantified in the model. Land application of

treated effluent and waste as irrigation for agricultural crops is becoming a prevalent

method of wastewater reuse. In these systems, the nitrogen response of the crop is

needed by engineers in the design process, since both the Florida Department of

Environmental Protection (FDEP) and the Environmental Protection Agency (EPA)

regulate wastewater application rates based upon nitrogen concentrations and the uptake

abilities of crops grown.

Many sources of data for this analysis can be found for various forage grasses and

locations around the world. This reservoir of information has different variables for the

crops studied including the following: applied nutrients (N, P, and K), water availability

(with/without irrigation and year to year variability), harvest interval, site specificity,

and plant partitioning. These different variables produce varying response curves.

Furthermore, the studies range from examining only dry matter yield response to applied

N to including N removal response to investigating the effects of nitrogen, phosphorus,

and potassium on dry matter yield and their respective removals. The logistic equation has

provided high correlation coefficients for estimation of growth and nutrient removal for








3

various grasses (Overman, 1990a; Reck, 1992; Overman el al., 1994a, 1994b). Because

of this, the form of the logistic will be studied to determine its broader applicability.

To better understand the logistic equation, consider the "rumor model". A room is

filled with 100 people and the doors are closed. No one is allowed to leave or come into

the room. The people in the room are milling around in random fashion. At the time the

doors are closed, five people know a certain rumor. The speed at which the rumor

spreads among the people present, or rate of transfer among the people, is dependent

upon how many people have heard it and how many have not heard it. The probability

that during an encounter with two people the rumor is spread is the probability that one

knows the rumor and one does not. These two events are mutually exclusive and

independent; that is, someone cannot know and not know the rumor at the same time. As

a result, their joint probability is the product of their individual probabilities. The

probability that the rumor is spread is the product of the probability a person knows and

the probability a person does not know. Since the number of people in the room is fixed,

the sum of those who know and do not know is constant as well. Furthermore, the

probability density function for the spreading rumor has a bell shaped distribution; that is,

the probability that the rumor is spread is the same when 5 people know and when 95

people know. The rate of transfer is equal but in opposite directions. At the beginning,

there are more people who have not heard the rumor; so the rate of transfer is increasing

until half of the people know. At that point the rate of transfer falls off since there are

fewer people to tell. The logistic equation applied to grass growth operates similarly. As

time was the independent variable for the rumor model, applied nitrogen is the

independent variable for the logistic crop growth model. Let the number of people who

know represent the amount of dry matter present at a particular moment in time. It

follows that the number of people who do not know represents the amount of dry matter







4

that still needs to be produced. As the number of people in the room was assumed

constant with the rumor model, the potential dry matter production of the grass is constant

as well: the sum of that already produced and that still to be produced. Also, the rate of

growth is a product of what has been produced and what is still to be produced, as the

rate of rumor transfer was a product of those who knew with those who did not know.

A nonlinear regression analysis of the variables involved will be used to determine

the values of the parameters for a logistic model of forage grass response to applied

nitrogen assuming different relationships among the model parameters. The results of this

analysis will be compared using analysis of variance for the different assumptions to

determine which is correct and if a simplification can be made. The result will be a form

of an equation for nutrient response to various forage grasses that will adequately describe

the relationship of dry matter yield, plant nutrient removal, and plant nutrient

concentration to applied nutrients. This model will assist engineers and planners in

estimating plant nutrient removal for various application rates.

The result of this analysis should be the production of a model for engineers and

managers to use when determining nutrient budgets under varying conditions. The model

will estimate seasonal totals and use a "black box approach". No attempt will be made to

model the physical mechanisms of the plant. Once many different studies have been

conducted, a survey of the parameters for various forage grasses might provide a useful

insight to the internal mechanisms of the plant, as well as connect the parameters to the

physical system, but this is not pursued in this study.








5



50 -------

e 40
*o * g r
S30

o 20









E 400










S20 -

15 -
10 -











5l
0'
0 I I I I
0 200 400 600 800 1000 1200

Applied Nitrogen, kg/ha


Figure 1-1 Response of dry matter yield, N removal, and N concentration for
bermudagrass as a function of applied N grown over two years at five
clipping intervals at Tifton, GA. Data from Prine and Burton (1956).
Symbols will be defined later in the text.














CHAPTER 2
LITERATURE REVIEW


Crop models have been used for many years and have many variations (Keen and

Spain, 1992; France and Thornley, 1984). Models have been used to describe the

relationship between various forage crop yields and other parameters including nitrogen

application, water availability, harvest interval, etc. These models have ranged from

simple polynomial models to varying degrees of nonlinear models. Jones et al. (1987)

have defined a model as a mathematical representation of a system, and modeling as the

process of developing that representation. Modeling is often confused with simulation.

Simulation includes the processes necessary for operationalizing the model, or solving the

model to mimic real system behavior (Jones et al., 1987). Before beginning a model or

simulation project the objectives of the project should be stated clearly. A clear definition

of the intended end-product and the intended users of the models that will be developed

should be included in the objective statement. In their book on mathematical models in

agriculture, France and Thornley (1984, p. 173) noted that,

There are three fairly distinct communities of people to whom a crop model or
simulator may be of value: first, farmers, including advisory and extension
services, who are primarily concerned with current production; second, applied
scientists, including agronomists and plant breeders, whose objective is to improve
the efficiency of production techniques, for the most part using current knowledge;
and third, research scientists, whose aim is to extend the bounds of present-day
knowledge. There is no way in which a single model can satisfy the differing
objectives of these three groups, which are, at least in part, incompatible.

Various mathematical models have been used to describe crop growth.

Polynomials are often used because of the simplicity in determining parameter values;

however, they have serious weaknesses. These weaknesses have been addressed in detail
6








7

(Freund and Littell, 1991). The exponential and Mitscherlich models have also been used

in describing crop growth (France and Thornley, 1984); however these models have

limitations as well. The exponential model suggests that at high levels of input, the

response approaches infinity, and at low levels approaches zero response. The

Mitscherlich model suggests that at low levels of input the response approaches negative

infinity, and at high levels approaches a maximum response. These two models describe

the response well at different extremes. A combination of the two could potentially

explain the response well over all ranges of input.

The logistic equation, first proposed by Verhulst and later popularized by Pearl

(Kingsland 1985), was first used as a population model. It has been used with high

correlation coefficients (R > 0.99) for nitrogen removal (uptake) and dry matter

production of various forage grasses (Reck, 1992; Overman et al., 1990a, 1990b, 1994a,

1994b; Overman and Blue, 1991). Overman (1995a) has developed the relationship of the

exponential and Mitscherlich models to the logistic, and a short summary of that

discussion will be included here. In differential form, the exponential model assumes that

the response of dry matter to applied N is proportional to the amount of dry matter

present. Furthermore, the exponential is suitable at low N and asymptotically approaches

zero along the negative N (reduced soil N) axis. In mathematical form,

lower N: dy/dN a y [2.1]

In differential form, the Mitscherlich model assumes that the response of dry matter to

applied N is proportional to the unfilled capacity, ym - y, of the system, where ym is the

maximum yield. As mentioned before, the Mitscherlich is suitable at high N and

asymptotically approaches a maximum yield along the positive N axis. In mathematical

form,

higher N: dy/dN a y, - y [2.2]

This leads us to assume a composite function








8

allN: dy/dN = k y (ym - y) [2.3]

This is the form of the logistic model where k is the N response coefficient. It is a
nonlinear first order differential equation. The logistic equation is well behaved; that is,

the function is continuous, smooth, and asymptotic at both ends of the range. Recall the

analogy to the rumor model. The product of the dry matter present (filled capacity) and

that still to be produced (unfilled) is present in the differential form as well. Also, the total

capacity of the system (y + ym - y = yn) is assumed constant. Furthermore, as Overman

(1995a) has pointed out, the logistic equation reduces to the exponential model at lower N

and to the Mitscherlich model at higher N. If the logistic is normalized by defining new

variables D = y/A and = cN - b and expanded by Taylor series, a quadratic term is not

included, suggesting a parabola would not be appropriate. Furthermore, the logistic is

approximated by a linear function extremely well in the middle of the range.

The logistic model exhibits sigmoid behavior and has three parameters as shown

below
A
Y +eb-cN [2.4]

Parameters A and c are scaling coefficients for yield and applied N, respectively.

Parameter b describes the reference state at N = 0. In the context of this model,

parameters A and c are constrained to be positive, but parameter b can be either positive,

zero, or negative. Parameter b equal to zero suggests that there is enough background

level of nitrogen in the soil to reach half of the maximum yield. Parameter b less than zero

suggests that there is enough nitrogen in the soil before fertilization to reach more than

halfway up the response curve. The model can also be normalized as shown


y 1
yAN [2.5]
and linearized as shown+e
and linearized as shown








9
A
In(- -1) = b-cN [2.6]

To determine if the data follow a logistic curve, A/y - 1 vs. N can be plotted on semilog

paper to check for linearity. If the plot is linear, then the data can be described well by the

logistic. The parameters can be determined by one of two methods: regression on

linearized data or nonlinear regression on original data (Downey and Overman, 1988).

There are advantages to using both methods. The linearization method provides an easy

procedure to estimate the parameters with a hand calculator. By this method, an estimate

of A can be obtained by examining a plot of the y versus N on linear paper. The curve will

appear to approach a maximum. This maximum is the estimate of A. Several attempts

may be required to optimize A. Estimates of b and c then follow from linear regression of

In(A/y - 1) vs. N (Draper and Smith, 1981). An example of this is shown in Figure 2-1.

The data for this analysis are taken from a study of dallisgrass [Paspalum dilatatum Poir.]

grown at Baton Rouge, Louisiana (Robinson et al., 1988). Linearized dry matter yield

and plant nitrogen removal are plotted. The linear trend suggests that these two responses

can be described well by the logistic equation. The fact that the lines are parallel suggests

that the c parameter for both dry matter yield and plant nitrogen removal are the same.

The figure also suggests that the b parameter value for plant nitrogen yield is larger (more

positive) than the b value for dry matter yield. The nonlinear regression method requires a

computer program written to perform the regression and statistical inference and

diagnostic information. The regression can be conducted with SAS for the simple case.

As more complex cases are needed, SAS is not easily programmed to perform the

statistical analysis. In the nonlinear regression, parameters are estimated using least

squares (Bates and Watts, 1988; Ratkowsky, 1983). Unfortunately, only the A parameter

can be explicitly solved. The b and c parameters need to be solved implicitly. Second

order Newton-Raphson iteration is used (Adby and Dempster, 1974). For additional








10

comparison among the two methods, the reader is directed to Overman et al. (1990a) for

further details.

The logistic equation exhibits symmetry suggesting conservation of something

(energy, momentum, charge, spin, etc.) (Mehra, 1994 p. 132). If this is true, what could

be conserved in the system defined? As shown earlier, it is the total capacity of the

system, y,. In Pearl's work with the logistic equation he later expressed "unutilized

potentialities" as the "amount still unused or unexpended in the given universe (or area) of

actual or potential resources for the support of growth" (Kingsland, 1985, p. 67). In

relation to this work, the total yield capacity consisting of the filled, y, and unfilled

capacity, y - y, is what is conserved. But what is the total yield capacity? It is y + y - y

= ym = A, where A is assumed constant. Pearl further noted, "The rate of growth,

therefore, was proportional to two quantities: the existing population and the difference

between existing and limiting populations" (Kingsland, 1985, p. 68). This is also

demonstrated in the differential form, Equation [2.3].

The logistic equation can also be written as
A
( [2.7]
1 + e(N-N)N' [2.7]

where N1/2 = b/c = nitrogen value for half maximum yield, and

N' = 1/c = characteristic nitrogen.

Recall Figure 2-1. The intercept on the vertical axis is eb. On the horizontal axis (y/A =

0.5), N = N12. Pearl also noted, "Symmetry meant that the inflection point came at the

halfway point of the curve, and that the saturation population was exactly twice the

population at the point of inflection" (Kingsland, 1985, p. 69). This is demonstrated in the

above equation. It has been pointed out by Hosmer and Lemeshow (1989, p. 38) that the

most important coefficient in the logistic equation is the response coefficient c.

In this analysis, three models will be used: simple logistic, extended logistic, and

triple logistic (or NPK). The simple logistic model is given by








11
A
S= b-cN [2.8]
1+e
where y = seasonal dry matter yield, Mg/ha;

N = applied N, kg/ha;

A = maximum seasonal dry matter yield, Mg/ha;

b = intercept parameter for yield;

c = N response coefficient, ha/kg.

The extended model relates dry matter and plant N responses to applied N and is based

upon three postulates:

1. Seasonal dry matter yield follows logistic response to applied N.

2. Seasonal plant N removal follows logistic response to applied N.

3. The N response coefficients are the same for both.

Postulate 1 follows from work done by Overman and Wilkinson (1992). Postulates I and

2 are possibly true independently of one another, while Postulate 3 implies quantitative

coupling between dry matter and plant N accumulation. Three additional results derive as

a consequence of these postulates:

1. Plant N concentration response to applied N is described by a ratio of logistic

functions.

2. Seasonal dry matter yield and plant N removal are related by a hyperbolic

equation.

3. Plant N concentration and plant N removal are linearly related.

Postulate 1 is the simple logistic model. Postulate 2 can be written as

A'
N, b'-c'N [2.9]
1+e
where Nu = seasonal plant N removal, kg/ha;

A' = maximum seasonal plant N removal, kg/ha;








12

b' = intercept parameter (plant N removal);

c' = N response coefficient (plant N removal), ha/kg.

Now by using Postulate 3, the system is constrained by assuming c = c'. Figure 2-1

suggests this is a logical assumption. Plant N concentration is defined as the ratio of

Equation [2.9] to Equation [2.8]. Note that the plant N "concentration" has been defined

as the ratio of two extensive variables (y and Nu), and therefore is not truly an intensive

variable (like ionic concentration, temperature, pressure, etc.). By combining Equations

[2.8] and [2.9] with c = c' and defining Nc = Nu/Y, the response to applied N is defined by

b-cN
(l+e )
N= Ne ( + ebcN) [2.10]

where Nc = average plant N concentration, g/kg, and

Ncm= maximum plant N concentration, g/kg.

This equation is well behaved for b' > b. It approaches a minimum at low values of

applied N and approaches a maximum at high values of applied N. If b' < b, then Equation

[2.10] is no longer well behaved at low levels of applied N. Furthermore, Equations [2.8]

and [2.9] can be rearranged to give the hyperbolic equation


Y *N,,
y = K, N [2.11]
K'+N,

where the parameters Ym and K are given by


A A
Y -1 b-' -~A [2.12]
1-e 1-e


A' A'
and K'= b - [2.13]








13

where Ab = b' - b. In order for Ym and K' to be positive, b' must be greater than b. These

relationships can be reduced to dimensionless form by dividing the dry matter yield by its

estimated maximum and the plant N removal by its estimated maximum. After doing this,

the parameters of the hyperbolic relationship are defined as


Y 1 1
Y" -b [2.14]
A--eb-b' - e-Ab [2.14]
A 1-e 1-e


K' 1 1
and K b'b 1 [2.15]
A e '- le -e 1


It should be noted that the parameters used to describe the dimensionless relationship of

seasonal dry matter to plant N removal (two measurable quantities) are only dependent

upon the Ab. Dimensionless plots have been useful tools for engineers in many fields.

Dimensionless plots were used to develop and determine dimensionless numbers, such as

the Reynolds number in hydraulic flow. For example, James Clerk Maxwell used a

dimensionless plot in the 1860s to describe the distribution of molecular velocities in a gas

(Segre, 1984). The greatest value of a dimensionless plot is the ability to collapse data

sets with different ranges onto the same scale for comparison. This also aids in the search

for possible simplification. Equation [2.11] can be rearranged to the form

N= (K'/Y,) + (1/Y,) Nu [2.16]

This equation predicts a linear relationship between plant N concentration and plant N

removal.

The extended triple logistic (NPK) model is given by


A
S(l+eb--cN +)( +ebp-cpP )(l+ebI-cK [2.17]







14

Nt = '- )()(l -ck) [2.18]
N" (1 +eb'-N )( +eb-cP)(1 + ebk-qK


p - cN b'- --- [2.19]


Ak
K Ak [2.20]
(l+eb -,N)( l+eb-cP)(l+e bk'-cqK


where the subscripts n, p, and k refer to applied N, P, and K, respectively. This model

assumes that the response to each nutrient is logistic and is treated as independently
applied. In statistical terms, the independence assumption suggests the joint response is
the product of the marginal responses (Mood et al., 1963). Several characteristics should

be noted. With fixed levels of P and K, the NPK model reduces to the simple logistic
model with a modified maximum parameter A. This model will allow for evaluation of the

effect of combinations of N, P, and K on dry matter production. The amount of nutrient
required to reach half of maximum yield is given by N12 = b,/cn, P1a = bp/cp, and K1/2 =
bk/ck. Negative values of b mean that before application the soil contained more than

enough nutrient to achieve half of maximum yield.
Throughout this analysis a model is viewed as a simplification of reality. Occam's

razor will be applied to simplify to the essentials required for description. Box (1976, p.

792) suggested this approach in a paper by stating,

Since all models are wrong the scientist cannot obtain a "correct" one by excessive
elaboration. On the contrary following William of Occam he should seek an
economical description of natural phenomena. Just as the ability to devise simple
but evocative models is the signature of the great scientist so overelaboration and
overparameterization is often the mark of mediocrity.
Evaluation and validation of the models will be conducted by dimensionless, scatter and
residual plots. In the literature, validation is defined as the process by which a simulation








15

model is compared to field data not used previously in the development or calibration

process (Jones et al., 1987). It should be noted that this study is not simulation, but true

modeling: mathematical representation of a system. Furthermore, the purpose of

validation is to determine if the model is sufficiently accurate for its application as defined

by the objectives (Jones et al., 1987). This leads back to a point mentioned earlier. It is

essential that the objectives be stated clearly from the beginning. Jones et al. (1987, p. 16)

also noted that common sense should prevail in validation of models, and that "a model

cannot be validated, it can only be invalidated," a point frequently emphasized in statistics.

Dimensionless plots will be used to determine if the form of the model is adequate to

validate the model. It is more important to determine if the model estimates adequately:

Does the model capture the.essence of what is trying to be accomplished. The scatter and

residual plots will be used to evaluate how well the model estimated the data, by looking

for biases or trends in the residuals. Box (1979, p. 2) noted the difference between

estimating and validation in a paper by stating

.. two different kinds of inferential process . . . . The first, used in estimating
parameters from data conditional on the truth of some tentative model, is
appropriately called Estimation. The second, used in checking whether, in the
light of the data, any model of the kind proposed is plausible, has been aptly named
by Cuthbert Daniel Criticism. While estimation should . .. employ . .. likelihood,
criticism needs a different approach. In practice, it is often best done in a rather
informal way by examination of residuals. ...

Box, Hunter, and Hunter (1978, p. 552) also discussed the importance of residuals and

other visual displays in evaluating data, especially when the work has been done with a

computer by noting,

Without computers most of the work done on nonlinear models would not be
feasible. However, the more sophisticated the model and the more elaborate the
techniques employed, the more important it is to submit complicated analyses to
surveillance by data plots, residual plots, and other visual displays. The moder
computer can make the plots itself, but graphs need not only be made but also to
be carefully examined and thought about. The data analyst must "fondle" the data.








16
Hand plotting used to be one to the ways this came about. The original data, as
well as the various plots, whether made by hand or by the computer, should be
mulled over. The experimenter's imagination, intuition, subject-matter knowledge,
and experience must interact. This interaction will often lead to new ideas that
may, in turn, lead to further analysis or experimentation.

While the objective of some model projects is to describe the system in a

fundamental way, this project has used Box's approach. The decision to use the logistic

equation was not based upon some insight into the plant process, but rather as an equation

that would describe the response well. The more prevalent approach in the literature is to

use compartmental models such as EPIC, CREAMS, GLEAMS, DRAINMOD, etc.

These models attempt to simulate the growth of a plant (or field) from planting to harvest

by breaking the soil and plant system into small compartments. The compartments keep

track of important state variables as water and other nutrients move through the system.

Common time steps include minute, hour, days and weeks. A model is written for each

plant process, such as respiration, transpiration, reproduction, etc. Common inputs

include weather data, (solar radiation, rainfall, degree-days), and soil characteristics. Plant

specific parameters are also needed. Although a thorough attempt has been made to

explain the growth behavior, the seasonal yields are not estimated well. An example of

this is demonstrated in a paper by Williams et al. (1989) where the highest correlation

coefficient in the scatter plots was 0.89 and the average correlation coefficient was 0.67.

It is the opinion of the author that the geometry and processes involved for a single plant

is much too complicated to be solved mechanisticly, much less to try to extrapolate to the

larger system. We are faced with the same problem as with the Navier-Stokes equation in

porous media--there is no hope of describing the geometric flow paths in the soil and

plant. This project is an attempt to estimate the seasonal totals well and provide a low

input and simple approach to crop modeling.








17



10
1) Dry Matter Yield
[E N Removal









5 0.1 0
(2 0.1 --""c =0.0055





0.01






0.0001
0 200 400 600 800 1000
Applied Nitrogen, kg/ha


Figure 2-1 Response of linearized dry matter yield and plant N removal to applied N
for dallisgrass [Paspalum dilatatum Poir.] grown at Baton Rouge, LA.
Original data from Robinson et al. (1988). Data are linearized using Eq.
[2.6] and A values of 15.60 and 431, respectively.














CHAPTER 3
METHODS AND MATERIALS


Analysis of Data

Depending upon the nature of the data, one of three different models will be used.

It should be noted that the three models are directly related to one another because they

are based upon the same mathematical form. For data sets where only plant dry matter

yield is recorded, the simple logistic model, Equation [2.8], will be used to describe plant

yield response to applied nitrogen. For data sets where both dry matter yield and plant N

removal are recorded, the extended logistic model, Equations [2.8] and [2.9], will be used

to describe the response to applied nitrogen. Finally, when varying amounts of nitrogen,

phosphorus, and potassium are applied and dry matter yield and nutrient removals

recorded, the extended triple logistic (or NPK) model, Equations [2.17] through [2.20],

are used to describe the response. Water availability and harvest interval (or cutting

frequency) will be related to the linear model parameter. For all three models, the

parameters, A, b, and c, are estimated using nonlinear regression (including second-order

Newton-Raphson method) on the data to minimize the error, E, given by


E = -(y, -,)2 [3.1]

where

E = error sum of squares,

y, = measured yield or N removal,
, = estimated yield or N removal from model,

i = observation number.
18








19
Correlation coefficients will be used to measure the fit of the data to the model. The

correlation coefficient is given by



R = - v[3.2]

An adjusted correlation coefficient is given by


R =I- y-y)-p) [3.3]


where n = total number of data points used, and

p = number of parameters estimated.
This quantity adjusts the correlation coefficient by the number of parameters in the model.

In the case of simple regression, Rdj = R. Because the adjusted correlation coefficient
accounts for the number of parameters in the model, it essentially deflates the R value

hence providing for a better evaluation of the fit.

A program was written in Pascal to estimate the parameters given the data and first
estimates of the b and c parameters using nonlinear regression and Newton-Raphson

iteration. The b and c parameters can be better estimated by the linearization method as
discussed in the previous chapter; however, the author has chosen to use nonlinear

regression because inference upon the parameters is more straightforward. If the

linearization technique had been employed, then the inference and summary statistics

would have been based upon the linearized data and not the actual data. This analysis will
include evaluation and validation of the form of the logistic model, to determine if the
model adequately describes the behavior of the data regardless of crop or site. To answer
this question, various grasses were studied. A list of these grasses and their common and

scientific names is given in Table 3-1.








20
Models to be Investigated


Simple Logistic Model for Dry Matter Yield

Analysis of variance, scatter plots and residual plots were used to evaluate this

model. The analysis of variance tests if the model can be simplified due to common

parameters under different conditions (that is, year, harvest interval, water availability,

etc.). There are three basic modes. In mode 1, all the data were analyzed together over

the various management factors (yield, N removal, years, harvest interval, irrigation, etc.)

and a common A, b, and c are found for the combined data. In mode 2, the data were fit

separately for each specific situation (yield, N removal, year, harvest interval, irrigated,

etc.), requiring a different A, b, and c parameter for each situation. In mode 3, the data

were analyzed together again and a different A is fit for each situation, but a common b

and c are found for the entire set of data. These different modes were then compared by

analysis of variance. They are compared by using an F test. The hypothesis that is being

tested is that one of the modes (or models) describes the data better. It is basically

examining for possible simplification.


Extended Logistic Model for Dry Matter Yield and Nitrogen Removal

These data sets are analyzed in a similar manner to the simple logistic with two

additional modes. For mode 4, the data are analyzed together and an individual A and b

are fitted to each situation with a common c for all. Mode 5 involves fitting an individual

A for each situation, a separate b for dry matter yield and plant N removal, and a common

c for all. These two additional modes are compared with the previous three modes by

analysis of variance to determine which scenario describes the data best. In some cases
the statistics suggest that the c parameter is not common. Instead of blindly using the








21
statistics, we further compare the mean sum of squares (MSS) for the two modes in

question. If there is not a large increase in MSS by simplifying the model (requiring fewer

parameters), we will accept that there is a common c. By examination of the plotted data,

it appears this supposition is valid.

As stated previously, the extended model is based upon three postulates. One of

the consequences of the postulates was that dry matter yield and plant N removal are

hyperbolically related and plant N concentration and plant N removal are linearly related.

To test these results, dry matter yield and N concentration are plotted against plant N

removal. The lines describing these relationships are dependent only upon the Ab = (b' -

b). Furthermore, the data can be normalized by dividing by the appropriate estimated

maximums. If the assumption of common Ab is true, all the normalized data will fall on

one line in a dimensionless plot. This is one way of testing the adequacy of the model to

fit the data and testing the postulates.


Extended Triple Logistic (NPK) Model for Dry Matter Yield and Nitrogen Removal

This model is developed on the assumption that response of dry matter yield and

plant nutrient removal to applied N, P, and K individually follow the extended logistic

model. Overman and Wilkinson (1995) first discussed this model and applied it to a

complete factorial. Dimensionless plots, scatter plots and residual plots are used to

evaluate the model.








22
Data Sets to be Investigated


Thorsby. Alabama: Bahiagrass and Bermudagrass

This study is from the Auburn Agricultural Experiment Station. The results have

been previously reported in a station bulletin (Evans el al., 1961). The results of field

tests and grazing trials conducted in Alabama to show the response of bermudagrass and

bahiagrass to nitrogen and irrigation on specific soil types were reported. The yields of

the grasses grown on Greenville fine sandy loam, Thorsby, AL, over three years (1957-

1959) were used in this analysis. Four nitrogen levels were included: 0, 168, 336, and

672 kg/ha. The data are listed in Table 3-2.


Baton Rouge. Louisiana: Dallisgrass

The data for this analysis were acquired from a study by Robinson et al. (1988).

Six nitrogen levels were included: 0, 56, 112, 224, 448, and 896 kg/ha. The data are

listed in Table 3-3.


Thorsby, Alabama: Bermudagrass

The data for this analysis were taken from a study by Doss et al. (1966). Two

harvest intervals were studied: 3.0 and 4.5 weeks. Six nitrogen levels were included: 0,

224, 448, 672, 1344, and 2016 kg/ha. The data are listed in Table 3-4.


Maryland: Bermudagrass

The data for this analysis were acquired from a study by Decker et al. (1971).

Five harvest intervals were studied: 3.2, 3.6, 4.3, 5.5, and 7.7 weeks. Six nitrogen levels

were included: 0, 112, 224, 448, 672, and 896 kg/ha. The data are listed in Table 3-5.








23
Tifton. Georgia: Bermudagrass


This data set was taken from a study by Prine and Burton (1956). Data from two

years were utilized: 1953, a wet year, and 1954, a dry year. Five harvest intervals were

studied: 2, 3, 4, 6, and 8 weeks. Five nitrogen levels were included: 0, 112, 336, 672,

and 1010 kg/ha. The data are listed in Table 3-6.


England: Ryegrass


This data set was taken from a study by Reid (1978). Twenty-one nitrogen levels

were included: 0, 28, 56, 84, 112, 140, 168, 196, 224, 252, 280, 308, 336, 392, 448, 504,

560, 616, 672, 784, and 896 kg/ha. The ryegrass was harvested at different growth

stages, resulting in three different number of cuttings per season: 10, 5, 3. The length of

the season was 26 weeks. It should be noted that the different number of cuttings

represent variable harvest intervals, namely 2.6, 5.2, and 8.67 weeks. As a result, we

expect this variable effect to appear in the results. The data are listed in Table 3-7.


Fayetteville. Arkansas: Bermudagrass


This data set was acquired from a study by Huneycutt et al. (1988). The

bermudagrass was grown over three years, with and without irrigation. Six nitrogen levels

were included: 0, 112, 224, 336, 448, 560, and 672 kg/ha. The data are listed in Table 3-

8.


Fayetteville. Arkansas: Tall Fescue

This data set was taken from the same study as above (Huneycutt et al., 1988).

The tall fescue was grown over three seasons, with and without irrigation. Six nitrogen








24
levels were included: 0, 112, 224, 336, 448, 560, and 672 kg/ha. The data are listed in

Table 3-9.


Eagle Lake. Texas: Bahiagrass and Bermudagrass

This data set was acquired from a study by Evers (1984). Both grasses were

grown over two years (1979 and 1980), and five nitrogen levels were included: 0, 84,

168, 252, and 336 kg/ha. The data are listed in Table 3-10.


North Carolina: Corn

These field experiments were conducted with corn at three North Carolina

Research Stations: Central Crops Research Station, Clayton, NC, on Dothan loamy sand

soils (fine loamy, siliceous thermic, Plinthic Paleudults), Lower Coastal Plain Tobacco

Research Station, Kinston, NC, on Goldsboro sandy loam soils (fine loamy, siliceous

thermic, Aquic Paleudults), and Tidewater Research Station, Plymouth, NC, on

Portsmouth very fine sandy loam soils (fine loamy, mixed thermic, Typic Umbraquults)

(Kamprath, 1986). The experiments were conducted from 1981 through 1984. The corn

grown at Clayton received irrigation. Each year the experiment was conducted in a

different field and for this reason averages were used in this study. The data from the

Dothan and Goldsboro soils are listed in Table 3-11. The data from the Portsmouth soil

are listed in Table 3-12. For all three sites, five nitrogen levels were included: 0, 56, 112,

168, and 224 kg/ha.


Florida: Bahiagrass


The data from this analysis were drawn from Blue (1987). The bahiagrass was

grown on two soils, an Entisol (Astatula sand) near Williston, Florida, and a Spodosol








25
(Myakka fine sand) near Gainesville, Florida. Entisol is typically a dry soil and Spodosol

is typically wet. Five nitrogen levels were included: 0, 100, 200, 300, and 400 kg/ha.

The data are listed in Table 3-13.


England: Ryegrass


The data for this analysis were taken from a study by Morrison et al. (1980) and is

listed in Table 3-14. Twenty different sites in England were used to grow the ryegrass.

Six different nitrogen levels were included in the study at each site: 0, 150, 300, 450, 600,

and 750 kg/ha. The reader is directed to the report for further information about the site

characteristics and weather data..


Tifton. Georgia: Rye

The data for this analysis were drawn from a study by Walker and Morey (1962).

Six levels of nitrogen (0, 45, 90, 135, 180, and 225), phosphorous (0, 20, 40, 60, 80, and

100), and potassium (0, 37, 74, 111, 148, and 185) were investigated. The data are listed

in Table 3-15.








26
Table 3-1. Common and Scientific Names of Grasses Studied.
Common Name Scientific Name
Bahiagrass Paspalum notatum Flugge
Bermudagrass Cynodon dactylon (L.) Pers.
Corn Zea mays L.
Dallisgrass Paspalum dilatatum Poir.
Rye Secale cereale
Ryegrass Lolum perenne L.
Tall Fescue Festuca arundinacea








27
Table 3-2. Seasonal Dry Matter Yield of Coastal Bermudagrass and Pensacola Bahiagrass
Grown on Greenville Fine Sandy Loam at Thorsby, Alabama.
Applied Nitrogen, kg/ha
Species Irrigation Year 0 168 336 672
-------------------------Mg/ha--------------------
Bermuda No 1957 3.74 12.52 17.41 22.32
1958 4.70 11.38 17.44 21.56
1959 3.40 9.04 13.36 20.44

Yes 1957 3.49 11.70 18.32 21.08
1958 4.33 10.75 17.89 22.61
1959 4.76 11.94 19.42 24.43

Bahia No 1957 4.56 11.70 16.39 20.32
1958 3.63 10.39 16.85 22.23
1959 2.67 8.19 14.14 19.62

Yes 1957 4.31 10.35 16.81 23.26
1958 3.24 9.25 15.69 22.33
1959 3.82 9.52 16.33 22.43
Source: Data from Evans et al. (1961).








28
Table 3-3. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Dallisgrass Grown at Baton Rouge, Louisiana.
Applied Nitrogen Dry Matter Yield N Removal N Concentration
kg/ha Mg/ha kg/ha g/kg
0 5.33 77 15.7
56 6.56 103 16.2
112 7.97 129 17.0
224 10.53 194 18.9
448 13.21 305 23.4
896 15.34 417 27.5
Source: Data from Robinson et al. (1988).








29
Table 3-4. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bermudagrass Grown at Thorsby, Alabama over 3.0 and 4.5 Week Harvest
Intervals (1961).
Applied Dry Matter Yield N Removal N Concentration
Nitrogen Mg/ha kg/ha g/kg
kg/ha 3.0 weeks 4.5 weeks 3.0 weeks 4.5 weeks 3.0 weeks 4.5 weeks
0 2.95 3.60 53 55 18.1 15.4
224 9.70 11.95 230 220 23.7 18.4
448 14.70 17.20 406 387 27.6 22.5
672 16.70 18.50 501 466 30.0 25.2
1344 17.50 19.95 560 549 32.0 27.5
2016 17.60 19.20 597 568 33.9 29.6
Source: Data from Doss et al. (1966).








30
Table 3-5. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Dallisgrass Grown in Maryland.
Harvest
Interval Applied Nitrogen Dry Matter Yield N Removal N Concentration
weeks kg/ha Mg/ha kg/ha g/kg
3.2 0 1.29 21.5 16.7
112 4.37 90.5 20.7
224 8.72 225 25.8
448 13.53 409 30.2
672 14.07 461 32.8
896 14.36 497 34.6

3.6 0 1.24 19.2 15.5
112 4.84 86.6 17.9
224 9.04 198 21.9
448 12.94 371 28.7
672 13.94 450 32.3
896 13.78 480 34.8

4.3 0 1.09 15.7 14.4
112 5.19 88.7 17.1
224 9.81 197 20.1
448 14.36 393 27.4
672 15.33 445 29.0
896 14.93 451 30.2

5.5 0 1.55 20.6 13.3
112 7.75 122 15.8
224 12.96 238 18.4
448 16.71 369 22.1
672 17.07 418 24.5
896 17.07 485 28.4

7.7 0 1.65 17.7 10.7
112 9.37 125 13.3
224 14.55 215 14.8
448 18.23 354 19.4
672 18.89 399 21.1
896 18.37 421 22.9
Source: Data from Decker el al. (1971).








31
Table 3-6. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bermudagrass Grown at Tifton, Georgia.
Year Harvest Interval Applied Nitrogen, kg/ha
weeks 0 112 336 672 1010
----------------------Dry Matter Yield, Mg/ha-------------------
1953 2 2.33 5.96 11.76 17.43 19.71
3 3.33 8.91 13.64 19.24 20.47
4 2.71 9.86 17.65 21.68 23.61
6 4.35 12.77 21.79 28.11 30.11
8 5.64 13.66 22.38 27.93 29.30

1954 2 0.76 2.69 6.83 7.84 8.62
3 0.94 3.65 7.39 9.90 9.99
4 1.08 4.55 9.45 11.13 11.49
6 1.30 6.16 11.60 13.57 14.13
8 1.93 6.45 12.23 15.86 16.22

-------------------------N Removal, kg/ha---------------------
1953 2 37 130 327 582 721
3 51 184 363 579 682
4 40 176 431 590 739
6 53 158 392 621 738
8 62 184 372 545 624

1954 2 17 70 224 299 320
3 15 75 208 294 364
4 19 80 233 323 344
6 21 92 261 293 390
8 26 100 223 325 417

-----------------------N Concentration, g/kg---------------------
1953 2 16.0 21.8 27.8 33.4 36.6
3 15.4 20.6 26.6 30.1 33.3
4 14.8 17.9 24.4 27.2 31.3
6 12.1 12.4 18.0 22.1 24.5
8 11.0 13.5 16.6 19.5 21.3

1954 2 22.6 26.0 32.8 38.2 37.1
3 16.0 20.5 28.1 29.7 36.4
4 17.5 17.5 24.7 29.0 29.9
6 16.4 14.9 22.5 21.6 27.6
8 13.5 15.5 18.2 20.5 25.7
Source: Data from Prine and Burton (1956).








32
Table 3-7. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Perennial Ryegrass Grown in England with a Different Number of Harvests
over the Season.
Dry Matter Yield N Removal N Concentration
Applied Cuttings Cuttings Cuttings
Nitrogen 10 5 3 10 5 3 10 5 3
kg/ha Mg/ha kg/ha g/kg
0 0.75 1.81 2.65 18 34 38 24.3 18.2 14.6
28 1.12 2.30 4.08 27 48 64 24.0 18.2 15.4
56 1.58 3.28 4.88 40 59 67 25.0 18.2 13.8
84 2.27 4.00 5.17 59 75 77 26.1 18.9 15.0
112 2.85 5.58 5.97 77 104 83 26.9 18.6 13.9
140 3.12 5.65 6.90 85 102 96 27.4 18.2 13.9
168 3.75 7.13 8.06 106 136 120 28.2 19.2 15.0
196 4.50 7.61 8.58 131 149 136 29.1 19.5 15.8
224 4.73 8.49 9.59 142 174 155 30.1 20.5 16.2
252 5.30 9.18 9.36 162 192 168 30.2 20.8 17.9
280 6.38 9.30 10.83 200 195 197 31.4 21.0 18.2
308 6.72 10.32 10.90 222 235 213 33.0 22.7 19.5
336 6.67 11.41 11.92 221 261 237 33.0 22.9 19.8
392 7.95 11.36 11.90 283 290 274 35.7 25.6 22.9
448 8.31 11.84 12.01 301 310 280 36.3 26.2 23.4
504 8.75 12.24 12.46 320 350 302 38.4 28.6 24.3
560 8.82 12.45 12.38 349 362 325 39.7 29.0 26.2
616 8.91 12.24 12.90 354 373 347 39.7 30.4 26.9
672 9.17 12.01 11.65 381 386 330 41.4 32.0 28.3
784 9.18 11.95 11.97 379 397 349 41.3 33.0 29.1
896 8.86 11.86 11.96 376 413 370 42.4 34.9 31.0
Source: Data from Reid (1978).








33
Table 3-8. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bermudagrass Grown at Fayetteville, Arkansas over Three Years, Irrigated
and Non-irrigated.
Applied Nitrogen, kg/ha
Year 0 112 224 336 448 560 672
Non-Irrigated
Dry Matter Yield 1983 1.95 6.88 11.52 13.88 16.39 16.70 17.71
Mg/ha 1984 2.35 8.43 11.59 12.71 16.50 16.48 16.01
1985 1.70 7.20 9.37 15.42 19.70 18.70 19.41
Irrigated
1983 4.53 9.12 14.50 20.80 21.90 23.63 23.87
1984 4.17 11.52 14.17 18.31 22.42 23.67 24.34
1985 1.75 8.32 12.55 18.56 20.11 21.36 23.18

Non-Irrigated
N Removal 1983 27 109 210 262 354 358 419
kg/ha 1984 29 136 210 266 370 393 379
1985 21 105 178 294 416 407 484
Irrigated
1983 67 156 278 379 466 537 512
1984 61 186 252 366 402 451 561
1985 25 129 229 333 376 458 523

Non-Irrigated
N Concentration 1983 13.8 15.8 18.2 18.9 21.6 21.4 23.7
g/kg 1984 12.2 16.2 18.0 21.0 22.4 23.8 23.7
1985 12.5 14.6 19.0 19.0 21.1 21.8 25.0
Irrigated
1983 14.7 17.1 19.2 18.2 21.3 22.7 21.4
1984 14.7 16.2 17.8 20.0 17.9 19.0 23.0
1985 14.1 15.5 18.2 17.9 18.7 21.4 22.6
Source: Data from Huneycutt et al. (1988).








34
Table 3-9. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Tall Fescue Grown at Fayetteville, Arkansas over Three Seasons, Irrigated
and Non-irrigated.
Applied Nitrogen, kg/ha
Year 0 112 224 336 448 560 672
Non-Irrigated
Dry Matter Yield 1981-2 2.51 5.47 8.97 10.31 12.33 11.21 11.86
Mg/ha 1982-3 1.86 5.00 4.64 7.11 7.67 7.40 8.32
1983-4 1.91 3.99 5.45 4.86 4.89 5.09 4.95
Irrigated
1981-2 3.92 7.85 10.90 13.36 14.84 15.33 15.53
1982-3 2.91 6.37 7.29 11.70 12.76 13.45 13.72
1983-4 3.77 7.91 10.49 14.59 15.92 17.15 17.15

Non-Irrigated
N Removal 1981-2 55 118 207 259 337 334 368
kg/ha 1982-3 36 97 90 174 196 206 229
1983-4 37 94 147 133 140 150 147
Irrigated
1981-2 84 166 248 344 399 424 442
1982-3 62 120 139 275 331 362 371
1983-4 75 162 237 352 390 414 447

Non-Irrigated
N Concentration 1981-2 21.9 21.6 23.1 25.1 27.4 29.8 31.0
g/kg 1982-3 19.2 19.4 19.4 24.5 25.6 27.8 27.5
1983-4 19.4 23.7 27.0 27.4 28.6 29.4 29.8
Irrigated
1981-2 21.4 21.1 22.7 25.8 26.9 27.7 28.5
1982-3 21.4 18.9 19.0 23.5 25.9 26.9 27.0
1983-4 19.8 20.5 22.6 24.2 24.5 24.2 26.1
Source: Data from Huneycutt et al. (1988).








35
Table 3-10. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bahiagrass and Bermudagrass Grown at Eagle Lake, Texas.
Applied Nitrogen, kg/ha
Year 0 84 168 252 336
Bermudagrass Dry Matter 1979 4.47 6.98 10.09 12.91 14.23
Mg/ha 1980 3.90 5.47 6.33 7.47 8.12

N Removal 1979 58 104 156 225 258
kg/ha 1980 62 102 116 155 178

N Concentration 1979 13.0 14.9 15.5 17.4 18.1
g/kg 1980 16.0 18.7 18.4 20.7 21.9

Bahiagrass Dry Matter 1979 4.38 6.02 7.71 10.10 10.40
Mg/ha 1980 2.95 3.99 5.17 5.30 6.32

N Removal 1979 55 88 123 173 186
kg/ha 1980 65 81 108 120 150

N Concentration 1979 12.6 14.6 16.0 17.1 17.9
g/kg 1980 22.0 20.2 20.9 22.7 23.7
Source: Data from Evers (1984).








36
Table 3-11. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Grain and Total Plant of Corn Grown on Dothan and Goldsboro Soils at
Clayton and Kinston, North Carolina, Respectively.
Applied Dry Matter
Nitrogen Yield N Removal N Concentration
Site Part kg/ha Mg/ha kg/ha g/kg
Dothan Grain 0 4.37 47 10.8
56 7.12 74 10.4
112 9.77 113 11.6
168 10.81 135 12.5
224 11.04 150 13.6

Total 0 9.09 61 6.7
56 14.67 92 6.3
112 18.35 135 7.4
168 19.11 166 8.7
224 19.90 188 9.4

Goldsboro Grain 0 3.00 32 10.7
56 5.47 62 11.3
112 6.93 87 12.6
168 7.46 101 13.5
224 7.57 107 14.1

Total 0 6.63 36 5.4
56 10.60 71 6.7
112 13.10 104 7.9
168 13.55 120 8.9
224 13.92 134 9.6
Source: Data from Kamprath (1986).








37
Table 3-12. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Grain and Total Plant of Corn Grown on Portsmouth Soil at Plymouth,
North Carolina.
Part Applied Nitrogen Dry Matter Yield N Removal N Concentration
kg/ha Mg/ha kg/ha g/kg
Grain 0 4.53 48 10.6
56 6.30 68 10.8
112 7.65 89 11.6
168 8.52 104 12.2
224 9.04 115 12.7

Total 0 8.93 60 6.7
56 11.63 83 7.1
112 13.39 101 7.5
168 14.68 121 8.2
224 15.10 133 8.8
Source: Data from Kamprath (1986).








38
Table 3-13. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bahiagrass Grown on Entisol and Spodosol Soils in Florida.
Applied Nitrogen Dry Matter Yield N Removal N Concentration
Soil kg/ha Mg/ha kg/ha g/kg
Entisol 0 1.59 18.7 11.8
100 5.02 68.7 13.7
200 7.97 121 15.2
300 9.92 166 16.8
400 10.54 185 17.6

Spodosol 0 4.40 49.6 11.3
100 8.67 105 12.1
200 13.76 185 13.4
300 17.07 248 14.5
400 18.28 291 15.9
Source: Data from Blue (1987).








39
Table 3-14. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Ryegrass Grown in England.
Applied Nitrogen Dry Matter Yield N Removal N Concentration
Site kg/ha Mg/ha kg/ha g/kg
5 0 6.08 157 25.8
150 8.45 206 24.4
300 11.76 327 27.8
450 12.94 389 30.1
600 12.94 432 33.4
750 12.75 458 35.9

6 0 4.42 102 23.1
150 8.74 216 24.7
300 11.88 354 29.8
450 13.16 471 35.8
600 12.84 490 38.2
750 12.99 534 41.1

7 0 1.64 46 28.0
150 4.95 140 28.3
300 7.50 245 32.7
450 7.82 285 36.4
600 8.29 321 38.7
750 7.59 309 40.7

8 0 0.64 12 18.8
150 5.28 105 19.9
300 9.57 252 26.3
450 11.43 347 30.4
600 11.95 413 34.6
750 12.09 464 38.4

9 0 5.65 136 24.1
150 10.02 263 26.2
300 12.10 350 28.9
450 14.26 464 32.5
600 13.89 476 34.3
750 13.29 474 35.7








40
Table 3-14--continued
Applied Nitrogen Dry Matter Yield N Removal N Concentration
Site kg/ha Mg/ha kg/ha g/kg
10 0 2.96 64 21.6
150 7.03 162 23.0
300 11.24 304 27.0
450 12.52 401 32.0
600 12.30 413 33.6
750 11.98 433 36.1

12 0 1.24 28 22.6
150 4.84 103 21.3
300 9.71 237 24.4
450 13.58 384 28.3
600 14.75 438 29.7
750 15.06 518 34.4

13 0 4.40 102 23.2
150 8.38 221 26.4
300 10.93 341 31.2
450 12.23 426 34.8
600 12.19 458 37.6
750 11.44 452 39.5

14 0 2.90 63 21.7
150 8.24 212 25.7
300 10.41 282 27.1
450 12.42 393 31.6
600 12.81 445 34.7
750 12.09 469 38.8

15 0 1.53 34 22.2
150 5.06 118 23.3
300 8.14 220 27.0
450 9.95 310 31.2
600 10.81 376 34.8
750 11.17 426 38.1








41
Table 3-14--continued
Applied Nitrogen Dry Matter Yield N Removal N Concentration
Site kg/ha Mg/ha kg/ha g/kg
16 0 4.10 88 21.5
150 8.69 206 23.7
300 11.92 336 28.2
450 12.70 420 33.1
600 12.35 447 36.2
750 11.92 448 37.6

17 0 0.69 11 15.9
150 4.83 96 19.9
300 9.05 230 25.4
450 10.74 315 29.3
600 11.07 367 33.2
750 10.91 384 35.2

19 0 1.35 18 13.3
150 3.41 80 23.5
300 5.64 153 27.1
450 6.09 190 31.2
600 6.15 215 35.0
750 5.95 215 36.1

20 0 2.85 58 20.4
150 7.09 146 20.6
300 11.28 275 24.4
450 12.39 337 27.2
600 15.06 473 31.4
750 14.10 468 33.2

22 0 1.61 33 20.5
150 5.30 123 23.2
300 8.67 241 27.8
450 9.98 323 32.4
600 10.09 348 34.5
750 10.06 370 36.8








42
Table 3-14--continued.
Applied Nitrogen Dry Matter Yield N Removal N Concentration
Site kg/ha Mg/ha kg/ha g/kg
23 0 1.98 40 20.2
150 5.69 128 22.5
300 8.55 231 27.0
450 10.08 317 31.4
600 9.92 333 33.6
750 9.66 353 36.5

25 0 3.94 91 23.1
150 7.17 174 24.3
300 10.12 285 28.2
450 12.24 391 31.9
600 12.20 437 35.8
750 12.14 462 38.1

26 0 1.82 36 19.8
150 4.89 103 21.1
300 7.62 193 25.3
450 10.23 309 30.2
600 10.69 350 32.7
750 10.61 374 35.2

27 0 2.75 62 22.5
150 6.17 155 25.1
300 8.59 235 27.4
450 9.91 325 32.8
600 10.51 376 35.8
750 9.93 390 38.3

28 0 0.82 16 19.5
150 4.75 100 21.1
300 7.89 198 25.1
450 10.27 291 28.3
600 11.48 363 31.6
750 12.03 411 34.2
Source: Data from Morrison et al. (1980).








43
Table 3-15. Seasonal Dry Matter Yields and Nutrient Response to Applied Nutrients for
Gator Rye at Tifton, Georgia.
N P K Y Nu Pu Ku Nc Pc Kc
kg/ha kg/ha kg/ha Mg/ha kg/ha kg/ha kg/ha g/kg g/kg g/kg
0 40 74 0.55 - - - . .
45 40 74 1.81 58 12.1 66 32.3 6.69 36.4
90 40 74 3.01 112 19.6 108 37.3 6.51 36.0
135 40 74 3.75 160 24.4 136 42.6 6.51 36.4
180 40 74 4.01 175 26.8 146 43.7 6.69 36.4
225 40 74 4.55 216 28.6 148 47.5 6.29 32.5

135 0 74 2.09 92 13.0 75 44.1 6.21 35.8
135 20 74 3.08 134 17.0 118 43.5 5.51 38.3
135 40 74 3.36 133 21.1 119 39.5 6.29 35.4
135 60 74 3.74 146 24.3 141 39.1 6.51 37.8
135 80 74 4.00 148 21.7 134 37.0 5.42 33.4
135 100 74 3.97 - - - - - .

135 40 0 2.87 112 27.6 62 39.0 9.61 21.7
135 40 37 3.18 116 15.3 90 36.6 4.81 28.3
135 40 74 3.74 140 24.3 133 37.3 6.51 35.5
135 40 111 3.56 134 18.8 130 37.5 5.29 36.4
135 40 148 3.96 160 24.2 150 40.4 6.12 37.8
135 40 185 4.11 162 31.2 159 39.4 7.60 38.8
Source: Data from Walker and Morey (1962).













CHAPTER 4
RESULTS AND DISCUSSION


Evaluation of the Simple Logistic Model



Thorsby. Alabama: Bermudagrass and Bahiagrass

Data from Evans et al. (1961) for bermudagrass and bahiagrass were used. First

the grasses will be analyzed separately, then they will be pooled to see if any of the

parameters are common for the two grasses. The analysis of variance for the Coastal

bermudagrass and Pensacola bahiagrass over years 1957-1959 is given in Tables 4-1 and

4-2, respectively (3 yrs x 1 grass x 2 irrigation x 4 N). In mode 1, a common (A, b, c) is

assumed to apply to Equation [2.8], while individual (A, b, c) for each equation is

assumed in mode 2. For the Coastal bermudagrass, comparison among modes 1 and 2 has

a variance ratio of F = 2.589/0.584 = 4.43 and is significant at the 5% confidence level.

This comparison tests if mode 2 better describes the data. A significant variance ratio

suggests that mode 2 is better. Comparison among modes 3 and 2 has a variance ratio of

F = 0.875/0.584 = 1.50 and is not significant. This comparison also tests if mode 2 better

describes the data. The non-significant variance ratio suggests that mode 3 describes the

data as well or better than mode 2. Because mode 3 requires less parameters, we will

select it at this point in the analysis following Occam's razor. The variance ratio is not

significant for any of the comparisons among modes 2, 3, 4, or 5, suggesting that mode 3,

an individual A for each combination of year and irrigation and a common b and c for all.

For the Pensacola bahiagrass, comparison among modes 1 and 2 has a variance ratio of


44








45
7.48 and is significant at the 2.5% confidence level. Comparison among modes 3 and 4

has a variance ratio of 9.49 and is significant at the 0.5% confidence level. Furthermore,

comparison among modes 4 and 5 has a variance ratio of 8.78 and is significant at the

0.5% confidence level. These tests suggest when the grasses are studied separately, a

single value can be used for c for all years and irrigation treatments to adequately describe

the data. Next the data from the two grasses are combined and the analysis of variance

data are presented in Table 4-3 (3 yrs x 2 grass x 2 irrigation x 4 N). A comparison of

modes 1 and 2 results in a variance ratio of 5.61 that is significant at the 0.5% confidence

level. Comparison of modes 3 and 4 results in a variance ratio of 3.27 that is significant at

the 1% confidence level. Comparison of modes 3 and 5 results in a variance ratio of 2.93

that is significant at the 5% confidence level. Comparison of modes 5 and 4 leads to a

variance ratio of 2.87 that is significant at the 2.5% confidence level. Comparison of

modes 4 and 6 leads to a variance ratio of 3.41 that is significant at the 1% confidence

level. Comparison of modes 5 and 6 results in a variance ratio of 3.74 that is significant at

the 5% confidence level. Comparisons among modes 3 and 7 and 4 and 7 result in

variance ratios of 5.57 and 2.75, respectively. Both comparisons are significant at the

2.5% confidence level. Based upon these comparisons, we can conclude that mode 7,

with individual A for each year, individual b for each grass, and common c describes the

data best. The overall correlation coefficient of 0.9949 and adjusted correlation

coefficient of 0.9927 are calculated for mode 7. The error analysis of the parameters are

shown in Table 4-4. The largest relative error (Istandard error/estimatel) was for the b

parameter. This is due in part to the small numbers involved and that nonlinear regression

on the logistic equation places more emphasis upon the maximum and less upon the

intercept. Still the largest relative error for this set of parameters was under 4%. Since
the range of A parameter values (largest - smallest) between years and irrigation schemes

is less than six, the data will be averaged over years. Overman et al. (1990a, 1990b) have








46
shown that averaging over years is appropriate, since variations due to water availability

and harvest interval appear in the linear parameter A. The averaged data are in Table 4-5

and the error analysis for the averaged data is in Table 4-6. The overall correlation

coefficient of 0.9981 and adjusted correlation coefficient of 0.9969 are calculated. Note

that the estimates of b and c are the same as for the unaveraged data, supporting what

Overman et al. (1990a, 1990b) have found previously. Results are shown in Figure 4-1,

where curves for Coastal bermudagrass and Pensacola bahiagrass dry matter, irrigated and

non-irrigated, are drawn from the following equations:

Coastal bermudagrass, non-irrigated: Y = 21.57/[1 + exp(1.39 - 0.0078N)] [4.1]

Coastal bermudagrass, irrigated: Y = 23.44/[1 + exp(1.39 - 0.0078N)] [4.2]

Pensacola bahiagrass, non-irrigated: Y =21.49/[1 + exp(1.57 - 0.0078N)] [4.3]

Pensacola bahiagrass, irrigated: Y = 22.73/[1 + exp(1.57 - 0.0078N)] [4.4]

The scatter and residual plots of seasonal dry matter are given in Figures 4-2 and

4-3. The mean and the �2 standard errors of the residuals are shown by the solid and

dashed horizontal lines, respectively. As shown in Figure 4-3, all of the data fit between

�2 standard errors with no apparent trend.


Evaluation of the Extended Logistic Model


Baton Rouge. LA: Dallisgrass

The data for this analysis are taken from Robinson et al. (1988). The analysis of
variance is shown in Table 4-7 (dry matter and N removal x 6 N). Comparison among

modes 1 and 2 results in a variance ratio of 3736 and is significant at the 0.1% level.

Comparison among modes 3 and 2 leads to a variance ratio of 127.8 which is also
significant at the 0.1% level. Mode 4, with individual A and b for dry matter and N

removal, and common c describes the data best, since F(l,6,95) = 5.99, and F(1,7,99.9) =








47
29.25. This outcome supports Postulate 3. The overall correlation coefficient and

adjusted correlation coefficient calculated from mode 4 are 0.9997 and 0.9995,

respectively. The error analysis for the parameters is shown in Table 4-8. Results are

shown in Figure 4-4, where curves for dry matter and plant N removal are drawn from

Y = 15.60/[1 + exp(0.58 - 0.0055N)] [4.5]

Nu = 430.7/[1 + exp(1.47 - 0.0055N)] [4.6]

From these results, it follows that plant N concentration, shown in Figure 4-4, is estimated

from

N, = 27.6 [1 + exp(1.47 - 0.0055N)]/[1 + exp(0.58 - 0.0055N)] [4.7]

As shown, the data are described well by these equations. It should be noted that the

plant N concentration data were not defined by regression techniques, but rather as a ratio

of plant N removal and dry matter yield.

From Postulate 3, the relationships between yield and plant N removal and

between plant N concentration and plant N removal, shown in Figure 4-5, are described by

Y = 26.5 NJ(300+N,) [4.8]

N, = 11.3 + 0.0378NK [4.9]

The intercept (11.3 g/kg) represents plant N concentration in a nitrogen deficient

environment with low plant N removal.

The form of this model can be validated by plotting the data in dimensionless form.

The data plotted in Figure 4-5 were strictly measured data; that is, the independent

variable applied N has been ignored. The results of Postulate 3 defined the curve drawn.

By dividing dry matter yield, plant N concentration and plant N removal by their

appropriate maximums, all of the data are collapsed onto the same dimensionless scale.

Furthermore, the curves drawn are dependent upon the Ab, (b' - b), term alone. This will

be more significant as larger, more complex data sets are investigated. The form of the








48

model can be validated in this manner. The dimensionless plot for dallisgrass is shown in

Figure 4-6, and validates the form of the model. The curves were drawn from

Y/A = 1.70 (NJA')/[0.70 + (NJA')] [4.10]

N/Ncm = 0.41 +0.59 (NJA') [4.11]

Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-

7 through 4-10.


Thorsby, AL: Bermudagrass

The data for this analysis are taken from Doss et al. (1966). The analysis of

variance is shown in Table 4-9 (dry matter and N removal x 2 At x 6 N). Comparison

among modes 1 and 2 results in a variance ratio of 402 and is significant at the 0.1% level.

Comparison among modes 3 and 2 leads to a variance ratio of 4.63 that is significant at

the 5% level. Comparison among modes 3 and 4 results in a variance ratio of 8.07 that is

significant at the 0.5% level. Mode 5, with individual A for yield and plant N removal at

both harvest intervals, b for dry matter and N removal, and common c describes the data

best, since F(5,12,95) = 3.11, and F(1,17,99.9) = 15.72. This outcome also supports

Postulate 3. The overall- correlation coefficient and adjusted correlation coefficient

calculated from mode 5 are 0.9983 and 0.9977, respectively. The error analysis for the

parameters is shown in Table 4-10. Results are shown in Figure 4-11, where curves for

dry matter and plant N removal at both harvest intervals, are drawn from

3.0 weeks: Y = 17.42/[1 + exp(1.27 - 0.0067N)] [4.12]

N,= 568.9/[1 + exp(2.02 - 0.0067N)] [4.13]

4.5 weeks: Y = 19.75/[1 + exp(1.27 - 0.0067N)] [4.14]

Nu= 543.5/[1 + exp(2.02 - 0.0067N)] [4.15]

From these results, it follows that plant N concentration, shown in Figure 4-11, is

estimated from








49

3.0 weeks: N,= 32.7 [1 + exp(2.02 - 0.0067N)]/[1 + exp(1.27 - 0.0067N)] [4.16]

4.5 weeks: N,= 27.5 [1 + exp(2.02 - 0.0067N)]/[1 + exp(1.27 - 0.0067N)] [4.17]

As shown, the data are described well by these equations.

From Postulate 3, the relationships between yield and plant N removal and

between plant N concentration and plant N removal, shown in Figure 4-12, are described

by

3.0 weeks: Y = 33.0 N.J(509 + N,) [4.18]

N,= 15.4 + 0.0303Nu [4.19]

4.5 weeks: Y = 37.4 N.J(487 + Nu) [4.20]

Nc= 13.0 +0.0267N, [4.21]

The dimensionless plot for bermudagrass is shown in Figure 4-13, and validates the

form of the model. The curves were drawn from

Y/A = 1.90 (NJA')/[0.90 + (NJ/A')] [4.22]

NJN,, = 0.47 +0.53 (NJ/A') [4.23]

Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-

14 through 4-17.


Maryland: Bermudagrass

The data for this analysis are taken from Decker et al. (1971). The analysis of

variance is shown in Table 4-11 (dry matter and N removal x 5 At x 6 N). Comparison

among modes 1 and 2 results in a variance ratio of 190.9 and is significant at the 0.1%

level. Comparison among modes 3 and 2 leads to a variance ratio of 4.06 that is

significant at the 0.1% level. Comparison among modes 3 and 4 results in a variance ratio

of 4.65 that is significant at the 0.1% level. Mode 5, with individual A for yield and plant

N removal at the five harvest intervals, b for dry matter and N removal, and common c

describes the data best, since F(17,30,95) =1.98, F(1,47,99.9) = 12.32, and the residual








50

sums of squares (RSS) for mode 5 is smaller than mode 4 while using less parameters.

Postulate 3 is supported again. The overall correlation coefficient and adjusted correlation

coefficient calculated from mode 5 are 0.9964 and 0.9955, respectively. The error

analysis for the parameters is shown in Table 4-12. Results are shown in Figure 4-18,

where curves for dry matter and plant N removal at all five harvest intervals, are drawn

from

3.2 weeks: Y = 13.90/[1 + exp(1.78 - 0.0112N)] [4.24]

N, = 469.5/[1 + exp(2.59 - 0.0112N)] [4.25]

3.6 weeks: Y = 13.65/[1 + exp(1.78 - 0.0112N)] [4.26]

N, = 444.7/[1 + exp(2.59 - 0.0112N)] [4.27]

4.3 weeks: Y = 14.93/[1 + exp(1.78 - 0.0112N)] [4.28]

N, = 440.2/[1 + exp(2.59 - 0.0112N)] [4.29]

5.5 weeks: Y = 17.56/[1 + exp(1.78 - 0.0112N)] [4.30]

N. = 444.0/[1 + exp(2.59 - 0.0112N)] [4.31]

7.7 weeks: Y = 19.34/[1 + exp(1.78 - 0.0112N)] [4.32]

N, = 409.7/[1 + exp(2.59 - 0.0112N)] [4.33]

From these results, it follows that plant N concentration, shown in Figure 4-18, is

estimated from

3.2 weeks: N, = 33.7 [1 + exp(2.59 - 0.0112N)]/[1 + exp(1.78 - 0.0112N)] [4.34]

3.6 weeks: Nc = 32.6 [1 + exp(2.59 - 0.0112N)]/[1 + exp(1.78 - 0.0112N)] [4.35]

4.3 weeks: N, = 29.5 [1 + exp(2.59 - 0.0112N)]/[1 + exp(1.78 - 0.0112N)] [4.36]

5.5 weeks: N, = 25.3 [1 + exp(2.59 - 0.0112N)]/[1 + exp(1.78 - 0.0112N)] [4.37]

7.7 weeks: N, = 21.2 [1 + exp(2.59 - 0.0112N)]/[1 + exp(1.78 - 0.0112N)] [4.38]

As shown, the data are described well by these equations.








51

From Postulate 3, the relationships between yield and plant N removal and

between plant N concentration and plant N removal, shown in Figure 4-19, are described

by

3.2 weeks: Y = 25.0 NJ(376 + N.) [4.39]

Nc= 15.0 + 0.0399N, [4.40]

3.6 weeks: Y = 24.6 NJ(356 + N.) [4.41]

Ne= 14.5 + 0.0407N, [4.42]

4.3 weeks: Y = 26.9 NJ(353 + N,) [4.43]

Nc= 13.1 + 0.0372N. [4.44]

5.5 weeks: Y = 31.6 N/(356 + N,) [4.45]

Nc= 11.2 + 0.0316N, [4.46]

7.7 weeks: Y = 34.8 N.J(328 + N,) [4.47]

Nc= 9.4 + 0.0287N, [4.48]

The dimensionless plot for bermudagrass is shown in Figure 4-20, and validates the

form of the model. The curves were drawn from

Y/A = 1.80 (N/A')/[0.80 + (NJA')] [4.49]

Nc/Ne, = 0.44 +0.56 (Nd/A') [4.50]

Notice that all of the data have collapsed onto one curve and line, respectively. This

implies that the Ab is constant and the same for each harvest interval. Scatter and residual

plots of dry matter yield and plant N removal are shown in Figures 4-21 through 4-24.

It appears from Table 4-12, that there is a linear relationship between A, estimated

maximum dry matter yield, and harvest interval. The relationship does not appear to be as

clear for A', estimated maximum plant N removal. Overman et al. (1990b) found that

both water availability and harvest interval could be linked linearly to A up to a six weeks

interval, after which senescence sets in, and lower plant leaves die and/or drop off,

reducing dry matter accumulation. The estimated maximums of dry matter yield and plant








52

N removal are plotted against harvest interval in Figure 4-25. Linear regression was

conducted on the estimated maximum dry matter yield, omitting the 7.7 week harvest

interval, resulting in the following relationship

A = 7.90 + 1.71At = 7.90 (1 + 0.22At) [4.51]

with a correlation coefficient of 0.9676. Linear regression was also conducted on the

estimated maximum plant N removal, omitting the 7.7 week harvest interval, resulting in

the following relationship

A' = 483.9 - 8.3At = 483.9 (1 - 0.017At) [4.52]

with a correlation coefficient of 0.6206. The small correlation coefficient and rather flat

response suggests that there is uncertainty in the nature of the relationship.


Tifton. GA: Bermudagrass

The data for this analysis are taken from Prine and Burton (1956). The analysis of

variance is shown in Table 4-13 (dry matter and N removal x 2 yrs x 5 At x 5 N).

Comparison among modes 1 and 2 results in a variance ratio of 81.2 that is significant at

the 0.1% level. Comparison among modes 3 and 2 leads to a variance ratio of 1.89 that is

significant at the 97.5 % level. Comparison among modes 3 and 4 results in a variance

ratio of 2.45 that is significant at the 0.5% level. Comparison among modes 3 and 5,

individual A for yield and plant N removal at the five harvest intervals and over both years,

b for dry matter and N removal at both years, and common c, leads to a variance ratio of

15.9 that is significant at the 0.1% level. Mode 6, individual A for yield and plant N

removal at the five harvest intervals and over both years, b for dry matter and N removal

over both years, and common c, describes the data best, since F(37,40,95) = 1.71,

F(1,77,99.9) = 11.71, F(18;59,95) = 1.78, and F(2,75,95) = 3.12. The difference due to

years (water availability) and harvest interval is explained by the A parameter. Postulate 3

is supported again. The overall correlation coefficient and adjusted correlation coefficient








53

calculated from mode 6 are 0.9941 and 0.9923, respectively. The error analysis for the

parameters is shown in Table 4-14. Results are shown in Figures 4-26 through 4-30 for

each clipping interval, where curves for dry matter and plant N removal at all five harvest

intervals, are drawn from

2 weeks, 1953: Y = 17.95/[1 + exp(1.47 - 0.0077N)] [4.53]

N, = 644.7/[1 + exp(2.15 - 0.0077N)] [4.54]
1954: Y = 8.40/[1 + exp(1.47 - 0.0077N)] [4.55]

N, = 324.5/[1 + exp(2.15 - 0.0077N)] [4.56]

3 weeks, 1953: Y = 19.88/[1 + exp(1.47 - 0.0077N)] [4.57]

N, = 641.8/[1 + exp(2.15 - 0.0077N)] [4.58]

1954: Y = 9.95/[1 + exp(1.47 - 0.0077N)] [4.59]

Nu = 337.5/[1 + exp(2.15 - 0.0077N)] [4.60]

4 weeks, 1953: Y = 23.15/[1 + exp(1.47 - 0.0077N)] [4.61]

N. = 687.2/[1 + exp(2.15 - 0.0077N)] [4.62]

1954: Y = 11.67/[1 + exp(1.47 - 0.0077N)] [4.63]

N, = 348.0/[1 + exp(2.15 - 0.0077N)] [4.64]

6 weeks, 1953: Y = 29.57/[1 + exp(1.47 - 0.0077N)] [4.65]

N. = 688.2/[1 + exp(2.15 - 0.0077N)] [4.66]

1954: Y = 14.37/[1 + exp(1.47 - 0.0077N)] [4.67]

Nu = 363.8/[1 + exp(2.15 - 0.0077N)] [4.68]

8 weeks, 1953: Y = 29.58/[1 + exp(1.47 - 0.0077N)] [4.69]

N, = 606.0/[1 + exp(2.15 - 0.0077N)] [4.70]

1954: Y = 16.24/[1 +exp(1.47 - 0.0077N)] [4.71]

N. = 379.5/[1 + exp(2.15 - 0.0077N)] [4.72]
From these results, it follows that plant N concentration, shown in Figures 4-26 through

4-30, is estimated from








54

2 weeks, 1953: Nc = 35.9 [1 + exp(2.15 - 0.0077N)]/[1 + exp(1.47 - 0.0077N)] [4.73]

1954: N, = 38.6 [1 + exp(2.15 - 0.0077N)]/[1 + exp(1.47 - 0.0077N)] [4.74]

3 weeks, 1953: Nc = 32.3 [1 + exp(2.15 - 0.0077N)]/[1 + exp(1.47 - 0.0077N)] [4.75]

1954: Nc = 33.9 [1 + exp(2.15 - 0.0077N)]/[1 + exp(1.47 - 0.0077N)] [4.76]

4 weeks, 1953: N, = 29.7 [1 + exp(2.15 - 0.0077N)]/[1 + exp(1.47 - 0.0077N)] [4.77]

1954: Nc = 29.8 [1 + exp(2.15 - 0.0077N)]/[1 + exp(1.47 - 0.0077N)] [4.78]

6 weeks, 1953: Nc = 23.3 [1 + exp(2.15 - 0.0077N)]/[1 + exp(1.47 - 0.0077N)] [4.79]

1954: N, = 25.3 [1 + exp(2.15 - 0.0077N)]/[1 + exp(1.47 - 0.0077N)] [4.80]

8 weeks, 1953: Nc = 20.5 [1 + exp(2.15 - 0.0077N)]/[1 + exp(1.47 - 0.0077N)] [4.81]

1954: Nc = 23.4 [1 + exp(2.15 - 0.0077N)]/[1 + exp(1.47 - 0.0077N)] [4.82]

As shown, the data are described well by these equations.

From Postulate 3, the relationships between yield and plant N removal and

between plant N concentration and plant N removal at each clipping interval over both

years, shown in Figures 4-31 through 4-35, are described by

2 weeks, 1953: Y = 36.4 NJ(662 + N,) [4.83]

N, = 18.2 + 0.0275Nu [4.84]

1954: Y = 17.0N /(333 +N,) [4.85]

N, = 19.6 + 0.0587N, [4.86]

3 weeks, 1953: Y = 40.3 NJ(659 + Nu) [4.87]

Nc = 16.4 + 0.0248Nu [4.88]

1954: Y = 20.2 NJ(347 + N,) [4.89]

N, = 17.2 + 0.0496N, [4.90]

4 weeks, 1953: Y = 46.9 NJ(706 + Nu) [4.91]

N, = 15.0 + 0.0213N. [4.92]

1954: Y = 23.7 NJ(357 + Nu) [4.93]

Nc = 15.1 + 0.0423N, [4.94]








55

6 weeks, 1953: Y = 59.9 Nd(707 + N.) [4.95]

N, = 11.8 + 0.0167N, [4.96]

1954: Y = 29.1 N./(374 + N,) [4.97]

N, = 12.8 + 0.0343N, [4.98]

8 weeks, 1953: Y = 60.0 NJ(622 + N,) [4.99]

Nc = 10.4 + 0.0167Nu [4.100]

1954: Y = 32.9 NJ(390 + N) [4.101]

Nc = 11.8 + 0.0304N, [4.102]

The dimensionless plot for bermudagrass is shown in Figure 4-36, and validates the

form of the model. Note that all of the data have been collapsed onto one graph, and the

data fall on one curve. The curves were drawn from

Y/A = 2.03 (NJA')/[1.03 + (NJA')] [4.103]

Nd/Nc = 0.51 +0.49 (N/A') [4.104]

This result supports the hypothesis that the Ab is common for all harvest intervals and

both years.

As with the Maryland data set, it appears from Table 4-14, that there is a linear

relationship between A, estimated maximum dry matter yield, and harvest interval. The

estimated maximums of dry matter yield and plant N removal are plotted against harvest

interval in Figure 4-37. Linear regression was conducted on the estimated maximum dry

matter yield for both years, omitting the 8 week harvest interval, resulting in the following

relationships

1953: A= 11.50 +2.97At = 11.50(1+0.26At) [4.105]

1954: A = 5.49+ 1.50At = 5.49(1 + 0.27At) [4.106]

with correlation coefficients of 0.9958 and 0.9986. It appears from these equations that

after the intercept has been factored out, the coefficient of At may be the same for both








56

years. Linear regression was also conducted on the estimated maximum plant N removal,

omitting the 8 week harvest interval, resulting in the following relationships

1953: A'= 628.0 + 12.6At = 628.0 (1+0.020At) [4.107]

1954: A'= 307.2 + 9.7At = 307.2 (1+0.032At) [4.108]

with correlation coefficients of 0.8407 and 0.9925. The response for the estimated plant

N removal is flat as with the Maryland data. The line for 1954 might fit better due to the

limited available water. Since it was a dry year, growth was limited and hence the plant

was growing as much as possible with the limited water. As with the estimated maximum

dry matter yield, it appears that the coefficient of At may be the same for both years.

Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-

38 through 4-41.


England: Ryegrass

The data for this analysis are taken from Reid (1978). It is important of remember

throughout this specific analysis that the ryegrass was harvested at particular stages of

growth. As a result, the harvest interval was variable. This result will present itself in the

results. The analysis of variance is shown in Table 4-15 (dry matter and N removal x 3

variable At x 21 N). As shown in the table, all of the comparisons are highly significant

(0. 1%). This is due in part to the large degrees of freedom. If comparison is made among

the mean sums of squares (MSS), mode 6 has the smallest MSS with the exception of

mode 2 (which fits each individually). Mode 6 is the preferred one since it requires less

parameters to estimate (11 versus 18). This choice is in accordance with the approach

common in physics: seeking the simplest model consistent with observation (Rothman,

1972), often cited as Occam's razor (Will, 1986). Mode 6 assumes an individual A for

each number of clippings, an individual b for dry matter yield, common b for plant N

removal, and common c for all. The overall correlation coefficient and adjusted








57

correlation coefficient calculated from mode 6 are 0.9938 and 0.9931, respectively. The

error analysis for the parameters is shown in Table 4-16. Results are shown in Figure 4-

42, where curves for dry matter and plant N removal are drawn from

10 clippings: Y = 9.42/[1 + exp(1.74 - 0.0080N)] [4.109]

N, = 377.0/[1 + exp(2.15 - 0.0080N)] [4.110]

5 clippings: Y = 12.72/[1 + exp(1.23 - 0.0080N)] [4.111]

N, = 403.5/[1 + exp(2.15 - 0.0080N)] [4.112]

3 clippings: Y = 12.75/[1 + exp(0.90 - 0.0080N)] [4.113]

N. = 363.0/[1 + exp(2.15 - 0.0080N)] [4.114]

From these results, it follows that plant N concentration, shown in Figure 4-42, is

estimated from

10 clippings: N, = 40.0 [1 + exp(2.15 - 0.0080N)]/[1 + exp(1.74 - 0.0080N)] [4.115]

5 clippings: Nc = 31.7 [1 + exp(2.15 - 0.0080N)]/[1 + exp(1.23 - 0.0080N)] [4.116]

3 clippings: Nc = 28.5 [1 + exp(2.15 - 0.0080N)]/[1 + exp(0.90 - 0.0080N)] [4.117]

As shown, the data are not described very well by these equations; however, the problem

most likely lies in the effect of variable harvest interval.

From Postulate 3, the relationships between yield and plant N removal and

between plant N concentration and plant N removal, shown in Figure 4-43, are described

by

10 clippings: Y = 28.0 NJ(744 + Nu) [4.118]

N, = 26.6 + 0.0357Nu [4.119]

5 clippings: Y = 21.1 NJ(267 + N,) [4.120]

Nc = 12.6 + 0.0473Nu [4.121]

3 clippings: Y = 17.9 NJ(146 + N,) [4.122]

Nc = 8.2 + 0.0560Nu [4.123]







58

The dimensionless plot for ryegrass is shown in Figure 4-44, and this graph also

demonstrates the difficulty of variable harvest interval. The curves were drawn from

10 clippings: Y/A = 2.97 (N/A')/[1.97 + (N/A')] [4.124]
Nc/Ncm = 0.66 +0.34 (NJA') [4.125]

5 clippings: Y/A = 1.66 (Nu/A')/[0.66 + (NJA')] [4.126]

NJ/Nc, = 0.40 +0.60 (Ni/A') [4.127]

3 clippings: Y/A = 1.40 (NJA')/[0.40 + (NJA')] [4.128]

Nc/Nm = 0.29 +0.71 (NJA') [4.129]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-

45 through 4-48. It appears that in Figure 4-47, there may be a trend of some kind among

the residuals. It is likely that the trend is a result of the variable harvest interval.

The growing season of the rye was 26 weeks. Using this length, an average

harvest interval can be found for each of the three clipping frequencies. The estimated

maximum dry matter yield and plant N removal can be plotted against the average harvest

intervals to see if the same relationship holds for this study. There are two possible

obstacles. First, there are only three harvest intervals to plot. For the Maryland and

Tifton studies, a linear relationship was found between the expected maximum dry matter

and harvest interval. With only three points, it is difficult if not impossible to determine
the "true" relationship. Furthermore, the relationship tends to drop off after a six week

interval (Figures 4-25 and 4-37). The third average harvest interval is 8.67 weeks, and

could possibly affect the results. Secondly, these are not actual harvest intervals, but

rather an average harvest interval over the growing season. The actual harvest intervals

are variable. The effect of the variable harvest interval was observed in the figures, by

deduction it is likely to arise here as well. The plot of the estimated maximum dry matter

yield and plant N removal versus the average harvest interval is presented in Figure 4-49.








59

Linear regression was conducted on the estimated maximum dry matter yield resulting in

the following relationship

A = 8.78 + 0.52At [4.130]

with a correlation coefficient of 0.8263. Linear regression was also conducted on the

estimated maximum plant N removal resulting in the following relationship

A'= 396.6 - 2.8At [4.131]

with a correlation coefficient of 0.4170.


Fayetteville. AR: Bermudagrass

The data for this analysis are taken from Huneycutt et al. (1988). The analysis of

variance is shown in Table 4-17 (dry matter and N removal x 3 yrs x 2 irrigation x 7 N).

Comparison among modes I and 2 results in a variance ratio of 170.3 that is significant at

the 0.1% level. Comparison among modes 3 and 2 leads to a variance ratio of 2.05 that is

significant at the 2.5% level. Comparison among modes 3 and 4 results in a variance ratio

of 3.22 that is significant at the 99.5 % level. Comparison among modes 3 and 5,

individual A for yield and plant N removal with and without irrigation for all three years, b

for dry matter and N removal with and without irrigation, and common c, leads to a

variance ratio of 5.21 and is significant at the 0.5% level. Mode 6, individual A for yield

and plant N removal with and without irrigation for all three years, b for dry matter and N

removal over both with and without irrigation and all years, and common c, describes the

data best, since F(21,48,95) = 1.78, F(1,69,99.9) = 11.81, and F(10,59,95) = 2.00. The

difference due to years and irrigation is explained by the A parameter. Postulate 3 is

supported again. The overall correlation coefficient and adjusted correlation coefficient

calculated from mode 6 are 0.9950 and 0.9939, respectively. The error analysis for the

parameters is shown in Table 4-18. Results are shown in Figure 4-50, where curves for

dry matter and plant N removal for all three years and irrigation, are drawn from








60

Non-irrigated, 1983: Y = 17.90/[1 + exp(1.50 - 0.0084N)] [4.132]

N,= 408.7/[1 + exp(2.04 - 0.0084N)] [4.133]

1984: Y = 17.37/[1 + exp(1.50 - 0.0084N)] [4.134]

Nu= 413.9/[1 + exp(2.04 - 0.0084N)] [4.135]

1985: Y = 19.61/[1 + exp(1.50 - 0.0084N)] [4.136]

N,= 459.3/[1 + exp(2.04 - 0.0084N)] [4.137]

Irrigated, 1983: Y =24.70/[1 + exp(1.50 - 0.0084N)] [4.138]

Nu= 554.3/[1 + exp(2.04 - 0.0084N)] [4.139]

1984: Y = 24.60/[1 + exp(1.50 - 0.0084N)] [4.140]

N,= 523.7/[1 + exp(2.04 - 0.0084N)] [4.141]

1985: Y = 22.58/[1 + exp(1.50 - 0.0084N)] [4.142]

Nu= 492.1/[1 + exp(2.04 - 0.0084N)] [4.143]

From these results, it follows that plant N concentration, shown in Figure 4-50, is

estimated from

Non-irrigated,1983: N = 22.8 [1 + exp(2.04 - 0.0084N)]/[1 + exp(1.50 - 0.0084N)][4.144]

1984: N,=23.8 [1 + exp(2.04 - 0.0084N)]/[1 + exp(1.50 - 0.0084N)][4.145]

1985: N,= 23.4 [1 + exp(2.04 - 0.0084N)]/[1 + exp(1.50 - 0.0084N)][4.146]

Irrigated, 1983: Nc= 22.4 [1 + exp(2.04 - 0.0084N)]/[1 + exp(1.50 - 0.0084N)][4.147]

1984: Nc= 21.3 [1 + exp(2.04 - 0.0084N)]/[1 + exp(1.50 - 0.0084N)][4.148]

1985: Nc= 21.8 [1 + exp(2.04 - 0.0084N)]/[1 + exp(1.50 - 0.0084N)][4.149]

As shown, the data are described relativity well by these equations.

From Postulate 3, the relationships between yield and plant N removal and

between plant N concentration and plant N removal at each clipping interval for all three

years with and without irrigation, shown in Figure 4-51, are described by
Non-irrigated,1983: Y = 42.9 N/(571 + Nu) [4.150]

Nc = 13.3 + 0.0233N, [4.151]








61

1984: Y = 41.6 NJ(578 + N,) [4.152]

N, = 13.9 + 0.0240N, [4.153]

1985: Y = 47.0 Nd(641 + N,) [4.154]

Nc = 13.6 + 0.0213N, [4.155]

Irrigated, 1983: Y = 59.2 NJ(774 + Nu) [4.156]

Nc = 13.1 + 0.0169Nu [4.157]

1984: Y = 59.0 N/(731 +N,) [4.158]

Nc = 12.4 + 0.0170N, [4.159]

1985: Y = 54.1 NJ(687 + N,) [4.160]

N, = 12.7 + 0.0185N, [4.161]

The dimensionless plot for bermudagrass is shown in Figure 4-52, and validates the

form of the model. Note that all of the data have been collapsed onto one graph, and the

data fall on the curve. The curves were drawn from

Y/A = 2.40 (NdA')/[1.40 + (N/A')] [4.162]

Nc/N,, = 0.58 +0.42 (NJA') [4.163]

Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-

53 through 4-56.

As done earlier, the data will be averaged over years, since variations due to water

availability appear in the linear parameter A. The averaged data are in Table 4-19 and the

analysis of variance is in Table 4-20 (dry matter and N removal x 1 yr x 2 irrigation x 7

N). Comparison among modes 1 and 2 results in a variance ratio of 373.4 that is

significant at the 0.1% level. Comparison among modes 3 and 2 leads to a non-significant

variance ratio of 2.72. Comparison among modes 3 and 4 results in a variance ratio of

6.11 that is significant at the 0.5% level. Mode 5, individual A for yield and plant N

removal with and without irrigation, b for dry matter and N removal, and common c,

accounts for all the significant variation, since F(5,16,95) = 2.85, F(1,21,99) = 8.02, and








62

F(2,19,95) = 3.52. The difference due to irrigation is explained by the A parameter.

Postulate 3 is supported again. The overall correlation coefficient and adjusted correlation

coefficient calculated from mode 5 was 0.9973 and 0.9965, respectively. The error

analysis for the parameters is shown in Table 4-21. Results are shown in Figure 4-57,

where curves for dry matter and plant N removal with and without irrigation averaged

over three years, are drawn from

Non-irrigated: Y = 18.63/[1 + exp(1.51 - 0.0084N)] [4.164]

N, = 435.7/[1 + exp(2.04 - 0.0084N)] [4.165]

Irrigated: Y = 24.40/[1 + exp(1.51 - 0.0084N)] [4.166]

N, = 534.8/[1 + exp(2.04 - 0.0084N)] [4.167]

The error analysis for the averaged data is in Table 4-21. Note that the estimates ofb and

c are the essentially the same as for the unaveraged data, supporting what Overman et al.

(1990a, 1990b) have found previously. From these results, it follows that plant N

concentration, shown in Figure 4-57, is estimated from

Non-irrigated: Ne= 23.4 [1 + exp(2.04 - 0.0084N)]/[1 + exp(1.51 - 0.0084N)] [4.168]

Irrigated: Ne= 21.9 [1 + exp(2.04 - 0.0084N)]/[1 + exp(1.51 - 0.0084N)] [4.169]

As shown, the data are described well by these equations.
From Postulate 3, the relationships between yield and plant N removal and

between plant N concentration and plant N removal at each clipping interval for all three

years with and without irrigation, shown in Figure 4-58, are described by

Non-irrigated: Y = 45.3 Nu/(623 + Nu) [4.170]

N, = 13.8 + 0.0221N, [4.171]

Irrigated: Y = 59.3 NJ(765 + N.) [4.172]

Nc = 12.9 + 0.0169N, [4.173]








63

The dimensionless plot for bermudagrass is shown in Figure 4-59, and validates the

form of the model. Note that all of the data have been collapsed onto one graph, and the

data fall on the curve reasonably well. The curves were drawn from

Y/A = 2.43 (NJA')/[1.43 + (NJA')] [4.174]

N/Nm,, = 0.59 +0.41 (NJA') [4.175]

Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-

60 through 4-63.


Fayetteville. AR: Tall Fescue

The data for this analysis are taken from Huneycutt et al. (1988). The analysis of

variance is shown in Table 4-22 (dry matter and N removal x 3 yrs x 2 irrigation x 7 N).

Comparison among modes 1 and 2 results in a variance ratio of 352 that is significant at

the 0.1% level. Comparison among modes 3 and 2 also leads to a variance ratio of 7.60

that is significant at the 0.1% level. Comparison among modes 3 and 4 results in a

variance ratio of 9.90 that is significant at the 0.1% level. Comparison among modes 3

and 5, individual A for yield and plant N removal with and without irrigation for all three

years, b for dry matter and N removal with and without irrigation, and common c, leads to

a variance ratio of 18.3 that is significant at the 0.1% level. Mode 6, individual A for yield

and plant N removal with and without irrigation for all three years, b for dry matter and N

removal over both with and without irrigation and all years, and common c, describes the

data best, since F(21,48,95) = 2.94, F(1,69,99.9) = 11.81, F(10,59,97.5) = 2.27, and

F(2,67,95) = 3.13. The difference due to years and irrigation is explained by the A

parameter. Postulate 3 is supported again. The overall correlation coefficient and

adjusted correlation coefficient calculated from mode 6 were 0.9956 and 0.9947,

respectively. The error analysis for the parameters is shown in Table 4-23. Results are








64

shown in Figure 4-64, where curves for dry matter and plant N removal for all three years

and irrigation, are drawn from

Non-irrigated, 1981-2: Y = 12.08/[1 + exp(0.92 - 0.0081N)] [4.176]

N,= 357.5/[1 + exp(1.47 - 0.0081N)] [4.177]

1982-3: Y = 8.01/[1 + exp(0.92- 0.0081N)] [4.178]

N.= 217.8/[1 + exp(1.47 - 0.0081N)] [4.179]

1983-4: Y = 5.71/[1 + exp(0.92 - 0.0081N)] [4.180]

Nu= 169.5/[1 + exp(1.47 - 0.0081N)] [4.181]

Irrigated, 1981-2: Y = 15.63/[1 + exp(0.92 - 0.0081N)] [4.182]

Nu= 443.7/[1 + exp(1.47 - 0.0081N)] [4.183]

1982-3: Y = 13.22/[1 + exp(0.92 - 0.0081N)] [4.184]

N,= 357.3/[1 + exp(1.47 - 0.0081N)] [4.185]

1983-4: Y = 16.81/[1 + exp(0.92 - 0.0081N)] [4.186]

Nu= 439.4/[1 + exp(1.47 - 0.0081N)] [4.187]

From these results, it follows that plant N concentration, shown in Figure 4-64, is

estimated from

Non-Irrigated:

1981-2: 29.6 [1 + exp(1.47 - 0.0081N)]/[1 + exp(0.92 - 0.0081N)] [4.188]

1982-3: 27.2 [1 + exp(1.47 - 0.0081N)]/[1 + exp(0.92 - 0.0081N)] [4.189]

1983-4: 29.7 [1 + exp(1.47 - 0.0081N)]/[1 + exp(0.92 - 0.0081N)] [4.190]

Irrigated:

1981-2: 28.4 [1 + exp(1.47 - 0.0081N)]/[1 + exp(0.92 - 0.0081N)] [4.191]

1982-3: 27.0 [1 + exp(1.47 - 0.0081N)]/[1 + exp(0.92 - 0.0081N)] [4.192]

1983-4: 26.1 [1 + exp(1.47 - 0.0081N)]/[1 + exp(0.92 - 0.0081N)] [4.193]

As shown, the data are described relativity well by these equations.








65

From Postulate 3, the relationships between yield and plant N removal and

between plant N concentration and plant N removal at each clipping interval for all three

years with and without irrigation, shown in Figure 4-65, are described by

Non-irrigated,1981-2: Y = 28.6 NJ(488 + Nu) [4.194]

N, = 17.1 + 0.0350N, [4.195]

1982-3: Y = 18.9 NJ(297 + N) [4.196]

Nc = 15.7 + 0.0528N, [4.197]

1983-4: Y = 13.5 N/(231 + N,) [4.198]

N, = 17.1+0.0741N. [4.199]

Irrigated, 1981-2: Y = 36.9 NJ(605 + N,) [4.200]

N, = 16.4 + 0.0271N, [4.201]

1982-3: Y = 31.2 NJ(487 + N,) [4.202]

Ne = 15.6 + 0.0320Nu [4.203]

1983-4: Y = 39.7 N,/(599 + Nu) [4.204]

N, = 15.1 + 0.0252N, [4.205]

The dimensionless plot for bermudagrass is shown in Figure 4-66, and validates the

form of the model. Note that all of the data have been collapsed onto one graph, and the

data fall on the curve. The curves were drawn from

Y/A = 2.36(NJA')/[1.36 + (NJA')] [4.206]

NJ/Nm = 0.58 +0.42 (NdA') [4.207]

Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-

67 through 4-70.

As done with the previous study, the data will be averaged over years, since

variations due to water availability appear in the linear parameter A. The averaged data

are in Table 4-24 and the analysis of variance is in Table 4-25 (dry matter and N removal x

1 yr x 2 irrigation x 7 N). Comparison among modes 1 and 2 results in a variance ratio of








66

904 that is significant at the 0.1% level. Comparison among modes 3 and 2 leads to a

variance ratio of 9.44 that is significant at the 0.1% level. Comparison among modes 3

and 4 results in a variance ratio of 18.4 that is significant at the 0.1% level. Mode 5,

individual A for yield and plant N removal with and without irrigation, b for dry matter

and N removal, and common c, accounts for all the significant variation, since F(5,16,95)

= 2.85, F(1,21,99.9) = 14.59, and F(2,19,95) = 3.52. The difference due to irrigation is

explained by the A parameter. Postulate 3 is supported again. The overall correlation

coefficient and adjusted correlation coefficient calculated from mode 5 were 0.9986 and

0.9982, respectively. The error analysis for the parameters is shown in Table 4-27.

Results are shown in Figure 4-71, where curves for dry matter and plant N removal with

and without irrigation averaged over three years, are drawn from

Non-irrigated: Y = 8.67/[1 + exp(0.99 - 0.008 N)] [4.208]

N, = 250.7/[1 + exp(1.53 - 0.0081N)] [4.209]

Irrigated: Y = 15.36/[1 + exp(0.99- 0.0081N)] [4.210]

Nu = 417.4/[1 + exp(1.53 - 0.008IN)] [4.211]

The error analysis for the averaged data is in Table 4-26. Note that although the estimates

of b and c are not quite the same as for the unaveraged data, the Ab (b'-b), is essentially

constant. From these results, it follows that plant N concentration, shown in Figure 4-71,

is estimated from

Non-irrigated: N,= 28.9 [1 + exp(1.53 - 0.0081N)]/[1 + exp(0.99 - 0.0081N)] [4.212]

Irrigated: Nc= 27.2 [1 + exp(1.53 - 0.0081N)]/[1 + exp(0.99 - 0.0081N)] [4.213]

As shown, the data are described well by these equations.

From Postulate 3, the relationships between yield and plant N removal and

between plant N concentration and plant N removal averaged over three years with and

without irrigation, shown in Figure 4-72, are described by

Non-irrigated: Y = 20.8 NJ(350 + Nu) [4.214]








67

Nc = 16.9 + 0.0481N, [4.215]

Irrigated: Y = 36.8 N/(583 + N,) [4.216]

Nc = 15.8 +0.0272N, [4.217]

The dimensionless plot for tall fescue is shown in Figure 4-73, and validates the

form of the model. Note that all of the data have been collapsed onto one graph, and the

data fall on the curve. The curves were drawn from

Y/A = 2.40 (NJA')/[1.40 + (NJA')] [4.218]

Nc/Ncm = 0.58 +0.42 (NJA') [4.219]

Notice the similarity between the above equations and Equations [4.174] and [4.175].

The Ab values are similar for both grasses. It is not clear if this is a coincidence or a result

of the system. Scatter and residual plots of dry matter yield and plant N removal are

shown in Figures 4-74 through 4-77.


Eagle Lake. TX: bermudagrass and bahiagrass

Data from Evans et al. (1961) for bermudagrass and bahiagrass were used. First

the grasses will be analyzed separately, then they will be pooled to see if any of the

parameters are common for the two grasses. The analysis of variance for the Coastal

bermudagrass over years 1979-1980 is shown in Table 4-27 (dry matter and N removal x

2 yrs x 5 N). Comparison among modes 1 and 2 results in a variance ratio of 419 that is

significant at the 0.1% level. Comparison among modes 3 and 2 leads to a variance ratio

of 9.22 that is significant at the 0.5% level. Comparison among modes 3 and 4 results in a

variance ratio of 14.6 that is significant at the 0.1% level. Comparison among modes 5

and 4 leads to a variance ratio of 10.4 that is significant at the 0.5% level. As a result,

mode 4, individual A and b for yield and plant N removal at each year and common c,

accounts for all the significant variation, since F(3,8,95) = 4.07, F(3,11,99.9) = 11.56, and

F(2,11,99.5) = 8.91. The analysis of variance for the Pensacola bahiagrass over years








68

1979-1980 is shown in Table 4-28 (dry matter and N removal x 2 yrs x 5 N). Comparison

among modes 1 and 2 results in variance ratio of 328 that is significant at the 0.1% level.

Comparison among modes 3 and 2 leads to a variance ratio of 9.54 that is significant at

the 99.5 level. Comparison among modes 3 and 4 results in a variance ratio of 12.3 that is

significant at the 0.1% level. Mode 5, individual A for yield and plant N removal for each

year, b for dry matter and N removal, and common c, accounts for all the significant

variation, since F(5,8,95) = 3.69, F(1,13,99.9) = 17.81, and F(2,11,95) = 3.98. Next the

data from the two grasses are combined and the analysis of variance data are presented in

Table 4-29 (dry matter and N removal x 2 grasses x 2 yrs x 5 N). A comparison of modes

I and 2 results in a variance ratio of 330 that is significant at the 0.1% confidence level.

Comparison of modes 3 and 2 leads to a variance ratio of 7.62, that is significant at the

0. 1% confidence level. Comparison of modes 3 and 4 results in a variance ratio of 9.66

that is significant at the 0.1% confidence level. Comparison of modes 3 and 5 results in a

variance ratio of 16.2 that is significant at the 0.1% confidence level. Comparison of

modes 5 and 4 leads to a variance ratio of 5.85 that is significant at the 0.1% confidence

level. Comparison of modes 3 and 6 leads to a variance ratio of 7.54 that is significant at

the 0.1% confidence level. Comparison of modes 6 and 4 results in a variance ratio of

6.58 that is significant at the 0.5% confidence level. Based upon these comparisons, we

can conclude that mode 4, with individual A and b for dry matter yield and plant N

removal for each year and grass and common c describes the data best. The overall

correlation coefficient of 0.9927 and adjusted correlation coefficient of 0.9876 were

calculated by mode 4. The statistical analysis might be affected by the close numbers for

yield and N removal at low values of applied N for both grasses. In the concern that fewer

parameters might be used to significantly account for the variation, the data will be

averaged over years to determine if an individual b can be used for yield and N removal
for both grasses. The averaged data are in Table 4-30 and the analysis of variance for the








69

averaged data is in Table 4-31. Comparison among modes 1 and 2 results in a variance

ratio of 792, that is significant at the 0.1% confidence interval. Comparison among modes

3 and 2 leads to a variance ratio of 7.79, that is significant at the 0.5% confidence level.

Comparison among modes 3 and 4 results in a variance ratio of 18.4, that is significant at

the 0.1% confidence level. Mode 5, individual A for dry matter yield and plant N removal

of each grass, individual b for dry matter yield and plant N removal, and common c,

accounts for all of the significant variation, since F(5,8,95) = 3.69, F(1,13,99.9) = 17.81,

and F(2,11,97.5) = 5.26. The two grasses have the same b and b', hence suggesting mode

5 of Table 4-29. The overall correlation coefficient of 0.9987 and adjusted correlation

coefficient of 0.9981 were calculated using mode 5 of the averaged data. Using mode 5

and the original data, the overall correlation coefficient of 0.9798 and adjusted correlation

coefficient of 0.9727 were calculated. The error analysis of the averaged and original data

is in Tables 4-32 and 4-33. Note that the estimates ofb and c are the essentially the same

as for the unaveraged data, suggesting this is a valid way to describe the data. Results are

shown in Figure 4-78 and 4-79 for bermudagrass and bahiagrass, respectively, where are

drawn from

Coastal bermudagrass, 1979: Y = 15.99/[1 + exp(0.57 - 0.0072N)] [4.220]

Nu= 310.9/[l + exp(1.07 - 0.0072N)] [4.221]

1980: Y = 9.89/[1 + exp(0.57 - 0.0072N)] [4.222]

Nu= 227.4/[1 + exp(1.07 - 0.0072N)] [4.223]

Pensacola bahiagrass, 1979: Y = 12.42/[1 + exp(0.57 - 0.0072N)] [4.224]

Nu= 237.2/[1 + exp(1.07 - 0.0072N)] [4.225]

1980: Y = 7.51/[1 + exp(0.57 - 0.0072N)] [4.226]

Nu= 191.3/[1 + exp(1.07 - 0.0072N)] [4.227]
From these results, it follows that plant N concentration, shown in Figures 4-78 and 4-79,

is estimated from








70

Bermudagrass, 1979: N,= 19.4 [1 + exp(1.07 - 0.0072N)]/[1 + exp(0.57 - 0.0072N)][4.228]

1980: Nc= 23.0 [1 + exp(1.07 - 0.0072N)]/[1 + exp(0.57 - 0.0072N)][4.229]

Bahiagrass, 1979: Ne= 19.1 [1 + exp(1.07 - 0.0072N)]/[1 + exp(0.57 - 0.0072N)][4.230]

1980: N,= 25.5 [1 + exp(1.07 - 0.0072N)]/[1 + exp(0.57 - 0.0072N)][4.231]

As shown, the data are described well by these equations.

From Postulate 3, the relationships between yield and plant N removal and

between plant N concentration and plant N removal over two years, shown in Figure 4-80

and 4-81, are described by

Bermudagrass, 1979: Y = 40.6 N/(479 + N,) [4.232]

Nc= 11.8 + 0.0246N, [4.233]

1980: Y = 25.1 NJ(351 + N) [4.234]

Nc= 13.9 + 0.0398N, [4.235]

Bahiagrass, 1979: Y = 31.6 NJ(366 + Nu) [4.236]

N,= 11.6 + 0.0317N, [4.237]

1980: Y= 19.1 NJ(295 +N) [4.238]

Ne= 15.4 + 0.0524N, [4.239]

The dimensionless plot for bermudagrass and bahiagrass is shown in Figure 4-82,

and validates the form of the model. Note that all of the data have been collapsed onto

one graph, and the data fall on the curve. The curves were drawn from

Y/A = 2.54 (NJA')/[1.54 + (NJA')] [4.240]

Nc/Nc, = 0.61 +0.39 (NJA') [4.241]

Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-

83 through 4-86.








71
Clayton and Kinston. NC: Corn


Data from Kamprath (1986) for corn were used. The analysis of variance for the

Dothan sandy loam is shown in Table 4-34 (dry matter and N removal x 2 components x 5

N). Comparisons among modes 1 and 2, modes 3 and 2, modes 3 and 4, and modes 3 and

5 result in variance ratios of 1686, 22.1, 49.8, and 39.0, which are all significant at the

0.1% confidence level. Comparison among modes 5 and 4 leads to a non-significant

variance ratio of 0.31. As a result, mode 5, individual A for grain and total plant,

individual b for yield and plant N removal, and common c, accounts for all the significant

variation, since F(5,8,97.5) = 4.82, F(1,13,99.9) = 17.81, and F(2,11,95) = 3.98. The

analysis of variance for the Goldsboro sandy loam is shown in Table 4-35 (dry matter and

N removal x 2 components x 5 N). As with the Dothan, comparisons among modes 1 and

2, modes 3 and 2, modes 3 and 4, and modes 3 and 5 result in variance ratios of 1920,

26.9, 35.7, and 83.2, which are all significant at the 0.1% confidence level. Comparison

among modes 5 and 4 leads to a non-significant variance ratio of 2.48. As a result, mode

5, individual A for grain and total plant, individual b for yield and plant N removal, and

common c, accounts for all the significant variation, since F(5,8,95) = 3.69, F(1,13,99.9) =

17.81, and F(2,11,95) = 3.98. Next the data from the two soils are combined and the

analysis of variance data are presented in Table 4-36 (dry matter and N removal x 2

components x 2 soils x 5 N). A comparison among modes 1 and 2 results in a variance

ratio of 1562 that is significant at the 0.1% confidence level. Comparison among modes

3 and 2 leads to a variance ratio of 24.8, that is significant at the 0.1% confidence level.

Comparison among modes 3 and 4 results in a variance ratio of 8.29 that is significant at

the 0.1% confidence level. Comparison among modes 3 and 5 results in a variance ratio

of 19.7 that is significant at the 0.1% confidence level. Comparison among modes 3 and 6

leads to a variance ratio of 67.9 that is significant at the 0.1% confidence level.

Comparison among modes 3 and 7 results in a variance ratio of 21.9, that is significant at








72

the 0.1% confidence level. Comparison among modes 6 and 7 leads to a variance ratio of

0.36, which is not significant. Mode 6, with individual A for dry matter yield and plant N

removal for both grain and total plant, b for dry matter yield and plant N removal, and

common c describes the data best, since F(2,27,95) = 3.35, and the MSS for mode 6 was

smaller than that of mode 5. The overall correlation coefficient of 0.9941 and adjusted

correlation coefficient of 0.9921 were calculated by mode 6. The error analysis for the

parameters is shown in Table 4-37. Results are shown in Figure 4-87 for both soils, grain

and total plant, where the curves are drawn from

Dothan, Grain: Y = 11.12/[1 + exp(0.27 - 0.0187N)] [4.242]

Nu = 151.7/[1 + exp(0.97 - 0.0187N)] [4.243]

Total: Y = 20.68/[1 + exp(0.27- 0.0187N)] [4.244]

N, = 187.2/[1 + exp(0.97 - 0.0187N)] [4.245]

Goldsboro, Grain: Y = 7.83/[1 + exp(0.27 - 0.0187N)] [4.246]
N, = 113.3/[1 + exp(0.97 - 0.0187N)] [4.247]

Total: Y = 14.70/[1 + exp(0.27 - 0.0187N)] [4.248]

N. = 136.5/[1 + exp(0.97 - 0.0187N)] [4.249]

From these results, it follows that plant N concentration, shown in Figure 4-87, is

estimated from

Dothan, Grain: Nc= 13.6 [1 + exp(0.97 - 0.0187N)]/[1 + exp(0.27 - 0.0187N)][4.250]

Total: Ne= 9.1 [1 + exp(0.97 - 0.0187N)]/[1 + exp(0.27 - 0.0187N)][4.251]

Goldsboro, Grain: N,= 14.5 [1 + exp(0.97 - 0.0187N)]/[1 + exp(0.27 - 0.0187N)][4.252]

Total: N,= 9.3 [1 + exp(0.97 - 0.0187N)]/[1 + exp(0.27 - 0.0187N)][4.253]

As shown, the data are described well by these equations.

From Postulate 3, the relationships between yield and plant N removal and

between plant N concentration and plant N removal for both soils, grain and total plant,

shown in Figure 4-88, are described by








73

Dothan, Grain: Y = 22.1 NJ(150 + N.) [4.254]

N,= 6.8 + 0.0453N, [4.255]

Total: Y = 41.1 NJ(185 + N,) [4.256]

Ne= 4.5 +0.0243N, [4.257]

Goldsboro, Grain: Y = 15.6 NJ(112 + N,) [4.258]

NC= 7.2 + 0.0643N, [4.259]

Total: Y = 29.2 NJ(135 + N,) [4.260]

N,= 4.6 + 0.0342N, [4.261]

The dimensionless plot for corn is shown in Figure 4-89, and validates the form of

the model. Note that all of the data have been collapsed onto one graph, and the data fall

on the curve. The curves were drawn from

Y/A = 1.99 (NJA')/[0.99 + (NJA')] [4.262]

Nc/Ncm = 0.50 +0.50 (N.JA') [4.263]

Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-

90 through 4-93.


Plymouth. NC: Corn

Data from Kamprath (1986) for corn were used. The analysis of variance for the

Plymouth very fine sandy loam is shown in Table 4-38 (dry matter and N removal x 2

components x 5 N). Comparisons among modes 1 and 2, modes 3 and 2, modes 3 and 4,

and modes 3 and 5 result in variance ratios of 5819, 32.5, 27.8, and 57.4, which are all

significant at the 0.1% confidence level. Comparison among modes 5 and 4 leads to a

non-significant variance ratio of 3.21. As a result, mode 5, individual A for grain and total

plant, individual b for yield and plant N removal, and common c, accounts for all the

significant variation, since F(5,8,97.5) = 4.82, F(1,13,99.9) = 17.81, and F(2,11,95) =

3.98. The overall correlation coefficient of 0.9997 and adjusted correlation coefficient of








74

0.9995 were calculated by mode 5. Results are shown in Figure 4-94 for grain and total

plant, where the curves are drawn from

Grain: Y = 9.48/[1 + exp(-0.065 - 0.0119N)] [4.264]

N, = 126.2/[1 + exp(0.46 - 0.0119N)] [4.265]

Total: Y = 16.60/[1 + exp(-0.065 - 0.0119N)] [4.266]

N, = 147.3/[1 + exp(0.46 - 0.0119N)] [4.267]

From these results, it follows that plant N concentration, shown in Figure 4-94, is

estimated from

Grain: N, = 13.3 [1 + exp(0.46 - 0.0119N)]/[1 + exp(-0.065 - 0.0119N)] [4.268]

Total: Nc = 8.9 [1 + exp(0.46 - 0.0119N)]/[1 + exp(-0.065 - 0.0119N)] [4.269]

As shown, the data are described well by these equations.

From Postulate 3, the relationships between yield and plant N removal and

between plant N concentration and plant N removal for grain and total plant, shown in

Figure 4-95, are described by

Grain: Y = 23.2 Nj(183 +Nu) [4.270]

Nc = 7.9 + 0.0431N, [4.271]

Total: Y = 40.6 NJ(213 +N,) [4.272]

N, = 5.2 + 0.0246N, [4.273]

The dimensionless plot for corn is shown in Figure 4-96, and validates the form of

the model. Note that all of the data have been collapsed onto one graph, and the data fall

on the curve. The curves were drawn from

Y/A = 2.45 (Nu/A')/[1.45 + (Nu/A')] [4.274]

Nc/Nc, = 0.59 +0.41 (NJA') [4.275]

Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-

97 through 4-100.








75
Florida: Bahiagrass

Data from Blue (1987) for bahiagrass were used. The analysis of variance for the

bahiagrass grown on the Entisol and Spodosol is shown in Table 4-40 (dry matter and N

removal x 2 soils x 5 N). Comparisons among modes 1 and 2 result in a variance ratio of

1792, that is significant at the 0.1% confidence level. Comparisons among modes 3 and 2

lead to a variance ratio of 12.24, that is significant at the 0.5% confidence level.

Comparisons among modes 3 and 4 lead to a variance ratio of 6.41, that is significant at

the 1% confidence level. Comparisons among modes 5 and 2 lead to a variance ratio of

10.72, that is significant at the 0.5% confidence level. Comparison among modes 5 and 4

leads to a variance ratio of 5.94, that is significant at the 2.5% confidence level. The

statistics suggest that mode 4, individual A for both soils, individual b for yield and plant

N removal, and common c, accounts for all the significant variation, since F(3,8,97.5) =

5.42, and F(3,11,99) = 6.22. This mode estimates the following b and c parameters:

Entisol, dry matter: b = 1.46 � 0.07

Spodosol, dry matter: b = 1.31 � 0.07

Entisol, N removal: b' = 1.89 � 0.08

Spodosol, N removal: b' = 1.83 � 0.08

c = 0.0118

while mode 5 estimates the following b and c parameters:

both soils, dry matter: b = 1.39 � 0.05

both soils, N removal: b' = 1.86 � 0.06

c = 0.0118

The estimates and their standard errors overlap, suggesting that mode 5 is correct.

Furthermore, since there are less parameters to estimate assuming the b values are not

affected by the soils, mode 5 will be used for estimation. The overall correlation

coefficient of 0.9985 and adjusted correlation coefficient of 0.9978 are calculated by mode








76

5. The error analysis for the parameters is shown in Table 4-41. Results are shown in

Figure 4-101 for both soils, where the curves are drawn from

Entisol: Y = 11.14/[1 +exp(1.39-0.0118N)] [4.276]

Nu = 311.0/[1 + exp(1.86 - 0.0118N)] [4.277]

Spodosol: Y = 19.39/[1 +exp(1.39 - 0.0118N)] [4.278]

N, = 201.5/[1 + exp(1.86 - 0.0118N)] [4.279]

From these results, it follows that plant N concentration, shown in Figure 4-101, is

estimated from

Entisol: N, =27.9 [1 + exp(1.86 - 0.0118N)]/[1 + exp(1.39 - 0.0118N)] [4.280]

Spodosol:N, = 10.4 [1 + exp(1.86 - 0.0118N)]/[1 + exp(1.39 - 0.0118N)] [4.281]

As shown, the data are described well by these equations.

From Postulate 3, the relationships between yield and plant N removal and

between plant N concentration and plant N removal for both soils, shown in Figure 4-102,

are described by

Entisol: Y = 29.7 NJ(518 + N.) [4.282]

Ne= 17.4 + 0.0337N, [4.283]

Spodosol: Y = 51.7 Nu/(336 + N,) [4.284]

Ne= 6.5 + 0.0193N. [4.285]

The dimensionless plot for bahiagrass is shown in Figure 4-103, and validates the

form of the model. Note that all of the data have been collapsed onto one graph, and the

data fall on the curve. The curves were drawn from

Y/A = 2.67 (NJA')/[1.67 + (NJA')] [4.286]

Ne/Nem = 0.63 +0.37 (NJA') [4.287]

Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-

104 through 4-107.








77
England: Ryegrass

The data for this analysis were taken from Morrison et al. (1980). The analysis of

variance is shown in Table 4-42 (dry matter and N removal x 20 sites x 6 N).. As shown

in the table, all of the comparisons are highly significant (0.1%). This is due in part to the

large degrees of freedom. If comparison is made among the mean sums of squares (MSS),

mode 4 has the smallest MSS with the exception of mode 2 (which fits each individually).

Mode 4 is the preferred mode since it requires less parameters to estimate (81 versus 120).

Mode 4 assumes an individual A and b for each site, dry matter yield and plant N removal

and common c for all. The overall correlation coefficient and adjusted correlation

coefficient calculated from mode 4 are 0.9935 and 0.9903, respectively. The error

analysis for the parameters is shown in Table 4-43. A comparison among estimated

maximum plant N concentrations and Ab's among sites is in Table 4-44. These variables

have been plotted in Figure 4-108, the mean and �2 standard errors of the estimates for all

sites are designated by a solid and dashed horizontal line, respectively. This suggests an

important result, the Ab is constant from site to site for a particular grass. If this is true,

then in conjunction with the postulates of the extended model, one less parameter would

need to be calculated. It also appears from Figure 4-108, that the ratio of estimated

maximum plant N removal to estimated maximum dry matter yield, or estimated maximum

plant N concentration is constant as well for a particular grass from site to site. Due to

the large number of sites, the results of the regression will not be plotted directly. The

dimensionless plot of ryegrass over all twenty sites is shown in Figure 4-109, with the

curves drawn assuming a constant Ab, namely the mean of the set, 0.83. The curves are

drawn by

Y/A = 1.77 (NJA')/[0.77 + (Nu/A')] [4.288]

NNcN = 0.44 +0.56 (NeJA') [4.289]




Full Text
Dry Matter Yield, Mg/ha Plant N Removal, kg/ha Plant N Concentration, g/kg
5
Figure 1-1 Response of dry matter yield, N removal, and N concentration for
bermudagrass as a function of applied N grown over two years at five
clipping intervals at Tifln, GA. Data from Prine and Burton (1956).
Symbols will be defined later in the text.


241
Nitrogen Removal, kg/ha
Figure 4-114 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for rye grown at Tifln, GA. Data from Walker and Morey
(1962). Curves drawn by Eq. [4 299] and [4.300],


201
20
18
16
t 14
s
H 12
o
S io
s
! *
B
o
-5
S 6
O Non-lrrigatcd
Irrigated
o
&

f)
x
6
rn
8
10
12
Measured Dr\ Matter Yield. Mg/ha
Figure 4-74 Scatter plot of seasonal dry matter yield for tall fescue averaged over three
years at Fayetteville, AR, with and without irrigation. Original data from
Huneycutt el al. (1988).


254
2. The logistic model is well behaved. The response is positive for the entire
domain of applied nitrogen The slope is greater than zero (dY/dN > 0) for all values of
N, i.e., it is monotone increasing. It asymptotically approaches a maximum at high N and
zero at low N.
3. As shown with various data sets, harvest interval, water availability and plant
partitioning can be accounted for in the linear parameter, A. This means that once the
relationship has been identified, the effects of different levels and combinations of these
factors on dry matter yield or nutrient removal response to applied nutrients can be
estimated. Had these factors been accounted for only by the nonlinear parameters b
and/or c, the task of simplification would have been greatly limited Furthermore, since
the effects are linear, the dependence upon yield and N removal can be factored into
product terms:
A = Aw Ah Ap [5.1]
where A = maximum yield or N removal parameter for Eq [2.8] or [2.9],
Aw = water availability coefficient.
Ah = harvest interval coefficient,
Ap = plant partitioning coefficient.
Because these effects fall into products, averaging over years only affects the A parameter.
Why was the A parameter linearly related to harvest interval? The answer lies in how the
plant grows between harvests. Overman et a/. (1989) have shown that between harvests
the plant follows a quadratic intrinsic growth function When these sawtooth growths for
multiple harvests are summed, the result is linear dependence of seasonal yield on harvest
interval. Further, if the harvest interval is too long, the quadratic trend falls off and this
effect appears in the sum as well as in the slow loss of linearity.
4. Dimensionless plots are a valuable tool in evaluating the form of a model
(Segr, 1984, p. 168). All data are normalized to the same scale, thereby removing the


115
Table 4-35. Analysis of Variance on Model Parameters for Corn Grown on
Goldsboro Sandy Loam at Kinston, NC, Both Grain and Total.
Mode
Parameters
Estimated
df
Residual Sum
of Squares
Mean Sum
of Squares
F
(1) Com A,b,c
3
17
34443.380
2026.081
-
(2) Ind A,b,c
12
8
15.942
1.993
-
(l)-(2)
9
34427.438
3825.271
1920**
(3) Ind A, Com b,c
6
14
337.763
24.126
-
(3)-(2)
6
321.821
53.637
26.9**
(4) Ind A, b, Com c
9
11
31.448
2.859
-
(4)-(2)
3
15.506
5.169
2.59
(3)-(4)
3
306.315
102.105
35.7**
(5) Ind A, Com c, Ind b
(dm and Nu)
7
13
45.635
3.510
_
(5)-(2)
5
29.694
5.939
2.98
(3)-(5)
1
292.128
292.128
83.2**
(5H4)
2
14.187
7.094
2.48
Source: Original data from Kamprath (1986).
** Significance level of 0.001
F(9, 8,99.9) =11.77
F(6, 8,99.9) =12.86
F(3, 8,95) =4.07
F(3,11,99.9) =11.56
F(5, 8,95) =3.69
F(l,13,99.9) =17.81
F(2,11,95) =3.98


Dry Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
171
0.0 0.2 0.4 0.6 0.8 1.0 1.2
N Removal/Estimatcd Maximum
Figure 4-44 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for three different number of cuttings over
the season for ryegrass grown at England. Original data from Reid (1978);
curves drawn from Eq. [4.124] through [4.129],


4-25 Analysis of Variance of Model Parameters for Tall Fescue Grown at
Fayetteville, AR, Averaged over Three Seasons, with and without Irrigation. 105
4-26 Error Analysis for Model Parameters of Tall Fescue Grown at Fayetteville,
AR, Averaged over Three Seasons, with and without Irrigation 106
4-27 Analysis of Variance for Bermudagrass Grown at Eagle Lake, Texas, over
Two Years 107
4-28 Analysis of Variance for Bahiagrass Grown at Eagle Lake, Texas, over Two
Years 108
4-29 Analysis of Variance for Bermudagrass and Bahiagrass Grown at Eagle
Lake, Texas, over Two Years 109
4-30 Averaged Seasonal Dry Matter Yield, Plant N Removal, and Plant N
Concentration for Bermudagrass and Bahiagrass Grown at Eagle Lake, TX.. 110
4-31 Analysis of Variance on Model Parameters for Bermudagrass and Bahiagrass
Grown at Eagle Lake, TX, Averaged to Estimate b and c Parameters Ill
4-32 Error Analysis for Model Parameters of Bermudagrass and Bahiagrass
Grown at Eagle Lake, TX Averaged over Years 112
4-33 Error Analysis for Model Parameters of Bermudagrass and Bahiagrass
Grown at Eagle Lake, TX over Two Years 113
4-34 Analysis of Variance on Model Parameters for Corn Grown on Dothan
Sandy Loam at Clayton, NC, Both Grain and Total 114
4-35 Analysis of Variance on Model Parameters for Corn Grown on Goldsboro
Sandy Loam at Kinston, NC, Both Grain and Total 115
4-36 Analysis of Variance on Model Parameters for Grain and Total Plant of Corn
Grown on Dothan and Goldsboro Sandy Loam at Clayton and Kinston, NC.. 116
4-37 Error Analysis of Model Parameters for Grain and Total Plant of Corn
Grown on Dothan and Goldsboro Sandy Loam at Clayton and Kinston, NC.. 117
4-38 Analysis of Variance on Model Parameters for Grain and Total Plant of Corn
Grown on Portsmouth Very Fine Sandy Loam at Plymouth, NC 118
4-39 Error Analysis for Model Parameters for Grain and Total Plant of Corn
Grown on Portsmouth Very Fine Sandy Loam at Plymouth, NC 119
x


7
(Freund and Littell, 1991). The exponential and Mitscherlich models have also been used
in describing crop growth (France and Thornley, 1984); however these models have
limitations as well. The exponential model suggests that at high levels of input, the
response approaches infinity, and at low levels approaches zero response. The
Mitscherlich model suggests that at low levels of input the response approaches negative
infinity, and at high levels approaches a maximum response. These two models describe
the response well at different extremes. A combination of the two could potentially
explain the response well over all ranges of input.
The logistic equation, first proposed by Verhulst and later popularized by Pearl
(Kingsland 1985), was first used as a population model. It has been used with high
correlation coefficients (R > 0.99) for nitrogen removal (uptake) and dry matter
production of various forage grasses (Reck, 1992; Overman et a/., 1990a, 1990b, 1994a,
1994b; Overman and Blue, 1991). Overman (1995a) has developed the relationship of the
exponential and Mitscherlich models to the logistic, and a short summary of that
discussion will be included here. In differential form, the exponential model assumes that
the response of dry matter to applied N is proportional to the amount of dry matter
present. Furthermore, the exponential is suitable at low N and asymptotically approaches
zero along the negative N (reduced soil N) axis. In mathematical form,
lower N: dy/dN ay [2.1]
In differential form, the Mitscherlich model assumes that the response of dry matter to
applied N is proportional to the unfilled capacity, ym y, of the system, where ym is the
maximum yield. As mentioned before, the Mitscherlich is suitable at high N and
asymptotically approaches a maximum yield along the positive N axis. In mathematical
form,
higher N: dy/dN a ym y
This leads us to assume a composite function
[2.2]


256
of regressing the two responses to a logistic with a common c. An alternative approach
would be to fit the logistic to the dry matter yield response to applied N and the hyperbolic
to the dry matter yield response to plant N removal This is a more fundamental approach,
since it involves fitting a relationship between two extensive variables: dry matter and
plant N removal. In this relationship, the effect of the independent variable, applied N, has
been removed and only the measured variables are included. This procedure would be
more difficult to compute since it would involve regressing data to two different functions
simultaneously. It is important to remember that the hyperbolic relationship is based upon
two measurable quantities, i.e. the independent variable (applied nitrogen) has been
removed from the picture From data set to data set, the same hyperbolic trend appears.
What physically in the system causes this relationship? Recall the dimensionless parameter
for the hyperbolic, Ym/A, represented by Equation [2.14] This describes the maximum
dry matter attained at high levels of plant N removal For the studies investigated in this
work, the value of this ratio ranges from 1 7 to 2.7, suggesting that approximately half of
the potential capacity of the system is being observed in the various experiments. For
Yni/A = 2, it follows from Equation [2.14] that Ab = 0.69. One possible explanation for
this lies in a paper coupling various plant components for forage production by Overman
(1995b). He has shown that the response of leaf area to C02 concentration is hyperbolic.
Assuming a current C02 concentration equal to the level in the atmosphere, 300
pmol/mol, it appears from Figure 13 in the article that for days after planting (DAP)
greater than 30, only a little more than half of the potential maximum leaf area is attained.
Although this has not directly linked dry matter production to C02 concentration,
production is directly related to leaf area. This suggests that at higher levels of
atmospheric C02 concentration would result in higher leaf areas, which would then result
in higher dry matter production


204
C3
00
a
>
o
a
u
o£
ed
3
ts
to
o
Oi
75
50
25
-25
-50
-75
O

Non-Irrigatcd
Irrigated
G

TT-O-
(1
()

jgJU
()
o
n
100
200 300
Predicted N Removal, kgdia
400
500
Figure 4-77 Residual plot of seasonal plant N removal for tall fescue averaged over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt et al. (1988). Solid line is mean and dashed lines are 2
standard errors.


ESTIMATION OF DRY MATTER AND NITROGEN REMOVAL
BY THE LOGISTIC EQUATION
By
DENISE MARIE WILSON
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1995
UNIVERSITY of FLORIDA LIBRARIES


Table 4-16. Error Analysis for Model Parameters of Ryegrass Grown at England, with
Three Different Numbers of Cuttings over the Season for 1969.
Component
Number of
Cuttings
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter
10
A, Mg/ha
9.42
0.105
0.011
5
A
12.72
0.127
0.010
3
A
12.75
0.121
0.009
N Removal
10
A, kg/ha
377.0
4.72
0.013
5
A
403.5
5.02
0.012
3
A
363.0
4.46
0.012
Dry Matter
10
b
1.74
0.046
0.026
5
b
1.23
0.044
0.036
3
b
0.90
0.044
0.049
N Removal
All
b
2.15
0.050
0.023
Both
All
c, ha/kg
0.0080
0.0001
0.013
Source: Original data from Reid (1978).


161
O 75 150 225 300 375 450 525 600 675 750
Nitrogen Removal, kg/ha
Figure 4-34 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a six week clipping interval over two years for bermudagrass
grown at Tifton, GA. Data from Prine and Burton (1956); curves drawn
from Eq. [4.95] through [4.98],


Dry Matter Yield/Estimated Maximum Nitrogen Concentration/Estimated Maximum
216
Nitrogen Removal/Estimatcd Maximum
Figure 4-89 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for grain and total plant of corn grown at
Clayton and Kinston, NC, on Dothan and Goldsboro soils, respectively.
Original data from Kamprath (1986); curves drawn from Eq. [4.262] and
[4.263],


119
Table 4-39. Error Analysis for Model Parameters for Grain and Total Plant of Corn
Grown on Portsmouth Very Fine Sandy Loam at Plymouth, NC.
Component
Part
Parameter
Estimate
Standard Error
Relative Error
Dry Matter Yield
Grain
A, Mg/ha
9.48
0.168
0.018
Total
A
16.60
0.293
0.018
N Removal
Grain
A, kg/ha
126.2
3.22
0.026
Total
A
147.3
3.76
0.026
Dry Matter Yield
Both
b
-0.065
0.0458
0.705
N Removal
Both
b
0.46
0.047
0.102
Both
Both
c, ha/kg
0.0119
0.0007
0.059
Source: Original data from Kamprath (1986).


80
Y = 12.5 Nu/(338 + Nu) [4.299]
Nc= 27.1 +0.0800NU [4.300]
y = ^s.sPu/-ivn + Pu) [4.301]
Pe = 6.4 0.0037PU [4.302]
Y = 7.28 K/(78.4 + Ku) [4.303]
Kc= 10.8 +0.1374KU [4.304]
The dimensionless plot for rye is shown in Figure 4-117, and validates the form of
the model. This plots supports the result that the Ab¡ is different for each nutrient. The
curves were drawn from
Y/A = 2.30(NU/A')/[1.30 + (NU/A')] [4.305]
Y/A = -49.5 (Pu/Ayt-50.5 + (Pu/A')] [4.306]
Y/A = 1.34(Ku/A')/[0.34 + (Ku/A')] [4.307]
Nc/Ncm = 0.57 + 0.43 (N./A) [4.308]
Pc/Pcm = 1.02-0.02 (Pu/A) [4.309]
Kc/Kem= 0.25 + 0.75 (Ku/A1) [4.310]
Scatter and residual plots of dry matter yield and plant nutrient removals are shown in
Figures 4-118 through 4-125.


97
Table 4-17. Analysis of Variance for Bermudagrass over Three Seasons, Irrigated and
Non-irrigated, Grown at
Fayett
eville, AR.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
81
2114817.76
26108.86
-
(2) Ind A,b,c
36
48
17913.02
373.19
-
(1)-(2)
33
2096904.75
63542.57
170.3**
(3) Ind A, Com b,c
14
70
34746.83
496.38
-
(3)-(2)
22
16833.81
765.17
2.05t
(4) Ind A,b Com c
25
59
21706.70
367.91
-
(4)-(2)
11
3793.68
344.88
0.92
(3)-(4)
11
13040.13
1185.47
3.22++
(5) Ind A, Com c, Ind
b (Irr, dm, Nu)
17
67
28178.64
420.58
(5)-(2)
19
10265.63
540.30
1.45
(3)-(5)
3
6568.19
2189.40
5.21++
(5)-(4)
8
6471.95
808.99
2.20+
(6) Ind A, Com c, Ind
b (dm, Nu)
15
69
26492.41
383.95
(6)-(2)
21
8579.40
408.54
1.09
(3)-(6)
1
8254.42
8254.42
21.5**
(6H4)
10
4785.72
478.57
1.30
Source: Original data from Huneycutt et al. (1988).
Significant at the 0.001 level
Significant at the 0.005 level
1 Significant at the 0.025 level
Significant at the 0.05 level
F(33,48,99.9)= 2.66
F(22,48,97.5)= 1.97
F( 11,48,95) = 1.99
F(11,59,99.5)= 2.82
F( 19,48,95) = 1.81
F( 3,67,99.5)= 4.68
F( 8,59,95) =2.10
F(21,48,95) = 1.78
F( 1,69,99.9)= 11.81
F( 10,59,95) =2.00


27
Table 3-2. Seasonal Dry Matter Yield of Coastal Bermudagrass and Pensacola Bahiagrass
Species
Irrigation
Year
0
Applied Nitrogen, kg/ha
168 336
672
Bermuda
No
1957
3.74
Mg/ha-
12.52
17.41
22.32
1958
4.70
11.38
17.44
21.56
1959
3.40
9.04
13.36
20.44
Yes
1957
3.49
11.70
18.32
21.08
1958
4.33
10.75
17.89
22.61
1959
4.76
11.94
19.42
24.43
Bahia
No
1957
4.56
11.70
16.39
20.32
1958
3.63
10.39
16.85
22.23
1959
2.67
8.19
14.14
19.62
Yes
1957
4.31
10.35
16.81
23.26
1958
3.24
9.25
15.69
22.33
1959
3.82
9.52
16.33
22.43
Source: Data from Evans et al. (1961).


This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepteehps partial fulfillment of the
requirements for the degree of Doctor of Philosophy
December 1995
M. Phillips
Dean, College of Engineering
Karen A. Holbrook
Dean, Graduate School


36
Table 3-11. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Grain and Total Plant of Corn Grown on Dothan and Goldsboro Soils at
Clayton and Kinston, North Carolina, Respectively.
Site Part
Applied
Nitrogen
kg/ha
Dry Matter
Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
Dothan Grain
0
4.37
47
10.8
56
7.12
74
10.4
112
9.77
113
11.6
168
10.81
135
12.5
224
11.04
150
13.6
Total
0
9.09
61
6.7
56
14.67
92
6.3
112
18.35
135
7.4
168
19.11
166
8.7
224
19.90
188
9.4
Goldsboro Grain
0
3.00
32
10.7
56
5.47
62
11.3
112
6.93
87
12.6
168
7.46
101
13.5
224
7.57
107
14.1
Total
0
6.63
36
5.4
56
10.60
71
6.7
112
13.10
104
7.9
168
13.55
120
8.9
224
13.92
134
9.6
Source: Data from Kamprath (1986).


Dry Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
156
Figure 4-29 Seasonal dry matter yield, plant N removal, and plant N concentration for a
six week clipping interval over two years for bermudagrass grown at
Tifln, GA. Data from Prine and Burton (1956); curves drawn from Eq.
[4.65] through [4.68], [4.79], and [4.80]


183
Figure 4-56 Residual plot of seasonal plant N removal for bermudagrass grown over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt el al. (1988). Solid line is mean and dashed lines are 2
standard errors.


13
where Ab = b' b. In order for Ym and K' to be positive, b' must be greater than b. These
relationships can be reduced to dimensionless form by dividing the dry matter yield by its
estimated maximum and the plant N removal by its estimated maximum. After doing this,
the parameters of the hyperbolic relationship are defined as
4 = 1 = \
A 1 eb~b' \-e~Ab
[2.14]
and
/C 1 1
A ~ eh'~h 1 1
[2.15]
It should be noted that the parameters used to describe the dimensionless relationship of
seasonal dry matter to plant N removal (two measurable quantities) are only dependent
upon the Ab. Dimensionless plots have been useful tools for engineers in many fields.
Dimensionless plots were used to develop and determine dimensionless numbers, such as
the Reynolds number in hydraulic flow. For example, James Clerk Maxwell used a
dimensionless plot in the 1860s to describe the distribution of molecular velocities in a gas
(Segr, 1984). The greatest value of a dimensionless plot is the ability to collapse data
sets with different ranges onto the same scale for comparison. This also aids in the search
for possible simplification. Equation [2.11] can be rearranged to the form
Nc = (K'/Ym) + (l/Ym)Nu [2.16]
This equation predicts a linear relationship between plant N concentration and plant N
removal.
The extended triple logistic (NPK) model is given by
(1 + eb'~cNX1 + e CfP)(1 + [2.17]


I dedicate this dissertation to my parents, Robert and Marian Shelton, my second-
parents, Brooks and Carol Wilson, and especially to my husband, Russell.


259
estimation purposes, the key is narrowing down the value for c. As mentioned before,
this is believed to be the most critical parameter. As with many engineering situations,
safety factors are always included. If the use of this model in a specific situation is to
estimate N removal for a field receiving municipal wastewater, it would be better to
underestimate the parameters in order to overdesign the system. By underestimating the
parameters, the seasonal estimates are underestimated. Because these are underestimated,
any design using these numbers will be conservative in nature. For example, if the
seasonal plant N removal was estimated to be 400 kg/ha and it really was 450 kg/ha, and
this value was used in a nutrient budget to determine the fate of nitrogen in a water reuse
system, the system would be designed to accommodate 400 kg/ha. Since the plant would
really be removing 450 kg/ha, even less nitrogen would make it into the aquifer than was
estimated. Similarly, the farmer would be producing more dry matter than he had
estimated. The only situation where underestimating the parameters could lead to
problems are those where a target plant N removal or dry matter yield is desired and the
equation is used to back calculate how much nitrogen should be applied. Although the c
parameter is important in estimation, the model is not very sensitive to this parameter. A
"large" c value results in a steeper slope at the midpoint of the curve; that is, the plant is
reaching maximum much faster. A "small" value of c results in a flatter slope at the
midpoint. As a consequence, small values of c should be used for conservative estimates.
A plot of the normalized logistic equation is shown in Figure 5-1 with b = 2.0 for all
curves and c values of 0.008, 0.010 and 0.012 ha/kg to demonstrate the sensitivity of the
parameter. At N = 200 kg/ha, a 20% error in c causes a 20% error in response. For c =
0.01, this is the point of maximum slope since N1/2 = b/c = 2/0.01 = 200. An example of
how this model could be used in practice is presented below:
Assume effluent from a municipality is available for water reuse, and further that
the average N application rate for the system is 500 kg/ha. The farmer wishes to


9
ln(-1 ) = b-cN [2.6]
y
To determine if the data follow a logistic curve, A/y 1 vs. N can be plotted on semilog
paper to check for linearity. If the plot is linear, then the data can be described well by the
logistic. The parameters can be determined by one of two methods: regression on
linearized data or nonlinear regression on original data (Downey and Overman, 1988).
There are advantages to using both methods. The linearization method provides an easy
procedure to estimate the parameters with a hand calculator. By this method, an estimate
of A can be obtained by examining a plot of the y versus N on linear paper. The curve will
appear to approach a maximum. This maximum is the estimate of A. Several attempts
may be required to optimize A. Estimates of b and c then follow from linear regression of
ln(A/y 1) vs. N (Draper and Smith, 1981). An example of this is shown in Figure 2-1.
The data for this analysis are taken from a study of dallisgrass [Paspalum dilatatum Poir ]
grown at Baton Rouge, Louisiana (Robinson et ai, 1988). Linearized dry matter yield
and plant nitrogen removal are plotted. The linear trend suggests that these two responses
can be described well by the logistic equation. The fact that the lines are parallel suggests
that the c parameter for both dry matter yield and plant nitrogen removal are the same.
The figure also suggests that the b parameter value for plant nitrogen yield is larger (more
positive) than the b value for dry matter yield. The nonlinear regression method requires a
computer program written to perform the regression and statistical inference and
diagnostic information. The regression can be conducted with SAS for the simple case.
As more complex cases are needed, SAS is not easily programmed to perform the
statistical analysis. In the nonlinear regression, parameters are estimated using least
squares (Bates and Watts, 1988; Ratkowsky, 1983). Unfortunately, only the A parameter
can be explicitly solved. The b and c parameters need to be solved implicitly. Second
order Newton-Raphson iteration is used (Adby and Dempster, 1974). For additional


196
Figure 4-69 Residual plot of seasonal dry matter yield for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation. Original data from
Huneycutt el al. (1988). Solid line is mean and dashed lines are 2 standard
errors.


4-89 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for grain and total plant of corn grown at
Clayton and Kinston, NC on Dothan and Goldsboro soils, respectively 216
4-90 Scatter plot of seasonal dry matter yield for grain and total plant of corn
grown at Clayton and Kinston, NC on Dothan and Goldsboro soils,
respectively 217
4-91 Scatter plot of seasonal plant N removal for grain and total plant of corn
grown at Clayton and Kinston, NC on Dothan and Goldsboro soils,
respectively 218
4-92 Residual plot of seasonal dry matter yield for grain and total plant of corn
grown at Clayton and Kinston, NC on Dothan and Goldsboro soils,
respectively 219
4-93 Residual plot of seasonal plant N removal for grain and total plant of corn
grown at Clayton and Kinston, NC on Dothan and Goldsboro soils,
respectively 220
4-94 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for grain and total plant of corn grown at
Plymouth, NC 221
4-95 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for grain and total plant of corn grown at Plymouth, NC 222
4-96 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for grain and total plant of corn grown at
Plymouth, NC 223
4-97 Scatter plot of seasonal dry matter yield for grain and total plant of corn
grown at Plymouth, NC 224
4-98 Scatter plot of seasonal plant N removal for grain and total plant of corn
grown at Plymouth, NC 225
4-99 Residual plot of seasonal dry matter yield for grain and total plant of corn
grown at Plymouth, NC 226
4-100 Residual plot of seasonal plant N removal for grain and total plant of corn
grown at Plymouth, NC 227
4-101 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bahiagrass grown on two soils in Florida 228
xix


Dry Matter Yield. Mg/ha N Removal, kg/ha N Concentration, g/kg
131
Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for dallisgrass grown at Baton Rouge, LA.
Data from Robinson et al. (1988); curves drawn from Eq. [4.5] through
[4.7].
Figure 4-4


106
Table 4-26. Error Analysis for Model Parameters of Tall Fescue Grown at
Fayetteville, AR, Averaged over Three Seasons, with and without Irrigation.
Component
Irrigation
Parameter
Estimate
Standard Error
Relative Error
Dry Matter
No
A, Mg/ha
8.67
0.15
0.017
Yes
A
15.36
0.27
0.018
N Removal
No
A, kg/ha
250.7
4.95
0.020
Yes
A
417.4
8.34
0.020
Dry Matter
Both
b
0.99
0.069
0.070
N Removal
Both
b
1.53
0.079
0.052
Both
Both
c, ha/kg
0.0081
0.0004
0.049
Source: Original data from Huneycutt et al. (1988).


Table 4-19. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration
for Bermudagrass Grown at Fayetteville, AR Averaged over Three Years.
Applied Nitrogen Dry Matter Yield N Removal N Concentration
kg/ha Mg/ha kg/ha g/kg
Non-Irrigated
0
2.00
26
12.80
112
7.50
117
15.52
224
10.83
199
18.45
336
14.00
274
19.63
448
17.53
380
21.71
560
17.29
386
22.35
672
17.71
428
Irrigated
24.11
0
3.48
51
14.51
112
9.65
157
16.27
224
13.74
253
18.40
336
19.22
359
18.72
448
21.48
415
19.31
560
22.89
482
21.07
672
23.80
532
22.35
Source: Original data from Huneycutt et al. (1988).


Diy Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
154
Figure 4-27 Seasonal dry matter yield, plant N removal, and plant N concentration for a
three week clipping interval over two years for bermudagrass grown at
Tifton, GA. Data from Prine and Burton (1956); curves drawn from Eq.
[4.57] through [4.60], [4.75], and [4.76],


258
to applied N. What in the system causes this phenomenon? At this point, it is not clear
what physiological process might explain this behavior. Additional investigation might
reveal a connection between this parameter and various effects: type of grass, water, soil
characteristics, etc. It also appears from the table that rye and corn have higher c values
than the other grasses. One possible explanation is that these grasses are annuals, while
the others are grown as perennials.
9. This analysis has focused on seasonal quantities. Although time is not
explicitly contained within the model, it is implicit within the A parameter. The maximum
parameter contains within it the accumulation of dry matter over time, as evidenced by its
relationship to harvest interval (Maryland and Tifton, GA, studies).
10. The models developed and analyzed in this study adequately describe seasonal
relationships. The focus has been to determine, by dimensionless plots, if the form of the
logistic adequately describes the behavior, not to determine parameter values. No attempt
has been made to develop a "cookbook" model for the general public, so that site specific
information, such as weather and soil properties, can be entered and automatically produce
estimates. Rather the approach as been to determine if the form is sufficient in describing
the relationship. If so, then the model can be used universally with little input.
Although this study has not developed a table of parameter values, reasonable
estimates can be obtained. Of the parameters, b is the least significant since it is the
response at N = 0. Typically for sandy soils the b estimate is high (low intercept) as a
result of low residual nitrogen in the soil. The A parameter can be estimated relatively
easily since it is an estimate of the maximum Very often in practice, high application rates
are not used since the efficiency of the system to remove N decreases as the maximum is
approached. The normal operating range is concentrated around the Nt/2 value. Recall
that this parameter is the amount of nitrogen applied to achieve half of the maximum yield.
This is also the point of maximum slope (incremental increase of yield with N). For


89
Table 4-9. Analysis of Variance of Model Parameters Used to Describe Bermudagrass
Grown at T
orsby, AL with Two Clipping Interva
s.
Mode
Parameters
Estimated
df
Residual Sum
of Squares
Mean Sum
of Squares
F
(1) Com A,b,c
3
21
1018614.201
48505.438
-
(2) Ind A,b,c
12
12
3368.695
280.725
-
(D-(2)
9
1015245.506
112805.056
402
(3) Ind A, Com b,c
6
18
11170.579
620.588
-
(3)-(2)
6
7801.885
1300.314
4.63+
(4) Ind A, b, Com c
9
15
4274.029
284.935
-
(4)-(2)
3
905.335
301.778
1.08
(3)-(4)
3
6896.550
2298.850
8.07++
(5) Ind A, Com c, Ind b
(dm and Nu)
7
17
4261.737
250.690
(5)-(2)
5
893.042
178.608
0.64
(3)-(5)
1
6908.842
6908.842
27.56
Source: Original data from Doss et al. (1966).
Significance level of 0.001
Significance level of 0.005
Significance level of 0.05
F(9,12,99.9) =7.48
F(6,12,95) =3.00
F(3,12,95) =3.89
F(3,15,99.5) =6.48
F(5,12,95) =3.11
F( 1,17,99.9) = 15.72


46
shown that averaging over years is appropriate, since variations due to water availability
and harvest interval appear in the linear parameter A. The averaged data are in Table 4-5
and the error analysis for the averaged data is in Table 4-6. The overall correlation
coefficient of 0.9981 and adjusted correlation coefficient of 0.9969 are calculated. Note
that the estimates of b and c are the same as for the unaveraged data, supporting what
Overman el al. (1990a, 1990b) have found previously. Results are shown in Figure 4-1,
where curves for Coastal bermudagrass and Pensacola bahiagrass dry matter, irrigated and
non-irrigated, are drawn from the following equations:
Coastal bermudagrass, non-irrigated: Y =21 57/[ 1 + exp(1.39 0.0078N)] [4.1]
Coastal bermudagrass, irrigated: Y =23.44/[l + exp(1.39 0.0078N)] [4.2]
Pensacola bahiagrass, non-irrigated: Y =21 49/[ 1 + exp(l .57 0.0078N)] [4.3]
Pensacola bahiagrass, irrigated: Y =22.73/[l + exp(1.57 0.0078N)] [4.4]
The scatter and residual plots of seasonal dry matter are given in Figures 4-2 and
4-3. The mean and the 2 standard errors of the residuals are shown by the solid and
dashed horizontal lines, respectively. As shown in Figure 4-3, all of the data fit between
2 standard errors with no apparent trend.
Evaluation of the Extended Logistic Model
Baton Rouge. LA: Dalliserass
The data for this analysis are taken from Robinson el al. (1988). The analysis of
variance is shown in Table 4-7 (dry matter and N removal x 6 N). Comparison among
modes 1 and 2 results in a variance ratio of 3736 and is significant at the 0.1% level.
Comparison among modes 3 and 2 leads to a variance ratio of 127.8 which is also
significant at the 0.1% level. Mode 4, with individual A and b for dry matter and N
removal, and common c describes the data best, since F( 1,6,95) = 5.99, and F( 1,7,99.9) =


95
Table 4-15. Analysis of Variance of Model Parameters on Ryegrass Grown at England,
with Three Different Numbers of Cuttings over the Season for 1969.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
123
1698556.455
13809.402
-
(2) Ind A,b,c
18
108
3703.415
34.291
-
(l)-(2)
15
1694853.040
112990.203
3295**
(3) Ind A, Com b,c
8
118
35982.858
304.939
-
(3)-(2)
10
32279.443
3227.944
94 r*
(4) Ind A,b Com c
13
113
6635.747
58.723
-
(4)-(2)
5
2932.332
586.466
17.1
(3)-(4)
5
29347.111
5869.422
o
o


(5) Ind A, Com c, Ind b
(y and Nu)
9
117
8196.505
70.056
(5)-(2)
9
4493.090
499.232
14.6**
(3)-(5)
1
27786.353
27786.353
397**
(5)-(4)
4
1560.758
390.189
6.64*
(6) Ind A, Com c, Ind b
(y) and Com b (Nu)
11
115
6050.656
52.614
(6)-(2)
7
2347.241
335.32
9.8"
(3)-(6)
3
29932.202
9977.401
190**
(5)-(6)
2
2145.849
1072.925
20.4
Source: Original data from Reid (1978).
Significant at the 0.001 level
F( 15,108,99.9) =2.81
F(10,108,99.9) =3.27
F( 5,108,99.9) =4.45
F( 5,113,99.9) =4.44
F( 9,108,99.9) =3.41
F( 1,117,99.9) =11.37
F( 4,113,99.9) =4.97
F( 7,108,99.9) =3.80
F( 3,115,99.9) =5.80
F( 2,115,99.9) =7.34


92
Table 4-12.
Error Analysis for Model Parameters of Bermudagrass
Maryland and Cut at Five Harvest Intervals.
Grown at
Component
Harvest
Interval
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter
3.2, weeks
A, Mg/ha
13.90
0.28
0.020
3.6
A
13.65
0.28
0.021
4.3
A
14.93
0.30
0.020
5.5
A
17.56
0.36
0.021
7.7
A
19.34
0.39
0.020
N Removal
3.2
A, kg/ha
469.5
10.4
0.022
3.6
A
444.7
9.9
0.022
4.3
A
440.2
9.8
0.022
5.5
A
444.0
9.9
0.022
7.7
A
409.7
9.0
0.022
Dry Matter
All
b
1.78
0.08
0.045
N Removal
All
b
2.59
0.10
0.039
Both
All
c, ha/kg
0.0112
0.0005
0.045
Source: Original data from Decker et al. (1971).


227
Predicted N Removal, kg/ha
Figure 4-100 Residual plot of seasonal plant N removal for grain and total plant of corn
grown at Plymouth, NC. Original data from Kamprath (1986). Solid line is
mean and dashed lines are 2 standard errors.


104
Table 4-24. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration
for Tall Fescue Grown at Fayetteville, AR, Averaged over Three Seasons.
Applied Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
0
2.09
-Non-Irrigated
43
20.57
112
4.82
103
21.37
224
6.35
148
23.31
336
7.43
189
25.44
448
8.30
225
27.11
560
7.90
230
29.11
672
8.38
248
29.59
0
3.53
Irrigated
74
20.96
112
7.38
149
20.19
224
9.56
208
21.76
336
13.22
324
24.51
448
14.51
373
25.71
560
15.31
400
26.13
672
15.47
420
27.15
Source: Original data from Huneycutt et al. (1988).


Dry Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
133
0.0 0.2 0.4 0.6 0.8 1.0 1.2
N Removal/Estimated Maximum
Figure 4-6 Dimensionless plot of dry matter and plant N concentration as a function of
plant N removal for dallisgrass grown at Baton Rouge, LA. Original data
from Robinson el al. (1988); curves drawn from Eq. [4.10] and [4.11],


224
20
18
16
t 14
s
? 12
£
10
2
| 8
T3
CJ
4 -
1 1 ~ T"
O Grain
r i
i r 1
1
Total
-
-

-
_

-
-
a

-
Q

0
-

-
i i i
_ _l L_
L 1 1
J
10
12
14
6 8
Measured Dry Matter Yield. Mg/lia
16
18 20
Figure 4-97 Scatter plot of seasonal dry matter yield for grain and total plant of com
grown at Plymouth, NC. Original data from Kamprath (1986).


245
S
2 .
o
t
C3
s
T3
6
C/l
W
()
()
0
j
2
()
(0_
. ' 0
()
<:>
< )
O'
00
Measured Dry Matter Yield, Mg/lia
Figure 4-118 Scatter plot of dry matter yield for rye grown at Tifln, GA. Original data
from Walker and Morey (1962).


Dry Matter Yield, Mg/ha N Removak kg/ha N Concentration, g/kg
153
Figure 4-26 Seasonal dry matter yield, plant N removal, and plant N concentration for a
two week clipping interval over two years for bermudagrass grown at
Tifln, GA. Data from Prine and Burton (1956); curves drawn from Eq.
[4.53] through [4.56], [4.73], and [4.74],


83
Table 4-3. Analysis of Variance of Model Parameters Used to Describe Coastal
Bermudagrass and Pensacola Bahiagrass Yield Response to Nitrogen at
Thorsby, Alabama, 1957-1959,
J 5 *
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
45
79.043
1.757
-
(2) Ind A,b,c
36
12
4.813
0.401
-
(l)-(2)
33
74.230
2.249
5.61++
(3) Ind A, Com b,c
14
34
26.727
0.786
-
(3)-(2)
22
21.914
0.996
2.48
(4) Ind A,b Com c
25
23
10.414
0.453
-
(4)-(2)
11
5.601
0.509
1.27
(3)-(4)
(5) Ind A, Com c, Ind b
11
16.313
1.483
3.27*
(type of grass,
irrigation)
17
31
20.825
0.672
-
(5)-(2)
19
16.012
0.843
2.10
(3)-(5)
3
5.902
1.967
2.93+
(5)-(4)
(6) Ind A, Com c, Ind b
8
10.411
1.301
2.87*
(irrigation)
15
33
25.854
0.783
-
(6)-(2)
21
21.041
1.002
2.50
(3)-(6)
1
0.873
0.873
1.11
(6)-(4)
10
15.440
1.544
3.41*
(6)-(5)
(7) Ind A, Com c, Ind b
2
5.029
2.415
3.74+
(type of grass)
15
33
22.868
0.693
-
(7)-(2)
21
18.055
0.860
2.14
(3)-(7)
1
3.859
3.589
5.57*
(7)-(4)
10
12.454
1.245
2.75*
(7)-(5)
2
2.043
1.021
1.52
Source: Original yield data from Evans et al. (1961).
Significant at the 0.005 level
Significant at the 0.01 level
* Signifcant at the 0.025 level
Significant at the 0.05 level
F(33,12,99.5)= 4.23 F(22,12,95)
F(11,12,95) =2.72 F(11,23,99)
F( 19,12,95) =2.55 F( 3,31,95)
F( 8,23,97.5)= 2.81 F(21,12,95)
F( 1,33,95) =4.14 F(10,23,99)
F( 2,31,95) =3.30
= 2.52
= 3.14
= 2.91
= 2.53
= 3.21
F(10,23,97.5)= 2.67


174
Predicted Dry Matter Yield, Mg/ha
Figure 4-47 Residual plot of seasonal dry matter yield for ryegrass grown at England
and cut different times over the season. Original data from Reid (1978).
Solid line is mean and dashed lines are 2 standard errors.


38
Table 3-13. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bahiagrass Grown on Entisol and Spodosol Soils in Florida.
Soil
Applied Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
Entisol
0
1.59
18.7
11.8
100
5.02
68.7
13.7
200
7.97
121
15.2
300
9.92
166
16.8
400
10.54
185
17.6
Spodosol
0
4.40
49.6
11.3
100
8.67
105
12.1
200
13.76
185
13.4
300
17.07
248
14.5
400
18.28
291
15.9
Source: Data from Blue (1987).


107
Table 4-27. Analysis of Variance for Bermudagrass Grown at Eagle Lake, Texas,
over Two Years.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
17
110513.95
6500.821
-
(2) Ind A,b,c
12
8
233.901
29.238
-
U)-(2)
9
110280.052
12253.339
419"
(3) Ind A, Com b,c
6
14
1851.481
132.249
-
(3)-(2)
6
1617.580
269.597
9.22++
(4) Ind A,b Com c
9
11
370.923
33.72
-
(4)-(2)
3
137.021
45.674
1.56
(3)-(4)
(5) Ind A, Com c, Ind b
3
1480.558
493.519
14.6"
(dm and Nu)
7
13
1069.846
82.296
-
(5)-(2)
5
835.945
167.189
5.72*
(3)-(5)
1
781.635
781.635
9.50*
(3)-(4)
2
698.923
349.462
10.4++
Source: Original data from Evers (1984).
Significant at the 0.001 level
Significant at the 0.005 level
f Significant at the 0.025 level
Significant at the 0.01 level
F(9, 8,99.9) =11.77
F(6, 8,99.5) =7.95
F(3, 8,95) =4.07
F(3,11,99.9) =11.56
F(5, 8,97.5) =4.82
F( 1,13,99) =9.07
F(2,11,99.5) =8.91


42
Table 3-14~continued.
Site
Applied Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
23
0
1.98
40
20.2
150
5.69
128
22.5
300
8.55
231
27.0
450
10.08
317
31.4
600
9.92
333
33.6
750
9.66
353
36.5
25
0
3.94
91
23.1
150
7.17
174
24.3
300
10.12
285
28.2
450
12.24
391
31.9
600
12.20
437
35.8
750
12.14
462
38.1
26
0
1.82
36
19.8
150
4.89
103
21.1
300
7.62
193
25.3
450
10.23
309
30.2
600
10.69
350
32.7
750
10.61
374
35.2
27
0
2.75
62
22.5
150
6.17
155
25.1
300
8.59
235
27.4
450
9.91
325
32.8
600
10.51
376
35.8
750
9.93
390
38.3
28
0
0.82
16
19.5
150
4.75
100
21.1
300
7.89
198
25.1
450
10.27
291
28.3
600
11.48
363
31.6
750
12.03
411
34.2
Source: Data from Morrison et al. (1980).


263
Table 5-2. A Summary of c and N1 Parameters from Various Studies.
Location Grass c_ N' Factors Included
Baton Rouge, LA Dallisgrass 0.0055
Fayetteville, AR
Tall Fescue
0.0081
Thorsby, AL
Bermudagrass
0.0067
Maryland
Bermudagrass
0.0112
Tifton, GA
Bermudagrass
0.0077
Fayetteville, AR
Bermudagrass
0.0084
Eagle Lake, TX
Bermudagrass
0.0072
Eagle Lake, TX
Bahiagrass
0.0072
Williston, FL
Bahiagrass
0.0118
Gainesville, FL
Bahiagrass
0.0118
England
Ryegrass
0.0080
Various in England
Ryegrass
0.0088
Tifton, GA
Rye
0.0234
Clayton, NC
Corn
0.0187
Kinston, NC
Corn
0.0187
Plymouth, NC
Corn
0.0119
180 None
120 Water Availability
150 Harvest Interval
89 Harvest Interval
130 Harvest Interval/Water Availably
120 Water Availability
140 Water Availability
140 Water Availability
85 None
85 None
125 Variable Harvest Interval
110 Many different locations
45 None
54 Grain and Total Plant
54 Grain and Total Plant
84 Grain and Total Plant


Drv Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
186
N Removal/Estimated Maximum
Figure 4-59 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for bermudagrass averaged over three
years at Fayetteville, AR, with and without irrigation. Original data from
Huneycutt el a!. (1988); curves drawn from Eq. [4.174] and [4.175],


172
Figure 4-45 Scatter plot of seasonal dry matter yield for ryegrass grown at England and
cut different times over the season. Data from Reid (1978).


243
Potassium Removal, kg/ha
Figure 4-116 Seasonal dry matter yield and plant K concentration as a function of plant
K removal for rye grown at Tifton, GA. Data from Walker and Morey
(1962). Curves drawn by Eq. [4.303] and [4.304],


57
correlation coefficient calculated from mode 6 are 0.9938 and 0.9931, respectively. The
error analysis for the parameters is shown in Table 4-16. Results are shown in Figure 4-
42, where curves for dry matter and plant N removal are drawn from
10 clippings: Y = 9.42/[l + exp(1.74 0.0080N)] [4.109]
Nu = 377.0/[l + exp(2.15 0.0080N)] [4.110]
5 clippings: Y = 12.72/[1 + exp(1.23 0.0080N)] [4.111]
Nu = 403.5/[l + exp(2.15 0.0080N)] [4.112]
3 clippings: Y = 12.75/[1 + exp(0.90 0.0080N)] [4.113]
Nu = 363.0/[l + exp(2.15 0.0080N)] [4.114]
From these results, it follows that plant N concentration, shown in Figure 4-42, is
estimated from
10 clippings: Nc = 40.0 [1 + exp(2.15 0.0080N)]/[1 + exp(1.74 0.0080N)] [4.115]
5 clippings: Nc = 31.7 [1 + exp(2.15 0.0080N)]/[1 + exp(1.23 0.0080N)] [4.116]
3 clippings: Nc = 28.5 [1 + exp(2.15 0.0080N)]/[1 + exp(0.90 0.0080N)] [4.117]
As shown, the data are not described very well by these equations; however, the problem
most likely lies in the effect of variable harvest interval.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal, shown in Figure 4-43, are described
by
10 clippings: Y = 28.0 N/(744 + Nu) [4.118]
No = 26.6 + 0.0357NU [4.119]
5 clippings: Y = 21.1 N/(267 + Nu) [4.120]
Nc = 12.6 + 0.0473NU [4.121]
3 clippings: Y = 17.9 Nu/( 146 + Nu) [4.122]
Nc = 8.2 + 0.0560NU [4.123]


159
O 100 200 300 400 500 600 700
Nitrogen Removal, kg/ha
Figure 4-32 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a three week clipping interval over two years for
bermudagrass grown at Tifln, GA. Data from Prine and Burton (1956);
curves drawn from Eq. [4.87] through [4.90],


4-10 Error Analysis of Model Parameters of Bermudagrass Grown at Thorsby,
AL 90
4-11 Analysis of Variance of Model Parameters for Bermudagrass Grown at
Maryland and Cut at Five Harvest Intervals 91
4-12 Error Analysis for Model Parameters of Bermudagrass Grown at Maryland
and Cut at Five Harvest Intervals 92
4-13 Analysis of Variance on Model Parameters for Bermudagrass Grown at
Tifton, GA, over Two Years and Cut at Five Different Harvest Intervals 93
4-14 Error Analysis for Model Parameters of Bermudagrass Grown at Tifton, GA
over Two Years and Cut at Five Different Harvest Intervals 94
4-15 Analysis of Variance of Model Parameters on Ryegrass Grown at England,
with Three Different Numbers of Cuttings over the Season for 1969 95
4-16 Error Analysis for Model Parameters of Ryegrass Grown at England, with
Three Different Numbers of Cuttings over the Season for 1969 96
4-17 Analysis of Variance for Bermudagrass over Three Seasons, Irrigated and
Non-irrigated, Grown at Fayetteville, AR 97
4-18 Error Analysis for Model Parameters of Bermudagrass Grown at
Fayetteville, AR, over Three Years, with and without Irrigation 98
4-19 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bermudagrass Grown at Fayetteville, AR. Averaged over Three Years 99
4-20 Analysis of Variance for Model Parameters for Bermudagrass Grown at
Fayetteville, AR, Averaged over Three Years, with and without Irrigation 100
4-21 Error Analysis for Model Parameters of Bermudagrass Grown at
Fayetteville, AR, Averaged over Three Years, with and without Irrigation 101
4-22 Analysis of Variance of Model Paramters for Tall Fescue Grown at
Fayetteville, AR, over Three Seasons, with and without Irrigation 102
4-23 Error Analysis for Model Parameters of Tall Fescue Grown at Fayetteville,
AR, over Three Seasons, with and without Irrigation 103
4-24 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Tall Fescue Grown at Fayetteville, AR, Averaged over Three Seasons 104
IX


142
Figure 4-15 Scatter plot of plant N removal for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals. Original data from Doss et al. (1966).


Table 4-25. Analysis of Variance of Model Parameters for Tall Fescue Grown at
Fayetteville, AR, Averaged overr
'hree Seasons, with and without
Irrigation.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
25
438490.94
17539.64
-
(2) Ind A,b,c
12
16
860.75
53.80
-
(l)-(2)
9
437630.19
48625.58
904**
(3) Ind A, Com b,c
6
22
3908.10
177.64
-
(3)-(2)
6
3047.35
507.89
9.44**
(4) Ind A,b Com c
9
19
999.28
52.59
-
(4)-(2)
3
138.53
46.18
0.86
(3)-(4)
3
2908.82
969.61
18.4**
(5) Ind A, Com c, Ind b
(dm, Nu)
7
21
1437.08
68.43
(5)-(2)
5
576.32
115.26
2.14
(3)-(5)
1
2471.02
2471.02
36.1**
(5)-(4)
2
437.80
218.90
4.16+
Source: Original data from Huneycutt el al. (1988).
Significant at the 0.001 level
Significant at the 0.05 level
F(9,16,99.9)
= 5.98
F(6,16,99.9)
= 6.81
F(3,16,95)
= 3.24
F(3,19,99.9)
= 8.28
F(5,16,95)
= 2.85
F( 1,21,99.9)
= 14.59
F(2,19,95)
= 3.52


CHAPTER 3
METHODS AND MATERIALS
Analysis of Data
Depending upon the nature of the data, one of three different models will be used.
It should be noted that the three models are directly related to one another because they
are based upon the same mathematical form. For data sets where only plant dry matter
yield is recorded, the simple logistic model, Equation [2.8], will be used to describe plant
yield response to applied nitrogen. For data sets where both dry matter yield and plant N
removal are recorded, the extended logistic model, Equations [2.8] and [2.9], will be used
to describe the response to applied nitrogen. Finally, when varying amounts of nitrogen,
phosphorus, and potassium are applied and dry matter yield and nutrient removals
recorded, the extended triple logistic (or NPK) model, Equations [2.17] through [2.20],
are used to describe the response. Water availability and harvest interval (or cutting
frequency) will be related to the linear model parameter. For all three models, the
parameters, A, b, and c, are estimated using nonlinear regression (including second-order
Newton-Raphson method) on the data to minimize the error, E, given by
[3.1]
where
E
error sum of squares,
measured yield or N removal,
estimated yield or N removal from model,
observation number.
18


74
0.9995 were calculated by mode 5. Results are shown in Figure 4-94 for grain and total
plant, where the curves are drawn from
Grain: Y = 9.48/[l + exp(-0.065 0.0119N)] [4.264]
Nu = 126.2/[1 + exp(0.46 0.0119N)] [4.265]
Total: Y = 16.60/[1+exp(-0.065 0.0119N)] [4.266]
Nu = 147.3/[ 1 + exp(0.46 0.0119N)] [4.267]
From these results, it follows that plant N concentration, shown in Figure 4-94, is
estimated from
Grain: Nc = 13.3 [1 + exp(0.46 0.0119N)]/[1 + exp(-0.065 0.0119N)] [4.268]
Total: Nc = 8.9 [1 + exp(0.46 0.0119N)]/[1 + exp(-0.065 0.0119N)] [4.269]
As shown, the data are described well by these equations.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal for grain and total plant, shown in
Figure 4-95, are described by
Grain: Y = 23.2 NJ( 183 + Nu) [4.270]
Nc = 7.9 + 0.0431NU [4.271]
Total: Y = 40.6 NJ(2\3 + Nu) [4.272]
Nc = 5.2 + 0.0246NU [4.273]
The dimensionless plot for corn is shown in Figure 4-96, and validates the form of
the model. Note that all of the data have been collapsed onto one graph, and the data fall
on the curve. The curves were drawn from
Y/A = 2.45 (Nu/A')/[ 1.45 + (Nu/A')] [4.274]
Nc/Ncm = 0.59+0.41 (Nu/A') [4.275]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
97 through 4-100.


108
Table 4-28. Analysis of Variance for Bahiagrass Grown at Eagle Lake, Texas, over
Two Years.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
17
68411.328
4024.196
-
(2) Ind A,b,c
12
8
184.755
23.094
-
(l)-(2)
9
68226.573
7580.73
328**
(3) Ind A, Com b,c
6
14
1507.231
107.659
-
(3)-(2)
6
1322.476
220.413
9.54++
(4) Ind A,b Com c
9
11
346.949
31.541
-
(4)-(2)
3
162.193
54.064
2.34
(3)-(4)
(5) Ind A, Com c, Ind b
3
1160.282
386.761
12.3**
(dm and Nu)
7
13
634.002
48.769
-
(5)-(2)
5
449.247
89.849
3.89+
(3)-(5)
1
873.229
873.229
17.9**
(5H4)
2
287.054
143.527
4.55+
Source: Original data from Evers (1984).
Significant at the 0.001 level
Significant at the 0.005 level
Significant at the 0.05 level
F(9, 8,99.9) =11.77
F(6, 8,99.5) =7.95
F(3, 8,95) =4.07
F(3,11,99.9) =11.56
F(5, 8,95) =3.69
F( 1,13,99.9) =17.81
F(2,11,95) =3.98


134
Figure 4-7 Scatter plot of dry matter yield for dallisgrass grown at Baton Rouge, LA.
Original data from Robinson el al. (1988).


116
Table 4-36. Analysis of Variance on Model Parameters for Grain and Total Plant of
Corn Grown on Dothan and Goldsboro Sandy Loam at Clayton and Kinston,
NC.
Mode
Parameters
Estimated
df
Residual Sum
of Squares
Mean Sum
of Squares
F
(1) Com A,b,c
3
37
101127.594
2733.178
-
(2) Ind A,b,c
24
16
49.314
3.082
-
(l)-(2)
21
101078.280
4813.251
1562*
(3) Ind A, Com b,c
10
30
1117.935
37.265
-
(3)-(2)
14
1068.620
76.33
24.8**
(4) Ind A, b, Com c
17
23
317.411
13.800
-
(4)-(2)
7
268.096
38.299
12.4*
(3)-(4)
7
800.524
114.361
8.29**
(5) Ind A, Com c, Ind b
(site: dm and Nu)
13
27
350.409
12.978
(5)-(2)
11
301.095
27.372
8.88**
(3)-(5)
3
767.526
255.842
19.7**
(5)-(4)
4
32.998
8.250
0.60
(6) Ind A, Com c, Ind b
(dm and Nu)
11
29
334.633
11.539
(6)-(2)
13
285.319
21.948
7.12**
(3)-(6)
1
783.302
783.302
67.9**
(6)-(4)
6
17.222
2.870
0.21
(7) Ind A, Com c, Ind b
(part: dm and Nu)
13
27
325.903
12.070
(7)-(2)
11
276.589
25.144
8.16**
(3)-(7)
3
792.032
264.011
21.9**
(7)-(4)
4
8.492
2.123
0.15
(6)-(7)
2
8.730
4.365
0.36
Source: Original data from Kamprath (1986).
** Significance level of 0.001
F(21,16,99.9)= 4.95
F( 14,16,99.9)= 5.35
F( 7,16,99.9)= 6.46
F( 7,23,99.9)= 5.33
F(11,16,99.9)= 5.67
F( 3,27,99.9)= 7.27
F( 4,23,95) =2.80
F( 1,29,99.9)= 13.39
F( 6,23,95) =2.53
F(13,16,99.9)= 5.44
F( 2,27,95) =3.35


190
Figure 4-63 Residual plot of seasonal plant N removal for bermudagrass averaged over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt el af. (1988). Solid line is mean and dashed lines are 2
standard errors.


129
Measured Dry Matter Yield, Mg/ha
Figure 4-2 Scatter plot of seasonal dry matter yield for Coastal bermudagrass and
Pensacola bahiagrass at Thorsby, AL. Original data from Evans et al.
(1961).


60
Non-irrigated, 1983: Y = 17.90/[1 + exp(1.50 0.0084N)] [4.132]
Nu=408.7/[1 + exp(2.04 0.0084N)] [4.133]
1984: Y = 17.37/[1 + exp(1.50 0.0084N)] [4.134]
N=413.9/[l + exp(2.04 0.0084N)] [4.135]
1985: Y = 19.61/[1 + exp(1.50 0.0084N)] [4.136]
Nu=459.3/[1 + exp(2.04 0.0084N)] [4.137]
Irrigated, 1983: Y = 24.70/[l + exp(1.50 0.0084N)] [4.138]
Nu=554.3/[1 + exp(2.04 0.0084N)] [4.139]
1984: Y =24.60/[l + exp(1.50 0.0084N)] [4.140]
N=523.7/[l + exp(2.04 0.0084N)] [4.141]
1985: Y = 22.58/[l + exp(1.50 0.0084N)] [4.142]
Nu= 492.1/[1 + exp(2.04 0.0084N)] [4.143]
From these results, it follows that plant N concentration, shown in Figure 4-50, is
estimated from
Non-irrigated, 1983: Nc=22.8 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.144]
1984: Nc=23.8 [1 + exp(2.04 0.0084N)]/[1 + exp( 1.50 0.0084N)][4.145]
1985: Nc= 23.4 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.146]
Irrigated, 1983: Nc=22.4 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.147]
1984: Nc=21.3 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.148]
1985: Nc=21.8 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.149]
As shown, the data are described relativity well by these equations.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal at each clipping interval for all three
years with and without irrigation, shown in Figure 4-51, are described by
Non-irrigated, 1983: Y = 42.9NU/(571 +NU)
N,
13.3 + 0.0233N,
[4.150]
[4.151]


Dry Matter Yield. Mg/ha N Removal, kg/ha N Concentration, g/kg
206
Applied Nitrogen, kg/lia
Figure 4-79 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bahiagrass grown over two years at Eagle
Lake, TX. Data from Evers (1984), curves drawn from Eq. [4.224]
through [4.227], [4 230], and [4 231 ]


158
Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a two week clipping interval over two years for
bermudagrass grown at Tifton, GA. Data from Prine and Burton (1956);
curves drawn from Eq. [4.83] through [4.86],
Figure 4-31


118
Table 4-38. Analysis of Variance on Model Parameters for Grain and Total Plant of
Corn Grown on Portsmouth Very Fine Sandy Loam at Plymouth NC.
Mode
Parameters
Estimated
df
Residual Sum
of Squares
Mean Sum
of Squares
F
(1) Com A,b,c
3
17
37148.680
2185.216
-
(2) Ind A,b,c
12
8
5.674
0.709
-
(1)-(2)
9
37143.006
4127.001
5819**
(3) Ind A, Com b,c
6
14
144.076
10.291
-
(3)-(2)
6
138.402
23.067
32.5**
(4) Ind A, b, Com c
9
11
16.808
1.528
-
(4)-(2)
3
11.134
3.711
5.23+
(3)-(4)
3
127.268
42.423
27.8**
(5) Ind A, Com c, Ind b
(dm and Nu)
7
13
26.613
2.047
(5)-(2)
5
20.939
4.188
5.91*
(3)-(5)
1
117.463
117.463
57.4
(5)-(4)
2
9.796
4.898
3.21
Source. Original data from Kamprath (1986).
** Significance level of 0.001
f Significance level of 0.025
Significance level of 0.05
F(9, 8,99.9) =11.77
F(6, 8,99.9) =12.86
F(3, 8,95) =4.07
F(3,11,99.9) =11.56
F(5, 8,97.5) =4.82
F( 1,13,99.9) =17.81
F(2,11,95) =3.98


61
Irrigated, 1983: Y = 59.2N/(774 + N)
1984: Y = 41.6 Nu/(578 + N)
1985: Y = 47.0 N/(641 + Nu)
1984: Y = 59.0N4731 +NU)
1985: Y = 54.1 N/(687 + Nu)
Nc = 13.9 + 0.0240NU
Nc = 13.6 + 0.0213NU
Nc = 12.7 + 0.0185NU
Nc 13.1 +0.0169NU
Nc = 12.4 +0.0170NU
[4.152]
[4.153]
[4.154]
[4.155]
[4.156]
[4.157]
[4.158]
[4.159]
[4.160]
[4.161]
The dimensionless plot for bermudagrass is shown in Figure 4-52, and validates the
form of the model. Note that all of the data have been collapsed onto one graph, and the
data fall on the curve. The curves were drawn from
Y/A = 2.40 (Nu/A')/[1.40 + (Nu/A')]
Nc/Ncn, = 0.58 +0.42 (Nu/A')
[4.162]
[4.163]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
53 through 4-56.
As done earlier, the data will be averaged over years, since variations due to water
availability appear in the linear parameter A. The averaged data are in Table 4-19 and the
analysis of variance is in Table 4-20 (dry matter and N removal x 1 yr x 2 irrigation x 7
N). Comparison among modes 1 and 2 results in a variance ratio of 373.4 that is
significant at the 0.1% level. Comparison among modes 3 and 2 leads to a non-significant
variance ratio of 2.72. Comparison among modes 3 and 4 results in a variance ratio of
6.11 that is significant at the 0.5% level. Mode 5, individual A for yield and plant N
removal with and without irrigation, b for dry matter and N removal, and common c,
accounts for all the significant variation, since F(5,16,95) = 2.85, F(l,21,99) = 8.02, and


15
model is compared to field data not used previously in the development or calibration
process (Jones el al., 1987). It should be noted that this study is not simulation, but true
modeling: mathematical representation of a system. Furthermore, the purpose of
validation is to determine if the model is sufficiently accurate for its application as defined
by the objectives (Jones et al., 1987). This leads back to a point mentioned earlier. It is
essential that the objectives be stated clearly from the beginning. Jones el al. (1987, p. 16)
also noted that common sense should prevail in validation of models, and that "a model
cannot be validated, it can only be invalidated," a point frequently emphasized in statistics.
Dimensionless plots will be used to determine if the form of the model is adequate to
validate the model. It is more important to determine if the model estimates adequately:
Does the model capture the essence of what is trying to be accomplished. The scatter and
residual plots will be used to evaluate how well the model estimated the data, by looking
for biases or trends in the residuals. Box (1979, p. 2) noted the difference between
estimating and validation in a paper by stating
. . two different kinds of inferential process .... The first, used in estimating
parameters from data conditional on the truth of some tentative model, is
appropriately called Estimation. The second, used in checking whether, in the
light of the data, any model of the kind proposed is plausible, has been aptly named
by Cuthbert Daniel Criticism. While estimation should . employ . likelihood,
criticism needs a different approach. In practice, it is often best done in a rather
informal way by examination of residuals. . .
Box, Hunter, and Hunter (1978, p. 552) also discussed the importance of residuals and
other visual displays in evaluating data, especially when the work has been done with a
computer by noting.
Without computers most of the work done on nonlinear models would not be
feasible. However, the more sophisticated the model and the more elaborate the
techniques employed, the more important it is to submit complicated analyses to
surveillance by data plots, residual plots, and other visual displays. The modern
computer can make the plots itself, but graphs need not only be made but also to
be carefully examined and thought about. The data analyst must "fondle" the data.


Dry Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
157
Figure 4-30 Seasonal dry matter yield, plant N removal, and plant N concentration for a
eight week clipping interval over two years for bermudagrass grown at
Tifln, GA. Data from Prine and Burton (1956); curves drawn from Eq.
[4.69] through [4.72], [4.81], and [4.82],


4-63 Residual plot of seasonal plant N removal for bermudagrass averaged over
three years at Fayetteville, AR, with and without irrigation 190
4-64 Response of seasonal dry matter yield, plant N removal and plant N
concentration for tall fescue grown over three years at Fayetteville, AR,
with and without irrigation 191
4-65 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for tall fescue grown over three years at Fayetteville, AR, with
and without irrigation 192
4-66 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for tall fescue grown over three years at
Fayetteville, AR, with and without irrigation 193
4-67 Scatter plot of seasonal dry matter yield for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation 194
4-68 Scatter plot of seasonal plant N removal for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation 195
4-69 Residual plot of seasonal dry matter yield for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation 196
4-70 Residual plot of seasonal plant N removal for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation 197
4-71 Response of seasonal dry matter yield, plant N removal and plant N
concentration for tall fescue averaged over three years at Fayetteville, AR,
with and without irrigation 198
4-72 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for tall fescue averaged over three years at Fayetteville, AR,
with and without irrigation 199
4-73 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for tall fescue averaged over three years
at Fayetteville, AR, with and without irrigation 200
4-74 Scatter plot of seasonal dry matter yield for tall fescue averaged over three
years at Fayetteville, AR, with and without irrigation 201
4-75 Scatter plot of seasonal plant N removal for tall fescue averaged over
three years at Fayetteville, AR, with and without irrigation 202
XVII


141
Figure 4-14 Scatter plot of dry matter yield for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals. Original data from Doss et al. (1966).


202
C3
OI)
T3
>
o
S
o
o
-o
o
-a
1
200 300
Measured N Removal, kg/lia
Figure 4-75 Scatter plot of seasonal plant N removal for tall fescue averaged over three
years at Fayetteville, AR, with and without irrigation. Original data from
Huneycutt el a!. (1988).


BIOGRAPHICAL SKETCH
Denise M. Wilson was bom May 2, 1970 in Tampa, Florida. She was very active in
the Girl Scouts of America while growing up and was awarded the Gold Award (equivalent to
Eagle Scout in Boy Scouts) in 1987. At Gaither High School, Denise was active in the Future
Farmers of America and Mu Alpha Theta (math honor society), competing in many contests
for both clubs. She graduated from Gaither High School in 1988. She enrolled at the
University of Florida the following fall. She received her Bachelor of Science in Engineering
(agricultural engineering) with high honors and a minor in Mathematics in May, 1992. After
having been awarded a National Science Foundation (NSF) Fellowship in April of the same
year, she began work in the Department of Agricultural Engineering towards a degree of
Doctor of Philosophy in August 1992. In August 1993, she was also enrolled in a Master of
Statistics degree program. She was married to Russell A. Wilson on June 20, 1992. In
September 1995, she began working with Camp Dresser & McKee, Inc where she plans on
becoming a professional engineer. At some point, she would like to teach mathematics,
statistics, or engineering.
269


Residual N Removal, kg/ha
197
Predicted N Removal, kg/lia
Figure 4-70 Residual plot of seasonal plant N removal for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation. Original data from
Huneycutt el at. (1988). Solid line is mean and dashed lines are 2 standard
errors.


Predicted N Removal, kg/ha
225
C
n
90
Measured N Removal, kg/ha
Figure 4-98 Scatter plot of seasonal plant N removal for grain and total plant of corn
grown at Plymouth, NC. Original data from Kamprath (1986).


This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepteehps partial fulfillment of the
requirements for the degree of Doctor of Philosophy
December 1995
M. Phillips
Dean, College of Engineering
Karen A. Holbrook
Dean, Graduate School


25
(Myakka fine sand) near Gainesville, Florida. Entisol is typically a dry soil and Spodosol
is typically wet. Five nitrogen levels were included: 0, 100, 200, 300, and 400 kg/ha.
The data are listed in Table 3-13.
England: Ryegrass
The data for this analysis were taken from a study by Morrison et a/. (1980) and is
listed in Table 3-14. Twenty different sites in England were used to grow the ryegrass.
Six different nitrogen levels were included in the study at each site: 0, 150, 300, 450, 600,
and 750 kg/ha. The reader is directed to the report for further information about the site
characteristics and weather data..
Tifln, Georgia: Rye
The data for this analysis were drawn from a study by Walker and Morey (1962).
Six levels of nitrogen (0, 45, 90, 135, 180, and 225), phosphorous (0, 20, 40, 60, 80, and
100), and potassium (0, 37, 74, 111, 148, and 185) were investigated. The data are listed
in Table 3-15.


150
i
c
Q
15
P
rs
CO

12 1
13
*>-
Vh
y
t A
C3 U
-1
-2
-3
-4
0
3.2 weeks

3.6 weeks
A
4.3 weeks
V
5.5 weeks
<0
7.7 weeks /S
V
V
.A
V
0
%
O
->zr
O
&
A

v
O
5 10 15
Predicted Dry Matter Yield, Mg/lia
20
25
Figure 4-23 Residual plot of dry matter yield for bermudagrass grown at Maryland and
cut at five harvest intervals. Original data from Decker et al. (1971). Solid
line is mean and dashed lines are 2 standard errors.


63
The dimensionless plot for bermudagrass is shown in Figure 4-59, and validates the
form of the model. Note that all of the data have been collapsed onto one graph, and the
data fall on the curve reasonably well. The curves were drawn from
Y/A = 2.43 (Nu/A')/[ 1.43 + (Nu/A')] [4.174]
Nc/Ncm= 0.59+0.41 (KM') [4.175]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
60 through 4-63.
Fayetteville, AR: Tall Fescue
The data for this analysis are taken from Huneycutt et al. (1988). The analysis of
variance is shown in Table 4-22 (dry matter and N removal x 3 yrs x 2 irrigation x 7 N).
Comparison among modes 1 and 2 results in a variance ratio of 352 that is significant at
the 0.1% level. Comparison among modes 3 and 2 also leads to a variance ratio of 7.60
that is significant at the 0.1% level. Comparison among modes 3 and 4 results in a
variance ratio of 9.90 that is significant at the 0.1% level. Comparison among modes 3
and 5, individual A for yield and plant N removal with and without irrigation for all three
years, b for dry matter and N removal with and without irrigation, and common c, leads to
a variance ratio of 18.3 that is significant at the 0.1% level. Mode 6, individual A for yield
and plant N removal with and without irrigation for all three years, b for dry matter and N
removal over both with and without irrigation and all years, and common c, describes the
data best, since F(21,48,95) = 2.94, F(l,69,99.9) = 11.81, F(10,59,97.5) = 2.27, and
F(2,67,95) = 3.13. The difference due to years and irrigation is explained by the A
parameter. Postulate 3 is supported again. The overall correlation coefficient and
adjusted correlation coefficient calculated from mode 6 were 0.9956 and 0.9947,
respectively. The error analysis for the parameters is shown in Table 4-23. Results are


229
Nitrogen Removal, kg/ha
Figure 4-102 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bahiagrass grown on two soils in Florida. Data from Blue
(1987); curves drawn from Eq. [4 282] through [4.285],


47
29.25. This outcome supports Postulate 3. The overall correlation coefficient and
adjusted correlation coefficient calculated from mode 4 are 0.9997 and 0.9995,
respectively. The error analysis for the parameters is shown in Table 4-8. Results are
shown in Figure 4-4, where curves for dry matter and plant N removal are drawn from
Y = 15.60/[1 + exp(0.58 0.0055N)] [4.5]
N = 430.7/[l + exp(1.47 0.0055N)] [4.6]
From these results, it follows that plant N concentration, shown in Figure 4-4, is estimated
from
Nc = 27.6 [1 + exp(1.47 0.0055N)]/[1 + exp(0.58 0.0055N)] [4.7]
As shown, the data are described well by these equations. It should be noted that the
plant N concentration data were not defined by regression techniques, but rather as a ratio
of plant N removal and dry matter yield.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal, shown in Figure 4-5, are described by
Y = 26.5 N/(300 + N) [4.8]
Nc = 11.3 + 0.0378NU [4.9]
The intercept (11.3 g/kg) represents plant N concentration in a nitrogen deficient
environment with low plant N removal.
The form of this model can be validated by plotting the data in dimensionless form.
The data plotted in Figure 4-5 were strictly measured data; that is, the independent
variable applied N has been ignored. The results of Postulate 3 defined the curve drawn.
By dividing dry matter yield, plant N concentration and plant N removal by their
appropriate mximums, all of the data are collapsed onto the same dimensionless scale.
Furthermore, the curves drawn are dependent upon the Ab, (b' b), term alone. This will
be more significant as larger, more complex data sets are investigated. The form of the


CHAPTER 5
SUMMARY AND CONCLUSIONS
Lately, environmental issues have become a popular topic in public discussion.
People are interested in controlling pollution in soil, water and air. Land application of
treated wastes and effluent is being used to help control contamination of surface and
groundwater of Florida A simple method is needed to aid engineers, farmers, and
managers in obtaining estimates of nutrient removal by various forage crops. This
research project has focused on modeling the seasonal production of seven different
forage grasses, at different locations and under different factors, such as water availability,
harvest interval and plant partitioning.
The simple logistic model, Equation [2.8], the extended logistic model, Equations
[2.8] and [2.9], and the extended triple logistic (NPK) model, Equations [2.17] through
[2.20], were found to adequately describe grass response to applied nutrients (R > 0.99).
Data from various studies were used to relate management parameters of water
availability, harvest interval, soil type, and crop species to the parameters of the equation.
Second-order Newton-Raphson method for nonlinear regression was used to optimize the
fit of the model to the data. Analysis of variance (ANOVA) was used to search for
simplification of the model in the form of common parameter values. The conclusions
from the analysis of the data from the various studies are presented below.
1. The logistic model is relatively simple to use. Once the parameters are known,
estimates can be computed on a basic hand calculator. The logistic model involves
analytical functions rather than finite difference (numerical techniques).
253


69
averaged data is in Table 4-31. Comparison among modes 1 and 2 results in a variance
ratio of 792, that is significant at the 0.1% confidence interval. Comparison among modes
3 and 2 leads to a variance ratio of 7.79, that is significant at the 0.5% confidence level.
Comparison among modes 3 and 4 results in a variance ratio of 18.4, that is significant at
the 0.1% confidence level. Mode 5, individual A for dry matter yield and plant N removal
of each grass, individual b for dry matter yield and plant N removal, and common c,
accounts for all of the significant variation, since F(5,8,95) = 3.69, F(l,13,99.9) = 17.81,
and F(2,l 1,97.5) = 5.26. The two grasses have the same b and b', hence suggesting mode
5 of Table 4-29. The overall correlation coefficient of 0.9987 and adjusted correlation
coefficient of 0.9981 were calculated using mode 5 of the averaged data. Using mode 5
and the original data, the overall correlation coefficient of 0.9798 and adjusted correlation
coefficient of 0.9727 were calculated. The error analysis of the averaged and original data
is in Tables 4-32 and 4-33. Note that the estimates of b and c are the essentially the same
as for the unaveraged data, suggesting this is a valid way to describe the data. Results are
shown in Figure 4-78 and 4-79 for bermudagrass and bahiagrass, respectively, where are
drawn from
Coastal bermudagrass, 1979: Y = 15.99/[1 + exp(0.57 0.0072N)] [4.220]
Nu= 310.9/[ 1 + exp(1.07 0.0072N)] [4.221]
1980: Y = 9.89/[l + exp(0.57 0.0072N)] [4.222]
N= 227.4/p + exp(1.07 0.0072N)] [4.223]
Pensacola bahiagrass, 1979: Y = 12.42/[1 + exp(0.57 0.0072N)] [4.224]
Nu= 237.2/p + exp(1.07 0.0072N)] [4.225]
1980: Y 7.51/[1 + exp(0.57 0.0072N)] [4.226]
N= 191.3/p + exp(1.07 0.0072N)] [4.227]
From these results, it follows that plant N concentration, shown in Figures 4-78 and 4-79,
is estimated from


Dry Matter Yield, Mg/ha P Remov al, kg/ha P Concentration, g/kg
239
Figure 4-112 Response of seasonal dry matter, plant P removal, and plant P
concentration to applied P for rye grown at Tifton, GA and fixed
application rates of 135 and 74 kg/ha of N and K, respectively. Data from
Walker and Morey (1962). Curves drawn by Eq. [4.292], [4.294], and
[4.297],


Clayton and Kinston. NC: Corn
71
Data from Kamprath (1986) for corn were used. The analysis of variance for the
Dothan sandy loam is shown in Table 4-34 (dry matter and N removal x 2 components x 5
N). Comparisons among modes 1 and 2, modes 3 and 2, modes 3 and 4, and modes 3 and
5 result in variance ratios of 1686, 22.1, 49.8, and 39.0, which are all significant at the
0.1% confidence level. Comparison among modes 5 and 4 leads to a non-significant
variance ratio of 0.31. As a result, mode 5, individual A for grain and total plant,
individual b for yield and plant N removal, and common c, accounts for all the significant
variation, since F(5,8,97.5) = 4.82, F(l, 13,99.9) = 17.81, and F(2,11,95) = 3.98. The
analysis of variance for the Goldsboro sandy loam is shown in Table 4-35 (dry matter and
N removal x 2 components x 5 N). As with the Dothan, comparisons among modes 1 and
2, modes 3 and 2, modes 3 and 4, and modes 3 and 5 result in variance ratios of 1920,
26.9, 35.7, and 83.2, which are all significant at the 0.1% confidence level. Comparison
among modes 5 and 4 leads to a non-significant variance ratio of 2.48. As a result, mode
5, individual A for grain and total plant, individual b for yield and plant N removal, and
common c, accounts for all the significant variation, since F(5,8,95) = 3.69, F(l,13,99.9) =
17.81, and F(2,11,95) = 3.98. Next the data from the two soils are combined and the
analysis of variance data are presented in Table 4-36 (dry matter and N removal x 2
components x 2 soils x 5 N). A comparison among modes 1 and 2 results in a variance
ratio of 1562 that is significant at the 0.1% confidence level. Comparison among modes
3 and 2 leads to a variance ratio of 24.8, that is significant at the 0.1% confidence level.
Comparison among modes 3 and 4 results in a variance ratio of 8.29 that is significant at
the 0.1% confidence level. Comparison among modes 3 and 5 results in a variance ratio
of 19.7 that is significant at the 0.1% confidence level. Comparison among modes 3 and 6
leads to a variance ratio of 67.9 that is significant at the 0.1% confidence level.
Comparison among modes 3 and 7 results in a variance ratio of 21.9, that is significant at


152
O 2 4 6 8 10
Harvest Interval, weeks
Figure 4-25 Estimated maximum dry matter yield and estimated maximum plant N
removal as a function of harvest interval for bermudagrass in Maryland.
Lines drawn from Eq. [4 51] and [4.52]


219
f.
2
£
V->
rj
ti
cd
a
a
3
"O
w5
o
-1
-2
O Dothan Grain
Dothan Total
A Goldsboro Grain
V Goldsboro Total
A
O
A
(i)
A
~Ar
V


a r-:V) '1
v
(>
V
V
n

5 10 15
Predicted Dry Matter Yield. Mg/ha
20
25
Figure 4-92 Residual plot of seasonal dry matter yield for grain and total plant of com
grown at Clayton and Kinston, NC, on Dothan and Goldsboro soils,
respectively. Original data from Kamprath (1986). Solid line is mean and
dashed lines are 2 standard errors.


Estimated N Removal, kg/ha
246
250
100 150
Measured N Removal, kg/lia
Figure 4-119 Scatter plot of plant N removal for rye grown at Tifton, GA. Original data
from Walker and Morey (1962).


114
Table 4-34. Analysis of Variance on Model Parameters for Corn Grown on Dothan
Sandy Loam at Clayton, Is
C, Both Grain and Tota
Mode
Parameters
Estimated
df
Residual Sum
of Squares
Mean Sum
of Squares
F
(1) Com A,b,c
3
17
63333.848
3725.520
-
(2) Ind A,b,c
12
8
33.372
4.172
-
(l)-(2)
9
63300.476
7033.386
1686**
(3) Ind A, Com b,c
6
14
586.733
41.910
-
(3)-(2)
6
553.360
92.227
22.1**
(4) Ind A, b, Com c
9
11
138.832
12.621
-
(4)-(2)
3
105.460
35.153
8.43*
(3)-(4)
3
447.901
149.300
49.8**
(5) Ind A, Com c, Ind b
(dm and Nu)
7
13
146.647
11.281
_
(5)-(2)
5
113.275
22.655
5.43*
(3)-(5)
1
440.086
440.086
39.0**
(5)-(4)
2
7.815
3.908
0.31
Source: Original data from Kamprath (1986).
** Significance level of 0.001
Significance level of 0.01
* Significance level of 0.025
F(9, 8,99.9) =11.77
F(6, 8,99.9) =12.86
F(3, 8,99) =7.59
F(3,11,99.9) =11.56
F(5, 8,97.5) =4.82
F( 1,13,99.9) =17.81
F(2,11,95) =3.98


53
calculated from mode 6 are 0.9941 and 0.9923, respectively. The error analysis for the
parameters is shown in Table 4-14. Results are shown in Figures 4-26 through 4-30 for
each clipping interval, where curves for dry matter and plant N removal at all five harvest
intervals, are drawn from
2 weeks, 1953: Y = 17.95/[1
Nu = 644.7/[l
1954: Y = 8.40/[l
Nu = 324.5/[l
3 weeks, 1953: Y = 19.88/[1
N = 641.8/[ 1
1954: Y = 9.95/[l
N = 337.5/[l
4 weeks, 1953: Y = 23.15/[1
Nu = 687.2/[l
1954: Y = 11.67/[1
N = 348.0/[l
6 weeks, 1953: Y = 29.57/[l
Nu = 688.2/[l
1954: Y = 14.37/[1
Nu = 363.8/[l
8 weeks, 1953: Y = 29.58/[l
Nu = 606.0/[l
1954: Y = 16.24/[1
N 379.5/[l
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(l .47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
[4.53]
[4.54]
[4.55]
[4.56]
[4.57]
[4.58]
[4.59]
[4.60]
[4.61]
[4.62]
[4.63]
[4.64]
[4.65]
[4.66]
[4.67]
[4.68]
[4.69]
[4.70]
[4.71]
[4.72]
From these results, it follows that plant N concentration, shown in Figures 4-26 through
4-30, is estimated from


Table 4-29. Analysis of Variance for Bermudagrass and Bahiagrass Grown at Eagle
Lake, Texas, over Two Years.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
37
181775.297
4912.846
-
(2) Ind A,b,c
24
16
418.657
26.166
-
(l)-(2)
21
181356.641
8636.031
330*
(3) Ind A, Com b,c
10
30
3209.170
106.972
-
(3)-(2)
14
2790.514
199.322
7.62**
(4) Ind A,b Com c
17
23
814.463
35.411
-
(4)-(2)
7
395.806
56.544
2.16
(3)-(4)
7
2394.707
342.101
9.66**
(5) Ind A, Com c, Ind b
(dm and Nu)
11
29
2058.255
70.971
(5)-(2)
13
1639.598
126.123
4.82++
(3)-(5)
1
1150.915
1150.915
16.2**
(5)-(4)
6
1243.792
207.299
5.85**
(6) Ind A, Com c, Ind b
(grass, dm and Nu)
13
27
1746.733
64.694
(6)-(2)
11
1328.076
120.730
4.61++
(3)-(6)
3
1462.437
487.479
7.54**
(6)-(4)
4
932.270
233.067
6.58++
(5H6)
2
311.522
155.761
2.41
Source: Original data from Evers (1984).
Significant at the 0.001 level
Significant at the 0.005 level
F(21,16,99.9) =
4.95
F( 14,16,99.9) =
5.35
F( 7,16,95) =
2.66
F( 7,23,99.9)=
5.33
F( 13,16,99.5) =
4.03
F( 1,29,99.9)=
13.39
F( 6,23,99.9)=
5.65
F(11,16,99.5) =
4.18
F( 3,27,99.9)=
7.27
F( 4,23,99.5)=
4.95
F( 2,27,95) =
3.35


2
adequately describe all the behavior, the task of identifying the specific relationship would
be greatly simplified. In theory, the form of the model should represent the physical
system beyond the range of data. The challenge is to identify patterns in the data (such as
Figure 1-1) and identify relationships to describe such data.
The objective of this project is to establish a model that provides reasonable
estimates of dry matter yield and nutrient removal given a nutrient application rate.
Studies from the literature will be used to document the fit of the model to numerous data
sets with varying factors. These estimates could be used by engineers, managers, and
regulators. The form of the equation should work regardless of forage or site. Water
availability and harvest interval should also be quantified in the model. Land application of
treated effluent and waste as irrigation for agricultural crops is becoming a prevalent
method of wastewater reuse. In these systems, the nitrogen response of the crop is
needed by engineers in the design process, since both the Florida Department of
Environmental Protection (FDEP) and the Environmental Protection Agency (EPA)
regulate wastewater application rates based upon nitrogen concentrations and the uptake
abilities of crops grown.
Many sources of data for this analysis can be found for various forage grasses and
locations around the world. This reservoir of information has different variables for the
crops studied including the following: applied nutrients (N, P, and K), water availability
(with/without irrigation and year to year variability), harvest interval, site specificity,
and plant partitioning. These different variables produce varying response curves.
Furthermore, the studies range from examining only dry matter yield response to applied
N to including N removal response to investigating the effects of nitrogen, phosphorus,
and potassium on dry matter yield and their respective removals. The logistic equation has
provided high correlation coefficients for estimation of growth and nutrient removal for


208
/A
A
A
A
VA'
100
Nitrogen Removal, kg/lia
Figure 4-81 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bahiagrass grown over two years at Eagle Lake, TX. Data
from Evers (1984); curves drawn from Eq. [4.236] through [4.239],


231
§>
S
2
13
£
<3
t¡
cl
O
2
o
-3
o
20
18
16
14
12
10
8
6
4
2
0

~r
i r i i
i 1 7
o
Enlisol
'

Spodosol
-

-

Q
0

-
0'
-
-
-
0

-
- o.
/
/
V

I
i i i i 1
J L
0 2 4 6 8 10 12 14 16 18 20
Measured Dry Matter Yield, Mg/ha
Figure 4-104 Scatter plot of seasonal dry matter yield for bahiagrass grown on two soils
in Florida. Original data from Blue (1987).


19
Correlation coefficients will be used to measure the fit of the data to the model. The
correlation coefficient is given by
[3.2]
An adjusted correlation coefficient is given by
[3.3]
where n = total number of data points used, and
p = number of parameters estimated.
This quantity adjusts the correlation coefficient by the number of parameters in the model.
In the case of simple regression, Radj = R. Because the adjusted correlation coefficient
accounts for the number of parameters in the model, it essentially deflates the R value
hence providing for a better evaluation of the fit.
A program was written in Pascal to estimate the parameters given the data and first
estimates of the b and c parameters using nonlinear regression and Newton-Raphson
iteration. The b and c parameters can be better estimated by the linearization method as
discussed in the previous chapter; however, the author has chosen to use nonlinear
regression because inference upon the parameters is more straightforward. If the
linearization technique had been employed, then the inference and summary statistics
would have been based upon the linearized data and not the actual data. This analysis will
include evaluation and validation of the form of the logistic model, to determine if the
model adequately describes the behavior of the data regardless of crop or site. To answer
this question, various grasses were studied. A list of these grasses and their common and
scientific names is given in Table 3-1.


144
T3
>
o
a
g
od
C3
3
O
100
75
50
25
S -25
-50
-75
-100
O

3.0 weeks
4.5 weeks

O
o

o
o
CD


O
100 200 300 400
Predicted N Removal, kg/lia
500
600
Figure 4-17 Residual plot of plant N removal for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals. Original data from Doss et al. (1966).
Solid line is mean and dashed lines are 2 standard errors.


65
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal at each clipping interval for all three
years with and without irrigation, shown in Figure 4-65, are described by
Non-irrigated,1981-2: Y
=
28.6 Nu/(488 + Nu)
[4.194]
Nc
=
17.1 +0.0350NU
[4.195]
1982-3: Y
=
18.9 N/(297 + Nu)
[4.196]
Nc
-
15.7 + 0.0528NU
[4.197]
1983-4: Y
=
13.5 Nu/(231 +NU)
[4.198]
Nc
=
17.1 +0.0741NU
[4.199]
Irrigated, 1981-2: Y
=
36.9 Nu/(605 + Nu)
[4.200]
Nc
=
16.4 + 0.0271NU
[4.201]
1982-3: Y
=
31.2 N/(487 + N)
[4.202]
Nc
-
15.6 + 0.0320NU
[4.203]
1983-4: Y
=
39.7 Nu/(599 + Nu)
[4.204]
Nc
-
15.1 +0.0252NU
[4.205]
The dimensionless plot for bermudagrass is shown in Figure 4-66, and validates the
form of the model. Note that all of the data have been collapsed onto one graph, and the
data fall on the curve. The curves were drawn from
Y/A = 2.36(NU/A')/[1.36 + (NU/A')] [4.206]
Nc/Ncm= 0.58+0.42 (Nu/A') [4.207]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
67 through 4-70.
As done with the previous study, the data will be averaged over years, since
variations due to water availability appear in the linear parameter A. The averaged data
are in Table 4-24 and the analysis of variance is in Table 4-25 (dry matter and N removal x
1 yr x 2 irrigation x 7 N). Comparison among modes 1 and 2 results in a variance ratio of


268
Reid, D. 1978. The effects of frequency of defoliation on yield response of a perennial
ryegrass sward to a wide range of N application rates. ./. of Agr. Science.
Cambridge. 90.447-457.
Robinson, D.L., KG. Wheat, N.L. Hubbert, M S. Henderson, and H.J. Savoy, Jr. 1988.
Dallisgrass yield, quality and nitrogen recovery responses to nitrogen and
phosphorus fertilizers. Commun. Soil Sci. Plan! Anal. 19:529-642.
Rothman, M A. 1972. Discovering the Natural Laws: The Experimental Basis of
Physics. Doubleday. New York.
Segr, E. 1984. From Falling Bodies to Radio Waves. W.H. Freeman and Co. New
York.
Walker, M.E. and D.D. Morey. 1962. Influence of rates of N, P, and K on forage and
grain production of Gator rye in South Georgia. Georgia Agr. Exp. Stn. Cir. N. S.
27. University of Georgia. Athens, GA.
Will, C M. 1986. Was Einstein Right? Putting General Relativity to the Test. Basic
Books. New York.
Williams, JR, C.A. Jones, JR Kiniry, and D A. Spanel. 1989. The EPIC crop growth
model. Trans. Am. Soc. Agr. Engr. 32:497-511.


79
45. The relative errors on the parameters are high due to the flat response to P and K,
making estimation of b and c parameters for these nutrients difficult since the data are high
on the curve. Also, some of the estimates of b are very close to zero. As the standard
errors are divided by these small numbers, they are artificially inflated. The RSS was
calculated to be 5895.76. The overall correlation coefficient and adjusted correlation
coefficient calculated from the regression are 0.9888 and 0.9860, respectively. Results are
shown in Figures 4-111 through 4-113 for the three nutrients, where the curves are drawn
from
5.43
y ~ + ^1.36-0.0225AT + g -0.14-0.0464 A ^ + g -0.91-0.0201 A ^
[4.292]
N
U
260
(1 + ^ 1.93-0.0225N + g-0.14-0.0464A + ^-0.91-0.0201* ^
[4.293]
Pu =
34
+ e 1.36-0.0225N ^ + ^-0.16-0.0464* + g-0.91-0.0201 A' ^
[4.294]
230
~ ^ j + e 1.36-0.0225W ^ j + ^-0.14-0.0464* ^ j + ^0.46-0.0201* ^ [4.295]
From these results, it follows that plant nutrient concentrations, shown in Figures 4-111
through 4-113, are estimated from
Nc = 47.9 [1 + exp(1.93 0.0225N)]/[1 + exp(1.36 0.0225N)] [4.296]
Pc = 6.26 [1 + exp(-0.16 0.0464P)]/[1 + exp(-0.14 0.0464P)] [4.297]
Kc = 42.4 [1 + exp(0.46 0.0201K)]/[1 + exp(-0.91 0.020IK)] [4.298]
As shown, the data are described well by these equations.
From Postulate 3, the relationships between yield and plant nutrient removals and
between plant nutrient concentrations and plant nutrient removals for all three nutrients,
shown in Figures 4-114 through 4-116, are described by


247
()
O
9' o
<~> C >
o
(I
<)
20 25 30
Measured P Removal, kg/ha
35 40
Figure 4-120 Scatter plot of plant P removal for rye grown at Tifton, G A. Original data
from Walker and Morey (1962).


Table 4-43. Error Analysis of Model Parameters for Seasonal Dry Matter Yield and
Plant N Removal of Ryegrass Grown on 20 Different Sites in England.
Site
Component Parameter
Estimate
Standard Error
Relative Error
5
Dry Matter A, Mg/ha
13.84
0.223
0.016
6
A
13.90
0.231
0.017
7
A
8.39
0.146
0.017
8
A
11.80
0.243
0.021
9
A
14.82
0.236
0.016
10
A
12.87
0.229
0.018
12
A
13.83
0.330
0.024
13
A
12.85
0.209
0.016
14
A
13.00
0.228
0.018
15
A
10.63
0.219
0.021
16
A
13.36
0.217
0.016
17
A
10.93
0.201
0.018
19
A
6.37
0.114
0.018
20
A
14.11
0.276
0.020
22
A
10.32
0.196
0.019
23
A
10.25
0.187
0.018
25
A
12.67
0.225
0.018
26
A
10.38
0.215
0.021
27
A
10.55
0.187
0.018
28
A
10.97
0.250
0.023
5
N Removal A, kg/ha
469.7
8.42
0.018
6
A
537.2
10.47
0.019
7
A
334.6
6.23
0.019
8
A
423.2
10.67
0.025
9
A
518.0
9.00
0.017
10
A
446.8
8.82
0.020
12
A
454.7
12.45
0.027
13
A
488.0
8.84
0.018
14
A
466.0
9.49
0.020
15
A
381.5
9.68
0.025
16
A
478.7
8.83
0.018
17
A
370.6
8.68
0.023
19
A
222.2
4.49
0.020
20
A
450.4
10.69
0.024
22
A
368.7
7.82
0.021
23
A
356.1
7.37
0.021
25
A
458.6
9.55
0.021
26
A
355.0
8.68
0.024
27
A
384.0
8.22
0.021
28
A
365.4
9.85
0.027


78
A hyperbolic regression was conducted on the dry matter yield/estimated maximum versus
N removal/estimated maximum plot. The results are shown in Figure 4-110, with the
curves drawn by
Y/A
+
oo
oo
o
""a
z
OO
OO
II
(Nu/A')]
[4.290]
Nc/Ncm
= 0.47+0.53 (Nu/A')
[4.291]
Ym
1 K' 1
This is an important
til
ai and A.
, Ab is calculated to be 0.76.
A \-e'Ab A' e^-X
result. Overman (1995a) has compared the probability density functions for the gaussian
and logistic functions normalized to unit area and inflection points. In the gaussian
distribution V2 arises as an integration constant to guarantee that the function will sum to
one. Similarly, 1.317 is the integration constant in the logistic distribution. The inverse of
this number is calculated to be 0.759. This was also the value of Ab found in the
hyperbolic regression above. Could there be something to this coincidence? Overman
(1995a) also showed in the same paper that the logistic equation could also be redefined
as
1 1
^=1 + e-i/U17=)+<,-0.759{
where <¡) = y/A and £ = cN b. At this point, it is not clear if this is just a surprising
coincidence or a result of some fundamental process of the system. Regardless, it is very
intriguing.
Evaluation of the Extended Triple Logistic (NPK) Model
Tifton, GA: Rye
Data from Walker and Morey (1962) were used. The parameters, their standard
errors, and relative errors resulting from the nonlinear regression are presented in Table 4-


Drv Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
179
0.0 0.2 0.4 0.6 0.8 1.0 1.2
N Removal/Estimaled Maximum
Figure 4-52 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for bermudagrass grown over three years
at Fayetteville, AR, with and without irrigation. Original data from
Huneycutt el a!. (1988); curves drawn from Eq.[4.162] and [4.163],


14
N
(1 + eb"'~c"N)(1 + ebf CfP)(1 + eb*~C*A')
[2.18]
P =
(1 + eb"-c"N XI + eh> CPXI + eb'~c>K)
[2.19]
K =
(\ + eb-~C-N X1 + eb CfPXI + ehl'~ClK)
[2.20]
where the subscripts n, p, and k refer to applied N, P, and K, respectively. This model
assumes that the response to each nutrient is logistic and is treated as independently
applied. In statistical terms, the independence assumption suggests the joint response is
the product of the marginal responses (Mood et al., 1963). Several characteristics should
be noted. With fixed levels of P and K, the NPK model reduces to the simple logistic
model with a modified maximum parameter A. This model will allow for evaluation of the
effect of combinations of N, P, and K on dry matter production. The amount of nutrient
required to reach half of maximum yield is given by Ni/2 = b/c, P1/2 = bp/cp, and K1/2 =
bk/ci<. Negative values of b mean that before application the soil contained more than
enough nutrient to achieve half of maximum yield.
Throughout this analysis a model is viewed as a simplification of reality. Occams
razor will be applied to simplify to the essentials required for description. Box (1976, p.
792) suggested this approach in a paper by stating,
Since all models are wrong the scientist cannot obtain a "correct" one by excessive
elaboration. On the contrary following William of Occam he should seek an
economical description of natural phenomena. Just as the ability to devise simple
but evocative models is the signature of the great scientist so overelaboration and
overparameterization is often the mark of mediocrity.
Evaluation and validation of the models will be conducted by dimensionless, scatter and
residual plots. In the literature, validation is defined as the process by which a simulation


162
Figure 4-35 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a eight week clipping interval over two years for
bermudagrass grown at Tifln, GA. Data from Prine and Burton (1956);
curves drawn from Eq. [4.99] through [4.102],


68
1979-1980 is shown in Table 4-28 (dry matter and N removal x 2 yrs x 5 N). Comparison
among modes 1 and 2 results in variance ratio of 328 that is significant at the 0.1% level.
Comparison among modes 3 and 2 leads to a variance ratio of 9.54 that is significant at
the 99.5 level. Comparison among modes 3 and 4 results in a variance ratio of 12.3 that is
significant at the 0.1% level. Mode 5, individual A for yield and plant N removal for each
year, b for dry matter and N removal, and common c, accounts for all the significant
variation, since F(5,8,95) = 3.69, F(l,13,99.9) = 17.81, and F(2,11,95) = 3.98. Next the
data from the two grasses are combined and the analysis of variance data are presented in
Table 4-29 (dry matter and N removal x 2 grasses x 2 yrs x 5 N). A comparison of modes
1 and 2 results in a variance ratio of 330 that is significant at the 0.1% confidence level.
Comparison of modes 3 and 2 leads to a variance ratio of 7.62, that is significant at the
0.1% confidence level. Comparison of modes 3 and 4 results in a variance ratio of 9.66
that is significant at the 0.1% confidence level. Comparison of modes 3 and 5 results in a
variance ratio of 16.2 that is significant at the 0.1% confidence level. Comparison of
modes 5 and 4 leads to a variance ratio of 5.85 that is significant at the 0.1% confidence
level. Comparison of modes 3 and 6 leads to a variance ratio of 7.54 that is significant at
the 0.1% confidence level. Comparison of modes 6 and 4 results in a variance ratio of
6.58 that is significant at the 0.5% confidence level. Based upon these comparisons, we
can conclude that mode 4, with individual A and b for dry matter yield and plant N
removal for each year and grass and common c describes the data best. The overall
correlation coefficient of 0.9927 and adjusted correlation coefficient of 0.9876 were
calculated by mode 4. The statistical analysis might be affected by the close numbers for
yield and N removal at low values of applied N for both grasses. In the concern that fewer
parameters might be used to significantly account for the variation, the data will be
averaged over years to determine if an individual b can be used for yield and N removal
for both grasses. The averaged data are in Table 4-30 and the analysis of variance for the


127
Table 4-45. Summary of Model Parameters, Standard Errors, and Relative Errors for
the Extended Triple Logistic (NPK) Model for Rye Grown at Tifton, GA.
Parameter
Dry Matter
N Removal
P Removal
K Removal
54301
260
34
230
A, kg/ha
1922
11.4
1.00
11.3
0.0353
0.044
0.029
0.049
1.36
1.93
1.36
1.36
b
0.139
0.166
0.139
0.139
0.102
0.086
0.102
0.102
-0.14
-0.14
-0.16
-0.14
bp
0.150
0.150
0.162
0.150
1.071
1.071
1.012
1.071
-0.91
-0.91
-0.91
0.46
bk
0.199
0.199
0.199
0.130
0.219
0.219
0.219
0.283
0.0225
0.0225
0.0225
0.0225
c, ha/kg
0.0021
0.0021
0.0021
0.0021
0.093
0.093
0.093
0.093
0.0464
0.0464
0.0464
0.0464
cp
0.0081
0.0081
0.0081
0.0081
0.175
0.175
0.175
0.175
0.0201
0.0201
0.0201
0.0201
Ck
0.0039
0.0039
0.0039
0.0039
0.194
0.194
0.194
0.194
Source: Original data from Walker and Morey (1962).
' Estimate
2 Standard Error
3 Relative Error


85
Table 4-5. Seasonal Dry Matter Yield for Coastal Bermudagrass and Pensacola
Bahiagrass Grown at Thorsby, Alabama Averaged over Years 1957-1959.
Type
Irrigation
0
Applied Nitrogen, kg/ha
168 336
672
Coastal Bermudagrass
No
3.95
10.98
16.07
21.44
Yes
4.19
11.46
18.54
22.71
Pensacola Bahiagrass
No
3.62
10.09
15.79
20.72
Yes
3.79
9.71
16.28
22.67
Source: Data from Evans el al. (1961).


Dry Matter Yield. Mg/ha N Concentration, g/kg
199
Figure 4-72 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for tall fescue averaged over three years at Fayetteville, AR,
with and without irrigation. Data from Huneycutt el al. (1988); curves
drawn from Eq. [4.214] through [4.217],


91
Table 4-11. Analysis of Variance of Model Parameters for Bermudagrass Grown at
Maryland and Cut at Five
Harvest Intervals.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
57
1419082.51
24896.18
-
(2) Ind A,b,c
30
30
8213.60
273.79
-
(l)-(2)
27
1410868.91
52254.40
190.9**
(3) Ind A, Com b,c
12
48
28200.40
587.51
-
(3)-(2)
18
19986.80
1110.38
4.06**
(4) Ind A,b Com c
21
39
13605.56
348.86
-
(4)-(2)
9
5391.96
599.11
2.19
(3)-(4)
9
14594.84
1621.65
4.65**
(5) Ind A (yr, At), Ind b
(y,Nu), Com c
13
47
13473.57
286.67
_
(5)-(2)
17
5259.97
309.41
1.13
(3)-(5)
1
14726.83
14726.83
51.4**
Source: Original data from Decker et al. (1971).
Significant at the 0.001 level
F(27,30,99.9)= 3.28
F(18,30,99.9)=3.58
F( 9,30,95) =2.21
F( 9,39,99.9)= 4.05
F( 17,30,95) = 1.98
F( 1,47,99.9)= 12.32


Dry Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
169
Figure 4-42 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for three different number of cuttings over the
season for ryegrass grown at England. Data from Reid (1978); curves
drawn from Eq. [4 109] through [4.117],


Tifln. Georgia: Bermudagrass
23
This data set was taken from a study by Prine and Burton (1956). Data from two
years were utilized. 1953, a wet year, and 1954, a dry year. Five harvest intervals were
studied: 2, 3, 4, 6, and 8 weeks. Five nitrogen levels were included: 0, 112, 336, 672,
and 1010 kg/ha. The data are listed in Table 3-6.
England: Ryegrass
This data set was taken from a study by Reid (1978). Twenty-one nitrogen levels
were included: 0, 28, 56, 84, 112, 140, 168, 196, 224, 252, 280, 308, 336, 392, 448, 504,
560, 616, 672, 784, and 896 kg/ha. The ryegrass was harvested at different growth
stages, resulting in three different number of cuttings per season: 10, 5, 3. The length of
the season was 26 weeks. It should be noted that the different number of cuttings
represent variable harvest intervals, namely 2.6, 5.2, and 8.67 weeks. As a result, we
expect this variable effect to appear in the results. The data are listed in Table 3-7.
Fayetteville. Arkansas: Bermudagrass
This data set was acquired from a study by Huneycutt et al. (1988). The
bermudagrass was grown over three years, with and without irrigation. Six nitrogen levels
were included: 0, 112, 224, 336, 448, 560, and 672 kg/ha. The data are listed in Table 3-
8.
Fayetteville, Arkansas: Tall Fescue
This data set was taken from the same study as above (Huneycutt el al., 1988).
The tall fescue was grown over three seasons, with and without irrigation. Six nitrogen


Predicted N Removal, kg/ha
211
300
250
200
150
100
50
O Bcrmudagrass 1979
Bemuidagrass 1980
A Bahiagrass 1979
V Bahiagrass 1980
/
A
O e
-V

0
$0
V
V
50
100
Measured N Removal, kg/lia
Figure 4-84 Scatter plot of seasonal plant N removal for bahiagrass and bermudagrass
grown over two years at Eagle Lake, TX. Original data from Evers
(1984).


24
levels were included: 0, 112, 224, 336, 448, 560, and 672 kg/ha. The data are listed in
Table 3-9.
Eagle Lake. Texas: Bahiagrass and Bermudagrass
This data set was acquired from a study by Evers (1984). Both grasses were
grown over two years (1979 and 1980), and five nitrogen levels were included: 0, 84,
168, 252, and 336 kg/ha. The data are listed in Table 3-10.
North Carolina: Corn
These field experiments were conducted with corn at three North Carolina
Research Stations: Central Crops Research Station, Clayton, NC, on Dothan loamy sand
soils (fine loamy, siliceous thermic, Plinthic Paleudults), Lower Coastal Plain Tobacco
Research Station, Kinston, NC, on Goldsboro sandy loam soils (fine loamy, siliceous
thermic, Aquic Paleudults), and Tidewater Research Station, Plymouth, NC, on
Portsmouth very fine sandy loam soils (fine loamy, mixed thermic, Typic Umbraquults)
(Kamprath, 1986). The experiments were conducted from 1981 through 1984. The corn
grown at Clayton received irrigation. Each year the experiment was conducted in a
different field and for this reason averages were used in this study. The data from the
Dothan and Goldsboro soils are listed in Table 3-11. The data from the Portsmouth soil
are listed in Table 3-12. For all three sites, five nitrogen levels were included: 0, 56, 112,
168, and 224 kg/ha.
Florida: Bahiagrass
The data from this analysis were drawn from Blue (1987). The bahiagrass was
grown on two soils, an Entisol (Astatula sand) near Williston, Florida, and a Spodosol


Linearized Response
17
Figure 2-1 Response of linearized dry matter yield and plant N removal to applied N
for dallisgrass [Paspalum dilatatum Poir ] grown at Baton Rouge, LA.
Original data from Robinson et al. (1988). Data are linearized using Eq.
[2.6] and A values of 15.60 and 431, respectively.


4-36 Dimensionless plot of seasonal dry matter yield and seasonal plant N
concentration versus seasonal plant N removal 163
4-37 Estimated mximums of seasonal dry matter yield and plant N removal as
a function of harvest interval for two years of bermudagrass grown at
Tifton, GA 164
4-38 Scatter plot of seasonal dry matter yield for bermudagrass grown over
two years at Tifton, GA and cut at five different harvest intervals 165
4-39 Scatter plot of seasonal plant N removal for bermudagrass grown over
two years at Tifton, GA and cut at five different harvest intervals 166
4-40 Residual plot of seasonal dry matter yield for bermudagrass grown over
two years at Tifton, GA and cut at five different harvest intervals 167
4-41 Residual plot of seasonal plant N removal for bermudagrass grown over
two years at Tifton, GA and cut at five different harvest intervals 168
4-42 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for three different number of cuttings over the
season for ryegrass grown at England 169
4-43 Seasonal dry matter yield and plant N removal as a function of plant N
concentration for three different number of cuttings over the season for
ryegrass grown at England 170
4-44 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for three different number of cuttings
over the season for ryegrass grown at England 171
4-45 Scatter plot of seasonal dry matter yield for ryegrass grown at England
and cut different times over the season 172
4-46 Scatter plot of seasonal plant N removal for ryegrass grown at England
and cut different times over the season 173
4-47 Residual plot of seasonal dry matter yield for ryegrass grown at England
and cut different times over the season 174
4-48 Residual plot of seasonal plant N removal for ryegrass grown at England
and cut different times over the season 175
4-49 Estimated mximums of seasonal dry matter yield and plant N removal as
a function of average harvest interval for ryegrass grown at England 176
xv


54
2 weeks, 1953: Nc
1954: Nc
3 weeks, 1953: Nc
1954: Nc
4 weeks, 1953: Nc
1954: Nc
6 weeks, 1953: Nc
1954: Nc
8 weeks, 1953: Nc
1954: Nc
35.9 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.73]
38.6 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.74]
32.3 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.75]
33.9 [1 +exp(2.15 0.0077N)]/[1 + exp(l .47 0.0077N)] [4.76]
29.7 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.77]
29.8 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.78]
23.3 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.79]
25.3 [1 +exp(2.15 0.0077N)]/[1 + exp(l .47 0.0077N)] [4.80]
20.5 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.81]
23.4 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.82]
As shown, the data are described well by these equations.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal at each clipping interval over both
years, shown in Figures 4-31 through 4-35, are described by
2 weeks, 1953: Y =
36.4 Nu/(662 + Nu)
[4.83]
Nc =
18.2 + 0.0275NU
[4.84]
1954: Y =
17.0Nu/(333 +NU)
[4.85]
Nc =
19.6 + 0.0587NU
[4.86]
3 weeks, 1953: Y =
40.3 Nu/(659 + Nu)
[4.87]
Nc =
16.4 + 0.0248NU
[4.88]
1954: Y =
20.2 Nu/(347 + Nu)
[4.89]
Nc =
17.2 + 0.0496NU
[4.90]
4 weeks, 1953: Y =
46.9 Nu/(706 + Nu)
[4.91]
Nc =
15.0 + 0.0213NU
[4.92]
1954: Y =
23.7 N/(357 + N)
[4.93]
Nc =
15.1 +0.0423NU
[4.94]


59
Linear regression was conducted on the estimated maximum dry matter yield resulting in
the following relationship
A = 8.78 + 0.52At [4.130]
with a correlation coefficient of 0.8263. Linear regression was also conducted on the
estimated maximum plant N removal resulting in the following relationship
A' = 396.6-2.8At [4.131]
with a correlation coefficient of 0.4170.
Fayetteville. AR: Bermudagrass
The data for this analysis are taken from Huneycutt et al. (1988). The analysis of
variance is shown in Table 4-17 (dry matter and N removal x 3 yrs x 2 irrigation x 7 N).
Comparison among modes 1 and 2 results in a variance ratio of 170.3 that is significant at
the 0.1% level. Comparison among modes 3 and 2 leads to a variance ratio of 2.05 that is
significant at the 2.5% level. Comparison among modes 3 and 4 results in a variance ratio
of 3.22 that is significant at the 99.5 % level. Comparison among modes 3 and 5,
individual A for yield and plant N removal with and without irrigation for all three years, b
for dry matter and N removal with and without irrigation, and common c, leads to a
variance ratio of 5.21 and is significant at the 0.5% level. Mode 6, individual A for yield
and plant N removal with and without irrigation for all three years, b for dry matter and N
removal over both with and without irrigation and all years, and common c, describes the
data best, since F(21,48,95) = 1.78, F(l,69,99.9) = 11.81, and F(10,59,95) = 2.00. The
difference due to years and irrigation is explained by the A parameter. Postulate 3 is
supported again. The overall correlation coefficient and adjusted correlation coefficient
calculated from mode 6 are 0.9950 and 0.9939, respectively. The error analysis for the
parameters is shown in Table 4-18. Results are shown in Figure 4-50, where curves for
dry matter and plant N removal for all three years and irrigation, are drawn from


Dry Matter Yield, Mg/ha Phosphorus Concentration, g/kg
242
Phosphorus Removal, kg/ha
Figure 4-115 Seasonal dry matter yield and plant P concentration as a function of plant P
removal for rye grown at Tifton, GA. Data from Walker and Morey
(1962). Curves drawn by Eq. [4.301 ] and [4.302],


4-115 Seasonal dry matter yield and plant P concentration as a function of plant
P removal for rye grown at Tifln, GA 242
4-116 Seasonal dry matter yield and plant K concentration as a function of plant
K removal for rye grown at Tifln, GA 243
4-117 Dimensionless plot of seasonal dry matter yield and plant nutrient
concentration as a function of plant nutrient removal for rye grown at
Tifton, GA 244
4-118 Scatter plot of dry matter yield for rye grown at Tifton, GA 245
4-119 Scatter plot of plant N removal for rye grown at Tifton, GA 246
4-120 Scatter plot of plant P removal for rye grown at Tifton, GA 247
4-121 Scatter plot of plant K removal for rye grown at Tifton, GA 248
4-122 Residual plot of dry matter yield for rye grown at Tifton, GA 249
4-123 Residual plot of plant N removal for rye grown at Tifton, GA 250
4-124 Residual plot of plant P removal for rye grown at Tifton, GA 251
4-125 Residual plot of plant K removal for rye grown at Tifton, GA 252
5-1 Sensitivity of logistic to the c parameter 264
xxi


1000
800
600
400
200
0
40
30
20
10
0
-37
164
1953
R = 0.8407
Q_
0'
o
.-0
1954
R = 0.9925
1953
R = 0.9958
o

1954
R = 0.9986
_L
4 6
Harvest Interval, weeks
10
Estimated mximums of seasonal dry matter yield and plant N removal as a
function of harvest interval for two years of bermudagrass grown at Tifton,
GA. Lines drawn from Eq. [4.105] through [4.108],


175
Predicted N Removal, kg/ha
Figure 4-48 Residual plot of seasonal plant N removal for ryegrass grown at England
and cut different times over the season. Original data from Reid (1978).
Solid line is mean and dashed lines are 2 standard errors.


Dry Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
221
Figure 4-94 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for grain and total plant of corn grown at
Plymouth, NC. Data from Kamprath (1986); curves drawn from Eq.
[4.264] through [4.269]


Dry Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
145
Figure 4-18 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bermudagrass grown at Maryland and cut at
five harvest intervals. Data from Decker et al. (1971); curves drawn from
Eq. [4.24] through [4.38],


125
Table 4-43continued
Site
Type
Parameter
Estimate
Standard Error
Relative Error
All
Both
c, ha/ka
0.0088
0.0001
0.011
Source: Original data from Morrison el al. (1980).


CHAPTER 2
LITERATURE REVIEW
Crop models have been used for many years and have many variations (Keen and
Spain, 1992; France and Thornley, 1984). Models have been used to describe the
relationship between various forage crop yields and other parameters including nitrogen
application, water availability, harvest interval, etc. These models have ranged from
simple polynomial models to varying degrees of nonlinear models. Jones et al (1987)
have defined a model as a mathematical representation of a system, and modeling as the
process of developing that representation. Modeling is often confused with simulation.
Simulation includes the processes necessary for operationalizing the model, or solving the
model to mimic real system behavior (Jones et al., 1987). Before beginning a model or
simulation project the objectives of the project should be stated clearly. A clear definition
of the intended end-product and the intended users of the models that will be developed
should be included in the objective statement. In their book on mathematical models in
agriculture, France and Thornley (1984, p. 173) noted that,
There are three fairly distinct communities of people to whom a crop model or
simulator may be of value: first, farmers, including advisory and extension
services, who are primarily concerned with current production; second, applied
scientists, including agronomists and plant breeders, whose objective is to improve
the efficiency of production techniques, for the most part using current knowledge;
and third, research scientists, whose aim is to extend the bounds of present-day
knowledge. There is no way in which a single model can satisfy the differing
objectives of these three groups, which are, at least in part, incompatible.
Various mathematical models have been used to describe crop growth.
Polynomials are often used because of the simplicity in determining parameter values;
however, they have serious weaknesses. These weaknesses have been addressed in detail


81
Table 4-1. Analysis of Variance of Model Parameters Used to Describe Coastal
Bermudagrass Yield Response
to Nil
rogen at Thorsby, Alabama, 1957-1959.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
21
42.347
2.017
-
(2) Ind A,b,c
18
6
3.507
0.584
-
(D-(2)
15
38.840
2.589
4.43+
(3) Ind A, Com b,c
8
16
12.267
0.766
-
(3)-(2)
10
8.750
0.875
1.50
(4) Ind A,b Com c
13
11
8.019
0.729
-
(4)-(2)
5
4.512
0.902
1.54
(3)-(4)
5
4.248
0.850
1.17
(5) Ind A, Com c, Ind b
(irrigation)
9
15
12.342
0.823
-
(5)-(2)
9
8.835
0.982
1.68
(5H4)
4
4.323
1.081
1.48
Source: Original yield data from Evans etal. (1961).
Significant at the 0.05 level
F(15, 6,95) =3.94
F(10, 6,95) =4.06
F( 5, 6,95) = 4.39
F( 5,11,95) =3.20
F( 9, 6,95) =4.10
F( 4,11,95) =3.36


151
C3
§
on
A
-a
>
o
S
s
Q
13
3
IS
w
0
125
G
1
3.2 weeks
r
i 1
100

A
3.6 weeks
4.3 weeks
-
V
5.5 weeks
75
- O
7.7 weeks
50
-
<>
25
V
G _
0
G
o A
o
)
&
£>
G
-25
IS
O Ov
-
-50
-
-75
-
-100
-125
100 200 300
Predicted N Removal, kg/ha
400
500
Figure 4-24 Residual plot of plant N removal for bermudagrass grown at Maryland and
cut at five harvest intervals. Original data from Decker et al. (1971). Solid
line is mean and dashed lines are 2 standard errors.


166
700
600
500
C3
*5
on
>
o
a
§
&
T3
6
a
J
+3
CO
UJ
400
300
200
100
o
i 1 1 i
1953, 2 weeks
1 1 7
A v /

1953, 3 weeks
/ D
A
1953. 4 weeks
A> -
V
1953, 6 weeks
O/
o
1953. 8 weeks

1954, 2 weeks
/

1954.3 weeks
/
A
1954. 4 weeks

1954, 6 weeks

1954, 8 weeks ^7
-
O / A
~
-
T
I*'
/
-
-
A'
-
A7
/ 1 1 1 1
j 1
100 200 300 400 500
Measured N Removal, kg/lia
600
700
Figure 4-39 Scatter plot of seasonal plant N removal for bermudagrass grown over two
years at Tifton, GA, and cut at five different harvest intervals. Original
data from Prine and Burton (1956).


21
statistics, we further compare the mean sum of squares (MSS) for the two modes in
question. If there is not a large increase in MSS by simplifying the model (requiring fewer
parameters), we will accept that there is a common c. By examination of the plotted data,
it appears this supposition is valid.
As stated previously, the extended model is based upon three postulates. One of
the consequences of the postulates was that dry matter yield and plant N removal are
hyperbolically related and plant N concentration and plant N removal are linearly related.
To test these results, dry matter yield and N concentration are plotted against plant N
removal. The lines describing these relationships are dependent only upon the Ab = (b' -
b). Furthermore, the data can be normalized by dividing by the appropriate estimated
mximums. If the assumption of common Ab is true, all the normalized data will fall on
one line in a dimensionless plot. This is one way of testing the adequacy of the model to
fit the data and testing the postulates.
Extended Triple Logistic (NPK) Model for Dry Matter Yield and Nitrogen Removal
This model is developed on the assumption that response of dry matter yield and
plant nutrient removal to applied N, P, and K individually follow the extended logistic
model. Overman and Wilkinson (1995) first discussed this model and applied it to a
complete factorial. Dimensionless plots, scatter plots and residual plots are used to
evaluate the model.


212
1
2
13
£
a
i o
£
c3
3
3 .i
o
Q
-2
-3
O Bcrmudagrass 1979
Bcmmdagrass 1980
A Baliiagrass 1979
V Baliiagrass 1980
D
V Dv v
A
( )
O
.A-
AQ
A
-<)--
i >
J I L I
6 8 10 12
Predicted Dry Matter Yield, Mg/ha
14
16
Figure 4-85 Residual plot of seasonal dry matter yield for bahiagrass and berntudagrass
grown over two years at Eagle Lake, TX. Original data from Evers
(1984). Solid line is mean and dashed lines are 2 standard errors.


34
Table 3-9. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Tall Fescue Grown at Fayetteville, Arkansas over Three Seasons, Irrigated
and Non-irrigated.
Year
0
112
Applied Nitrogen, kg/ha
224 336 448
560
672
Dry Matter Yield
1981-2
2.51
5.47
Non-irrigated
8.97 10.31 12.33
11.21
11.86
Mg/ha
1982-3
1.86
5.00
4.64
7.11
7.67
7.40
8.32
1983-4
1.91
3.99
5.45
4.86
4.89
5.09
4.95
1981-2
3.92
7.85
10.90
Irrigated
13.36
14.84
15.33
15.53
1982-3
2.91
6.37
7.29
11.70
12.76
13.45
13.72
1983-4
3.77
7.91
10.49
14.59
15.92
17.15
17.15
N Removal
1981-2
55
118
Non-irrigated
207 259 337
334
368
kg/ha
1982-3
36
97
90
174
196
206
229
1983-4
37
94
147
133
140
150
147
1981-2
84
166
248
Irrigated
344
399
424
442
1982-3
62
120
139
275
331
362
371
1983-4
75
162
237
352
390
414
447
N Concentration
1981-2
21.9
21.6
Non-irrigated
23.1 25.1 27.4
29.8
31.0
g/kg
1982-3
19.2
19.4
19.4
24.5
25.6
27.8
27.5
1983-4
19.4
23.7
27.0
27.4
28.6
29.4
29.8
1981-2
21.4
21.1
22.7
Irrigated
25.8
26.9
27.7
28.5
1982-3
21.4
18.9
19.0
23.5
25.9
26.9
27.0
1983-4
19.8
20.5
22.6
24.2
24.5
24.2
26.1
Source: Data from Huneycutt et al. (1988).


Table 4-10. Error Analysis of Model Parameters of Bermudagrass Grown at
Thorsby, AL.
Type
Clipping
Interval
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter Yield
3.0 weeks
A, Mg/ha
17.42
0.248
0.014
4.0 weeks
A
19.75
0.279
0.014
N Removal
3.0 weeks
A, kg/ha
568.9
8.83
0.016
4.0 weeks
A
543.5
8.45
0.016
Both
DM
b
1.27
0.076
0.060
Both
Nu
b
2.02
0.095
0.047
Both
Both
c, ha/kg
0.0067
0.0003
0.045
Source: Original data from Doss et al. (1966).


52
N removal are plotted against harvest interval in Figure 4-25. Linear regression was
conducted on the estimated maximum dry matter yield, omitting the 7.7 week harvest
interval, resulting in the following relationship
A = 7.90+ 1.71 At = 7.90(1 +0.22At) [4.51]
with a correlation coefficient of 0.9676. Linear regression was also conducted on the
estimated maximum plant N removal, omitting the 7.7 week harvest interval, resulting in
the following relationship
A'= 483.9-8.3At = 483.9 (1 0.017At) [4.52]
with a correlation coefficient of 0.6206. The small correlation coefficient and rather flat
response suggests that there is uncertainty in the nature of the relationship.
Tifln. GA: Bermudattrass
The data for this analysis are taken from Prine and Burton (1956). The analysis of
variance is shown in Table 4-13 (dry matter and N removal x 2 yrs x 5 At x 5 N).
Comparison among modes 1 and 2 results in a variance ratio of 81.2 that is significant at
the 0.1% level. Comparison among modes 3 and 2 leads to a variance ratio of 1.89 that is
significant at the 97.5 % level. Comparison among modes 3 and 4 results in a variance
ratio of 2.45 that is significant at the 0.5% level. Comparison among modes 3 and 5,
individual A for yield and plant N removal at the five harvest intervals and over both years,
b for dry matter and N removal at both years, and common c, leads to a variance ratio of
15.9 that is significant at the 0.1% level. Mode 6, individual A for yield and plant N
removal at the five harvest intervals and over both years, b for dry matter and N removal
over both years, and common c, describes the data best, since F(37,40,95) = 1.71,
F( 1,77,99.9) = 11.71, F(18,59,95) = 1.78, and F(2,75,95) = 3.12. The difference due to
years (water availability) and harvest interval is explained by the A parameter. Postulate 3
is supported again. The overall correlation coefficient and adjusted correlation coefficient


30
Table 3-5. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Dallisgrass Grown in Maryland.
Harvest
Interval
weeks
Applied Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
3.2
0
1.29
21.5
16.7
112
4.37
90.5
20.7
224
8.72
225
25.8
448
13.53
409
30.2
672
14.07
461
32.8
896
14.36
497
34.6
3.6
0
1.24
19.2
15.5
112
4.84
86.6
17.9
224
9.04
198
21.9
448
12.94
371
28.7
672
13.94
450
32.3
896
13.78
480
34.8
4.3
0
1.09
15.7
14.4
112
5.19
88.7
17.1
224
9.81
197
20.1
448
14.36
393
27.4
672
15.33
445
29.0
896
14.93
451
30.2
5.5
0
1.55
20.6
13.3
112
7.75
122
15.8
224
12.96
238
18.4
448
16.71
369
22.1
672
17.07
418
24.5
896
17.07
485
28.4
7.7
0
1.65
17.7
10.7
112
9.37
125
13.3
224
14.55
215
14.8
448
18.23
354
19.4
672
18.89
399
21.1
896
18.37
421
22.9
Source: Data from Decker et al. (1971).


182
Figure 4-55 Residual plot of seasonal dry matter yield for bermudagrass grown over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt el al. (1988) Solid line is mean and dashed lines are 2
standard errors.


Data Sets to be Investigated
22
Thorsbv. Alabama: Bahiagrass and Bermudagrass
This study is from the Auburn Agricultural Experiment Station. The results have
been previously reported in a station bulletin (Evans el al., 1961). The results of field
tests and grazing trials conducted in Alabama to show the response of bermudagrass and
bahiagrass to nitrogen and irrigation on specific soil types were reported. The yields of
the grasses grown on Greenville fine sandy loam, Thorsby, AL, over three years (1957-
1959) were used in this analysis. Four nitrogen levels were included: 0, 168, 336, and
672 kg/ha. The data are listed in Table 3-2.
Baton Rouge. Louisiana: Dallisgrass
The data for this analysis were acquired from a study by Robinson et al. (1988).
Six nitrogen levels were included: 0, 56, 112, 224, 448, and 896 kg/ha. The data are
listed in Table 3-3.
Thorsbv. Alabama: Bermudagrass
The data for this analysis were taken from a study by Doss el al. (1966). Two
harvest intervals were studied: 3.0 and 4.5 weeks. Six nitrogen levels were included: 0,
224, 448, 672, 1344, and 2016 kg/ha. The data are listed in Table 3-4.
Maryland: Bermudagrass
The data for this analysis were acquired from a study by Decker et al. (1971).
Five harvest intervals were studied: 3.2, 3.6, 4.3, 5.5, and 7.7 weeks. Six nitrogen levels
were included. 0, 112, 224, 448, 672, and 896 kg/ha. The data are listed in Table 3-5.


LIST OF REFERENCES
Adby, P R. and MAH Dempster. 1974. Introduction to Optimization Methods. John
Wiley & Sons, Inc. New York.
Allhands, M.N., S.A. Allick, A R. Overman, W.G. Leseman, and W. Vidak. 1995.
Municipal water reuse at Tallahassee, Florida. Trans. Am. Soc. Agr. Engr. 38:411-
418.
Bates, D M. and D.G. Watts. 1988. Nonlinear Regression Analysis and Its Applications.
John Wiley & Sons, Inc. New York.
Blue, W.G. 1987. Response of Pensacola bahiagrass (Paspalum notatum Fliigge) to
fertilizer nitrogen on an Entisol and a Spodosol in north Florida. Soil and Crop
Sci. Soc. Fla. Proc. 47:135-139.
Box, G.E.P. 1976. Science and statistics. J. Amer. Slat. Assoc. 71:791-799.
Box, G.E.P. 1979. Some problems of statistics and everyday life. J. Amer. Slat. Assoc.
74:1-4.
Box, G.E.P., W.G. Hunter, and J.S. Hunter. 1978. Statistics for Experimenters: An
Introduction to Design, Data Analysis, and Model Building. John Wiley & Sons,
Inc. New York.
Decker, A M., R.W. Hemkin, JR Miller, N.A. Clark, and A.V. Okorie. 1971. Nitrogen
fertilization, harvest management, and utilization of 'Midland' bermudagrass
(Cynodon dactylon Pers ). University of Maryland Agricultural Experiment
Station Bulletin 487. University of Maryland. College Park, MD.
Doss, B.D., D A. Ashley, O.L. Bennet, and R.M. Patterson. 1966. Interactions of soil
moisture, nitrogen, and clipping frequency on yield and nitrogen content of Coastal
bermudagrass. Agron. J. 58:510-512.
Downey, D and A.R Overman 1988. Simulation models for bahiagrass. Agricultural
Engineering Department, University of Florida. Gainesville, FL.
Draper, N.R. and H. Smith. 1981. Applied Regression Analysis. John Wiley & Sons.
New York.
265


180
Measured Dry Matter Yield. Mg/ha
Figure 4-53
Scatter plot of seasonal dry matter yield for bermudagrass grown over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt el al. (1988).


167
I)
f 2
33
13
£
i-i
u
S o
£
Q
13
3
3 -2
CO ^
4>
Q
o
1953, 2 weeks

1953,3 weeks
A
1953, 4 weeks
- V
1953, 6 weeks
o
1953, 8 weeks

1954, 2 weeks

1954,3 weeks
A
1954, 4 w eeks
T
1954, 6 weeks

1954, 8 weeks
V
ha
Hi
-4
-6
-8
o
A

A
V
o
V
T A
()

Aa
V
A
$

O
10 15 20
Predicted Dry Matter Yields, Mg/ha
25
30
Figure 4-40 Residual plot of seasonal dry matter yield for bermudagrass grown over
two years at Tifton, GA, and cut at five different harvest intervals. Original
data from Prine and Burton (1956). Solid line is mean and dashed lines are
2 standard errors.


86
Table 4-6. Error Analysis for Model Parameters on Averaged Dry Matter Yield of
Coastal Bermudagrass and Pensacola Bahiagrass Grown at Thorsby,
Alabama.
Type
Irrigation
Parameter
Estimate
Standard
Error
Relative
Error
Coastal Bermudagrass
No
A, Mg/ha
21.57
0,359
0.017
Yes
A
23.44
0.388
0.017
Pensacola Bahiagrass
No
A
21.49
0.369
0.017
Yes
A
22.73
0.395
0.017
Coastal Bermudagrass
Both
b
1.39
0.056
0.040
Pensacola Bahiagrass
Both
b
1.57
0.059
0.038
Both
Both
c, ha/kg
0.0078
0.0003
0.038
Source: Original data from Evans et al. (1961).


250
100
75
50
25
o
S
2 0
CO
3
-25
C/J
-50 -
-75 -
-100
O
ri
co
O o O
o
o
O Qj
- 0 '
50 100 150
Estimated N Removal, kg/lia
200
250
Figure 4-123 Residual plot of plant N removal for rye grown at Tifln, GA. Original
data from Walker and Morey (1962). Solid line is mean and dashed lines
are 2 standard errors.


Drv Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
200
1.2 1 1 i
0.4
O Non-Irrigated
Irrigated
0.0 1 1 1 J
0.0 0.2 0.4 0.6 0.8 1.0 1.2
N Removal/Estimatcd Maximum
Figure 4-73 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for tall fescue averaged over three years at
Fayetteville, AR, with and without irrigation. Original data from
Huneycutt etol. (1988); curves drawn from Eq. [4.218] and [4.219],


149
O 100 200 300 400 500
Measured N Removal, kg/ha
Figure 4-22 Scatter plot of plant N removal for bermudagrass grown at Maryland and
cut at five harvest intervals. Original data from Decker et al. (1971).


Dry Matter Yield. Mg/ha N Removal, kg/ha N Concentration, g/kg
198
Figure 4-71 Response of seasonal dry matter yield, plant N removal and plant N
concentration for tall fescue averaged over three years at Fayetteville, AR,
with and without irrigation Data from Huneycutt el al. (1988); curves
drawn from Eq. [4.208] through [4.213],


Dry Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
228
Applied Nitrogen, kg/ha
Figure 4-101 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bahiagrass grown on two soils in Florida.
Data from Blue (1987); curves drawn from Eq. [4.276] through [4.281],



PAGE 1

ESTIMATION OF DRY MATTER AND NITROGEN REMOVAL BY THE LOGISTIC EQUATION By DENISE MARIE WILSON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1995 UNIVERSITY OF FLORIDA LIBRARIES — «

PAGE 2

Copyright 1995 by Denise M. Wilson

PAGE 3

I dedicate this dissertation to my parents, Robert and Marian Shelton, my parents, Brooks and Carol Wilson, and especially to my husband, Russell.

PAGE 4

ACKNOWLEDGMENTS First, I would like to thank the many field plant-soil science researchers throughout the world for having well-designed experiments resulting in great data. Developing and maintaining a field plot experiment are no easy task and I wish to recognize and thank those people. I would like to thank my husband, Russell, for the many hours of brainstorming and editing. He was a stabilizing influence during a swift three years of graduate work, and I thank him for the encouragement he provided during my weak moments. I would like to thank my parents, Robert and Marian Shelton, for the many sacrifices that were done in order that I might be able to attend college in the first place. Furthermore, I would like to thank them for instilling in me a sense of honor, a hard work ethic, and determination that I could accomplish anything. I would like to thank my major professor and advisor over the last five years, Dr. Allen Overman. He instilled in me a sense of professionalism and always demanded high quality work. I am thankful for the opportunity I was given the summer of 1991 to work with him, since this work helped establish the foundation for my graduate work. I would like to give my appreciation to the members of my committee, Dr. Larry Bagnall, Dr. Stanley Wilkinson, Dr. Paul Chadik, and Dr. Frank Martin for giving their time. I would like to give special thanks to Dr. Frank Martin for also serving as my committee chair on my concurrent master's program in statistics. Your wisdom, guidance, and encouragement were greatly appreciated. I would also like to thank the National Science Foundation for selecting me as a fellow for the period of August 1992 to August 1995. The stipend and cost of education allowance iv

PAGE 5

enabled me to concentrate solely on my studies. I considered it a high honor to have been selected for this award out of the thousands of applicants in science and engineering. The fellowship provided me with a freedom to plot my own course and research project. V

PAGE 6

TABLE OF CONTENTS page ACKNOWLEDGMENTS iv LIST OF TABLES vii LIST OF FIGURES xii ABSTRACT xxii CHAPTERS 1 INTRODUCTION 1 2 LITERATURE REVIEW 6 3 MATERIALS AND METHODS 18 Analysis of Data 18 Models to be Investigated 20 Data Sets to be Investigated 21 4 RESULTS AND DISCUSSION 44 Evaluation of the Simple Logistic Model 44 Evaluation of the Extended Logistic Model 46 Evaluation of the Extended Triple Logistic (NPK) Model 78 5 SUMMARY AND CONCLUSIONS 253 REFERENCES 265 BIOGRAPHICAL SKETCH 269 vi

PAGE 7

LIST OF TABLES Table page 3-1 Common and Scientific Names of Grasses Studied 26 3-2 Seasonal Dry Matter Yield of Coastal Bermudagrass and Pensacola Bahiagrass Grown on Greenville Fine Sandy Loam at Thorsby, Alabama 27 3-3 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Dallisgrass Grown at Baton Rouge, Louisiana 28 3-4 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Bermudagrass Grown at Thorsby, Alabama over 3.0 and 4.5 Week Harvest Intervals (1961) 29 3-5 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Dallisgrass Grown in Maryland 30 3-6 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Bermudagrass Grown at Tifton, Georgia 31 3-7 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Perennial Ryegrass Grown in England with a Different Number of Harvests over the Season .' 32 3-8 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Bermudagrass Grown at Fayetteville, Arkansas over Three Years, Irrigated and Non-irrigated 33 3-9 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Tall Fescue Grown at Fayetteville, Arkansas over Three Seasons, Irrigated and Non-irrigated 34 3-10 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Bahiagrass and Bermudagrass Grown at Eagle Lake, Texas 35 3-1 1 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Grain and Total Plant of Corn Grown on Dothan and Goldsboro Soils at Clayton and Kinston, North Carolina, Respectively 36 vii

PAGE 8

3-12 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Grain and Total Plant of Corn Grown on Portsmouth Soil at Plymouth, North Carolina 37 3-13 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Bahiagrass Grown on Entisol and Spodosol Soils in Florida 38 3-14 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Ryegrass Grown in England 39 315 Seasonal Dry Matter Yields and Nutrient Response to Applied Nutrients for Gator Rye at Tifton, Georgia 43 41 Analysis of Variance of Model Parameters Used to Describe Coastal Bermudagrass Yield Response to Nitrogen at Thorsby, Alabama, 19571959 81 4-2 Analysis of Variance of Model Parameters Used to Describe Pensacola Bahiagrass Yield Response to Nitrogen at Thorsby, Alabama, 1957-1959 82 4-3 Analysis of Variance of Model Parameters Used to Describe Coastal Bermudagrass and Pensacola Bahiagrass Yield Response to Nitrogen at Thorsby, Alabama, 1957-1959 83 4-4 Error Analysis for Model Parameters of Coastal Bermudagrass and Pensacola Bahiagrass Grown at Thorsby, Alabama 84 4-5 Seasonal Dry Matter Yield for Coastal Bermudagrass and Pensacola Bahiagrass Grown at Thorsby, Alabama Averaged over Years 1957-1959 85 4-6 Error Analysis for Model Parameters on Averaged Dry Matter Yield of Coastal Bermudagrass and Pensacola Bahiagrass Grown at Thorsby, Alabama 86 4-7 Analysis of Variance of Model Parameters Used to Describe Dallisgrass Grown at Baton Rouge, LA 87 4-8 Error Analysis for Model Parameters of Dallisgrass Grown at Baton Rouge, LA 88 4-9 Analysis of Variance of Model Parameters Used to Describe Bermudagrass Grown at Thorsby, AL with Two Clipping Intervals 89 viii

PAGE 9

4-10 Error Analysis of Model Parameters of Bermudagrass Grown at Thorsby, AL 90 4-11 Analysis of Variance of Model Parameters for Bermudagrass Grown at Maryland and Cut at Five Harvest Intervals 91 41 2 Error Analysis for Model Parameters of Bermudagrass Grown at Maryland and Cut at Five Harvest Intervals 92 4-13 Analysis of Variance on Model Parameters for Bermudagrass Grown at Tifton, GA, over Two Years and Cut at Five Different Harvest Intervals 93 4-14 Error Analysis for Model Parameters of Bermudagrass Grown at Tifton, GA over Two Years and Cut at Five Different Harvest Intervals 94 4-15 Analysis of Variance of Model Parameters on Ryegrass Grown at England, with Three Different Numbers of Cuttings over the Season for 1969 95 41 6 Error Analysis for Model Parameters of Ryegrass Grown at England, with Three Different Numbers of Cuttings over the Season for 1969 96 4-17 Analysis of Variance for Bermudagrass over Three Seasons, Irrigated and Non-irrigated, Grown at Fayetteville, AR 97 4-18 Error Analysis for Model Parameters of Bermudagrass Grown at Fayetteville, AR, over Three Years, with and without Irrigation 98 41 9 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Bermudagrass Grown at Fayetteville, AR. Averaged over Three Years 99 4-20 Analysis of Variance for Model Parameters for Bermudagrass Grown at Fayetteville, AR, Averaged over Three Years, with and without Irrigation 100 4-21 Error Analysis for Model Parameters of Bermudagrass Grown at Fayetteville, AR, Averaged over Three Years, with and without Irrigation 101 4-22 Analysis of Variance of Model Paramters for Tall Fescue Grown at Fayetteville, AR, over Three Seasons, with and without Irrigation 102 4-23 Error Analysis for Model Parameters of Tall Fescue Grown at Fayetteville, AR, over Three Seasons, with and without Irrigation 103 4-24 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Tall Fescue Grown at Fayetteville, AR, Averaged over Three Seasons 104 ix

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4-25 Analysis of Variance of Model Parameters for Tall Fescue Grown at Fayetteville, AR, Averaged over Three Seasons, with and without Irrigation. 105 4-26 Error Analysis for Model Parameters of Tall Fescue Grown at Fayetteville, AR, Averaged over Three Seasons, with and without Irrigation 106 4-27 Analysis of Variance for Bermudagrass Grown at Eagle Lake, Texas, over Two Years 107 4-28 Analysis of Variance for Bahiagrass Grown at Eagle Lake, Texas, over Two Years 108 4-29 Analysis of Variance for Bermudagrass and Bahiagrass Grown at Eagle Lake, Texas, over Two Years 109 4-30 Averaged Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Bermudagrass and Bahiagrass Grown at Eagle Lake, TX. . 110 4-3 1 Analysis of Variance on Model Parameters for Bermudagrass and Bahiagrass Grown at Eagle Lake, TX, Averaged to Estimate b and c Parameters Ill 4-32 Error Analysis for Model Parameters of Bermudagrass and Bahiagrass Grown at Eagle Lake, TX Averaged over Years 1 12 4-33 Error Analysis for Model Parameters of Bermudagrass and Bahiagrass Grown at Eagle Lake, TX over Two Years 113 4-34 Analysis of Variance on Model Parameters for Corn Grown on Dothan Sandy Loam at Clayton, NC, Both Grain and Total 114 4-35 Analysis of Variance on Model Parameters for Corn Grown on Goldsboro Sandy Loam at Kinston, NC, Both Grain and Total 115 4-36 Analysis of Variance on Model Parameters for Grain and Total Plant of Corn Grown on Dothan and Goldsboro Sandy Loam at Clayton and Kinston, NC. . 1 16 4-37 Error Analysis of Model Parameters for Grain and Total Plant of Corn Grown on Dothan and Goldsboro Sandy Loam at Clayton and Kinston, NC. . 117 4-38 Analysis of Variance on Model Parameters for Grain and Total Plant of Corn Grown on Portsmouth Very Fine Sandy Loam at Plymouth, NC 118 4-39 Error Analysis for Model Parameters for Grain and Total Plant of Corn Grown on Portsmouth Very Fine Sandy Loam at Plymouth, NC 119 x

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4-40 Analysis of Variance on Model Parameters for Bahiagrass Grown on Two Soils: an Entisol and Spodosol at Williston and Gainesville, FL, Respectively 120 4-4 1 Error Analysis for Model Parameters for Bahiagrass Grown on Two Soils: an Entisol and Spodosol at Williston and Gainesville, FL, Respectively 121 4-42 Analysis of Variance on Model Parameters for Seasonal Dry Matter Yield and Plant N Removal of Ryegrass Grown on 20 Different Sites in England. ... 122 4-43 Error Analysis of Model Parameters for Seasonal Dry Matter Yield and Plant N Removal of Ryegrass Grown on 20 Different Sites in England 123 4-44 Summary of Model Parameters for Ryegrass in England 126 445 Summary of Model Parameters, Standard Errors, and Relative Errors for the Extended Triple Logistic (NPK) Model for Rye Grown at Tifton, GA 127 51 A Summary of the Ab Parameter for Various Studies 262 5-2 A Summary of c and N' Parameters from Various Studies 263 xi

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LIST OF FIGURES Figure page 11 Response of dry matter yield, N removal, and N concentration for bermudagrass as a function of applied N grown over two years at five clipping intervals at Tifton, GA 5 21 Response of linearized dry matter yield and plant N removal to applied N for dallisgrass [Paspalum dilatatum Poir ] grown at Baton Rouge, LA 17 4-1 Response of seasonal dry matter yield for Coastal bermudagrass and Pensacola bahiagrass to applied N at Thorsby, AL 128 4-2 Scatter plot of seasonal dry matter yield for Coastal bermudagrass and Pensacola bahiagrass at Thorsby, AL 129 4-3 Residual plot of seasonal dry matter yield for Coastal bermudagrass and Pensacola bahiagrass at Thorsby, AL 130 4-4 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for dallisgrass grown at Baton Rouge, LA 131 4-5 Seasonal dry matter yield and plant N concentration as a function of plant N removal for dallisgrass grown at Baton Rouge, LA 132 4-6 Dimensionless plot of dry matter and plant N concentration as a function of plant N removal for dallisgrass grown at Baton Rouge, LA 133 4-7 Scatter plot of dry matter yield for dallisgrass grown at Baton Rouge, LA. .. 134 4-8 Scatter plot of plant N removal for dallisgrass grown at Baton Rouge, LA. .. 135 4-9 Residual plot of dry matter yield for dallisgrass grown at Baton Rouge, LA 136 4-10 Residual plot of plant N removal for dallisgrass grown at Baton Rouge, LA 137 xii

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41 1 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for bermudagrass grown at Thorsby, AL and cut at two harvest intervals 138 4-12 Seasonal dry matter yield and plant N concentration as a function of N removal for bermudagrass grown at Thorsby, AL and cut at two harvest intervals 139 4-13 Dimensionless plot of dry matter and plant N concentration as a function of plant N removal for bermudagrass grown at Thorsby, AL and cut at two harvest intervals 140 4-14 Scatter plot of dry matter yield for bermudagrass grown at Thorsby, AL and cut at two harvest intervals 141 41 5 Scatter plot of plant N removal for bermudagrass grown at Thorsby, AL and cut at two harvest intervals 142 41 6 Residual plot of dry matter yield for bermudagrass grown at Thorsby, AL and cut at two harvest intervals 143 41 7 Residual plot of plant N removal for bermudagrass grown at Thorsby, AL and cut at two harvest intervals 144 4-18 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for bermudagrass grown at Maryland and cut at five harvest intervals 145 4-19 Seasonal dry matter yield and plant N concentration as a function of plant N removal for bermudagrass grown at Maryland and cut at five harvest intervals 146 4-20 Dimensionless plot of dry matter and plant N concentration as a function of plant N removal for bermudagrass grown at Maryland and cut at five harvest intervals 147 4-2 1 Scatter plot of dry matter yield for bermudagrass grown at Maryland and cut at five harvest intervals 148 4-22 Scatter plot of plant N removal for bermudagrass grown at Maryland and cut at five harvest intervals 149 4-23 Residual plot of dry matter yield for bermudagrass grown at Maryland and cut at five harvest intervals 150 xiii

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4-24 Residual plot of plant N removal for bermudagrass grown at Maryland and cut at five harvest intervals 151 4-25 Estimated maximum dry matter yield and estimated maximum plant N removal as a function of harvest interval for bermudagrass in Maryland 152 4-26 Seasonal dry matter yield, plant N removal, and plant N concentration for a two week clipping interval over two years for bermudagrass grown at Tifton, GA 153 4-27 Seasonal dry matter yield, plant N removal, and plant N concentration for a three week clipping interval over two years for bermudagrass grown at Tifton, GA 154 4-28 Seasonal dry matter yield, plant N removal, and plant N concentration for a four week clipping interval over two years for bermudagrass grown at Tifton, GA : 155 4-29 Seasonal dry matter yield, plant N removal, and plant N concentration for a six week clipping interval over two years for bermudagrass grown at Tifton, GA 156 4-30 Seasonal dry matter yield, plant N removal, and plant N concentration for a eight week clipping interval over two years for bermudagrass grown at Tifton, GA 157 4-3 1 Seasonal dry matter yield and plant N concentration as a function of plant N removal for a two week clipping interval over two years for bermudagrass grown at Tifton, GA 158 4-32 Seasonal dry matter yield and plant N concentration as a function of plant N removal for a three week clipping interval over two years for bermudagrass grown at Tifton, GA 159 4-33 Seasonal dry matter yield and plant N concentration as a function of plant N removal for a' four week clipping interval over two years for bermudagrass grown at Tifton, GA 160 4-34 Seasonal dry matter yield and plant N concentration as a function of plant N removal for a six week clipping interval over two years for bermudagrass grown at Tifton, GA 161 4-35 Seasonal dry matter yield and plant N concentration as a function of plant N removal for a eight week clipping interval over two years for bermudagrass grown at Tifton, GA 162 xiv

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4-36 Dimensionless plot of seasonal dry matter yield and seasonal plant N concentration versus seasonal plant N removal 163 4-37 Estimated maximums of seasonal dry matter yield and plant N removal as a function of harvest interval for two years of bermudagrass grown at Tifton, GA 164 4-38 Scatter plot of seasonal dry matter yield for bermudagrass grown over two years at Tifton, GA and cut at five different harvest intervals 165 4-39 Scatter plot of seasonal plant N removal for bermudagrass grown over two years at Tifton, GA and cut at five different harvest intervals 166 4-40 Residual plot of seasonal dry matter yield for bermudagrass grown over two years at Tifton, GA and cut at five different harvest intervals 167 4-41 Residual plot of seasonal plant N removal for bermudagrass grown over two years at Tifton, GA and cut at five different harvest intervals 168 4-42 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for three different number of cuttings over the season for ryegrass grown at England 169 4-43 Seasonal dry matter yield and plant N removal as a function of plant N concentration for three different number of cuttings over the season for ryegrass grown at England 1 70 4-44 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for three different number of cuttings over the season for ryegrass grown at England 171 4-45 Scatter plot of seasonal dry matter yield for ryegrass grown at England and cut different times over the season 1 72 4-46 Scatter plot of seasonal plant N removal for ryegrass grown at England and cut different times over the season 173 4-47 Residual plot of seasonal dry matter yield for ryegrass grown at England and cut different times over the season 174 4-48 Residual plot of seasonal plant N removal for ryegrass grown at England and cut different times over the season 175 4-49 Estimated maximums of seasonal dry matter yield and plant N removal as a function of average harvest interval for ryegrass grown at England 176 xv

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4-50 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for bermudagrass grown over three years at Fayetteville, AR, with and without irrigation 177 4-5 1 Seasonal dry matter yield and plant N concentration as a function of plant N removal for bermudagrass grown over three years at Fayetteville, AR, with and without irrigation 178 4-52 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for bermudagrass grown over three years at Fayetteville, AR, with and without irrigation 179 4-53 Scatter plot of seasonal dry matter yield for bermudagrass grown over three years at Fayetteville, AR, with and without irrigation 180 4-54 Scatter plot of seasonal plant N removal for bermudagrass grown over three years at Fayetteville, AR, with and without irrigation 181 4-55 Residual plot of seasonal dry matter yield for bermudagrass grown over three years at Fayetteville, AR, with and without irrigation 182 4-56 Residual plot of seasonal plant N removal for bermudagrass grown over three years at Fayetteville, AR, with and without irrigation 183 4-57 Response of seasonal dry matter yield, plant N removal and plant N concentration for bermudagrass averaged over three years at Fayetteville, AR, with and without irrigation 1 84 4-58 Seasonal dry matter yield and plant N concentration as a function of plant N removal for bermudagrass averaged over three years at Fayetteville, AR, with and without irrigation 185 4-59 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for bermudagrass averaged over three years at Fayetteville, AR, with and without irrigation 186 4-60 Scatter plot of seasonal dry matter yield for bermudagrass averaged over three years at Fayetteville, AR, with and without irrigation 187 4-6 1 Scatter plot of seasonal plant N removal for bermudagrass averaged over three years at Fayetteville, AR, with and without irrigation 188 4-62 Residual plot of seasonal dry matter yield for bermudagrass averaged over three years at Fayetteville, AR, with and without irrigation 189 xvi

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4-63 Residual plot of seasonal plant N removal for bermudagrass averaged over three years at Fayetteville, AR, with and without irrigation 190 4-64 Response of seasonal dry matter yield, plant N removal and plant N concentration for tall fescue grown over three years at Fayetteville, AR, with and without irrigation 191 4-65 Seasonal dry matter yield and plant N concentration as a function of plant N removal for tall fescue grown over three years at Fayetteville, AR, with and without irrigation 192 4-66 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for tall fescue grown over three years at Fayetteville, AR, with and without irrigation 193 4-67 Scatter plot of seasonal dry matter yield for tall fescue grown over three years at Fayetteville, AR, with and without irrigation 194 4-68 Scatter plot of seasonal plant N removal for tall fescue grown over three years at Fayetteville, AR, with and without irrigation 195 4-69 Residual plot of seasonal dry matter yield for tall fescue grown over three years at Fayetteville, AR, with and without irrigation 196 4-70 Residual plot of seasonal plant N removal for tall fescue grown over three years at Fayetteville, AR, with and without irrigation 197 4-71 Response of seasonal dry matter yield, plant N removal and plant N concentration for tall fescue averaged over three years at Fayetteville, AR, with and without irrigation 198 4-72 Seasonal dry matter yield and plant N concentration as a function of plant N removal for tall fescue averaged over three years at Fayetteville, AR, with and without irrigation 199 4-73 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for tall fescue averaged over three years at Fayetteville, AR, with and without irrigation 200 4-74 Scatter plot of seasonal dry matter yield for tall fescue averaged over three years at Fayetteville, AR, with and without irrigation 201 4-75 Scatter plot of seasonal plant N removal for tall fescue averaged over three years at Fayetteville, AR, with and without irrigation 202 xvii

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4-76 Residual plot of seasonal dry matter yield for tall fescue averaged over three years at Fayetteville, AR, with and without irrigation 203 4-77 Residual plot of seasonal plant N removal for tall fescue averaged over three years at Fayetteville, AR, with and without irrigation 204 4-78 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for bermudagrass grown over two years at Eagle Lake, TX 205 4-79 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for bahiagrass grown over two years at Eagle Lake, TX 206 4-80 Seasonal dry matter yield and plant N concentration as a function of plant N removal for bermudagrass grown over two years at Eagle Lake, TX 207 4-81 Seasonal dry matter yield and plant N concentration as a function of plant N removal for bahiagrass grown over two years at Eagle Lake, TX 208 4-82 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for bahiagrass and bermudagrass grown over two years at Eagle Lake, TX 209 4-83 Scatter plot of seasonal dry matter yield for bahiagrass and bermudagrass grown over two years at Eagle Lake, TX 210 4-84 Scatter plot of seasonal plant N removal for bahiagrass and bermudagrass grown over two years at Eagle Lake, TX 211 4-85 Residual plot of seasonal dry matter yield for bahiagrass and bermudagrass grown over two years at Eagle Lake, TX 212 4-86 Residual plot of seasonal plant N removal for bahiagrass and bermudagrass grown over two years at Eagle Lake, TX 213 4-87 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for grain and total plant of corn grown at Clayton and Kinston, NC on Dothan and Goldsboro soils, respectively 214 4-88 Seasonal dry matter yield and plant N concentration as a function of plant N removal for grain and total plant of corn grown at Clayton and Kinston, NC on Dothan and Goldsboro soils, respectively 215 xviii

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4-89 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for grain and total plant of corn grown at Clayton and Kinston, NC on Dothan and Goldsboro soils, respectively 216 4-90 Scatter plot of seasonal dry matter yield for grain and total plant of corn grown at Clayton and Kinston, NC on Dothan and Goldsboro soils, respectively 217 4-9 1 Scatter plot of seasonal plant N removal for grain and total plant of corn grown at Clayton and Kinston, NC on Dothan and Goldsboro soils, respectively 218 4-92 Residual plot of seasonal dry matter yield for grain and total plant of corn grown at Clayton and Kinston, NC on Dothan and Goldsboro soils, respectively 219 4-93 Residual plot of seasonal plant N removal for grain and total plant of corn grown at Clayton and Kinston, NC on Dothan and Goldsboro soils, respectively 220 4-94 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for grain and total plant of corn grown at Plymouth, NC 221 4-95 Seasonal dry matter yield and plant N concentration as a function of plant N removal for grain and total plant of corn grown at Plymouth, NC 222 4-96 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for grain and total plant of corn grown at Plymouth, NC 223 4-97 Scatter plot of seasonal dry matter yield for grain and total plant of corn grown at Plymouth, NC 224 4-98 Scatter plot of seasonal plant N removal for grain and total plant of corn grown at Plymouth, NC 225 4-99 Residual plot of seasonal dry matter yield for grain and total plant of corn grown at Plymouth, NC 226 41 00 Residual plot of seasonal plant N removal for grain and total plant of corn grown at Plymouth, NC 227 4-101 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for bahiagrass grown on two soils in Florida 228 xix

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4-102 Seasonal dry matter yield and plant N concentration as a function of plant N removal for bahiagrass grown on two soils in Florida 229 41 03 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for bahiagrass grown on two soils in Florida 230 41 04 Scatter plot of seasonal dry matter yield for bahiagrass grown on two soils in Florida 23 1 4-105 Scatter plot of seasonal plant N removal for bahiagrass grown on two soils in Florida 232 4-106 Residual plot of seasonal dry matter yield for bahiagrass grown on two soils in Florida 233 4-107 Residual plot of seasonal plant N removal for bahiagrass grown on two soils in Florida 234 41 08 Plot of the mean and ±2 standard errors of A/ A and Ab for twenty sites in England 235 41 09 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for twenty sites in England 236 4-110 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for twenty sites in England 237 4-111 Response of seasonal dry matter, plant N removal, and plant N concentration to applied N for rye grown at Tifton, GA and fixed application rates of 40 and 74 kg/ha of P and K, respectively 238 4-112 Response of seasonal dry matter, plant P removal, and plant P concentration to applied P for rye grown at Tifton, GA and fixed application rates of 135 and 74 kg/ha of N and K, respectively 239 4-113 Response of seasonal dry matter, plant K removal, and plant K concentration to applied K for rye grown at Tifton, GA and fixed application rates of 135 and 40 kg/ha of N and P, respectively 240 4-114 Seasonal dry matter yield and plant N concentration as a function of plant N removal for rye grown at Tifton, GA 241 xx

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4-115 Seasonal dry matter yield and plant P concentration as a function of plant P removal for rye grown at Tifton, GA 242 4-116 Seasonal dry matter yield and plant K concentration as a function of plant K removal for rye grown at Tifton, GA 243 4-117 Dimensionless plot of seasonal dry matter yield and plant nutrient concentration as a function of plant nutrient removal for rye grown at Tifton, GA 244 4-118 Scatter plot of dry matter yield for rye grown at Tifton, GA 245 4-119 Scatter plot of plant N removal for rye grown at Tifton, GA 246 41 20 Scatter plot of plant P removal for rye grown at Tifton, GA 247 4-121 Scatter plot of plant K removal for rye grown at Tifton, GA 248 41 22 Residual plot of dry matter yield for rye grown at Tifton, GA 249 41 23 Residual plot of plant N removal for rye grown at Tifton, GA 250 4-1 24 Residual plot of plant P removal for rye grown at Tifton, GA 25 1 41 25 Residual plot of plant K removal for rye grown at Tifton, GA 252 51 Sensitivity of logistic to the c parameter 264 xxi

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ESTIMATION OF DRY MATTER AND NITROGEN REMOVAL BY THE LOGISTIC EQUATION By Denise Marie Wilson December 1995 Chairman: Allen R. Overman, Ph.D. Major Department: Agricultural and Biological Engineering Department Environmental issues are shaping today's world. Land application of treated wastes and effluent is often done to manage the excess nutrients. Forage grasses are grown to accomplish two objectives: remove the nutrients from the waste or effluent and to produce food for livestock. Engineers, regulators, and managers use nutrient budgets in the design of systems to reduce the chance of pollution. This analysis was performed to establish a form of a model that would describe forage grass response to applied nutrients and provide reasonable estimates. Forms of the logistic equation were used to relate dry matter yield and plant N removal to applied nitrogen for seven different forage grasses. Many different factors, such as water availability, harvest interval, and plant partitioning, were investigated by examining thirteen studies in the literature. The logistic equation has shown a high correlation between grass yields and applied nitrogen. It is the purpose of this analysis to show that the form of the logistic xxii

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equation adequately describes dry matter yield and plant N removal response to applied nitrogen. The parameters of the models are determined using nonlinear regression. Analysis of variance is used to search for simplification in the form of common parameter values. The results of the analysis showed that the logistic model is well behaved and relatively simple to use. Harvest interval, water availability and plant partitioning can be accounted for in the linear parameter. Dimensionless plots are a valuable tool in evaluating the form of a model. The logistic equation exhibits symmetry suggesting conservation of something. For this use of the equation, the total capacity of the system is conserved. The assumptions of the extended model suggest a hyperbolic relationship between dry matter yield and plant N removal. This relationship was observed throughout the analysis. Although parameter values cannot be determined without prior experimentation, estimates for the parameters can be assumed. If parameter values are needed before any investigation, they should be underestimated in order to overdesign the system. Furthermore, for large error in the b, b', and c parameters, the seasonal estimate is not affected greatly (<15% error). xxiii

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CHAPTER 1 INTRODUCTION In today's society, there is an increasing focus on environmental issues. Concerns are being raised about pollution in soil, water and air. One of the major pollution concerns in Florida is nitrate and phosphate contamination in the aquifer and lakes, respectively. Engineers are using nutrient budgets to ensure that excess nutrients are not applied and to help control and eliminate contamination. Often nutrients are applied to forage grasses as treated wastes or effluent. Many municipalities are using water reuse systems as a way to remove high levels of nutrients from treated reclaimed water (Allhands el al., 1995). This process is beneficial to two main parties, those who wish to clean the water and return it to the aquifer and those who benefit from the addition of the nutrients to their system. Since a large amount of the required nutrients is applied in the water, less fertilizer is needed. The effluent is often applied to forage grasses, golf courses and lawns. A simple procedure is needed to assist engineers in estimating and predicting for various forage grasses the amount of nutrient removed and dry matter produced given a specific nutrient application rate. This research project primarily deals with various forage grasses and their nitrogen response curves. Consider the response data presented in Figure 1-1. The data were taken from a study by Prine and Burton (1956). Dry matter yield, plant N removal, and plant N concentration was recorded for bermudagrass [Cynodon dactylon] grown at Tifton, GA for two years at five harvest intervals. As expected, there is a relationship between yield, N removal, and N concentration to applied N, but what exactly is the relationship? Linear, hyperbolic, quadratic, sigmoid? If one form of an equation would 1

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2 adequately describe all the behavior, the task of identifying the specific relationship would be greatly simplified. In theory, the form of the model should represent the physical system beyond the range of data. The challenge is to identify patterns in the data (such as Figure 1-1) and identify relationships to describe such data. The objective of this project is to establish a model that provides reasonable estimates of dry matter yield and nutrient removal given a nutrient application rate. Studies from the literature will be used to document the fit of the model to numerous data sets with varying factors. These estimates could be used by engineers, managers, and regulators. The form of the equation should work regardless of forage or site. Water availability and harvest interval should also be quantified in the model. Land application of treated effluent and waste as irrigation for agricultural crops is becoming a prevalent method of wastewater reuse. In these systems, the nitrogen response of the crop is needed by engineers in the design process, since both the Florida Department of Environmental Protection (FDEP) and the Environmental Protection Agency (EPA) regulate wastewater application rates based upon nitrogen concentrations and the uptake abilities of crops grown. Many sources of data for this analysis can be found for various forage grasses and locations around the world. This reservoir of information has different variables for the crops studied including the following: applied nutrients (N, P, and K), water availability (with/without irrigation and year to year variability), harvest interval, site specificity, and plant partitioning. These different variables produce varying response curves. Furthermore, the studies range from examining only dry matter yield response to applied N to including N removal response to investigating the effects of nitrogen, phosphorus, and potassium on dry matter yield and their respective removals. The logistic equation has provided high correlation coefficients for estimation of growth and nutrient removal for

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3 various grasses (Overman, 1990a; Reck, 1992; Overman et al, 1994a, 1994b). Because of this, the form of the logistic will be studied to determine its broader applicability. To better understand the logistic equation, consider the "rumor model". A room is filled with 100 people and the doors are closed. No one is allowed to leave or come into the room. The people in the room are milling around in random fashion. At the time the doors are closed, five people know a certain rumor. The speed at which the rumor spreads among the people present, or rate of transfer among the people, is dependent upon how many people have heard it and how many have not heard it. The probability that during an encounter with two people the rumor is spread is the probability that one knows the rumor and one does not. These two events are mutually exclusive and independent; that is, someone cannot know and not know the rumor at the same time. As a result, their joint probability is the product of their individual probabilities. The probability that the rumor is spread is the product of the probability a person knows and the probability a person does not know. Since the number of people in the room is fixed, the sum of those who know and do not know is constant as well. Furthermore, the probability density function for the spreading rumor has a bell shaped distribution; that is, the probability that the rumor is spread is the same when 5 people know and when 95 people know. The rate of transfer is equal but in opposite directions. At the beginning, there are more people who have not heard the rumor; so the rate of transfer is increasing until half of the people know. At that point the rate of transfer falls off since there are fewer people to tell. The logistic equation applied to grass growth operates similarly. As time was the independent variable for the rumor model, applied nitrogen is the independent variable for the logistic crop growth model. Let the number of people who know represent the amount of dry matter present at a particular moment in time. It follows that the number of people who do not know represents the amount of dry matter

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that still needs to be produced. As the number of people in the room was assumed constant with the rumor model, the potential dry matter production of the grass is constant as well: the sum of that already produced and that still to be produced. Also, the rate of growth is a product of what has been produced and what is still to be produced, as the rate of rumor transfer was a product of those who knew with those who did not know. A nonlinear regression analysis of the variables involved will be used to determine the values of the parameters for a logistic model of forage grass response to applied nitrogen assuming different relationships among the model parameters. The results of this analysis will be compared using analysis of variance for the different assumptions to determine which is correct and if a simplification can be made. The result will be a form of an equation for nutrient response to various forage grasses that will adequately describe the relationship of dry matter yield, plant nutrient removal, and plant nutrient concentration to applied nutrients. This model will assist engineers and planners in estimating plant nutrient removal for various application rates. The result of this analysis should be the production of a model for engineers and managers to use when determining nutrient budgets under varying conditions. The model will estimate seasonal totals and use a "black box approach". No attempt will be made to model the physical mechanisms of the plant. Once many different studies have been conducted, a survey of the parameters for various forage grasses might provide a useful insight to the internal mechanisms of the plant, as well as connect the parameters to the physical system, but this is not pursued in this study.

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5 1200 Applied Nitrogen, kg/ha Figure 1-1 Response of dry matter yield, N removal, and N concentration for bermudagrass as a function of applied N grown over two years at five clipping intervals at Tifton, GA. Data from Prine and Burton (1956). Symbols will be defined later in the text.

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CHAPTER 2 LITERATURE REVIEW Crop models have been used for many years and have many variations (Keen and Spain, 1992; France and Thornley, 1984). Models have been used to describe the relationship between various forage crop yields and other parameters including nitrogen application, water availability, harvest interval, etc. These models have ranged from simple polynomial models to varying degrees of nonlinear models. Jones et al. (1987) have defined a model as a mathematical representation of a system, and modeling as the process of developing that representation. Modeling is often confused with simulation. Simulation includes the processes necessary for operationalizing the model, or solving the model to mimic real system behavior (Jones et al., 1987). Before beginning a model or simulation project the objectives of the project should be stated clearly. A clear definition of the intended end-product and the intended users of the models that will be developed should be included in the objective statement. In their book on mathematical models in agriculture, France and Thornley (1984, p. 173) noted that, There are three fairly distinct communities of people to whom a crop model or simulator may be of value: first, farmers, including advisory and extension services, who are primarily concerned with current production; second, applied scientists, including agronomists and plant breeders, whose objective is to improve the efficiency of production techniques, for the most part using current knowledge; and third, research scientists, whose aim is to extend the bounds of present-day knowledge. There is no way in which a single model can satisfy the differing objectives of these three groups, which are, at least in part, incompatible. Various mathematical models have been used to describe crop growth. Polynomials are often used because of the simplicity in determining parameter values; however, they have serious weaknesses. These weaknesses have been addressed in detail 6

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7 (Freund and Littell, 1991). The exponential and Mitscherlich models have also been used in describing crop growth (France and Thornley, 1984); however these models have limitations as well. The exponential model suggests that at high levels of input, the response approaches infinity, and at low levels approaches zero response. The Mitscherlich model suggests that at low levels of input the response approaches negative infinity, and at high levels approaches a maximum response. These two models describe the response well at different extremes. A combination of the two could potentially explain the response well over all ranges of input. The logistic equation, first proposed by Verhulst and later popularized by Pearl (Kingsland 1985), was first used as a population model. It has been used with high correlation coefficients (R > 0.99) for nitrogen removal (uptake) and dry matter production of various forage grasses (Reck, 1992; Overman el a/., 1990a, 1990b, 1994a, 1994b; Overman and Blue, 1991). Overman (1995a) has developed the relationship of the exponential and Mitscherlich models to the logistic, and a short summary of that discussion will be included here. In differential form, the exponential model assumes that the response of dry matter to applied N is proportional to the amount of dry matter present. Furthermore, the exponential is suitable at low N and asymptotically approaches zero along the negative N (reduced soil N) axis. In mathematical form, lower N: dy/dN ay [2.1] In differential form, the Mitscherlich model assumes that the response of dry matter to applied N is proportional to the unfilled capacity, y m y, of the system, where y m is the maximum yield. As mentioned before, the Mitscherlich is suitable at high N and asymptotically approaches a maximum yield along the positive N axis. In mathematical form, higher N: dy/dN a y m y [2.2] This leads us to assume a composite function

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8 allN: dy/dN = k y (y m y) [2.3] This is the form of the logistic model where k is the N response coefficient. It is a nonlinear first order differential equation. The logistic equation is well behaved; that is, the function is continuous, smooth, and asymptotic at both ends of the range. Recall the analogy to the rumor model. The product of the dry matter present (filled capacity) and that still to be produced (unfilled) is present in the differential form as well. Also, the total capacity of the system (y + y m y = y m ) is assumed constant. Furthermore, as Overman ( 1 995a) has pointed out, the logistic equation reduces to the exponential model at lower N and to the Mitscherlich model at higher N. If the logistic is normalized by defining new variables O = y/A and £ = cN b and expanded by Taylor series, a quadratic term is not included, suggesting a parabola would not be appropriate. Furthermore, the logistic is approximated by a linear function extremely well in the middle of the range. The logistic model exhibits sigmoid behavior and has three parameters as shown below Parameters A and c are scaling coefficients for yield and applied N, respectively. Parameter b describes the reference state at N = 0. In the context of this model, parameters A and c are constrained to be positive, but parameter b can be either positive, zero, or negative. Parameter b equal to zero suggests that there is enough background level of nitrogen in the soil to reach half of the maximum yield. Parameter b less than zero suggests that there is enough nitrogen in the soil before fertilization to reach more than halfway up the response curve. The model can also be normalized as shown y__ A \ + e' 1 b-cN [2.5] and linearized as shown

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9 \n(--\) = b-cN [2.6] y To determine if the data follow a logistic curve, A/y 1 vs. N can be plotted on semilog paper to check for linearity. If the plot is linear, then the data can be described well by the logistic. The parameters can be determined by one of two methods: regression on linearized data or nonlinear regression on original data (Downey and Overman, 1988). There are advantages to using both methods. The linearization method provides an easy procedure to estimate the parameters with a hand calculator. By this method, an estimate of A can be obtained by examining a plot of the y versus N on linear paper. The curve will appear to approach a maximum. This maximum is the estimate of A. Several attempts may be required to optimize A. Estimates of b and c then follow from linear regression of ln(A/y 1) vs. N (Draper and Smith, 1981). An example of this is shown in Figure 2-1. The data for this analysis are taken from a study of dallisgrass [Paspalum dilatatum Poir.] grown at Baton Rouge, Louisiana (Robinson et al., 1988). Linearized dry matter yield and plant nitrogen removal are plotted. The linear trend suggests that these two responses can be described well by the logistic equation. The fact that the lines are parallel suggests that the c parameter for both dry matter yield and plant nitrogen removal are the same. The figure also suggests that the b parameter value for plant nitrogen yield is larger (more positive) than the b value for dry matter yield. The nonlinear regression method requires a computer program written to perform the regression and statistical inference and diagnostic information. The regression can be conducted with SAS for the simple case. As more complex cases are needed, SAS is not easily programmed to perform the statistical analysis. In the nonlinear regression, parameters are estimated using least squares (Bates and Watts, 1988; Ratkowsky, 1983) . Unfortunately, only the A parameter can be explicitly solved. The b and c parameters need to be solved implicitly. Second order Newton-Raphson iteration is used (Adby and Dempster, 1974). For additional

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10 comparison among the two methods, the reader is directed to Overman et al. (1990a) for further details. The logistic equation exhibits symmetry suggesting conservation of something (energy, momentum, charge, spin, etc.) (Mehra, 1994 p. 132). If this is true, what could be conserved in the system defined? As shown earlier, it is the total capacity of the system, y m . In Pearl's work with the logistic equation he later expressed "unutilized potentialities" as the "amount still unused or unexpended in the given universe (or area) of actual or potential resources for the support of growth" (Kingsland, 1985, p. 67). In relation to this work, the total yield capacity consisting of the filled, y, and unfilled capacity, y m y, is what is conserved. But what is the total yield capacity? It is y + y,„ y = y m = A, where A is assumed constant. Pearl further noted, "The rate of growth, therefore, was proportional to two quantities: the existing population and the difference between existing and limiting populations" (Kingsland, 1985, p. 68). This is also demonstrated in the differential form, Equation [2.3]. The logistic equation can also be written as A y ~ i +e t N »-*V N ' [2 7] where N1/2 = b/c = nitrogen value for half maximum yield, and N' = 1/c = characteristic nitrogen. Recall Figure 2-1. The intercept on the vertical axis is e b . On the horizontal axis (y/A = 0.5), N = N1/2. Pearl also noted, "Symmetry meant that the inflection point came at the halfway point of the curve, and that the saturation population was exactly twice the population at the point of inflection" (Kingsland, 1985, p. 69). This is demonstrated in the above equation. It has been pointed out by Hosmer and Lemeshow (1989, p. 38) that the most important coefficient in the logistic equation is the response coefficient c. In this analysis, three models will be used: simple logistic, extended logistic, and triple logistic (or NPK). The simple logistic model is given by

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11 \ + e where y = seasonal dry matter yield, Mg/ha; N = applied. N, kg/ha; A = maximum seasonal dry matter yield, Mg/ha; b = intercept parameter for yield; c = N response coefficient, ha/kg. The extended model relates dry matter and plant N responses to applied N and is based upon three postulates: 1 . Seasonal dry matter yield follows logistic response to applied N. 2. Seasonal plant N removal follows logistic response to applied N. 3 . The N response coefficients are the same for both. Postulate 1 follows from work done by Overman and Wilkinson (1992). Postulates 1 and 2 are possibly true independently of one another, while Postulate 3 implies quantitative coupling between dry matter and plant N accumulation. Three additional results derive as a consequence of these postulates: 1 . Plant N concentration response to applied N is described by a ratio of logistic functions. 2. Seasonal dry matter yield and plant N removal are related by a hyperbolic equation. 3. Plant N concentration and plant N removal are linearly related. Postulate 1 is the simple logistic model. Postulate 2 can be written as 1 + e where N„ = seasonal plant N removal, kg/ha; A' = maximum seasonal plant N removal, kg/ha;

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12 b' = intercept parameter (plant N removal); c' = N response coefficient (plant N removal), ha/kg. Now by using Postulate 3, the system is constrained by assuming c = c'. Figure 2-1 suggests this is a logical assumption. Plant N concentration is defined as the ratio of Equation [2.9] to Equation [2.8]. Note that the plant N "concentration" has been defined as the ratio of two extensive variables (y and N„), and therefore is not truly an intensive variable (like ionic concentration, temperature, pressure, etc.). By combining Equations [2.8] and [2.9] with c = c' and defining Nc = Nu/Y, the response to applied N is defined by Nc = Ncm \l + ^) [21 ° ] where N c = average plant N concentration, g/kg, and N cm = maximum plant N concentration, g/kg. This equation is well behaved for b' > b. It approaches a minimum at low values of applied N and approaches a maximum at high values of applied N. If b' < b, then Equation [2. 10] is no longer well behaved at low levels of applied N. Furthermore, Equations [2.8] and [2.9] can be rearranged to give the hyperbolic equation Y *N y= Jo+NT [211] where the parameters Y m and K are given by A A Y m ~ , b-b' ~ , -Ab [2 12] 1 e 1 e and K ' = 7^~i = ^~i [213]

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13 where Ab = b' b. In order for Y m and K' to be positive, b' must be greater than b. These relationships can be reduced to dimensionless form by dividing the dry matter yield by its estimated maximum and the plant N removal by its estimated maximum. After doing this, the parameters of the hyperbolic relationship are defined as Y„, 1 1 [2.14] e" " \ e' It should be noted that the parameters used to describe the dimensionless relationship of seasonal dry matter to plant N removal (two measurable quantities) are only dependent upon the Ab. Dimensionless plots have been useful tools for engineers in many fields. Dimensionless plots were used to develop and determine dimensionless numbers, such as the Reynolds number in hydraulic flow. For example, James Clerk Maxwell used a dimensionless plot in the 1 860s to describe the distribution of molecular velocities in a gas (Segre, 1984). The greatest value of a dimensionless plot is the ability to collapse data sets with different ranges onto the same scale for comparison. This also aids in the search for possible simplification. Equation [2.11] can be rearranged to the form N c = (K7Y m ) + (l/Y m )N u [2.16] This equation predicts a linear relationship between plant N concentration and plant N removal. The extended triple logistic (NPK) model is given by

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14 p L [2 191 "~(l + e^)(l + ^'-^)(l + e^ A ') v \2 201 " (i + ^-^)(i + ^-^)(i + ^'-^') 1 ' J where the subscripts n, p, and k refer to applied N, P, and K, respectively. This model assumes that the response to each nutrient is logistic and is treated as independently applied. In statistical terms, the independence assumption suggests the joint response is the product of the marginal responses (Mood et al., 1963). Several characteristics should be noted. With fixed levels of P and K, the NPK model reduces to the simple logistic model with a modified maximum parameter A. This model will allow for evaluation of the effect of combinations of N, P, and K on dry matter production. The amount of nutrient required to reach half of maximum yield is given by Ni /2 = b„/c„, Pt/2 = bp/c p , and Km = bk/c k . Negative values of b mean that before application the soil contained more than enough nutrient to achieve half of maximum yield. Throughout this analysis a model is viewed as a simplification of reality. Occam's razor will be applied to simplify to the essentials required for description. Box (1976, p. 792) suggested this approach in a paper by stating, Since all models are wrong the scientist cannot obtain a "correct" one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist so overelaboration and overparameterization is often the mark of mediocrity. Evaluation and validation of the models will be conducted by dimensionless, scatter and residual plots. In the literature, validation is defined as the process by which a simulation

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15 model is compared to field data not used previously in the development or calibration process (Jones et ai, 1987). It should be noted that this study is not simulation, but true modeling: mathematical representation of a system. Furthermore, the purpose of validation is to determine if the model is sufficiently accurate for its application as defined by the objectives (Jones et ai, 1987). This leads back to a point mentioned earlier. It is essential that the objectives be stated clearly from the beginning. Jones et ai (1987, p. 16) also noted that common sense should prevail in validation of models, and that "a model cannot be validated, it can only be invalidated," a point frequently emphasized in statistics. Dimensionless plots will be used to determine if the form of the model is adequate to validate the model. It is more important to determine if the model estimates adequately: Does the model capture the. essence of what is trying to be accomplished. The scatter and residual plots will be used to evaluate how well the model estimated the data, by looking for biases or trends in the residuals. Box (1979, p. 2) noted the difference between estimating and validation in a paper by stating . . . two different kinds of inferential process .... The first, used in estimating parameters from data conditional on the truth of some tentative model, is appropriately called Estimation. The second, used in checking whether, in the light of the data, any model of the kind proposed is plausible, has been aptly named by Cuthbert Daniel Criticism. While estimation should . . . employ . . . likelihood, criticism needs a different approach. In practice, it is often best done in a rather informal way by examination of residuals. . . . Box, Hunter, and Hunter (1978, p. 552) also discussed the importance of residuals and other visual displays in evaluating data, especially when the work has been done with a computer by noting. Without computers most of the work done on nonlinear models would not be feasible. However, the more sophisticated the model and the more elaborate the techniques employed, the more important it is to submit complicated analyses to surveillance by data plots, residual plots, and other visual displays. The modern computer can make the plots itself, but graphs need not only be made but also to be carefully examined and thought about. The data analyst must "fondle" the data.

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16 Hand plotting used to be one to the ways this came about. The original data, as well as the various plots, whether made by hand or by the computer, should be mulled over. The experimenter's imagination, intuition, subject-matter knowledge, and experience must interact. This interaction will often lead to new ideas that may, in turn, lead to further analysis or experimentation. While the objective of some model projects is to describe the system in a fundamental way, this project has used Box's approach. The decision to use the logistic equation was not based upon some insight into the plant process, but rather as an equation that would describe the response well. The more prevalent approach in the literature is to use compartmental models such as EPIC, CREAMS, GLEAMS, DRAINMOD, etc. These models attempt to simulate the growth of a plant (or field) from planting to harvest by breaking the soil and plant system into small compartments. The compartments keep track of important state variables as water and other nutrients move through the system. Common time steps include minute, hour, days and weeks. A model is written for each plant process, such as respiration, transpiration, reproduction, etc. Common inputs include weather data, (solar radiation, rainfall, degree-days), and soil characteristics. Plant specific parameters are also needed. Although a thorough attempt has been made to explain the growth behavior, the seasonal yields are not estimated well. An example of this is demonstrated in a paper by Williams et al. (1989) where the highest correlation coefficient in the scatter plots was 0.89 and the average correlation coefficient was 0.67. It is the opinion of the author that the geometry and processes involved for a single plant is much too complicated to be solved mechanisticly, much less to try to extrapolate to the larger system. We are faced with the same problem as with the Navier-Stokes equation in porous media— there is no hope of describing the geometric flow paths in the soil and plant. This project is an attempt to estimate the seasonal totals well and provide a low input and simple approach to crop modeling.

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17 O Dry Matter Yield 0.0001 1 — — 1 ' — 1 — — ' 1 0 200 400 600 800 1000 Applied Nitrogen, kg/lia Figure 2-1 Response of linearized dry matter yield and plant N removal to applied N for dallisgrass [Paspalum dilatatum Poir ] grown at Baton Rouge, LA. Original data from Robinson et al. (1988). Data are linearized using Eq. [2.6] and A values of 15.60 and 43 1, respectively.

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CHAPTER 3 METHODS AND MATERIALS Analysis of Data Depending upon the nature of the data, one of three different models will be used. It should be noted that the three models are directly related to one another because they are based upon the same mathematical form. For data sets where only plant dry matter yield is recorded, the simple logistic model, Equation [2.8], will be used to describe plant yield response to applied nitrogen. For data sets where both dry matter yield and plant N removal are recorded, the extended logistic model, Equations [2.8] and [2.9], will be used to describe the response to applied nitrogen. Finally, when varying amounts of nitrogen, phosphorus, and potassium are applied and dry matter yield and nutrient removals recorded, the extended triple logistic (or NPK) model, Equations [2.17] through [2.20], are used to describe the response. Water availability and harvest interval (or cutting frequency) will be related to the linear model parameter. For all three models, the parameters, A, b, and c, are estimated using nonlinear regression (including second-order Newton-Raphson method) on the data to minimize the error, E, given by [3.1] where E error sum of squares, measured yield or N removal, estimated yield or N removal from model, observation number. 18

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19 Correlation coefficients will be used to measure the fit of the data to the model. The correlation coefficient is given by [3.2] An adjusted correlation coefficient is given by X(y-y) 2 /(»-p) Z(y-y) 2 /(»-V [3.3] where n = total number of data points used, and p = number of parameters estimated. This quantity adjusts the correlation coefficient by the number of parameters in the model. In the case of simple regression, R adj = R. Because the adjusted correlation coefficient accounts for the number of parameters in the model, it essentially deflates the R value hence providing for a better evaluation of the fit. A program was written in Pascal to estimate the parameters given the data and first estimates of the b and c parameters using nonlinear regression and Newton-Raphson iteration. The b and c parameters can be better estimated by the linearization method as discussed in the previous chapter; however, the author has chosen to use nonlinear regression because inference upon the parameters is more straightforward. If the linearization technique had been employed, then the inference and summary statistics would have been based upon the linearized data and not the actual data. This analysis will include evaluation and validation of the form of the logistic model, to determine if the model adequately describes the behavior of the data regardless of crop or site. To answer this question, various grasses were studied. A list of these grasses and their common and scientific names is given in Table 3-1.

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Models to be Investigated 20 Simple Logistic Model for Dry Matter Yield Analysis of variance, scatter plots and residual plots were used to evaluate this model. The analysis of variance tests if the model can be simplified due to common parameters under different conditions (that is, year, harvest interval, water availability, etc.). There are three basic modes. In mode 1, all the data were analyzed together over the various management factors (yield, N removal, years, harvest interval, irrigation, etc.) and a common A, b, and c are found for the combined data. In mode 2, the data were fit separately for each specific situation (yield, N removal, year, harvest interval, irrigated, etc.), requiring a different A, b, and c parameter for each situation. In mode 3, the data were analyzed together again and a different A is fit for each situation, but a common b and c are found for the entire set of data. These different modes were then compared by analysis of variance. They are compared by using an F test. The hypothesis that is being tested is that one of the modes (or models) describes the data better. It is basically examining for possible simplification. Extended Logistic Model for Dry Matter Yield and Nitrogen Removal These data sets are analyzed in a similar manner to the simple logistic with two additional modes. For mode 4, the data are analyzed together and an individual A and b are fitted to each situation with a common c for all. Mode 5 involves fitting an individual A for each situation, a separate b for dry matter yield and plant N removal, and a common c for all. These two additional modes are compared with the previous three modes by analysis of variance to determine which scenario describes the data best. In some cases the statistics suggest that the c parameter is not common. Instead of blindly using the

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21 statistics, we further compare the mean sum of squares (MSS) for the two modes in question. If there is not a large increase in MSS by simplifying the model (requiring fewer parameters), we will accept that there is a common c. By examination of the plotted data, it appears this supposition is valid. As stated previously, the extended model is based upon three postulates. One of the consequences of the postulates was that dry matter yield and plant N removal are hyperbolically related and plant N concentration and plant N removal are linearly related. To test these results, dry matter yield and N concentration are plotted against plant N removal. The lines describing these relationships are dependent only upon the Ab = (b' b). Furthermore, the data can be normalized by dividing by the appropriate estimated maximums. If the assumption of common Ab is true, all the normalized data will fall on one line in a dimensionless plot. This is one way of testing the adequacy of the model to fit the data and testing the postulates. Extended Triple Logistic (NPK) Model for Dry Matter Yield and Nitrogen Removal This model is developed on the assumption that response of dry matter yield and plant nutrient removal to applied N, P, and K individually follow the extended logistic model. Overman and Wilkinson (1995) first discussed this model and applied it to a complete factorial. Dimensionless plots, scatter plots and residual plots are used to evaluate the model.

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Data Sets to be Investigated 22 Thorsby. Alabama: Bahiagrass and Bermudagrass This study is from the Auburn Agricultural Experiment Station. The results have been previously reported in a station bulletin (Evans et al., 1961). The results of field tests and grazing trials conducted in Alabama to show the response of bermudagrass and bahiagrass to nitrogen and irrigation on specific soil types were reported. The yields of the grasses grown on Greenville fine sandy loam, Thorsby, AL, over three years (19571959) were used in this analysis. Four nitrogen levels were included: 0, 168, 336, and 672 kg/ha. The data are listed in Table 3-2. Baton Rouge. Louisiana: Dallisgrass The data for this analysis were acquired from a study by Robinson et al. (1988). Six nitrogen levels were included: 0, 56, 112, 224, 448, and 896 kg/ha. The data are listed in Table 3-3. Thorsby, Alabama: Bermudagrass The data for this analysis were taken from a study by Doss et al. (1966). Two harvest intervals were studied: 3.0 and 4.5 weeks. Six nitrogen levels were included: 0, 224, 448, 672, 1344, and 2016 kg/ha. The data are listed in Table 3-4. Maryland: Bermudagrass The data for this analysis were acquired from a study by Decker et al. (1971). Five harvest intervals were studied: 3.2, 3.6, 4.3, 5.5, and 7.7 weeks. Six nitrogen levels were included: 0, 1 12, 224, 448, 672, and 896 kg/ha. The data are listed in Table 3-5.

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23 Tifton, Georgia: Bermudagrass This data set was taken from a study by Prine and Burton (1956). Data from two years were utilized: 1953, a wet year, and 1954, a dry year. Five harvest intervals were studied: 2, 3, 4, 6, and 8 weeks. Five nitrogen levels were included: 0, 112, 336, 672, and 1010 kg/ha. The data are listed in Table 3-6. England: Ryegrass This data set was taken from a study by Reid (1978). Twenty-one nitrogen levels were included: 0, 28, 56, 84, 112, 140, 168, 196, 224, 252, 280, 308, 336, 392, 448, 504, 560, 616, 672, 784, and 896 kg/ha. The ryegrass was harvested at different growth stages, resulting in three different number of cuttings per season: 10, 5, 3. The length of the season was 26 weeks. It should be noted that the different number of cuttings represent variable harvest intervals, namely 2.6, 5.2, and 8.67 weeks. As a result, we expect this variable effect to appear in the results. The data are listed in Table 3-7. Fayetteville, Arkansas: Bermudagrass This data set was acquired from a study by Huneycutt et al. (1988). The bermudagrass was grown over three years, with and without irrigation. Six nitrogen levels were included: 0, 1 12, 224, 336, 448, 560, and 672 kg/ha. The data are listed in Table 38. Favetteville. Arkansas: Tall Fescue This data set was taken from the same study as above (Huneycutt et al, 1988). The tall fescue was grown over three seasons, with and without irrigation. Six nitrogen

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24 levels were included: 0, 112, 224, 336, 448, 560, and 672 kg/ha. The data are listed in Table 3-9. Eaule Lake. Texas: Bahiagrass and Bermudagrass This data set was acquired from a study by Evers (1984). Both grasses were grown over two years (1979 and 1980), and five nitrogen levels were included: 0, 84, 168, 252, and 336 kg/ha. The data are listed in Table 3-10. North Carolina: Corn These field experiments were conducted with corn at three North Carolina Research Stations: Central Crops Research Station, Clayton, NC, on Dothan loamy sand soils (fine loamy, siliceous thermic, Plinthic Paleudults), Lower Coastal Plain Tobacco Research Station, Kinston, NC, on Goldsboro sandy loam soils (fine loamy, siliceous thermic, Aquic Paleudults), and Tidewater Research Station, Plymouth, NC, on Portsmouth very fine sandy loam soils (fine loamy, mixed thermic, Typic Umbraquults) (Kamprath, 1986). The experiments were conducted from 1981 through 1984. The corn grown at Clayton received irrigation. Each year the experiment was conducted in a different field and for this reason averages were used in this study. The data from the Dothan and Goldsboro soils are listed in Table 3-11. The data from the Portsmouth soil are listed in Table 3-12. For all three sites, five nitrogen levels were included: 0, 56, 1 12, 168, and 224 kg/ha. Florida: Bahiagrass The data from this analysis were drawn from Blue (1987). The bahiagrass was grown on two soils, an Entisol (Astatula sand) near Williston, Florida, and a Spodosol

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25 (Myakka fine sand) near Gainesville, Florida. Entisol is typically a dry soil and Spodosol is typically wet. Five nitrogen levels were included: 0, 100, 200, 300, and 400 kg/ha. The data are listed in Table 3-13. England: Ryegrass The data for this analysis were taken from a study by Morrison et al. (1980) and is listed in Table 3-14. Twenty different sites in England were used to grow the ryegrass. Six different nitrogen levels were included in the study at each site: 0, 150, 300, 450, 600, and 750 kg/ha. The reader is directed to the report for further information about the site characteristics and weather data.. Tifton. Georgia: Rye The data for this analysis were drawn from a study by Walker and Morey (1962). Six levels of nitrogen (0, 45, 90, 135, 180, and 225), phosphorous (0, 20, 40, 60, 80, and 100), and potassium (0, 37, 74, 1 1 1, 148, and 185) were investigated. The data are listed in Table 3-15.

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Table 31 . Common and Scientific Names of Grasses Studied. Common Name Scientific Name Bahiaerass Paspalum notation Fliigge Bermudagrass Cynodon dactylon (L.) Pers. Corn Zea mays L. Dallisgrass Paspalum dilatation Poir. Rye Secale cereale Ryegrass Lolum perenne L. Tall Fescue Festuca arundinacea

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27 Table 3-2. Seasonal Dry Matter Yield of Coastal Bermudagrass and Pensacola Bahiagrass Grown on Greenville Fine Sandy Loam at Thorsby, Alabama. Applied Nitrogen, kg/ha Species Irrigation Year 0 168 336 672 Mg/ha Bermuda No 1957 3.74 12.52 17.41 22.32 1958 4.70 11.38 17.44 21.56 1959 3.40 9.04 13.36 20.44 Yes 1957 1958 1959 3.49 4.33 4.76 11.70 10.75 11.94 18.32 17.89 19.42 21.08 22.61 24.43 Bahia No 1957 1958 1959 4.56 3.63 2.67 11.70 10.39 8.19 16.39 16.85 14.14 20.32 22.23 19.62 Yes 1957 4 31 10.35 16.81 23.26 1958 3.24 9.25 15.69 22.33 1959 3.82 9.52 16.33 22.43 Source: Data from Evans et al. (1961).

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28 Table 3-3. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Dallisgrass Grown at Baton Rouge, Louisiana. Applied Nitrogen Dry Matter Yield N Removal N Concentration kg/ha Mg/ha kg/ha 0 5.33 77 15.7 56 6.56 103 16.2 112 7.97 129 17.0 224 10.53 194 18.9 448 13.21 305 23.4 896 15.34 417 27.5 Source: Data from Robinson et al. (1988).

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29 Table 3-4. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Bermudagrass Grown at Thorsby, Alabama over 3.0 and 4.5 Week Harvest Intervals (1961). Applied Dry Matter Yield N Removal N Concentration Nitrogen Mg/ha kg/ha g/kg kg/ha 3.0 weeks 4.5 weeks 3.0 weeks 4.5 weeks 3.0 weeks 4.5 weeks 0 2.95 3.60 53 55 18.1 15.4 224 9.70 11.95 230 220 23.7 18.4 448 14.70 17.20 406 387 27.6 22.5 672 16.70 18.50 501 466 30.0 25.2 1344 17.50 19.95 560 549 32.0 27.5 2016 17.60 19.20 597 568 33.9 29.6 Source: Data from Doss et al. (1966).

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Table 3-5. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Dallisgrass Grown in Maryland. Harvest Interval Applied Nitrogen Dry Matter Yield N Removal N Concentration weeks kg/ha Mg/ha kg/ha g/kg 3.2 0 1.29 21.5 16.7 112 4.37 90.5 20.7 224 8.72 225 25.8 448 13.53 409 30.2 672 14.07 461 32.8 896 14.36 497 34.6 3.6 0 1.24 19.2 15.5 112 4.84 86.6 17 9 224 9.04 198 21.9 448 12.94 371 28.7 672 13.94 450 32.3 896 13.78 480 34.8 4.3 0 1.09 15.7 14.4 112 5.19 88.7 17.1 224 9.81 197 20.1 448 14.36 393 27.4 672 15.33 445 29.0 896 14.93 451 30.2 5.5 0 1.55 20.6 13.3 112 7.75 122 15.8 224 12.96 238 18.4 448 16.71 369 22.1 672 17.07 418 24.5 896 17.07 485 28.4 7.7 0 1.65 17.7 10.7 112 9.37 125 13.3 224 14.55 215 14.8 448 18.23 354 19.4 672 18.89 399 21.1 896 18.37 421 22.9 Source: Data from Decker etal.( 1 97 1 ).

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Table 3-6. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Bermudagrass Grown at Tifton, Georgia. Year Harvest Interval Applied Nitrogen, kg/1 ia WCCM) u 112 336 672 1010 Dry Matter Yield, Mg/ha 1953 2 2.33 5.96 11.76 17.43 19.71 3 3.33 8.91 13.64 19.24 20.47 4 2.71 9.86 17.65 21 68 23.61 6 4.35 12.77 21.79 ?R 1 1 30.11 8 5.64 13.66 22.38 77 Q1 29.30 1954 2 0.76 2.69 6.83 7 R4 8.62 3 0.94 3.65 7.39 9.90 9.99 4 1.08 4.55 9.45 11.13 11.49 6 1.30 6.16 11.60 1 T S7 14.13 8 1.93 6.45 12.23 15.86 16.22 N Removal, kg/ha1953 2 37 130 327 582 721 3 51 184 363 579 682 4 40 176 431 590 739 6 53 158 392 621 738 8 62 184 372 545 624 1954 2 17 70 224 299 320 3 15 75 208 294 364 4 19 80 233 323 344 6 21 92 261 293 390 8 26 100 223 325 417 N Concentration, g/kg 1953 2 16.0 21.8 27.8 33.4 36.6 3 15.4 20.6 26.6 30.1 33.3 4 14.8 17.9 24.4 27.2 31.3 6 12.1 12.4 18.0 22.1 24.5 8 11.0 13.5 16.6 19.5 21.3 1954 2 22.6 26.0 32.8 38.2 37.1 3 16.0 20.5 28.1 29.7 36.4 4 17.5 17.5 24.7 29.0 29.9 6 16.4 14.9 22.5 21.6 27.6 8 13.5 15.5 18.2 20.5 25.7 Source: Data from Prine and Burton (1956).

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32 Table 3-7. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Perennial Ryegrass Grown in England with a Different Number of Harvests over the Season. Dry Matter Yield N Removal N Concentration Applied Cuttings Cuttings Cuttings Nitrogen 10 5 3 10 5 3 10 5 3 kg/ha Mg/ha kg/ha 0 0.75 1.81 2.65 18 34 38 24.3 18.2 14 6 28 1.12 2.30 4.08 27 48 64 24.0 18.2 15.4 56 1.58 3.28 4.88 40 59 67 25.0 18.2 13.8 84 2.27 4.00 5.17 59 75 77 26.1 18.9 15.0 112 2.85 5.58 5.97 77 104 83 26.9 18.6 13 9 140 3.12 5.65 6.90 85 102 96 27.4 18.2 13.9 168 3.75 7.13 8.06 106 136 120 28.2 19.2 15.0 196 4.50 7.61 8.58 131 149 136 29.1 19.5 15.8 224 4.73 8.49 9.59 142 174 155 30.1 20.5 16.2 252 5.30 9.18 9.36 162 192 168 30.2 20.8 17.9 280 6.38 9.30 10.83 200 195 197 31.4 21.0 18.2 308 6.72 10.32 10.90 222 235 213 33.0 22.7 19.5 336 6.67 11.41 11.92 221 261 237 33.0 22.9 19.8 392 7.95 11.36 11.90 283 290 274 35.7 25.6 22.9 448 8.31 11.84 12.01 301 310 280 36.3 26.2 23.4 504 8.75 12.24 12.46 320 350 302 38.4 28.6 24.3 560 8.82 12.45 12.38 349 362 325 39.7 29.0 26.2 616 8.91 12.24 12.90 354 373 347 39.7 30.4 26.9 672 9.17 12.01 11.65 381 386 330 41.4 32.0 28.3 784 9.18 11.95 11.97 379 397 349 41.3 33.0 29.1 896 8.86 11.86 11.96 376 413 370 42.4 34.9 31.0 Source: Data from Reid (1978).

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Table 3-8. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Bermudagrass Grown at Fayetteville, Arkansas over Three Years, Irrigated and Non-irrigated. Year 0 112 Applied Nitrogen, kg/ha 224 336 448 560 672 Non-Irrigated Dry Matter Yield 1983 1.95 6.88 11.52 13.88 16.39 16.70 17.71 Mg/ha 1984 2.35 8.43 11.59 12.71 16.50 16.48 16.01 1985 1.70 7.20 9.37 15.42 19.70 18.70 19.41 Irrigated 1983 4.53 9.12 14.50 20.80 21.90 23.63 23.87 1984 4.17 11.52 14.17 18.31 22.42 23.67 24.34 1985 1.75 8.32 12.55 18.56 20.11 21.36 23.18 Non-Irrigated N Removal 1983 27 109 210 262 354 358 419 kg/ha 1984 29 136 210 266 370 393 379 1985 21 105 178 294 416 407 484 Irrigated 1983 67 156 278 379 466 537 512 1984 61 186 252 366 402 451 561 1985 25 129 229 333 376 458 523 Non-Irrigated N Concentration 1983 13.8 15.8 18.2 18.9 21.6 21.4 23.7 g/kg 1984 12.2 16.2 18.0 21.0 22.4 23.8 23.7 1985 12.5 14.6 19.0 19.0 21.1 21.8 25.0 Irrigated 1983 14.7 17.1 19.2 18.2 21.3 22.7 21.4 1984 14.7 16.2 17.8 20.0 17.9 19.0 23.0 1985 14.1 15.5 18.2 17.9 18.7 21.4 22.6 Source: Data from Huneycutt et al. (1988).

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34 Table 3-9. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Tall Fescue Grown at Fayetteville, Arkansas over Three Seasons, Irrigated and Non-irrigated. Year 0 112 Applied Nitrogen, kg/ha 224 336 448 560 672 Non-Irrigated Dry Matter Yield 1981-2 2.51 5.47 8.97 10.31 12.33 11.21 11.86 Mg/ha 1982-3 1.86 5.00 4.64 7.11 7.67 7.40 8.32 1983-4 1.91 3.99 5.45 4.86 4.89 5.09 4.95 Irrigated 1981-2 3.92 7.85 10.90 13.36 14.84 15.33 15.53 1982-3 2.91 6.37 7.29 11.70 12.76 13.45 13.72 1983-4 3.77 7.91 10.49 14.59 15.92 17.15 17.15 Non-Irrigated N Removal 1981-2 55 118 207 259 337 334 368 kg/ha 1982-3 36 97 90 174 196 206 229 1983-4 37 94 147 133 140 150 147 Irrigated 1981-2 84 166 248 344 399 424 442 1982-3 62 120 139 275 331 362 371 1983-4 75 162 237 352 390 414 447 Non-Irrigated N Concentration 1981-2 21.9 21.6 23.1 25.1 27.4 29.8 31.0 g/kg 1982-3 19.2 19.4 19.4 24.5 25.6 27.8 27.5 1983-4 19.4 23.7 27.0 27.4 28.6 29.4 29.8 Irrigated 1981-2 21.4 21.1 22.7 25.8 26.9 27.7 28.5 1982-3 21.4 18.9 19.0 23.5 25.9 26.9 27.0 1983-4 19.8 20.5 22.6 24.2 24.5 24.2 26.1 Source: Data from Huneycutt et al. (1988).

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35 Table 3-10. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Bahiagrass and Bermudagrass Grown at Eagle Lake, Texas. Year 0 Applied Nitrogen, kg/ha 84 168 252 336 Bermudagrass Dry Matter 1979 4.47 6.98 10.09 12 91 1 7 1 14.23 Mg/na 1980 3.90 5.47 6.33 7.47 8.12 N Removal 1979 58 104 1 56 ??5 258 kg/ha 1980 62 102 116 155 178 N Concentration 1979 13.0 14.9 1 5 5 1 7 4 18.1 it g/kg 1980 16.0 18.7 18.4 20.7 21.9 Bahiagrass Dry Matter 1979 4.38 6.02 7.71 10.10 10.40 Mg/ha 1980 2.95 3.99 5.17 5.30 6.32 N Removal 1979 55 88 123 173 186 kg/ha 1980 65 SI 108 120 150 N Concentration 1979 12.6 14.6 16.0 17.1 17.9 g/kg 1980 22.0 20.2 20.9 22.7 23.7 Source: Data from Evers (1984).

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36 Table 3-11. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Grain and Total Plant of Corn Grown on Dothan and Goldsboro Soils at Clayton and Kinston, North Carolina, Respectively. Applied Dry Matter Nitrogen Yield N Removal N Concentration Site Part kg/ha Mg/ha kg/ha Dothan Grain 0 4.37 47 10.8 56 7. 12 74 10.4 112 9.77 113 11.6 168 10.81 135 12.5 224 11.04 150 13.6 Total 0 9.09 61 6.7 c c 56 14.67 92 6.3 112 18.35 135 7.4 168 19.11 166 8.7 224 19.90 188 9.4 Goldsboro Grain 0 3.00 32 10.7 56 5.47 62 11.3 112 6.93 87 12.6 168 7.46 101 13.5 224 7.57 107 14.1 Total 0 6.63 36 5.4 56 10.60 71 6.7 112 13.10 104 7.9 168 13.55 120 8.9 224 13.92 134 9.6 Source: Data from Kamprath (1986).

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Table 3-12. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Grain and Total Plant of Corn Grown on Portsmouth Soil at Plymouth, North Carolina. Part Applied Nitrogen Dry Matter Yield N Removal N Concentration kg/ha Mg/ha kg/ha g/kg Grain 0 4.53 48 10.6 56 6.30 68 10.8 112 7.65 89 11.6 168 8.52 104 12.2 224 9.04 115 12.7 Total 0 8.93 60 6.7 56 11.63 83 7.1 112 13.39 101 7.5 168 14.68 121 8.2 224 15.10 133 8.8 Source: Data from Kamprath (1986).

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38 Table 3-13. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Bahiagrass Grown on Entisol and Spodosol Soils in Florida. Applied Nitrogen Dry Matter Yield N Removal N Concentration Soil kg/ha Mg/ha kg/ha e/ke sL S Entisol 0 1.59 18.7 1 1.8 100 5.02 68.7 13.7 200 7.97 121 15.2 300 9 92 166 1
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39 Table 3-14. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Ryegrass Grown in England. Applied Nitrogen Dry Matter Yield N Removal N Concentration Site kg/ha Mg/ha kg/ha g/kg 5 0 6.08 157 25.8 150 8.45 206 24.4 300 11.76 327 27.8 450 12.94 389 30.1 600 12.94 432 33.4 750 12.75 458 35.9 6 0 4.42 102 23.1 150 8.74 216 24.7 300 11.88 354 29.8 450 13.16 471 35.8 600 12.84 490 38.2 750 12.99 534 41.1 7 0 1.64 46 28.0 150 4.95 140 28.3 300 7.50 245 32.7 450 7.82 285 36.4 600 8.29 321 38.7 750 7.59 309 40.7 8 0 0.64 12 18.8 150 5.28 105 19.9 300 9.57 252 26.3 450 11.43 347 30.4 600 11.95 413 34.6 750 12.09 464 38.4 9 0 5.65 136 24.1 150 10.02 263 26.2 300 12.10 350 28.9 450 14.26 464 32.5 600 13.89 476 34.3 750 13.29 474 35.7

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Table 3-14-continued Applied Nitrogen Dry Matter Yield N Removal N Concentration kg/ha Mg/ha kg/ha g/ke 0 2.96 64 21.6 150 7.03 162 23.0 300 11.24 304 27.0 450 12.52 401 32.0 600 12.30 413 33.6 750 11.98 433 36.1 12 [3 0 150 300 450 600 750 0 150 300 450 600 750 1.24 4.84 9.71 13.58 14.75 15.06 4.40 8.38 10.93 12.23 12.19 11.44 28 103 237 384 438 518 102 221 341 426 458 452 22.6 21.3 24.4 28.3 29.7 34.4 23.2 26.4 31.2 34.8 37.6 39.5 14 15 0 150 300 450 600 750 0 150 300 450 600 750 2.90 8.24 10.41 12.42 12.81 12.09 1.53 5.06 8.14 9.95 10.81 11.17 63 212 282 393 445 469 34 118 220 310 376 426 21.7 25.7 27.1 31.6 34.7 38.8 22.2 23.3 27.0 31.2 34.8 38.1

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Table 3-14— continued Applied Nitrogen Dry Matter Yield N Removal N Concentration Site kg/ha Mg/ha kg/ha g/kg 16 0 4.10 88 21.5 150 8.69 206 23.7 300 11.92 336 28.2 450 12.70 420 33.1 600 12.35 447 36.2 750 11.92 448 37.6 17 0 150 300 450 600 750 19 0 150 300 450 600 750 0.69 4.83 9.05 10.74 11.07 10.91 1.35 3.41 5.64 6.09 6.15 5.95 1 1 96 230 315 367 384 18 80 153 190 215 215 15.9 19.9 25.4 29.3 33.2 35.2 13.3 23.5 27.1 31.2 35.0 36.1 20 0 150 300 450 600 750 22 0 150 300 450 600 750 2.85 7.09 11.28 12.39 15.06 14.10 1.61 5.30 8.67 9.98 10.09 10.06 58 146 275 337 473 468 33 123 241 323 348 370 20.4 20.6 24.4 27.2 31.4 33.2 20.5 23.2 27.8 32.4 34.5 36.8

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Table 3-14--continued. Applied Nitrogen Dry Matter Yield N Removal N Concentration Site kg/ha Mg/ha kg/ha g/kg 23 0 1.98 40 20.2 150 5.69 128 22.5 300 8.55 231 27.0 450 10.08 317 31.4 600 9.92 333 33.6 750 9.66 353 36.5 25 0 3.94 91 23.1 150 7.17 174 24.3 300 10.12 285 28.2 450 12.24 391 31.9 600 12.20 437 35.8 750 12.14 462 38.1 26 0 1.82 36 19.8 150 4.89 103 21.1 300 7.62 193 25.3 450 10.23 309 30.2 600 10.69 350 32.7 750 10.61 374 35.2 27 0 2.75 62 22.5 150 6.17 155 25.1 300 8.59 235 27.4 450 9.91 325 32.8 600 10.51 376 35.8 750 9.93 390 38.3 28 0 0.82 16 19.5 150 4.75 100 21.1 300 7.89 198 25.1 450 10.27 291 28.3 600 11.48 363 31.6 750 12.03 41J 34.2 Source: Data from Morrison et al. (1980).

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43 Table 3-15. Seasonal Dry Matter Yields and Nutrient Response to Applied Nutrients for Gator Rye at Tifton, Georgia. N P K Y Nu Pu Ku Nc Pc Kc kg/ha kg/ha kg/ha Mg/ha kg/ha IcoVha IV li/ 1 Id t(t/|ia 1 la g/Kg g/kg g/kg 0 40 74 0.55 45 40 74 1.81 58 12.1 66 32.3 6.69 36.4 90 40 74 3.01 112 19.6 108 37.3 6.51 36.0 135 40 74 3.75 160 24.4 136 42.6 6.51 36.4 180 40 74 4.01 175 26.8 146 43.7 6.69 36.4 225 40 74 4.55 216 28.6 148 47.5 6.29 32.5 135 0 74 2.09 92 13.0 75 44.1 6.21 35.8 135 20 74 3.08 134 17.0 118 43.5 5.51 38.3 135 40 74 3.36 133 21.1 119 39.5 6.29 35.4 135 60 74 3.74 146 24.3 141 39.1 6.51 37.8 135 80 74 4.00 148 21.7 134 37.0 5.42 33.4 135 100 74 3.97 135 40 0 2.87 112 27.6 62 39.0 9.61 21.7 135 40 37 3.18 116 15.3 90 36.6 4.81 28.3 135 40 74 3.74 140 24.3 133 37.3 6.51 35.5 135 40 111 3.56 134 18.8 130 37.5 5.29 36.4 135 40 148 3.96 160 24.2 150 40.4 6.12 37.8 135 40 185 4.11 162 31.2 159 39.4 7.60 38.8 Source: Data from Walker and Morey (1962).

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CHAPTER 4 RESULTS AND DISCUSSION Evaluation of the Simple Logistic Model Thorsby. Alabama: Bermudagrass and Bahiagrass Data from Evans et al. (1961) for bermudagrass and bahiagrass were used. First the grasses will be analyzed separately, then they will be pooled to see if any of the parameters are common for the two grasses. The analysis of variance for the Coastal bermudagrass and Pensacola bahiagrass over years 1957-1959 is given in Tables 4-1 and 4-2, respectively (3 yrs x 1 grass x 2 irrigation x 4 N). In mode 1, a common (A, b, c) is assumed to apply to Equation [2.8], while individual (A, b, c) for each equation is assumed in mode 2. For the Coastal bermudagrass, comparison among modes 1 and 2 has a variance ratio of F = 2.589/0.584 = 4.43 and is significant at the 5% confidence level. This comparison tests if mode 2 better describes the data. A significant variance ratio suggests that mode 2 is better. Comparison among modes 3 and 2 has a variance ratio of F = 0.875/0.584 = 1.50 and is not significant. This comparison also tests if mode 2 better describes the data. The non-significant variance ratio suggests that mode 3 describes the data as well or better than mode 2. Because mode 3 requires less parameters, we will select it at this point in the analysis following Occam's razor. The variance ratio is not significant for any of the comparisons among modes 2, 3, 4, or 5, suggesting that mode 3, an individual A for each combination of year and irrigation and a common b and c for all. For the Pensacola bahiagrass, comparison among modes 1 and 2 has a variance ratio of 44

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45 7.48 and is significant at the 2.5% confidence level. Comparison among modes 3 and 4 has a variance ratio of 9.49 and is significant at the 0.5% confidence level. Furthermore, comparison among modes 4 and 5 has a variance ratio of 8.78 and is significant at the 0.5% confidence level. These tests suggest when the grasses are studied separately, a single value can be used for c for all years and irrigation treatments to adequately describe the data. Next the data from the two grasses are combined and the analysis of variance data are presented in Table 4-3 (3 yrs x 2 grass x 2 irrigation x 4 N). A comparison of modes 1 and 2 results in a variance ratio of 5.61 that is significant at the 0.5% confidence level. Comparison of modes 3 and 4 results in a variance ratio of 3 .27 that is significant at the 1% confidence level. Comparison of modes 3 and 5 results in a variance ratio of 2.93 that is significant at the 5% confidence level. Comparison of modes 5 and 4 leads to a variance ratio of 2.87 that is significant at the 2.5% confidence level. Comparison of modes 4 and 6 leads to a variance ratio of 3.41 that is significant at the 1% confidence level. Comparison of modes 5 and 6 results in a variance ratio of 3.74 that is significant at the 5% confidence level. Comparisons among modes 3 and 7 and 4 and 7 result in variance ratios of 5.57 and 2.75, respectively. Both comparisons are significant at the 2.5% confidence level. Based upon these comparisons, we can conclude that mode 7, with individual A for each year, individual b for each grass, and common c describes the data best. The overall correlation coefficient of 0.9949 and adjusted correlation coefficient of 0.9927 are calculated for mode 7. The error analysis of the parameters are shown in Table 4-4. The largest relative error (|standard error/estimate|) was for the b parameter. This is due in part to the small numbers involved and that nonlinear regression on the logistic equation places more emphasis upon the maximum and less upon the intercept. Still the largest relative error for this set of parameters was under 4%. Since the range of A parameter values (largest smallest) between years and irrigation schemes is less than six, the data will be averaged over years. Overman et al. (1990a, 1990b) have

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46 shown that averaging over years is appropriate, since variations due to water availability and harvest interval appear in the linear parameter A. The averaged data are in Table 4-5 and the error analysis for the averaged data is in Table 4-6. The overall correlation coefficient of 0.9981 and adjusted correlation coefficient of 0.9969 are calculated. Note that the estimates of b and c are the same as for the unaveraged data, supporting what Overman et a/. (1990a, 1990b) have found previously. Results are shown in Figure 4-1, where curves for Coastal bermudagrass and Pensacola bahiagrass dry matter, irrigated and non-irrigated, are drawn from the following equations: Coastal bermudagrass, non-irrigated: Y =21.57/[1 + exp(1.39 0.0078N)] [4.1] Coastal bermudagrass, irrigated: Y = 23.44/[l + exp(1.39 0.0078N)] [4.2] Pensacola bahiagrass, non-irrigated: Y =21.49/[1 + exp(1.57 0.0078N)] [4.3] Pensacola bahiagrass, irrigated: Y =22.73/[l + exp(1.57 0.0078N)] [4.4] The scatter and residual plots of seasonal dry matter are given in Figures 4-2 and 4-3. The mean and the ±2 standard errors of the residuals are shown by the solid and dashed horizontal lines, respectively. As shown in Figure 4-3, all of the data fit between ±2 standard errors with no apparent trend. Evaluation of the Extended Logistic Model Baton Rouge. LA: Dallisgrass The data for this analysis are taken from Robinson el al. (1988). The analysis of variance is shown in Table 4-7 (dry matter and N removal x 6 N). Comparison among modes 1 and 2 results in a variance ratio of 3736 and is significant at the 0.1% level. Comparison among modes 3 and 2 leads to a variance ratio of 127.8 which is also significant at the 0.1% level. Mode 4, with individual A and b for dry matter and N removal, and common c describes the data best, since F(l,6,95) = 5.99, and F(l, 7,99.9) =

PAGE 70

47 29.25. This outcome supports Postulate 3. The overall correlation coefficient and adjusted correlation coefficient calculated from mode 4 are 0.9997 and 0.9995, respectively. The error analysis for the parameters is shown in Table 4-8. Results are shown in Figure 4-4, where curves for dry matter and plant N removal are drawn from Y = 15.60/[1 + exp(0.58 0.0055N)] [4.5] N u = 430.7/[l + exp(1.47 0.0055N)] [4.6] From these results, it follows that plant N concentration, shown in Figure 4-4, is estimated from N c 27.6 [1 + exp(1.47 0.0055N)]/[1 + exp(0.58 0.0055N)] [4.7] As shown, the data are described well by these equations. It should be noted that the plant N concentration data were not defined by regression techniques, but rather as a ratio of plant N removal and dry matter yield. From Postulate 3, the relationships between yield and plant N removal and between plant N concentration and plant N removal, shown in Figure 4-5, are described by Y = 26.5 N„/(300 + N„) [4.8] N c = 11.3 + 0.0378N„ [4.9] The intercept (11.3 g/kg) represents plant N concentration in a nitrogen deficient environment with low plant N removal. The form of this model can be validated by plotting the data in dimensionless form. The data plotted in Figure 4-5 were strictly measured data; that is, the independent variable applied N has been ignored. The results of Postulate 3 defined the curve drawn. By dividing dry matter yield, plant N concentration and plant N removal by their appropriate maximums, all of the data are collapsed onto the same dimensionless scale. Furthermore, the curves drawn are dependent upon the Ab, (b' b), term alone. This will be more significant as larger, more complex data sets are investigated. The form of the

PAGE 71

48 model can be validated in this manner. The dimensionless plot for dallisgrass is shown in Figure 4-6, and validates the form of the model. The curves were drawn from Y/A = l.70(N,/A , )/[0.70 + (N u /A*)] [4.10] N c /N cm = 0.41 +0.59 (KM') [4.1 1] Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 47 through 4-10. Thorsby. AL: Bermudagrass The data for this analysis are taken from Doss ei al. (1966). The analysis of variance is shown in Table 4-9 (dry matter and N removal x 2 At x 6 N). Comparison among modes 1 and 2 results in a variance ratio of 402 and is significant at the 0. 1% level. Comparison among modes 3 and 2 leads to a variance ratio of 4.63 that is significant at the 5% level. Comparison among modes 3 and 4 results in a variance ratio of 8.07 that is significant at the 0.5% level. Mode 5, with individual A for yield and plant N removal at both harvest intervals, b for dry matter and N removal, and common c describes the data best, since F(5,12,95) = 3.11, and F(l, 17,99.9) = 15.72. This outcome also supports Postulate 3. The overall correlation coefficient and adjusted correlation coefficient calculated from mode 5 are 0.9983 and 0.9977, respectively. The error analysis for the parameters is shown in Table 4-10. Results are shown in Figure 4-11, where curves for dry matter and plant N removal at both harvest intervals, are drawn from 3.0 weeks: Y= 17.42/[1 + exp(1.27 0.0067N)] [4.12] N u = 568.9/[l + exp(2.02 0.0067N)] [4. 13] 4.5 weeks: Y= 19.75/[1 + exp(1.27 0.0067N)] [4.14] N u = 543.5/[l + exp(2.02 0.0067N)] [4.15] From these results, it follows that plant N concentration, shown in Figure 4-11, is estimated from

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49 3.0 weeks: N c = 32.7 [1 + exp(2.02 0.0067N)]/[1 + exp(1.27 0.0067N)] [4.16] 4.5 weeks: N c = 27.5 [1 + exp(2.02 0.0067N)]/[1 + exp(1.27 0.0067N)] [4.17] As shown, the data are described well by these equations. From Postulate 3, the relationships between yield and plant N removal and between plant N concentration and plant N removal, shown in Figure 4-12, are described by 3.0weeks:Y= 33.0 N„/(509 + N u ) [4.18] N c = 15.4 + 0.0303N U [4.19] 4.5 weeks: Y = 37.4 N„/(487 + N u ) [4.20] N c = 13.0 + 0.0267N U [4.21] The dimensionless plot for bermudagrass is shown in Figure 4-13, and validates the form of the model. The curves were drawn from Y/A = l^OCK/A'ytO^O + CH/A')] [4.22] N c /N cm = 0.47 +0.53 (NJA') [4.23] Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 414 through 4-17. Maryland: Bermudagrass The data for this analysis are taken from Decker et al. (1971). The analysis of variance is shown in Table 4-1 1 (dry matter and N removal x 5 At x 6 N). Comparison among modes 1 and 2 results in a variance ratio of 190.9 and is significant at the 0.1% level. Comparison among modes 3 and 2 leads to a variance ratio of 4.06 that is significant at the 0.1% level. Comparison among modes 3 and 4 results in a variance ratio of 4.65 that is significant at the 0. 1% level. Mode 5, with individual A for yield and plant N removal at the five harvest intervals, b for dry matter and N removal, and common c describes the data best, since F(17,30,95) =1.98, F(l,47,99.9) = 12.32, and the residual

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50 sums of squares (RSS) for mode 5 is smaller than mode 4 while using less parameters. Postulate 3 is supported again. The overall correlation coefficient and adjusted correlation coefficient calculated from mode 5 are 0.9964 and 0.9955, respectively. The error analysis for the parameters is shown in Table 4-12. Results are shown in Figure 4-18, where curves for dry matter and plant N removal at all five harvest intervals, are drawn from 3 2 weeks' Y = 13 90/n + exod 78 0 01 12NVI \4 241 N u = 469.5/[l + exp(2.59 0.01 12N)] T4 7^1 3.6 weeks: Y = 13.65/[1 +exp(1.78-0.0112N)] N u = 444.7/[l + exp(2.59 0.01 12N)] \4 271 L ' J 4.3 weeks: Y = 14.93/[1 +exp(1.78-0.0112N)] \4 281 N u = 440.2/[l + exp(2.59 0.01 12N)] \4 291 L J 5.5 weeks: Y = 17.56/[1 +exp(1.78-0.0112N)] r A O AT [4.30] N u = 444.0/[l + exp(2.59 0.01 12N)] [4.31] 7.7 weeks: Y = 19.34/[1 +exp(1.78-0.0112N)] [4.32] N u = 409.7/[l + exp(2.59 0.01 12N)] [4.33] From these results, it follows that plant N concentration, shown in Figure 4-18, is estimated from 3.2 weeks: N c = 33.7 [1 +exp(2.590.01 12N)]/[1 +exp(l. 780.01 12N)] [4.34] 3.6 weeks: N c = 32.6 [1 + exp(2.59 0.01 12N)]/[1 + exp(1.78 0.01 12N)] [4.35] 4.3 weeks: N c = 29.5 [1 +exp(2.59-0.0112N)]/[l +exp(1.780.01 12N)] [4.36] 5.5 weeks: N c = 25.3 [1 +exp(2.59-0.0112N)]/[l + exp(1.780.01 12N)] [4.37] 7.7weeks: N c = 21.2 [1 + exp(2.59 0.01 12N)]/[1 + exp(l. 78 0.01 12N)] [4.38] As shown, the data are described well by these equations.

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51 From Postulate 3, the relationships between yield and plant N removal and between plant N concentration and plant N removal, shown in Figure 4-19, are described by 3.2 weeks: Y — r\ XT Kin ^ i xt \ Zj.V INu/^J /O + JN U J [4.39J M — JN C i r a _i_ a niOOM i j.u + u.UjyyjNu XA AIW 3.6 weeks: Y — 1A Ik M //"l^A 4XT \ TA A 1 1 [4.41J N c = 14.5 + 0.0407N U [4.42] 4.3 weeks: Y = 26.9 N„/(353 + N u ) [4.43] N c = 13.1 +0.0372N U [4.44] 5.5 weeks: Y = 31.6Nu/(356 + N u ) [4.45] N c = 11.2 + 0.03 16N U [4.46] 7.7 weeks: Y = 34.8 N„/(328 + N u ) [4.47] N c = 9.4 + 0.0287N U [4.48] The dimensionless plot for bermudagrass is shown in Figure 4-20, and validates the form of the model. The curves were drawn from Y/A = l.SOtNu/A'ytO.SO + CNu/A')] [4.49] N c /N cm = 0.44 +0.56 (K/A) [4.50] Notice that all of the data have collapsed onto one curve and line, respectively. This implies that the Ab is constant and the same for each harvest interval. Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-21 through 4-24. It appears from Table 4-12, that there is a linear relationship between A, estimated maximum dry matter yield, and harvest interval. The relationship does not appear to be as clear for A, estimated maximum plant N removal. Overman et a/. (1990b) found that both water availability and harvest interval could be linked linearly to A up to a six weeks interval, after which senescence sets in, and lower plant leaves die and/or drop off, reducing dry matter accumulation. The estimated maximums of dry matter yield and plant

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52 N removal are plotted against harvest interval in Figure 4-25. Linear regression was conducted on the estimated maximum dry matter yield, omitting the 7.7 week harvest interval, resulting in the following relationship A = 7.90+ 1.71At 7.90(1 + 0.22At) [4.51] with a correlation coefficient of 0.9676. Linear regression was also conducted on the estimated maximum plant N removal, omitting the 7.7 week harvest interval, resulting in the following relationship A'= 483.98.3At = 483.9 (1 0.01 7At) [4.52] with a correlation coefficient of 0.6206. The small correlation coefficient and rather flat response suggests that there is uncertainty in the nature of the relationship. Tifton. GA: Bermudagrass The data for this analysis are taken from Prine and Burton (1956). The analysis of variance is shown in Table 4-13 (dry matter and N removal x 2 yrs x 5 At x 5 N). Comparison among modes 1 and 2 results in a variance ratio of 81.2 that is significant at the 0. 1% level. Comparison among modes 3 and 2 leads to a variance ratio of 1 .89 that is significant at the 97.5 % level. Comparison among modes 3 and 4 results in a variance ratio of 2.45 that is significant at the 0.5% level. Comparison among modes 3 and 5, individual A for yield and plant N removal at the five harvest intervals and over both years, b for dry matter and N removal at both years, and common c, leads to a variance ratio of 15.9 that is significant at the 0.1% level. Mode 6, individual A for yield and plant N removal at the five harvest intervals and over both years, b for dry matter and N removal over both years, and common c, describes the data best, since F(37,40,95) = 1.71, F(l,77,99.9) = 11.71, F(18,59,95) = 1.78, and F(2,75,95) = 3.12. The difference due to years (water availability) and harvest interval is explained by the A parameter. Postulate 3 is supported again. The overall correlation coefficient and adjusted correlation coefficient

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53 calculated from mode 6 are 0.9941 and 0.9923, respectively. The error analysis for the parameters is shown in Table 4-14. Results are shown in Figures 4-26 through 4-30 for each clipping interval, where curves for dry matter and plant N removal at all five harvest intervals, are drawn from 2 weeks 1953' Y 17 95/ri + exnfl 47 0 0077N)1 [4 531 644. 7/[l + exDC2 1 5 0.0077N)] [4 541 L 'J 1954: Y 8.40/[l + exp(1.47 0.0077N)] T4.551 N„ 324. 5/[] + exo(2 1 5 0.0077N)] T4 561 3 weeks, 1953: Y 19.88/[1 + exp(1.47 0.0077N)] T4 571 N u 641 8/[l + exp(2.15 0.0077N)] \4 581 L J 1954: Y 9.95/[] + exp(1.47 0.0077N)] T4 591 N„ 337.5/[l + exp(2.15 0.0077N)] r4.601 L J 4 weeks, 1953: Y 23.15/[1 + exp(1.47 0.0077N)] r4.6ii N u 687.2/[l + exp(2.15 0.0077N)] [4.62] L J 1954: Y 11.67/[1 + exp(1.47 0.0077N)] T4 631 N u 348.0/[l + exp(2.15 0.0077N)] [4.641 L J 6 weeks, 1953: Y 29.57/[] + exp(1.47 0.0077N)] [4.65] Nu 688.2/[l + exp^z. i j 0.0077N)] [4.66] 1954: Y 14.37/[1 + exp(1.47 0.0077N)] [4.67] N u 363.8/[l + exp(2.15 0.0077N)] [4.68] 8 weeks, 1953: Y 29.58/[l + exp(1.47 0.0077N)] [4.69] N u 606.0/[l + exp(2.15 0.0077N)] [4.70] 1954: Y 16.24/[1 + exp(1.47 0.0077N)] [4.71] N u 379.5/[l + exp(2.15 0.0077N)] [4.72] From these results, it follows that plant N concentration, shown in Figures 4-26 through 4-30, is estimated from

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54 Z, WCCIVS, 17JJ. N = 35 9 "1 + pyW2 1 5 0 0077NWN + exoCl 47 \J . \J\J 1 1 1 l / 1/ 1 1 1 WAUI 1 . I / 0 0077NM \4 731 1954 N = 38 6 n + exDf2 15 0 0077NYI/H + exDH 47 0 0077N)1 [4 741 1 wppIcs 1953 N = 32 3 "1 + exni2 15 0 0077NWH + exDCl 47 \J . \J \J / / 1 1 / If 11 ' V/\L/l 1.1/ 0 0077N)1 T4 751 L ' J 1954N = 33.9 1 + exDf2 15 0 0077NYI/n + exDd 47 0.0077N)] \4 761 4 wppWq 1QSV N = 29 7 n + exnf2 1 5 0 0077NWn + exDH 47 \J .\J\J 1 / 1 ^ # 1/ 1 l 1 vAUl 1 . I / 0 00771^1 v/ . w # / 1 ™ / 1 T4 771 1954 N = 29 R n + pxnf2 1 5 0 0077NYI/N + exnH 47 \J .\J\J 1 / 1 l /J/ 1 1 1 1 . ~ / 0 0077IsM 14 781 [t. / OJ \J WTClVO, 1 7 J *j . N -23 3 n + exnC2 1 5 0 0077NYI/M + exnd 47 0 0077N)1 14 791 1954: N c = 25.3 ;i + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.80] 8 weeks, 1953: N c = 20.5 [1 +exp(2.15 0.0077N)]/[1 +exp(1.47 0.0077N)] [4.81] 1954: N c = 23.4 :1 + exp(2.15 -0.0077N)]/[1 +exp(1.47 0.0077N)] [4.82] As shown, the data are described well by these equations. From Postulate 3, the relationships between yield and plant N removal and between plant N concentration and plant N removal at each clipping interval over both years, shown in Figures 4-31 through 4-35, are described by 2 weeks, 1953: Y = 36.4 N„/(662 + N u ) [4.83] N c = 18.2 + 0.0275N U [4.84] 1954: Y = 17.0N„/(333 +N U ) [4.85] N c = 19.6 + 0.0587N U [4.86] 3 weeks, 1953: Y = 40.3 Nu/(659 + N u ) [4.87] N c = 16.4 + 0.0248N U [4.88] 1954: Y = 20.2Nu/(347 + N u ) [4.89] N c = 17.2 + 0.0496N U [4.90] 4 weeks, 1953: Y = 46.9 N„/(706 + N u ) [4.91] N c = 15.0 + 0.0213N U [4.92] 1954: Y = 23.7N„/(357 + N u ) [4.93] N c = 15.1 +0.0423N„ [4.94]

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55 6 wppIc^ 1 QSl ' Y = 59 9 N JH07 + K.) [4 951 N = 118 + 0 0167N T4 961 1954 Y = 29 1 N /G74 + N1 \4 971 N = 19 8 + 0 0343N 14 981 8 weeks 1953' Y = 60 0 N../(622 + N„1 14 991 N c = 10.4 + 0.0 167N U [4.100] 1954: Y = 32.9 N„/(390 + N u ) [4.101] N c = 11.8 + 0.0304N U [4.102] The dimensionless plot for bermudagrass is shown in Figure 4-36, and validates the form of the model. Note that all of the data have been collapsed onto one graph, and the data fall on one curve. The curves were drawn from Y/A = 2.03(N U /A)/[1.03 + (N U /A)] [4.103] N c /N cm = 0.5 1 +0.49 (KM') [4. 104] This result supports the hypothesis that the Ab is common for all harvest intervals and both years. As with the Maryland data set, it appears from Table 4-14, that there is a linear relationship between A, estimated maximum dry matter yield, and harvest interval. The estimated maximums of dry matter yield and plant N removal are plotted against harvest interval in Figure 4-37. Linear regression was conducted on the estimated maximum dry matter yield for both years, omitting the 8 week harvest interval, resulting in the following relationships 1953: A= 11.50 + 2.97At = 1 1.50 (1 + 0.26At) [4.105] 1954: A= 5.49+ 1.50At = 5.49 (1 + 0.27At) [4.106] with correlation coefficients of 0.9958 and 0.9986. It appears from these equations that after the intercept has been factored out, the coefficient of At may be the same for both

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56 years. Linear regression was also conducted on the estimated maximum plant N removal, omitting the 8 week harvest interval, resulting in the following relationships 1953: A'= 628.0+ 12.6At = 628.0 (l+0.020At) [4.107] 1954: A'= 307.2+ 9.7At = 307.2 (l+0.032At) [4.108] with correlation coefficients of 0.8407 and 0.9925. The response for the estimated plant N removal is flat as with the Maryland data. The line for 1954 might fit better due to the limited available water. Since it was a dry year, growth was limited and hence the plant was growing as much as possible with the limited water. As with the estimated maximum dry matter yield, it appears that the coefficient of At may be the same for both years. Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 438 through 4-41. England: Ryegrass The data for this analysis are taken from Reid (1978). It is important of remember throughout this specific analysis that the ryegrass was harvested at particular stages of growth. As a result, the harvest interval was variable. This result will present itself in the results. The analysis of variance is shown in Table 4-15 (dry matter and N removal x 3 variable At x 21 N). As shown in the table, all of the comparisons are highly significant (0. 1%). This is due in part to the large degrees of freedom. If comparison is made among the mean sums of squares (MSS), mode 6 has the smallest MSS with the exception of mode 2 (which fits each individually). Mode 6 is the preferred one since it requires less parameters to estimate (11 versus 18). This choice is in accordance with the approach common in physics: seeking the simplest model consistent with observation (Rothman, 1972), often cited as Occam's razor (Will, 1986). Mode 6 assumes an individual A for each number of clippings, an individual b for dry matter yield, common b for plant N removal, and common c for all. The overall correlation coefficient and adjusted

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57 correlation coefficient calculated from mode 6 are 0.9938 and 0.9931, respectively. The error analysis for the parameters is shown in Table 4-16. Results are shown in Figure 442, where curves for dry matter and plant N removal are drawn from 10 clippings: Y = 9.42/[l + exp(1.74 0.0080N)] [4.109] N u = 377.0/[l + exp(2.15 0.0080N)] [4.110] 5 clippings: Y = 12.72/[1 + exp(1.23 0.0080N)] [4.1 1 1] N u = 403.5/[l + exp(2.15 0.0080N)] [4.112] 3 clippings: Y = 12.75/[1 + exp(0.90 0.0080N)] [4.113] N u = 363.0/[l + exp(2.15 0.0080N)] [4.114] From these results, it follows that plant N concentration, shown in Figure 4-42, is estimated from 10 clippings: N c = 40.0 [1 + exp(2.15 0.0080N)]/[1 + exp(1.74 0.0080N)] [4.115] 5 clippings: N c = 31.7 [1 + exp(2.15 0.0080N)]/[1 + exp(1.23 0.0080N)] [4.116] 3 clippings: N c = 28.5 [1 + exp(2.15 0.0080N)]/[1 + exp(0.90 0.0080N)] [4.117] As shown, the data are not described very well by these equations; however, the problem most likely lies in the effect of variable harvest interval. From Postulate 3, the relationships between yield and plant N removal and between plant N concentration and plant N removal, shown in Figure 4-43, are described by 10 clippings: Y = 28.0 Nu/(744 + N u ) [4.118] N c = 26.6 + 0.0357N U [4.119] 5 clippings: Y = 21.1 N„/(267 + N u ) [4.120] N c = 12.6 + 0.0473N U [4.121] 3 clippings: Y = 17.9 NJ( 146 + N„) [4.122] N c = 8.2 + 0.0560N u [4.123]

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The dimensionless plot for ryegrass is shown in Figure 4-44, and this graph also demonstrates the difficulty of variable harvest interval. The curves were drawn from 10 clippings: Y/A = 2.97 (N u /A')/[ 1.97 + (N u /A')] [4.124] N c /N cm = 0.66 +0.34 (Nu/A) [4.125] 5 clippings: Y/A = 1 .66 (N„/A')/[0.66 + (N„/A')] [4.126] Nc/N cm = 0.40 +0.60 (N„/A) [4.127] 3 clippings: Y/A = 1 .40 (NJA')/[0A0 + (NJA')] [4.128] N c /N cm = 0.29 +0.71 (N u /A) [4. 129] Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 445 through 4-48. It appears that in Figure 4-47, there may be a trend of some kind among the residuals. It is likely that the trend is a result of the variable harvest interval. The growing season of the rye was 26 weeks. Using this length, an average harvest interval can be found for each of the three clipping frequencies. The estimated maximum dry matter yield and plant N removal can be plotted against the average harvest intervals to see if the same relationship holds for this study. There are two possible obstacles. First, there are only three harvest intervals to plot. For the Maryland and Tifton studies, a linear relationship was found between the expected maximum dry matter and harvest interval. With only three points, it is difficult if not impossible to determine the "true" relationship. Furthermore, the relationship tends to drop off after a six week interval (Figures 4-25 and 4-37). The third average harvest interval is 8.67 weeks, and could possibly affect the results. Secondly, these are not actual harvest intervals, but rather an average harvest interval over the growing season. The actual harvest intervals are variable. The effect of the variable harvest interval was observed in the figures, by deduction it is likely to arise here as well. The plot of the estimated maximum dry matter yield and plant N removal versus the average harvest interval is presented in Figure 4-49.

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59 Linear regression was conducted on the estimated maximum dry matter yield resulting in the following relationship A = 8.78 + 0.52At [4.130] with a correlation coefficient of 0.8263. Linear regression was also conducted on the estimated maximum plant N removal resulting in the following relationship A'= 396.6 -2.8At [4.131] with a correlation coefficient of 0.4170. Fayetteville. AR: Bermudagrass The data for this analysis are taken from Huneycutt et al. (1988). The analysis of variance is shown in Table 4-17 (dry matter and N removal x 3 yrs x 2 irrigation x 7 N). Comparison among modes 1 and 2 results in a variance ratio of 170.3 that is significant at the 0.1% level. Comparison among modes 3 and 2 leads to a variance ratio of 2.05 that is significant at the 2.5% level. Comparison among modes 3 and 4 results in a variance ratio of 3.22 that is significant at the 99.5 % level. Comparison among modes 3 and 5, individual A for yield and plant N removal with and without irrigation for all three years, b for dry matter and N removal with and without irrigation, and common c, leads to a variance ratio of 5.21 and is significant at the 0.5% level. Mode 6, individual A for yield and plant N removal with and without irrigation for all three years, b for dry matter and N removal over both with and without irrigation and all years, and common c, describes the data best, since F(21,48,95) = 1.78, F(l,69,99.9) = 11.81, and F(10,59,95) = 2.00. The difference due to years and irrigation is explained by the A parameter. Postulate 3 is supported again. The overall correlation coefficient and adjusted correlation coefficient calculated from mode 6 are 0.9950 and 0.9939, respectively. The error analysis for the parameters is shown in Table 4-18. Results are shown in Figure 4-50, where curves for dry matter and plant N removal for all three years and irrigation, are drawn from

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60 Non-irrigated, 1983: Y = 17.90/[1 + exp(1.50 0.0084N)] [4.132] N u =408.7/[1 + exp(2.04 0.0084N)] [4.133] 1984: Y = 17.37/[1 + exp(1.50 0.0084N)] [4.134] N u =413.9/[1 + exp(2.04 0.0084N)] [4.135] 1985: Y = 19.61/[1 + exp(1.50 0.0084N)] [4.136] N u =459.3/[1 + exp(2.04 0.0084N)] [4.137] Irrigated, 1983: Y =24.70/[l + exp(1.50 0.0084N)] [4.138] N u =554.3/[1 +exp(2.04 0.0084N)] [4.139] 1984: Y =24.60/[l + exp(1.50 0.0084N)] [4.140] N u =523.7/[1 + exp(2.04 0.0084N)] [4.141] 1985: Y =22.58/[l + exp(1.50 0.0084N)] [4.142] N u = 492. 1/[1 + exp(2.04 0.0084N)] [4. 143] From these results, it follows that plant N concentration, shown in Figure 4-50, is estimated from Non-irrigated, 1983: N c =22.8 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.144] 1984: N c =23.8 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.145] 1985: N c =23.4 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.146] Irrigated, 1983: N c =22.4 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.147] 1984: N c =21.3 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.148] 1985: N c =21.8 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.149] As shown, the data are described relativity well by these equations. From Postulate 3, the relationships between yield and plant N removal and between plant N concentration and plant N removal at each clipping interval for all three years with and without irrigation, shown in Figure 4-51, are described by Non-irrigated, 1 983 : Y = 42.9 N u /(571 + N u ) [4.150] N c = 13.3 + 0.0233N U [4.151]

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61 1984: Y = 41.6 N„/(578 + N u ) [4.152] N c = 13.9 + 0.0240N U [4.153] 1985: Y = 47.0N„/(641 +N U ) [4.154] N c = 13.6 + 0.02 13N U [4.155] Irrigated, 1983: Y = 59.2 NJ(174 + N„) [4.156] N c = 13.1 +0.0169N U [4.157] 1984: Y = 59.0 Nu/(731 +N U ) [4.158] N c = 12.4 + 0.0 170N U [4.159] 1985: Y = 54.1 N„/(687 + N u ) [4.160] N c = 12.7 + 0.0185R. [4.161] The dimensionless plot for bermudagrass is shown in Figure 4-52, and validates the form of the model. Note that all of the data have been collapsed onto one graph, and the data fall on the curve. The curves were drawn from Y/A = 2.40(N U /A , )/[1.40 + (N U /A')] [4.162] Nc/Nc m = 0.58 +0.42 (Nu/A) [4.163] Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 453 through 4-56. As done earlier, the data will be averaged over years, since variations due to water availability appear in the linear parameter A. The averaged data are in Table 4-19 and the analysis of variance is in Table 4-20 (dry matter and N removal x 1 yr x 2 irrigation x 7 N). Comparison among modes 1 and 2 results in a variance ratio of 373.4 that is significant at the 0.1% level. Comparison among modes 3 and 2 leads to a non-significant variance ratio of 2.72. Comparison among modes 3 and 4 results in a variance ratio of 6.11 that is significant at the 0.5% level. Mode 5, individual A for yield and plant N removal with and without irrigation, b for dry matter and N removal, and common c, accounts for all the significant variation, since F(5, 16,95) = 2.85, F(l,21,99) = 8.02, and

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62 F(2, 19,95) = 3.52. The difference due to irrigation is explained by the A parameter. Postulate 3 is supported again. The overall correlation coefficient and adjusted correlation coefficient calculated from mode 5 was 0.9973 and 0.9965, respectively. The error analysis for the parameters is shown in Table 4-21. Results are shown in Figure 4-57, where curves for dry matter and plant N removal with and without irrigation averaged over three years, are drawn from Non-irrigated: Y = 18.63/[1 + exp(1.51 0.0084N)] [4.164] N u = 435.7/[l + exp(2.04 0.0084N)] [4. 165] Irrigated: Y = 24.40/[l + exp(1.51 0.0084N)] [4.166] N u = 534.8/[l + exp(2.04 0.0084N)] [4.167] The error analysis for the averaged data is in Table 4-21. Note that the estimates of b and c are the essentially the same as for the unaveraged data, supporting what Overman et al. (1990a, 1990b) have found previously. From these results, it follows that plant N concentration, shown in Figure 4-57, is estimated from Non-irrigated: N c = 23.4 [1 + exp(2.04 0.0084N)]/[1 + exp(1.51 0.0084N)] [4.168] Irrigated: N c = 21.9 [1 + exp(2.04 0.0084N)]/[1 + exp(1.51 0.0084N)] [4.169] As shown, the data are described well by these equations. From Postulate 3, the relationships between yield and plant N removal and between plant N concentration and plant N removal at each clipping interval for all three years with and without irrigation, shown in Figure 4-58, are described by Non-irrigated: Y = 45.3 N u /(623 + N u ) [4.170] N c = 13.8 + 0.0221N U [4.171] Irrigated: Y = 59.3 N„/(765 + N u ) [4.172] N c = 12.9 + 0.0 169N U [4.173]

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63 The dimensionless plot for bermudagrass is shown in Figure 4-59, and validates the form of the model. Note that all of the data have been collapsed onto one graph, and the data fall on the curve reasonably well. The curves were drawn from Y/A = 2.43(N U /A')/[1.43 + (N U /A')] [4.174] Nc/N cm = 0.59 +0.41 (NJA!) [4. 175] Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 460 through 4-63. Fayetteville. AR: Tall Fescue The data for this analysis are taken from Huneycutt et al. (1988). The analysis of variance is shown in Table 4-22 (dry matter and N removal x 3 yrs x 2 irrigation x 7 N). Comparison among modes 1 and 2 results in a variance ratio of 352 that is significant at the 0. 1% level. Comparison among modes 3 and 2 also leads to a variance ratio of 7.60 that is significant at the 0.1% level. Comparison among modes 3 and 4 results in a variance ratio of 9.90 that is significant at the 0.1% level. Comparison among modes 3 and 5, individual A for yield and plant N removal with and without irrigation for all three years, b for dry matter and N removal with and without irrigation, and common c, leads to a variance ratio of 18.3 that is significant at the 0. 1% level. Mode 6, individual A for yield and plant N removal with and without irrigation for all three years, b for dry matter and N removal over both with and without irrigation and all years, and common c, describes the data best, since F(21,48,95) = 2.94, F(l,69,99.9) = 11.81, F(10,59,97.5) = 2.27, and F(2,67,95) = 3.13. The difference due to years and irrigation is explained by the A parameter. Postulate 3 is supported again. The overall correlation coefficient and adjusted correlation coefficient calculated from mode 6 were 0.9956 and 0.9947, respectively. The error analysis for the parameters is shown in Table 4-23. Results are

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64 shown in Figure 4-64, where curves for dry matter and plant N removal for all three years and irrigation, are drawn from Non-irrigated, 1981-2: Y = 1? 08/n + exnfO 92 0 0081NYI [4 1761 N u = 357 S/r 1 + exnfl 47 0 0081NYI \j .\j\jkj 1 1 1 ij [4 1771 1982-3: Y = 8 01 /N + exnfO 92 0 0081NY1 T4 1781 N u = 217 8/n + exofl 47 0 0081NYI [4 1791 L 1 ' J 1983-4: Y = 71/n + exnfO 92 0 0081NYI ill ij [4.180] N = 169 VM + exnd 47 0 0081NYI v.V/W 111 ft T4 1811 Irrigated, 1981-2: Y = 15.63/[1 + exp(0.92 0.008 IN)] [4.182] N u = 443.7/[l + exp(1.470.008 IN)] [4.183] 1982-3: Y = 13.22/[1 +exp(0.920.008 IN)] [4.184] N u = 357.3/[l + exp(1.470.0081N)] [4.185] 1983-4: Y = 16.81/[1 +exp(0.920.008 IN)] [4.186] N u = 439.4/[l + exp(1.470.0081N)] [4.187] From these results, it follows that plant N concentration, shown in Figure 4-64, is estimated from Non-Irrigated: 19812: 29.6 [1 + exp(1.47 0.0081N)]/[1 + exp(0.92 0.0081N)] [4.188] 19823: 27.2 [1 + exp(1.47 0.0081N)]/[1 + exp(0.92 0.0081N)] [4.189] 19834: 29.7 [1 + exp(1.47 0.0081N)]/[1 + exp(0.92 0.0081N)] [4.190] Irrigated: 19812: 28.4 [1 + exp(1.47 0.0081N)]/[1 + exp(0.92 0.0081N)] [4.191] 19823: 27.0 [1 + exp(1.47 0.0081N)]/[1 + exp(0.92 0.008 IN)] [4.192] 19834: 26.1 [1 +exp(l. 47 -0.008 1N)]/[1 + exp(0.92 0.008 IN)] [4.193] As shown, the data are described relativity well by these equations.

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65 From Postulate 3, the relationships between yield and plant N removal and between plant N concentration and plant N removal at each clipping interval for all three years with and without irrigation, shown in Figure 4-65, are described by Mnn irriontprl 1081-91NOI1 11 1 lgalCU, I7OI v = T4 1941 1N C 17 14-0 01 SON 14 1951 1Q89 1V = i — \A 1961 N = 1 S 7 40 0S9RN T4 1971 108^ 4 V = i 1 J.J l^i|/^Z.J 1 ' i^u/ \4 1981 N c = 17.1 +0.0741N U [4.199] Irrigated, 1981-2: Y = 36.9 N„/(605 + N u ) [4.200] N c = 16.4 + 0.0271N U [4.201] 1982-3: Y 31.2N„/(487 + N u ) [4.202] N c = 15.6 + 0.0320N U [4.203] 1983-4: Y = 39.7 Nu/(599 + N u ) [4.204] N c = 15.1 +0.0252N U [4.205] The dimensionless plot for bermudagrass is shown in Figure 4-66, and validates the form of the model. Note that all of the data have been collapsed onto one graph, and the data fall on the curve. The curves were drawn from Y/A = 2.36(N U /A')/[1.36 + (N U /A')] [4.206] N c /N cm = 0.58 +0.42 (N„/A) [4.207] Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 467 through 4-70. As done with the previous study, the data will be averaged over years, since variations due to water availability appear in the linear parameter A. The averaged data are in Table 4-24 and the analysis of variance is in Table 4-25 (dry matter and N removal x 1 yr x 2 irrigation x 7 N). Comparison among modes 1 and 2 results in a variance ratio of

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66 904 that is significant at the 0.1% level. Comparison among modes 3 and 2 leads to a variance ratio of 9.44 that is significant at the 0.1% level. Comparison among modes 3 and 4 results in a variance ratio of 18.4 that is significant at the 0.1% level. Mode 5, individual A for yield and plant N removal with and without irrigation, b for dry matter and N removal, and common c, accounts for all the significant variation, since F(5, 16,95) = 2.85, F(l,21,99.9) = 14.59, and F(2,19,95) = 3.52. The difference due to irrigation is explained by the A parameter. Postulate 3 is supported again. The overall correlation coefficient and adjusted correlation coefficient calculated from mode 5 were 0.9986 and 0.9982, respectively. The error analysis for the parameters is shown in Table 4-27. Results are shown in Figure 4-71, where curves for dry matter and plant N removal with and without irrigation averaged over three years, are drawn from Non-irrigated: Y = 8.67/[l + exp(0.99 0.008 IN)] [4.208] N u = 250.7/[l + exp(1.53 0.0081N)] [4.209] Irrigated: Y = 15.36/[1 + exp(0.99 0.008 IN)] [4.210] N u = 417.4/[1 +exp(1.53 -0.0081N)] [4.211] The error analysis for the averaged data is in Table 4-26. Note that although the estimates of b and c are not quite the same as for the unaveraged data, the Ab (b'-b), is essentially constant. From these results, it follows that plant N concentration, shown in Figure 4-71, is estimated from Non-irrigated: N c = 28.9 [1 + exp(1.53 0.008 1N)]/[1 + exp(0.99 0.0081N)] [4.212] Irrigated: N c = 27.2 [1 + exp(1.53 0.0081N)]/[1 + exp(0.99 0.0081N)] [4.213] As shown, the data are described well by these equations. From Postulate 3, the relationships between yield and plant N removal and between plant N concentration and plant N removal averaged over three years with and without irrigation, shown in Figure 4-72, are described by Non-irrigated: Y = 20.8 Nu/(350 + N u ) [4.214]

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67 N c = 16.9 + 0.048 1N U [4.215] Irrigated: Y = 36.8N„/(583 +N U ) [4.216] N c = 15.8 + 0.0272N, u [4.217] The dimensionless plot for tall fescue is shown in Figure 4-73, and validates the form of the model. Note that all of the data have been collapsed onto one graph, and the data fall on the curve. The curves were drawn from Notice the similarity between the above equations and Equations [4.174] and [4.175]. The Ab values are similar for both grasses. It is not clear if this is a coincidence or a result of the system. Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-74 through 4-77. Eagle Lake. TX: bermudagrass and bahiagrass Data from Evans et al. (1961) for bermudagrass and bahiagrass were used. First the grasses will be analyzed separately, then they will be pooled to see if any of the parameters are common for the two grasses. The analysis of variance for the Coastal bermudagrass over years 1979-1980 is shown in Table 4-27 (dry matter and N removal x 2 yrs x 5 N). Comparison among modes 1 and 2 results in a variance ratio of 419 that is significant at the 0. 1% level. Comparison among modes 3 and 2 leads to a variance ratio of 9.22 that is significant at the 0.5% level. Comparison among modes 3 and 4 results in a variance ratio of 14.6 that is significant at the 0.1% level. Comparison among modes 5 and 4 leads to a variance ratio of 10.4 that is significant at the 0.5% level. As a result, mode 4, individual A and b for yield and plant N removal at each year and common c, accounts for all the significant variation, since F(3,8,95) = 4.07, F(3,l 1,99.9) = 1 1.56, and F(2, 11,99.5) = 8.91. The analysis of variance for the Pensacola bahiagrass over years Y/A = 2.40(N U /A')/[1.40 + (N U /A')] [4.218] N c /N cm = 0.58 +0.42 (Nu/A') [4.219]

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68 1979-1980 is shown in Table 4-28 (dry matter and N removal x 2 yrs x 5 N). Comparison among modes 1 and 2 results in variance ratio of 328 that is significant at the 0.1% level. Comparison among modes 3 and 2 leads to a variance ratio of 9.54 that is significant at the 99.5 level. Comparison among modes 3 and 4 results in a variance ratio of 12.3 that is significant at the 0.1% level. Mode 5, individual A for yield and plant N removal for each year, b for dry matter and N removal, and common c, accounts for all the significant variation, since F(5,8,95) = 3.69, F(l, 13,99.9) 17.81, and F(2,ll,95) = 3.98. Next the data from the two grasses are combined and the analysis of variance data are presented in Table 4-29 (dry matter and N removal x 2 grasses x 2 yrs x 5 N). A comparison of modes 1 and 2 results in a variance ratio of 330 that is significant at the 0.1% confidence level. Comparison of modes 3 and 2 leads to a variance ratio of 7.62, that is significant at the 0. 1% confidence level. Comparison of modes 3 and 4 results in a variance ratio of 9.66 that is significant at the 0. 1% confidence level. Comparison of modes 3 and 5 results in a variance ratio of 16.2 that is significant at the 0.1% confidence level. Comparison of modes 5 and 4 leads to a variance ratio of 5.85 that is significant at the 0.1% confidence level. Comparison of modes 3 and 6 leads to a variance ratio of 7.54 that is significant at the 0.1% confidence level. Comparison of modes 6 and 4 results in a variance ratio of 6.58 that is significant at the 0.5% confidence level. Based upon these comparisons, we can conclude that mode 4, with individual A and b for dry matter yield and plant N removal for each year and grass and common c describes the data best. The overall correlation coefficient of 0.9927 and adjusted correlation coefficient of 0.9876 were calculated by mode 4. The statistical analysis might be affected by the close numbers for yield and N removal at low values of applied N for both grasses. In the concern that fewer parameters might be used to significantly account for the variation, the data will be averaged over years to determine if an individual b can be used for yield and N removal for both grasses. The averaged data are in Table 4-30 and the analysis of variance for the

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69 averaged data is in Table 4-3 1 . Comparison among modes 1 and 2 results in a variance ratio of 792, that is significant at the 0. 1% confidence interval. Comparison among modes 3 and 2 leads to a variance ratio of 7.79, that is significant at the 0.5% confidence level. Comparison among modes 3 and 4 results in a variance ratio of 18.4, that is significant at the 0. 1% confidence level. Mode 5, individual A for dry matter yield and plant N removal of each grass, individual b for dry matter yield and plant N removal, and common c, accounts for all of the significant variation, since F(5,8,95) = 3.69, F(l, 13,99.9) = 17.81, and F(2,l 1,97.5) = 5.26. The two grasses have the same b and b', hence suggesting mode 5 of Table 4-29. The overall correlation coefficient of 0.9987 and adjusted correlation coefficient of 0.9981 were calculated using mode 5 of the averaged data. Using mode 5 and the original data, the overall correlation coefficient of 0.9798 and adjusted correlation coefficient of 0.9727 were calculated. The error analysis of the averaged and original data is in Tables 4-32 and 4-33. Note that the estimates of b and c are the essentially the same as for the unaveraged data, suggesting this is a valid way to describe the data. Results are shown in Figure 4-78 and 4-79 for bermudagrass and bahiagrass, respectively, where are drawn from Coastal bermudagrass, 1979: Y = 15.99/[1 +exp(0.57 0.0072N)] [4.220] N u = 310.9/[1 + exp(1.07 0.0072N)] [4.221] 1980: Y = 9.89/[l + exp(0.57 0.0072N)] [4.222] N u = 227.4/[l +exp(1.07 0.0072N)] [4.223] Pensacola bahiagrass, 1979: Y = 12.42/[1 +exp(0.57 0.0072N)] [4.224] N u = 237.2/[l +exp(1.07 0.0072N)] [4.225] 1980: Y = 7.51/[1 +exp(0.57 0.0072N)] [4.226] N u = 191.3/[1 +exp(1.07 0.0072N)] [4.227] From these results, it follows that plant N concentration, shown in Figures 4-78 and 4-79, is estimated from

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70 Bermudagrass, 1979: N c = 19.4 [1 + exp(1.07 0.0072N)]/[1 + exp(0.57 0.0072N)][4.228] 1980: N c = 23.0 [1 + exp(1.07 0.0072N)]/[1 + exp(0.57 0.0072N)][4.229] Bahiagrass, 1979: N c = 19.1 [1 + exp(1.07 0.0072N)]/[1 + exp(0.57 0.0072N)][4.230] 1980: N c = 25.5 [1 + exp(1.07 0.0072N)]/[1 + exp(0.57 0.0072N)][4.231] As shown, the data are described well by these equations. From Postulate 3, the relationships between yield and plant N removal and between plant N concentration and plant N removal over two years, shown in Figure 4-80 and 4-8 1 , are described by Bermudagrass, 1979: Y = 40.6 N„/(479 + N u ) [4.232] N c = 11.8 + 0.0246N U [4.233] 1980: Y = 25.1 Nu/(351 +N U ) [4.234] N c = 13.9 + 0.0398N U [4.235] Bahiagrass, 1979: Y = 31.6N„/(366 + N u ) [4.236] N c = 11.6 + 0.03 17N„ [4.237] 1980: Y = 19.1 Nu/(295 +N U ) [4.238] N c = 15.4 + 0.0524N U [4.239] The dimensionless plot for bermudagrass and bahiagrass is shown in Figure 4-82, and validates the form of the model. Note that all of the data have been collapsed onto one graph, and the data fall on the curve. The curves were drawn from Y/A = 2.54(N U /A , )/[1.54 + (N„/A')] [4.240] Nc/N cm = 0.61 +0.39 (Nu/A') [4.241] Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 483 through 4-86.

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71 Clayton and Kinston. NC: Corn Data from Kamprath (1986) for corn were used. The analysis of variance for the Dothan sandy loam is shown in Table 4-34 (dry matter and N removal x 2 components x 5 N). Comparisons among modes 1 and 2, modes 3 and 2, modes 3 and 4, and modes 3 and 5 result in variance ratios of 1686, 22.1, 49.8, and 39.0, which are all significant at the 0.1% confidence level. Comparison among modes 5 and 4 leads to a non-significant variance ratio of 0.31. As a result, mode 5, individual A for grain and total plant, individual b for yield and plant N removal, and common c, accounts for all the significant variation, since F(5,8,97.5) = 4.82, F(l, 13,99.9) = 17.81, and F(2,ll,95) = 3.98. The analysis of variance for the Goldsboro sandy loam is shown in Table 4-35 (dry matter and N removal x 2 components x 5 N). As with the Dothan, comparisons among modes 1 and 2, modes 3 and 2, modes 3 and 4, and modes 3 and 5 result in variance ratios of 1920, 26.9, 35.7, and 83.2, which are all significant at the 0.1% confidence level. Comparison among modes 5 and 4 leads to a non-significant variance ratio of 2.48. As a result, mode 5, individual A for grain and total plant, individual b for yield and plant N removal, and common c, accounts for all the significant variation, since F(5,8,95) = 3.69, F(l, 13,99.9) = 17.81, and F(2, 1 1,95) = 3.98. Next the data from the two soils are combined and the analysis of variance data are presented in Table 4-36 (dry matter and N removal x 2 components x 2 soils x 5 N). A comparison among modes 1 and 2 results in a variance ratio of 1562 that is significant at the 0.1% confidence level. Comparison among modes 3 and 2 leads to a variance ratio of 24.8, that is significant at the 0.1% confidence level. Comparison among modes 3 and 4 results in a variance ratio of 8.29 that is significant at the 0.1% confidence level. Comparison among modes 3 and 5 results in a variance ratio of 1 9.7 that is significant at the 0. 1% confidence level. Comparison among modes 3 and 6 leads to a variance ratio of 67.9 that is significant at the 0.1% confidence level. Comparison among modes 3 and 7 results in a variance ratio of 21.9, that is significant at

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72 the 0. 1% confidence level. Comparison among modes 6 and 7 leads to a variance ratio of 0.36, which is not significant. Mode 6, with individual A for dry matter yield and plant N removal for both grain and total plant, b for dry matter yield and plant N removal, and common c describes the data best, since F(2,27,95) = 3.35, and the MSS for mode 6 was smaller than that of mode 5. The overall correlation coefficient of 0.9941 and adjusted correlation coefficient of 0.9921 were calculated by mode 6. The error analysis for the parameters is shown in Table 4-37. Results are shown in Figure 4-87 for both soils, grain and total plant, where the curves are drawn from Dothan, Grain: Y = 11.1 2/[ 1 + exp(0.270.0187N)] [4.242] K, = 151.7/[1 +exp(0.97•0.0187N)] [4.243] Total: Y = 20.68/[l +exp(0.270.0187N)] [4.244] N u = 187.2/[1 +exp(0.970.0187N)] [4.245] Goldsboro, Grain: Y = 7.83/[l+exp(0.270.0187N)] [4.246] N u = 113.3/[1 +exp(0.970.0187N)] [4.247] Total: Y = 14.70/[1 + exp(0.27 0.0187N)] [4.248] N u = 136.5/[1 +exp(0.970.0187N)] [4.249] From these results, it follows that plant N concentration, shown in Figure 4-87, is estimated from Dothan, Grain: N c = 13.6 [1 + exp(0.97 0.0187N)]/[1 + exp(0.27 0.01 87N)] [4.250] Total: N c = 9.1 [1 + exp(0.97 0.0187N)]/[1 + exp(0.27 0.0 187N)] [4.251] Goldsboro, Grain: N c = 14.5 [1 + exp(0.97 0.0187N)]/[1 + exp(0.27 0.01 87N)] [4.252] Total: N c = 9.3 [1 + exp(0.97 0.0187N)]/[1 + exp(0.27 0.0187N)][4.253] As shown, the data are described well by these equations. From Postulate 3, the relationships between yield and plant N removal and between plant N concentration and plant N removal for both soils, grain and total plant, shown in Figure 4-88, are described by

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73 Dnthan Grain' Y = 22 1 N./n50 + N„) [4.254] N = 6 8 + 0 0453N.. [4.255] L ' J TotalY= 41 1N/n85+N.^ [4 2561 L 1 • J N = 4S + 0 0241N [4 2571 L J Gnlrkhnrn GrainY= 15 6 N /(l 12 + N-1 VjUIUjUUI vJ, VJI Cull. 1 l J ,\J i^u/^i i 1 i^u/ [4.2581 M = 7 7 + 0 0643N T4 259] Total: Y= 29.2 N„/( 135 + N u ) |4.zoUJ N c = 4.6 + 0.0342N U [4.261] The dimensionless plot for corn is shown in Figure 4-89, and validates the form of the model. Note that all of the data have been collapsed onto one graph, and the data fall on the curve. The curves were drawn from Y/A = l^CK/A'ytO^ + CNu/A')] [4.262] Ne/Ncn, = 0.50 +0.50 (Nu/A) [4.263] Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 490 through 4-93. Plymouth. NC: Corn Data from Kamprath (1986) for corn were used. The analysis of variance for the Plymouth very fine sandy loam is shown in Table 4-38 (dry matter and N removal x 2 components x 5 N). Comparisons among modes 1 and 2, modes 3 and 2, modes 3 and 4, and modes 3 and 5 result in variance ratios of 5819, 32.5, 27.8, and 57.4, which are all significant at the 0.1% confidence level. Comparison among modes 5 and 4 leads to a non-significant variance ratio of 3.21. As a result, mode 5, individual A for grain and total plant, individual b for yield and plant N removal, and common c, accounts for all the significant variation, since F(5,8,97.5) = 4.82, F(l, 13,99.9) = 17.81, and F(2,l 1,95) = 3.98. The overall correlation coefficient of 0.9997 and adjusted correlation coefficient of

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74 0.9995 were calculated by mode 5. Results are shown in Figure 4-94 for grain and total plant, where the curves are drawn from Grain: Y = 9.48/[l + exp(-0.065 0.01 19N)] [4.264] N u = 126.2/[1 + exp(0.46 0.01 19N)] [4.265] Total: Y = 16.60/[1 + exp(-0.065 0.01 19N)] [4.266] N u = 147.3/[1 + exp(0.46 0.01 19N)] [4.267] From these results, it follows that plant N concentration, shown in Figure 4-94, is estimated from Grain: N c = 13.3 [1 + exp(0.46 0.0119N)]/[1 + exp(-0.065 0.01 19N)] [4.268] Total: N c = 8.9 [1 + exp(0.46 0.01 19N)]/[1 + exp(-0.065 0.01 19N)] [4.269] As shown, the data are described well by these equations. From Postulate 3, the relationships between yield and plant N removal and between plant N concentration and plant N removal for grain and total plant, shown in Figure 4-95, are described by Grain: Y = 23.2 Nu/( 183 + N u ) [4.270] N c = 7.9 + 0.043 1N U [4.271] Total: Y = 40.6 N„/(213 + N u ) [4.272] N c = 5.2 + 0.0246N U [4.273] The dimensionless plot for corn is shown in Figure 4-96, and validates the form of the model. Note that all of the data have been collapsed onto one graph, and the data fall on the curve. The curves were drawn from Y/A = 2.45(N U /A')/[1.45 + (N U /A')] [4.274] N c /N cm = 0.59+0.41 (K/A') [4.275] Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 497 through 4-100.

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75 Florida: Bahiagrass Data from Blue (1987) for bahiagrass were used. The analysis of variance for the bahiagrass grown on the Entisol and Spodosol is shown in Table 4-40 (dry matter and N removal x 2 soils x 5 N). Comparisons among modes 1 and 2 result in a variance ratio of 1 792, that is significant at the 0. 1% confidence level. Comparisons among modes 3 and 2 lead to a variance ratio of 12.24, that is significant at the 0.5% confidence level. Comparisons among modes 3 and 4 lead to a variance ratio of 6.41, that is significant at the 1% confidence level. Comparisons among modes 5 and 2 lead to a variance ratio of 10.72, that is significant at the 0.5% confidence level. Comparison among modes 5 and 4 leads to a variance ratio of 5.94, that is significant at the 2.5% confidence level. The statistics suggest that mode 4, individual A for both soils, individual b for yield and plant N removal, and common c, accounts for all the significant variation, since F(3,8,97.5) = 5.42, and F(3,l 1,99) = 6.22. This mode estimates the following b and c parameters: Entisol, dry matter: b =1.46 ± 0.07 Spodosol, dry matter: b =1.31 ± 0.07 Entisol, N removal: b' = 1.89 ± 0.08 Spodosol, N removal: b' = 1.83 ± 0.08 c = 0.0118 while mode 5 estimates the following b and c parameters: both soils, dry matter: b = 1.39 ± 0.05 both soils, N removal: b' = 1.86 ± 0.06 c = 0.0118 The estimates and their standard errors overlap, suggesting that mode 5 is correct. Furthermore, since there are less parameters to estimate assuming the b values are not affected by the soils, mode 5 will be used for estimation. The overall correlation coefficient of 0.9985 and adjusted correlation coefficient of 0.9978 are calculated by mode

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76 5. The error analysis for the parameters is shown in Table 4-41. Results are shown in Figure 4-101 for both soils, where the curves are drawn from Entisol: Y = 1 1.14/[1 + exp(1.39 0.01 18N)] [4.276] N u = 311.0/[1 +exp(1.86-0.0118N)] [4.277] Spodosol: Y = 19.39/[1 + exp(1.39 0.01 18N)] [4.278] N u = 201.5/[1 +exp(1.86-0.0118N)] [4.279] From these results, it follows that plant N concentration, shown in Figure 4-101, is estimated from Entisol: N c =27.9 [1 + exp(l. 86 0.01 18N)]/[1 + exp(l. 39 0.01 18N)] [4.280] Spodosol:N c = 10.4 [1 + exp(l. 86 0.01 18N)]/[1 + exp(1.39 0.01 18N)] [4.281] As shown, the data are described well by these equations. From Postulate 3, the relationships between yield and plant N removal and between plant N concentration and plant N removal for both soils, shown in Figure 4-102, are described by Entisol: Y= 29.7 N„/(518 + N u ) [4.282] N c = 17.4 + 0.0337N U [4.283] Spodosol: Y = 51.7 N u /(336 + N u ) [4.284] N c = 6.5 + 0.0193N u [4.285] The dimensionless plot for bahiagrass is shown in Figure 4-103, and validates the form of the model. Note that all of the data have been collapsed onto one graph, and the data fall on the curve. The curves were drawn from Y/A = 2.67(N U /A')/[1.67 + (N U /A')] [4.286] Wo m = 0.63 +0.37 (Nu/A') [4.287] Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4104 through 4-107.

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77 England: Ryegrass The data for this analysis were taken from Morrison et al. (1980). The analysis of variance is shown in Table 4-42 (dry matter and N removal x 20 sites x 6 N).. As shown in the table, all of the comparisons are highly significant (0.1%). This is due in part to the large degrees of freedom. If comparison is made among the mean sums of squares (MSS), mode 4 has the smallest MSS with the exception of mode 2 (which fits each individually). Mode 4 is the preferred mode since it requires less parameters to estimate (81 versus 120). Mode 4 assumes an individual A and b for each site, dry matter yield and plant N removal and common c for all. The overall correlation coefficient and adjusted correlation coefficient calculated from mode 4 are 0.9935 and 0.9903, respectively. The error analysis for the parameters is shown in Table 4-43. A comparison among estimated maximum plant N concentrations and Ab's among sites is in Table 4-44. These variables have been plotted in Figure 4-108, the mean and ±2 standard errors of the estimates for all sites are designated by a solid and dashed horizontal line, respectively. This suggests an important result, the Ab is constant from site to site for a particular grass. If this is true, then in conjunction with the postulates of the extended model, one less parameter would need to be calculated. It also appears from Figure 4-108, that the ratio of estimated maximum plant N removal to estimated maximum dry matter yield, or estimated maximum plant N concentration is constant as well for a particular grass from site to site. Due to the large number of sites, the results of the regression will not be plotted directly. The dimensionless plot of ryegrass over all twenty sites is shown in Figure 4-109, with the curves drawn assuming a constant Ab, namely the mean of the set, 0.83. The curves are drawn by Y/A = 1.77(N u /A')/[0.77 + (N u /A)] [4.288] N c /N cm = 0.44 +0.56 (N„/A) [4.289]

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78 A hyperbolic regression was conducted on the dry matter yield/estimated maximum versus N removal/estimated maximum plot. The results are shown in Figure 4-110, with the curves drawn by Y/A 1.88 (N./AW.88 + OVA 1 )] [4.290] Nc/N cm = 0.47 +0.53 (K/A') [4.291] Y 1 K' 1 Since — = rr and — = — r, , Ab is calculated to be 0.76. This is an important A \-e~ Ab A' result. Overman (1995a) has compared the probability density functions for the gaussian and logistic functions normalized to unit area and inflection points. In the gaussian distribution V2 arises as an integration constant to guarantee that the function will sum to one. Similarly, 1.317 is the integration constant in the logistic distribution. The inverse of this number is calculated to be 0.759. This was also the value of Ab found in the hyperbolic regression above. Could there be something to this coincidence? Overman (1995a) also showed in the same paper that the logistic equation could also be redefined as , 1 1 where = y/A and £ = cN b. At this point, it is not clear if this is just a surprising coincidence or a result of some fundamental process of the system. Regardless, it is very intriguing. Evaluation of the Extended Triple Logistic (NPK) Model Tifton, GA: Rye Data from Walker and Morey (1962) were used. The parameters, their standard errors, and relative errors resulting from the nonlinear regression are presented in Table 4-

PAGE 102

79 45. The relative errors on the parameters are high due to the flat response to P and K, making estimation of b and c parameters for these nutrients difficult since the data are high on the curve. Also, some of the estimates of b are very close to zero. As the standard errors are divided by these small numbers, they are artificially inflated. The RSS was calculated to be 5895.76. The overall correlation coefficient and adjusted correlation coefficient calculated from the regression are 0.9888 and 0.9860, respectively. Results are shown in Figures 4-111 through 4-113 for the three nutrients, where the curves are drawn from 5^43 y = /i l „ 136-0.0225^ V1 , „-0.14-0.0464/> vl , -0.9 1-0.0201 A' x (l + e )(l + e )U + e ) [4.292] 260 N » = ,, , ,1.93-0.0225^ V1 , -0.14-0.0464P V1 , -0.91-0.0201AT x [4.293] (l + e )(l + e X 1 + e ) 34 (1 +e U*WUp« )(1 + e -U.!0-U.U4 M r )(1 + e [4.294] 230^ K» ~ n , 1.36-0.0225JV vi , .-0.14-0.0464F V1 , „ 0.46-0.0201 K x [4.295] From these results, it follows that plant nutrient concentrations, shown in Figures 4-111 through 4-113, are estimated from N c = 47.9 [1 + exp(1.93 0.0225N)]/[1 + exp(1.36 0.0225N)] [4.296] P c = 6.26 [1 + exp(-0.16 0.0464P)]/[1 + exp(-0.14 0.0464P)] [4.297] Kc = 42.4 [1 + exp(0.46 0.0201K)]/[1 + exp(-0.91 0.0201K)] [4.298] As shown, the data are described well by these equations. From Postulate 3, the relationships between yield and plant nutrient removals and between plant nutrient concentrations and plant nutrient removals for all three nutrients, shown in Figures 4-114 through 4-116, are described by

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80 Y= 12.5Nu/(338 +N u ) [4.299] N c = 27.1 +0.0800N U [4.300] Y= -268.8 P.A-1717 + Pu) [4.301] P c = 6.4 0.0037P U [4.302] Y = 7.28 K„/(78.4 + K u ) [4.303] 1^= 10.8 + 0.1374K U [4.304] The dimensionless plot for rye is shown in Figure 4-117, and validates the form of the model. This plots supports the result that the Ab ; is different for each nutrient. The curves were drawn from Y/A = 2.30(N U /A , )/[1.30 + (N U /A')] [4.305] Y/A = -49.5(P u /A*)/[-50.5 + (P u /A')] [4.306] Y/A = 1.34(K u /A')/[0.34 + (K u /A')] [4.307] Nc/N cm = 0.57 + 0.43 (NJA!) [4.308] Pc/Pcm = 1.020.02 (Pu/A 1 ) [4.309] KJK cm = 0.25 + 0.75 (WA') [4.310] Scatter and residual plots of dry matter yield and plant nutrient removals are shown in Figures 4-118 through 4-125.

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Table 4-1. Analysis of Variance of Model Parameters Used to Describe Coastal Bermudagrass Yield Response to Nil rogen at Thorsby, Alabama, 1957-1959. Parameters Residual Sums Mean Sums Mode estimated df of Squares of Squares F (l)Com A,b,c 3 21 42.347 2.017 (2) Ind A,b,c 18 6 3.507 0.584 0M2) 15 38.840 2.589 4.43 + (3) Ind A, Com b,c 8 16 12.267 0.766 (3)-(2) 10 8.750 0.875 1.50 (4) Ind A,b Com c 13 11 8.019 0.729 (4)-(2) 5 4.512 0.902 1.54 (3)-(4) 5 4.248 0.850 1.17 (5) Ind A Com c, Ind b (irrigation) 9 15 12.342 0.823 (5)-(2) 9 8.835 0.982 1.68 (5)-(4) 4 4.323 1.081 1.48 Source: Original yield data from Evans et al. (1961). Significant at the 0.05 level F(15, 6,95) =3.94 F(10, 6,95) =4.06 F( 5, 6,95) = 4.39 F( 5,11,95) =3.20 F( 9, 6,95) =4.10 F( 4,11,95) =3.36

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Table 4-2. Analysis of Variance of Model Parameters Used to Describe Pensacola Bahiagrass Yield Response 1 o Nitrogen at Thorsby, Alabama, 1957-1959. Parameters Residual Sums Mean Sums Mode estimated df of Squares of Squares F (l)Com A,b,c 3 21 25.712 1.224 (2) Ind A,b,c 18 6 1.306 0.218 d)-(2) 15 24.406 1.627 7.48* (3) Ind A, Com b,c 8 16 9.942 0.621 (3)-(2) 10 8.636 0.864 3.97 (4) Ind A,b Com c 13 11 1.872 0.170 (4)-(2) 5 0.567 0.113 0.52 (3)-(4) 5 8.070 1.614 9.49 ++ (5) Ind A, Com c, Ind b (irrigation) 9 15 7.844 0.523 (5)-(2) 9 6.539 0.727 3.34 (3)-(5) 1 2.098 2.098 4.01 (5M4) 4 5.972 1.493 8.78 ++ Source: Original yield data from Evans et al. (1961). Significant at the 0.005 level f Significant at the 0.025 level F(15, 6,97.5)= 5.27 F(10, 6,95) =4.06 F( 5, 6,95) =4.39 F( 5,1 1,99.5)= 6.42 F( 9, 6,95) =4.10 F( 1,15,95) =4.54 F( 4,1 1,99.5)= 6.88

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83 Table 4-3. Analysis of Variance of Model Parameters Used to Describe Coastal Bermudagrass and Pensacola Bahiagrass Yield Response to Nitrogen at J 5 " , — Parameters Residual Sums Mean Sums Mode A* J estimated df of Squares of Squares F (1) Com A,b,c 3 45 79.043 1.757 (2) Ind A,b,c 36 12 4.813 0.401 0X2) 33 74.230 2.249 5. 61 (3) Ind A, Com b,c 14 34 26.727 0.786 (3)-(2) 22 21.914 0.996 2.48 (4) Ind A,b Com c 25 23 10.414 0.453 (4)-(2) 11 5.601 0.509 1.27 (3)-(4) 11 16.313 1.483 _ _ * 3.27 (5) Ind A, Com c, Ind b (type of grass, 17 31 20.825 0.672 irrigation) (5)-(2) 19 16.012 0.843 2.10 (3H5) 3 5.902 1.967 2.93 (5)-(4) 8 10.411 1.301 2.87 t (6) Ind A, Com c, Ind b (irrigation) 15 33 25.854 0.783 (6)-(2) 21 21.041 1.002 2.50 (3)-(6) 1 0.873 0.873 1.11 (6)-(4) 10 15.440 1.544 3.41 (6)-(5) 2 5.029 2.415 3.74 + (7) Ind A, Com c, Ind b (type of grass) 15 33 22.868 0.693 (7)-(2) 21 18.055 0.860 2.14 (3H7) 1 3.859 3.589 5.57 ! (7)-(4) 10 12.454 1.245 2.75 f (7H5) 2 2.043 1.021 1.52 Source: Original yield data from Evans et al. (1961). ' f Significant at the 0.005 level Significant at the 0.01 level + Signifcant at the 0.025 level Significant at the 0.05 level F(33, 12,99.5)= 4.23 F(22, 12,95) =2.52 F( 11,12,95) =2.72 F(l 1,23,99) =3.14 F( 19,12,95) =2.55 F( 3,31,95) =2.91 F( 8,23,97.5)= 2.81 F(2 1,12,95) =2.53 F( 1,33,95) =4.14 F(10,23,99) =3.21 F( 2,31,95) =3.30 F(10,23,97.5)=2.67

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84 Table 4-4. Error Analysis for Model Parameters of Coastal Bermudagrass and Pensacola Bahiagrass Grown at Thorsby, Alabama. Grass Irrigation Year Parameter Estimate Standard Error Relative Error Coastal Bermudagrass No 1957 A, Mg/ha 22.96 0.510 0.022 1958 A 22.37 0.497 0.022 1959 A 19.29 0.434 0.022 Yes 1957 A 22.44 0.496 0.022 1958 A 22.92 0.511 0.022 1959 A 24.89 0.554 0.022 Pensacola Bahiagrass No 1957 A 22.00 0.502 0.023 1958 A 22.85 0.526 0.023 1959 A 19.55 0.452 0.023 Yes 1957 A 23.48 0.542 0.023 1958 A 22.08 0.512 0.023 1959 A 22.54 0.521 0.023 Coastal Bermudagrass Both all b 1.39 0.0503 0.036 Pensacola Bahiagrass Both all b 1.58 0.0533 0.034 Both Both all c, ha/kg 0.0078 0.0002 0.026 Source: Original yield data from Evans et al. (1961).

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85 Table 4-5. Seasonal Dry Matter Yield for Coastal Bermudagrass and Pensacola Bahiagrass Grown at Thorsby, Alabama Averaged over Years 1957-1959. Applied Nitrogen, kg/ha Type Irrigation 0 168 336 672 Coastal Bermudagrass No 3.95 10.98 16.07 21.44 Yes 4.19 11.46 18.54 22.71 Pensacola Bahiagrass No 3.62 10.09 15.79 20.72 Yes 3.79 9.71 16.28 22.67 Source: Data from Evans el at. (1961).

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86 Table 4-6. Error Analysis for Model Parameters on Averaged Dry Matter Yield of Coastal Bermudagrass and Pensacola Bahiagrass Grown at Thorsby, Alabama. iype Trnoatinn 11 1 igaiiuii rai aiiivivi pctimatP Standard Error Relative Error Coastal Bermudagrass No A, Mg/ha 21.57 0.359 0.017 Yes A 23.44 0.388 0.017 Pensacola Bahiagrass No A 21.49 0.369 0.017 Yes A 22.73 0.395 0.017 Coastal Bermudagrass Both b 1.39 0.056 0.040 Pensacola Bahiagrass Both b 1.57 0.059 0.038 Both Both c, ha/kg 0.0078 0.0003 0.038 Source: Original data from Evans et al. (1961).

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Table 4-7. Analysis of Variance of Model Parameters Used to Describe Dallisgrass Grown at Baton Rouge, LA. Parameters -i Residual Sums Mean Sums Mode estimated df of Squares of Squares r (l)Com A,b,c 3 9 154655.03 17183.89 ~ (2) Ind A,b,c 6 6 82.72 13.79 (D-(2) 3 154572.31 51524.10 3736" (3) Ind A, Com b,c 4 8 3606.97 450.87 (3)-(2) 2 3524.25 1762.12 127.8" (4) Ind A,b Com c 5 7 122.54 17.51 (4)-(2) 1 39.82 39.82 2.89 (3H4) 1 3484.43 3484.43 199.0" Source: Original data from Robinson et al. (1988). "Significant at the 0.001 level F(3,6,99.9) 23.70 F(2,6,99.9) 27.00 F( 1,6,95) = 5.99 F( 1,7,99.9) = 29.25

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Table 4-8. Error Analysis for Model Parameters of Dallisgrass Grown at Baton Rouge, LA. Component Parameter Estimate Error R elativp Error Dry Matter A, Mg/ha 15.60 0.16 0.010 N Removal A, kg/ha 430.7 5.6 0.013 Dry Matter b 0.58 0.010 0.017 N Removal b 1.47 0.007 0.005 Both c, ha/kg 0.0055 0.0002 0.036 Source: Original data from Robinson et al. (1988).

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Table 4-9. Analysis of Variance of Model Parameters Used to Describe Bermudagrass Grown at T lorsby, AL with Two Clipping Interva s. Parameters Residual Sum Mean Sum Mode Estimated df of Squares of Squares F (l)Com A,b,c 3 21 1018614.201 48505.438 (2) Ind A,b,c 12 12 3368.695 280.725 (0-(2) 9 1015245.506 112805.056 402" (3) Ind A, Com b,c 6 18 11170.579 620.588 (3)-(2) 6 7801.885 1300.314 4.63 + (4) Ind A, b, Com c 9 15 4274.029 284.935 (4)-(2) 3 905.335 301.778 1.08 (3)-(4) 3 6896.550 2298.850 8.07 ++ (5) Ind A, Com c, Ind b (dm and Nu) 7 17 4261.737 250.690 (5)-(2) 5 893.042 178.608 0.64 (3H5) 1 6908.842 6908.842 27.56" Source: Original data from Doss et al. (1966). Significance level of 0.001 Significance level of 0.005 Significance level of 0.05 F(9, 12,99.9) =7.48 F(6, 12,95) =3.00 F(3, 12,95) =3.89 F(3, 15,99.5) =6.48 F(5, 12,95) =3.11 F( 1,1 7,99.9) = 15.72

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Table 4-10. Error Analysis of Model Parameters of Bermudagrass Grown at Thorsby, AL. Clipping Standard Relative Type Interval Parameter Estimate Error Error Dry Matter Yield 3.0 weeks A, Mg/ha 17.42 0.248 0.014 4.0 weeks A 19.75 0.279 0.014 N Removal 3.0 weeks A, kg/ha 568.9 8.83 0.016 4.0 weeks A 543.5 8.45 0.016 Both DM b 1.27 0.076 0.060 Both Nu b 2.02 0.095 0.047 Both Both c, ha/kg 0.0067 0.0003 0.045 Source: Original data from Doss et al. (1966).

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Table 4-11. Analysis of Variance of Model Parameters for Bermudagrass Grown at Maryland and Cut at Five harvest Intervals. Parameters Residual Sums Mean Sums Mode estimated df of Squares of Squares F (l)Com A,b,c 3 57 1419082.51 24896.18 (2) Ind A,b,c 30 30 8213.60 273.79 (l)-(2) 27 1410868.91 52254.40 190.9" (3) Ind A, Com b,c 12 48 28200.40 587.51 (3)-(2) 18 19986.80 1110.38 4.06 (4) Ind A,b Com c 21 39 13605.56 348.86 (4)-(2) 9 5391.96 599.11 2.19 (3)-(4) 9 14594.84 1621.65 4.65" (5) Ind A (yr, At), Ind b (y,Nu), Com c 13 47 13473.57 286.67 (5)-(2) 17 5259.97 309.41 1.13 (3)-(5) 1 14726.83 14726.83 51.4" Source: Original data from Decker et al. (1971). Significant at the 0.00 1 level F(27,30,99.9)=3.28 F(18,30,99.9)=3.58 F( 9,30,95) =2.21 F( 9,39,99.9)= 4.05 F( 17,30,95) =1.98 F( 1,47,99.9)= 12.32

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Table 4-12. Error Analysis for Model Parameters of Bermudagrass Grown at Maryland and Cut at Five Harvest Intervals. Harvest Standard Relative Component Interval Parameter Estimate Error Error Dry Matter 3.2, weeks A, Mg/ha 13.90 0.28 0.020 3.6 A 13.65 0.28 0.021 4.3 A 14.93 0.30 0.020 5.5 A 17.56 0.36 0.021 7.7 A 19.34 0.39 0.020 N Removal 3.2 A, kg/ha 469.5 10.4 0.022 3.6 A 444.7 9.9 0.022 4.3 A 440.2 9.8 0.022 5.5 A 444.0 9.9 0.022 7.7 A 409.7 9.0 0.022 Dry Matter All b 1.78 0.08 0.045 N Removal All b 2.59 0.10 0.039 Both All c, ha/kg 0.0112 0.0005 0.045 Source: Original data from Decker et al. (1971).

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93 Table 4-13. Analysis of Variance on Model Parameters for Bermudagrass Grown at Tifton, GA, over Two Years and Cut at Five Different Harvest Intervals. — — "5 1 Parameters Residual Sums Mean Sums \ t 1 Mode estimated dt ot Squares ot Squares c r ( 1 ) Com A,b,c 3 97 3326183.05 34290.55 (2) Ind Ab,c 60 40 OOC 1 C /^A / lz.yz (l)-(2) J / Jzy /666. J6 c."7Qc.i en j loj j. 80 o 1 .z (3) Ind A, Com b,c 22 78 79680.46 1 AO 1 C yl 1021.54 (3)-(2) 35 M 163. / / 1 1A A /lO 134o.4z 1 QOt i .sy (4) Ind Ab Com c A t 41 59 A A COO 1 44528.61 TO / 54. /z (4)-(2) 1 a iy i £Ai 1 no 1601 i.yz »4z. /J 110 1 . lo / A\ (3)-(4) 1 o iy 3M jI.oj 1 ojU. 1U Z.4j (5) Ind A (yr, At), Ind b (yr,y,Nu), Com c 0 £ ZJ "7C IJ fiO 4oO /O.oZ ikAQ no (5)-(2) 35 20160.13 C O/; AA 576.00 A O 1 0.81 (3)-(5) -*> 3 31003.64 10334.55 15. y /" C \ / A\ (5)-(4) 16 A 1 AO 11 4148.21 orn O/C zjy.z6 A 1/1 0.34 (6) Ind A (yr, At), Ind b (y,Nu), Com c 23 77 50414.37 654.73 (6)-(2) 37 21897.68 591.83 0.83 (3)-(6) 1 29266.09 29266.09 44.7 (6)-(4) 18 5885.76 326.99 0.43 (6)-(5) 2 1737.55 868.78 1.34 Source: Original data from Prine and Burton (1956). Significant at the 0.001 level f+ Significant at the 0.005 level f Significant at the 0.025 level F(57,40,99.9)=2.59 F(38,40,97.5)= 1.89 F( 19,40,95) = 1.85 F( 19,59,99.5)= 2.42 F(35,40,95) = 1.72 F( 3,75,99.9)= 6.01 F( 16,59,95) = 1.82 F(37,40,95) = 1.71 F( 1,77,99.9)= 11.71 F( 18,59,95) = 1.78 F( 2,75,95) =3.12

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Table 4-14. Error Analysis for Model Parameters of Bermudagrass Grown at Tifton, GA over Two Years and Cut at Five Different Harvest Intervals. Harvest Standard Relative Year Component Interval Parameter Estimate Error Error 1953 Dry Matter 2, weeks A, Mg/ha 17.95 0.52 0.029 3 A 19.88 0.57 0.029 4 A 23.15 0.66 0.029 6 A 29.57 0.85 0.029 8 A 29.58 0.85 0.029 1954 Dry Matter 2 A 8.40 0.24 0.029 3 A 9.95 0.28 0.028 4 A 11.67 0.33 0.028 6 A 14.37 0.41 0.029 8 A 16.24 0.46 0.028 1953 N Removal 2 A, kg/ha 644.7 20.4 0.032 3 A 641.8 20.2 0.031 4 A 687.2 21.5 0.031 o A A £88 0 Ooo.Z Z 1 .0 ft m i 8 A 606.0 19.0 0.031 1954 N Removal 2 A 324.5 10.1 0.031 j A j j i.j 1 u.o ft ft s 1 4 A 348.0 10.9 0.031 6 A 363.8 11.3 0.031 8 A 379.5 11.9 0.031 Both Dry Matter All b 1.47 0.060 0.041 Both N Removal All b 2.15 0.077 0.036 Both Both All c, ha/kg 0.0077 0.0003 0.039 Source: Original data from Prine and Burton (1956).

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95 Table 4-15. Analysis of Variance of Model Parameters on Ryegrass Grown at England, with Three Different Numbers of Cuttings over the Season for 1969. Parameters Residual Sums Mean Sums Mode estimated df of Squares of Squares F (l)Com A,b,c 3 123 1698556.455 13809.402 (2) Ind A,b,c 18 108 3703.415 34.291 0H2) 15 1694853.040 112990.203 3295 (3) Ind A, Com b,c 8 118 35982.858 304.939 (3)-(2) 10 32279.443 3227.944 94.1 (4) Ind A,b Com c 13 113 6635.747 58.723 (4)-(2) 5 2932.332 586.466 17.1 (3)-(4) 5 29347.111 5869.422 100 (5) Ind A Com c, Ind b (y and Nu) 9 117 8196.505 70.056 (5)-(2) 9 4493.090 499.232 14.6 (3)-(5) 1 27786.353 27786.353 397 (5)-(4) 4 1560.758 390.189 6.64" (6) Ind A, Com c, Ind b (y) and Com b (Nu) 11 115 6050.656 52.614 (6)-(2) 7 2347.241 335.32 9.8(3)-(6) 3 29932.202 9977.401 190" (5H6) 2 2145.849 1072.925 20.4" Source: Original data from Reid (1978). Significant at the 0.001 level F( 15, 108,99.9) = 2.81 F( 0,108,99.9) = 3.27 F( 5,108,99.9) = 4.45 F( 5,113,99.9) = 4.44 F( 9,108,99.9) = 3.41 F( 1,117,99.9) = 11.37 F( 4,113,99.9) = 4.97 F( 7,108,99.9) = 3.80 F( 3,115,99.9) = 5.80 F( 2,115,99.9) = 7.34

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96 Table 4-16. Error Analysis for Model Parameters of Ryegrass Grown at England, with Three Different Numbers of Cuttings over the Season for 1969. Number of Standard Relative rnmnnnpnt V/VJlllL/l/UVlll Cuttines V U I III l^yJ Parameter Estimate Error Error Dry Matter 10 A, Mg/ha 9.42 0.105 0.011 5 A 12.72 0.127 0.010 3 A 12.75 0.121 0.009 N Removal 10 A, kg/ha 377.0 4.72 0.013 5 A 403.5 5.02 0.012 3 A 363.0 4.46 0.012 Dry Matter 10 b 1.74 0.046 0.026 5 b 1.23 0.044 0.036 3 b 0.90 0.044 0.049 N Removal All b 2.15 0.050 0.023 Both All c, ha/kg 0.0080 0.0001 0.013 Source: Original data from Reid (1978).

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Table 4-17. Analysis of Variance for Bermudagrass over Three Seasons, Irrigated and Non-irrigated, Grown at Fayett eville, AR. Parameters Residual Sums Mean Sums Mode estimated df of Squares of Squares F (l)Com A,b,c 3 81 2114817.76 26108.86 (2) Ind A,b,c 36 48 17913.02 373.19 0M2) 33 2096904.75 63542.57 170.3** (3) Ind A, Com b,c 14 70 34746.83 496.38 (3)-(2) 22 16833.81 765.17 2.05 t (4) Ind A,b Com c 25 59 21706.70 367.91 (4)-(2) 11 3793.68 344.88 0.92 (3)-(4) 11 13040.13 1185.47 3.22 ++ (5) Ind A, Com c, Ind b (Irr, dm, Nu) 17 67 28178.64 420.58 (5>(2) 19 10265.63 540.30 1.45 (3)-(5) 3 6568.19 2189.40 5.21 ++ (5)-(4) 8 6471.95 808.99 2.20 + (6) Ind A Com c, Ind b (dm, Nu) 15 69 26492.41 383.95 (6)-(2) 21 8579.40 408.54 1.09 (3)-(6) 1 8254.42 8254.42 21.5** (6)-(4) 10 4785.72 478.57 1.30 Source: Original data from Huneycutt et al. (1988). Significant at the 0.001 level ' ' Significant at the 0.005 level t Significant at the 0.025 level Significant at the 0.05 level F(33,48,99.9)=2.66 1.97 1.99 2.S2 1.81 4.68 2.10 I. 78 II. 81 2.00 F(22,48,97.5)= F( 11,48,95) F(l 1,59,99.5)= F( 19,48,95) = F( 3,67,99.5)= F( 8,59,95) = F(2 1,48,95) = F( 1,69,99.9)= F( 10,59,95) =

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Table 4-18. Error Analysis for Model Parameters of Bermudagrass Grown at Fayetteville, AR, over Three Years, with and without Irrigation. Standard Relative Type Irrigation Year Parameter Estimate Error Error Dry Matter No 1983 A, Mg/ha 17.90 0.41 0.023 1984 A 17.37 0.39 0.022 1985 A 19.61 0.45 0.023 Yes 1983 A 24.70 0.56 0.023 1 CiO A 1984 A OA A A yj.jo 1985 A 22.58 0.51 0.023 N Removal No 1983 A, kg/ha 408.7 10.4 0.025 1 CiQ A A A A 1 1 Q in ^ U.UZ J 1985 A 459.3 11.8 0.026 Yes 1983 A 554.3 14.1 0.025 1984 A 523.7 13.4 0.026 1985 A 492.1 12.6 0.026 Dry Matter Both all b 1.50 0.063 0.042 N Removal Both all b 2.04 0.072 0.035 Both Both all c, ha/kg 0.0084 0.0003 0.036 Source: Original data from Huneycutt et al. (1988).

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Table 4-19. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Bermudagrass Grown at Fayetteville, AR Averaged over Three Years. Applied Nitrogen Dry Matter Yield N Removal N Concentration kg/ha Mg/ha kg/ha g/kg Non-Irrigated 0 2.00 26 12.80 112 7.50 117 15.52 224 10.83 199 18.45 336 14.00 274 19.63 448 17.53 380 21.71 560 17.29 386 22.35 672 17.71 428 24.11 Irrigated 0 3.48 51 14.51 112 9.65 157 16.27 224 13.74 253 18.40 336 19.22 359 18.72 448 21.48 415 19.31 560 22.89 482 21.07 672 23.80 532 22.35 Source: Original data from Huneycutt el al. (1988).

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100 Table 4-20. Analysis of Variance for Model Parameters for Bermudagrass Grown at Fayetteville, AR, Averaged over Three Years, with and without Irrigation. Parameters Residual Sums Mean Sums Mode estimated df of Squares of Squares F (l)Com A,b,c 3 25 695955.49 27838.22 (2) Ind A,b,c 12 16 3297.76 206.11 (1H2) 9 692657.73 76961.97 373.4 (3) Ind A Com b,c 6 22 6659.06 302.68 (3)-(2) 6 3361.30 560.22 2.72 (4) Ind A,b Com c 9 19 3898.95 205.21 (4)-(2) 3 601.19 200.40 0.97 (3)-(4) 3 3760.11 1253.37 6.11 ++ (5) Ind A Com c, Ind b (dm, Nu) 7 21 4745.07 225.96 (5)-(2) 5 1447.31 289.46 1.40 PH5) 1 1913.99 1913.99 8.47* (5)-(4) 2 846.12 423.06 2.06 Source: Original data from Huneycutt et al. (1988). Significant at the 0.001 level ++ Significant at the 0.005 level Significant at the 0.01 level F(9, 16,99.9) =5.98 F(6, 16,95) =2.74 F(3,16,95) =3.24 F(3, 19,99.5) =5.92 F(5,16,95) =2.85 F( 1,2 1,99) =8.02 F(2, 19,95) =3.52

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Table 4-21. Error Analysis for Model Parameters of Bermudagrass Grown at Fayetteville, AR, Averaged over Three Years, with and without Irrigation. Standard Relative ^oiiiponciu Irrigation Parameter Estimate Error Error Dry Matter No A, Mg/ha 1 8 l o.OJ yj.jj Yes A 24.40 0.45 0.018 N Removal No A, kg/ha 435.7 9.36 0.021 Yes A 534.8 11.58 0.022 Dry Matter Both b 1.51 0.076 0.050 N Removal Both b 2.04 0.087 0.043 Both Both c, ha/kg 0.0084 0.0004 0.048 Source: Original data from Huneycutt et al. (1988).

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Table 4-22. Analysis of Variance of Model Parameters for Tall Fescue Grown at Fayetteville, AR, over Three Seasons, with and wi thout Irrigation. Parameters Residual Sums Mean Sums Mode estimated df of Squares of Squares F (l)Com A,b,c 3 81 1401206.34 17298.84 (2) Ind A,b,c 36 48 5767.59 120.16 (D-(2) 33 1395438.76 42286.023 ** 352 (3) Ind A, Com b,c 14 70 25848.47 369.26 (3)-(2) 22 20080.88 912.77 7.60" (4) Ind A,b Com c 25 59 9085.81 154.00 (4)-(2) 11 3318.22 301.66 2.51 f (3)-(4) 11 16762.66 1523.88 9.90" (5) Ind A, Com c, Ind b (Irr, dm, Nu) 17 67 14215.61 212.17 (5)-(2) 19 8448.03 444.63 3.70" (3)-(5) 3 11632.86 3877.62 18.3" (5)-(4) 8 5129.80 641.23 4.16" (6) Ind A Com c, Ind b (dm, Nu) 69 14425.24 209.06 (6)-(2) 21 8657.66 412.27 3.43" (3)-(6) 1 11423.23 11423.23 565" (6)-(4) 10 5339.44 533.94 3.47 f (6)-(5) 2 209.63 104.82 0.49 Source: Original data from Huneycutt et al. (1988). Significant at the 0.001 level f Significant at the 0.025 level F(33,48,99.9)=2.66 F(22,48,99.9)=2.91 F(l 1,48,97.5)= 2.27 F(l 1,59,99.9)= 3.43 F( 19,48,99.9)= 3.02 F( 3,67,99.9)= 6.09 F( 8,59,99.9)= 3.88 F(21,48,99.9)=2.94 F( 1,69,99.9)= 11.81 F(10,59,97.5)=2.27 F( 2,67,95) =3.13

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103 Table 4-23. Error Analysis for Model Parameters of Tall Fescue Grown at Component Irrigation Year Parameter Estimate Standard Error Relative Error Dry Matter No 1981-2 A, Mg/ha 12.08 0.38 0.031 1982-3 A 8.01 0.25 0.031 1983-4 A 5.71 0.17 0.030 Yes 1981-2 A 15.63 0.49 0.031 1982-3 A 13.22 0.42 0.032 1983-4 A I*. 16.81 0.53 0.032 N Removal No 1981-2 A, kg/ha 357.5 12.5 0.035 1982-3 A 217.8 7.6 0.035 1983-4 A 169.5 5.7 0.034 Yes 1981-2 A 443.7 15.4 0.035 1982-3 A 357.3 12.6 0.035 1983-4 A 439.4 15.3 0.035 Dry Matter Both all b 0.92 0.084 0.091 N Removal Both all b 1.47 0.095 0.065 Both Both all c, ha/kg 0.0081 0.0005 0.062 Source: Original data from Huneycutt et al. (1988).

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104 Table 4-24. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Tall Fescue Grown at Fayetteville, AR, Averaged over Three Se asons. Applied Nitrogen Dry Matter Yield N Removal N Concentration kg/ha Mg/ha kg/ha g/kg Non-Irrigated 0 2.09 43 20.57 112 4.82 103 21.37 224 6.35 148 23.31 336 7.43 189 25.44 448 8.30 225 27.11 560 7.90 230 29.11 672 8.38 248 29.59 Irrigated 0 3.53 74 20.96 112 7.38 149 20.19 224 9.56 208 21.76 336 13.22 324 24.51 448 14.51 373 25.71 560 15.31 400 26.13 672 15.47 420 27.15 Source: Original data from Huneycutt et al. (1988).

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Table 4-25. Analysis of Variance of Model Parameters for Tall Fescue Grown at Fayetteville, AR, Averaged over Three Seasons, with and without Irrigation. Parameters — _ 7 Residual Sums Mean Sums Mode estimated df of Squares of Squares F ( 1 ) Com A,b,c 3 25 438490.94 17539.64 ~ (2) Ind A,b,c 12 16 860.75 53.80 0M2) 9 437630.19 48625.58 904 (3) Ind A, Com b,c 6 22 3908.10 177.64 (3H2) 6 3047.35 507.89 9.44 (4) Ind A,b Com c 9 19 999.28 52.59 ~ (4)-(2) 3 138.53 46.18 0.86 (3)-(4) 3 2908.82 969.61 18.4" (5) Ind A, Com c, Indb (dm, Nu) 7 21 1437.08 68.43 (5>(2) 5 576.32 115.26 2.14 (3H5) 1 2471.02 2471.02 36. r (5)-(4) 2 437.80 218.90 4.16 + Source: Original data from Huneycutt et al. (1988). Significant at the 0.001 level Significant at the 0.05 level F(9, 16,99.9) =5.98 F(6, 16,99.9) =6.81 F(3, 16,95) =3.24 F(3, 19,99.9) =8.28 F(5,16,95) =2.85 F(l,21,99.9) =14.59 F(2, 19,95) =3.52

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106 Table 4-26. Error Analysis for Model Parameters of Tall Fescue Grown at Fayetteville, AR, Averaged over Three Seasons, with and without Irrigation . Component Irrigation Parameter Estimate Standard Error Relative Error Dry Matter No A, Mg/ha 8.67 0.15 0.017 Yes A 15.36 0.27 0.018 N Removal No A, kg/ha 250.7 4.95 0.020 Yes A 417.4 8.34 0.020 Dry Matter Both b 0.99 0.069 0.070 N Removal Both b 1.53 0.079 0.052 Both Both c, ha/kg 0.0081 0.0004 0.049 Source: Original data from Huneycutt et al. (1988).

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107 Table 4-27. Analysis of Variance for Bermudagrass Grown at Eagle Lake, Texas, over Two Years. Parameters Residual Sums Mean Sums Mode estimated df of Squares of Squares F (l)Com A,b,c 3 17 110513.95 6500.821 (2) Ind A,b,c 12 8 233.901 29.238 0M2) 9 110280.052 12253.339 419 (3) Ind A, Com b,c 6 14 1851.481 132.249 (3)-(2) 6 1617.580 269.597 9.22 (4) Ind A,b Com c 9 11 370.923 33.72 (4)-(2) 3 137.021 45.674 1.56 (3)-(4) 3 1480.558 493.519 14.6" (5) Ind A, Com c, Ind b (dm and Nu) 7 13 1069.846 82.296 (5)-(2) 5 835.945 167.189 5.72 f (3)-(5) 1 781.635 781.635 9.50* (5)-(4) 2 698.923 349.462 10.4 ++ Source: Original data from Evers (1984). Significant at the 0.001 level ++ Significant at the 0.005 level y Significant at the 0.025 level Significant at the 0.01 level F(9, 8,99.9) =11.77 F(6, 8,99.5) =7.95 F(3, 8,95) =4.07 F(3,l 1,99.9) =11.56 F(5, 8,97.5) =4.82 F( 1,13,99) =9.07 F(2, 11,99.5) =8.91

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108 Table 4-28. Analysis of Variance for Bahiagrass Grown at Eagle Lake, Texas, over Two Years. Parameters Residual Sums Mean Sums Mode estimated dt of Squares or Squares r (l)Com A,b,c 3 17 68411.328 4024.196 (2) Ind A,b,c 12 8 184.755 23.094 0)-(2) 9 68226.573 7580.73 TOO** 328 (3) Ind A, Com b,c 6 14 1 507.231 107.659 (3)-(2) 6 1322.476 220.413 9.54 (4) Ind A,b Com c 9 11 346.949 31.541 (4)-(2) 3 162.193 54.064 2.34 (3)-(4) 3 1160.282 386.761 12.3" (5) Ind A, Com c, Ind b (dm and Nu) 7 13 634.002 48.769 (5)-(2) 5 449.247 89.849 3.89 + (3)-(5) 1 873.229 873.229 17.9" (5)-(4) 2 287.054 143.527 4.55 + Source: Original data from Evers (1984). Significant at the 0.001 level ++ Significant at the 0.005 level Significant at the 0.05 level F(9, 8,99.9) =11.77 F(6, 8,99.5) =7.95 F(3, 8,95) =4.07 F(3, 11,99.9) =11.56 F(5, 8,95) =3.69 F(l, 13,99.9) =17.81 F(2, 11,95) =3.98

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109 Table 4-29. Analysis of Variance for Bermudagrass and Bahiagrass Grown at Eagle Lake, Texas, over Two Years. Parameters Residual Sums Mean Sums Mode estimated df of Squares of Squares F (l)Com A,b,c 3 37 181775.297 4912.846 (2) Ind A,b,c 24 16 418.657 26.166 (0-(2) 21 181356.641 8636.031 mm 330 (3) Ind A, Com b,c 10 30 3209.170 106.972 (3)-(2) 14 2790.514 199.322 7.62" (4) Ind A,b Com c 17 23 814.463 35.411 (4)-(2) 7 395.806 56.544 2.16 (3)-(4) 7 2394.707 342.101 mm 9.66 (5) Ind A, Com c, Ind b (dm and Nu) 11 29 2058.255 70.971 (5)-(2) 13 1639.598 126.123 4.82 ++ (3)-(5) 1 1150.915 1150.915 16.2" (5)-(4) 6 1243.792 207.299 5.85" (6) Ind A, Com c, Ind b (grass, dm and Nu) 13 27 1746.733 64.694 (6)-(2) 11 1328.076 120.730 4.6r (3H6) 3 1462.437 487.479 7.54" (6)-(4) 4 932.270 233.067 6.58 ++ (5)-(6) 2 311.522 155.761 2.41 Source: Original data from Evers (1984). Significant at the 0.001 level ++ Significant at the 0.005 level F(2 1,1 6,99.9)= 4.95 F( 14, 16,99.9)= 5.35 F( 7,16,95) = 2.66 F( 7,23,99.9)= 5.33 F(13,16,99.5)= 4.03 F( 1,29,99.9)= 13.39 F( 6,23,99.9)= 5.65 F(l 1,16,99.5)= 4.18 F( 3,27,99.9)= 7.27 F( 4,23,99.5)= 4.95 F( 2,27,95) = 3.35

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1 10 Table 4-30. Averaged Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for Bermudagrass and Bahiagrass Grown at Eagle L ake, TX. Applied Nitrogen Dry Matter Yield 1 N Removal 2 N Concentration kg/ha Mg/ha kg/ha g/kg . Coastal bermudagrass 0 4.05 60 14.81 84 6.07 103 16.97 168 7.91 136 17.19 252 9.62 190 19.75 336 10.83 218 20.13 — Pensacola bahiagrass— 0 3.79 60 15.83 84 5.17 84.5 16.34 168 6.37 115.5 18.13 252 7.27 146.5 20.15 336 8.17 168 20.56 Source: Original data from Evers (1984). 1 Averaged over years 1978-1980. 2 Averaged over years 1979-1980.

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Table 4-31. Analysis of Variance on Model Parameters for Bermudagrass and Bahiagrass Grown at Eagle Lake, TX, Averaged to Estimate b and c Parameters. Parameters T* * J 1 C Residual Sums Mean Sums Mode A.' J estimated dr or Squares or Squares r (l)Com A,b,c 3 17 86802.271 5106.016 (2) Ind A,b,c 12 8 97.371 12.171 0M2) 9 86704.900 9633.878 792 (3) Ind A, Com b,c 6 14 666.562 47.612 (3)-(2) 6 569.191 94.865 7.79 (4) Ind A,b Com c 9 11 110.820 10.075 (4)-(2) 3 13.449 4.483 0.37 (3)-(4) 3 555.742 185.247 18.4 (5) Ind A, Com c, Ind b (dm and Nu) 7 13 251.797 19.369 (5H2) 5 154.426 30.885 2.54 (3)-(5) 1 414.765 414.765 21.4" (5H4) 2 140.977 70.488 7.00 f Source: Original data from Evers (1984). Significant at the 0.001 level ++ Significant at the 0.005 level 1 Significant at the 0.025 level F(9, 8,99.9) =11.77 F(6, 8,99.5) =7.95 F(3, 8,95) =4.07 F(3,l 1,99.9) =11.56 F(5, 8,95) =3.69 F(l, 13,99.9) =17.81 F(2,l 1,97.5) =5.26

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Table 4-32. Error Analysis for Model Parameters of Bermudagrass and Bahiagrass Grown at Eagle Lake, TX Averaged over Years. Standard Relative Component Grass Parameter Estimate Error Error Dry Matter Yield bermudagrass A, Mg/ha 12.16 0.442 0.036 bahiagrass A 9.56 0.346 0.036 N Removal bermudagrass A, kg/ha 271.2 14.7 0.054 bahiagrass A 216.0 11.7 0.054 Dry Matter Yield Both b 0.55 0.062 0.113 N Removal Both b 1.11 0.071 0.064 c, ha/kg 0.0072 0.0005 0.069 Both Both Source: Original data from Evers (1984).

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113 Table 4-33. Error Analysis for Model Parameters of Bermudagrass and Bahiagrass Grown at Eagle Lake, TX over Two Years. Component Grass Year Parameter Estimate Standard Error Relative Error Yield bermudagrass 1979 A, Mg/ha 15.99 0.820 0.051 1 080 A r\ y . oy 0 051 bahiagrass 1979 A 12.42 0.634 0.051 1980 A 7.51 0.382 0.051 iy IxCUlUVdl UCI IllUUclgl nos 1070 17/7 rV, Kg/ 1 la lino 11 61 o 07i 1980 A 227.4 16.49 0.073 bahiagrass 1979 A 237.2 17.18 0.072 1980 A 191.3 13.87 0.073 Yield Both All b 0.57 0.082 0.144 N Removal Both All b 1.07 0.094 0.088 Both Both All c, ha/kg 0.0072 0.0007 0.097 Source: Original data from Evers (1984).

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Table 4-34. Analysis of Variance on Model Parameters for Corn Grown on Dothan Sandy Loam at Clayton, Is C, Both Grain and Tota Parameters Residual Sum Mean Sum Mode Estimated df of Squares of Squares F (l)Com A,b,c 3 17 63333.848 3725.520 (2) Ind A,b,c 12 8 33.372 4.172 (l)-(2) 9 63300.476 7033.386 1686 (3) Ind A, Com b,c 6 14 586.733 41.910 (3)-(2) 6 553.360 92.227 ** 22.1 (4) Ind A, b, Com c 9 11 138.832 12.621 (4)-(2) 3 105.460 35.153 • 8.43 (3)-(4) 3 447.901 149.300 49.8" (5) Ind A, Com c, Ind b (dm and Nu) 7 13 146.647 11.281 (5)-(2) 5 113.275 22.655 5.43 f (3)-(5) 1 440.086 440.086 39.0" (5)-(4) 2 7.815 3.908 0.31 Source: Original data from Kamprath (1986). Significance level of 0.001 Significance level of 0.01 t Significance level of 0.025 F(9, 8,99.9) =11.77 F(6, 8,99.9) =12.86 F(3, 8,99) =7.59 F(3,l 1,99.9) =11.56 F(5, 8,97.5) =4.82 F(l, 13,99.9) =17.81 F(2,ll,95) =3.98

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115 Table 4-35. Analysis of Variance on Model Parameters for Corn Grown on Goldsboro Sandy Loam at Kinston, NC, Both Grain and Total. Parameters Residual Sum Mean Sum Mode Estimated df or Squares or Squares b (l)Com A,b,c 3 17 34443.380 2026.081 (2) Ind A,b,c 12 8 15.942 1.993 (l)-(2) 9 34427.438 3825.271 1920 (3) Ind A, Com b,c 6 14 337.763 24.126 (3)-(2) 6 321.821 53.637 _ , _** 26.9 (4) Ind A, b, Com c 9 11 31.448 2.859 (4)-(2) 3 15.506 5.169 2.59 (3)-(4) 3 306.315 102.105 _ 35.7 (5) Ind A Com c, Ind b (dm and Nu) 7 13 45.635 3.510 (5)-(2) 5 29.694 5.939 2.98 (3)-(5) 1 292.128 292.128 83.2" (5)-(4) 2 14.187 7.094 2.48 Source: Original data from Kamprath (1986). Significance level of 0.001 F(9, 8,99.9) =11.77 F(6, 8,99.9) =12.86 F(3, 8,95) =4.07 F(3, 11,99.9) =11.56 F(5, 8,95) =3.69 F(l, 13,99.9) =17.81 F(2, 11,95) =3.98

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116 Table 4-36. Analysis of Variance on Model Parameters for Grain and Total Plant of Corn Grown on Dothan and Goldsboro Sandy Loam at Clayton and Kinston, NC. Parameters Residual Sum Mean Sum Mode Estimated df of Squares of Squares F (l)Com A,b,c 3 37 101127.594 2733.178 (2) Ind A,b,c 24 16 49.314 3.082 (l)-(2) 21 101078.280 4813.251 1562 (3) Ind A, Com b,c 10 30 1117.935 37.265 (3)-(2) 14 1068.620 76.33 ** 24.8 (4) Ind A, b, Com c 17 23 317.411 13.800 (4)-(2) 7 268.096 38.299 12.4 (3)-(4) 7 800.524 114.361 *• 8.29 (5) Ind A, Com c, Ind b (site: dm and Nu) 13 27 350.409 12.978 (5H2) 11 301.095 27.372 8.88" (3>(5) 3 767.526 255.842 19.7" (5H4) 4 32.998 8.250 0.60 (6) Ind A, Com c, Ind b (dm and Nu) 11 29 334.633 11.539 (6)-(2) 13 285.319 21.948 ** 7.12 (3)-(6) 1 783.302 783.302 67.9 (6)-(4) 6 17.222 2.870 0.21 (7) Ind A, Com c, Ind b (part: dm and Nu) 13 27 325.903 12.070 (7)-(2) 11 276.589 25.144 8.16" (3)-(7) 3 792.032 264.011 21.9" (7)-(4) 4 8.492 2.123 0.15 (6)-(7) 2 8.730 4.365 0.36 Source: Original data from Kamprath (1986). " Significance level of 0.001 F(21,16,99.9)=4.95 F(14, 16,99.9)= 5.35 F( 7,16,99.9)= 6.46 F( 7,23,99.9)= 5.33 F(l 1,16,99.9)= 5.67 F( 3,27,99.9)= 7.27 F( 4,23,95) =2.80 F( 1,29,99.9)= 13.39 F( 6,23,95) =2.53 F(13,16,99.9)=5.44 F( 2,27,95) =3.35

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117 Table 4-37. Error Analysis of Model Parameters for Grain and Total Plant of Corn Grown on Dothan and Goldsboro Sandy Loam at Clayton and Kinston, NC. Standard Relative Component Site Part Parameter Estimate Error Error Dry Matter Yield Dothan Grain A, ivig/na 11 19 1 1 . 1 z n i U. 1 JO 0 014 Total A 20.68 0.288 0.014 Goldsboro Grain A 7.83 0.109 0.014 total A 1 A 70 14. I\J O 904 0 014 N Removal Dothan Grain A, kg/ha 151.7 2.71 0.018 Total A 187.2 3.35 0.018 Goldsboro Grain A 113.3 2.00 0.018 Total A 136.5 2.42 0.018 Dry Matter Yield Both Both b 0.27 0.040 0.148 N Removal Both Both b 0.97 0.044 0.045 Both Both Both c, ha/kg 0.0187 0.0007 0.037 Source: Original data from Kamprath (1986).

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118 Table 4-38. Analysis of Variance on Model Parameters for Grain and Total Plant of Corn Grown on Portsmouth Very Fine Sandy Loam at Plymouth NC. Parameters Residual Sum Mean Sum Mode Estimated df of Squares or Squares r (l)Com A,b,c 3 17 37148.680 2185.216 (2) Ind A,b,c 12 8 5.674 0.709 0)-(2) 9 37143.006 4127.001 5819 (3) Ind A, Com b,c 6 14 144.076 10.291 (3)-(2) 6 138.402 23.067 32.5 (4) Ind A b, Com c 9 11 16.808 1.528 (4)-(2) 3 11.134 3.711 5.23 + (3)-(4) 3 127.268 42.423 27.8" (5) Ind A, Com c, Ind b (dm and Nu) 7 13 26.613 2.047 (5)-(2) 5 20.939 4.188 5.91 f (3)-(5) 1 117.463 117.463 57.4" (5)-(4) 2 9.796 4.898 3.21 Source: Original data from Kamprath (1986). " Significance level of 0.001 + Significance level of 0.025 Significance level of 0.05 F(9, 8,99.9) =11.77 F(6, 8,99.9) =12.86 F(3, 8,95) =4.07 F(3,l 1,99.9) =11.56 F(5, 8,97.5) =4.82 F(l, 13,99.9) =17.81 F(2,ll,95) =3.98

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119 Table 4-39. Error Analysis for Model Parameters for Grain and Total Plant of Corn Grown on Portsmouth Very Fine Sandy Loam at Plymouth, NC. Component Part Parameter Estimate Standard Error Relative Error Uly lvlallcl I 1C1U vJl dill A Ma/no /\, IVlg/Ila 0 48 0 1 f,R DDIS Total A 16.60 0.293 0.018 N Removal Grain A, kg/ha 126.2 3.22 0.026 Total A 147.3 3.76 0.026 Dry Matter Yield Both b -0.065 0.0458 0.705 N Removal Both b 0.46 0.047 0.102 Both Both c, ha/kg 0.0119 0.0007 0.059 Source: Original data from Kamprath (1986).

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Table 4-40. Analysis of Variance on Model Parameters for Bahiagrass Grown on Two Soils: an Entisol and Spodosol at Williston and Gainesville, FL, Respectively. Parameters Residual Sum Mean Sum Mode Estimated df of Squares of Squares F (l)Com A,b,c 3 17 127269.607 7486.447 (2) Ind A,b,c 12 8 63.094 7.887 (l)-(2) 9 127206.513 14134.057 1792" (3) Ind A, Com b,c 6 14 642.094 45.864 (3)-(2) 6 579.000 96.500 12.24 ++ (4) Ind A, b, Com c 9 11 233.578 21.234 (4)-(2) 3 170.484 56.828 7.21* (3)-(4) 3 408.514 136.171 6.41* (5) Ind A Com c, Ind b (dm and Nu) 7 13 485.676 37.360 (5)-(2) 5 422.582 84.516 10.72^ (3)-(5) 1 156.418 156.418 4.19 (5)-(4) 2 252.098 126.049 5.94 f Source: Original data from Blue (1987). Significant at the 0.001 level Significant at the 0.005 level Significant at the 0.01 level f Significant at the 0.025 level F(9, 8,99.9) =11.77 F(6, 8,99.5) =7.95 F(3, 8,97.5) =5.42 F(3, 11,99) =6.22 F(5, 8,99.5) =8.30 F(l, 13,95) =4.67 F(2, 11,97.5) =5.26

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121 Table 4-41. Error Analysis for Model Parameters for Bahiagrass Grown on Two Soils: an Entisol and Spodosol at Williston and Gainesville, FL, Respectively. Standard Relative Component Soil Parameter Estimate Error Error Dry Matter Yield Entisol A, Mg/ha 11.14 0.176 0.016 Spodosol A 19.39 0.307 0.016 N Removal Entisol A, kg/ha 311.0 5.84 0.019 Spodosol A 201.5 3.76 0.019 Dry Matter Yield Both b 1.39 0.053 0.038 N Removal Both b 1.86 0.060 0.032 Both Both c, ha/kg 0.0118 0.0004 0.034 Source: Original data from Blue (1987).

PAGE 145

122 Table 4-42. Analysis of Variance on Model Parameters for Seasonal Dry Matter Yield and Plant N Removal of Ryegrass Grown on 20 Different Sites in England. Parameters Residual Sums Mean Sums Mode estimated df of Squares of Squares F (l)Com A,b,c 3 237 5659492.221 23879.714 (2) Ind A,b,c 120 120 12902.124 107.518 (D-(2) 117 5646590.097 48261.454 449 (3) Ind A, Com b,c 42 198 130080.207 656.971 (3)-(2) 78 117178.083 1502.283 14.0 (4) Ind A, b, Com c 81 159 32739.137 205.907 (4)-(2) 39 19837.013 508.641 4.73" (3)-(4) 39 97341.070 2495.925 12.1" (5) Ind A Com c, Ind b (dm and Nu) 43 197 90106.618 457.364 (5)-(2) 79 77204.494 977.272 9.09" (3)-(5) 1 39973.589 39973.589 87.4" (5)-(4) 38 57367.481 1509.671 7.33" Source: Original data from Morrison et al. (1980). " Significant at the 0.001 level F(l 17,120,99.9) = 1.77 F( 78,120,99.9)= 1.87 F( 39, 120,99.9) = 2. 12 F( 39,159,99.9) = 2.06 F( 79,120,99.9) = 1.87 F( 1,197,99.9)= 11.16 F( 38,159,99.9) =2.07

PAGE 146

123 Table 4-43. Error Analysis of Model Parameters for Seasonal Dry Matter Yield and Plant N Removal of Ryegrass Grown on 20 Different Sites in England. Site Component Parameter Estimate Standard Error Relative Error 5 Dry Matter A, Mg/ha 13.84 0.223 0.016 6 A 13.90 0.231 0.017 7 A 8.39 0.146 0.017 8 A 11.80 0.243 0.021 9 A 14.82 0.236 0.016 10 A 12.87 0.229 0.018 12 A 13.83 0.330 0.024 13 A 12.85 0.209 0.016 14 A 13.00 0.228 0.018 15 A 10.63 0.219 0.021 16 A 13.36 0.217 0.016 17 A 10.93 0.201 0.018 19 A 6.37 0.114 0.018 20 A 14.11 0.276 0.020 22 A 10.32 0.196 0.019 23 A 10.25 0.187 0.018 25 A 12.67 0.225 0.018 26 A 10.38 0.215 0.021 27 A 10.55 0.187 0.018 28 A 10.97 0.250 0.023 5 N Removal A, kg/ha 469.7 8.42 0.018 6 A 537.2 10.47 0.019 7 A 334.6 6.23 0.019 8 A 423.2 10.67 0.025 9 A 518.0 9.00 0.017 10 A 446.8 8.82 0.020 12 A 454.7 12.45 0.027 13 A 488.0 8.84 0.018 14 A 466.0 9.49 0.020 15 A 381.5 9.68 0.025 16 A 478.7 8.83 0.018 17 A 370.6 8.68 0.023 19 A 222.2 4.49 0.020 20 A 450.4 10.69 0.024 22 A 368.7 7.82 0.021 23 A 356.1 7.37 0.021 25 A 458.6 9.55 0.021 26 A 355.0 8.68 0.024 27 A 384.0 8.22 0.021 28 A 365.4 9.85 0.027

PAGE 147

124 Table 4-43 —continued Site Component Parameter Estimate Standard Error Relative Error 5 Dry Matter b 0.34 0.117 0.344 6 b 0.65 0.106 0.163 7 b 0.98 0.099 0.101 8 b 1.68 0.101 0.060 9 b 0.38 0.110 0.289 10 b 1.04 0.102 0.098 12 b 2.10 0.110 0.052 13 b 0.52 0.108 0.208 14 b 0.93 0.103 0.111 15 b 1.60 0.107 0.067 16 b 0.58 0.103 0.178 17 b 1.64 0.090 0.055 19 b 1.09 0.101 0.093 20 b 1.38 0.108 0.078 22 b 1.31 0.102 0.078 23 b 1.14 0.102 0.089 25 b 0.91 0.111 0.122 26 b 1.59 0.109 0.069 27 b 0.96 0.106 0.110 28 b 1.92 0.111 0.058 5 N Removal b 1.19 0.121 0.102 6 b 1.68 0.113 0.067 7 b 1.57 0.106 0.068 8 b 2.55 0.117 0.046 9 b 1.13 0.113 0.100 10 b 1.80 0.108 0.060 12 b 2.77 0.121 0.044 13 b 1.39 0.110 0.079 14 b 1.78 0.117 0.066 15 b 2.47 0.123 0.050 16 b 1.50 0.109 0.073 17 b 2.39 0.111 0.046 19 b 1.90 0.107 0.056 20 b 2.29 0.123 0.054 22 b 2.03 0.110 0.054 23 b 1.93 0.111 0.058 25 b 1.86 0.119 0.064 26 b 2.45 0.118 0.048 27 b 1.91 0.120 0.063 28 b 2.69 0.124 0.046

PAGE 148

125 Table 4-43~continued Site Type Parameter Estimate Standard Error Relative Error All Both c, ha/ka 0.0088 0.0001 0.011 Source: Original data from Morrison et al. (1980).

PAGE 149

126 Table 4-44. Summary of Model Parameters for Ryegrass in England. R 1 A A„ A„/A b b„ Ab Site cm Mg/ha kg/ha g/kg 5 31.0 13.84 469.7 33.9 0.34 1.19 0.85 6 33.2 13.90 537.2 38.7 0.65 1.68 1.03 7 29.8 8.39 334.6 39.9 0.98 1.57 0.59 8 32.5 11.80 423.2 35.9 1.68 2.55 0.87 9 46.0 14.82 518.0 35.0 0.38 1.13 0.75 10 52.0 12.87 446.8 34.7 1.04 1.80 0.76 12 47.0 13.83 454.7 32.9 2.10 2.77 0.67 13 40.0 12.85 488.0 38.0 0.52 1.39 0.87 14 32.2 13.00 466.0 35.9 0.93 1.78 0.85 15 28.6 10.63 381.5 35.9 1.60 2.47 0.87 16 35.5 13.36 478.7 35.8 0.58 1.50 0.92 17 33.4 10.93 370.6 33.9 1.64 2.39 0.75 19 23.6 6.37 222.2 34.9 1.09 1.90 0.81 20 33.7 14.11 450.4 31.9 1.38 2.29 0.91 22 32.3 10.32 368.7 35.7 1.31 2.03 0.72 23 30.2 10.25 356.1 34.7 1.14 1.93 0.79 25 25.5 12.67 458.6 36.2 0.91 1.86 0.95 26 31.3 10.38 355.0 34.2 1.59 2.45 0.86 27 29.6 10.55 384.0 36.4 0.96 1.91 0.95 28 27.9 10.97 365.4 33.3 1.92 2.69 0.77 c ha/kg 0.0088 20 sites 35.4 ±1.9 0.83 ±0.10 Source: Original data from Morrison ef al. (1980). 1 Annual rainfall.

PAGE 150

127 Table 4-45. Summary of Model Parameters, Standard Errors, and Relative Errors for the Extended Triple Logistic (NPK) Model for Rye Grown at Tifton, GA. Parameter Dry Matter N Removal P Removal K Removal 5430 1 ?6f) 34 930 A, kg/ha 1 Q9 2 1 11.4 1.00 11.3 0.035 3 0.044 0.029 0.049 1.36 1 en i .yj 1 16 1 . J u b„ u. i jy 0.166 0.139 0.139 0.102 0.086 0.102 0.102 -0.14 0 14 -U. IU 0 14 -V) . 1 H bp U. 1 J\J 0.150 0.162 0.150 1.071 1.071 1.012 1.071 -0.91 -0 91 -0 91 0 46 bk 0 1QQ u. i yy 0.199 0.199 0.130 0.219 0.219 0.219 0.283 0.0225 c„, ha/kg 0.0021 0.0021 0.0021 0.093 0.093 0.093 0.093 0.0464 0.0464 0.0464 0.0464 0.0081 0.0081 0.0081 0.0081 0.175 0.175 0.175 0.175 0.0201 0.0201 0.0201 0.0201 Ck 0.0039 0.0039 0.0039 0.0039 0.194 0.194 0.194 0.194 Source: Original data from Walker and Morey (1962). 1 Estimate 2 Standard Error ? Relative Error

PAGE 151

128 Figure 4-1 Response of seasonal dry matter yield for Coastal bermudagrass and Pensacola bahiagrass to applied N at Thorsby, AL. Data from Evans et al. (1961); curves drawn from Eq. [4.1] through [4.4].

PAGE 152

129 0 5 10 15 20 25 Measured Dry Mailer Yield, Mg/lia Figure 4-2 Scatter plot of seasonal dry matter yield for Coastal bermudagrass and Pensacola bahiagrass at Thorsby, AL. Original data from Evans et al. (1961).

PAGE 153

130 w I 2 y — i o i -i O Coastal bermudagrass, Non-irrigated Coastal bermudagrass, Irrigated A Pensacola bahiagrass. Non-irrigated V Pensacola bahiagrass, irrigated o v A A 0 A n V 5 10 15 Estimated Dry Matter Yield, Mg/lia 20 25 Figure 4-3 Residual plot of seasonal dry matter yield for Coastal bermudagrass and Pensacola bahiagrass at Thorsby, AL. Original data from Evans et al. (1961). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 154

131 0 200 400 600 800 1000 Applied Nitrogen, kg/ha Figure 4-4 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for dallisgrass grown at Baton Rouge, LA. Data from Robinson et al (1988); curves drawn from Eq. [4.5] through [4.7].

PAGE 155

132 Figure 4-5 Seasonal dry matter yield and plant N concentration as a function of plant N removal for dallisgrass grown at Baton Rouge, LA. Data from Robinson etal. (1988); curves drawn from Eq. [4.8] and [4.9].

PAGE 156

133 1.2 I 1 1 1 1 r 0.0 0.2 0.4 0.6 0.8 1.0 1.2 N Removal/Estimated Maximum Figure 4-6 Dimensionless plot of dry matter and plant N concentration as a function of plant N removal for dallisgrass grown at Baton Rouge, LA. Original data from Robinson et al. (1988); curves drawn from Eq. [4. 10] and [4. 1 1].

PAGE 157

0 5 10 15 20 Measured Dry Matter Yield, Mg/ha ;ure 4-7 Scatter plot of dry matter yield for dallisgrass grown at Baton Rouge Original data from Robinson el al. (1988).

PAGE 158

135 0 50 100 150 200 250 300 350 400 450 Measured N Removal, kg/ha Figure 4-8 Scatter plot of plant N removal for dallisgrass grown at Baton Rouge, LA. Original data from Robinson et at. (1988).

PAGE 159

136 O ... o o -QO 5 10 15 Predicted Dry Mailer Yield, Mg/lia 20 Figure 4-9 Residual plot of dry matter yield for dallisgrass grown at Baton Rouge, LA. Original data from Robinson et al. (1988). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 160

137 O o o o o o 50 100 150 200 250 300 Predicted N Removal, kg/ha 350 400 450 Figure 4-10 Residual plot of plant N removal for dallisgrass grown at Baton Rouge, LA. Original data from Robinson et al. (1988). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 161

138 Figure 4-11 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for bermudagrass grown at Thorsby, AL and cut at two harvest intervals. Data from Doss et al. (1966); curves drawn from Eq. [4. 12] through [4. 1 7].

PAGE 162

139 Figure 4-12 Seasonal dry matter yield and plant N concentration as a function of N removal for bermudagrass grown at Thorsby, AL and cut at two harvest intervals. Data from Doss et al (1966); curves drawn from Eq. [4.18] through [4.21].

PAGE 163

140 Figure 4-13 Dimensionless plot of dry matter and plant N concentration as a function of plant N removal for bermudagrass grown at Thorsby, AL and cut at two harvest intervals. Original data from Doss et al. (1966); curves drawn fromEq. [4.22] and [4.23].

PAGE 164

141 Figure 4-14 Scatter plot of dry matter yield for bermudagrass grown at Thorsby, AL and cut at two harvest intervals. Original data from Doss et al. (1966).

PAGE 165

142 Figure 4-15 Scatter plot of plant N removal for bermudagrass grown at Thorsby, AL and cut at two harvest intervals. Original data from Doss el al. (1966).

PAGE 166

143 1 O 3.0 weeks 4.5 weeks JD 6 — -e 10 15 20 25 Predicted Dry Matter Yield, Mg/lia Figure 4-16 Residual plot of dry matter yield for bermudagrass grown at Thorsby, AL and cut at two harvest intervals. Original data from Doss et al. (1966). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 167

144 100 75 0 3.0 weeks 4.5 weeks 50 25 i« ~25 O 8 O O ED n o -50 -75 100 100 200 300 400 Predicted N Removal, kg/ha 500 600 Figure 4-17 Residual plot of plant N removal for bermudagrass grown at Thorsby, AL and cut at two harvest intervals. Original data from Doss et al. (1966). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 168

145 40 0 200 400 600 800 1000 Applied Nitrogen, kg/ha Figure 4-18 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for bermudagrass grown at Maryland and cut at five harvest intervals. Data from Decker el al. (1971); curves drawn from Eq. [4.24] through [4.38].

PAGE 169

146 Nitrogen Removal, kg/ha Figure 4-19 Seasonal dry matter yield and plant N concentration as a function of plant N removal for bermudagrass grown at Maryland and cut at five harvest intervals. Data from Decker et al. (1971); curves drawn from Eq. [4.39] through [4.48].

PAGE 170

147 1 -a a I .2 fa I U T3 g S g 0.4 0.6 0.8 N Removal/Eslimalcd Maximum Figure 4-20 Dimensionless plot of dry matter and plant N concentration as a function of plant N removal for bermudagrass grown at Maryland and cut at five harvest intervals. Original data from Decker el al. (1971); curves drawn from Eq. [4.49] and [4.50].

PAGE 171

148 0 5 10 15 20 Dry Matter Yield, Mg/lia Figure 4-21 Scatter plot of dry matter yield for bermudagrass grown at Maryland and cut at five harvest intervals. Original data from Decker et al. (1971).

PAGE 172

149 Figure 4-22 Scatter plot of plant N removal for bermudagrass grown at Maryland and cut at five harvest intervals. Original data from Decker et al. (1971).

PAGE 173

150 2 1 > u i o a 3 •p a 0 A V 3.2 weeks 3.6 weeks 4.3 weeks 5.5 weeks 7.7 weeks V Q A O -O-AO ) A V7 o 10 15 Predicted Dry Matter Yield, Mg/ha 20 25 Figure 4-23 Residual plot of dry matter yield for bermudagrass grown at Maryland and cut at five harvest intervals. Original data from Decker et al. (1971). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 174

151 o i 3.2 weeks 1 1 1 3.6 weeks A 4.3 weeks V 5.5 weeks — <•> \/ 7.7 weeks 0 V 0 _ <> £ o — s — £> o O °v i i i i 100 200 300 Predicted N Removal, kg/ha 400 500 re 4-24 Residual plot of plant N removal for bermudagrass grown at Maryland and cut at five harvest intervals. Original data from Decker et al. (1971). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 175

152 ,ure4-25 Estimated maximum dry matter yield and estimated maximum plant N removal as a function of harvest interval for bermudagrass in Maryland. Lines drawn from Eq. [4.51] and [4.52].

PAGE 176

153 50 0 200 400 600 800 1000 1200 Applied Nilrogcn, kg/ha ;ure 4-26 Seasonal dry matter yield, plant N removal, and plant N concentration for a two week clipping interval over two years for bermudagrass grown at Tifton, GA. Data from Prine and Burton (1956); curves drawn from Eq. [4.53] through [4.56], [4.73], and [4.74].

PAGE 177

154 50 0 200 400 600 800 1000 1200 Applied Nitrogen, kg/ha Figure 4-27 Seasonal dry matter yield, plant N removal, and plant N concentration for a three week clipping interval over two years for bermudagrass grown at Tifton, GA. Data from Prine and Burton (1956); curves drawn from Eq. [4.57] through [4.60], [4.75], and [4.76].

PAGE 178

155 Figure 4-28 Seasonal dry matter yield, plant N removal, and plant N concentration for a four week clipping interval over two years for bermudagrass grown at Tifton, GA. Data from Prine and Burton ( 1 956); curves drawn from Eq. [4.61] through [4.64], [4.77], and [4.78].

PAGE 179

156 Figure 4-29 Seasonal dry matter yield, plant N removal, and plant N concentration for a six week clipping interval over two years for bermudagrass grown at Tifton, GA. Data from Prine and Burton (1956); curves drawn from Eq. [4.65] through [4.68], [4.79], and [4.80].

PAGE 180

157 400 600 800 Applied Nitrogen, kg/ha 1200 Figure 4-30 Seasonal dry matter yield, plant N removal, and plant N concentration for a eight week clipping interval over two years for bermudagrass grown at Tifton, GA. Data from Prine and Burton (1956); curves drawn from Eq. [4.69] through [4.72], [4.81], and [4.82].

PAGE 181

158 Figure 4-3 1 Seasonal dry matter yield and plant N concentration as a function of plant N removal for a two week clipping interval over two years for bermudagrass grown at Tifton, GA. Data from Prine and Burton (1956); curves drawn from Eq. [4.83] through [4.86].

PAGE 182

159 60 on U> 50 0 100 200 300 400 500 600 700 Nitrogen Removal, kg/ha Figure 4-32 Seasonal dry matter yield and plant N concentration as a function of plant N removal for a three week clipping interval over two years for bermudagrass grown at Tifton, GA. Data from Prine and Burton (1956); curves drawn from Eq. [4.87] through [4.90]. i

PAGE 183

160 Figure 4-33 Seasonal dry matter yield and plant N concentration as a function of plant N removal for a four week clipping interval over two years for bermudagrass grown at Tifton, GA. Data from Prine and Burton (1956); curves drawn from Eq. [4.91] through [4.94].

PAGE 184

161 re 4-34 Seasonal dry matter yield and plant N concentration as a function of plant N removal for a six week clipping interval over two years for bermudagrass grown at Tifton, GA. Data from Prine and Burton (1956); curves drawn fromEq. [4.95] through [4.98].

PAGE 185

162 70 I 1 r— — 1 i i 1 r 60 60 g bo 50 _ 0 75 150 225 300 375 450 525 600 675 Nitrogen Removal, kg/ha Figure 4-35 Seasonal dry matter yield and plant N concentration as a function of plant N removal for a eight week clipping interval over two years for bermudagrass grown at Tifton, GA. Data from Prine and Burton (1956); curves drawn from Eq. [4.99] through [4.102].

PAGE 186

163 1.2 s I -a o £ 3 pa § C o U 1.0 Z 0.2 0.0 0.4 0.6 0.8 o 1953, 2 weeks 1953, 3 weeks A 1953, 4 weeks V 1953, 6 weeks 1953, 8 weeks • 1954, 2 weeks 1954, 3 weeks A 1954, 4 weeks T 1954, 6 weeks 1954, 8 weeks i 1.0 1.2 N Removal/Estimated Maximum Figure 4-36 Dimensionless plot of seasonal dry matter yield and seasonal plant N concentration versus seasonal plant N removal. Original data from Prine and Burton (1956); curves drawn from Eq. [4.103] and [4.104].

PAGE 187

164 Figure 4-37 Estimated maximums of seasonal dry matter yield and plant N removal as a function of harvest interval for two years of bermudagrass grown at Tifton, GA. Lines drawn from Eq. [4. 105] through [4. 108].

PAGE 188

165 O 1953, 2 weeks 1953, 3 weeks A 1953, 4 weeks V 1953, 6 weeks <> 1953, 8 weeks • 1954, 2 weeks 1954, 3 weeks A 1954, 4 weeks T 1954, 6 weeks 1954, 8 weeks O T V<2> 10 15 20 25 30 Measured Dry Matter Yield, Mg/lia Figure 4-38 Scatter plot of seasonal dry matter yield for bermudagrass grown over two years at Tifton, GA, and cut at five different harvest intervals. Original data from Prine and Burton ( 1 956).

PAGE 189

166 700 600 500 400 300 O V T 1953, 1953, 1953. 1953, 1953. 1954, 1954, 1954. 1954, 1954. 2 weeks 3 weeks 4 weeks 6 weeks 8 weeks 2 weeks 3 weeks 4 weeks 6 weeks 8 weeks Q 200 100 100 200 300 400 500 600 700 Measured N Removal, kg/ha Figure 4-39 Scatter plot of seasonal plant N removal for bermudagrass grown over two years at Tifton, GA, and cut at five different harvest intervals. Original data from Prine and Burton ( 1 956).

PAGE 190

167 1 1 O 1953, 2 weeks 1953, 3 weeks A 1953, 4 weeks V 1953, 6 weeks O 1953, 8 weeks • 1954, 2 weeks 1954, 3 weeks A 1954, 4 weeks 1954, 6 weeks 1954, 8 weeks <•> 10 15 20 25 30 Predicted Dry Matter Yields, Mg/lia Figure 4-40 Residual plot of seasonal dry matter yield for bermudagrass grown over two years at Tifton, GA, and cut at five different harvest intervals. Original data from Prine and Burton (1956). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 191

168 an > o S5 3 IS o 250 200 150 100 50 -50 -100 -150 -200 -250 o 1953. 2 weeks 1 1 1953. 3 weeks A t wccks V 1953, 6 weeks O 1953, 8 weeks • 1954, 2 weeks 1954, 3 weeks A 1954. 4 weeks T 1954, 6 weeks 1954, 8 weeks -O •a A 1 o <& *^ 0 A i i i 200 400 Estimated N Removal, kg/lia 600 800 Figure 4-41 Residual plot of seasonal plant N removal for bermudagrass grown over two years at Tifton, GA, and cut at five different harvest intervals. Original data from Prine and Burton (1956). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 192

169 50 1000 Applied Nitrogen, kg/ha Figure 4-42 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for three different number of cuttings over the season for ryegrass grown at England. Data from Reid (1978); curves drawn from Eq. [4.109] through [4.1 17].

PAGE 193

170 ;ure 4-43 Seasonal dry matter yield and plant N removal as a function of plant N concentration for three different number of cuttings over the season for ryegrass grown at England. Data from Reid (1978); curves drawn from Eq. [4.118] through [4.123].

PAGE 194

171 1.2 ,f] 0.6 0.2 0.0 1.0 0.8 0.6 O 10 clippings 5 clippings A 3 clippings 0.4 0.6 0.8 N Renioval/Eslimalcd Maximum 1.0 1.2 Figure 4-44 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for three different number of cuttings over the season for ryegrass grown at England. Original data from Reid (1978); curves drawn from Eq. [4. 1 24] through [4. 1 29].

PAGE 195

172 O 10 clippings 5 clippings A 3 clippings o 0 A 0 n o 4 6 8 10 Measured Dry Matter Yield. Mg/ha 12 14 Figure 4-45 Scatter plot of seasonal dry matter yield for ryegrass grown at England and cut different times over the season. Data from Reid (1978).

PAGE 196

173 O 10 clippings 5 clippings A 3 clippings A A3 i 75 150 225 300 Measured N Removal, kg/ha 375 450 Figure 4-46 Scatter plot of seasonal plant N removal for ryegrass grown at England and cut different times over the season. Data from Reid (1978).

PAGE 197

174 O 10 clippings 5 clippings A 3 clippings D%___. FJ O ^ o o A A A "A" 3 6 9 Predicted Dry Matter Yield, Mg/ha 12 15 Figure 4-47 Residual plot of seasonal dry matter yield for ryegrass grown at England and cut different times over the season. Original data from Reid (1978). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 198

175 1 1 r O 10 clippings 5 clippings A 3 clippings -GO O O o o o o o o o A A --A A AO O A© -O— 50 100 150 200 250 300 Predicted N Removal, kg/ha 350 400 450 Figure 4-48 Residual plot of seasonal plant N removal for ryegrass grown at England and cut different times over the season. Original data from Reid (1978). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 199

Figure 4-49 Estimated maximums of seasonal dry matter yield and plant N removal as a function of average harvest interval for ryegrass grown at England. Lines drawn from Eq. [4. 1 30] and [4. 131].

PAGE 200

177 30 | 1 1 r 1 1 1 r 0 100 200 300 400 500 600 700 800 Applied Nitrogen, kg/ha Figure 4-50 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for bermudagrass grown over three years at Fayetteville, AR, with and without irrigation. Data from Huneycutt et al. (1988); curves drawn from Eq. [4.132] through [4.149].

PAGE 201

178 30 0 100 200 300 400 500 600 Nitrogen Removal, kg/ha Figure 4-5 1 Seasonal dry matter yield and plant N concentration as a function of plant N removal for bermudagrass grown over three years at Fayetteville, AR, with and without irrigation. Data from Huneycutt et al. (1988); curves drawn from Eq.[4.150] through [4.161].

PAGE 202

179 1 s T3 CJ M £ •a 1.2 1.0 0.8 0.6 8 04 1= o * 0.2 0.0 C3 -a o 1.0 0.8 O Non-Irrigaled 1983 Non-lrrigatcd 1984 A Non-Irrigated 1985 V Irrigated 1983 O Irrigated 1984 O Irrigated 1985 ^ 0.6 fc 0.4 s § 0.2 s ().() 0.0 0.2 0.4 0.6 0.8 N Removal/Estimated Maximum 1.0 1.2 Figure 4-52 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for bermudagrass grown over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt et al. (1988); curves drawn from Eq.[4.162] and [4.163].

PAGE 203

180 30 25 i 20 2 5 > i 15 -a a £ 10 1 1 O Non-lrri gated 1983 Non-Irrigated 1984 A Non-Irrigated 1985 V Irrigated 1983 <$> Irrigated 1984 0 Irrigated 1985 A u alM i I 0 0 5 10 15 20 Measured Dry Matter Yield, Mg/ha 25 30 Figure 4-53 Scatter plot of seasonal dry matter yield for bermudagrass grown over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt et al. (1988).

PAGE 204

181 600 500 | sn J2 13 > o S S -a o -a £ 400 300 200 O Non-Irrigated 1983 Non-Irrigated 1984 A Non-Irrigated 1985 V Irrigated 1983 O Irrigated 1984 0 Irrigated 1985 V V A A i i A V i i A 100 ) v 100 200 300 400 Measured N Removal, kg/ha 500 600 Figure 4-54 Scatter plot of seasonal plant N removal for bermudagrass grown over three years at Fayettevilie, AR, with and without irrigation. Original data from Huneycutt el al. (1988).

PAGE 205

182 10 15 20 Predicted Dry Matter Yield. Mg/ha Figure 4-55 Residual plot of seasonal dry matter yield for bermudagrass grown over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt el al. (1988). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 206

183 200 300 400 Predicted N Removal, kg/ha 600 Figure 4-56 Residual plot of seasonal plant N removal for bermudagrass grown over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt el al. (1988). Solid line is mean and dashed lines are ±2 standard errors.

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184 40 30 500 400 25 Non-Irrigated Irrigated 0 .0 o Non-Irrigated Irrigated -Bn u Non-Irrigated 300 400 500 Applied Nitrogen, kg/ha 600 700 800 Figure 4-57 Response of seasonal dry matter yield, plant N removal and plant N concentration for bermudagrass averaged over three years at Fayetteville, AR, with and without irrigation. Data from Huneycutt ef a/. (1988); curves drawn from Eq. [4. 164] through [4. 169].

PAGE 208

185 40 Figure 4-58 Seasonal dry matter yield and plant N concentration as a function of plant N removal for bermudagrass averaged over three years at Fayetteville, AR, with and without irrigation. Data from Huneycutt el al. (1988); curves drawn from Eq. [4.170] through [4.173].

PAGE 209

186 1.2 1 -a o a B B § u C o U 1.0 0.8 0.6 0.4 LI O Z 0.2 ().() 1.0 -a 0.8 0.6 0.4 0.2 0.0 Q o O Non-Irrigated Irrigated 0.0 0.2 0.4 0.6 0.8 N Removal/Estimated Maximum 1.0 1.2 Figure 4-59 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for bermudagrass averaged over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt el a/. (1988); curves drawn from Eq. [4.174] and [4.175].

PAGE 210

187 30 25 20 s 22 1 15 T3 '-3 10 0 1 1 Non-Irrigated — i — i i / Irrigated 0 Q } -J 0 i i i i i 10 15 20 Measured Dry Matter Yield. Mg/lia 25 30 Figure 4-60 Scatter plot of seasonal dry matter yield for bermudagrass averaged over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt el al (1988).

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188 O Non-Irrigated Irrigated 0 1 1 0 o j6 i i o 100 200 300 400 Measured N Removal, kg/ha 500 600 Figure 4-6 1 Scatter plot of seasonal plant N removal for bermudagrass averaged over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt el al. (1988)

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189 f 5 u H ID a b Q "3 3 -g ' 55 a PC 10 15 20 Predicted Dry Matter Yield. Mg/ha Figure 4-62 Residual plot of seasonal dry matter yield for bermudagrass averaged over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt el al. (1988). Solid line is mean and dashed lines are ±2 standard errors.

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190 100 7? O Non-Irrigated Irrigated 50 100 200 300 400 Predicted N Removal, kg/ha Figure 4-63 Residual plot of seasonal plant N removal for bermudagrass averaged over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt el al. (1988). Solid line is mean and dashed lines are ±2 standard errors.

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191 300 400 500 Applied Nitrogen, kg/ha 800 Figure 4-64 Response of seasonal dry matter yield, plant N removal and plant N concentration for tall fescue grown over three years at Fayetteville, AR, with and without irrigation. Data from Huneycutt el al. (1988); curves drawn from Eq. [4.176] through [4.193].

PAGE 215

192 Figure 4-65 Seasonal dry matter yield and plant N concentration as a function of plant N removal for tall fescue grown over three years at Fayetteville, AR, with and without irrigation. Data from Huneycutt et al. (1988); curves drawn from Eq. [4.194] through [4.205],

PAGE 216

193 s 3 s § T3 O S 1.2 1.0 0.8 " 0 6 o C3 fi C o a C o U 0.4 £ 0.2 0.0 3 § 1 s 2 § 2 1.0 0.8 0.6 0.4 0.2 0.0 <•> Non-Irrigalcd 1981-2 Non-Irrigated 1982-3 Non-Irrigated 1983-4 Irrigated 1981-2 Irrigated 1982-3 m Irrigated 1983-4 i r i I' m / i i i 0.0 0.2 0.4 0.6 0.8 Nitrogen Removal/Estimated Maximum 1.0 1.2 Figure 4-66 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for tall fescue grown over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt el al. (1988); curves drawn from Eq. [4.206] and [4.207].

PAGE 217

194 i U u ti r3 T3 U -a u 20 18 16 14 12 10 8 6 4 2 0 O Non-Irrigated 1981 -2 Non-Irrigated 1982-3 A Non-Irrigated 1983-4 V Irrigated 1981-2 <> Irrigated 1982-3 0 Irrigated 1983-4 <$>, J > A A *5 v 0 10 12 6 8 10 12 14 Measured Dry Matter Yield. Mg/Iia 16 IK 20 Figure 4-67 Scatter plot of seasonal dry matter yield for tall fescue grown over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt eta/. (1988).

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195 O El A V O o Non-Irrigated 1981-2 Non-lrrigatcd 1982-3 Non-Irrigated 1983-4 Irrigated 1981-2 Irrigated 1982-3 Irrigated 1983-4 0* 05* (•) l-i" 1 1 A A A 100 200 300 Measured N Removal, kg/ha 400 500 Figure 4-68 Scatter plot of seasonal plant N removal for tall fescue grown over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt el at. (1988).

PAGE 219

196 O Non-Irrigated 1981-2 Non-Irrigated IQ82-3 A Non-Irrigated 1983-4 V Irrigated 1981-2 Irrigated 1982-3 0 Irrigated 1983-4 5 10 15 Predicted Dry Matter Yield. Mg/ha 20 Figure 4-69 Residual plot of seasonal dry matter yield for tall fescue grown over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt et al. ( 1 988). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 220

197 SI) > o I a -g s 200 300 Predicted N Removal, kg/lia Figure 4-70 Residual plot of seasonal plant N removal for tall fescue grown over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt ei al. (1988). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 221

198 40 30 N 10 500 400 300 F3 Irrigated B ^ 0Non-Irrigated 300 400 500 600 Applied Nitrogen, kg/ha 800 Figure 4-7 1 Response of seasonal dry matter yield, plant N removal and plant N concentration for tall fescue averaged over three years at Fayetteville, AR, with and without irrigation. Data from Huneycutt et al. (1988); curves drawn from Eq. [4.208] through [4.213].

PAGE 222

199 15 32 13 H o s 1— 1 — t 1 Irrigated EK'^ Non-Irrigalcd 1 i i > 100 200 300 Nitrogen Removal, kg/ha 400 500 Figure 4-72 Seasonal dry matter yield and plant N concentration as a function of plant N removal for tall fescue averaged over three years at Fayetteville, AR, with and without irrigation. Data from Huneycutt et al. (1988); curves drawn from Eq. [4.214] through [4.217].

PAGE 223

200 1.2 S 3 S 1 -a S 03 E 1.0 0.8 0.6 N O 0 © $5h c a o c o U 8 -fc| +-» CO £ u o c a 0.4 0.2 0.0 1.0 0.8 0.6 0.4 0.2 ().() 0.0 i ' Non-Irrigated Irrigated 0.2 0.4 0.6 0.8 N Removal/Estimated Maximum 1.0 Figure 4-73 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for tall fescue averaged over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt etal. (1988); curves drawn from Eq. [4.218] and [4.219].

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201 i i Non-lrrigatcd Irrigated -i — — r 1 r ED o 6 i i ID ED _i i_ 8 10 „i., 12 14 16 Measured Dry Matter Yield. Mg/lia 18 20 Figure 4-74 Scatter plot of seasonal dry matter yield for tall fescue averaged over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt et al. (1988).

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202 500 O Non-Irrigated Irrigated 400 300 -o % 200 100 El n no o Q d 0 () 0 100 200 300 Measured N Removal, kgflia 400 500 Figure 4-75 Scatter plot of seasonal plant N removal for tall fescue averaged over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt el al. (1988).

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203 c Q "a 3 O Non-Irrigated Irrigated 32 < r >' € 0 = O I l 9 -2 0 5 10 15 20 Predicted Dry Matter Yield. Mg/ha Figure 4-76 Residual plot of seasonal dry matter yield for tall fescue averaged over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt et al ( 1 988). Solid line is mean and dashed lines are ±2 standard errors.

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204 75 50 « 25 on O Non-Irrigated Irrigated G <& ( ) 0 n " GT-Q ( O -25 -50 -75 0 100 200 300 400 500 Predicted N Removal, kg/ha Figure 4-77 Residual plot of seasonal plant N removal for tall fescue averaged over three years at Fayetteville, AR, with and without irrigation. Original data from Huneycutt ei al. (1988). Solid line is mean and dashed lines are ±2 standard errors.

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205 40 _i i i i i i 1 1 50 100 150 200 250 300 350 400 Applied Nitrogen, kg/ha Figure 4-78 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for bermudagrass grown over two years at Eagle Lake, TX. Data from Evers (1984); curves drawn from Eq. [4.220] through [4.223], [4.228] and [4.229].

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206 Figure 4-79 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for bahiagrass grown over two years at Eagle Lake, TX. Data from Evers (1984); curves drawn from Eq. [4.224] through [4.227], [4.230], and [4.231]..

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207 Figure 4-80 Seasonal dry matter yield and plant N concentration as a function of plant N removal for bermudagrass grown over two years at Eagle Lake, TX. Data from Evers (1984); curves drawn from Eq. [4.232] through [4.235].

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208 Figure 4-81 Seasonal dry matter yield and plant N concentration as a function of plant N removal for bahiagrass grown over two years at Eagle Lake, TX. Data from Evers (1984); curves drawn from Eq. [4.236] through [4.239].

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209 1.2 1.0 1 S 0.8 -a a | 0.6 c 0 03 c o o c o o 0.4 £ 0.2 OA v n 0.0 1.0 S 3 Q Bcmuidagrass 1979 Bermudagrass 1980 A Bahiagrass 1979 V Bahiagrass 1980 a o.8 S3 a o.6 a £ 0.4 a 0.2 A0 .A ^7 0.0 0.0 0.2 0.4 0.6 N Removal/Estimated Maximum 0.8 1.0 Figure 4-82 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for bahiagrass and bermudagrass grown over two years at Eagle Lake, TX. Original data from Evers (1984); curves drawn from Eq. [4.240] and [4.241].

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210 O Bennudagrass 1979 Bennudagrass 1980 A Bahiagrass 1979 V Bahiagrass 1980 Q -A i • i A O A O v H A V *7 / /o i 6 9 12 15 Measured Dry Matter Yield. Mg/ha Figure 4-83 Scatter plot of seasonal dry matter yield for bahiagrass and bermudagrass grown over two years at Eagle Lake, TX. Original data from Evers (1984).

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211 300 250 O A V Bcmnidagrass 1979 Benmidagrass 1980 Bahiagrass 1979 Bahiagrass 1980 O 2 200 1* "3 > o £ § -a o -a o 150 100 O o A V o LI A ( > A 50 50 100 150 200 Measured N Removal, kg/lia 250 300 Figure 4-84 Scatter plot of seasonal plant N removal for bahiagrass and bermudagrass grown over two years at Eagle Lake, TX. Original data from Evers (1984).

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212 S 1 tJ u i o C3 I ., u o2 o 1 1 1 1 1 Bcrmudagrass 1979 r Bcrniudagrass 1980 Bahiagrass 1979 V RnliiioriQC IQ80 V „ A 1 A O — 6 f-QT ~ V o 1 1 1 1 1 i 6 8 10 12 14 16 Predicted Dry Matter Yield, Mg/ha Figure 4-85 Residual plot of seasonal dry matter yield for bahiagrass and bermudagrass grown over two years at Eagle Lake, TX. Original data from Evers (1984). Solid line is mean and dashed lines are ±2 standard errors.

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213 1 I — 1 1 i o Bcrmudagrass 1979 Bcrmudagrass 1980 A V Bahiagrass 1980 D A (J o V V r i v 1 1 a A ^ v m V o O 1 1 1 . ... i 1 50 100 150 200 Predicted N Removal, kg/lia 250 300 Figure 4-86 Residual plot of seasonal plant N removal for bahiagrass and bermudagrass grown over two years at Eagle Lake, TX. Original data from Evers (1984). Solid line is mean and dashed lines are ±2 standard errors.

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214 0 50 100 150 200 250 300 Applied Nitrogen, kg/ha Figure 4-87 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for grain and total plant of corn grown at Clayton and Kinston, NC, on Dothan and Goldsboro soils, respectively. Data from Kamprath (1986); curves drawn from Eq. [4.242] through [4.253].

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215 Figure 4-88 Seasonal dry matter yield and plant N concentration as a function of plant N removal for grain and total plant of corn grown at Clayton and Kinston, NC, on Dothan and Goldsboro soils, respectively. Data from Kamprath (1986); curves drawn from Eq. [4.254] through [4.261].

PAGE 239

216 1.2 3 l.o "8 .1 B O 1 c o a o o d 0 C3 s o o 9 0.6 0.4 0.2 0.0 1.0 0.8 0.6 (J All A r O Dothan Grain Dothan Total A Goldsboro Grain \7 Goldsboro Total era ("•) 0.4 0.6 0.8 Nitrogen Removal/Estimated Maximum 1.0 1.2 Figure 4-89 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for grain and total plant of corn grown at Clayton and Kinston, NC, on Dothan and Goldsboro soils, respectively. Original data from Kamprath (1986); curves drawn from Eq. [4.262] and [4.263],

PAGE 240

217 25 20 •g 15 O Dothan Grain Dothan Tolal A Goldsboro Grain V Goldsboro Total V 10 n 4" 10 15 DryMatter Yield. Mg/ha 20 25 Figure 4-90 Scatter plot of seasonal dry matter yield for grain and total plant of corn grown at Clayton and Kinston, NC, on Dothan and Goldsboro soils, respectively. Original data from Kamprath ( 1 986).

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218 200 O Dotlian Grain Dotlian Tolal A Goldsboro Grain V Goldsboro Total 150 | on •a > o S T3 O u -5 100 50 \7 6 v i i A 57 C 1 A O A o A 50 100 N Removal, kg/lia 150 200 Figure 4-91 Scatter plot of seasonal plant N removal for grain and total plant of corn grown at Clayton and Kinston, NC, on Dothan and Goldsboro soils, respectively. Original data from Kamprath (1986).

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219 (J 1 r Dothan Grain 1 ' ' 1 n LyUUIilll 1 Uull A Goldsboro Grain V Goldsboro Total A M V V A A A % 0 V ( : > V V B 0 5 10 15 20 25 Predicted Dry Matter Yield. Mg/lia Figure 4-92 Residual plot of seasonal dry matter yield for grain and total plant of corn grown at Clayton and Kinston, NC, on Dothan and Goldsboro soils, respectively. Original data from Kamprath ( 1 986). Solid line is mean and dashed lines are ±2 standard errors.

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220 30 20 O Dollian Grain Dolhan Total A Goldsboro Grain V Goldsboro Total ft > o Z 1 3 10 10 -20 -30 0 50 100 150 200 Predicted N Removal, kg/ha Figure 4-93 Residual plot of seasonal plant N removal for grain and total plant of corn grown at Clayton and Kinston, NC, on Dothan and Goldsboro soils, respectively. Original data from Kamprath ( 1 986). Solid line is mean and dashed lines are ±2 standard errors.

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221 20 0 50 100 150 200 250 300 Applied Nitrogen, kg/ha Figure 4-94 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for grain and total plant of corn grown at Plymouth, NC. Data from Kamprath (1986); curves drawn from Eq. [4.264] through [4.269].

PAGE 245

222 20 Nitrogen Removal, kg/ha Figure 4-95 Seasonal dry matter yield and plant N concentration as a function of plant N removal for grain and total plant of corn grown at Plymouth, NC. Data from Kamprath (1986); curves drawn from Eq. [4.270] through [4.273].

PAGE 246

223 o.o 0.2 0.4 0.6 0.8 Nitrogen Removal/Estimated Maximum Figure 4-96 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for grain and total plant of corn grown at Plymouth, NC. Original data from Kamprath (1986); curves drawn from Eq. [4.274] and [4.275].

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224 20 18 16 14 1 12 € io 2 T3 £ 6 Oh ~i 1 1 1 1 1 ' r O Grain Total 0 o J L 0 n 12 i_ 14 6 8 10 Measured Dry Matter Yield. Mg/ha 16 18 20 Figure 4-97 Scatter plot of seasonal dry matter yield for grain and total plant of corn grown at Plymouth, NC. Original data from Kamprath (1986).

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225 150 r O Grain Total 120 on a" I 90 6 E0 0 T3 U t3 60 30 Q 06 0 L 30 60 90 Measured N Removal, kg/ha 120 150 re 4-98 Scatter plot of seasonal plant N removal for grain and total plant of corn grown at Plymouth, NC. Original data from Kamprath (1986).

PAGE 249

226 i o 1 0 Grain Total i i _ 1 1 0 0 1 1 [ 1 i 1 • 1 3 6 9 12 Predicted Dry Matter Yield. Mg/ha 15 18 Figure 4-99 Residual plot of seasonal dry matter yield for grain and total plant of corn grown at Plymouth, NC. Original data from Kamprath ( 1 986). Solid line is mean and dashed lines are ±2 standard errors.

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227 30 60 90 Predicted N Removal, kg/ha 120 re 4-100 Residual plot of seasonal plant N removal for grain and total plant of corn grown at Plymouth, NC. Original data from Kamprath (1986). Solid line is mean and dashed lines are ±2 standard errors.

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228 30 . i 1 0 1 J 1 1 0 100 200 300 400 500 Applied Nitrogen, kg/ha re 4-101 Response of seasonal dry matter yield, plant N removal, and plant N concentration to applied N for bahiagrass grown on two soils in Florida. Data from Blue (1987); curves drawn from Eq. [4.276] through [4.281].

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229 25 0 50 100 150 200 250 300 350 400 Nitrogen Removal, kg/ha Figure 4-102 Seasonal dry matter yield and plant N concentration as a function of plant N removal for bahiagrass grown on two soils in Florida. Data from Blue (1987); curves drawn from Eq. [4.282] through [4.285].

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230 1.2 g 1.0 I ^ 0.8 T3 I 1 0.6 fT LJ I 04 I £ 0.2 O Entisol Spodosol 0.0 1.0 0.8 0.6 0.4 i s b Q 0.2 0 0.0 0.0 0.2 0.4 0.6 N Removal/Estimated Maximum 0.8 1.0 re 4-103 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for bahiagrass grown on two soils in Florida. Original data from Blue (1987); curves drawn from Eq. [4.286] and [4.287].

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231 20 18 16 14 O Entisol Spodosol 1 12 3 10 T3 0 ( 0 ~i 1 r 0, / _1_ 10 L 12 4 6 8 10 12 14 Measured Dry Matter Yield. Mg/ha 18 20 Figure 4104 Scatter plot of seasonal dry matter yield for bahiagrass grown on two soils in Florida. Original data from Blue ( 1 987).

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232 300 250 | 200 on J* i z £ 100 50 0 Enlisol 1 ] Spodosol 9 0 Q/ 9 Q 50 100 150 200 Measured N Removal, kg/ha 250 300 Figure 4-105 Scatter plot of seasonal plant N removal for bahiagrass grown on two soils in Florida. Original data from Blue ( 1 987).

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233 i 1 1 r O Entisol Spodosol « 0 I & I 0 7j --GL n 0 4 6 8 10 12 14 16 Predicted Dry Matter Yield. Mg/ha Figure 4-106 Residual plot of seasonal dry matter yield for bahiagrass grown on two soils in Florida. Original data from Blue (1987). Solid line is mean and dashed lines are ±2 standard errors.

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234 40 30 20 £ 10 i i ° i s -lo pi -20 -30 1 — O Entisol Spodosol o O — Q o H — o -40 J 50 100 150 200 250 Predicted N Removal, kg/ha 300 350 jure 4-107 Residual plot of seasonal plant N removal for bahiagrass grown on two soils in Florida. Original data from Blue (1987). Solid line is mean and dashed lines are ±2 standard errors.

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235 1.50 1.25 1.00 0.75 0.50 0.25 "Oo o IT 0 0 Q oo o o o 0 0 0.00 50 40 30 20 ,0_ i-i ( ) ( . >f , ( )< )(•) r o () ~0~ ( ? ) 0 -Oo 10 10 J 15 Site L_ 20 25 30 Figure 4-108 Plot of the mean and ±2 standard errors of A n /A and Ab for twenty sites in England.

PAGE 259

236 1.2 N Removal/Estimated Maximum re 4-109 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for twenty sites in England. Original data from Morrison el al. (1980); curves drawn from Eq. [4.288] and [4.289].

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237 1.2 I 1.0 0.8 M 0.6 1 °4 1 U Z 0.2 0.0 1.0 s "8 — -* CO .9 -*-» § s "3 >l_ 1 0.8 0.6 0.4 0.2 0.0 **X XX mil Hyperbolic Regression: A b = 0.7607 r 0.9793 0.0 0.2 0.4 0.6 0.8 N Removal/Estimated Maximum 1.2 re 4-1 10 Dimensionless plot of seasonal dry matter yield and plant N concentration as a function of plant N removal for twenty sites in England. Original data from Morrison el a/. (1980); curves drawn from Eq. [4.290] and [4.291] with Ab=0.76.

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238 60 0 50 100 150 200 250 Applied Nitrogen, kg/ha ;ure4-lll Response of seasonal dry matter, plant N removal, and plant N concentration to applied N for rye grown at Tifton, GA and fixed application rates of 40 and 74 kg/ha of P and K, respectively. Data from Walker and Morey (1962). Curves drawn by Eq. [4.292], [4.293] and [4.296].

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239 2d 40 60 80 Applied Phosphorus, kg/ha 100 re 4-1 12 Response of seasonal dry matter, plant P removal, and plant P concentration to applied P for rye grown at Tifton, GA and fixed application rates of 135 and 74 kg/ha of N and K, respectively. Data from Walker and Morey (1962). Curves drawn by Eq. [4.292], [4.294], and [4.297],

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240 50 SiQ | I I I 1 1 1 1 0 25 50 75 100 125 150 175 200 Applied Potassium, kg/ha re 4-1 13 Response of seasonal dry matter, plant K removal, and plant K concentration to applied K for rye grown at Tifton, GA and fixed application rates of 135 and 40 kg/ha of N and P, respectively. Data from Walker and Morey (1962). Curves drawn by Eq. [4.292], [4.295], and [4.298].

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241 50 100 150 200 250 Nitrogen Removal, kg/ha Figure 4-114 Seasonal dry matter yield and plant N concentration as a function of plant N removal for rye grown at Tifton, GA. Data from Walker and Morey (1962). Curves drawn by Eq. [4.299] and [4.300].

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242 Figure 4-115 Seasonal dry matter yield and plant P concentration as a function of plant P removal for rye grown at Tifton, GA. Data from Walker and Morey (1962). Curves drawn by Eq. [4.301] and [4.302].

PAGE 266

243 50 i i i 0 25 50 75 100 125 150 175 200 Potassium Removal, kg/ha Figure 4-116 Seasonal dry matter yield and plant K concentration as a function of plant K removal for rye grown at Tifton, GA. Data from Walker and Morey (1962). Curves drawn by Eq. [4.303] and [4.304],

PAGE 267

244 1.2 Nutrient Removal/Estimated Maximum ;ure4-117 Dimensionless plot of seasonal dry matter yield and plant nutrient concentration as a function of plant nutrient removal for rye grown at Tifton, GA. Original data from Walker and Morey ( 1 962). Curves drawn by Eq. [4.305] through [4.310].

PAGE 268

245 s 1 T3 CO ( I 0 '6. ( i o 6 M Q Q' 0© o I 2 3 Measured Dry Matter Yield, Mg/ha Figure 4-118 Scatter plot of dry matter yield for rye grown at Tifton, GA. from Walker and Morey (1962). Original data

PAGE 269

246 250 200 150 3 i T3 U 3 100 a 1 o o 0 0 o O 0 O 50 50 100 150 Measured N Removal, kg/ha 200 250 Figure 4-119 Scatter plot of plant N removal for rye grown at Tifton, GA. Original data from Walker and Morey (1962).

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247 40 35 30 5 25 q 20 a. 1 15 10 Q o 9 O G G 6 o o i 1 10 15 _j 20 „i 25 30 35 40 Measured P Removal, kg/ha Figure 4-120 Scatter plot of plant P removal for rye grown at Tifton, GA. Original data from Walker and Morey ( 1 962).

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248 25 50 75 100 125 Measured K Removal, kg/ha 50 200 Figure 4-121 Scatter plot of plant K removal for rye grown at Tifton, GA. Original data from Walker and Morey ( 1 962).

PAGE 272

249 1.25 1.00 0.75 0.50 0.25 22 > S cq b Q 1 a m M a; 0.00 -0.25 -0.50 -0.75 -1.00 -1.25 0 -oo o ° o Q G O © 0 0 1 2 3 Estimated Dry Matter Yield. Mg/lia jure 4122 Residual plot of dry matter yield for rye grown at Tifton, GA. Original data from Walker and Morey ( 1 962). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 273

250 100 150 Estimated N Removal, kg/ha ure4-123 Residual plot of plant N removal for rye grown at Tifton, GA. Original data from Walker and Morey (1962). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 274

251 ( i O G O o o o <(•) (0 o < > o 5 10 15 20 Estimated P Removal, kg/lia 25 30 ;ure4-124 Residual plot of plant P removal for rye grown at Tifton, GA. Original data from Walker and Morey (1962). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 275

252 45 30 2 15 i 1 | 0 o o o o o o o o o o O o & -15 --0--30 -45 0 25 50 75 100 125 150 Estimated K Removal, kg/ha 1 75 200 Figure 41 25 Residual plot of plant K removal for rye grown at Tifton, GA. Original data from Walker and Morey (1962). Solid line is mean and dashed lines are ±2 standard errors.

PAGE 276

CHAPTER 5 SUMMARY AND CONCLUSIONS Lately, environmental issues have become a popular topic in public discussion. People are interested in controlling pollution in soil, water and air. Land application of treated wastes and effluent is being used to help control contamination of surface and groundwater of Florida. A simple method is needed to aid engineers, farmers, and managers in obtaining estimates of nutrient removal by various forage crops. This research project has focused on modeling the seasonal production of seven different forage grasses, at different locations and under different factors, such as water availability, harvest interval and plant partitioning. The simple logistic model, Equation [2.8], the extended logistic model, Equations [2.8] and [2.9], and the extended triple logistic (NPK) model, Equations [2.17] through [2.20], were found to adequately describe grass response to applied nutrients (R > 0.99). Data from various studies were used to relate management parameters of water availability, harvest interval, soil type, and crop species to the parameters of the equation. Second-order Newton-Raphson method for nonlinear regression was used to optimize the fit of the model to the data. Analysis of variance (ANOVA) was used to search for simplification of the model in the form of common parameter values. The conclusions from the analysis of the data from the various studies are presented below. 1 . The logistic model is relatively simple to use. Once the parameters are known, estimates can be computed on a basic hand calculator. The logistic model involves analytical functions rather than finite difference (numerical techniques). 253

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254 2. The logistic model is well behaved. The response is positive for the entire domain of applied nitrogen. The slope is greater than zero (dY/dN > 0) for all values of N, i.e., it is monotone increasing. It asymptotically approaches a maximum at high N and zero at low N. 3. As shown with various data sets, harvest interval, water availability and plant partitioning can be accounted for in the linear parameter, A. This means that once the relationship has been identified, the effects of different levels and combinations of these factors on dry matter yield or nutrient removal response to applied nutrients can be estimated. Had these factors been accounted for only by the nonlinear parameters b and/or c, the task of simplification would have been greatly limited. Furthermore, since the effects are linear, the dependence upon yield and N removal can be factored into product terms: A = A w *A h *A p [5.1] where A = maximum yield or N removal parameter for Eq. [2.8] or [2.9], A w = water availability coefficient. Ah = harvest interval coefficient, A p = plant partitioning coefficient. Because these effects fall into products, averaging over years only affects the A parameter. Why was the A parameter linearly related to harvest interval? The answer lies in how the plant grows between harvests. Overman et al. ( 1 989) have shown that between harvests the plant follows a quadratic intrinsic growth function. When these sawtooth growths for multiple harvests are summed, the result is linear dependence of seasonal yield on harvest interval. Further, if the harvest interval is too long, the quadratic trend falls off and this effect appears in the sum as well as in the slow loss of linearity. 4. Dimensionless plots are a valuable tool in evaluating the form of a model (Segre, 1984, p. 168). All data are normalized to the same scale, thereby removing the

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255 units. After normalizing, the curve and line drawn are only dependent upon the Ab value of the logistic model. In cases such as the Prine and Burton (1956) study (bermudagrass grown at Tifton, GA), the model could be simplified since all the different combinations of factors had common Ab and c parameters as evidenced by the dimensionless plot (Figure 4-36). If after normalizing the data there had been no distinct relationship, it would be difficult to separate or identify the effects of different factors. In many different aspects of science and engineering, dimensionless plots are valuable tools in examining the underlying relationship between two variables and in collapsing data with different ranges onto the same scale. 5. The logistic equation exhibits symmetry suggesting that something in the system is conserved. In the case of these models, it is the total capacity of the system. A, the estimated maximum. But how exactly does this occur? Recall the rumor model. The total capacity of the system is analogous to the number of people in the room. The number of people in the room remains constant as the rumor is spread throughout the room. Furthermore, the transfer rate of the rumor is a product of two entities, the people who know the rumor and those who still need to be told. Similarly to the logistic crop growth system, the total capacity of the system is the sum of the dry matter that has been produced (y) and that which still can be produced (A-y). Response to increased input (N) is a product of two things: that which has already been produced and that which can be produced. As shown with Equation [2.3], the differential form of the logistic is directly proportional to the amount present and to the amount left to be produced. 6. The extended model was developed assuming the logistic equation fits both dry matter and plant N removal and further that the nitrogen response coefficient, c, was common for both. The hyperbolic relationship between dry matter yield and plant N removal resulted from these assumptions. One could ask: What combination of these assumptions is more fundamental? The original set of assumptions resulted from the ease

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256 of regressing the two responses to a logistic with a common c. An alternative approach would be to fit the logistic to the dry matter yield response to applied N and the hyperbolic to the dry matter yield response to plant N removal. This is a more fundamental approach, since it involves fitting a relationship between two extensive variables: dry matter and plant N removal. In this relationship, the effect of the independent variable, applied N, has been removed and only the measured variables are included. This procedure would be more difficult to compute since it would involve regressing data to two different functions simultaneously. It is important to remember that the hyperbolic relationship is based upon two measurable quantities, i.e. the independent variable (applied nitrogen) has been removed from the picture. From data set to data set, the same hyperbolic trend appears. What physically in the system causes this relationship? Recall the dimensionless parameter for the hyperbolic, YJA, represented by Equation [2.14]. This describes the maximum dry matter attained at high levels of plant N removal. For the studies investigated in this work, the value of this ratio ranges from 1 7 to 2.7, suggesting that approximately half of the potential capacity of the system is being observed in the various experiments. For Y„,/A = 2, it follows from Equation [2.14] that Ab = 0.69. One possible explanation for this lies in a paper coupling various plant components for forage production by Overman (1995b). He has shown that the response of leaf area to C0 2 concentration is hyperbolic. Assuming a current C0 2 concentration equal to the level in the atmosphere, 300 p.mol/mol, it appears from Figure 13 in the article that for days after planting (DAP) greater than 30, only a little more than half of the potential maximum leaf area is attained. Although this has not directly linked dry matter production to CO2 concentration, production is directly related to leaf area. This suggests that at higher levels of atmospheric C0 2 concentration would result in higher leaf areas, which would then result in higher dry matter production.

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257 7. The England study (Morrison et a/., 1980) suggests that Ab and N cm may be constant for a particular grass. A compilation of Ab values for the various studies examined in this work is summarized in Table 5-1. The mean, standard deviation, and relative error of Ab is also shown in the table. It appears from the table that the two grasses grown at Fayetteville, AR have the same Ab. Also the two grasses grown at Eagle Lake, TX have the same Ab. Corn grown at two sites with similar soils (Clayton and Kinston, NC) has the same Ab as well. Finally, bahiagrass grown on two different soils in Florida (Entisol and Spodosol) has the same Ab. What does this mean? Based upon the data presented in the table, it appears that Ab may be associated with specific grasses. Recall that the b parameter is a measure of the amount of available nitrogen in the soil before nutrient application (response at N = 0). Although no clear connection between the b parameters and types of grasses or measurable soil properties exists at this point, further investigation might reveal the nature of the relationship. 8. A compilation of c and N' parameter values for the various studies examined in this work is summarized in Table 5-2. Recall that N' = 1/c. It appears from the table that the two grasses grown at Fayetteville, AR probably have the same c. Also the two grasses grown at Eagle Lake, TX have the same c. Corn grown at two sites with similar soils (Clayton and Kinston, NC) has the same c as well. Bahiagrass grown on two different soils in Florida (Entisol and Spodosol) has the same c. Finally, for the twenty sites in England where ryegrass was grown, they all had the same c. The c values for bermudagrass grown at different sites are not the same; however, they all lie within a relatively small range (0.0067 0.01 12). The c values compare well with those reported in the literature for other data sets (Overman and Wilkinson, 1992, 1995). What does this mean? It is not clear what effects are incorporated in the c parameter, but the table suggests that this parameter might be related to type of grass. One possible clue is that c is the same for both dry matter response to applied nitrogen and plant N removal response

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258 to applied N. What in the system causes this phenomenon? At this point, it is not clear what physiological process might explain this behavior. Additional investigation might reveal a connection between this parameter and various effects: type of grass, water, soil characteristics, etc. It also appears from the table that rye and corn have higher c values than the other grasses. One possible explanation is that these grasses are annuals, while the others are grown as perennials. 9. This analysis has focused on seasonal quantities. Although time is not explicitly contained within the model, it is implicit within the A parameter. The maximum parameter contains within it the accumulation of dry matter over time, as evidenced by its relationship to harvest interval (Maryland and Tifton, GA, studies). 10. The models developed and analyzed in this study adequately describe seasonal relationships. The focus has been to determine, by dimensionless plots, if the form of the logistic adequately describes the behavior, not to determine parameter values. No attempt has been made to develop a "cookbook" model for the general public, so that site specific information, such as weather and soil properties, can be entered and automatically produce estimates. Rather the approach as been to determine if the form is sufficient in describing the relationship. If so, then the model can be used universally with little input. Although this study has not developed a table of parameter values, reasonable estimates can be obtained. Of the parameters, b is the least significant since it is the response at N = 0. Typically for sandy soils the b estimate is high (low intercept) as a result of low residual nitrogen in the soil. The A parameter can be estimated relatively easily since it is an estimate of the maximum. Very often in practice, high application rates are not used since the efficiency of the system to remove N decreases as the maximum is approached. The normal operating range is concentrated around the N l 2 value. Recall that this parameter is the amount of nitrogen applied to achieve half of the maximum yield. This is also the point of maximum slope (incremental increase of yield with N). For

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259 estimation purposes, the key is narrowing down the value for c. As mentioned before, this is believed to be the most critical parameter. As with many engineering situations, safety factors are always included. If the use of this model in a specific situation is to estimate N removal for a field receiving municipal wastewater, it would be better to underestimate the parameters in order to overdesign the system. By underestimating the parameters, the seasonal estimates are underestimated. Because these are underestimated, any design using these numbers will be conservative in nature. For example, if the seasonal plant N removal was estimated to be 400 kg/ha and it really was 450 kg/ha, and this value was used in a nutrient budget to determine the fate of nitrogen in a water reuse system, the system would be designed to accommodate 400 kg/ha. Since the plant would really be removing 450 kg/ha, even less nitrogen would make it into the aquifer than was estimated. Similarly, the farmer would be producing more dry matter than he had estimated. The only situation where underestimating the parameters could lead to problems are those where a target plant N removal or dry matter yield is desired and the equation is used to back calculate how much nitrogen should be applied. Although the c parameter is important in estimation, the model is not very sensitive to this parameter. A "large" c value results in a steeper slope at the midpoint of the curve; that is, the plant is reaching maximum much faster. A "small" value of c results in a flatter slope at the midpoint. As a consequence, small values of c should be used for conservative estimates. A plot of the normalized logistic equation is shown in Figure 5-1 with b = 2.0 for all curves and c values of 0.008, 0.010 and 0.012 ha/kg to demonstrate the sensitivity of the parameter. At N = 200 kg/ha, a 20% error in c causes a 20% error in response. For c = 0.01, this is the point of maximum slope since Ni /2 = b/c = 2/0.01 = 200. An example of how this model could be used in practice is presented below: Assume effluent from a municipality is available for water reuse, and further that the average N application rate for the system is 500 kg/ha. The farmer wishes to

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260 grow bermudagrass for hay. The soil in the available area is typically sandy and has very low levels of N. The effluent will be applied as irrigation, hence water availability should not be a problem. Based upon these assumptions, parameter values could be estimated based on analysis completed in this work on bermudagrass as the following: A = 20 Mg/ha, A' = 500 kg/ha, b = 1 .4, b' = 2.0, and c = 0.008 ha/kg. The seasonal estimates of dry matter production and plant N removal for the bermudagrass are 18.6 Mg/ha and 440 kg/ha. If the estimate of c was off by +20 % or -20%, then the estimates of dry matter production would be off by 4% and 8% and the estimates for plant N removal would be off by 7% and 1 3%, respectively. Similarly, if the estimates of b and b' were off by +20 % or 20%, then the estimates of dry matter production would be off by 2% and 2% and the estimates for plant N removal would be off by 6% and 4%, respectively. Further, any percentage error in A or A would result in the same percentage in the estimates since they are scalar multiples of the logistic equation (Eq. [2.4]). As demonstrated by this hypothetical example for large error (20%) in the b, b', and c parameters, the seasonal estimate is not affected greatly (<15% error). The A parameter needs to be estimated relatively well since any error in it will demonstrate itself in the seasonal estimates; however, an experienced farmer should be able to obtain a reasonable estimate for the expected maximum yield (A). This analysis has not attempted to describe the physical mechanisms of the soilplant-air system. Instead it was based upon an analysis of trends in forage grass response to applied nutrients. The intent was to develop a simple procedure that engineers, regulators, and managers could use in determining nutrient budgets, crop yields, and plant N removal. Gell-Mann (1994, p. 343) has written, "Humanity will be much better off when the reward structure is altered so that selection pressures on careers favors sorting out information as well as its acquisition." This project has been an attempt to follow

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261 Gell-Mann's approach. Furthermore, it is the author's opinion that if more energy was directed toward analyzing the currently available data as well as designing and implementing additional experiments, not only might a physical basis for the parameters be revealed, but science might progress and advance as well.. 1 I . The extended triple logistic (NPK) model is a product of extended logistic models for three different nutrients. This is a simple form to work with since if two of the three nutrient rates are known or constant, the extended triple logistic simplifies to the extended logistic. This is a result of the assumption of independent effects. It could have been just as logical to assume the joint response to the three nutrients appeared in a single exponential: This would have been much more complicated to analyze. Some interactions do exist between these three nutrients (Jones et a/., 1991); however, the triple logistic appears to have accounted for these small interactions since it fits rye at Tifton, GA, and bermudagrass at Watkinsville, GA (Overman and Wilkinson, 1995) well (R > 0.9).

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Table 5-1 . A Summary of the Ab Parameter for Various Studies. Location Grass Ab Factors Included Baton Rouge, LA Dallisgrass 0.89 None Fayetteville, AR Tall Fescue 0.55 Water Availability TIlArCm/ A I i iiui ^uy, ' »l DCI IllUUdUl 0 75 Harvest Intprval Maryland Bermudagrass 0.81 Harvest Interval Tifton, GA Bermudagrass 0.68 Harvest Interval and Water Availablity rdyciicviiic, /\t\. Del IllUUdgl d»3 o 54 \A/iitf*r Avmlanilitv vv aid /av ai lain u i y Eagle Lake, TX Bermudagrass 0.50 Water Availability Eagle Lake, TX Bahiagrass 0.50 Water Availability Williston, FL Bahiagrass 0.47 None Gainesville, FL Bahiagrass 0.47 None Various in England Ryegrass 0.83 Many different locations 1 lllUIl, VJ/A R \/p rvyc 0 S7 nunc Clayton, NC Corn 0.70 Grain and Total Plant Kinston, NC Corn 0.70 Grain and Total Plant Plymouth, NC Corn 0.52 Grain and Total Plant Average 0.63 Standard deviation ±0.14 Relative error 0.22

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263 Table 5-2. A Summary of c and N' Parameters from Various Studies. Location Grass c N' Factors Included Baton Rouge, LA uamsgrass U.UUjj 1 so None Fayetteville, AR Tall Fescue 0.0081 120 Water Availability Thorsby, AL Bermudagrass 0.0067 150 Harvest Interval Maryland Bermudagrass 0.0112 89 Harvest Interval Tifton, GA Bermudagrass 0.0077 130 Harvest Interval/Water Availablity Fayetteville, AR Bermudagrass U.UUo4 I zu water Avaiiauiiuy Eagle Lake, TX Bermudagrass 0.0072 140 Water Availability Eagle Lake, TX Bahiagrass 0.0072 140 Water Availability Williston, FL Bahiagrass 0.0118 85 None Gainesville, FL Bahiagrass 0.0118 85 None England Ryegrass U.UUoU 1 L j vanaoie Harvest interval Various in England Ryegrass 0.0088 110 Many different locations Tifton, GA Rye 0.0234 45 None Clayton, NC Corn 0.0187 54 Grain and Total Plant Kinston, NC Corn 0.0187 54 Grain and Total Plant Plymouth, NC Corn 0.0119 84 Grain and Total Plant

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264 1.2 b = 2.0 1.0 c = 0.008 c = 0.010 c0.012 0.8 U 0.6 N •-3 0.4 0.2 / / / / • / / // / 7 / / / / / / / // 0.0 0 200 400 600 800 Applied Nitrogen, kg/lia Figure 5-1 Sensitivity of logistic to the c parameter. Curves drawn with the values of b and c shown above and using Eq. [2.5].

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LIST OF REFERENCES Adby, P R and M A H Dempster. 1974. Introduction to Optimization Methods. John Wiley & Sons, Inc. New York. Allhands, MR, S.A. Allick, A R. Overman, W.G. Leseman, and W. Vidak. 1995. Municipal water reuse at Tallahassee, Florida. Tram. Am. Soc. Agr. Engr. 38:41 1418. Bates, D M. and D.G. Watts. 1988. Nonlinear Regression Analysis and Its Applications. John Wiley & Sons, Inc. New York. Blue, W.G. 1987. Response of Pensacola bahiagrass (Paspalum notatum Fliigge) to fertilizer nitrogen on an Entisol and a Spodosol in north Florida. SMI and Crop Sci. Soc. Fla. Proc. 47:135-139. Box, GEP 1976. Science and statistics. J. Amer. Slat. Assoc. 71:791-799. Box, GEP. 1979. Some problems of statistics and everyday life. J. Amer. Stat. Assoc. 74:1-4. Box, GEP., W.G. Hunter, and J.S. Hunter. 1978. Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building. John Wiley & Sons, Inc. New York. Decker, A.M., R.W. Hemkin, J R. Miller, N.A. Clark, and A.V. Okorie. 1971. Nitrogen fertilization, harvest management, and utilization of 'Midland' bermudagrass (Cynodon dactylon Pers ). University of Maryland Agricultural Experiment Station Bulletin 487. University of Maryland. College Park, MD. Doss, B.D., D A. Ashley, O.L. Bennet, and R.M. Patterson. 1966. Interactions of soil moisture, nitrogen, and clipping frequency on yield and nitrogen content of Coastal bermudagrass. Agron.J. 58:510-512. Downey, D. and A.R. Overman. 1988. Simulation models for bahiagrass. Agricultural Engineering Department, University of Florida. Gainesville, FL. Draper, N R. and H. Smith. 1981. Applied Regression Analysis. John Wiley & Sons. New York. 265

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266 Evans, EM, L.E Ensminger, B.D. Doss, and O.L. Bennett. 1961 Nitrogen and moisture requirements of Coastal bermudagrass and Pensacola bahiagrass. Alabama Agric. Exp. Stn., Bull. 337. Auburn, AL. Evers, G.W. 1984. Effect of nitrogen fertilizer, clovers, and weed control on Coastal bermudagrass and Pensacola bahiagrass in southeast Texas. Texas Agric. Exp. Stn. Bulletin MP1546. College Station, TX. France, J. and J.H.M. Thornley. 1984. Mathematical Models in Agriculture. Butterworth & Co. Ltd. London. Freund, R.J. and R.C. Littell. 1991. SAS System for Regression. SAS Institute. Cary, NC. Gell-Mann, M. 1994. The Quark and the Jaguar. W.H. Freeman and Co. New York. Hosmer, D.W. and S. Lemeshow. 1989. Applied Logistic Regression. John Wiley & Sons. New York. Huneycutt. H.J., CP. West, and J.M. Phillips. 1988. Responses of bermudagrass, tall fescue and tall fescue-clover to broiler litter and commercial fertilizer. Arkansas Agric. Exp. Stn. Bull. 913. Fayetteville, AR. Jones, J.W., J.W. Mishoe, and K.J. Boote. 1987. Introduction to simulation and modeling. Food and Fertilizer Technology Center. Tech. Bui. No. 100. Taiwan. Jones, Jr., J. B , B. Wolf, and H A. Mills. 1991. Plant Analysis Handbook. Micro-Macro Publishing, Inc. Athens, GA. Kamprath, E L. 1986. Nitrogen studies with corn on Coastal plain soils. Tech. Bull. 282. North Carolina Agric. Res. Serv, North Carolina State Univ. Raleigh, NC. Keen, R E. and J.D. Spain. 1992. Computer Simulation in Biology: A BASIC Introduction. John Wiley & Sons, Inc. New York. Kingsland, S.E. 1985. Modeling Nature. University of Chicago Press. Chicago. Mehra, J. 1994. The Beat of a Different Drum: The Life and Science of Richard Feynman. Oxford Univ. Press. New York. Mood, A.M., F A. Graybill, and D C. Boes. 1963. Introduction to the Theory of Statistics. McGraw-Hill, Inc. New York.

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267 Morrison, J., M.V. Jackson, and P E Sparrow. 1980. The response of perennial ryegrass to fertilizer nitrogen in relation to climate and soil. Grassland Research Institute Technical Report No 27. Berkshire, England. Overman, A.R. 1995a. Rational basis for the logistic model for forage grasses. ./. Plant Nutrition. 18:995-1012. Overman, A.R. 1995b. Coupling among applied, soil, root, and top components for forage crop production. Commun. Soil Sci. Plant Anal. 26: 1 179-1202. Overman, A.R., E.A. Angley, and S R. Wilkinson. 1989. A phenomenological model of Coastal bermudagrass production. Agr. Sys. 29:137-148. Overman, A.R., F.G. Martin, and S R. Wilkinson. 1990a. A logistic equation for yield response of forage grass to nitrogen. Commun. Soil Sci. Plant Anal. 2 1 : 595-609. Overman, A.R., C.R. Neff, S R. Wilkinson, and F.G. Martin. 1990b. Water, harvest interval, and applied nitrogen effects on forage yield of bermudagrass and bahiagrass. Agron.J. 82:1011-1016. Overman, A.R. and S R. Wilkinson. 1992. Model evaluation for perennial grasses in the southern United States. Agron. J. 84:523-529. Overman, A.R., and S R. Wilkinson. 1995. Extended logistic model of forage grass response to applied nitrogen, phosphorus, and potassium. Trans. Am. Soc. Agr. Engr. 38:103-108. Overman, A.R , S R. Wilkinson, and D M. Wilson. 1994a. An extended model of forage grass response to applied nitrogen. Agron. J. 86:61 7-620. Overman, A.R., D M. Wilson, and E.J. Kamprath. 1994b. Estimation of yield and nitrogen removal by corn. Agron. ./. 86:1012-1016. Prine, G.M. and G.W. Burton. 1956. The effect of nitrogen rate and clipping frequency upon the yield, protein content and certain morphological characteristics of Coastal bermudagrass [Cynodon dactylon, (L.) Pers.]. Agron.J. 48:296-301. Ratkowsky, D A. 1983. Nonlinear Regression Modeling. Marcel Dekker New York. Reck, W.R. 1992. Logistic equation for use in estimating applied nitrogen effects on corn yields. Master's Thesis University of Florida. Gainesville, FL.

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268 Reid, D. 1978. The effects of frequency of defoliation on yield response of a perennial ryegrass sward to a wide range of N application rates. J. of Agr. Science. Cambridge. 90:447-457. Robinson, D.L., K G. Wheat, N.L. Hubbert, M.S. Henderson, and H.J. Savoy, Jr. 1988. Dallisgrass yield, quality and nitrogen recovery responses to nitrogen and phosphorus fertilizers. Commun. Soil Sci. Plant Anal. 19:529-642. Rothman, M A. 1972. Disc* vering the Natural Laws: The Experimental Basis of Physics. Doubleday. New York. Segre, E. 1984. From Falling Bodies to Radio Waves. W.H. Freeman and Co. New York. Walker, M E. and D.D. Morey. 1962. Influence of rates of N, P, and K on forage and grain production of Gator rye in South Georgia. Georgia Agr. Exp. Stn. Cir. N. S. 27. University of Georgia. Athens, GA. Will, CM. 1986. Was Einstein Right? Putting General Relativity to the Test. Basic Books. New York. Williams, J R., C.A. Jones, J R. Kiniry, and D A. Spanel. 1989. The EPIC crop growth model. Trans. Am. Soc. Agr. Engr. 32:497-51 1.

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BIOGRAPHICAL SKETCH Denise M. Wilson was born May 2, 1970 in Tampa, Florida. She was very active in the Girl Scouts of America while growing up and was awarded the Gold Award (equivalent to Eagle Scout in Boy Scouts) in 1987. At Gaither High School, Denise was active in the Future Farmers of America and Mu Alpha Theta (math honor society), competing in many contests for both clubs. She graduated from Gaither High School in 1988. She enrolled at the University of Florida the following fall. She received her Bachelor of Science in Engineering (agricultural engineering) with high honors and a minor in Mathematics in May, 1992. After having been awarded a National Science Foundation (NSF) Fellowship in April of the same year, she began work in the Department of Agricultural Engineering towards a degree of Doctor of Philosophy in August 1992. In August 1993, she was also enrolled in a Master of Statistics degree program. She was married to Russell A. Wilson on June 20, 1992. In September 1995, she began working with Camp Dresser & McKee, Inc where she plans on becoming a professional engineer. At some point, she would like to teach mathematics, statistics, or engineering.

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. lien R. Overman, Chair Professor of Agricultural and Biological Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. larry^O. Bagnalr Professor of Agricultural and Biological Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. <4iJoj^ — Stanley RyWilkinson Soil Scientist, USDA-Agricultural Research Service I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Frank G. Martin Professor of Statistics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Paul A. Chadik Assistant Professor of Environmental Engineering Sciences

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This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was acceptechas partial fulfillment of the requirements for the degree of Doctor of Philosophy December 1995 TInfred M. Phillips Dean, College of Engineering Karen A. Holbrook Dean, Graduate School

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This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was acceptechas partial fulfillment of the requirements for the degree of Doctor of Philosophy December 1995 TInfred M. Phillips Dean, College of Engineering Karen A. Holbrook Dean, Graduate School


enabled me to concentrate solely on my studies. I considered it a high honor to have been
selected for this award out of the thousands of applicants in science and engineering. The
fellowship provided me with a freedom to plot my own course and research project.
v


LIST OF FIGURES
Figure page
1-1 Response of dry matter yield, N removal, and N concentration for
bermudagrass as a function of applied N grown over two years at five
clipping intervals at Tifton, GA 5
2-1 Response of linearized dry matter yield and plant N removal to applied N
for dallisgrass [Paspalum dilatatum Poir ] grown at Baton Rouge, LA 17
4-1 Response of seasonal dry matter yield for Coastal bermudagrass and
Pensacola bahiagrass to applied N at Thorsby, AL 128
4-2 Scatter plot of seasonal dry matter yield for Coastal bermudagrass and
Pensacola bahiagrass at Thorsby, AL 129
4-3 Residual plot of seasonal dry matter yield for Coastal bermudagrass and
Pensacola bahiagrass at Thorsby, AL 130
4-4 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for dallisgrass grown at Baton Rouge, LA 131
4-5 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for dallisgrass grown at Baton Rouge, LA 132
4-6 Dimensionless plot of dry matter and plant N concentration as a function
of plant N removal for dallisgrass grown at Baton Rouge, LA 133
4-7 Scatter plot of dry matter yield for dallisgrass grown at Baton Rouge, LA. .. 134
4-8 Scatter plot of plant N removal for dallisgrass grown at Baton Rouge, LA... 135
4-9 Residual plot of dry matter yield for dallisgrass grown at Baton Rouge,
LA 136
4-10 Residual plot of plant N removal for dallisgrass grown at Baton Rouge,
LA 137
xn


58
The dimensionless plot for ryegrass is shown in Figure 4-44, and this graph also
demonstrates the difficulty of variable harvest interval. The curves were drawn from
10 clippings: Y/A = 2.97 (Nu/A')/[ 1.97 + (Nu/A')] [4.124]
Nc/Ncm= 0.66+0.34 (Nu/A1) [4.125]
5 clippings: Y/A = 1.66 (K/AW0.66 + (Nu/A)] [4.126]
Nc/Ncm= 0.40+0.60 (Nu/A') [4.127]
3 clippings: Y/A = 1.40 (Nu/A'VfO^O + (Nu/A')] [4.128]
Nc/Ncm = 0.29 +0.71 (Nu/A') [4.129]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
45 through 4-48. It appears that in Figure 4-47, there may be a trend of some kind among
the residuals. It is likely that the trend is a result of the variable harvest interval.
The growing season of the rye was 26 weeks. Using this length, an average
harvest interval can be found for each of the three clipping frequencies. The estimated
maximum dry matter yield and plant N removal can be plotted against the average harvest
intervals to see if the same relationship holds for this study. There are two possible
obstacles. First, there are only three harvest intervals to plot. For the Maryland and
Tifton studies, a linear relationship was found between the expected maximum dry matter
and harvest interval. With only three points, it is difficult if not impossible to determine
the "true" relationship. Furthermore, the relationship tends to drop off after a six week
interval (Figures 4-25 and 4-37). The third average harvest interval is 8.67 weeks, and
could possibly affect the results. Secondly, these are not actual harvest intervals, but
rather an average harvest interval over the growing season. The actual harvest intervals
are variable. The effect of the variable harvest interval was observed in the figures, by
deduction it is likely to arise here as well. The plot of the estimated maximum dry matter
yield and plant N removal versus the average harvest interval is presented in Figure 4-49.


194
1
2
Z2
i>
£

3
a
2
T3
o
*3
p
20

1 -i r
o
Non-lrrigalcd l)8l-2
18
_

Non-Irri gated 1982-3
A
Non-lrrigatcd 1983-4
V
Irrigated 1981-2
16
-
o
Irrigated 1982-3
0
Irrigated 1983-4
14
-
12
-
10
O
8
-
0
w
6
-
J
4
-
/ A
2
-
V
n
.
i i i
0 2 4 6 8
Measured Dry Matter Yield. Mg/ha
Figure 4-67 Scatter plot of seasonal dry matter yield for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation Original data from
Huneycutt et al. (1988).


28
Table 3-3. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Dallisgrass Grown at Baton Rouge, Louisiana.
Applied Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
0
5.33
77
15.7
56
6.56
103
16.2
112
7.97
129
17.0
224
10.53
194
18.9
448
13.21
305
23.4
896
15.34
417
27.5
Source: Data from Robinson et al. (1988).


168
250
200
150
100
50
73
>
o
S
o
0
o
3
-a
5¡ -50
o
-100
-150
-200
-250
o
1953. 2 weeks

1953.3 weeks
A
1953,4 weeks
V
1953, 6 weeks
o
1953,8 weeks

1954, 2 weeks
-
1954, 3 weeks

1954,4 weeks
T
1954, 6 weeks

1954,8 weeks
4
^ T
A '*
V
(A
o

--O-

$
A
O
A1
S2
V
O
O O v
A
200 400
Estimated N Removal, kg/lia
600
800
Figure 4-41 Residual plot of seasonal plant N removal for bermudagrass grown over
two years at Tifton, GA, and cut at five different harvest intervals. Original
data from Prine and Burton (1956). Solid line is mean and dashed lines are
2 standard errors.


Residual K Removal, kg/ha
252
0 25 50 75 100 125 150 175 200
Estimated K Removal, kg/lia
Figure 4-125 Residual plot of plant K removal for rye grown at Tifton, GA. Original
data from Walker and Morey (1962). Solid line is mean and dashed lines
are 2 standard errors.


Normalized Response
264
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1
b = 2.0
c = 0.008
c = 0.010
c = 0.012
_
/ /
/ /
/ / /
// /
/ / /
7 / /
// /
-
/7 / /
//
/ / /
///
// /
\
1

S'
1
i J
0 200 400 600 800
Applied Nitrogen, kg/lia
Figure 5-1 Sensitivity of logistic to the c parameter. Curves drawn with the values of
b and c shown above and using Eq [2.5].


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
M A fan
in R. Overman, Chair
Professor of Agricultural and Biological
Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
). Bagnair
Professor of Agricultural and Biological
Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Stanley Rl/Wilkinson
Soil Scientist, USD A-Agricultural
Research Service
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Frank G. Martin
Professor of Statistics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Paul A. Chadik
Assistant Professor of Environmental
Engineering Sciences


235
1.50
1.25
1.00
< 0.75
0.50
0.25
0.00
50
40
<
30
20
10
0 5 10 15 20 25 30
Site
Figure 4-108 Plot of the mean and 2 standard errors of An/A and Ab for twenty sites in
England.


45
7.48 and is significant at the 2.5% confidence level. Comparison among modes 3 and 4
has a variance ratio of 9.49 and is significant at the 0.5% confidence level. Furthermore,
comparison among modes 4 and 5 has a variance ratio of 8.78 and is significant at the
0.5% confidence level. These tests suggest when the grasses are studied separately, a
single value can be used for c for all years and irrigation treatments to adequately describe
the data. Next the data from the two grasses are combined and the analysis of variance
data are presented in Table 4-3 (3 yrs x 2 grass x 2 irrigation x 4 N). A comparison of
modes 1 and 2 results in a variance ratio of 5.61 that is significant at the 0.5% confidence
level. Comparison of modes 3 and 4 results in a variance ratio of 3.27 that is significant at
the 1% confidence level. Comparison of modes 3 and 5 results in a variance ratio of 2.93
that is significant at the 5% confidence level. Comparison of modes 5 and 4 leads to a
variance ratio of 2.87 that is significant at the 2.5% confidence level. Comparison of
modes 4 and 6 leads to a variance ratio of 3.41 that is significant at the 1% confidence
level Comparison of modes 5 and 6 results in a variance ratio of 3.74 that is significant at
the 5% confidence level. Comparisons among modes 3 and 7 and 4 and 7 result in
variance ratios of 5.57 and 2.75, respectively. Both comparisons are significant at the
2.5% confidence level. Based upon these comparisons, we can conclude that mode 7,
with individual A for each year, individual b for each grass, and common c describes the
data best. The overall correlation coefficient of 0.9949 and adjusted correlation
coefficient of 0.9927 are calculated for mode 7. The error analysis of the parameters are
shown in Table 4-4. The largest relative error (¡standard error/estimate|) was for the b
parameter. This is due in part to the small numbers involved and that nonlinear regression
on the logistic equation places more emphasis upon the maximum and less upon the
intercept. Still the largest relative error for this set of parameters was under 4%. Since
the range of A parameter values (largest smallest) between years and irrigation schemes
is less than six, the data will be averaged over years. Overman et al. (1990a, 1990b) have


76
5. The error analysis for the parameters is shown in Table 4-41. Results are shown in
Figure 4-101 for both soils, where the curves are drawn from
Entisol: Y = 11.14/[1 + exp(1.39 0.0118N)] [4.276]
N = 311.0/[1 + exp(1.86 0.0118N)] [4.277]
Spodosol: Y = 19.39/[1 + exp(1.39 0.0118N)] [4.278]
Nu = 201.5/p +exp(1.86-0.0118N)] [4.279]
From these results, it follows that plant N concentration, shown in Figure 4-101, is
estimated from
Entisol: Nc =27.9 [1 + exp(l.86 0.0118N)]/[1 + exp(1.39 0.0118N)] [4.280]
Spodosol: Nc =10.4 [1 + exp(1.86 0.0118N)]/[1 + exp(1.39 0.0118N)] [4.281]
As shown, the data are described well by these equations.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal for both soils, shown in Figure 4-102,
are described by
Entisol: Y= 29.7 N/(518 + Nu) [4.282]
Nc= 17.4 + 0.0337NU [4.283]
Spodosol: Y = 51.7 Nu/(336 + Nu) [4.284]
Nc= 6.5 + 0.0193NU [4.285]
The dimensionless plot for bahiagrass is shown in Figure 4-103, and validates the
form of the model. Note that all of the data have been collapsed onto one graph, and the
data fall on the curve. The curves were drawn from
Y/A = 2.67 (Nu/A')/[1.67 + (Nu/A')] [4.286]
Nc/Ncm = 0.63 +0.37 (Nu/A1) [4.287]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
104 through 4-107.


Dry Matter Yield. Mg/ha N Removal, kg/ha N Concentration, g/kg
191
Figure 4-64 Response of seasonal dry matter yield, plant N removal and plant N
concentration for tall fescue grown over three years at Fayetteville, AR,
with and without irrigation. Data from Huneycutt el al (1988); curves
drawn from Eq [4.176] through [4 193],


121
Table 4-41. Error Analysis for Model Parameters for Bahiagrass Grown on Two
Soils: an Entisol and Spodosol at Williston and Gainesville, FL,
Component
Soil
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter Yield
Entisol
A, Mg/ha
11.14
0.176
0.016
Spodosol
A
19.39
0.307
0.016
N Removal
Entisol
A, kg/ha
311.0
5.84
0.019
\
Spodosol
A
201.5
3.76
0.019
Dry Matter Yield
Both
b
1.39
0.053
0.038
N Removal
Both
b
1.86
0.060
0.032
Both
Both
c, ha/kg
0.0118
0.0004
0.034
Source: Original data from Blue (1987).


67
Nc =
16.9 + 0.0481NU
[4.215]
Irrigated.
Y =
36.8 N/(583 + N)
[4.216]
Nc =
15.8 + 0.0272NU
[4.217]
The dimensionless plot for tall fescue is shown in Figure 4-73, and validates the
form of the model. Note that all of the data have been collapsed onto one graph, and the
data fall on the curve. The curves were drawn from
Y/A = 2.40 (Nu/A')/[1.40 + (Nu/A')] [4.218]
Nc/Ncm 0.58 +0.42 (Nu/A*) [4.219]
Notice the similarity between the above equations and Equations [4.174] and [4.175],
The Ab values are similar for both grasses. It is not clear if this is a coincidence or a result
of the system. Scatter and residual plots of dry matter yield and plant N removal are
shown in Figures 4-74 through 4-77.
Eagle Lake. TX: bermudagrass and bahiagrass
Data from Evans et al. (1961) for bermudagrass and bahiagrass were used. First
the grasses will be analyzed separately, then they will be pooled to see if any of the
parameters are common for the two grasses. The analysis of variance for the Coastal
bermudagrass over years 1979-1980 is shown in Table 4-27 (dry matter and N removal x
2 yrs x 5 N). Comparison among modes 1 and 2 results in a variance ratio of 419 that is
significant at the 0.1% level. Comparison among modes 3 and 2 leads to a variance ratio
of 9.22 that is significant at the 0.5% level. Comparison among modes 3 and 4 results in a
variance ratio of 14.6 that is significant at the 0.1% level. Comparison among modes 5
and 4 leads to a variance ratio of 10.4 that is significant at the 0.5% level. As a result,
mode 4, individual A and b for yield and plant N removal at each year and common c,
accounts for all the significant variation, since F(3,8,95) = 4.07, F(3,l 1,99.9) = 11.56, and
F(2,11,99.5) = 8.91. The analysis of variance for the Pensacola bahiagrass over years


CHAPTER 1
INTRODUCTION
In today's society, there is an increasing focus on environmental issues. Concerns
are being raised about pollution in soil, water and air. One of the major pollution concerns
in Florida is nitrate and phosphate contamination in the aquifer and lakes, respectively.
Engineers are using nutrient budgets to ensure that excess nutrients are not applied and to
help control and eliminate contamination. Often nutrients are applied to forage grasses as
treated wastes or effluent. Many municipalities are using water reuse systems as a way to
remove high levels of nutrients from treated reclaimed water (Allhands et al., 1995). This
process is beneficial to two main parties, those who wish to clean the water and return it
to the aquifer and those who benefit from the addition of the nutrients to their system.
Since a large amount of the required nutrients is applied in the water, less fertilizer is
needed. The effluent is often applied to forage grasses, golf courses and lawns. A simple
procedure is needed to assist engineers in estimating and predicting for various forage
grasses the amount of nutrient removed and dry matter produced given a specific nutrient
application rate.
This research project primarily deals with various forage grasses and their nitrogen
response curves. Consider the response data presented in Figure 1-1. The data were
taken from a study by Prine and Burton (1956). Dry matter yield, plant N removal, and
plant N concentration was recorded for bermudagrass [CynoJon dactylon] grown at
Tifln, GA, for two years at five harvest intervals. As expected, there is a relationship
between yield, N removal, and N concentration to applied N, but what exactly is the
relationship? Linear, hyperbolic, quadratic, sigmoid? If one form of an equation would
1


94
Table 4-14. Error Analysis for Model Parameters of Bermudagrass Grown at Tifton,
GA over Two Years and Cut at Five Different Harvest Intervals.
Year
Component
Harvest
Interval
Parameter
Estimate
Standard
Error
Relative
Error
1953
Dry Matter
2, weeks
A, Mg/ha
17.95
0.52
0.029
3
A
19.88
0.57
0.029
4
A
23.15
0.66
0.029
6
A
29.57
0.85
0.029
8
A
29.58
0.85
0.029
1954
Dry Matter
2
A
8.40
0.24
0.029
3
A
9.95
0.28
0.028
4
A
11.67
0.33
0.028
6
A
14.37
0.41
0.029
8
A
16.24
0.46
0.028
1953
N Removal
2
A, kg/ha
644.7
20.4
0.032
3
A
641.8
20.2
0.031
4
A
687.2
21.5
0.031
6
A
688.2
21.6
0.031
8
A
606.0
19.0
0.031
1954
N Removal
2
A
324.5
10.1
0.031
3
A
337.5
10.6
0.031
4
A
348.0
10.9
0.031
6
A
363.8
11.3
0.031
8
A
379.5
11.9
0.031
Both
Dry Matter
All
b
1.47
0.060
0.041
Both
N Removal
All
b
2.15
0.077
0.036
Both
Both
All
c, ha/kg
0.0077
0.0003
0.039
Source: Original data from Prine and Burton (1956).


Dry Matter Yield/Estimated Maximum Nitrogen Concentration/Estimated Maximum
223
Nitrogen Removal/Estinialed Maximum
Figure 4-96 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for grain and total plant of corn grown at
Plymouth, NC. Original data from Kamprath (1986); curves drawn from
Eq. [4.274] and [4 275]


37
Table 3-12. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Grain and Total Plant of Corn Grown on Portsmouth Soil at Plymouth,
North Carolina.
Part
Applied Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
Grain
0
4.53
48
10.6
56
6.30
68
10.8
112
7.65
89
11.6
168
8.52
104
12.2
224
9.04
115
12.7
Total
0
8.93
60
6.7
56
11.63
83
7.1
112
13.39
101
7.5
168
14.68
121
8.2
224
15.10
133
8.8
Source: Data from Kamprath (1986).


220
Figure 4-93 Residual plot of seasonal plant N removal for grain and total plant of corn
grown at Clayton and Kinston, NC, on Dothan and Goldsboro soils,
respectively. Original data from Kamprath (1986). Solid line is mean and
dashed lines are 2 standard errors.


103
Table 4-23. Error Analysis for Model Parameters of Tall Fescue Grown at
Fayetteville, AR, over Three Seasons, with and without Irrigation.
Component
Irrigation
Year
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter
No
1981-2
A, Mg/ha
12.08
0.38
0.031
1982-3
A
8.01
0.25
0.031
1983-4
A
5.71
0.17
0.030
Yes
1981-2
A
15.63
0.49
0.031
1982-3
A
13.22
0.42
0.032
1983-4
A
16.81
0.53
0.032
N Removal
No
1981-2
A, kg/ha
357.5
12.5
0.035
1982-3
A
217.8
7.6
0.035
1983-4
A
169.5
5.7
0.034
Yes
1981-2
A
443.7
15.4
0.035
1982-3
A
357.3
12.6
0.035
1983-4
A
439.4
15.3
0.035
Dry Matter
Both
all
b
0.92
0.084
0.091
N Removal
Both
all
b
1.47
0.095
0.065
Both
Both
all
c, ha/kg
0.0081
0.0005
0.062
Source: Original data from Huneycutt etal. (1988).


8
all N: dy/dN = k y (ym y) [2.3]
This is the form of the logistic model where k is the N response coefficient. It is a
nonlinear first order differential equation. The logistic equation is well behaved; that is,
the function is continuous, smooth, and asymptotic at both ends of the range. Recall the
analogy to the rumor model. The product of the dry matter present (filled capacity) and
that still to be produced (unfilled) is present in the differential form as well. Also, the total
capacity of the system (y + ym y = ym) is assumed constant. Furthermore, as Overman
(1995a) has pointed out, the logistic equation reduces to the exponential model at lower N
and to the Mitscherlich model at higher N. If the logistic is normalized by defining new
variables O = y/A and ^ = cN b and expanded by Taylor series, a quadratic term is not
included, suggesting a parabola would not be appropriate. Furthermore, the logistic is
approximated by a linear function extremely well in the middle of the range.
The logistic model exhibits sigmoid behavior and has three parameters as shown
below
y-
\ + e
b-cN
[2.4]
Parameters A and c are scaling coefficients for yield and applied N, respectively.
Parameter b describes the reference state at N = 0. In the context of this model,
parameters A and c are constrained to be positive, but parameter b can be either positive,
zero, or negative. Parameter b equal to zero suggests that there is enough background
level of nitrogen in the soil to reach half of the maximum yield. Parameter b less than zero
suggests that there is enough nitrogen in the soil before fertilization to reach more than
halfway up the response curve. The model can also be normalized as shown
y 1
A \ + eb~cN
[2.5]
and linearized as shown


Drv Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
236
N Removal/Esiimated Maximum
Figure 4-109 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for twenty sites in England. Original data
from Morrison el al. (1980); curves drawn from Eq. [4.288] and [4.289],


43
Table 3-15. Seasonal Dry Matter Yields and Nutrient Response to Applied Nutrients for
Gator Rye at Tifton, Georgia.
N
kg/ha
P
kg/ha
K
kg/ha
Y
Mg/ha
Nu
kg/ha
Pu
kg/ha
Ku
kg/ha
Nc
g/kg
Pc
g/kg
Kc
g/kg
0
40
74
0.55
-
-
-
45
40
74
1.81
58
12.1
66
32.3
6.69
36.4
90
40
74
3.01
112
19.6
108
37.3
6.51
36.0
135
40
74
3.75
160
24.4
136
42.6
6.51
36.4
180
40
74
4.01
175
26.8
146
43.7
6.69
36.4
225
40
74
4.55
216
28.6
148
47.5
6.29
32.5
135
0
74
2.09
92
13.0
75
44.1
6.21
35.8
135
20
74
3.08
134
17.0
118
43.5
5.51
38.3
135
40
74
3.36
133
21.1
119
39.5
6.29
35.4
135
60
74
3.74
146
24.3
141
39.1
6.51
37.8
135
80
74
4.00
148
21.7
134
37.0
5.42
33.4
135
100
74
3.97
-
-
-
-
-
-
135
40
0
2.87
112
27.6
62
39.0
9.61
21.7
135
40
37
3.18
116
15.3
90
36.6
4.81
28.3
135
40
74
3.74
140
24.3
133
37.3
6.51
35.5
135
40
111
3.56
134
18.8
130
37.5
5.29
36.4
135
40
148
3.96
160
24.2
150
40.4
6.12
37.8
135
40
185
4.11
162
31.2
159
39.4
7.60
38.8
Source: Data from Walker and Morey (1962).


Dry Matter Yield, Mg/ha N Concentration, g/kg
185
Figure 4-58 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bermudagrass averaged over three years at Fayetteville, AR,
with and without irrigation Data from Huneycutt et aL (1988); curves
drawn from Eq. [4.170] through [4.173],


73
Goldsboro, Grain: Y= 15.6 N/(l 12 + Nu)
Dothan, Grain: Y= 22.1 N/( 150 + Nu)
Total: Y = 41.1 Nu/(185+Nu)
Total: Y = 29.2N,/(135+NU)
Nc= 6.8 + 0.0453NU
Nc= 4.5 + 0.0243K,
Nc= 7.2 + 0.0643K,
Nc= 4.6 + 0.0342NU
[4.254]
[4.255]
[4.256]
[4.257]
[4.258]
[4.259]
[4.260]
[4.261]
The dimensionless plot for com is shown in Figure 4-89, and validates the form of
the model. Note that all of the data have been collapsed onto one graph, and the data fall
on the curve. The curves were drawn from
[4.262]
[4.263]
Y/A = 1.99 (Nu/A')/[0.99 + (N^A')]
Nc/Ncm = 0.50+0.50 (Nu/A')
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
90 through 4-93.
Plymouth. NC: Corn
Data from Kamprath (1986) for corn were used. The analysis of variance for the
Plymouth very fine sandy loam is shown in Table 4-38 (dry matter and N removal x 2
components x 5 N). Comparisons among modes 1 and 2, modes 3 and 2, modes 3 and 4,
and modes 3 and 5 result in variance ratios of 5819, 32.5, 27.8, and 57.4, which are all
significant at the 0.1% confidence level. Comparison among modes 5 and 4 leads to a
non-significant variance ratio of 3.21. Asa result, mode 5, individual A for grain and total
plant, individual b for yield and plant N removal, and common c, accounts for all the
significant variation, since F(5,8,97.5) = 4.82, F( 1,13,99.9) = 17.81, and F(2,11,95) =
3.98. The overall correlation coefficient of 0.9997 and adjusted correlation coefficient of


102
Table 4-22. Analysis of Variance of Model Parameters for Tall Fescue Grown at
Fayetteville,
AR, over Three Seasons, with and wi
thout Irrigation.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
81
1401206.34
17298.84
-
(2) Ind A,b,c
36
48
5767.59
120.16
-
(D-(2)
33
1395438.76
42286.023
352
(3) Ind A, Com b,c
14
70
25848.47
369.26
-
(3)-(2)
22
20080.88
912.77
7.60
(4) Ind A,b Com c
25
59
9085.81
154.00
-
(4)-(2)
11
3318.22
301.66
2.5 lf
(3)-(4)
11
16762.66
1523.88
9.90
(5) Ind A, Com c, Ind b
(Irr, dm, Nu)
17
67
14215.61
212.17
.
(5)-(2)
19
8448.03
444.63
3.70
(3)-(5)
3
11632.86
3877.62
18.3
(5)-(4)
8
5129.80
641.23
4.16
(6) Ind A, Com c, Ind b
(dm, Nu)
15
69
14425.24
209.06
(6)-(2)
21
8657.66
412.27
3.43
(3)-(6)
1
11423.23
11423.23
565
(6)-(4)
10
5339.44
533.94
3.47f
(6)-(5)
2
209.63
104.82
0.49
Source: Original data from Huneycutt et al. (1988).
Significant at the 0.001 level
f Significant at the 0.025 level
F(33,48,99.9)= 2.66
F(22,48,99.9)= 2.91
F(11,48,97.5)= 2.27
F(11,59,99.9)= 3.43
F( 19,48,99.9)= 3.02
F( 3,67,99.9)= 6.09
F( 8,59,99.9)= 3.88
F(21,48,99.9)= 2.94
F( 1,69,99.9)= 11.81
F( 10,59,97.5)=2.27
F( 2,67,95) =3.13


124
Table 4-43-continued
Site Component
Parameter
Estimate
Standard Error
Relative Error
5 Dry Matter
b
0.34
0.117
0.344
6
b
0.65
0.106
0.163
7
b
0.98
0.099
0.101
8
b
1.68
0.101
0.060
9
b
0.38
0.110
0.289
10
b
1.04
0.102
0.098
12
b
2.10
0.110
0.052
13
b
0.52
0.108
0.208
14
b
0.93
0.103
0.111
15
b
1.60
0.107
0.067
16
b
0.58
0.103
0.178
17
b
1.64
0.090
0.055
19
b
1.09
0.101
0.093
20
b
1.38
0.108
0.078
22
b
1.31
0.102
0.078
23
b
1.14
0.102
0.089
25
b
0.91
0.111
0.122
26
b
1.59
0.109
0.069
27
b
0.96
0.106
0.110
28
b
1.92
0.111
0.058
5 N Removal
b
1.19
0.121
0.102
6
b
1.68
0.113
0.067
7
b
1.57
0.106
0.068
8
b
2.55
0.117
0.046
9
b
1.13
0.113
0.100
10
b
1.80
0.108
0.060
12
b
2.77
0.121
0.044
13
b
1.39
0.110
0.079
14
b
1.78
0.117
0.066
15
b
2.47
0.123
0.050
16
b
1.50
0.109
0.073
17
b
2.39
0.111
0.046
19
b
1.90
0.107
0.056
20
b
2.29
0.123
0.054
22
b
2.03
0.110
0.054
23
b
1.93
0.111
0.058
25
b
1.86
0.119
0.064
26
b
2.45
0.118
0.048
27
b
1.91
0.120
0.063
28
b
2.69
0.124
0.046


Dry Matter Yield, Mg/ha Nitrogen Concentration, g/kg
215
Figure 4-88 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for grain and total plant of com grown at Clayton and Kinston,
NC, on Dothan and Goldsboro soils, respectively. Data from Kamprath
(1986); curves drawn from Eq. [4.254] through [4.261],


Residual Dry Matter Yield, Mg/ha
136
1.5
1.0
0.5
0.0
-0.5
O
o
0)
o
o
o
-1.0 -
-1.5
5 10 15
Predicted Dry' Matter Yield, Mg/ha
20
Figure 4-9 Residual plot of dry matter yield for dallisgrass grown at Baton Rouge,
LA. Original data from Robinson el al. (1988). Solid line is mean and
dashed lines are 2 standard errors.


187
0
0

10 15
Measured Dry Matter Yield. Mg/ha
Figure 4-60 Scatter plot of seasonal dry matter yield for bermudagrass averaged over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt el al. (1988).


4
that still needs to be produced. As the number of people in the room was assumed
constant with the rumor model, the potential dry matter production of the grass is constant
as well: the sum of that already produced and that still to be produced. Also, the rate of
growth is a product of what has been produced and what is still to be produced, as the
rate of rumor transfer was a product of those who knew with those who did not know.
A nonlinear regression analysis of the variables involved will be used to determine
the values of the parameters for a logistic model of forage grass response to applied
nitrogen assuming different relationships among the model parameters. The results of this
analysis will be compared using analysis of variance for the different assumptions to
determine which is correct and if a simplification can be made. The result will be a form
of an equation for nutrient response to various forage grasses that will adequately describe
the relationship of dry matter yield, plant nutrient removal, and plant nutrient
concentration to applied nutrients. This model will assist engineers and planners in
estimating plant nutrient removal for various application rates.
The result of this analysis should be the production of a model for engineers and
managers to use when determining nutrient budgets under varying conditions. The model
will estimate seasonal totals and use a "black box approach". No attempt will be made to
model the physical mechanisms of the plant. Once many different studies have been
conducted, a survey of the parameters for various forage grasses might provide a useful
insight to the internal mechanisms of the plant, as well as connect the parameters to the
physical system, but this is not pursued in this study.


143
1
2
2
"o
£

g
B
C3
J3
- -1

o
0 -
-2 -
G

3.0 weeks
4.5 weeks
O


TJ
-GO:
@


--O-
10 15
Predicted Dry Matter Yield, Mg/ha
20
25
Figure 4-16 Residual plot of dry matter yield for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals. Original data from Doss et al. (1966).
Solid line is mean and dashed lines are 2 standard errors.


87
Table 4-7. Analysis of Variance of Model Parameters Used to Describe Dallisgrass
Grown at Baton Rouge, LA.]
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
9
154655.03
17183.89
-
(2) Ind A,b,c
6
6
82.72
13.79
-
(1 M2)
3
154572.31
51524.10
3736**
(3) Ind A, Com b,c
4
8
3606.97
450.87
-
(3)-(2)
2
3524.25
1762.12
127.8**
(4) Ind A,b Com c
5
7
122.54
17.51
-
(4)-(2)
1
39.82
39.82
2.89
(3)-(4)
1
3484.43
3484.43
199.0**
Source: Original data from Robinson et al. (1988).
Significant at the 0.001 level
F(3,6,99.9) 23.70
F(2,6,99.9) = 27.00
F( 1,6,95) = 5.99
F( 1,7,99.9) = 29.25


Drv Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
163
N Removal/Estimaled Maximum
Figure 4-36 Dimensionless plot of seasonal dry matter yield and seasonal plant N
concentration versus seasonal plant N removal. Original data from Prine
and Burton (1956); curves drawn from Eq. [4.103] and [4.104],


75
Florida: Bahiagrass
Data from Blue (1987) for bahiagrass were used. The analysis of variance for the
bahiagrass grown on the Entisol and Spodosol is shown in Table 4-40 (dry matter and N
removal x 2 soils x 5 N). Comparisons among modes 1 and 2 result in a variance ratio of
1792, that is significant at the 0.1% confidence level. Comparisons among modes 3 and 2
lead to a variance ratio of 12.24, that is significant at the 0.5% confidence level.
Comparisons among modes 3 and 4 lead to a variance ratio of 6.41, that is significant at
the 1% confidence level. Comparisons among modes 5 and 2 lead to a variance ratio of
10.72, that is significant at the 0.5% confidence level. Comparison among modes 5 and 4
leads to a variance ratio of 5.94, that is significant at the 2.5% confidence level. The
statistics suggest that mode 4, individual A for both soils, individual b for yield and plant
N removal, and common c, accounts for all the significant variation, since F(3,8,97.5) =
5.42, and F(3,l 1,99) = 6.22. This mode estimates the following b and c parameters:
Entisol, dry matter: b = 1.46 0.07
Spodosol, dry matter: b =1.31 0.07
Entisol, N removal: b' = 1.89 0.08
Spodosol, N removal: b' =1.83 0.08
c = 0.0118
while mode 5 estimates the following b and c parameters:
both soils, dry matter: b = 1.39 0.05
both soils, N removal: b' = 1.86 0.06
c = 0.0118
The estimates and their standard errors overlap, suggesting that mode 5 is correct.
Furthermore, since there are less parameters to estimate assuming the b values are not
affected by the soils, mode 5 will be used for estimation. The overall correlation
coefficient of 0.9985 and adjusted correlation coefficient of 0.9978 are calculated by mode


117
Table 4-37. Error Analysis of Model Parameters for Grain and Total Plant of Corn
Grown on Dothan and Goldsboro Sandy Loam at Clayton and Kinston, NC.
Component
Site
Part
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter Yield
Dothan
Grain
A, Mg/ha
11.12
0.156
0.014
Total
A
20.68
0.288
0.014
Goldsboro
Grain
A
7.83
0.109
0.014
Total
A
14.70
0.204
0.014
N Removal
Dothan
Grain
A, kg/ha
151.7
2.71
0.018
Total
A
187.2
3.35
0.018
Goldsboro
Grain
A
113.3
2.00
0.018
Total
A
136.5
2.42
0.018
Dry Matter Yield
Both
Both
b
0.27
0.040
0.148
N Removal
Both
Both
b
0.97
0.044
0.045
Both
Both
Both
c, ha/kg
0.0187
0.0007
0.037
Source: Original data from Kamprath (1986).


Dry Matter Yield. Mg/ha N Removal, kg/ha N Concentration, g/kg
205
40
30
20
10
0
250
200
150
100
50
0
15
10
5
0
0 50 100 150 200 250 300 350 400
Applied Nitrogen, kg/ha
n
o
(>

rv
FT
Figure 4-78 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bermudagrass grown over two years at
Eagle Lake, TX. Data from Evers (1984); curves drawn from Eq. [4.220]
through [4.223], [4.228] and [4 229]


Residual P Removal, kg/ha
251
0 5 10 15 20 25 30
Estimated P Removal, kg/lia
Figure 4-124 Residual plot of plant P removal for rye grown at Tifton, GA. Original
data from Walker and Morey (1962). Solid line is mean and dashed lines
are 2 standard errors.


165
30
25
a
*s
an
2
2
£
u
U
b
Q
T3
o
W
20
15
10
o
1953, 2 weeks

1953,3 weeks
A
1953,4 weeks
V
1953, 6 weeks
o
1953,8 weeks

1954, 2 weeks

1954, 3 weeks

1954, 4 weeks
T
1954, 6 weeks

1954,8 weeks
r
&
X
/


/
o
o if
/
T VO
Q
V/^
a a
10
15
20
25
30
Measured Dry Matter Yield, Mg/ha
Figure 4-38 Scatter plot of seasonal dry matter yield for bermudagrass grown over two
years at Tifln, GA, and cut at five different harvest intervals. Original
data from Prine and Burton (1956).


Drv Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
193
4:i *
(;i Non-lrrigatcd 1981-2
Non-Irrigated 1982-3
A Non-lrrigatcd 1983-4
Irrigated 1981-2
Irrigated 1982-3
m Irrigated 1983-4
Q
i:)
0.8
1.0
1.2
Nitrogen Removal/Estimated Maximum
Figure 4-66 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for tall fescue grown over three years at
Fayetteville, AR, with and without irrigation. Original data from
Huneycutt et al. (1988); curves drawn from Eq. [4.206] and [4.207],


98
Table 4-18. Error Analysis for Model Parameters of Bermudagrass Grown at
Fayetteville, AR, over Three Years, with and without Irrigation.
Type
Irrigation
Year
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter
No
1983
A, Mg/ha
17.90
0.41
0.023
1984
A
17.37
0.39
0.022
1985
A
19.61
0.45
0.023
Yes
1983
A
24.70
0.56
0.023
1984
A
24.60
0.56
0.023
1985
A
22.58
0.51
0.023
N Removal
No
1983
A, kg/ha
408.7
10.4
0.025
1984
A
413.9
10.5
0.025
1985
A
459.3
11.8
0.026
Yes
1983
A
554.3
14.1
0.025
1984
A
523.7
13.4
0.026
1985
A
492.1
12.6
0.026
Dry Matter
Both
all
b
1.50
0.063
0.042
N Removal
Both
all
b
2.04
0.072
0.035
Both
Both
all
c, ha/kg
0.0084
0.0003
0.036
Source: Original data from Huneycutt etal. (1988).


137
Figure 4-10 Residual plot of plant N removal for dallisgrass grown at Baton Rouge,
LA. Original data from Robinson el at. (1988). Solid line is mean and
dashed lines are 2 standard errors.


ESTIMATION OF DRY MATTER AND NITROGEN REMOVAL
BY THE LOGISTIC EQUATION
By
DENISE MARIE WILSON
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1995
UNIVERSITY of FLORIDA LIBRARIES

Copyright 1995
by
Denise M. Wilson

I dedicate this dissertation to my parents, Robert and Marian Shelton, my second-
parents, Brooks and Carol Wilson, and especially to my husband, Russell.

ACKNOWLEDGMENTS
First, I would like to thank the many field plant-soil science researchers throughout
the world for having well-designed experiments resulting in great data. Developing and
maintaining a field plot experiment are no easy task and I wish to recognize and thank
those people.
I would like to thank my husband, Russell, for the many hours of brainstorming and
editing. He was a stabilizing influence during a swift three years of graduate work, and I thank
him for the encouragement he provided during my weak moments.
I would like to thank my parents, Robert and Marian Shelton, for the many sacrifices
that were done in order that I might be able to attend college in the first place. Furthermore, I
would like to thank them for instilling in me a sense of honor, a hard work ethic, and
determination that I could accomplish anything.
I would like to thank my major professor and advisor over the last five years, Dr. Allen
Overman. He instilled in me a sense of professionalism and always demanded high quality
work. I am thankful for the opportunity I was given the summer of 1991 to work with him,
since this work helped establish the foundation for my graduate work.
I would like to give my appreciation to the members of my committee, Dr. Larry
Bagnall, Dr. Stanley Wilkinson, Dr. Paul Chadik, and Dr. Frank Martin for giving their time. I
would like to give special thanks to Dr. Frank Martin for also serving as my committee chair on
my concurrent master's program in statistics. Your wisdom, guidance, and encouragement
were greatly appreciated.
I would also like to thank the National Science Foundation for selecting me as a fellow
for the period of August 1992 to August 1995. The stipend and cost of education allowance
IV

enabled me to concentrate solely on my studies. I considered it a high honor to have been
selected for this award out of the thousands of applicants in science and engineering. The
fellowship provided me with a freedom to plot my own course and research project.
v

TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iv
LIST OF TABLES vii
LIST OF FIGURES xii
ABSTRACT xxii
CHAPTERS
1 INTRODUCTION 1
2 LITERATURE REVIEW 6
3 MATERIALS AND METHODS 18
Analysis of Data 18
Models to be Investigated 20
Data Sets to be Investigated 21
4 RESULTS AND DISCUSSION 44
Evaluation of the Simple Logistic Model 44
Evaluation of the Extended Logistic Model 46
Evaluation of the Extended Triple Logistic (NPK) Model 78
5 SUMMARY AND CONCLUSIONS 253
REFERENCES 265
BIOGRAPHICAL SKETCH 269
vi

LIST OF TABLES
Table page
3-1 Common and Scientific Names of Grasses Studied 26
3-2 Seasonal Dry Matter Yield of Coastal Bermudagrass and Pensacola Bahiagrass
Grown on Greenville Fine Sandy Loam at Thorsby, Alabama 27
3-3 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Dallisgrass Grown at Baton Rouge, Louisiana 28
3-4 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bermudagrass Grown at Thorsby, Alabama over 3.0 and 4.5 Week Harvest
Intervals (1961) 29
3-5 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Dallisgrass Grown in Maryland 30
3-6 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bermudagrass Grown at Tifln, Georgia 31
3-7 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Perennial Ryegrass Grown in England with a Different Number of Harvests
over the Season 32
3-8 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bermudagrass Grown at Fayetteville, Arkansas over Three Years, Irrigated and
Non-irrigated 33
3-9 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Tall Fescue Grown at Fayetteville, Arkansas over Three Seasons, Irrigated and
Non-irrigated 34
3-10 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bahiagrass and Bermudagrass Grown at Eagle Lake, Texas 35
3-11 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Grain and Total Plant of Com Grown on Dothan and Goldsboro Soils at Clayton
and Kinston, North Carolina, Respectively 36
vii

3-12 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Grain and Total Plant of Com Grown on Portsmouth Soil at Plymouth, North
Carolina 37
3-13 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bahiagrass Grown on Entisol and Spodosol Soils in Florida 38
3-14 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Ryegrass Grown in England 39
3-15 Seasonal Dry Matter Yields and Nutrient Response to Applied Nutrients for
Gator Rye at Tifton, Georgia 43
4-1 Analysis of Variance of Model Parameters Used to Describe Coastal
Bermudagrass Yield Response to Nitrogen at Thorsby, Alabama, 1957-
1959 81
4-2 Analysis of Variance of Model Parameters Used to Describe Pensacola
Bahiagrass Yield Response to Nitrogen at Thorsby, Alabama, 1957-1959 82
4-3 Analysis of Variance of Model Parameters Used to Describe Coastal
Bermudagrass and Pensacola Bahiagrass Yield Response to Nitrogen at
Thorsby, Alabama, 1957-1959 83
4-4 Error Analysis for Model Parameters of Coastal Bermudagrass and
Pensacola Bahiagrass Grown at Thorsby, Alabama 84
4-5 Seasonal Dry Matter Yield for Coastal Bermudagrass and Pensacola
Bahiagrass Grown at Thorsby, Alabama Averaged over Years 1957-1959 85
4-6 Error Analysis for Model Parameters on Averaged Dry Matter Yield of
Coastal Bermudagrass and Pensacola Bahiagrass Grown at Thorsby,
Alabama 86
4-7 Analysis of Variance of Model Parameters Used to Describe Dallisgrass
Grown at Baton Rouge, LA 87
4-8 Error Analysis for Model Parameters of Dallisgrass Grown at Baton Rouge,
LA 88
4-9 Analysis of Variance of Model Parameters Used to Describe Bermudagrass
Grown at Thorsby, AL with Two Clipping Intervals 89
viu

4-10 Error Analysis of Model Parameters of Bermudagrass Grown at Thorsby,
AL 90
4-11 Analysis of Variance of Model Parameters for Bermudagrass Grown at
Maryland and Cut at Five Harvest Intervals 91
4-12 Error Analysis for Model Parameters of Bermudagrass Grown at Maryland
and Cut at Five Harvest Intervals 92
4-13 Analysis of Variance on Model Parameters for Bermudagrass Grown at
Tifton, GA, over Two Years and Cut at Five Different Harvest Intervals 93
4-14 Error Analysis for Model Parameters of Bermudagrass Grown at Tifton, GA
over Two Years and Cut at Five Different Harvest Intervals 94
4-15 Analysis of Variance of Model Parameters on Ryegrass Grown at England,
with Three Different Numbers of Cuttings over the Season for 1969 95
4-16 Error Analysis for Model Parameters of Ryegrass Grown at England, with
Three Different Numbers of Cuttings over the Season for 1969 96
4-17 Analysis of Variance for Bermudagrass over Three Seasons, Irrigated and
Non-irrigated, Grown at Fayetteville, AR 97
4-18 Error Analysis for Model Parameters of Bermudagrass Grown at
Fayetteville, AR, over Three Years, with and without Irrigation 98
4-19 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bermudagrass Grown at Fayetteville, AR. Averaged over Three Years 99
4-20 Analysis of Variance for Model Parameters for Bermudagrass Grown at
Fayetteville, AR, Averaged over Three Years, with and without Irrigation 100
4-21 Error Analysis for Model Parameters of Bermudagrass Grown at
Fayetteville, AR, Averaged over Three Years, with and without Irrigation 101
4-22 Analysis of Variance of Model Paramters for Tall Fescue Grown at
Fayetteville, AR, over Three Seasons, with and without Irrigation 102
4-23 Error Analysis for Model Parameters of Tall Fescue Grown at Fayetteville,
AR, over Three Seasons, with and without Irrigation 103
4-24 Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Tall Fescue Grown at Fayetteville, AR, Averaged over Three Seasons 104
IX

4-25 Analysis of Variance of Model Parameters for Tall Fescue Grown at
Fayetteville, AR, Averaged over Three Seasons, with and without Irrigation. 105
4-26 Error Analysis for Model Parameters of Tall Fescue Grown at Fayetteville,
AR, Averaged over Three Seasons, with and without Irrigation 106
4-27 Analysis of Variance for Bermudagrass Grown at Eagle Lake, Texas, over
Two Years 107
4-28 Analysis of Variance for Bahiagrass Grown at Eagle Lake, Texas, over Two
Years 108
4-29 Analysis of Variance for Bermudagrass and Bahiagrass Grown at Eagle
Lake, Texas, over Two Years 109
4-30 Averaged Seasonal Dry Matter Yield, Plant N Removal, and Plant N
Concentration for Bermudagrass and Bahiagrass Grown at Eagle Lake, TX.. 110
4-31 Analysis of Variance on Model Parameters for Bermudagrass and Bahiagrass
Grown at Eagle Lake, TX, Averaged to Estimate b and c Parameters Ill
4-32 Error Analysis for Model Parameters of Bermudagrass and Bahiagrass
Grown at Eagle Lake, TX Averaged over Years 112
4-33 Error Analysis for Model Parameters of Bermudagrass and Bahiagrass
Grown at Eagle Lake, TX over Two Years 113
4-34 Analysis of Variance on Model Parameters for Corn Grown on Dothan
Sandy Loam at Clayton, NC, Both Grain and Total 114
4-35 Analysis of Variance on Model Parameters for Corn Grown on Goldsboro
Sandy Loam at Kinston, NC, Both Grain and Total 115
4-36 Analysis of Variance on Model Parameters for Grain and Total Plant of Corn
Grown on Dothan and Goldsboro Sandy Loam at Clayton and Kinston, NC.. 116
4-37 Error Analysis of Model Parameters for Grain and Total Plant of Corn
Grown on Dothan and Goldsboro Sandy Loam at Clayton and Kinston, NC.. 117
4-38 Analysis of Variance on Model Parameters for Grain and Total Plant of Corn
Grown on Portsmouth Very Fine Sandy Loam at Plymouth, NC 118
4-39 Error Analysis for Model Parameters for Grain and Total Plant of Corn
Grown on Portsmouth Very Fine Sandy Loam at Plymouth, NC 119
x

4-40 Analysis of Variance on Model Parameters for Bahiagrass Grown on Two
Soils: an Entisol and Spodosol at Williston and Gainesville, FL,
Respectively 120
4-41 Error Analysis for Model Parameters for Bahiagrass Grown on Two Soils:
an Entisol and Spodosol at Williston and Gainesville, FL, Respectively 121
4-42 Analysis of Variance on Model Parameters for Seasonal Dry Matter Yield
and Plant N Removal of Ryegrass Grown on 20 Different Sites in England.... 122
4-43 Error Analysis of Model Parameters for Seasonal Dry Matter Yield and
Plant N Removal of Ryegrass Grown on 20 Different Sites in England 123
4-44 Summary of Model Parameters for Ryegrass in England 126
4-45 Summary of Model Parameters, Standard Errors, and Relative Errors for the
Extended Triple Logistic (NPK) Model for Rye Grown at Tifton, GA 127
5-1 A Summary of the Ab Parameter for Various Studies 262
5-2 A Summary of c and N' Parameters from Various Studies 263
xi

LIST OF FIGURES
Figure page
1-1 Response of dry matter yield, N removal, and N concentration for
bermudagrass as a function of applied N grown over two years at five
clipping intervals at Tifton, GA 5
2-1 Response of linearized dry matter yield and plant N removal to applied N
for dallisgrass [Paspalum dilatatum Poir ] grown at Baton Rouge, LA 17
4-1 Response of seasonal dry matter yield for Coastal bermudagrass and
Pensacola bahiagrass to applied N at Thorsby, AL 128
4-2 Scatter plot of seasonal dry matter yield for Coastal bermudagrass and
Pensacola bahiagrass at Thorsby, AL 129
4-3 Residual plot of seasonal dry matter yield for Coastal bermudagrass and
Pensacola bahiagrass at Thorsby, AL 130
4-4 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for dallisgrass grown at Baton Rouge, LA 131
4-5 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for dallisgrass grown at Baton Rouge, LA 132
4-6 Dimensionless plot of dry matter and plant N concentration as a function
of plant N removal for dallisgrass grown at Baton Rouge, LA 133
4-7 Scatter plot of dry matter yield for dallisgrass grown at Baton Rouge, LA. .. 134
4-8 Scatter plot of plant N removal for dallisgrass grown at Baton Rouge, LA... 135
4-9 Residual plot of dry matter yield for dallisgrass grown at Baton Rouge,
LA 136
4-10 Residual plot of plant N removal for dallisgrass grown at Baton Rouge,
LA 137
xn

4-11 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bermudagrass grown at Thorsby, AL and
cut at two harvest intervals 138
4-12 Seasonal dry matter yield and plant N concentration as a function of N
removal for bermudagrass grown at Thorsby, AL and cut at two harvest
intervals 139
4-13 Dimensionless plot of dry matter and plant N concentration as a function
of plant N removal for bermudagrass grown at Thorsby, AL and cut at
two harvest intervals 140
4-14 Scatter plot of dry matter yield for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals 141
4-15 Scatter plot of plant N removal for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals 142
4-16 Residual plot of dry matter yield for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals 143
4-17 Residual plot of plant N removal for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals 144
4-18 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bermudagrass grown at Maryland and cut
at five harvest intervals 145
4-19 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bermudagrass grown at Maryland and cut at five harvest
intervals 146
4-20 Dimensionless plot of dry matter and plant N concentration as a function
of plant N removal for bermudagrass grown at Maryland and cut at five
harvest intervals 147
4-21 Scatter plot of dry matter yield for bermudagrass grown at Maryland and
cut at five harvest intervals 148
4-22 Scatter plot of plant N removal for bermudagrass grown at Maryland and
cut at five harvest intervals 149
4-23 Residual plot of dry matter yield for bermudagrass grown at Maryland and
cut at five harvest intervals 150
xiii

4-24 Residual plot of plant N removal for bermudagrass grown at Maryland
and cut at five harvest intervals 151
4-25 Estimated maximum dry matter yield and estimated maximum plant N
removal as a function of harvest interval for bermudagrass in Maryland 152
4-26 Seasonal dry matter yield, plant N removal, and plant N concentration for
a two week clipping interval over two years for bermudagrass grown at
Tifton, GA 153
4-27 Seasonal dry matter yield, plant N removal, and plant N concentration for
a three week clipping interval over two years for bermudagrass grown at
Tifton, GA 154
4-28 Seasonal dry matter yield, plant N removal, and plant N concentration for
a four week clipping interval over two years for bermudagrass grown at
Tifton, GA : 155
4-29 Seasonal dry matter yield, plant N removal, and plant N concentration for
a six week clipping interval over two years for bermudagrass grown at
Tifton, GA 156
4-30 Seasonal dry matter yield, plant N removal, and plant N concentration for
a eight week clipping interval over two years for bermudagrass grown at
Tifton, GA 157
4-31 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a two week clipping interval over two years for
bermudagrass grown at Tifton, GA 158
4-32 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a three week clipping interval over two years for
bermudagrass grown at Tifton, GA 159
4-33 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a four week clipping interval over two years for
bermudagrass grown at Tifton, GA 160
4-34 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a six week clipping interval over two years for
bermudagrass grown at Tifton, GA 161
4-35 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a eight week clipping interval over two years for
bermudagrass grown at Tifton, GA 162
xiv

4-36 Dimensionless plot of seasonal dry matter yield and seasonal plant N
concentration versus seasonal plant N removal 163
4-37 Estimated mximums of seasonal dry matter yield and plant N removal as
a function of harvest interval for two years of bermudagrass grown at
Tifton, GA 164
4-38 Scatter plot of seasonal dry matter yield for bermudagrass grown over
two years at Tifton, GA and cut at five different harvest intervals 165
4-39 Scatter plot of seasonal plant N removal for bermudagrass grown over
two years at Tifton, GA and cut at five different harvest intervals 166
4-40 Residual plot of seasonal dry matter yield for bermudagrass grown over
two years at Tifton, GA and cut at five different harvest intervals 167
4-41 Residual plot of seasonal plant N removal for bermudagrass grown over
two years at Tifton, GA and cut at five different harvest intervals 168
4-42 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for three different number of cuttings over the
season for ryegrass grown at England 169
4-43 Seasonal dry matter yield and plant N removal as a function of plant N
concentration for three different number of cuttings over the season for
ryegrass grown at England 170
4-44 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for three different number of cuttings
over the season for ryegrass grown at England 171
4-45 Scatter plot of seasonal dry matter yield for ryegrass grown at England
and cut different times over the season 172
4-46 Scatter plot of seasonal plant N removal for ryegrass grown at England
and cut different times over the season 173
4-47 Residual plot of seasonal dry matter yield for ryegrass grown at England
and cut different times over the season 174
4-48 Residual plot of seasonal plant N removal for ryegrass grown at England
and cut different times over the season 175
4-49 Estimated mximums of seasonal dry matter yield and plant N removal as
a function of average harvest interval for ryegrass grown at England 176
xv

4-50 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bermudagrass grown over three years at
Fayetteville, AR, with and without irrigation 177
4-51 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bermudagrass grown over three years at Fayetteville, AR,
with and without irrigation 178
4-52 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for bermudagrass grown over three years
at Fayetteville, AR, with and without irrigation 179
4-53 Scatter plot of seasonal dry matter yield for bermudagrass grown over
three years at Fayetteville, AR, with and without irrigation 180
4-54 Scatter plot of seasonal plant N removal for bermudagrass grown over
three years at Fayetteville, AR, with and without irrigation 181
4-55 Residual plot of seasonal dry matter yield for bermudagrass grown over
three years at Fayetteville, AR, with and without irrigation 182
4-56 Residual plot of seasonal plant N removal for bermudagrass grown over
three years at Fayetteville, AR, with and without irrigation 183
4-57 Response of seasonal dry matter yield, plant N removal and plant N
concentration for bermudagrass averaged over three years at Fayetteville,
AR, with and without irrigation 184
4-58 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bermudagrass averaged over three years at Fayetteville,
AR, with and without irrigation 185
4-59 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for bermudagrass averaged over three
years at Fayetteville, AR, with and without irrigation 186
4-60 Scatter plot of seasonal dry matter yield for bermudagrass averaged over
three years at Fayetteville, AR, with and without irrigation 187
4-61 Scatter plot of seasonal plant N removal for bermudagrass averaged over
three years at Fayetteville, AR, with and without irrigation 188
4-62 Residual plot of seasonal dry matter yield for bermudagrass averaged over
three years at Fayetteville, AR, with and without irrigation 189
XVI

4-63 Residual plot of seasonal plant N removal for bermudagrass averaged over
three years at Fayetteville, AR, with and without irrigation 190
4-64 Response of seasonal dry matter yield, plant N removal and plant N
concentration for tall fescue grown over three years at Fayetteville, AR,
with and without irrigation 191
4-65 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for tall fescue grown over three years at Fayetteville, AR, with
and without irrigation 192
4-66 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for tall fescue grown over three years at
Fayetteville, AR, with and without irrigation 193
4-67 Scatter plot of seasonal dry matter yield for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation 194
4-68 Scatter plot of seasonal plant N removal for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation 195
4-69 Residual plot of seasonal dry matter yield for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation 196
4-70 Residual plot of seasonal plant N removal for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation 197
4-71 Response of seasonal dry matter yield, plant N removal and plant N
concentration for tall fescue averaged over three years at Fayetteville, AR,
with and without irrigation 198
4-72 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for tall fescue averaged over three years at Fayetteville, AR,
with and without irrigation 199
4-73 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for tall fescue averaged over three years
at Fayetteville, AR, with and without irrigation 200
4-74 Scatter plot of seasonal dry matter yield for tall fescue averaged over three
years at Fayetteville, AR, with and without irrigation 201
4-75 Scatter plot of seasonal plant N removal for tall fescue averaged over
three years at Fayetteville, AR, with and without irrigation 202
XVII

4-76 Residual plot of seasonal dry matter yield for tall fescue averaged over
three years at Fayetteville, AR, with and without irrigation 203
4-77 Residual plot of seasonal plant N removal for tall fescue averaged over
three years at Fayetteville, AR, with and without irrigation 204
4-78 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bermudagrass grown over two years at
Eagle Lake, TX 205
4-79 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bahiagrass grown over two years at Eagle
Lake, TX 206
4-80 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bermudagrass grown over two years at Eagle Lake, TX 207
4-81 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bahiagrass grown over two years at Eagle Lake, TX 208
4-82 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for bahiagrass and bermudagrass grown
over two years at Eagle Lake, TX 209
4-83 Scatter plot of seasonal dry matter yield for bahiagrass and bermudagrass
grown over two years at Eagle Lake, TX 210
4-84 Scatter plot of seasonal plant N removal for bahiagrass and bermudagrass
grown over two years at Eagle Lake, TX 211
4-85 Residual plot of seasonal dry matter yield for bahiagrass and bermudagrass
grown over two years at Eagle Lake, TX 212
4-86 Residual plot of seasonal plant N removal for bahiagrass and
bermudagrass grown over two years at Eagle Lake, TX 213
4-87 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for grain and total plant of corn grown at
Clayton and Kinston, NC on Dothan and Goldsboro soils, respectively 214
4-88 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for grain and total plant of corn grown at Clayton and Kinston,
NC on Dothan and Goldsboro soils, respectively 215
xviii

4-89 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for grain and total plant of corn grown at
Clayton and Kinston, NC on Dothan and Goldsboro soils, respectively 216
4-90 Scatter plot of seasonal dry matter yield for grain and total plant of corn
grown at Clayton and Kinston, NC on Dothan and Goldsboro soils,
respectively 217
4-91 Scatter plot of seasonal plant N removal for grain and total plant of corn
grown at Clayton and Kinston, NC on Dothan and Goldsboro soils,
respectively 218
4-92 Residual plot of seasonal dry matter yield for grain and total plant of corn
grown at Clayton and Kinston, NC on Dothan and Goldsboro soils,
respectively 219
4-93 Residual plot of seasonal plant N removal for grain and total plant of corn
grown at Clayton and Kinston, NC on Dothan and Goldsboro soils,
respectively 220
4-94 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for grain and total plant of corn grown at
Plymouth, NC 221
4-95 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for grain and total plant of corn grown at Plymouth, NC 222
4-96 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for grain and total plant of corn grown at
Plymouth, NC 223
4-97 Scatter plot of seasonal dry matter yield for grain and total plant of corn
grown at Plymouth, NC 224
4-98 Scatter plot of seasonal plant N removal for grain and total plant of corn
grown at Plymouth, NC 225
4-99 Residual plot of seasonal dry matter yield for grain and total plant of corn
grown at Plymouth, NC 226
4-100 Residual plot of seasonal plant N removal for grain and total plant of corn
grown at Plymouth, NC 227
4-101 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bahiagrass grown on two soils in Florida 228
xix

4-102 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bahiagrass grown on two soils in Florida 229
4-103 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for bahiagrass grown on two soils in
Florida 230
4-104 Scatter plot of seasonal dry matter yield for bahiagrass grown on two soils
in Florida 231
4-105 Scatter plot of seasonal plant N removal for bahiagrass grown on two
soils in Florida 232
4-106 Residual plot of seasonal dry matter yield for bahiagrass grown on two
soils in Florida 233
4-107 Residual plot of seasonal plant N removal for bahiagrass grown on two
soils in Florida 234
4-108 Plot of the mean and 2 standard errors of A/A and Ab for twenty sites in
England 235
4-109 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for twenty sites in England i 236
4-110 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for twenty sites in England 237
4-111 Response of seasonal dry matter, plant N removal, and plant N
concentration to applied N for rye grown at Tifln, GA and fixed
application rates of 40 and 74 kg/ha of P and K, respectively 238
4-112 Response of seasonal dry matter, plant P removal, and plant P
concentration to applied P for rye grown at Tifln, GA and fixed
application rates of 135 and 74 kg/ha of N and K, respectively 239
4-113 Response of seasonal dry matter, plant K removal, and plant K
concentration to applied K for rye grown at Tifln, GA and fixed
application rates of 135 and 40 kg/ha of N and P, respectively 240
4-114 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for rye grown at Tifln, GA 241
xx

4-115 Seasonal dry matter yield and plant P concentration as a function of plant
P removal for rye grown at Tifln, GA 242
4-116 Seasonal dry matter yield and plant K concentration as a function of plant
K removal for rye grown at Tifln, GA 243
4-117 Dimensionless plot of seasonal dry matter yield and plant nutrient
concentration as a function of plant nutrient removal for rye grown at
Tifton, GA 244
4-118 Scatter plot of dry matter yield for rye grown at Tifton, GA 245
4-119 Scatter plot of plant N removal for rye grown at Tifton, GA 246
4-120 Scatter plot of plant P removal for rye grown at Tifton, GA 247
4-121 Scatter plot of plant K removal for rye grown at Tifton, GA 248
4-122 Residual plot of dry matter yield for rye grown at Tifton, GA 249
4-123 Residual plot of plant N removal for rye grown at Tifton, GA 250
4-124 Residual plot of plant P removal for rye grown at Tifton, GA 251
4-125 Residual plot of plant K removal for rye grown at Tifton, GA 252
5-1 Sensitivity of logistic to the c parameter 264
xxi

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ESTIMATION OF DRY MATTER AND NITROGEN REMOVAL
BY THE LOGISTIC EQUATION
By
Denise Marie Wilson
December 1995
Chairman: Allen R. Overman, Ph D.
Major Department: Agricultural and Biological Engineering Department
Environmental issues are shaping today's world. Land application of treated
wastes and effluent is often done to manage the excess nutrients. Forage grasses are
grown to accomplish two objectives: remove the nutrients from the waste or effluent and
to produce food for livestock. Engineers, regulators, and managers use nutrient budgets
in the design of systems to reduce the chance of pollution. This analysis was performed to
establish a form of a model that would describe forage grass response to applied nutrients
and provide reasonable estimates.
Forms of the logistic equation were used to relate dry matter yield and plant N
removal to applied nitrogen for seven different forage grasses. Many different factors,
such as water availability, harvest interval, and plant partitioning, were investigated by
examining thirteen studies in the literature.
The logistic equation has shown a high correlation between grass yields and
applied nitrogen. It is the purpose of this analysis to show that the form of the logistic
XXII

equation adequately describes dry matter yield and plant N removal response to applied
nitrogen. The parameters of the models are determined using nonlinear regression.
Analysis of variance is used to search for simplification in the form of common parameter
values.
The results of the analysis showed that the logistic model is well behaved and
relatively simple to use. Harvest interval, water availability and plant partitioning can be
accounted for in the linear parameter. Dimensionless plots are a valuable tool in
evaluating the form of a model. The logistic equation exhibits symmetry suggesting
conservation of something. For this use of the equation, the total capacity of the system is
conserved. The assumptions of the extended model suggest a hyperbolic relationship
between dry matter yield and plant N removal. This relationship was observed throughout
the analysis. Although parameter values cannot be determined without prior
experimentation, estimates for the parameters can be assumed. If parameter values are
needed before any investigation, they should be underestimated in order to overdesign the
system. Furthermore, for large error in the b, b', and c parameters, the seasonal estimate is
not affected greatly (<15% error).
XXlll

CHAPTER 1
INTRODUCTION
In today's society, there is an increasing focus on environmental issues. Concerns
are being raised about pollution in soil, water and air. One of the major pollution concerns
in Florida is nitrate and phosphate contamination in the aquifer and lakes, respectively.
Engineers are using nutrient budgets to ensure that excess nutrients are not applied and to
help control and eliminate contamination. Often nutrients are applied to forage grasses as
treated wastes or effluent. Many municipalities are using water reuse systems as a way to
remove high levels of nutrients from treated reclaimed water (Allhands et al., 1995). This
process is beneficial to two main parties, those who wish to clean the water and return it
to the aquifer and those who benefit from the addition of the nutrients to their system.
Since a large amount of the required nutrients is applied in the water, less fertilizer is
needed. The effluent is often applied to forage grasses, golf courses and lawns. A simple
procedure is needed to assist engineers in estimating and predicting for various forage
grasses the amount of nutrient removed and dry matter produced given a specific nutrient
application rate.
This research project primarily deals with various forage grasses and their nitrogen
response curves. Consider the response data presented in Figure 1-1. The data were
taken from a study by Prine and Burton (1956). Dry matter yield, plant N removal, and
plant N concentration was recorded for bermudagrass [CynoJon dactylon] grown at
Tifln, GA, for two years at five harvest intervals. As expected, there is a relationship
between yield, N removal, and N concentration to applied N, but what exactly is the
relationship? Linear, hyperbolic, quadratic, sigmoid? If one form of an equation would
1

2
adequately describe all the behavior, the task of identifying the specific relationship would
be greatly simplified. In theory, the form of the model should represent the physical
system beyond the range of data. The challenge is to identify patterns in the data (such as
Figure 1-1) and identify relationships to describe such data.
The objective of this project is to establish a model that provides reasonable
estimates of dry matter yield and nutrient removal given a nutrient application rate.
Studies from the literature will be used to document the fit of the model to numerous data
sets with varying factors. These estimates could be used by engineers, managers, and
regulators. The form of the equation should work regardless of forage or site. Water
availability and harvest interval should also be quantified in the model. Land application of
treated effluent and waste as irrigation for agricultural crops is becoming a prevalent
method of wastewater reuse. In these systems, the nitrogen response of the crop is
needed by engineers in the design process, since both the Florida Department of
Environmental Protection (FDEP) and the Environmental Protection Agency (EPA)
regulate wastewater application rates based upon nitrogen concentrations and the uptake
abilities of crops grown.
Many sources of data for this analysis can be found for various forage grasses and
locations around the world. This reservoir of information has different variables for the
crops studied including the following: applied nutrients (N, P, and K), water availability
(with/without irrigation and year to year variability), harvest interval, site specificity,
and plant partitioning. These different variables produce varying response curves.
Furthermore, the studies range from examining only dry matter yield response to applied
N to including N removal response to investigating the effects of nitrogen, phosphorus,
and potassium on dry matter yield and their respective removals. The logistic equation has
provided high correlation coefficients for estimation of growth and nutrient removal for

3
various grasses (Overman, 1990a; Reck, 1992; Overman et al., 1994a, 1994b). Because
of this, the form of the logistic will be studied to determine its broader applicability.
To better understand the logistic equation, consider the "rumor model". A room is
filled with 100 people and the doors are closed. No one is allowed to leave or come into
the room. The people in the room are milling around in random fashion. At the time the
doors are closed, five people know a certain rumor. The speed at which the rumor
spreads among the people present, or rate of transfer among the people, is dependent
upon how many people have heard it and how many have not heard it. The probability
that during an encounter with two people the rumor is spread is the probability that one
knows the rumor and one does not. These two events are mutually exclusive and
independent; that is, someone cannot know and not know the rumor at the same time. As
a result, their joint probability is the product of their individual probabilities. The
probability that the rumor is spread is the product of the probability a person knows and
the probability a person does not know. Since the number of people in the room is fixed,
the sum of those who know and do not know is constant as well. Furthermore, the
probability density function for the spreading rumor has a bell shaped distribution; that is,
the probability that the rumor is spread is the same when 5 people know and when 95
people know. The rate of transfer is equal but in opposite directions. At the beginning,
there are more people who have not heard the rumor; so the rate of transfer is increasing
until half of the people know. At that point the rate of transfer falls off since there are
fewer people to tell. The logistic equation applied to grass growth operates similarly. As
time was the independent variable for the rumor model, applied nitrogen is the
independent variable for the logistic crop growth model. Let the number of people who
know represent the amount of dry matter present at a particular moment in time. It
follows that the number of people who do not know represents the amount of dry matter

4
that still needs to be produced. As the number of people in the room was assumed
constant with the rumor model, the potential dry matter production of the grass is constant
as well: the sum of that already produced and that still to be produced. Also, the rate of
growth is a product of what has been produced and what is still to be produced, as the
rate of rumor transfer was a product of those who knew with those who did not know.
A nonlinear regression analysis of the variables involved will be used to determine
the values of the parameters for a logistic model of forage grass response to applied
nitrogen assuming different relationships among the model parameters. The results of this
analysis will be compared using analysis of variance for the different assumptions to
determine which is correct and if a simplification can be made. The result will be a form
of an equation for nutrient response to various forage grasses that will adequately describe
the relationship of dry matter yield, plant nutrient removal, and plant nutrient
concentration to applied nutrients. This model will assist engineers and planners in
estimating plant nutrient removal for various application rates.
The result of this analysis should be the production of a model for engineers and
managers to use when determining nutrient budgets under varying conditions. The model
will estimate seasonal totals and use a "black box approach". No attempt will be made to
model the physical mechanisms of the plant. Once many different studies have been
conducted, a survey of the parameters for various forage grasses might provide a useful
insight to the internal mechanisms of the plant, as well as connect the parameters to the
physical system, but this is not pursued in this study.

Dry Matter Yield, Mg/ha Plant N Removal, kg/ha Plant N Concentration, g/kg
5
Figure 1-1 Response of dry matter yield, N removal, and N concentration for
bermudagrass as a function of applied N grown over two years at five
clipping intervals at Tifln, GA. Data from Prine and Burton (1956).
Symbols will be defined later in the text.

CHAPTER 2
LITERATURE REVIEW
Crop models have been used for many years and have many variations (Keen and
Spain, 1992; France and Thornley, 1984). Models have been used to describe the
relationship between various forage crop yields and other parameters including nitrogen
application, water availability, harvest interval, etc. These models have ranged from
simple polynomial models to varying degrees of nonlinear models. Jones et al (1987)
have defined a model as a mathematical representation of a system, and modeling as the
process of developing that representation. Modeling is often confused with simulation.
Simulation includes the processes necessary for operationalizing the model, or solving the
model to mimic real system behavior (Jones et al., 1987). Before beginning a model or
simulation project the objectives of the project should be stated clearly. A clear definition
of the intended end-product and the intended users of the models that will be developed
should be included in the objective statement. In their book on mathematical models in
agriculture, France and Thornley (1984, p. 173) noted that,
There are three fairly distinct communities of people to whom a crop model or
simulator may be of value: first, farmers, including advisory and extension
services, who are primarily concerned with current production; second, applied
scientists, including agronomists and plant breeders, whose objective is to improve
the efficiency of production techniques, for the most part using current knowledge;
and third, research scientists, whose aim is to extend the bounds of present-day
knowledge. There is no way in which a single model can satisfy the differing
objectives of these three groups, which are, at least in part, incompatible.
Various mathematical models have been used to describe crop growth.
Polynomials are often used because of the simplicity in determining parameter values;
however, they have serious weaknesses. These weaknesses have been addressed in detail

7
(Freund and Littell, 1991). The exponential and Mitscherlich models have also been used
in describing crop growth (France and Thornley, 1984); however these models have
limitations as well. The exponential model suggests that at high levels of input, the
response approaches infinity, and at low levels approaches zero response. The
Mitscherlich model suggests that at low levels of input the response approaches negative
infinity, and at high levels approaches a maximum response. These two models describe
the response well at different extremes. A combination of the two could potentially
explain the response well over all ranges of input.
The logistic equation, first proposed by Verhulst and later popularized by Pearl
(Kingsland 1985), was first used as a population model. It has been used with high
correlation coefficients (R > 0.99) for nitrogen removal (uptake) and dry matter
production of various forage grasses (Reck, 1992; Overman et a/., 1990a, 1990b, 1994a,
1994b; Overman and Blue, 1991). Overman (1995a) has developed the relationship of the
exponential and Mitscherlich models to the logistic, and a short summary of that
discussion will be included here. In differential form, the exponential model assumes that
the response of dry matter to applied N is proportional to the amount of dry matter
present. Furthermore, the exponential is suitable at low N and asymptotically approaches
zero along the negative N (reduced soil N) axis. In mathematical form,
lower N: dy/dN ay [2.1]
In differential form, the Mitscherlich model assumes that the response of dry matter to
applied N is proportional to the unfilled capacity, ym y, of the system, where ym is the
maximum yield. As mentioned before, the Mitscherlich is suitable at high N and
asymptotically approaches a maximum yield along the positive N axis. In mathematical
form,
higher N: dy/dN a ym y
This leads us to assume a composite function
[2.2]

8
all N: dy/dN = k y (ym y) [2.3]
This is the form of the logistic model where k is the N response coefficient. It is a
nonlinear first order differential equation. The logistic equation is well behaved; that is,
the function is continuous, smooth, and asymptotic at both ends of the range. Recall the
analogy to the rumor model. The product of the dry matter present (filled capacity) and
that still to be produced (unfilled) is present in the differential form as well. Also, the total
capacity of the system (y + ym y = ym) is assumed constant. Furthermore, as Overman
(1995a) has pointed out, the logistic equation reduces to the exponential model at lower N
and to the Mitscherlich model at higher N. If the logistic is normalized by defining new
variables O = y/A and ^ = cN b and expanded by Taylor series, a quadratic term is not
included, suggesting a parabola would not be appropriate. Furthermore, the logistic is
approximated by a linear function extremely well in the middle of the range.
The logistic model exhibits sigmoid behavior and has three parameters as shown
below
y-
\ + e
b-cN
[2.4]
Parameters A and c are scaling coefficients for yield and applied N, respectively.
Parameter b describes the reference state at N = 0. In the context of this model,
parameters A and c are constrained to be positive, but parameter b can be either positive,
zero, or negative. Parameter b equal to zero suggests that there is enough background
level of nitrogen in the soil to reach half of the maximum yield. Parameter b less than zero
suggests that there is enough nitrogen in the soil before fertilization to reach more than
halfway up the response curve. The model can also be normalized as shown
y 1
A \ + eb~cN
[2.5]
and linearized as shown

9
ln(-1 ) = b-cN [2.6]
y
To determine if the data follow a logistic curve, A/y 1 vs. N can be plotted on semilog
paper to check for linearity. If the plot is linear, then the data can be described well by the
logistic. The parameters can be determined by one of two methods: regression on
linearized data or nonlinear regression on original data (Downey and Overman, 1988).
There are advantages to using both methods. The linearization method provides an easy
procedure to estimate the parameters with a hand calculator. By this method, an estimate
of A can be obtained by examining a plot of the y versus N on linear paper. The curve will
appear to approach a maximum. This maximum is the estimate of A. Several attempts
may be required to optimize A. Estimates of b and c then follow from linear regression of
ln(A/y 1) vs. N (Draper and Smith, 1981). An example of this is shown in Figure 2-1.
The data for this analysis are taken from a study of dallisgrass [Paspalum dilatatum Poir ]
grown at Baton Rouge, Louisiana (Robinson et ai, 1988). Linearized dry matter yield
and plant nitrogen removal are plotted. The linear trend suggests that these two responses
can be described well by the logistic equation. The fact that the lines are parallel suggests
that the c parameter for both dry matter yield and plant nitrogen removal are the same.
The figure also suggests that the b parameter value for plant nitrogen yield is larger (more
positive) than the b value for dry matter yield. The nonlinear regression method requires a
computer program written to perform the regression and statistical inference and
diagnostic information. The regression can be conducted with SAS for the simple case.
As more complex cases are needed, SAS is not easily programmed to perform the
statistical analysis. In the nonlinear regression, parameters are estimated using least
squares (Bates and Watts, 1988; Ratkowsky, 1983). Unfortunately, only the A parameter
can be explicitly solved. The b and c parameters need to be solved implicitly. Second
order Newton-Raphson iteration is used (Adby and Dempster, 1974). For additional

10
comparison among the two methods, the reader is directed to Overman et al. (1990a) for
further details.
The logistic equation exhibits symmetry suggesting conservation of something
(energy, momentum, charge, spin, etc.) (Mehra, 1994 p. 132). If this is true, what could
be conserved in the system defined? As shown earlier, it is the total capacity of the
system, ym. In Pearls work with the logistic equation he later expressed "unutilized
potentialities" as the "amount still unused or unexpended in the given universe (or area) of
actual or potential resources for the support of growth" (Kingsland, 1985, p. 67). In
relation to this work, the total yield capacity consisting of the filled, y, and unfilled
capacity, ym y, is what is conserved. But what is the total yield capacity? It is y + y, y
= ym = A, where A is assumed constant. Pearl further noted, "The rate of growth,
therefore, was proportional to two quantities: the existing population and the difference
between existing and limiting populations" (Kingsland, 1985, p. 68). This is also
demonstrated in the differential form, Equation [2.3],
The logistic equation can also be written as
A
y~ i + elNv*-NVN' [2J1
where N1/2 = b/c = nitrogen value for half maximum yield, and
N' = 1/c = characteristic nitrogen.
Recall Figure 2-1. The intercept on the vertical axis is eb. On the horizontal axis (y/A =
0.5), N = N1/2. Pearl also noted, "Symmetry meant that the inflection point came at the
halfway point of the curve, and that the saturation population was exactly twice the
population at the point of inflection" (Kingsland, 1985, p. 69). This is demonstrated in the
above equation. It has been pointed out by Hosmer and Lemeshow (1989, p. 38) that the
most important coefficient in the logistic equation is the response coefficient c.
In this analysis, three models will be used: simple logistic, extended logistic, and
triple logistic (or NPK). The simple logistic model is given by

11
A
[2.8]
where y = seasonal dry matter yield, Mg/ha;
N = applied N, kg/ha;
A = maximum seasonal dry matter yield, Mg/ha;
b = intercept parameter for yield;
c = N response coefficient, ha/kg.
The extended model relates dry matter and plant N responses to applied N and is based
upon three postulates:
1. Seasonal dry matter yield follows logistic response to applied N.
2. Seasonal plant N removal follows logistic response to applied N.
3. The N response coefficients are the same for both.
Postulate 1 follows from work done by Overman and Wilkinson (1992). Postulates 1 and
2 are possibly true independently of one another, while Postulate 3 implies quantitative
coupling between dry matter and plant N accumulation. Three additional results derive as
a consequence of these postulates:
1. Plant N concentration response to applied N is described by a ratio of logistic
functions.
2. Seasonal dry matter yield and plant N removal are related by a hyperbolic
equation.
3. Plant N concentration and plant N removal are linearly related.
Postulate 1 is the simple logistic model. Postulate 2 can be written as
[2.9]
where
Nu = seasonal plant N removal, kg/ha;
A' = maximum seasonal plant N removal, kg/ha;

12
b' = intercept parameter (plant N removal);
c' = N response coefficient (plant N removal), ha/kg.
Now by using Postulate 3, the system is constrained by assuming c = c'. Figure 2-1
suggests this is a logical assumption. Plant N concentration is defined as the ratio of
Equation [2.9] to Equation [2.8], Note that the plant N "concentration" has been defined
as the ratio of two extensive variables (y and Nu), and therefore is not truly an intensive
variable (like ionic concentration, temperature, pressure, etc.). By combining Equations
[2.8] and [2.9] with c = c' and defining Nc = Nu/Y, the response to applied N is defined by
7Y, =
(l+e^)
[2.10]
(\ + e )
where Nc = average plant N concentration, g/kg, and
Ncm= maximum plant N concentration, g/kg.
This equation is well behaved for b' > b. It approaches a minimum at low values of
applied N and approaches a maximum at high values of applied N. If b' < b, then Equation
[2 10] is no longer well behaved at low levels of applied N. Furthermore, Equations [2.8]
and [2.9] can be rearranged to give the hyperbolic equation
y, *
K'+N
where the parameters Ym and K are given by
k,=
1 e
b-b'
K'=
A'
7T7
[2.11]
[2.12]
and
[2.13]

13
where Ab = b' b. In order for Ym and K' to be positive, b' must be greater than b. These
relationships can be reduced to dimensionless form by dividing the dry matter yield by its
estimated maximum and the plant N removal by its estimated maximum. After doing this,
the parameters of the hyperbolic relationship are defined as
4 = 1 = \
A 1 eb~b' \-e~Ab
[2.14]
and
/C 1 1
A ~ eh'~h 1 1
[2.15]
It should be noted that the parameters used to describe the dimensionless relationship of
seasonal dry matter to plant N removal (two measurable quantities) are only dependent
upon the Ab. Dimensionless plots have been useful tools for engineers in many fields.
Dimensionless plots were used to develop and determine dimensionless numbers, such as
the Reynolds number in hydraulic flow. For example, James Clerk Maxwell used a
dimensionless plot in the 1860s to describe the distribution of molecular velocities in a gas
(Segr, 1984). The greatest value of a dimensionless plot is the ability to collapse data
sets with different ranges onto the same scale for comparison. This also aids in the search
for possible simplification. Equation [2.11] can be rearranged to the form
Nc = (K'/Ym) + (l/Ym)Nu [2.16]
This equation predicts a linear relationship between plant N concentration and plant N
removal.
The extended triple logistic (NPK) model is given by
(1 + eb'~cNX1 + e CfP)(1 + [2.17]

14
N
(1 + eb"'~c"N)(1 + ebf CfP)(1 + eb*~C*A')
[2.18]
P =
(1 + eb"-c"N XI + eh> CPXI + eb'~c>K)
[2.19]
K =
(\ + eb-~C-N X1 + eb CfPXI + ehl'~ClK)
[2.20]
where the subscripts n, p, and k refer to applied N, P, and K, respectively. This model
assumes that the response to each nutrient is logistic and is treated as independently
applied. In statistical terms, the independence assumption suggests the joint response is
the product of the marginal responses (Mood et al., 1963). Several characteristics should
be noted. With fixed levels of P and K, the NPK model reduces to the simple logistic
model with a modified maximum parameter A. This model will allow for evaluation of the
effect of combinations of N, P, and K on dry matter production. The amount of nutrient
required to reach half of maximum yield is given by Ni/2 = b/c, P1/2 = bp/cp, and K1/2 =
bk/ci<. Negative values of b mean that before application the soil contained more than
enough nutrient to achieve half of maximum yield.
Throughout this analysis a model is viewed as a simplification of reality. Occams
razor will be applied to simplify to the essentials required for description. Box (1976, p.
792) suggested this approach in a paper by stating,
Since all models are wrong the scientist cannot obtain a "correct" one by excessive
elaboration. On the contrary following William of Occam he should seek an
economical description of natural phenomena. Just as the ability to devise simple
but evocative models is the signature of the great scientist so overelaboration and
overparameterization is often the mark of mediocrity.
Evaluation and validation of the models will be conducted by dimensionless, scatter and
residual plots. In the literature, validation is defined as the process by which a simulation

15
model is compared to field data not used previously in the development or calibration
process (Jones el al., 1987). It should be noted that this study is not simulation, but true
modeling: mathematical representation of a system. Furthermore, the purpose of
validation is to determine if the model is sufficiently accurate for its application as defined
by the objectives (Jones et al., 1987). This leads back to a point mentioned earlier. It is
essential that the objectives be stated clearly from the beginning. Jones el al. (1987, p. 16)
also noted that common sense should prevail in validation of models, and that "a model
cannot be validated, it can only be invalidated," a point frequently emphasized in statistics.
Dimensionless plots will be used to determine if the form of the model is adequate to
validate the model. It is more important to determine if the model estimates adequately:
Does the model capture the essence of what is trying to be accomplished. The scatter and
residual plots will be used to evaluate how well the model estimated the data, by looking
for biases or trends in the residuals. Box (1979, p. 2) noted the difference between
estimating and validation in a paper by stating
. . two different kinds of inferential process .... The first, used in estimating
parameters from data conditional on the truth of some tentative model, is
appropriately called Estimation. The second, used in checking whether, in the
light of the data, any model of the kind proposed is plausible, has been aptly named
by Cuthbert Daniel Criticism. While estimation should . employ . likelihood,
criticism needs a different approach. In practice, it is often best done in a rather
informal way by examination of residuals. . .
Box, Hunter, and Hunter (1978, p. 552) also discussed the importance of residuals and
other visual displays in evaluating data, especially when the work has been done with a
computer by noting.
Without computers most of the work done on nonlinear models would not be
feasible. However, the more sophisticated the model and the more elaborate the
techniques employed, the more important it is to submit complicated analyses to
surveillance by data plots, residual plots, and other visual displays. The modern
computer can make the plots itself, but graphs need not only be made but also to
be carefully examined and thought about. The data analyst must "fondle" the data.

16
Hand plotting used to be one to the ways this came about. The original data, as
well as the various plots, whether made by hand or by the computer, should be
mulled over. The experimenter's imagination, intuition, subject-matter knowledge,
and experience must interact. This interaction will often lead to new ideas that
may, in turn, lead to further analysis or experimentation.
While the objective of some model projects is to describe the system in a
fundamental way, this project has used Boxs approach. The decision to use the logistic
equation was not based upon some insight into the plant process, but rather as an equation
that would describe the response well. The more prevalent approach in the literature is to
use compartmental models such as EPIC, CREAMS, GLEAMS, DRAINMOD, etc.
These models attempt to simulate the growth of a plant (or field) from planting to harvest
by breaking the soil and plant system into small compartments. The compartments keep
track of important state variables as water and other nutrients move through the system.
Common time steps include minute, hour, days and weeks. A model is written for each
plant process, such as respiration, transpiration, reproduction, etc. Common inputs
include weather data, (solar radiation, rainfall, degree-days), and soil characteristics. Plant
specific parameters are also needed. Although a thorough attempt has been made to
explain the growth behavior, the seasonal yields are not estimated well. An example of
this is demonstrated in a paper by Williams et a!. (1989) where the highest correlation
coefficient in the scatter plots was 0.89 and the average correlation coefficient was 0.67.
It is the opinion of the author that the geometry and processes involved for a single plant
is much too complicated to be solved mechanisticly, much less to try to extrapolate to the
larger system. We are faced with the same problem as with the Navier-Stokes equation in
porous mediathere is no hope of describing the geometric flow paths in the soil and
plant. This project is an attempt to estimate the seasonal totals well and provide a low
input and simple approach to crop modeling.

Linearized Response
17
Figure 2-1 Response of linearized dry matter yield and plant N removal to applied N
for dallisgrass [Paspalum dilatatum Poir ] grown at Baton Rouge, LA.
Original data from Robinson et al. (1988). Data are linearized using Eq.
[2.6] and A values of 15.60 and 431, respectively.

CHAPTER 3
METHODS AND MATERIALS
Analysis of Data
Depending upon the nature of the data, one of three different models will be used.
It should be noted that the three models are directly related to one another because they
are based upon the same mathematical form. For data sets where only plant dry matter
yield is recorded, the simple logistic model, Equation [2.8], will be used to describe plant
yield response to applied nitrogen. For data sets where both dry matter yield and plant N
removal are recorded, the extended logistic model, Equations [2.8] and [2.9], will be used
to describe the response to applied nitrogen. Finally, when varying amounts of nitrogen,
phosphorus, and potassium are applied and dry matter yield and nutrient removals
recorded, the extended triple logistic (or NPK) model, Equations [2.17] through [2.20],
are used to describe the response. Water availability and harvest interval (or cutting
frequency) will be related to the linear model parameter. For all three models, the
parameters, A, b, and c, are estimated using nonlinear regression (including second-order
Newton-Raphson method) on the data to minimize the error, E, given by
[3.1]
where
E
error sum of squares,
measured yield or N removal,
estimated yield or N removal from model,
observation number.
18

19
Correlation coefficients will be used to measure the fit of the data to the model. The
correlation coefficient is given by
[3.2]
An adjusted correlation coefficient is given by
[3.3]
where n = total number of data points used, and
p = number of parameters estimated.
This quantity adjusts the correlation coefficient by the number of parameters in the model.
In the case of simple regression, Radj = R. Because the adjusted correlation coefficient
accounts for the number of parameters in the model, it essentially deflates the R value
hence providing for a better evaluation of the fit.
A program was written in Pascal to estimate the parameters given the data and first
estimates of the b and c parameters using nonlinear regression and Newton-Raphson
iteration. The b and c parameters can be better estimated by the linearization method as
discussed in the previous chapter; however, the author has chosen to use nonlinear
regression because inference upon the parameters is more straightforward. If the
linearization technique had been employed, then the inference and summary statistics
would have been based upon the linearized data and not the actual data. This analysis will
include evaluation and validation of the form of the logistic model, to determine if the
model adequately describes the behavior of the data regardless of crop or site. To answer
this question, various grasses were studied. A list of these grasses and their common and
scientific names is given in Table 3-1.

Models to be Investigated
20
Simple Logistic Model for Dry Matter Yield
Analysis of variance, scatter plots and residual plots were used to evaluate this
model. The analysis of variance tests if the model can be simplified due to common
parameters under different conditions (that is, year, harvest interval, water availability,
etc ). There are three basic modes. In mode 1, all the data were analyzed together over
the various management factors (yield, N removal, years, harvest interval, irrigation, etc.)
and a common A, b, and c are found for the combined data. In mode 2, the data were fit
separately for each specific situation (yield, N removal, year, harvest interval, irrigated,
etc ), requiring a different A, b, and c parameter for each situation. In mode 3, the data
were analyzed together again and a different A is fit for each situation, but a common b
and c are found for the entire set of data. These different modes were then compared by
analysis of variance. They are compared by using an F test. The hypothesis that is being
tested is that one of the modes (or models) describes the data better. It is basically
examining for possible simplification.
Extended Logistic Model for Dry Matter Yield and Nitrogen Removal
These data sets are analyzed in a similar manner to the simple logistic with two
additional modes. For mode 4, the data are analyzed together and an individual A and b
are fitted to each situation with a common c for all. Mode 5 involves fitting an individual
A for each situation, a separate b for dry matter yield and plant N removal, and a common
c for all. These two additional modes are compared with the previous three modes by
analysis of variance to determine which scenario describes the data best. In some cases
the statistics suggest that the c parameter is not common. Instead of blindly using the

21
statistics, we further compare the mean sum of squares (MSS) for the two modes in
question. If there is not a large increase in MSS by simplifying the model (requiring fewer
parameters), we will accept that there is a common c. By examination of the plotted data,
it appears this supposition is valid.
As stated previously, the extended model is based upon three postulates. One of
the consequences of the postulates was that dry matter yield and plant N removal are
hyperbolically related and plant N concentration and plant N removal are linearly related.
To test these results, dry matter yield and N concentration are plotted against plant N
removal. The lines describing these relationships are dependent only upon the Ab = (b' -
b). Furthermore, the data can be normalized by dividing by the appropriate estimated
mximums. If the assumption of common Ab is true, all the normalized data will fall on
one line in a dimensionless plot. This is one way of testing the adequacy of the model to
fit the data and testing the postulates.
Extended Triple Logistic (NPK) Model for Dry Matter Yield and Nitrogen Removal
This model is developed on the assumption that response of dry matter yield and
plant nutrient removal to applied N, P, and K individually follow the extended logistic
model. Overman and Wilkinson (1995) first discussed this model and applied it to a
complete factorial. Dimensionless plots, scatter plots and residual plots are used to
evaluate the model.

Data Sets to be Investigated
22
Thorsbv. Alabama: Bahiagrass and Bermudagrass
This study is from the Auburn Agricultural Experiment Station. The results have
been previously reported in a station bulletin (Evans el al., 1961). The results of field
tests and grazing trials conducted in Alabama to show the response of bermudagrass and
bahiagrass to nitrogen and irrigation on specific soil types were reported. The yields of
the grasses grown on Greenville fine sandy loam, Thorsby, AL, over three years (1957-
1959) were used in this analysis. Four nitrogen levels were included: 0, 168, 336, and
672 kg/ha. The data are listed in Table 3-2.
Baton Rouge. Louisiana: Dallisgrass
The data for this analysis were acquired from a study by Robinson et al. (1988).
Six nitrogen levels were included: 0, 56, 112, 224, 448, and 896 kg/ha. The data are
listed in Table 3-3.
Thorsbv. Alabama: Bermudagrass
The data for this analysis were taken from a study by Doss el al. (1966). Two
harvest intervals were studied: 3.0 and 4.5 weeks. Six nitrogen levels were included: 0,
224, 448, 672, 1344, and 2016 kg/ha. The data are listed in Table 3-4.
Maryland: Bermudagrass
The data for this analysis were acquired from a study by Decker et al. (1971).
Five harvest intervals were studied: 3.2, 3.6, 4.3, 5.5, and 7.7 weeks. Six nitrogen levels
were included. 0, 112, 224, 448, 672, and 896 kg/ha. The data are listed in Table 3-5.

Tifln. Georgia: Bermudagrass
23
This data set was taken from a study by Prine and Burton (1956). Data from two
years were utilized. 1953, a wet year, and 1954, a dry year. Five harvest intervals were
studied: 2, 3, 4, 6, and 8 weeks. Five nitrogen levels were included: 0, 112, 336, 672,
and 1010 kg/ha. The data are listed in Table 3-6.
England: Ryegrass
This data set was taken from a study by Reid (1978). Twenty-one nitrogen levels
were included: 0, 28, 56, 84, 112, 140, 168, 196, 224, 252, 280, 308, 336, 392, 448, 504,
560, 616, 672, 784, and 896 kg/ha. The ryegrass was harvested at different growth
stages, resulting in three different number of cuttings per season: 10, 5, 3. The length of
the season was 26 weeks. It should be noted that the different number of cuttings
represent variable harvest intervals, namely 2.6, 5.2, and 8.67 weeks. As a result, we
expect this variable effect to appear in the results. The data are listed in Table 3-7.
Fayetteville. Arkansas: Bermudagrass
This data set was acquired from a study by Huneycutt et al. (1988). The
bermudagrass was grown over three years, with and without irrigation. Six nitrogen levels
were included: 0, 112, 224, 336, 448, 560, and 672 kg/ha. The data are listed in Table 3-
8.
Fayetteville, Arkansas: Tall Fescue
This data set was taken from the same study as above (Huneycutt el al., 1988).
The tall fescue was grown over three seasons, with and without irrigation. Six nitrogen

24
levels were included: 0, 112, 224, 336, 448, 560, and 672 kg/ha. The data are listed in
Table 3-9.
Eagle Lake. Texas: Bahiagrass and Bermudagrass
This data set was acquired from a study by Evers (1984). Both grasses were
grown over two years (1979 and 1980), and five nitrogen levels were included: 0, 84,
168, 252, and 336 kg/ha. The data are listed in Table 3-10.
North Carolina: Corn
These field experiments were conducted with corn at three North Carolina
Research Stations: Central Crops Research Station, Clayton, NC, on Dothan loamy sand
soils (fine loamy, siliceous thermic, Plinthic Paleudults), Lower Coastal Plain Tobacco
Research Station, Kinston, NC, on Goldsboro sandy loam soils (fine loamy, siliceous
thermic, Aquic Paleudults), and Tidewater Research Station, Plymouth, NC, on
Portsmouth very fine sandy loam soils (fine loamy, mixed thermic, Typic Umbraquults)
(Kamprath, 1986). The experiments were conducted from 1981 through 1984. The corn
grown at Clayton received irrigation. Each year the experiment was conducted in a
different field and for this reason averages were used in this study. The data from the
Dothan and Goldsboro soils are listed in Table 3-11. The data from the Portsmouth soil
are listed in Table 3-12. For all three sites, five nitrogen levels were included: 0, 56, 112,
168, and 224 kg/ha.
Florida: Bahiagrass
The data from this analysis were drawn from Blue (1987). The bahiagrass was
grown on two soils, an Entisol (Astatula sand) near Williston, Florida, and a Spodosol

25
(Myakka fine sand) near Gainesville, Florida. Entisol is typically a dry soil and Spodosol
is typically wet. Five nitrogen levels were included: 0, 100, 200, 300, and 400 kg/ha.
The data are listed in Table 3-13.
England: Ryegrass
The data for this analysis were taken from a study by Morrison et a/. (1980) and is
listed in Table 3-14. Twenty different sites in England were used to grow the ryegrass.
Six different nitrogen levels were included in the study at each site: 0, 150, 300, 450, 600,
and 750 kg/ha. The reader is directed to the report for further information about the site
characteristics and weather data..
Tifln, Georgia: Rye
The data for this analysis were drawn from a study by Walker and Morey (1962).
Six levels of nitrogen (0, 45, 90, 135, 180, and 225), phosphorous (0, 20, 40, 60, 80, and
100), and potassium (0, 37, 74, 111, 148, and 185) were investigated. The data are listed
in Table 3-15.

Table 3-1. Common and Scientific Names of Grasses Studied.
Common Name
Scientific Name
Bahiagrass
Bermudagrass
Corn
Dallisgrass
Rye
Ryegrass
Tall Fescue
Paspalum notatum Fliigge
Cynodon dactyl on (L.) Pers.
Zea mays L.
Paspalum dilatatum Poir.
Secale cerea/e
Lolum perenne L.
Festuca arundinacea

27
Table 3-2. Seasonal Dry Matter Yield of Coastal Bermudagrass and Pensacola Bahiagrass
Species
Irrigation
Year
0
Applied Nitrogen, kg/ha
168 336
672
Bermuda
No
1957
3.74
Mg/ha-
12.52
17.41
22.32
1958
4.70
11.38
17.44
21.56
1959
3.40
9.04
13.36
20.44
Yes
1957
3.49
11.70
18.32
21.08
1958
4.33
10.75
17.89
22.61
1959
4.76
11.94
19.42
24.43
Bahia
No
1957
4.56
11.70
16.39
20.32
1958
3.63
10.39
16.85
22.23
1959
2.67
8.19
14.14
19.62
Yes
1957
4.31
10.35
16.81
23.26
1958
3.24
9.25
15.69
22.33
1959
3.82
9.52
16.33
22.43
Source: Data from Evans et al. (1961).

28
Table 3-3. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Dallisgrass Grown at Baton Rouge, Louisiana.
Applied Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
0
5.33
77
15.7
56
6.56
103
16.2
112
7.97
129
17.0
224
10.53
194
18.9
448
13.21
305
23.4
896
15.34
417
27.5
Source: Data from Robinson et al. (1988).

29
Table 3-4. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bermudagrass Grown at Thorsby, Alabama over 3.0 and 4.5 Week Harvest
Intervals (1961),
Applied
Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
3.0 weeks
4.5 weeks
3.0 weeks
4.5 weeks
3.0 weeks
4.5 weeks
0
2.95
3.60
53
55
18.1
15.4
224
9.70
11.95
230
220
23.7
18.4
448
14.70
17.20
406
387
27.6
22.5
672
16.70
18.50
501
466
30.0
25.2
1344
17.50
19.95
560
549
32.0
27.5
2016
17.60
19.20
597
568
33.9
29.6
Source: Data from Doss et al. (1966).

30
Table 3-5. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Dallisgrass Grown in Maryland.
Harvest
Interval
weeks
Applied Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
3.2
0
1.29
21.5
16.7
112
4.37
90.5
20.7
224
8.72
225
25.8
448
13.53
409
30.2
672
14.07
461
32.8
896
14.36
497
34.6
3.6
0
1.24
19.2
15.5
112
4.84
86.6
17.9
224
9.04
198
21.9
448
12.94
371
28.7
672
13.94
450
32.3
896
13.78
480
34.8
4.3
0
1.09
15.7
14.4
112
5.19
88.7
17.1
224
9.81
197
20.1
448
14.36
393
27.4
672
15.33
445
29.0
896
14.93
451
30.2
5.5
0
1.55
20.6
13.3
112
7.75
122
15.8
224
12.96
238
18.4
448
16.71
369
22.1
672
17.07
418
24.5
896
17.07
485
28.4
7.7
0
1.65
17.7
10.7
112
9.37
125
13.3
224
14.55
215
14.8
448
18.23
354
19.4
672
18.89
399
21.1
896
18.37
421
22.9
Source: Data from Decker et al. (1971).

31
Table 3-6. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bermudagrass Grown at Tifton, Georgia.
Year
Harvest Interval
weeks
0
Applied Nitrogen, kg/ha
112 336 672
1010
1953
2
2.33
5.96
11.76
17.43
19.71
3
3.33
8.91
13.64
19.24
20.47
4
2.71
9.86
17.65
21.68
23.61
6
4.35
12.77
21.79
28.11
30.11
8
5.64
13.66
22.38
27.93
29.30
1954
2
0.76
2.69
6.83
7.84
8.62
3
0.94
3.65
7.39
9.90
9.99
4
1.08
4.55
9.45
11.13
11.49
6
1.30
6.16
11.60
13.57
14.13
8
1.93
6.45
12.23
15.86
16.22
N Removal, kg/ha
1953
2
37
130
327
582
721
3
51
184
363
579
682
4
40
176
431
590
739
6
53
158
392
621
738
8
62
184
372
545
624
1954
2
17
70
224
299
320
3
15
75
208
294
364
4
19
80
233
323
344
6
21
92
261
293
390
8
26
100
223
325
417
N
Concentration, g/kg
1953
2
16.0
21.8
27.8
33.4
36.6
3
15.4
20.6
26.6
30.1
33.3
4
14.8
17.9
24.4
27.2
31.3
6
12.1
12.4
18.0
22.1
24.5
8
11.0
13.5
16.6
19.5
21.3
1954
2
22.6
26.0
32.8
38.2
37.1
3
16.0
20.5
28.1
29.7
36.4
4
17.5
17.5
24.7
29.0
29.9
6
16.4
14.9
22.5
21.6
27.6
8
13.5
15.5
18.2
20.5
25.7
Source: Data from Prine and Burton (1956).

32
Table 3-7. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Perennial Ryegrass Grown in England with a Different Number of Harvests
over the Season.
Dry Matter Yield
N Removal
N Concentration
Applied
Cuttings
Cuttings
Cuttings
Nitrogen
10 5 3
10 5 3
10 5 3
kg/ha
Mg/ha
kg/ha
R/kg
0
0.75
1.81
2.65
18
34
38
24.3
18.2
14.6
28
1.12
2.30
4.08
27
48
64
24.0
18.2
15.4
56
1.58
3.28
4.88
40
59
67
25.0
18.2
13.8
84
2.27
4.00
5.17
59
75
77
26.1
18.9
15.0
112
2.85
5.58
5.97
77
104
83
26.9
18.6
13.9
140
3.12
5.65
6.90
85
102
96
27.4
18.2
13.9
168
3.75
7.13
8.06
106
136
120
28.2
19.2
15.0
196
4.50
7.61
8.58
131
149
136
29.1
19.5
15.8
224
4.73
8.49
9.59
142
174
155
30.1
20.5
16.2
252
5.30
9.18
9.36
162
192
168
30.2
20.8
17.9
280
6.38
9.30
10.83
200
195
197
31.4
21.0
18.2
308
6.72
10.32
10.90
222
235
213
33.0
22.7
19.5
336
6.67
11.41
11.92
221
261
237
33.0
22.9
19.8
392
7.95
11.36
11.90
283
290
274
35.7
25.6
22.9
448
8.31
11.84
12.01
301
310
280
36.3
26.2
23.4
504
8.75
12.24
12.46
320
350
302
38.4
28.6
24.3
560
8.82
12.45
12.38
349
362
325
39.7
29.0
26.2
616
8.91
12.24
12.90
354
373
347
39.7
30.4
26.9
672
9.17
12.01
11.65
381
386
330
41.4
32.0
28.3
784
9.18
11.95
11.97
379
397
349
41.3
33.0
29.1
896
8.86
11.86
11.96
376
413
370
42.4
34.9
31.0
Source: Data from Reid (1978).

33
Table 3-8. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bermudagrass Grown at Fayetteville, Arkansas over Three Years, Irrigated
and Non-irrigated.
Year
0
112
Applied Nitrogen, kg/ha
224 336 448
560
672
Dry Matter Yield
1983
1.95
6.88
Non-irrigated
11.52 13.88 16.39
16.70
17.71
Mg/ha
1984
2.35
8.43
11.59
12.71
16.50
16.48
16.01
1985
1.70
7.20
9.37
15.42
19.70
18.70
19.41
1983
4.53
9.12
14.50
Irrigated
20.80
21.90
23.63
23.87
1984
4.17
11.52
14.17
18.31
22.42
23.67
24.34
1985
1.75
8.32
12.55
18.56
20.11
21.36
23.18
N Removal
1983
27
109
Non-irrigated
210 262 354
358
419
kg/ha
1984
29
136
210
266
370
393
379
1985
21
105
178
294
416
407
484
1983
67
156
278
Irrigated
379
466
537
512
1984
61
186
252
366
402
451
561
1985
25
129
229
333
376
458
523
N Concentration
1983
13.8
15.8
Non-irrigated
18.2 18.9 21.6
21.4
23.7
g/kg
1984
12.2
16.2
18.0
21.0
22.4
23.8
23.7
1985
12.5
14.6
19.0
19.0
21.1
21.8
25.0
1983
14.7
17.1
19.2
Irrigated
18.2
21.3
22.7
21.4
1984
14.7
16.2
17.8
20.0
17.9
19.0
23.0
1985
14.1
15.5
18.2
17.9
18.7
21.4
22.6
Source: Data from Huneycutt et al. (1988).

34
Table 3-9. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Tall Fescue Grown at Fayetteville, Arkansas over Three Seasons, Irrigated
and Non-irrigated.
Year
0
112
Applied Nitrogen, kg/ha
224 336 448
560
672
Dry Matter Yield
1981-2
2.51
5.47
Non-irrigated
8.97 10.31 12.33
11.21
11.86
Mg/ha
1982-3
1.86
5.00
4.64
7.11
7.67
7.40
8.32
1983-4
1.91
3.99
5.45
4.86
4.89
5.09
4.95
1981-2
3.92
7.85
10.90
Irrigated
13.36
14.84
15.33
15.53
1982-3
2.91
6.37
7.29
11.70
12.76
13.45
13.72
1983-4
3.77
7.91
10.49
14.59
15.92
17.15
17.15
N Removal
1981-2
55
118
Non-irrigated
207 259 337
334
368
kg/ha
1982-3
36
97
90
174
196
206
229
1983-4
37
94
147
133
140
150
147
1981-2
84
166
248
Irrigated
344
399
424
442
1982-3
62
120
139
275
331
362
371
1983-4
75
162
237
352
390
414
447
N Concentration
1981-2
21.9
21.6
Non-irrigated
23.1 25.1 27.4
29.8
31.0
g/kg
1982-3
19.2
19.4
19.4
24.5
25.6
27.8
27.5
1983-4
19.4
23.7
27.0
27.4
28.6
29.4
29.8
1981-2
21.4
21.1
22.7
Irrigated
25.8
26.9
27.7
28.5
1982-3
21.4
18.9
19.0
23.5
25.9
26.9
27.0
1983-4
19.8
20.5
22.6
24.2
24.5
24.2
26.1
Source: Data from Huneycutt et al. (1988).

35
Table 3-10. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bahiagrass and Bermudagrass Grown at Eagle Lake, Texas.
Year
0
Applied Nitrogen, kg/ha
84 168 252
336
Bermudagrass
Dry Matter
1979
4.47
6.98
10.09
12.91
14.23
Mg/ha
1980
3.90
5.47
6.33
7.47
8.12
N Removal
1979
58
104
156
225
258
kg/ha
1980
62
102
116
155
178
N Concentration
1979
13.0
14.9
15.5
17.4
18.1
g/kg
1980
16.0
18.7
18.4
20.7
21.9
Bahiagrass
Dry Matter
1979
4.38
6.02
7.71
10.10
10.40
Mg/ha
1980
2.95
3.99
5.17
5.30
6.32
N Removal
1979
55
88
123
173
186
kg/ha
1980
65
81
108
120
150
N Concentration
1979
12.6
14.6
16.0
17.1
17.9
1980
22.0
20.2
20.9
22.7
23.7
Source: Data from Evers (1984).

36
Table 3-11. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Grain and Total Plant of Corn Grown on Dothan and Goldsboro Soils at
Clayton and Kinston, North Carolina, Respectively.
Site Part
Applied
Nitrogen
kg/ha
Dry Matter
Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
Dothan Grain
0
4.37
47
10.8
56
7.12
74
10.4
112
9.77
113
11.6
168
10.81
135
12.5
224
11.04
150
13.6
Total
0
9.09
61
6.7
56
14.67
92
6.3
112
18.35
135
7.4
168
19.11
166
8.7
224
19.90
188
9.4
Goldsboro Grain
0
3.00
32
10.7
56
5.47
62
11.3
112
6.93
87
12.6
168
7.46
101
13.5
224
7.57
107
14.1
Total
0
6.63
36
5.4
56
10.60
71
6.7
112
13.10
104
7.9
168
13.55
120
8.9
224
13.92
134
9.6
Source: Data from Kamprath (1986).

37
Table 3-12. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Grain and Total Plant of Corn Grown on Portsmouth Soil at Plymouth,
North Carolina.
Part
Applied Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
Grain
0
4.53
48
10.6
56
6.30
68
10.8
112
7.65
89
11.6
168
8.52
104
12.2
224
9.04
115
12.7
Total
0
8.93
60
6.7
56
11.63
83
7.1
112
13.39
101
7.5
168
14.68
121
8.2
224
15.10
133
8.8
Source: Data from Kamprath (1986).

38
Table 3-13. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Bahiagrass Grown on Entisol and Spodosol Soils in Florida.
Soil
Applied Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
Entisol
0
1.59
18.7
11.8
100
5.02
68.7
13.7
200
7.97
121
15.2
300
9.92
166
16.8
400
10.54
185
17.6
Spodosol
0
4.40
49.6
11.3
100
8.67
105
12.1
200
13.76
185
13.4
300
17.07
248
14.5
400
18.28
291
15.9
Source: Data from Blue (1987).

39
Table 3-14. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration for
Ryegrass Grown in England.
Site
Applied Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
5
0
6.08
157
25.8
150
8.45
206
24.4
300
11.76
327
27.8
450
12.94
389
30.1
600
12.94
432
33.4
750
12.75
458
35.9
6
0
4.42
102
23.1
150
8.74
216
24.7
300
11.88
354
29.8
450
13.16
471
35.8
600
12.84
490
38.2
750
12.99
534
41.1
7
0
1.64
46
28.0
150
4.95
140
28.3
300
7.50
245
32.7
450
7.82
285
36.4
600
8.29
321
38.7
750
7.59
309
40.7
8
0
0.64
12
18.8
150
5.28
105
19.9
300
9.57
252
26.3
450
11.43
347
30.4
600
11.95
413
34.6
750
12.09
464
38.4
9
0
5.65
136
24.1
150
10.02
263
26.2
300
12.10
350
28.9
450
14.26
464
32.5
600
13.89
476
34.3
750
13.29
474
35.7

40
Table 3-14continued
Site
Applied Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
10
0
2.96
64
21.6
150
7.03
162
23.0
300
11.24
304
27.0
450
12.52
401
32.0
600
12.30
413
33.6
750
11.98
433
36.1
12
0
1.24
28
22.6
150
4.84
103
21.3
300
9.71
237
24.4
450
13.58
384
28.3
600
14.75
438
29.7
750
15.06
518
34.4
13
0
4.40
102
23.2
150
8.38
221
26.4
300
10.93
341
31.2
450
12.23
426
34.8
600
12.19
458
37.6
750
11.44
452
39.5
14
0
2.90
63
21.7
150
8.24
212
25.7
300
10.41
282
27.1
450
12.42
393
31.6
600
12.81
445
34.7
750
12.09
469
38.8
15
0
1.53
34
22.2
150
5.06
118
23.3
300
8.14
220
27.0
450
9.95
310
31.2
600
10.81
376
34.8
750
11.17
426
38.1

41
Table 3-14continued
Site
Applied Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
16
0
4.10
88
21.5
150
8.69
206
23.7
300
11.92
336
28.2
450
12.70
420
33.1
600
12.35
447
36.2
750
11.92
448
37.6
17
0
0.69
11
15.9
150
4.83
96
19.9
300
9.05
230
25.4
450
10.74
315
29.3
600
11.07
367
33.2
750
10.91
384
35.2
19
0
1.35
18
13.3
150
3.41
80
23.5
300
5.64
153
27.1
450
6.09
190
31.2
600
6.15
215
35.0
750
5.95
215
36.1
20
0
2.85
58
20.4
150
7.09
146
20.6
300
11.28
275
24.4
450
12.39
337
27.2
600
15.06
473
31.4
750
14.10
468
33.2
22
0
1.61
33
20.5
150
5.30
123
23.2
300
8.67
241
27.8
450
9.98
323
32.4
600
10.09
348
34.5
750
10.06
370
36.8

42
Table 3-14~continued.
Site
Applied Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
23
0
1.98
40
20.2
150
5.69
128
22.5
300
8.55
231
27.0
450
10.08
317
31.4
600
9.92
333
33.6
750
9.66
353
36.5
25
0
3.94
91
23.1
150
7.17
174
24.3
300
10.12
285
28.2
450
12.24
391
31.9
600
12.20
437
35.8
750
12.14
462
38.1
26
0
1.82
36
19.8
150
4.89
103
21.1
300
7.62
193
25.3
450
10.23
309
30.2
600
10.69
350
32.7
750
10.61
374
35.2
27
0
2.75
62
22.5
150
6.17
155
25.1
300
8.59
235
27.4
450
9.91
325
32.8
600
10.51
376
35.8
750
9.93
390
38.3
28
0
0.82
16
19.5
150
4.75
100
21.1
300
7.89
198
25.1
450
10.27
291
28.3
600
11.48
363
31.6
750
12.03
411
34.2
Source: Data from Morrison et al. (1980).

43
Table 3-15. Seasonal Dry Matter Yields and Nutrient Response to Applied Nutrients for
Gator Rye at Tifton, Georgia.
N
kg/ha
P
kg/ha
K
kg/ha
Y
Mg/ha
Nu
kg/ha
Pu
kg/ha
Ku
kg/ha
Nc
g/kg
Pc
g/kg
Kc
g/kg
0
40
74
0.55
-
-
-
45
40
74
1.81
58
12.1
66
32.3
6.69
36.4
90
40
74
3.01
112
19.6
108
37.3
6.51
36.0
135
40
74
3.75
160
24.4
136
42.6
6.51
36.4
180
40
74
4.01
175
26.8
146
43.7
6.69
36.4
225
40
74
4.55
216
28.6
148
47.5
6.29
32.5
135
0
74
2.09
92
13.0
75
44.1
6.21
35.8
135
20
74
3.08
134
17.0
118
43.5
5.51
38.3
135
40
74
3.36
133
21.1
119
39.5
6.29
35.4
135
60
74
3.74
146
24.3
141
39.1
6.51
37.8
135
80
74
4.00
148
21.7
134
37.0
5.42
33.4
135
100
74
3.97
-
-
-
-
-
-
135
40
0
2.87
112
27.6
62
39.0
9.61
21.7
135
40
37
3.18
116
15.3
90
36.6
4.81
28.3
135
40
74
3.74
140
24.3
133
37.3
6.51
35.5
135
40
111
3.56
134
18.8
130
37.5
5.29
36.4
135
40
148
3.96
160
24.2
150
40.4
6.12
37.8
135
40
185
4.11
162
31.2
159
39.4
7.60
38.8
Source: Data from Walker and Morey (1962).

CHAPTER 4
RESULTS AND DISCUSSION
Evaluation of the Simple Logistic Model
Thorsbv. Alabama: Bermudagrass and Bahiagrass
Data from Evans et al. (1961) for bermudagrass and bahiagrass were used. First
the grasses will be analyzed separately, then they will be pooled to see if any of the
parameters are common for the two grasses. The analysis of variance for the Coastal
bermudagrass and Pensacola bahiagrass over years 1957-1959 is given in Tables 4-1 and
4-2, respectively (3 yrs x 1 grass x 2 irrigation x 4 N). In mode 1, a common (A, b, c) is
assumed to apply to Equation [2.8], while individual (A, b, c) for each equation is
assumed in mode 2. For the Coastal bermudagrass, comparison among modes 1 and 2 has
a variance ratio of F = 2.589/0.584 = 4.43 and is significant at the 5% confidence level.
This comparison tests if mode 2 better describes the data. A significant variance ratio
suggests that mode 2 is better. Comparison among modes 3 and 2 has a variance ratio of
F = 0.875/0.584 = 1.50 and is not significant. This comparison also tests if mode 2 better
describes the data. The non-significant variance ratio suggests that mode 3 describes the
data as well or better than mode 2. Because mode 3 requires less parameters, we will
select it at this point in the analysis following Occam's razor. The variance ratio is not
significant for any of the comparisons among modes 2, 3, 4, or 5, suggesting that mode 3,
an individual A for each combination of year and irrigation and a common b and c for all.
For the Pensacola bahiagrass, comparison among modes 1 and 2 has a variance ratio of
44

45
7.48 and is significant at the 2.5% confidence level. Comparison among modes 3 and 4
has a variance ratio of 9.49 and is significant at the 0.5% confidence level. Furthermore,
comparison among modes 4 and 5 has a variance ratio of 8.78 and is significant at the
0.5% confidence level. These tests suggest when the grasses are studied separately, a
single value can be used for c for all years and irrigation treatments to adequately describe
the data. Next the data from the two grasses are combined and the analysis of variance
data are presented in Table 4-3 (3 yrs x 2 grass x 2 irrigation x 4 N). A comparison of
modes 1 and 2 results in a variance ratio of 5.61 that is significant at the 0.5% confidence
level. Comparison of modes 3 and 4 results in a variance ratio of 3.27 that is significant at
the 1% confidence level. Comparison of modes 3 and 5 results in a variance ratio of 2.93
that is significant at the 5% confidence level. Comparison of modes 5 and 4 leads to a
variance ratio of 2.87 that is significant at the 2.5% confidence level. Comparison of
modes 4 and 6 leads to a variance ratio of 3.41 that is significant at the 1% confidence
level Comparison of modes 5 and 6 results in a variance ratio of 3.74 that is significant at
the 5% confidence level. Comparisons among modes 3 and 7 and 4 and 7 result in
variance ratios of 5.57 and 2.75, respectively. Both comparisons are significant at the
2.5% confidence level. Based upon these comparisons, we can conclude that mode 7,
with individual A for each year, individual b for each grass, and common c describes the
data best. The overall correlation coefficient of 0.9949 and adjusted correlation
coefficient of 0.9927 are calculated for mode 7. The error analysis of the parameters are
shown in Table 4-4. The largest relative error (¡standard error/estimate|) was for the b
parameter. This is due in part to the small numbers involved and that nonlinear regression
on the logistic equation places more emphasis upon the maximum and less upon the
intercept. Still the largest relative error for this set of parameters was under 4%. Since
the range of A parameter values (largest smallest) between years and irrigation schemes
is less than six, the data will be averaged over years. Overman et al. (1990a, 1990b) have

46
shown that averaging over years is appropriate, since variations due to water availability
and harvest interval appear in the linear parameter A. The averaged data are in Table 4-5
and the error analysis for the averaged data is in Table 4-6. The overall correlation
coefficient of 0.9981 and adjusted correlation coefficient of 0.9969 are calculated. Note
that the estimates of b and c are the same as for the unaveraged data, supporting what
Overman el al. (1990a, 1990b) have found previously. Results are shown in Figure 4-1,
where curves for Coastal bermudagrass and Pensacola bahiagrass dry matter, irrigated and
non-irrigated, are drawn from the following equations:
Coastal bermudagrass, non-irrigated: Y =21 57/[ 1 + exp(1.39 0.0078N)] [4.1]
Coastal bermudagrass, irrigated: Y =23.44/[l + exp(1.39 0.0078N)] [4.2]
Pensacola bahiagrass, non-irrigated: Y =21 49/[ 1 + exp(l .57 0.0078N)] [4.3]
Pensacola bahiagrass, irrigated: Y =22.73/[l + exp(1.57 0.0078N)] [4.4]
The scatter and residual plots of seasonal dry matter are given in Figures 4-2 and
4-3. The mean and the 2 standard errors of the residuals are shown by the solid and
dashed horizontal lines, respectively. As shown in Figure 4-3, all of the data fit between
2 standard errors with no apparent trend.
Evaluation of the Extended Logistic Model
Baton Rouge. LA: Dalliserass
The data for this analysis are taken from Robinson el al. (1988). The analysis of
variance is shown in Table 4-7 (dry matter and N removal x 6 N). Comparison among
modes 1 and 2 results in a variance ratio of 3736 and is significant at the 0.1% level.
Comparison among modes 3 and 2 leads to a variance ratio of 127.8 which is also
significant at the 0.1% level. Mode 4, with individual A and b for dry matter and N
removal, and common c describes the data best, since F( 1,6,95) = 5.99, and F( 1,7,99.9) =

47
29.25. This outcome supports Postulate 3. The overall correlation coefficient and
adjusted correlation coefficient calculated from mode 4 are 0.9997 and 0.9995,
respectively. The error analysis for the parameters is shown in Table 4-8. Results are
shown in Figure 4-4, where curves for dry matter and plant N removal are drawn from
Y = 15.60/[1 + exp(0.58 0.0055N)] [4.5]
N = 430.7/[l + exp(1.47 0.0055N)] [4.6]
From these results, it follows that plant N concentration, shown in Figure 4-4, is estimated
from
Nc = 27.6 [1 + exp(1.47 0.0055N)]/[1 + exp(0.58 0.0055N)] [4.7]
As shown, the data are described well by these equations. It should be noted that the
plant N concentration data were not defined by regression techniques, but rather as a ratio
of plant N removal and dry matter yield.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal, shown in Figure 4-5, are described by
Y = 26.5 N/(300 + N) [4.8]
Nc = 11.3 + 0.0378NU [4.9]
The intercept (11.3 g/kg) represents plant N concentration in a nitrogen deficient
environment with low plant N removal.
The form of this model can be validated by plotting the data in dimensionless form.
The data plotted in Figure 4-5 were strictly measured data; that is, the independent
variable applied N has been ignored. The results of Postulate 3 defined the curve drawn.
By dividing dry matter yield, plant N concentration and plant N removal by their
appropriate mximums, all of the data are collapsed onto the same dimensionless scale.
Furthermore, the curves drawn are dependent upon the Ab, (b' b), term alone. This will
be more significant as larger, more complex data sets are investigated. The form of the

48
model can be validated in this manner. The dimensionless plot for dallisgrass is shown in
Figure 4-6, and validates the form of the model. The curves were drawn from
Y/A = 1.70 (Nu/A')/[0.70 + (Nu/A')] [4.10]
Nc/Ncm = 0.41 +0.59 (Nu/A') [4.11]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
7 through 4-10.
Thorsby. AL: Bermudaerass
The data for this analysis are taken from Doss et al. (1966). The analysis of
variance is shown in Table 4-9 (dry matter and N removal x 2 At x 6 N). Comparison
among modes 1 and 2 results in a variance ratio of 402 and is significant at the 0.1% level.
Comparison among modes 3 and 2 leads to a variance ratio of 4.63 that is significant at
the 5% level. Comparison among inodes 3 and 4 results in a variance ratio of 8.07 that is
significant at the 0.5% level. Mode 5, with individual A for yield and plant N removal at
both harvest intervals, b for dry matter and N removal, and common c describes the data
best, since F(5,12,95) = 3.11, and F(l,17,99.9) = 15.72. This outcome also supports
Postulate 3. The overall correlation coefficient and adjusted correlation coefficient
calculated from mode 5 are 0.9983 and 0.9977, respectively. The error analysis for the
parameters is shown in Table 4-10. Results are shown in Figure 4-11, where curves for
dry matter and plant N removal at both harvest intervals, are drawn from
3.0 weeks: Y= 17.42/[1 + exp(1.27 0.0067N)] [4.12]
N= 568.9/[l + exp(2.02 0.0067N)] [4.13]
4.5 weeks: Y = 19.75/[1 + exp(1.27 0.0067N)] [4.14]
Nu= 543.5/[l + exp(2.02 0.0067N)] [4.15]
From these results, it follows that plant N concentration, shown in Figure 4-11, is
estimated from

49
3.0 weeks: Nc= 32.7 [1 + exp(2.02 0.0067N)]/[1 + exp(1.27 0.0067N)] [4.16]
4.5 weeks: Nc= 27.5 [1 + exp(2.02 0.0067N)]/[1 + exp(1.27 0.0067N)] [4.17]
As shown, the data are described well by these equations.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal, shown in Figure 4-12, are described
by
3.0 weeks: Y = 33.0 Nu/(509 + Nu) [4.18]
Nc= 15.4 + 0.0303NU [4.19]
4.5 weeks: Y = 37.4 N/(487 + Nu) [4.20]
Nc= 13.0 + 0.0267NU [4.21]
The dimensionless plot for bermudagrass is shown in Figure 4-13, and validates the
form of the model. The curves were drawn from
Y/A = l^OK/A'ytO.OO + tNu/A)] [4.22]
Nc/Ncm = 0.47 +0.53 (H/A') [4.23]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
14 through 4-17.
Maryland: Bermudagrass
The data for this analysis are taken from Decker el al. (1971). The analysis of
variance is shown in Table 4-11 (dry matter and N removal x 5 At x 6 N). Comparison
among modes 1 and 2 results in a variance ratio of 190.9 and is significant at the 0.1%
level. Comparison among modes 3 and 2 leads to a variance ratio of 4.06 that is
significant at the 0.1% level. Comparison among modes 3 and 4 results in a variance ratio
of 4.65 that is significant at the 0.1% level. Mode 5, with individual A for yield and plant
N removal at the five harvest intervals, b for dry matter and N removal, and common c
describes the data best, since F(17,30,95) =1.98, F(l,47,99.9) = 12.32, and the residual

50
sums of squares (RSS) for mode 5 is smaller than mode 4 while using less parameters.
Postulate 3 is supported again. The overall correlation coefficient and adjusted correlation
coefficient calculated from mode 5 are 0.9964 and 0.9955, respectively. The error
analysis for the parameters is shown in Table 4-12. Results are shown in Figure 4-18,
where curves for dry matter and plant N removal at all five harvest intervals, are drawn
from
3.2 weeks: Y = 13.90/[1 + exp(1.78 0.0112N)] [4.24]
Nu = 469.5/[l + exp(2.59 0.0112N)] [4.25]
3.6 weeks: Y = 13.65/[1 + exp(1.78 0.0112N)] [4.26]
Nu = 444.7/[l + exp(2.59 0.0112N)] [4.27]
4.3 weeks: Y 14.93/[1 + exp(1.78 0.0112N)] [4.28]
Nu = 440.2/[l + exp(2.59 0.0112N)] [4.29]
5.5 weeks: Y = 17.56/[1 + exp( 1.78 0.0112N)] [4.30]
N = 444.0/[l + exp(2.59 0.0112N)] [4.31]
7.7 weeks: Y = 19.34/[1 + exp(1.78 0.0112N)] [4.32]
Nu = 409.7/[l + exp(2.59 0.0112N)] [4.33]
From these results, it follows that plant N concentration, shown in Figure 4-18, is
estimated from
3.2 weeks: Nc = 33.7 [1 + exp(2.59 0.0112N)]/[1 + exp(l.78 0.0112N)] [4.34]
3.6 weeks: Nc = 32.6 [1 + exp(2.59 0.0112N)]/[1 + exp(l.78 0.0112N)] [4.35]
4.3 weeks: Nc = 29.5 [1 + exp(2.59 0.0112N)]/[1 + exp(l.78 0.0112N)] [4.36]
5.5 weeks: Nc = 25.3 [1 + exp(2.59 0.0112N)]/[1 + exp(l.78 0.0112N)] [4.37]
7.7 weeks: Nc = 21.2 [1 + exp(2.59 0.0112N)]/[1 + exp(l.78 0.0112N)] [4.38]
As shown, the data are described well by these equations.

51
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant
by
N removal, shown in Figure 4-19, are described
3.2 weeks: Y =
25.0 Nu/(376 + N)
[4.39]
Nc=
15.0 + 0.0399NU
[4.40]
3.6 weeks: Y =
24.6Nu/(356 + Nu)
[4.41]
Nc =
14.5 + 0.0407NU
[4.42]
4.3 weeks: Y =
26.9 K,/(353 + N)
[4.43]
Nc =
13.1 +0.0372NU
[4.44]
5.5 weeks: Y =
31.6Nu/(356 + Nu)
[4.45]
Nc=
11.2 +0.0316NU
[4.46]
7.7 weeks: Y =
34.8 Nu/(328 + Nu)
[4.47]
Nc=
9.4 + 0.0287NU
[4.48]
The dimensionless plot for bermudagrass is shown in Figure 4-20, and validates the
form of the model. The curves were drawn from
Y/A = 1.80(Nu/A)/[0.80 + (Nu/A')] [4.49]
Nc/Ncm = 0.44+0.56 (Nu/A') [4.50]
Notice that all of the data have collapsed onto one curve and line, respectively. This
implies that the Ab is constant and the same for each harvest interval. Scatter and residual
plots of dry matter yield and plant N removal are shown in Figures 4-21 through 4-24.
It appears from Table 4-12, that there is a linear relationship between A, estimated
maximum dry matter yield, and harvest interval. The relationship does not appear to be as
clear for A', estimated maximum plant N removal. Overman et al. (1990b) found that
both water availability and harvest interval could be linked linearly to A up to a six weeks
interval, after which senescence sets in, and lower plant leaves die and/or drop off,
reducing dry matter accumulation. The estimated mximums of dry matter yield and plant

52
N removal are plotted against harvest interval in Figure 4-25. Linear regression was
conducted on the estimated maximum dry matter yield, omitting the 7.7 week harvest
interval, resulting in the following relationship
A = 7.90+ 1.71 At = 7.90(1 +0.22At) [4.51]
with a correlation coefficient of 0.9676. Linear regression was also conducted on the
estimated maximum plant N removal, omitting the 7.7 week harvest interval, resulting in
the following relationship
A'= 483.9-8.3At = 483.9 (1 0.017At) [4.52]
with a correlation coefficient of 0.6206. The small correlation coefficient and rather flat
response suggests that there is uncertainty in the nature of the relationship.
Tifln. GA: Bermudattrass
The data for this analysis are taken from Prine and Burton (1956). The analysis of
variance is shown in Table 4-13 (dry matter and N removal x 2 yrs x 5 At x 5 N).
Comparison among modes 1 and 2 results in a variance ratio of 81.2 that is significant at
the 0.1% level. Comparison among modes 3 and 2 leads to a variance ratio of 1.89 that is
significant at the 97.5 % level. Comparison among modes 3 and 4 results in a variance
ratio of 2.45 that is significant at the 0.5% level. Comparison among modes 3 and 5,
individual A for yield and plant N removal at the five harvest intervals and over both years,
b for dry matter and N removal at both years, and common c, leads to a variance ratio of
15.9 that is significant at the 0.1% level. Mode 6, individual A for yield and plant N
removal at the five harvest intervals and over both years, b for dry matter and N removal
over both years, and common c, describes the data best, since F(37,40,95) = 1.71,
F( 1,77,99.9) = 11.71, F(18,59,95) = 1.78, and F(2,75,95) = 3.12. The difference due to
years (water availability) and harvest interval is explained by the A parameter. Postulate 3
is supported again. The overall correlation coefficient and adjusted correlation coefficient

53
calculated from mode 6 are 0.9941 and 0.9923, respectively. The error analysis for the
parameters is shown in Table 4-14. Results are shown in Figures 4-26 through 4-30 for
each clipping interval, where curves for dry matter and plant N removal at all five harvest
intervals, are drawn from
2 weeks, 1953: Y = 17.95/[1
Nu = 644.7/[l
1954: Y = 8.40/[l
Nu = 324.5/[l
3 weeks, 1953: Y = 19.88/[1
N = 641.8/[ 1
1954: Y = 9.95/[l
N = 337.5/[l
4 weeks, 1953: Y = 23.15/[1
Nu = 687.2/[l
1954: Y = 11.67/[1
N = 348.0/[l
6 weeks, 1953: Y = 29.57/[l
Nu = 688.2/[l
1954: Y = 14.37/[1
Nu = 363.8/[l
8 weeks, 1953: Y = 29.58/[l
Nu = 606.0/[l
1954: Y = 16.24/[1
N 379.5/[l
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(l .47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
+ exp(1.47 0.0077N)]
+ exp(2.15 0.0077N)]
[4.53]
[4.54]
[4.55]
[4.56]
[4.57]
[4.58]
[4.59]
[4.60]
[4.61]
[4.62]
[4.63]
[4.64]
[4.65]
[4.66]
[4.67]
[4.68]
[4.69]
[4.70]
[4.71]
[4.72]
From these results, it follows that plant N concentration, shown in Figures 4-26 through
4-30, is estimated from

54
2 weeks, 1953: Nc
1954: Nc
3 weeks, 1953: Nc
1954: Nc
4 weeks, 1953: Nc
1954: Nc
6 weeks, 1953: Nc
1954: Nc
8 weeks, 1953: Nc
1954: Nc
35.9 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.73]
38.6 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.74]
32.3 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.75]
33.9 [1 +exp(2.15 0.0077N)]/[1 + exp(l .47 0.0077N)] [4.76]
29.7 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.77]
29.8 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.78]
23.3 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.79]
25.3 [1 +exp(2.15 0.0077N)]/[1 + exp(l .47 0.0077N)] [4.80]
20.5 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.81]
23.4 [1 + exp(2.15 0.0077N)]/[1 + exp(1.47 0.0077N)] [4.82]
As shown, the data are described well by these equations.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal at each clipping interval over both
years, shown in Figures 4-31 through 4-35, are described by
2 weeks, 1953: Y =
36.4 Nu/(662 + Nu)
[4.83]
Nc =
18.2 + 0.0275NU
[4.84]
1954: Y =
17.0Nu/(333 +NU)
[4.85]
Nc =
19.6 + 0.0587NU
[4.86]
3 weeks, 1953: Y =
40.3 Nu/(659 + Nu)
[4.87]
Nc =
16.4 + 0.0248NU
[4.88]
1954: Y =
20.2 Nu/(347 + Nu)
[4.89]
Nc =
17.2 + 0.0496NU
[4.90]
4 weeks, 1953: Y =
46.9 Nu/(706 + Nu)
[4.91]
Nc =
15.0 + 0.0213NU
[4.92]
1954: Y =
23.7 N/(357 + N)
[4.93]
Nc =
15.1 +0.0423NU
[4.94]

55
6 weeks, 1953:
Y = 59.9 N/(707 + Nu)
[4.95]
Nc = 11.8 + 0.0167NU
[4.96]
1954:
Y = 29.1 Nu/(374 + Nu)
[4.97]
Nc = 12.8 + 0.0343NU
[4.98]
8 weeks, 1953:
Y = 60.0 N,/(622 + Nu)
[4.99]
Nc = 10.4 + 0.0167NU
[4.100]
1954:
Y = 32.9 Nu/(390 + N)
[4.101]
Nc = 11.8 + 0.0304NU
[4.102]
The dimensionless plot for bermudagrass is shown in Figure 4-36, and validates the
form of the model. Note that all of the data have been collapsed onto one graph, and the
data fall on one curve. The curves were drawn from
Y/A = 2.03(NU/A')/[1.03 + (NU/A')] [4.103]
Nc/Ncm = 0.51 +0.49 (N/A') [4.104]
This result supports the hypothesis that the Ab is common for all harvest intervals and
both years.
As with the Maryland data set, it appears from Table 4-14, that there is a linear
relationship between A, estimated maximum dry matter yield, and harvest interval. The
estimated mximums of dry matter yield and plant N removal are plotted against harvest
interval in Figure 4-37. Linear regression was conducted on the estimated maximum dry
matter yield for both years, omitting the 8 week harvest interval, resulting in the following
relationships
1953: A = 11.50 + 2.97At = 11.50 (1 + 0.26At) [4.105]
1954: A = 5.49 + 1.50At = 5.49 (1 + 0.27At) [4.106]
with correlation coefficients of 0.9958 and 0.9986. It appears from these equations that
after the intercept has been factored out, the coefficient of At may be the same for both

56
years. Linear regression was also conducted on the estimated maximum plant N removal,
omitting the 8 week harvest interval, resulting in the following relationships
1953: A'= 628.0+ 12.6At = 628.0 (l+0.020At) [4.107]
1954: A' = 307.2+ 9.7At = 307.2 (l+0.032At) [4.108]
with correlation coefficients of 0.8407 and 0.9925. The response for the estimated plant
N removal is flat as with the Maryland data. The line for 1954 might fit better due to the
limited available water. Since it was a dry year, growth was limited and hence the plant
was growing as much as possible with the limited water. As with the estimated maximum
dry matter yield, it appears that the coefficient of At may be the same for both years.
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
38 through 4-41.
England: Ryegrass
The data for this analysis are taken from Reid (1978). It is important of remember
throughout this specific analysis that the ryegrass was harvested at particular stages of
growth. As a result, the harvest interval was variable. This result will present itself in the
results. The analysis of variance is shown in Table 4-15 (dry matter and N removal x 3
variable At x 21 N). As shown in the table, all of the comparisons are highly significant
(0.1 %). This is due in part to the large degrees of freedom. If comparison is made among
the mean sums of squares (MSS), mode 6 has the smallest MSS with the exception of
mode 2 (which fits each individually). Mode 6 is the preferred one since it requires less
parameters to estimate (11 versus 18). This choice is in accordance with the approach
common in physics: seeking the simplest model consistent with observation (Rothman,
1972), often cited as Occam's razor (Will, 1986). Mode 6 assumes an individual A for
each number of clippings, an individual b for dry matter yield, common b for plant N
removal, and common c for all. The overall correlation coefficient and adjusted

57
correlation coefficient calculated from mode 6 are 0.9938 and 0.9931, respectively. The
error analysis for the parameters is shown in Table 4-16. Results are shown in Figure 4-
42, where curves for dry matter and plant N removal are drawn from
10 clippings: Y = 9.42/[l + exp(1.74 0.0080N)] [4.109]
Nu = 377.0/[l + exp(2.15 0.0080N)] [4.110]
5 clippings: Y = 12.72/[1 + exp(1.23 0.0080N)] [4.111]
Nu = 403.5/[l + exp(2.15 0.0080N)] [4.112]
3 clippings: Y = 12.75/[1 + exp(0.90 0.0080N)] [4.113]
Nu = 363.0/[l + exp(2.15 0.0080N)] [4.114]
From these results, it follows that plant N concentration, shown in Figure 4-42, is
estimated from
10 clippings: Nc = 40.0 [1 + exp(2.15 0.0080N)]/[1 + exp(1.74 0.0080N)] [4.115]
5 clippings: Nc = 31.7 [1 + exp(2.15 0.0080N)]/[1 + exp(1.23 0.0080N)] [4.116]
3 clippings: Nc = 28.5 [1 + exp(2.15 0.0080N)]/[1 + exp(0.90 0.0080N)] [4.117]
As shown, the data are not described very well by these equations; however, the problem
most likely lies in the effect of variable harvest interval.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal, shown in Figure 4-43, are described
by
10 clippings: Y = 28.0 N/(744 + Nu) [4.118]
No = 26.6 + 0.0357NU [4.119]
5 clippings: Y = 21.1 N/(267 + Nu) [4.120]
Nc = 12.6 + 0.0473NU [4.121]
3 clippings: Y = 17.9 Nu/( 146 + Nu) [4.122]
Nc = 8.2 + 0.0560NU [4.123]

58
The dimensionless plot for ryegrass is shown in Figure 4-44, and this graph also
demonstrates the difficulty of variable harvest interval. The curves were drawn from
10 clippings: Y/A = 2.97 (Nu/A')/[ 1.97 + (Nu/A')] [4.124]
Nc/Ncm= 0.66+0.34 (Nu/A1) [4.125]
5 clippings: Y/A = 1.66 (K/AW0.66 + (Nu/A)] [4.126]
Nc/Ncm= 0.40+0.60 (Nu/A') [4.127]
3 clippings: Y/A = 1.40 (Nu/A'VfO^O + (Nu/A')] [4.128]
Nc/Ncm = 0.29 +0.71 (Nu/A') [4.129]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
45 through 4-48. It appears that in Figure 4-47, there may be a trend of some kind among
the residuals. It is likely that the trend is a result of the variable harvest interval.
The growing season of the rye was 26 weeks. Using this length, an average
harvest interval can be found for each of the three clipping frequencies. The estimated
maximum dry matter yield and plant N removal can be plotted against the average harvest
intervals to see if the same relationship holds for this study. There are two possible
obstacles. First, there are only three harvest intervals to plot. For the Maryland and
Tifton studies, a linear relationship was found between the expected maximum dry matter
and harvest interval. With only three points, it is difficult if not impossible to determine
the "true" relationship. Furthermore, the relationship tends to drop off after a six week
interval (Figures 4-25 and 4-37). The third average harvest interval is 8.67 weeks, and
could possibly affect the results. Secondly, these are not actual harvest intervals, but
rather an average harvest interval over the growing season. The actual harvest intervals
are variable. The effect of the variable harvest interval was observed in the figures, by
deduction it is likely to arise here as well. The plot of the estimated maximum dry matter
yield and plant N removal versus the average harvest interval is presented in Figure 4-49.

59
Linear regression was conducted on the estimated maximum dry matter yield resulting in
the following relationship
A = 8.78 + 0.52At [4.130]
with a correlation coefficient of 0.8263. Linear regression was also conducted on the
estimated maximum plant N removal resulting in the following relationship
A' = 396.6-2.8At [4.131]
with a correlation coefficient of 0.4170.
Fayetteville. AR: Bermudagrass
The data for this analysis are taken from Huneycutt et al. (1988). The analysis of
variance is shown in Table 4-17 (dry matter and N removal x 3 yrs x 2 irrigation x 7 N).
Comparison among modes 1 and 2 results in a variance ratio of 170.3 that is significant at
the 0.1% level. Comparison among modes 3 and 2 leads to a variance ratio of 2.05 that is
significant at the 2.5% level. Comparison among modes 3 and 4 results in a variance ratio
of 3.22 that is significant at the 99.5 % level. Comparison among modes 3 and 5,
individual A for yield and plant N removal with and without irrigation for all three years, b
for dry matter and N removal with and without irrigation, and common c, leads to a
variance ratio of 5.21 and is significant at the 0.5% level. Mode 6, individual A for yield
and plant N removal with and without irrigation for all three years, b for dry matter and N
removal over both with and without irrigation and all years, and common c, describes the
data best, since F(21,48,95) = 1.78, F(l,69,99.9) = 11.81, and F(10,59,95) = 2.00. The
difference due to years and irrigation is explained by the A parameter. Postulate 3 is
supported again. The overall correlation coefficient and adjusted correlation coefficient
calculated from mode 6 are 0.9950 and 0.9939, respectively. The error analysis for the
parameters is shown in Table 4-18. Results are shown in Figure 4-50, where curves for
dry matter and plant N removal for all three years and irrigation, are drawn from

60
Non-irrigated, 1983: Y = 17.90/[1 + exp(1.50 0.0084N)] [4.132]
Nu=408.7/[1 + exp(2.04 0.0084N)] [4.133]
1984: Y = 17.37/[1 + exp(1.50 0.0084N)] [4.134]
N=413.9/[l + exp(2.04 0.0084N)] [4.135]
1985: Y = 19.61/[1 + exp(1.50 0.0084N)] [4.136]
Nu=459.3/[1 + exp(2.04 0.0084N)] [4.137]
Irrigated, 1983: Y = 24.70/[l + exp(1.50 0.0084N)] [4.138]
Nu=554.3/[1 + exp(2.04 0.0084N)] [4.139]
1984: Y =24.60/[l + exp(1.50 0.0084N)] [4.140]
N=523.7/[l + exp(2.04 0.0084N)] [4.141]
1985: Y = 22.58/[l + exp(1.50 0.0084N)] [4.142]
Nu= 492.1/[1 + exp(2.04 0.0084N)] [4.143]
From these results, it follows that plant N concentration, shown in Figure 4-50, is
estimated from
Non-irrigated, 1983: Nc=22.8 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.144]
1984: Nc=23.8 [1 + exp(2.04 0.0084N)]/[1 + exp( 1.50 0.0084N)][4.145]
1985: Nc= 23.4 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.146]
Irrigated, 1983: Nc=22.4 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.147]
1984: Nc=21.3 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.148]
1985: Nc=21.8 [1 + exp(2.04 0.0084N)]/[1 + exp(1.50 0.0084N)][4.149]
As shown, the data are described relativity well by these equations.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal at each clipping interval for all three
years with and without irrigation, shown in Figure 4-51, are described by
Non-irrigated, 1983: Y = 42.9NU/(571 +NU)
N,
13.3 + 0.0233N,
[4.150]
[4.151]

61
Irrigated, 1983: Y = 59.2N/(774 + N)
1984: Y = 41.6 Nu/(578 + N)
1985: Y = 47.0 N/(641 + Nu)
1984: Y = 59.0N4731 +NU)
1985: Y = 54.1 N/(687 + Nu)
Nc = 13.9 + 0.0240NU
Nc = 13.6 + 0.0213NU
Nc = 12.7 + 0.0185NU
Nc 13.1 +0.0169NU
Nc = 12.4 +0.0170NU
[4.152]
[4.153]
[4.154]
[4.155]
[4.156]
[4.157]
[4.158]
[4.159]
[4.160]
[4.161]
The dimensionless plot for bermudagrass is shown in Figure 4-52, and validates the
form of the model. Note that all of the data have been collapsed onto one graph, and the
data fall on the curve. The curves were drawn from
Y/A = 2.40 (Nu/A')/[1.40 + (Nu/A')]
Nc/Ncn, = 0.58 +0.42 (Nu/A')
[4.162]
[4.163]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
53 through 4-56.
As done earlier, the data will be averaged over years, since variations due to water
availability appear in the linear parameter A. The averaged data are in Table 4-19 and the
analysis of variance is in Table 4-20 (dry matter and N removal x 1 yr x 2 irrigation x 7
N). Comparison among modes 1 and 2 results in a variance ratio of 373.4 that is
significant at the 0.1% level. Comparison among modes 3 and 2 leads to a non-significant
variance ratio of 2.72. Comparison among modes 3 and 4 results in a variance ratio of
6.11 that is significant at the 0.5% level. Mode 5, individual A for yield and plant N
removal with and without irrigation, b for dry matter and N removal, and common c,
accounts for all the significant variation, since F(5,16,95) = 2.85, F(l,21,99) = 8.02, and

62
F(2,19,95) = 3.52. The difference due to irrigation is explained by the A parameter.
Postulate 3 is supported again. The overall correlation coefficient and adjusted correlation
coefficient calculated from mode 5 was 0.9973 and 0.9965, respectively. The error
analysis for the parameters is shown in Table 4-21. Results are shown in Figure 4-57,
where curves for dry matter and plant N removal with and without irrigation averaged
over three years, are drawn from
Non-irrigated: Y =
18.63/[ 1 + exp(1.51 0.0084N)]
[4.164]
Nu =
435.7/[l + exp(2.04 0.0084N)]
[4.165]
Irrigated:
Y =
24.40/[l + exp(1.51 0.0084N)]
[4.166]
Nu =
534.8/[l + exp(2.04 0.0084N)]
[4.167]
The error analysis for the averaged data is in Table 4-21. Note that the estimates of b and
c are the essentially the same as for the unaveraged data, supporting what Overman et al.
(1990a, 1990b) have found previously. From these results, it follows that plant N
concentration, shown in Figure 4-57, is estimated from
Non-irrigated: Nc= 23.4 [1 + exp(2.04 0.0084N)]/[1 + exp(1.51 0.0084N)] [4.168]
Irrigated: Nc= 21.9 [1 + exp(2.04 0.0084N)]/[1 + exp(1.51 0.0084N)] [4.169]
As shown, the data are described well by these equations.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal at each clipping interval for all three
years with and without irrigation, shown in Figure 4-58, are described by
Non-irrigated: Y = 45.3 N/(623 + Nu) [4.170]
Nc = 13.8 + 0.0221NU [4.171]
Irrigated: Y = 59.3 Nu/(765 + Nu) [4.172]
[4.173]
Nc = 12.9 +0.0169NU

63
The dimensionless plot for bermudagrass is shown in Figure 4-59, and validates the
form of the model. Note that all of the data have been collapsed onto one graph, and the
data fall on the curve reasonably well. The curves were drawn from
Y/A = 2.43 (Nu/A')/[ 1.43 + (Nu/A')] [4.174]
Nc/Ncm= 0.59+0.41 (KM') [4.175]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
60 through 4-63.
Fayetteville, AR: Tall Fescue
The data for this analysis are taken from Huneycutt et al. (1988). The analysis of
variance is shown in Table 4-22 (dry matter and N removal x 3 yrs x 2 irrigation x 7 N).
Comparison among modes 1 and 2 results in a variance ratio of 352 that is significant at
the 0.1% level. Comparison among modes 3 and 2 also leads to a variance ratio of 7.60
that is significant at the 0.1% level. Comparison among modes 3 and 4 results in a
variance ratio of 9.90 that is significant at the 0.1% level. Comparison among modes 3
and 5, individual A for yield and plant N removal with and without irrigation for all three
years, b for dry matter and N removal with and without irrigation, and common c, leads to
a variance ratio of 18.3 that is significant at the 0.1% level. Mode 6, individual A for yield
and plant N removal with and without irrigation for all three years, b for dry matter and N
removal over both with and without irrigation and all years, and common c, describes the
data best, since F(21,48,95) = 2.94, F(l,69,99.9) = 11.81, F(10,59,97.5) = 2.27, and
F(2,67,95) = 3.13. The difference due to years and irrigation is explained by the A
parameter. Postulate 3 is supported again. The overall correlation coefficient and
adjusted correlation coefficient calculated from mode 6 were 0.9956 and 0.9947,
respectively. The error analysis for the parameters is shown in Table 4-23. Results are

64
shown in Figure 4-64, where curves for dry matter and plant N removal for all three years
and irrigation, are drawn from
Non-irrigated, 1981-2: Y = 12.08/[1 + exp(0.92 0.008IN)] [4.176]
Nu= 357.5/[l + exp(1.47 0.0081N)] [4.177]
1982-3: Y = 8.01/[1 + exp(0.92 0.0081N)] [4.178]
N= 217.8/p + exp(1.47 0.0081N)] [4.179]
1983-4: Y = 5.71/[1 + exp(0.92 0.008IN)] [4.180]
Nu= 169.5/[1 + exp(1.47 0.0081N)] [4.181]
Irrigated, 1981-2: Y= 15.63/[1 + exp(0.92 0.0081N)] [4.182]
Nu= 443.7/[l + exp(1.47 0.008IN)] [4.183]
1982-3: Y = 13.22/[1 + exp(0.92 0.0081N)] [4.184]
Nu= 357.3/[l + exp(1.47 0.0081N)] [4.185]
1983-4: Y = 16.81/[1 + exp(0.92 0.0081N)] [4.186]
N= 439.4/[l + exp(1.47 0.0081N)] [4.187]
From these results, it follows that plant N concentration, shown in Figure 4-64, is
estimated from
Non-irrigated:
1981-2: 29.6 [1 + exp(1.47 0.0081N)]/[1 + exp(0.92 0.0081N)] [4.188]
1982-3: 27.2 [1 + exp(1.47 0.008lN)]/[ 1 + exp(0.92 0.0081N)] [4.189]
1983-4: 29.7 [1 + exp(1.47 0.0081N)]/[1 + exp(0.92 0.0081N)] [4.190]
Irrigated:
1981-2: 28.4 [1 + exp(1.47 0.0081N)]/[1 + exp(0.92 0.0081N)] [4.191]
1982-3: 27.0 [1 + exp(l .47 0.0081N)]/[1 + exp(0.92 0.008IN)] [4.192]
1983-4: 26.1 [1 +exp(l.47-0.0081N)]/[1 + exp(0.92 0.008 IN)] [4.193]
As shown, the data are described relativity well by these equations.

65
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal at each clipping interval for all three
years with and without irrigation, shown in Figure 4-65, are described by
Non-irrigated,1981-2: Y
=
28.6 Nu/(488 + Nu)
[4.194]
Nc
=
17.1 +0.0350NU
[4.195]
1982-3: Y
=
18.9 N/(297 + Nu)
[4.196]
Nc
-
15.7 + 0.0528NU
[4.197]
1983-4: Y
=
13.5 Nu/(231 +NU)
[4.198]
Nc
=
17.1 +0.0741NU
[4.199]
Irrigated, 1981-2: Y
=
36.9 Nu/(605 + Nu)
[4.200]
Nc
=
16.4 + 0.0271NU
[4.201]
1982-3: Y
=
31.2 N/(487 + N)
[4.202]
Nc
-
15.6 + 0.0320NU
[4.203]
1983-4: Y
=
39.7 Nu/(599 + Nu)
[4.204]
Nc
-
15.1 +0.0252NU
[4.205]
The dimensionless plot for bermudagrass is shown in Figure 4-66, and validates the
form of the model. Note that all of the data have been collapsed onto one graph, and the
data fall on the curve. The curves were drawn from
Y/A = 2.36(NU/A')/[1.36 + (NU/A')] [4.206]
Nc/Ncm= 0.58+0.42 (Nu/A') [4.207]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
67 through 4-70.
As done with the previous study, the data will be averaged over years, since
variations due to water availability appear in the linear parameter A. The averaged data
are in Table 4-24 and the analysis of variance is in Table 4-25 (dry matter and N removal x
1 yr x 2 irrigation x 7 N). Comparison among modes 1 and 2 results in a variance ratio of

66
904 that is significant at the 0.1% level. Comparison among modes 3 and 2 leads to a
variance ratio of 9.44 that is significant at the 0.1% level. Comparison among modes 3
and 4 results in a variance ratio of 18.4 that is significant at the 0.1% level. Mode 5,
individual A for yield and plant N removal with and without irrigation, b for dry matter
and N removal, and common c, accounts for all the significant variation, since F(5,16,95)
= 2.85, F(l,21,99.9) = 14.59, and F(2,19,95) = 3.52. The difference due to irrigation is
explained by the A parameter. Postulate 3 is supported again. The overall correlation
coefficient and adjusted correlation coefficient calculated from mode 5 were 0.9986 and
0.9982, respectively. The error analysis for the parameters is shown in Table 4-27.
Results are shown in Figure 4-71, where curves for dry matter and plant N removal with
and without irrigation averaged over three years, are drawn from
Non-irrigated: Y = 8.67/[l + exp(0.99 0.0081N)] [4.208]
Nu = 250.7/[l + exp(1.53 0.0081N)] [4.209]
Irrigated. Y = 15.36/[1 + exp(0.99 0.0081N)] [4.210]
Nu = 417.4/[ 1 + exp(l .53 0.0081N)] [4.211]
The error analysis for the averaged data is in Table 4-26. Note that although the estimates
of b and c are not quite the same as for the unaveraged data, the Ab (b'-b), is essentially
constant. From these results, it follows that plant N concentration, shown in Figure 4-71,
is estimated from
Non-irrigated: Nc= 28.9 [1 + exp(1.53 0.0081N)]/[1 + exp(0.99 0.0081N)] [4.212]
Irrigated: Nc= 27.2 [1 + exp(1.53 0.0081N)]/[1 + exp(0.99 0.0081N)] [4.213]
As shown, the data are described well by these equations.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal averaged over three years with and
without irrigation, shown in Figure 4-72, are described by
Non-irrigated: Y
20.8 Nu/(350 + Nu)
[4.214]

67
Nc =
16.9 + 0.0481NU
[4.215]
Irrigated.
Y =
36.8 N/(583 + N)
[4.216]
Nc =
15.8 + 0.0272NU
[4.217]
The dimensionless plot for tall fescue is shown in Figure 4-73, and validates the
form of the model. Note that all of the data have been collapsed onto one graph, and the
data fall on the curve. The curves were drawn from
Y/A = 2.40 (Nu/A')/[1.40 + (Nu/A')] [4.218]
Nc/Ncm 0.58 +0.42 (Nu/A*) [4.219]
Notice the similarity between the above equations and Equations [4.174] and [4.175],
The Ab values are similar for both grasses. It is not clear if this is a coincidence or a result
of the system. Scatter and residual plots of dry matter yield and plant N removal are
shown in Figures 4-74 through 4-77.
Eagle Lake. TX: bermudagrass and bahiagrass
Data from Evans et al. (1961) for bermudagrass and bahiagrass were used. First
the grasses will be analyzed separately, then they will be pooled to see if any of the
parameters are common for the two grasses. The analysis of variance for the Coastal
bermudagrass over years 1979-1980 is shown in Table 4-27 (dry matter and N removal x
2 yrs x 5 N). Comparison among modes 1 and 2 results in a variance ratio of 419 that is
significant at the 0.1% level. Comparison among modes 3 and 2 leads to a variance ratio
of 9.22 that is significant at the 0.5% level. Comparison among modes 3 and 4 results in a
variance ratio of 14.6 that is significant at the 0.1% level. Comparison among modes 5
and 4 leads to a variance ratio of 10.4 that is significant at the 0.5% level. As a result,
mode 4, individual A and b for yield and plant N removal at each year and common c,
accounts for all the significant variation, since F(3,8,95) = 4.07, F(3,l 1,99.9) = 11.56, and
F(2,11,99.5) = 8.91. The analysis of variance for the Pensacola bahiagrass over years

68
1979-1980 is shown in Table 4-28 (dry matter and N removal x 2 yrs x 5 N). Comparison
among modes 1 and 2 results in variance ratio of 328 that is significant at the 0.1% level.
Comparison among modes 3 and 2 leads to a variance ratio of 9.54 that is significant at
the 99.5 level. Comparison among modes 3 and 4 results in a variance ratio of 12.3 that is
significant at the 0.1% level. Mode 5, individual A for yield and plant N removal for each
year, b for dry matter and N removal, and common c, accounts for all the significant
variation, since F(5,8,95) = 3.69, F(l,13,99.9) = 17.81, and F(2,11,95) = 3.98. Next the
data from the two grasses are combined and the analysis of variance data are presented in
Table 4-29 (dry matter and N removal x 2 grasses x 2 yrs x 5 N). A comparison of modes
1 and 2 results in a variance ratio of 330 that is significant at the 0.1% confidence level.
Comparison of modes 3 and 2 leads to a variance ratio of 7.62, that is significant at the
0.1% confidence level. Comparison of modes 3 and 4 results in a variance ratio of 9.66
that is significant at the 0.1% confidence level. Comparison of modes 3 and 5 results in a
variance ratio of 16.2 that is significant at the 0.1% confidence level. Comparison of
modes 5 and 4 leads to a variance ratio of 5.85 that is significant at the 0.1% confidence
level. Comparison of modes 3 and 6 leads to a variance ratio of 7.54 that is significant at
the 0.1% confidence level. Comparison of modes 6 and 4 results in a variance ratio of
6.58 that is significant at the 0.5% confidence level. Based upon these comparisons, we
can conclude that mode 4, with individual A and b for dry matter yield and plant N
removal for each year and grass and common c describes the data best. The overall
correlation coefficient of 0.9927 and adjusted correlation coefficient of 0.9876 were
calculated by mode 4. The statistical analysis might be affected by the close numbers for
yield and N removal at low values of applied N for both grasses. In the concern that fewer
parameters might be used to significantly account for the variation, the data will be
averaged over years to determine if an individual b can be used for yield and N removal
for both grasses. The averaged data are in Table 4-30 and the analysis of variance for the

69
averaged data is in Table 4-31. Comparison among modes 1 and 2 results in a variance
ratio of 792, that is significant at the 0.1% confidence interval. Comparison among modes
3 and 2 leads to a variance ratio of 7.79, that is significant at the 0.5% confidence level.
Comparison among modes 3 and 4 results in a variance ratio of 18.4, that is significant at
the 0.1% confidence level. Mode 5, individual A for dry matter yield and plant N removal
of each grass, individual b for dry matter yield and plant N removal, and common c,
accounts for all of the significant variation, since F(5,8,95) = 3.69, F(l,13,99.9) = 17.81,
and F(2,l 1,97.5) = 5.26. The two grasses have the same b and b', hence suggesting mode
5 of Table 4-29. The overall correlation coefficient of 0.9987 and adjusted correlation
coefficient of 0.9981 were calculated using mode 5 of the averaged data. Using mode 5
and the original data, the overall correlation coefficient of 0.9798 and adjusted correlation
coefficient of 0.9727 were calculated. The error analysis of the averaged and original data
is in Tables 4-32 and 4-33. Note that the estimates of b and c are the essentially the same
as for the unaveraged data, suggesting this is a valid way to describe the data. Results are
shown in Figure 4-78 and 4-79 for bermudagrass and bahiagrass, respectively, where are
drawn from
Coastal bermudagrass, 1979: Y = 15.99/[1 + exp(0.57 0.0072N)] [4.220]
Nu= 310.9/[ 1 + exp(1.07 0.0072N)] [4.221]
1980: Y = 9.89/[l + exp(0.57 0.0072N)] [4.222]
N= 227.4/p + exp(1.07 0.0072N)] [4.223]
Pensacola bahiagrass, 1979: Y = 12.42/[1 + exp(0.57 0.0072N)] [4.224]
Nu= 237.2/p + exp(1.07 0.0072N)] [4.225]
1980: Y 7.51/[1 + exp(0.57 0.0072N)] [4.226]
N= 191.3/p + exp(1.07 0.0072N)] [4.227]
From these results, it follows that plant N concentration, shown in Figures 4-78 and 4-79,
is estimated from

70
Bermudagrass, 1979: Nc= 19.4 [1 + exp(1.07 0.0072N)]/[1 + exp(0.57 0.0072N)][4.228]
1980: Nc=23.0 [1 + exp(1.07 0.0072N)]/[1 + exp(0.57 0.0072N)][4.229]
Bahiagrass, 1979: Nc= 19.1 [1 + exp(1.07 0.0072N)]/[1 + exp(0.57 0.0072N)][4.230]
1980: Nc=25.5 [1 + exp(1.07 0.0072N)]/[1 + exp(0.57 0.0072N)][4.231]
As shown, the data are described well by these equations.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal over two years, shown in Figure 4-80
and 4-81, are described by
Bermudagrass, 1979:
Y = 40.6 Ny/(479 + Nu)
[4.232]
Nc= 11.8 + 0.0246NU
[4.233]
1980:
Y = 25.1 Nu/(351 + Nu)
[4.234]
Nc= 13.9 + 0.0398NU
[4.235]
Bahiagrass, 1979:
Y = 31.6Nu/(366 + Nu)
[4.236]
Nc= 11.6 +0.0317N
[4.237]
1980:
Y = 19.1 Nu/(295 +NU)
[4.238]
Nc= 15.4 + 0.0524NU
[4.239]
The dimensionless plot for bermudagrass and bahiagrass is shown in Figure 4-82,
and validates the form of the model. Note that all of the data have been collapsed onto
one graph, and the data fall on the curve. The curves were drawn from
Y/A = 2.54(NU/A')/[1.54 + (NU/A')] [4.240]
Nc/Ncm = 0.61 +0.39 (Nu/A) [4.241]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
83 through 4-86.

Clayton and Kinston. NC: Corn
71
Data from Kamprath (1986) for corn were used. The analysis of variance for the
Dothan sandy loam is shown in Table 4-34 (dry matter and N removal x 2 components x 5
N). Comparisons among modes 1 and 2, modes 3 and 2, modes 3 and 4, and modes 3 and
5 result in variance ratios of 1686, 22.1, 49.8, and 39.0, which are all significant at the
0.1% confidence level. Comparison among modes 5 and 4 leads to a non-significant
variance ratio of 0.31. As a result, mode 5, individual A for grain and total plant,
individual b for yield and plant N removal, and common c, accounts for all the significant
variation, since F(5,8,97.5) = 4.82, F(l, 13,99.9) = 17.81, and F(2,11,95) = 3.98. The
analysis of variance for the Goldsboro sandy loam is shown in Table 4-35 (dry matter and
N removal x 2 components x 5 N). As with the Dothan, comparisons among modes 1 and
2, modes 3 and 2, modes 3 and 4, and modes 3 and 5 result in variance ratios of 1920,
26.9, 35.7, and 83.2, which are all significant at the 0.1% confidence level. Comparison
among modes 5 and 4 leads to a non-significant variance ratio of 2.48. As a result, mode
5, individual A for grain and total plant, individual b for yield and plant N removal, and
common c, accounts for all the significant variation, since F(5,8,95) = 3.69, F(l,13,99.9) =
17.81, and F(2,11,95) = 3.98. Next the data from the two soils are combined and the
analysis of variance data are presented in Table 4-36 (dry matter and N removal x 2
components x 2 soils x 5 N). A comparison among modes 1 and 2 results in a variance
ratio of 1562 that is significant at the 0.1% confidence level. Comparison among modes
3 and 2 leads to a variance ratio of 24.8, that is significant at the 0.1% confidence level.
Comparison among modes 3 and 4 results in a variance ratio of 8.29 that is significant at
the 0.1% confidence level. Comparison among modes 3 and 5 results in a variance ratio
of 19.7 that is significant at the 0.1% confidence level. Comparison among modes 3 and 6
leads to a variance ratio of 67.9 that is significant at the 0.1% confidence level.
Comparison among modes 3 and 7 results in a variance ratio of 21.9, that is significant at

72
the 0.1% confidence level. Comparison among modes 6 and 7 leads to a variance ratio of
0.36, which is not significant. Mode 6, with individual A for dry matter yield and plant N
removal for both grain and total plant, b for dry matter yield and plant N removal, and
common c describes the data best, since F(2,27,95) = 3.35, and the MSS for mode 6 was
smaller than that of mode 5. The overall correlation coefficient of 0.9941 and adjusted
correlation coefficient of 0.9921 were calculated by mode 6. The error analysis for the
parameters is shown in Table 4-37. Results are shown in Figure 4-87 for both soils, grain
and total plant, where the curves are drawn from
Dothan, Grain: Y = 11.12/[1 + exp(0.27 0.0187N)] [4.242]
Nu = 151.7/p+exp(0.97-0.0187N)] [4.243]
Total: Y = 20.68/[l + exp(0.27 0.0187N)] [4.244]
Nu = 187.2/p + exp(0.97 0.0187N)] [4.245]
Goldsboro, Grain: Y = 7.83/[l + exp(0.27 0.0187N)] [4.246]
Nu = 113.3/[1 + exp(0.97 0.0187N)] [4.247]
Total: Y = 14.70/[1 + exp(0.27 0.0187N)] [4.248]
Nu = 136.5/p + exp(0.97 0.0187N)] [4.249]
From these results, it follows that plant N concentration, shown in Figure 4-87, is
estimated from
Dothan, Grain: Nc= 13.6 [1 + exp(0.97 0.0187N)]/[1 + exp(0.27 0.0187N)][4.250]
Total: Nc= 9.1 [1 + exp(0.97 0.0187N)]/[1 + exp(0.27 0.0187N)][4.251]
Goldsboro, Grain: Nc= 14.5 [1 + exp(0.97 0.0187N)]/[1 + exp(0.27 0.0187N)][4.252]
Total: Nc= 9.3 [1 + exp(0.97 0.0187N)]/[1 + exp(0.27 0.0187N)][4.253]
As shown, the data are described well by these equations.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal for both soils, grain and total plant,
shown in Figure 4-88, are described by

73
Goldsboro, Grain: Y= 15.6 N/(l 12 + Nu)
Dothan, Grain: Y= 22.1 N/( 150 + Nu)
Total: Y = 41.1 Nu/(185+Nu)
Total: Y = 29.2N,/(135+NU)
Nc= 6.8 + 0.0453NU
Nc= 4.5 + 0.0243K,
Nc= 7.2 + 0.0643K,
Nc= 4.6 + 0.0342NU
[4.254]
[4.255]
[4.256]
[4.257]
[4.258]
[4.259]
[4.260]
[4.261]
The dimensionless plot for com is shown in Figure 4-89, and validates the form of
the model. Note that all of the data have been collapsed onto one graph, and the data fall
on the curve. The curves were drawn from
[4.262]
[4.263]
Y/A = 1.99 (Nu/A')/[0.99 + (N^A')]
Nc/Ncm = 0.50+0.50 (Nu/A')
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
90 through 4-93.
Plymouth. NC: Corn
Data from Kamprath (1986) for corn were used. The analysis of variance for the
Plymouth very fine sandy loam is shown in Table 4-38 (dry matter and N removal x 2
components x 5 N). Comparisons among modes 1 and 2, modes 3 and 2, modes 3 and 4,
and modes 3 and 5 result in variance ratios of 5819, 32.5, 27.8, and 57.4, which are all
significant at the 0.1% confidence level. Comparison among modes 5 and 4 leads to a
non-significant variance ratio of 3.21. Asa result, mode 5, individual A for grain and total
plant, individual b for yield and plant N removal, and common c, accounts for all the
significant variation, since F(5,8,97.5) = 4.82, F( 1,13,99.9) = 17.81, and F(2,11,95) =
3.98. The overall correlation coefficient of 0.9997 and adjusted correlation coefficient of

74
0.9995 were calculated by mode 5. Results are shown in Figure 4-94 for grain and total
plant, where the curves are drawn from
Grain: Y = 9.48/[l + exp(-0.065 0.0119N)] [4.264]
Nu = 126.2/[1 + exp(0.46 0.0119N)] [4.265]
Total: Y = 16.60/[1+exp(-0.065 0.0119N)] [4.266]
Nu = 147.3/[ 1 + exp(0.46 0.0119N)] [4.267]
From these results, it follows that plant N concentration, shown in Figure 4-94, is
estimated from
Grain: Nc = 13.3 [1 + exp(0.46 0.0119N)]/[1 + exp(-0.065 0.0119N)] [4.268]
Total: Nc = 8.9 [1 + exp(0.46 0.0119N)]/[1 + exp(-0.065 0.0119N)] [4.269]
As shown, the data are described well by these equations.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal for grain and total plant, shown in
Figure 4-95, are described by
Grain: Y = 23.2 NJ( 183 + Nu) [4.270]
Nc = 7.9 + 0.0431NU [4.271]
Total: Y = 40.6 NJ(2\3 + Nu) [4.272]
Nc = 5.2 + 0.0246NU [4.273]
The dimensionless plot for corn is shown in Figure 4-96, and validates the form of
the model. Note that all of the data have been collapsed onto one graph, and the data fall
on the curve. The curves were drawn from
Y/A = 2.45 (Nu/A')/[ 1.45 + (Nu/A')] [4.274]
Nc/Ncm = 0.59+0.41 (Nu/A') [4.275]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
97 through 4-100.

75
Florida: Bahiagrass
Data from Blue (1987) for bahiagrass were used. The analysis of variance for the
bahiagrass grown on the Entisol and Spodosol is shown in Table 4-40 (dry matter and N
removal x 2 soils x 5 N). Comparisons among modes 1 and 2 result in a variance ratio of
1792, that is significant at the 0.1% confidence level. Comparisons among modes 3 and 2
lead to a variance ratio of 12.24, that is significant at the 0.5% confidence level.
Comparisons among modes 3 and 4 lead to a variance ratio of 6.41, that is significant at
the 1% confidence level. Comparisons among modes 5 and 2 lead to a variance ratio of
10.72, that is significant at the 0.5% confidence level. Comparison among modes 5 and 4
leads to a variance ratio of 5.94, that is significant at the 2.5% confidence level. The
statistics suggest that mode 4, individual A for both soils, individual b for yield and plant
N removal, and common c, accounts for all the significant variation, since F(3,8,97.5) =
5.42, and F(3,l 1,99) = 6.22. This mode estimates the following b and c parameters:
Entisol, dry matter: b = 1.46 0.07
Spodosol, dry matter: b =1.31 0.07
Entisol, N removal: b' = 1.89 0.08
Spodosol, N removal: b' =1.83 0.08
c = 0.0118
while mode 5 estimates the following b and c parameters:
both soils, dry matter: b = 1.39 0.05
both soils, N removal: b' = 1.86 0.06
c = 0.0118
The estimates and their standard errors overlap, suggesting that mode 5 is correct.
Furthermore, since there are less parameters to estimate assuming the b values are not
affected by the soils, mode 5 will be used for estimation. The overall correlation
coefficient of 0.9985 and adjusted correlation coefficient of 0.9978 are calculated by mode

76
5. The error analysis for the parameters is shown in Table 4-41. Results are shown in
Figure 4-101 for both soils, where the curves are drawn from
Entisol: Y = 11.14/[1 + exp(1.39 0.0118N)] [4.276]
N = 311.0/[1 + exp(1.86 0.0118N)] [4.277]
Spodosol: Y = 19.39/[1 + exp(1.39 0.0118N)] [4.278]
Nu = 201.5/p +exp(1.86-0.0118N)] [4.279]
From these results, it follows that plant N concentration, shown in Figure 4-101, is
estimated from
Entisol: Nc =27.9 [1 + exp(l.86 0.0118N)]/[1 + exp(1.39 0.0118N)] [4.280]
Spodosol: Nc =10.4 [1 + exp(1.86 0.0118N)]/[1 + exp(1.39 0.0118N)] [4.281]
As shown, the data are described well by these equations.
From Postulate 3, the relationships between yield and plant N removal and
between plant N concentration and plant N removal for both soils, shown in Figure 4-102,
are described by
Entisol: Y= 29.7 N/(518 + Nu) [4.282]
Nc= 17.4 + 0.0337NU [4.283]
Spodosol: Y = 51.7 Nu/(336 + Nu) [4.284]
Nc= 6.5 + 0.0193NU [4.285]
The dimensionless plot for bahiagrass is shown in Figure 4-103, and validates the
form of the model. Note that all of the data have been collapsed onto one graph, and the
data fall on the curve. The curves were drawn from
Y/A = 2.67 (Nu/A')/[1.67 + (Nu/A')] [4.286]
Nc/Ncm = 0.63 +0.37 (Nu/A1) [4.287]
Scatter and residual plots of dry matter yield and plant N removal are shown in Figures 4-
104 through 4-107.

England: Ryegrass
77
The data for this analysis were taken from Morrison el al. (1980). The analysis of
variance is shown in Table 4-42 (dry matter and N removal x 20 sites x 6 N).. As shown
in the table, all of the comparisons are highly significant (0.1%). This is due in part to the
large degrees of freedom. If comparison is made among the mean sums of squares (MSS),
mode 4 has the smallest MSS with the exception of mode 2 (which fits each individually).
Mode 4 is the preferred mode since it requires less parameters to estimate (81 versus 120).
Mode 4 assumes an individual A and b for each site, dry matter yield and plant N removal
and common c for all. The overall correlation coefficient and adjusted correlation
coefficient calculated from mode 4 are 0.9935 and 0.9903, respectively. The error
analysis for the parameters is shown in Table 4-43. A comparison among estimated
maximum plant N concentrations and Ab's among sites is in Table 4-44. These variables
have been plotted in Figure 4-108, the mean and 2 standard errors of the estimates for all
sites are designated by a solid and dashed horizontal line, respectively. This suggests an
important result, the Ab is constant from site to site for a particular grass. If this is true,
then in conjunction with the postulates of the extended model, one less parameter would
need to be calculated. It also appears from Figure 4-108, that the ratio of estimated
maximum plant N removal to estimated maximum dry matter yield, or estimated maximum
plant N concentration is constant as well for a particular grass from site to site. Due to
the large number of sites, the results of the regression will not be plotted directly. The
dimensionless plot of ryegrass over all twenty sites is shown in Figure 4-109, with the
curves drawn assuming a constant Ab, namely the mean of the set, 0.83. The curves are
drawn by
Y/A = 1.77 (Nu/A')/[0.77 + (Nu/A')]
Nc/Ncm = 0.44+0.56 (Nu/A')
[4.288]
[4.289]

78
A hyperbolic regression was conducted on the dry matter yield/estimated maximum versus
N removal/estimated maximum plot. The results are shown in Figure 4-110, with the
curves drawn by
Y/A
+
oo
oo
o
""a
z
OO
OO
II
(Nu/A')]
[4.290]
Nc/Ncm
= 0.47+0.53 (Nu/A')
[4.291]
Ym
1 K' 1
This is an important
til
ai and A.
, Ab is calculated to be 0.76.
A \-e'Ab A' e^-X
result. Overman (1995a) has compared the probability density functions for the gaussian
and logistic functions normalized to unit area and inflection points. In the gaussian
distribution V2 arises as an integration constant to guarantee that the function will sum to
one. Similarly, 1.317 is the integration constant in the logistic distribution. The inverse of
this number is calculated to be 0.759. This was also the value of Ab found in the
hyperbolic regression above. Could there be something to this coincidence? Overman
(1995a) also showed in the same paper that the logistic equation could also be redefined
as
1 1
^=1 + e-i/U17=)+<,-0.759{
where <¡) = y/A and £ = cN b. At this point, it is not clear if this is just a surprising
coincidence or a result of some fundamental process of the system. Regardless, it is very
intriguing.
Evaluation of the Extended Triple Logistic (NPK) Model
Tifton, GA: Rye
Data from Walker and Morey (1962) were used. The parameters, their standard
errors, and relative errors resulting from the nonlinear regression are presented in Table 4-

79
45. The relative errors on the parameters are high due to the flat response to P and K,
making estimation of b and c parameters for these nutrients difficult since the data are high
on the curve. Also, some of the estimates of b are very close to zero. As the standard
errors are divided by these small numbers, they are artificially inflated. The RSS was
calculated to be 5895.76. The overall correlation coefficient and adjusted correlation
coefficient calculated from the regression are 0.9888 and 0.9860, respectively. Results are
shown in Figures 4-111 through 4-113 for the three nutrients, where the curves are drawn
from
5.43
y ~ + ^1.36-0.0225AT + g -0.14-0.0464 A ^ + g -0.91-0.0201 A ^
[4.292]
N
U
260
(1 + ^ 1.93-0.0225N + g-0.14-0.0464A + ^-0.91-0.0201* ^
[4.293]
Pu =
34
+ e 1.36-0.0225N ^ + ^-0.16-0.0464* + g-0.91-0.0201 A' ^
[4.294]
230
~ ^ j + e 1.36-0.0225W ^ j + ^-0.14-0.0464* ^ j + ^0.46-0.0201* ^ [4.295]
From these results, it follows that plant nutrient concentrations, shown in Figures 4-111
through 4-113, are estimated from
Nc = 47.9 [1 + exp(1.93 0.0225N)]/[1 + exp(1.36 0.0225N)] [4.296]
Pc = 6.26 [1 + exp(-0.16 0.0464P)]/[1 + exp(-0.14 0.0464P)] [4.297]
Kc = 42.4 [1 + exp(0.46 0.0201K)]/[1 + exp(-0.91 0.020IK)] [4.298]
As shown, the data are described well by these equations.
From Postulate 3, the relationships between yield and plant nutrient removals and
between plant nutrient concentrations and plant nutrient removals for all three nutrients,
shown in Figures 4-114 through 4-116, are described by

80
Y = 12.5 Nu/(338 + Nu) [4.299]
Nc= 27.1 +0.0800NU [4.300]
y = ^s.sPu/-ivn + Pu) [4.301]
Pe = 6.4 0.0037PU [4.302]
Y = 7.28 K/(78.4 + Ku) [4.303]
Kc= 10.8 +0.1374KU [4.304]
The dimensionless plot for rye is shown in Figure 4-117, and validates the form of
the model. This plots supports the result that the Ab¡ is different for each nutrient. The
curves were drawn from
Y/A = 2.30(NU/A')/[1.30 + (NU/A')] [4.305]
Y/A = -49.5 (Pu/Ayt-50.5 + (Pu/A')] [4.306]
Y/A = 1.34(Ku/A')/[0.34 + (Ku/A')] [4.307]
Nc/Ncm = 0.57 + 0.43 (N./A) [4.308]
Pc/Pcm = 1.02-0.02 (Pu/A) [4.309]
Kc/Kem= 0.25 + 0.75 (Ku/A1) [4.310]
Scatter and residual plots of dry matter yield and plant nutrient removals are shown in
Figures 4-118 through 4-125.

81
Table 4-1. Analysis of Variance of Model Parameters Used to Describe Coastal
Bermudagrass Yield Response
to Nil
rogen at Thorsby, Alabama, 1957-1959.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
21
42.347
2.017
-
(2) Ind A,b,c
18
6
3.507
0.584
-
(D-(2)
15
38.840
2.589
4.43+
(3) Ind A, Com b,c
8
16
12.267
0.766
-
(3)-(2)
10
8.750
0.875
1.50
(4) Ind A,b Com c
13
11
8.019
0.729
-
(4)-(2)
5
4.512
0.902
1.54
(3)-(4)
5
4.248
0.850
1.17
(5) Ind A, Com c, Ind b
(irrigation)
9
15
12.342
0.823
-
(5)-(2)
9
8.835
0.982
1.68
(5H4)
4
4.323
1.081
1.48
Source: Original yield data from Evans etal. (1961).
Significant at the 0.05 level
F(15, 6,95) =3.94
F(10, 6,95) =4.06
F( 5, 6,95) = 4.39
F( 5,11,95) =3.20
F( 9, 6,95) =4.10
F( 4,11,95) =3.36

82
Table 4-2. Analysis of Variance of Model Parameters Used to Describe Pensacola
Bahiagrass Yield Response
o Nitrogen at Thorsby,
Alabama, 1957-1959.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
21
25.712
1.224
-
(2) Ind A,b,c
18
6
1.306
0.218
-
(l)-(2)
15
24.406
1.627
7.48f
(3) Ind A, Com b,c
8
16
9.942
0.621
-
(3)-(2)
10
8.636
0.864
3.97
(4) Ind A,b Com c
13
11
1.872
0.170
-
(4)-(2)
5
0.567
0.113
0.52
(3)-(4)
5
8.070
1.614
9.49++
(5) Ind A, Com c, Ind b
(irrigation)
9
15
7.844
0.523
-
(5)-(2)
9
6.539
0.727
3.34
(3)-(5)
1
2.098
2.098
4.01
(5)-(4)
4
5.972
1.493
8.78++
Source: Original yield data from Evans el al. (1961).
Significant at the 0.005 level
f Significant at the 0.025 level
F( 15, 6,97.5)= 5.27
F(10, 6,95) =4.06
F( 5, 6,95) =4.39
F( 5,11,99.5)= 6.42
F( 9, 6,95) =4.10
F( 1,15,95) =4.54
F( 4,11,99.5)= 6.88

83
Table 4-3. Analysis of Variance of Model Parameters Used to Describe Coastal
Bermudagrass and Pensacola Bahiagrass Yield Response to Nitrogen at
Thorsby, Alabama, 1957-1959,
J 5 *
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
45
79.043
1.757
-
(2) Ind A,b,c
36
12
4.813
0.401
-
(l)-(2)
33
74.230
2.249
5.61++
(3) Ind A, Com b,c
14
34
26.727
0.786
-
(3)-(2)
22
21.914
0.996
2.48
(4) Ind A,b Com c
25
23
10.414
0.453
-
(4)-(2)
11
5.601
0.509
1.27
(3)-(4)
(5) Ind A, Com c, Ind b
11
16.313
1.483
3.27*
(type of grass,
irrigation)
17
31
20.825
0.672
-
(5)-(2)
19
16.012
0.843
2.10
(3)-(5)
3
5.902
1.967
2.93+
(5)-(4)
(6) Ind A, Com c, Ind b
8
10.411
1.301
2.87*
(irrigation)
15
33
25.854
0.783
-
(6)-(2)
21
21.041
1.002
2.50
(3)-(6)
1
0.873
0.873
1.11
(6)-(4)
10
15.440
1.544
3.41*
(6)-(5)
(7) Ind A, Com c, Ind b
2
5.029
2.415
3.74+
(type of grass)
15
33
22.868
0.693
-
(7)-(2)
21
18.055
0.860
2.14
(3)-(7)
1
3.859
3.589
5.57*
(7)-(4)
10
12.454
1.245
2.75*
(7)-(5)
2
2.043
1.021
1.52
Source: Original yield data from Evans et al. (1961).
Significant at the 0.005 level
Significant at the 0.01 level
* Signifcant at the 0.025 level
Significant at the 0.05 level
F(33,12,99.5)= 4.23 F(22,12,95)
F(11,12,95) =2.72 F(11,23,99)
F( 19,12,95) =2.55 F( 3,31,95)
F( 8,23,97.5)= 2.81 F(21,12,95)
F( 1,33,95) =4.14 F(10,23,99)
F( 2,31,95) =3.30
= 2.52
= 3.14
= 2.91
= 2.53
= 3.21
F(10,23,97.5)= 2.67

84
Table 4-4. Error Analysis for Model Parameters of Coastal Bermudagrass and
Pensacola Bahiagrass Grown at Thorsby, Alabama.
Grass
Irrigation
Year
Parameter
Estimate
Standard
Error
Relative
Error
Coastal Bermudagrass
No
1957
A, Mg/ha
22.96
0.510
0.022
1958
A
22.37
0.497
0.022
1959
A
19.29
0.434
0.022
Yes
1957
A
22.44
0.496
0.022
1958
A
22.92
0.511
0.022
1959
A
24.89
0.554
0.022
Pensacola Bahiagrass
No
1957
A
22.00
0.502
0.023
1958
A
22.85
0.526
0.023
1959
A
19.55
0.452
0.023
Yes
1957
A
23.48
0.542
0.023
1958
A
22.08
0.512
0.023
1959
A
22.54
0.521
0.023
Coastal Bermudagrass
Both
all
b
1.39
0.0503
0.036
Pensacola Bahiagrass
Both
all
b
1.58
0.0533
0.034
Both
Both
all
c, ha/kg
0.0078
0.0002
0.026
Source: Original yield data from Evans et al. (1961).

85
Table 4-5. Seasonal Dry Matter Yield for Coastal Bermudagrass and Pensacola
Bahiagrass Grown at Thorsby, Alabama Averaged over Years 1957-1959.
Type
Irrigation
0
Applied Nitrogen, kg/ha
168 336
672
Coastal Bermudagrass
No
3.95
10.98
16.07
21.44
Yes
4.19
11.46
18.54
22.71
Pensacola Bahiagrass
No
3.62
10.09
15.79
20.72
Yes
3.79
9.71
16.28
22.67
Source: Data from Evans el al. (1961).

86
Table 4-6. Error Analysis for Model Parameters on Averaged Dry Matter Yield of
Coastal Bermudagrass and Pensacola Bahiagrass Grown at Thorsby,
Alabama.
Type
Irrigation
Parameter
Estimate
Standard
Error
Relative
Error
Coastal Bermudagrass
No
A, Mg/ha
21.57
0,359
0.017
Yes
A
23.44
0.388
0.017
Pensacola Bahiagrass
No
A
21.49
0.369
0.017
Yes
A
22.73
0.395
0.017
Coastal Bermudagrass
Both
b
1.39
0.056
0.040
Pensacola Bahiagrass
Both
b
1.57
0.059
0.038
Both
Both
c, ha/kg
0.0078
0.0003
0.038
Source: Original data from Evans et al. (1961).

87
Table 4-7. Analysis of Variance of Model Parameters Used to Describe Dallisgrass
Grown at Baton Rouge, LA.]
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
9
154655.03
17183.89
-
(2) Ind A,b,c
6
6
82.72
13.79
-
(1 M2)
3
154572.31
51524.10
3736**
(3) Ind A, Com b,c
4
8
3606.97
450.87
-
(3)-(2)
2
3524.25
1762.12
127.8**
(4) Ind A,b Com c
5
7
122.54
17.51
-
(4)-(2)
1
39.82
39.82
2.89
(3)-(4)
1
3484.43
3484.43
199.0**
Source: Original data from Robinson et al. (1988).
Significant at the 0.001 level
F(3,6,99.9) 23.70
F(2,6,99.9) = 27.00
F( 1,6,95) = 5.99
F( 1,7,99.9) = 29.25

Table 4-8. Error Analysis for Model Parameters of Dallisgrass Grown at Baton
Rouge, LA.
Component
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter
A, Mg/ha
15.60
0.16
0.010
N Removal
A, kg/ha
430.7
5.6
0.013
Dry Matter
b
0.58
0.010
0.017
N Removal
b
1.47
0.007
0.005
Both
c, ha/kg
0.0055
0.0002
0.036
Source: Original data from Robinson et al. (1988).

89
Table 4-9. Analysis of Variance of Model Parameters Used to Describe Bermudagrass
Grown at T
orsby, AL with Two Clipping Interva
s.
Mode
Parameters
Estimated
df
Residual Sum
of Squares
Mean Sum
of Squares
F
(1) Com A,b,c
3
21
1018614.201
48505.438
-
(2) Ind A,b,c
12
12
3368.695
280.725
-
(D-(2)
9
1015245.506
112805.056
402
(3) Ind A, Com b,c
6
18
11170.579
620.588
-
(3)-(2)
6
7801.885
1300.314
4.63+
(4) Ind A, b, Com c
9
15
4274.029
284.935
-
(4)-(2)
3
905.335
301.778
1.08
(3)-(4)
3
6896.550
2298.850
8.07++
(5) Ind A, Com c, Ind b
(dm and Nu)
7
17
4261.737
250.690
(5)-(2)
5
893.042
178.608
0.64
(3)-(5)
1
6908.842
6908.842
27.56
Source: Original data from Doss et al. (1966).
Significance level of 0.001
Significance level of 0.005
Significance level of 0.05
F(9,12,99.9) =7.48
F(6,12,95) =3.00
F(3,12,95) =3.89
F(3,15,99.5) =6.48
F(5,12,95) =3.11
F( 1,17,99.9) = 15.72

Table 4-10. Error Analysis of Model Parameters of Bermudagrass Grown at
Thorsby, AL.
Type
Clipping
Interval
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter Yield
3.0 weeks
A, Mg/ha
17.42
0.248
0.014
4.0 weeks
A
19.75
0.279
0.014
N Removal
3.0 weeks
A, kg/ha
568.9
8.83
0.016
4.0 weeks
A
543.5
8.45
0.016
Both
DM
b
1.27
0.076
0.060
Both
Nu
b
2.02
0.095
0.047
Both
Both
c, ha/kg
0.0067
0.0003
0.045
Source: Original data from Doss et al. (1966).

91
Table 4-11. Analysis of Variance of Model Parameters for Bermudagrass Grown at
Maryland and Cut at Five
Harvest Intervals.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
57
1419082.51
24896.18
-
(2) Ind A,b,c
30
30
8213.60
273.79
-
(l)-(2)
27
1410868.91
52254.40
190.9**
(3) Ind A, Com b,c
12
48
28200.40
587.51
-
(3)-(2)
18
19986.80
1110.38
4.06**
(4) Ind A,b Com c
21
39
13605.56
348.86
-
(4)-(2)
9
5391.96
599.11
2.19
(3)-(4)
9
14594.84
1621.65
4.65**
(5) Ind A (yr, At), Ind b
(y,Nu), Com c
13
47
13473.57
286.67
_
(5)-(2)
17
5259.97
309.41
1.13
(3)-(5)
1
14726.83
14726.83
51.4**
Source: Original data from Decker et al. (1971).
Significant at the 0.001 level
F(27,30,99.9)= 3.28
F(18,30,99.9)=3.58
F( 9,30,95) =2.21
F( 9,39,99.9)= 4.05
F( 17,30,95) = 1.98
F( 1,47,99.9)= 12.32

92
Table 4-12.
Error Analysis for Model Parameters of Bermudagrass
Maryland and Cut at Five Harvest Intervals.
Grown at
Component
Harvest
Interval
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter
3.2, weeks
A, Mg/ha
13.90
0.28
0.020
3.6
A
13.65
0.28
0.021
4.3
A
14.93
0.30
0.020
5.5
A
17.56
0.36
0.021
7.7
A
19.34
0.39
0.020
N Removal
3.2
A, kg/ha
469.5
10.4
0.022
3.6
A
444.7
9.9
0.022
4.3
A
440.2
9.8
0.022
5.5
A
444.0
9.9
0.022
7.7
A
409.7
9.0
0.022
Dry Matter
All
b
1.78
0.08
0.045
N Removal
All
b
2.59
0.10
0.039
Both
All
c, ha/kg
0.0112
0.0005
0.045
Source: Original data from Decker et al. (1971).

93
Table 4-13. Analysis of Variance on Model Parameters for Bermudagrass Grown at
Tifton, GA, over Two Years and Cut at Five Different Harvest Intervals.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
97
3326183.05
34290.55
-
(2) Ind A,b,c
60
40
28516.69
712.92
-
(l)-(2)
57
3297666.36
57853.80
81.2*
(3) Ind A, Com b,c
22
78
79680.46
1021.54
-
(3)-(2)
38
51163.77
1346.42
1.89t
(4) Ind A,b Com c
41
59
44528.61
754.72
-
(4)-(2)
19
16011.92
842.73
1.18
(3)-(4)
19
35151.85
1850.10
2.45++
(5) Ind A (yr, At), Ind b
(yr,y,Nu), Com c
25
75
48676.82
649.02
.
(5)-(2)
35
20160.13
576.00
0.81
(3)-(5)
3
31003.64
10334.55
15.9
(5)-(4)
16
4148.21
259.26
0.34
(6) Ind A (yr, At), Ind
b (y,Nu), Com c
23
77
50414.37
654.73
_
(6)-(2)
37
21897.68
591.83
0.83
(3)-(6)
1
29266.09
29266.09
44.7
(6)-(4)
18
5885.76
326.99
0.43
(6H5)
2
1737.55
868.78
1.34
Source: Original data from Prine and Burton (1956).
Significant at the 0.001 level
Significant at the 0.005 level
f Significant at the 0.025 level
F(57,40,99.9)=2.59
F(3 8,40,97.5)= 1.89
F( 19,40,95) = 1.85
F( 19,59,99.5) = 2.42
F(35,40,95) = 1.72
F( 3,75,99.9)= 6.01
F( 16,59,95) = 1.82
F(37,40,95) = 1.71
F( 1,77,99.9)= 11.71
F( 18,59,95) = 1.78
F( 2,75,95) =3.12

94
Table 4-14. Error Analysis for Model Parameters of Bermudagrass Grown at Tifton,
GA over Two Years and Cut at Five Different Harvest Intervals.
Year
Component
Harvest
Interval
Parameter
Estimate
Standard
Error
Relative
Error
1953
Dry Matter
2, weeks
A, Mg/ha
17.95
0.52
0.029
3
A
19.88
0.57
0.029
4
A
23.15
0.66
0.029
6
A
29.57
0.85
0.029
8
A
29.58
0.85
0.029
1954
Dry Matter
2
A
8.40
0.24
0.029
3
A
9.95
0.28
0.028
4
A
11.67
0.33
0.028
6
A
14.37
0.41
0.029
8
A
16.24
0.46
0.028
1953
N Removal
2
A, kg/ha
644.7
20.4
0.032
3
A
641.8
20.2
0.031
4
A
687.2
21.5
0.031
6
A
688.2
21.6
0.031
8
A
606.0
19.0
0.031
1954
N Removal
2
A
324.5
10.1
0.031
3
A
337.5
10.6
0.031
4
A
348.0
10.9
0.031
6
A
363.8
11.3
0.031
8
A
379.5
11.9
0.031
Both
Dry Matter
All
b
1.47
0.060
0.041
Both
N Removal
All
b
2.15
0.077
0.036
Both
Both
All
c, ha/kg
0.0077
0.0003
0.039
Source: Original data from Prine and Burton (1956).

95
Table 4-15. Analysis of Variance of Model Parameters on Ryegrass Grown at England,
with Three Different Numbers of Cuttings over the Season for 1969.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
123
1698556.455
13809.402
-
(2) Ind A,b,c
18
108
3703.415
34.291
-
(l)-(2)
15
1694853.040
112990.203
3295**
(3) Ind A, Com b,c
8
118
35982.858
304.939
-
(3)-(2)
10
32279.443
3227.944
94 r*
(4) Ind A,b Com c
13
113
6635.747
58.723
-
(4)-(2)
5
2932.332
586.466
17.1
(3)-(4)
5
29347.111
5869.422
o
o


(5) Ind A, Com c, Ind b
(y and Nu)
9
117
8196.505
70.056
(5)-(2)
9
4493.090
499.232
14.6**
(3)-(5)
1
27786.353
27786.353
397**
(5)-(4)
4
1560.758
390.189
6.64*
(6) Ind A, Com c, Ind b
(y) and Com b (Nu)
11
115
6050.656
52.614
(6)-(2)
7
2347.241
335.32
9.8"
(3)-(6)
3
29932.202
9977.401
190**
(5)-(6)
2
2145.849
1072.925
20.4
Source: Original data from Reid (1978).
Significant at the 0.001 level
F( 15,108,99.9) =2.81
F(10,108,99.9) =3.27
F( 5,108,99.9) =4.45
F( 5,113,99.9) =4.44
F( 9,108,99.9) =3.41
F( 1,117,99.9) =11.37
F( 4,113,99.9) =4.97
F( 7,108,99.9) =3.80
F( 3,115,99.9) =5.80
F( 2,115,99.9) =7.34

Table 4-16. Error Analysis for Model Parameters of Ryegrass Grown at England, with
Three Different Numbers of Cuttings over the Season for 1969.
Component
Number of
Cuttings
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter
10
A, Mg/ha
9.42
0.105
0.011
5
A
12.72
0.127
0.010
3
A
12.75
0.121
0.009
N Removal
10
A, kg/ha
377.0
4.72
0.013
5
A
403.5
5.02
0.012
3
A
363.0
4.46
0.012
Dry Matter
10
b
1.74
0.046
0.026
5
b
1.23
0.044
0.036
3
b
0.90
0.044
0.049
N Removal
All
b
2.15
0.050
0.023
Both
All
c, ha/kg
0.0080
0.0001
0.013
Source: Original data from Reid (1978).

97
Table 4-17. Analysis of Variance for Bermudagrass over Three Seasons, Irrigated and
Non-irrigated, Grown at
Fayett
eville, AR.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
81
2114817.76
26108.86
-
(2) Ind A,b,c
36
48
17913.02
373.19
-
(1)-(2)
33
2096904.75
63542.57
170.3**
(3) Ind A, Com b,c
14
70
34746.83
496.38
-
(3)-(2)
22
16833.81
765.17
2.05t
(4) Ind A,b Com c
25
59
21706.70
367.91
-
(4)-(2)
11
3793.68
344.88
0.92
(3)-(4)
11
13040.13
1185.47
3.22++
(5) Ind A, Com c, Ind
b (Irr, dm, Nu)
17
67
28178.64
420.58
(5)-(2)
19
10265.63
540.30
1.45
(3)-(5)
3
6568.19
2189.40
5.21++
(5)-(4)
8
6471.95
808.99
2.20+
(6) Ind A, Com c, Ind
b (dm, Nu)
15
69
26492.41
383.95
(6)-(2)
21
8579.40
408.54
1.09
(3)-(6)
1
8254.42
8254.42
21.5**
(6H4)
10
4785.72
478.57
1.30
Source: Original data from Huneycutt et al. (1988).
Significant at the 0.001 level
Significant at the 0.005 level
1 Significant at the 0.025 level
Significant at the 0.05 level
F(33,48,99.9)= 2.66
F(22,48,97.5)= 1.97
F( 11,48,95) = 1.99
F(11,59,99.5)= 2.82
F( 19,48,95) = 1.81
F( 3,67,99.5)= 4.68
F( 8,59,95) =2.10
F(21,48,95) = 1.78
F( 1,69,99.9)= 11.81
F( 10,59,95) =2.00

98
Table 4-18. Error Analysis for Model Parameters of Bermudagrass Grown at
Fayetteville, AR, over Three Years, with and without Irrigation.
Type
Irrigation
Year
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter
No
1983
A, Mg/ha
17.90
0.41
0.023
1984
A
17.37
0.39
0.022
1985
A
19.61
0.45
0.023
Yes
1983
A
24.70
0.56
0.023
1984
A
24.60
0.56
0.023
1985
A
22.58
0.51
0.023
N Removal
No
1983
A, kg/ha
408.7
10.4
0.025
1984
A
413.9
10.5
0.025
1985
A
459.3
11.8
0.026
Yes
1983
A
554.3
14.1
0.025
1984
A
523.7
13.4
0.026
1985
A
492.1
12.6
0.026
Dry Matter
Both
all
b
1.50
0.063
0.042
N Removal
Both
all
b
2.04
0.072
0.035
Both
Both
all
c, ha/kg
0.0084
0.0003
0.036
Source: Original data from Huneycutt etal. (1988).

Table 4-19. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration
for Bermudagrass Grown at Fayetteville, AR Averaged over Three Years.
Applied Nitrogen Dry Matter Yield N Removal N Concentration
kg/ha Mg/ha kg/ha g/kg
Non-Irrigated
0
2.00
26
12.80
112
7.50
117
15.52
224
10.83
199
18.45
336
14.00
274
19.63
448
17.53
380
21.71
560
17.29
386
22.35
672
17.71
428
Irrigated
24.11
0
3.48
51
14.51
112
9.65
157
16.27
224
13.74
253
18.40
336
19.22
359
18.72
448
21.48
415
19.31
560
22.89
482
21.07
672
23.80
532
22.35
Source: Original data from Huneycutt et al. (1988).

100
Table 4-20. Analysis of Variance for Model Parameters for Bermudagrass Grown at
Fayetteville, AR, Averaged over Three Years, with and without Irrigation.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
25
695955.49
27838.22
-
(2) Ind A,b,c
12
16
3297.76
206.11
-
(l)-(2)
9
692657.73
76961.97
373.4
(3) Ind A, Com b,c
6
22
6659.06
302.68
-
(3)-(2)
6
3361.30
560.22
2.72
(4) Ind A,b Com c
9
19
3898.95
205.21
-
(4)-(2)
3
601.19
200.40
0.97
(3)-(4)
3
3760.11
1253.37
e.ir
(5) Ind A, Com c,
Ind b (dm, Nu)
7
21
4745.07
225.96
(5)-(2)
5
1447.31
289.46
1.40
(3)-(5)
1
1913.99
1913.99
8.47*
(5)~(4)
2
846.12
423.06
2.06
Source: Original data from Huneycutt et al. (1988).
Significant at the 0.001 level
Significant at the 0.005 level
Significant at the 0.01 level
F(9,16,99.9)
= 5.98
F(6,16,95)
= 2.74
F(3,16,95)
= 3.24
F(3,19,99.5)
= 5.92
F(5,16,95)
= 2.85
F( 1,21,99)
= 8.02
F(2,19,95)
= 3.52

101
Table 4-21. Error Analysis for Model Parameters of Bermudagrass Grown at
Fayetteville, AR, Averaged over Three Years, with and without Irrigation.
Component
Irrigation
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter
No
A, Mg/ha
18.63
0.35
0.019
Yes
A
24.40
0.45
0.018
N Removal
No
A, kg/ha
435.7
9.36
0.021
Yes
A
534.8
11.58
0.022
Dry Matter
Both
b
1.51
0.076
0.050
N Removal
Both
b
2.04
0.087
0.043
Both
Both
c, ha/kg
0.0084
0.0004
0.048
Source: Original data from Huneycutt et al. (1988).

102
Table 4-22. Analysis of Variance of Model Parameters for Tall Fescue Grown at
Fayetteville,
AR, over Three Seasons, with and wi
thout Irrigation.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
81
1401206.34
17298.84
-
(2) Ind A,b,c
36
48
5767.59
120.16
-
(D-(2)
33
1395438.76
42286.023
352
(3) Ind A, Com b,c
14
70
25848.47
369.26
-
(3)-(2)
22
20080.88
912.77
7.60
(4) Ind A,b Com c
25
59
9085.81
154.00
-
(4)-(2)
11
3318.22
301.66
2.5 lf
(3)-(4)
11
16762.66
1523.88
9.90
(5) Ind A, Com c, Ind b
(Irr, dm, Nu)
17
67
14215.61
212.17
.
(5)-(2)
19
8448.03
444.63
3.70
(3)-(5)
3
11632.86
3877.62
18.3
(5)-(4)
8
5129.80
641.23
4.16
(6) Ind A, Com c, Ind b
(dm, Nu)
15
69
14425.24
209.06
(6)-(2)
21
8657.66
412.27
3.43
(3)-(6)
1
11423.23
11423.23
565
(6)-(4)
10
5339.44
533.94
3.47f
(6)-(5)
2
209.63
104.82
0.49
Source: Original data from Huneycutt et al. (1988).
Significant at the 0.001 level
f Significant at the 0.025 level
F(33,48,99.9)= 2.66
F(22,48,99.9)= 2.91
F(11,48,97.5)= 2.27
F(11,59,99.9)= 3.43
F( 19,48,99.9)= 3.02
F( 3,67,99.9)= 6.09
F( 8,59,99.9)= 3.88
F(21,48,99.9)= 2.94
F( 1,69,99.9)= 11.81
F( 10,59,97.5)=2.27
F( 2,67,95) =3.13

103
Table 4-23. Error Analysis for Model Parameters of Tall Fescue Grown at
Fayetteville, AR, over Three Seasons, with and without Irrigation.
Component
Irrigation
Year
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter
No
1981-2
A, Mg/ha
12.08
0.38
0.031
1982-3
A
8.01
0.25
0.031
1983-4
A
5.71
0.17
0.030
Yes
1981-2
A
15.63
0.49
0.031
1982-3
A
13.22
0.42
0.032
1983-4
A
16.81
0.53
0.032
N Removal
No
1981-2
A, kg/ha
357.5
12.5
0.035
1982-3
A
217.8
7.6
0.035
1983-4
A
169.5
5.7
0.034
Yes
1981-2
A
443.7
15.4
0.035
1982-3
A
357.3
12.6
0.035
1983-4
A
439.4
15.3
0.035
Dry Matter
Both
all
b
0.92
0.084
0.091
N Removal
Both
all
b
1.47
0.095
0.065
Both
Both
all
c, ha/kg
0.0081
0.0005
0.062
Source: Original data from Huneycutt etal. (1988).

104
Table 4-24. Seasonal Dry Matter Yield, Plant N Removal, and Plant N Concentration
for Tall Fescue Grown at Fayetteville, AR, Averaged over Three Seasons.
Applied Nitrogen
kg/ha
Dry Matter Yield
Mg/ha
N Removal
kg/ha
N Concentration
g/kg
0
2.09
-Non-Irrigated
43
20.57
112
4.82
103
21.37
224
6.35
148
23.31
336
7.43
189
25.44
448
8.30
225
27.11
560
7.90
230
29.11
672
8.38
248
29.59
0
3.53
Irrigated
74
20.96
112
7.38
149
20.19
224
9.56
208
21.76
336
13.22
324
24.51
448
14.51
373
25.71
560
15.31
400
26.13
672
15.47
420
27.15
Source: Original data from Huneycutt et al. (1988).

Table 4-25. Analysis of Variance of Model Parameters for Tall Fescue Grown at
Fayetteville, AR, Averaged overr
'hree Seasons, with and without
Irrigation.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
25
438490.94
17539.64
-
(2) Ind A,b,c
12
16
860.75
53.80
-
(l)-(2)
9
437630.19
48625.58
904**
(3) Ind A, Com b,c
6
22
3908.10
177.64
-
(3)-(2)
6
3047.35
507.89
9.44**
(4) Ind A,b Com c
9
19
999.28
52.59
-
(4)-(2)
3
138.53
46.18
0.86
(3)-(4)
3
2908.82
969.61
18.4**
(5) Ind A, Com c, Ind b
(dm, Nu)
7
21
1437.08
68.43
(5)-(2)
5
576.32
115.26
2.14
(3)-(5)
1
2471.02
2471.02
36.1**
(5)-(4)
2
437.80
218.90
4.16+
Source: Original data from Huneycutt el al. (1988).
Significant at the 0.001 level
Significant at the 0.05 level
F(9,16,99.9)
= 5.98
F(6,16,99.9)
= 6.81
F(3,16,95)
= 3.24
F(3,19,99.9)
= 8.28
F(5,16,95)
= 2.85
F( 1,21,99.9)
= 14.59
F(2,19,95)
= 3.52

106
Table 4-26. Error Analysis for Model Parameters of Tall Fescue Grown at
Fayetteville, AR, Averaged over Three Seasons, with and without Irrigation.
Component
Irrigation
Parameter
Estimate
Standard Error
Relative Error
Dry Matter
No
A, Mg/ha
8.67
0.15
0.017
Yes
A
15.36
0.27
0.018
N Removal
No
A, kg/ha
250.7
4.95
0.020
Yes
A
417.4
8.34
0.020
Dry Matter
Both
b
0.99
0.069
0.070
N Removal
Both
b
1.53
0.079
0.052
Both
Both
c, ha/kg
0.0081
0.0004
0.049
Source: Original data from Huneycutt et al. (1988).

107
Table 4-27. Analysis of Variance for Bermudagrass Grown at Eagle Lake, Texas,
over Two Years.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
17
110513.95
6500.821
-
(2) Ind A,b,c
12
8
233.901
29.238
-
U)-(2)
9
110280.052
12253.339
419"
(3) Ind A, Com b,c
6
14
1851.481
132.249
-
(3)-(2)
6
1617.580
269.597
9.22++
(4) Ind A,b Com c
9
11
370.923
33.72
-
(4)-(2)
3
137.021
45.674
1.56
(3)-(4)
(5) Ind A, Com c, Ind b
3
1480.558
493.519
14.6"
(dm and Nu)
7
13
1069.846
82.296
-
(5)-(2)
5
835.945
167.189
5.72*
(3)-(5)
1
781.635
781.635
9.50*
(3)-(4)
2
698.923
349.462
10.4++
Source: Original data from Evers (1984).
Significant at the 0.001 level
Significant at the 0.005 level
f Significant at the 0.025 level
Significant at the 0.01 level
F(9, 8,99.9) =11.77
F(6, 8,99.5) =7.95
F(3, 8,95) =4.07
F(3,11,99.9) =11.56
F(5, 8,97.5) =4.82
F( 1,13,99) =9.07
F(2,11,99.5) =8.91

108
Table 4-28. Analysis of Variance for Bahiagrass Grown at Eagle Lake, Texas, over
Two Years.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
17
68411.328
4024.196
-
(2) Ind A,b,c
12
8
184.755
23.094
-
(l)-(2)
9
68226.573
7580.73
328**
(3) Ind A, Com b,c
6
14
1507.231
107.659
-
(3)-(2)
6
1322.476
220.413
9.54++
(4) Ind A,b Com c
9
11
346.949
31.541
-
(4)-(2)
3
162.193
54.064
2.34
(3)-(4)
(5) Ind A, Com c, Ind b
3
1160.282
386.761
12.3**
(dm and Nu)
7
13
634.002
48.769
-
(5)-(2)
5
449.247
89.849
3.89+
(3)-(5)
1
873.229
873.229
17.9**
(5H4)
2
287.054
143.527
4.55+
Source: Original data from Evers (1984).
Significant at the 0.001 level
Significant at the 0.005 level
Significant at the 0.05 level
F(9, 8,99.9) =11.77
F(6, 8,99.5) =7.95
F(3, 8,95) =4.07
F(3,11,99.9) =11.56
F(5, 8,95) =3.69
F( 1,13,99.9) =17.81
F(2,11,95) =3.98

Table 4-29. Analysis of Variance for Bermudagrass and Bahiagrass Grown at Eagle
Lake, Texas, over Two Years.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
37
181775.297
4912.846
-
(2) Ind A,b,c
24
16
418.657
26.166
-
(l)-(2)
21
181356.641
8636.031
330*
(3) Ind A, Com b,c
10
30
3209.170
106.972
-
(3)-(2)
14
2790.514
199.322
7.62**
(4) Ind A,b Com c
17
23
814.463
35.411
-
(4)-(2)
7
395.806
56.544
2.16
(3)-(4)
7
2394.707
342.101
9.66**
(5) Ind A, Com c, Ind b
(dm and Nu)
11
29
2058.255
70.971
(5)-(2)
13
1639.598
126.123
4.82++
(3)-(5)
1
1150.915
1150.915
16.2**
(5)-(4)
6
1243.792
207.299
5.85**
(6) Ind A, Com c, Ind b
(grass, dm and Nu)
13
27
1746.733
64.694
(6)-(2)
11
1328.076
120.730
4.61++
(3)-(6)
3
1462.437
487.479
7.54**
(6)-(4)
4
932.270
233.067
6.58++
(5H6)
2
311.522
155.761
2.41
Source: Original data from Evers (1984).
Significant at the 0.001 level
Significant at the 0.005 level
F(21,16,99.9) =
4.95
F( 14,16,99.9) =
5.35
F( 7,16,95) =
2.66
F( 7,23,99.9)=
5.33
F( 13,16,99.5) =
4.03
F( 1,29,99.9)=
13.39
F( 6,23,99.9)=
5.65
F(11,16,99.5) =
4.18
F( 3,27,99.9)=
7.27
F( 4,23,99.5)=
4.95
F( 2,27,95) =
3.35

110
Table 4-30. Averaged Seasonal Dry Matter Yield, Plant N Removal, and Plant N
Concentration for Bermudagrass and Bahiagrass Grown at Eagle Lake, TX.
Applied Nitrogen
kg/ha
Dry Matter Yield1 N Removal2
Mg/ha kg/ha
N Concentration
g/kg
0
4.05
Coastal bermudagrass-
60
14.81
84
6.07
103
16.97
168
7.91
136
17.19
252
9.62
190
19.75
336
10.83
218
20.13
0
3.79
Pensacola bahiagrass-
60
15.83
84
5.17
84.5
16.34
168
6.37
115.5
18.13
252
7.27
146.5
20.15
336
8.17
168
20.56
Source: Original data from Evers (1984).
1 Averaged over years 1978-1980.
2 Averaged over years 1979-1980.

Ill
Table 4-31. Analysis of Variance on Model Parameters for Bermudagrass and
Bahiagrass Grown at Eagle Lake, TX, Averaged to Estimate b and c
Parameters.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
17
86802.271
5106.016
-
(2) Ind A,b,c
12
8
97.371
12.171
-
(l)-(2)
9
86704.900
9633.878
792**
(3) Ind A, Com b,c
6
14
666.562
47.612
-
(3)-(2)
6
569.191
94.865
7.79++
(4) Ind A,b Com c
9
11
110.820
10.075
-
(4)-(2)
3
13.449
4.483
0.37
(3)-(4)
(5) Ind A, Com c, Ind b
3
555.742
185.247
18.4
(dm and Nu)
7
13
251.797
19.369
-
(5)-(2)
5
154.426
30.885
2.54
(3)-(5)
1
414.765
414.765
21.4
(5)~(4)
2
140.977
70.488
7.00*
Source: Original data from Evers (1984).
Significant at the 0.001 level
Significant at the 0.005 level
f Significant at the 0.025 level
F(9, 8,99.9) =11.77
F(6, 8,99.5) =7.95
F(3, 8,95) =4.07
F(3,11,99.9) =11.56
F(5, 8,95) =3.69
F( 1,13,99.9) =17.81
F(2,l 1,97.5) =5.26

Table 4-32. Error Analysis for Model Parameters of Bermudagrass and Bahiagrass
Grown at Eagle Lake, TX Averaged over Years.
Component
Grass
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter Yield
bermudagrass
A, Mg/ha
12.16
0.442
0.036
bahiagrass
A
9.56
0.346
0.036
N Removal
bermudagrass
A, kg/ha
271.2
14.7
0.054
bahiagrass
A
216.0
11.7
0.054
Dry Matter Yield
Both
b
0.55
0.062
0.113
N Removal
Both
b
1.11
0.071
0.064
Both
Both
c, ha/kg
0.0072
0.0005
0.069
Source: Original data from Evers (1984).

1
Table 4-33. Error Analysis for Model Parameters of Bermudagrass and Bahiagrass
Grown at Eagle Lake, TX over Two Years.
Component
Grass
Year
Parameter
Estimate
Standard
Error
Relative
Error
Yield
bermudagrass
1979
A, Mg/ha
15.99
0.820
0.051
1980
A
9.89
0.503
0.051
bahiagrass
1979
A
12.42
0.634
0.051
1980
A
7.51
0.382
0.051
N Removal
bermudagrass
1979
A, kg/ha
310.9
22.61
0.073
1980
A
227.4
16.49
0.073
bahiagrass
1979
A
237.2
17.18
0.072
1980
A
191.3
13.87
0.073
Yield
Both
All
b
0.57
0.082
0.144
N Removal
Both
All
b
1.07
0.094
0.088
Both
Both
All
c, ha/kg
0.0072
0.0007
0.097
Source: Original data from Evers (1984).

114
Table 4-34. Analysis of Variance on Model Parameters for Corn Grown on Dothan
Sandy Loam at Clayton, Is
C, Both Grain and Tota
Mode
Parameters
Estimated
df
Residual Sum
of Squares
Mean Sum
of Squares
F
(1) Com A,b,c
3
17
63333.848
3725.520
-
(2) Ind A,b,c
12
8
33.372
4.172
-
(l)-(2)
9
63300.476
7033.386
1686**
(3) Ind A, Com b,c
6
14
586.733
41.910
-
(3)-(2)
6
553.360
92.227
22.1**
(4) Ind A, b, Com c
9
11
138.832
12.621
-
(4)-(2)
3
105.460
35.153
8.43*
(3)-(4)
3
447.901
149.300
49.8**
(5) Ind A, Com c, Ind b
(dm and Nu)
7
13
146.647
11.281
_
(5)-(2)
5
113.275
22.655
5.43*
(3)-(5)
1
440.086
440.086
39.0**
(5)-(4)
2
7.815
3.908
0.31
Source: Original data from Kamprath (1986).
** Significance level of 0.001
Significance level of 0.01
* Significance level of 0.025
F(9, 8,99.9) =11.77
F(6, 8,99.9) =12.86
F(3, 8,99) =7.59
F(3,11,99.9) =11.56
F(5, 8,97.5) =4.82
F( 1,13,99.9) =17.81
F(2,11,95) =3.98

115
Table 4-35. Analysis of Variance on Model Parameters for Corn Grown on
Goldsboro Sandy Loam at Kinston, NC, Both Grain and Total.
Mode
Parameters
Estimated
df
Residual Sum
of Squares
Mean Sum
of Squares
F
(1) Com A,b,c
3
17
34443.380
2026.081
-
(2) Ind A,b,c
12
8
15.942
1.993
-
(l)-(2)
9
34427.438
3825.271
1920**
(3) Ind A, Com b,c
6
14
337.763
24.126
-
(3)-(2)
6
321.821
53.637
26.9**
(4) Ind A, b, Com c
9
11
31.448
2.859
-
(4)-(2)
3
15.506
5.169
2.59
(3)-(4)
3
306.315
102.105
35.7**
(5) Ind A, Com c, Ind b
(dm and Nu)
7
13
45.635
3.510
_
(5)-(2)
5
29.694
5.939
2.98
(3)-(5)
1
292.128
292.128
83.2**
(5H4)
2
14.187
7.094
2.48
Source: Original data from Kamprath (1986).
** Significance level of 0.001
F(9, 8,99.9) =11.77
F(6, 8,99.9) =12.86
F(3, 8,95) =4.07
F(3,11,99.9) =11.56
F(5, 8,95) =3.69
F(l,13,99.9) =17.81
F(2,11,95) =3.98

116
Table 4-36. Analysis of Variance on Model Parameters for Grain and Total Plant of
Corn Grown on Dothan and Goldsboro Sandy Loam at Clayton and Kinston,
NC.
Mode
Parameters
Estimated
df
Residual Sum
of Squares
Mean Sum
of Squares
F
(1) Com A,b,c
3
37
101127.594
2733.178
-
(2) Ind A,b,c
24
16
49.314
3.082
-
(l)-(2)
21
101078.280
4813.251
1562*
(3) Ind A, Com b,c
10
30
1117.935
37.265
-
(3)-(2)
14
1068.620
76.33
24.8**
(4) Ind A, b, Com c
17
23
317.411
13.800
-
(4)-(2)
7
268.096
38.299
12.4*
(3)-(4)
7
800.524
114.361
8.29**
(5) Ind A, Com c, Ind b
(site: dm and Nu)
13
27
350.409
12.978
(5)-(2)
11
301.095
27.372
8.88**
(3)-(5)
3
767.526
255.842
19.7**
(5)-(4)
4
32.998
8.250
0.60
(6) Ind A, Com c, Ind b
(dm and Nu)
11
29
334.633
11.539
(6)-(2)
13
285.319
21.948
7.12**
(3)-(6)
1
783.302
783.302
67.9**
(6)-(4)
6
17.222
2.870
0.21
(7) Ind A, Com c, Ind b
(part: dm and Nu)
13
27
325.903
12.070
(7)-(2)
11
276.589
25.144
8.16**
(3)-(7)
3
792.032
264.011
21.9**
(7)-(4)
4
8.492
2.123
0.15
(6)-(7)
2
8.730
4.365
0.36
Source: Original data from Kamprath (1986).
** Significance level of 0.001
F(21,16,99.9)= 4.95
F( 14,16,99.9)= 5.35
F( 7,16,99.9)= 6.46
F( 7,23,99.9)= 5.33
F(11,16,99.9)= 5.67
F( 3,27,99.9)= 7.27
F( 4,23,95) =2.80
F( 1,29,99.9)= 13.39
F( 6,23,95) =2.53
F(13,16,99.9)= 5.44
F( 2,27,95) =3.35

117
Table 4-37. Error Analysis of Model Parameters for Grain and Total Plant of Corn
Grown on Dothan and Goldsboro Sandy Loam at Clayton and Kinston, NC.
Component
Site
Part
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter Yield
Dothan
Grain
A, Mg/ha
11.12
0.156
0.014
Total
A
20.68
0.288
0.014
Goldsboro
Grain
A
7.83
0.109
0.014
Total
A
14.70
0.204
0.014
N Removal
Dothan
Grain
A, kg/ha
151.7
2.71
0.018
Total
A
187.2
3.35
0.018
Goldsboro
Grain
A
113.3
2.00
0.018
Total
A
136.5
2.42
0.018
Dry Matter Yield
Both
Both
b
0.27
0.040
0.148
N Removal
Both
Both
b
0.97
0.044
0.045
Both
Both
Both
c, ha/kg
0.0187
0.0007
0.037
Source: Original data from Kamprath (1986).

118
Table 4-38. Analysis of Variance on Model Parameters for Grain and Total Plant of
Corn Grown on Portsmouth Very Fine Sandy Loam at Plymouth NC.
Mode
Parameters
Estimated
df
Residual Sum
of Squares
Mean Sum
of Squares
F
(1) Com A,b,c
3
17
37148.680
2185.216
-
(2) Ind A,b,c
12
8
5.674
0.709
-
(1)-(2)
9
37143.006
4127.001
5819**
(3) Ind A, Com b,c
6
14
144.076
10.291
-
(3)-(2)
6
138.402
23.067
32.5**
(4) Ind A, b, Com c
9
11
16.808
1.528
-
(4)-(2)
3
11.134
3.711
5.23+
(3)-(4)
3
127.268
42.423
27.8**
(5) Ind A, Com c, Ind b
(dm and Nu)
7
13
26.613
2.047
(5)-(2)
5
20.939
4.188
5.91*
(3)-(5)
1
117.463
117.463
57.4
(5)-(4)
2
9.796
4.898
3.21
Source. Original data from Kamprath (1986).
** Significance level of 0.001
f Significance level of 0.025
Significance level of 0.05
F(9, 8,99.9) =11.77
F(6, 8,99.9) =12.86
F(3, 8,95) =4.07
F(3,11,99.9) =11.56
F(5, 8,97.5) =4.82
F( 1,13,99.9) =17.81
F(2,11,95) =3.98

119
Table 4-39. Error Analysis for Model Parameters for Grain and Total Plant of Corn
Grown on Portsmouth Very Fine Sandy Loam at Plymouth, NC.
Component
Part
Parameter
Estimate
Standard Error
Relative Error
Dry Matter Yield
Grain
A, Mg/ha
9.48
0.168
0.018
Total
A
16.60
0.293
0.018
N Removal
Grain
A, kg/ha
126.2
3.22
0.026
Total
A
147.3
3.76
0.026
Dry Matter Yield
Both
b
-0.065
0.0458
0.705
N Removal
Both
b
0.46
0.047
0.102
Both
Both
c, ha/kg
0.0119
0.0007
0.059
Source: Original data from Kamprath (1986).

120
Table 4-40. Analysis of Variance on Model Parameters for Bahiagrass Grown on
Two Soils: an Entisol and Spodosol at Williston and Gainesville, FL,
Respectively.^^
Mode
Parameters
Estimated
df
Residual Sum
of Squares
Mean Sum
of Squares
F
(1) Com A,b,c
3
17
127269.607
7486.447
-
(2) Ind A,b,c
12
8
63.094
7.887
-
(l)-(2)
9
127206.513
14134.057
1792
(3) Ind A, Com b,c
6
14
642.094
45.864
-
(3)-(2)
6
579.000
96.500
12.24++
(4) Ind A, b, Com c
9
11
233.578
21.234
-
(4)-(2)
3
170.484
56.828
7.21t
(3)-(4)
3
408.514
136.171
6.41*
(5) Ind A, Com c, Ind b
(dm and Nu)
7
13
485.676
37.360
.
(5)-(2)
5
422.582
84.516
10.72++
(3)-(5)
1
156.418
156.418
4.19
(5)~(4)
2
252.098
126.049
5.94*
Source: Original data from Blue (1987).
Significant at the 0.001 level
Significant at the 0.005 level
Significant at the 0.01 level
1 Significant at the 0.025 level
F(9, 8,99.9) =11.77
F(6, 8,99.5) =7.95
F(3, 8,97.5) =5.42
F(3,11,99) =6.22
F(5, 8,99.5) =8.30
F( 1,13,95) =4.67
F(2,11,97.5) =5.26

121
Table 4-41. Error Analysis for Model Parameters for Bahiagrass Grown on Two
Soils: an Entisol and Spodosol at Williston and Gainesville, FL,
Component
Soil
Parameter
Estimate
Standard
Error
Relative
Error
Dry Matter Yield
Entisol
A, Mg/ha
11.14
0.176
0.016
Spodosol
A
19.39
0.307
0.016
N Removal
Entisol
A, kg/ha
311.0
5.84
0.019
\
Spodosol
A
201.5
3.76
0.019
Dry Matter Yield
Both
b
1.39
0.053
0.038
N Removal
Both
b
1.86
0.060
0.032
Both
Both
c, ha/kg
0.0118
0.0004
0.034
Source: Original data from Blue (1987).

122
Table 4-42. Analysis of Variance on Model Parameters for Seasonal Dry Matter
Yield and Plant N Removal of Ryegrass Grown on 20 Different Sites in
England.
Mode
Parameters
estimated
df
Residual Sums
of Squares
Mean Sums
of Squares
F
(1) Com A,b,c
3
237
5659492.221
23879.714
-
(2) Ind A,b,c
120
120
12902.124
107.518
-
(l)-(2)
117
5646590.097
48261.454
449
(3) Ind A, Com b,c
42
198
130080.207
656.971
-
(3)-(2)
78
117178.083
1502.283
14.0
(4) Ind A, b, Com c
81
159
32739.137
205.907
-
(4)-(2)
39
19837.013
508.641
4.73
(3)-(4)
39
97341.070
2495.925
12.1"
(5) Ind A, Com c, Ind b
(dm and Nu)
43
197
90106.618
457.364
(5)-(2)
79
77204.494
977.272
9.09
(3)-(5)
1
39973.589
39973.589
87.4
(5)-(4)
38
57367.481
1509.671
7.33
Source: Original data from Morrison et a/. (1980).
** Significant at the 0.001 level
F(117,120,99.9) = 1.77
F( 78,120,99.9) = 1.87
F( 39,120,99.9) = 2.12
F( 39,159,99.9) = 2.06
F( 79,120,99.9) = 1.87
F( 1,197,99.9)= 11.16
F( 38,159,99.9) = 2.07

Table 4-43. Error Analysis of Model Parameters for Seasonal Dry Matter Yield and
Plant N Removal of Ryegrass Grown on 20 Different Sites in England.
Site
Component Parameter
Estimate
Standard Error
Relative Error
5
Dry Matter A, Mg/ha
13.84
0.223
0.016
6
A
13.90
0.231
0.017
7
A
8.39
0.146
0.017
8
A
11.80
0.243
0.021
9
A
14.82
0.236
0.016
10
A
12.87
0.229
0.018
12
A
13.83
0.330
0.024
13
A
12.85
0.209
0.016
14
A
13.00
0.228
0.018
15
A
10.63
0.219
0.021
16
A
13.36
0.217
0.016
17
A
10.93
0.201
0.018
19
A
6.37
0.114
0.018
20
A
14.11
0.276
0.020
22
A
10.32
0.196
0.019
23
A
10.25
0.187
0.018
25
A
12.67
0.225
0.018
26
A
10.38
0.215
0.021
27
A
10.55
0.187
0.018
28
A
10.97
0.250
0.023
5
N Removal A, kg/ha
469.7
8.42
0.018
6
A
537.2
10.47
0.019
7
A
334.6
6.23
0.019
8
A
423.2
10.67
0.025
9
A
518.0
9.00
0.017
10
A
446.8
8.82
0.020
12
A
454.7
12.45
0.027
13
A
488.0
8.84
0.018
14
A
466.0
9.49
0.020
15
A
381.5
9.68
0.025
16
A
478.7
8.83
0.018
17
A
370.6
8.68
0.023
19
A
222.2
4.49
0.020
20
A
450.4
10.69
0.024
22
A
368.7
7.82
0.021
23
A
356.1
7.37
0.021
25
A
458.6
9.55
0.021
26
A
355.0
8.68
0.024
27
A
384.0
8.22
0.021
28
A
365.4
9.85
0.027

124
Table 4-43-continued
Site Component
Parameter
Estimate
Standard Error
Relative Error
5 Dry Matter
b
0.34
0.117
0.344
6
b
0.65
0.106
0.163
7
b
0.98
0.099
0.101
8
b
1.68
0.101
0.060
9
b
0.38
0.110
0.289
10
b
1.04
0.102
0.098
12
b
2.10
0.110
0.052
13
b
0.52
0.108
0.208
14
b
0.93
0.103
0.111
15
b
1.60
0.107
0.067
16
b
0.58
0.103
0.178
17
b
1.64
0.090
0.055
19
b
1.09
0.101
0.093
20
b
1.38
0.108
0.078
22
b
1.31
0.102
0.078
23
b
1.14
0.102
0.089
25
b
0.91
0.111
0.122
26
b
1.59
0.109
0.069
27
b
0.96
0.106
0.110
28
b
1.92
0.111
0.058
5 N Removal
b
1.19
0.121
0.102
6
b
1.68
0.113
0.067
7
b
1.57
0.106
0.068
8
b
2.55
0.117
0.046
9
b
1.13
0.113
0.100
10
b
1.80
0.108
0.060
12
b
2.77
0.121
0.044
13
b
1.39
0.110
0.079
14
b
1.78
0.117
0.066
15
b
2.47
0.123
0.050
16
b
1.50
0.109
0.073
17
b
2.39
0.111
0.046
19
b
1.90
0.107
0.056
20
b
2.29
0.123
0.054
22
b
2.03
0.110
0.054
23
b
1.93
0.111
0.058
25
b
1.86
0.119
0.064
26
b
2.45
0.118
0.048
27
b
1.91
0.120
0.063
28
b
2.69
0.124
0.046

125
Table 4-43continued
Site
Type
Parameter
Estimate
Standard Error
Relative Error
All
Both
c, ha/ka
0.0088
0.0001
0.011
Source: Original data from Morrison el al. (1980).

126
Table 4-44. Summary of Model Parameters for Ryegrass in England.
Site
R1
cm
A
Mg/ha
An
kg/ha
A/A
g/kg
b
b
Ab
c
ha/kg
5
31.0
13.84
469.7
33.9
0.34
1.19
0.85
0.0088
6
33.2
13.90
537.2
38.7
0.65
1.68
1.03
7
29.8
8.39
334.6
39.9
0.98
1.57
0.59
8
32.5
11.80
423.2
35.9
1.68
2.55
0.87
9
46.0
14.82
518.0
35.0
0.38
1.13
0.75
10
52.0
12.87
446.8
34.7
1.04
1.80
0.76
12
47.0
13.83
454.7
32.9
2.10
2.77
0.67
13
40.0
12.85
488.0
38.0
0.52
1.39
0.87
14
32.2
13.00
466.0
35.9
0.93
1.78
0.85
15
28.6
10.63
381.5
35.9
1.60
2.47
0.87
16
35.5
13.36
478.7
35.8
0.58
1.50
0.92
17
33.4
10.93
370.6
33.9
1.64
2.39
0.75
19
23.6
6.37
222.2
34.9
1.09
1.90
0.81
20
33.7
14.11
450.4
31.9
1.38
2.29
0.91
22
32.3
10.32
368.7
35.7
1.31
2.03
0.72
23
30.2
10.25
356.1
34.7
1.14
1.93
0.79
25
25.5
12.67
458.6
36.2
0.91
1.86
0.95
26
31.3
10.38
355.0
34.2
1.59
2.45
0.86
27
29.6
10.55
384.0
36.4
0.96
1.91
0.95
28
27.9
10.97
365.4
33.3
1.92
2.69
0.77
20 sites
35.4
0.83
1.9
0.10
Source: Original data from Morrison et al. (1980).
1 Annual rainfall.

127
Table 4-45. Summary of Model Parameters, Standard Errors, and Relative Errors for
the Extended Triple Logistic (NPK) Model for Rye Grown at Tifton, GA.
Parameter
Dry Matter
N Removal
P Removal
K Removal
54301
260
34
230
A, kg/ha
1922
11.4
1.00
11.3
0.0353
0.044
0.029
0.049
1.36
1.93
1.36
1.36
b
0.139
0.166
0.139
0.139
0.102
0.086
0.102
0.102
-0.14
-0.14
-0.16
-0.14
bp
0.150
0.150
0.162
0.150
1.071
1.071
1.012
1.071
-0.91
-0.91
-0.91
0.46
bk
0.199
0.199
0.199
0.130
0.219
0.219
0.219
0.283
0.0225
0.0225
0.0225
0.0225
c, ha/kg
0.0021
0.0021
0.0021
0.0021
0.093
0.093
0.093
0.093
0.0464
0.0464
0.0464
0.0464
cp
0.0081
0.0081
0.0081
0.0081
0.175
0.175
0.175
0.175
0.0201
0.0201
0.0201
0.0201
Ck
0.0039
0.0039
0.0039
0.0039
0.194
0.194
0.194
0.194
Source: Original data from Walker and Morey (1962).
' Estimate
2 Standard Error
3 Relative Error

Coastal Bermudagrass Dry- Matter Yield, Mg/ha Pensacola Bahiagrass Dry Matter Yield, Mg/ha
128
Applied Nitrogen, kg/ha
Figure 4-1 Response of seasonal dry matter yield for Coastal bermudagrass and
Pensacola bahiagrass to applied N at Thorsby, AL. Data from Evans el al.
(1961); curves drawn from Eq. [4.1] through [4.4],

129
Measured Dry Matter Yield, Mg/ha
Figure 4-2 Scatter plot of seasonal dry matter yield for Coastal bermudagrass and
Pensacola bahiagrass at Thorsby, AL. Original data from Evans et al.
(1961).

130
1h
5
CJ
2
9
a
3
-a
'55
2
1
0
-2
O Coastal bermudagrass, Non-irrigatcd
Coastal bermudagrass, Irrigated
A Pensacola bahiagrass, Non-irrigatcd
V Pensacola bahiagrass, irrigated

O
M
$
v
A
A


0 5 10 15 20 25
Estimated Dry Matter Yield, Mg/ha
Figure 4-3 Residual plot of seasonal dry matter yield for Coastal bermudagrass and
Pensacola bahiagrass at Thorsby, AL. Original data from Evans et al.
(1961). Solid line is mean and dashed lines are 2 standard errors.

Dry Matter Yield. Mg/ha N Removal, kg/ha N Concentration, g/kg
131
Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for dallisgrass grown at Baton Rouge, LA.
Data from Robinson et al. (1988); curves drawn from Eq. [4.5] through
[4.7].
Figure 4-4

Dry Matter Yield, Mg/ha Nitrogen Concentration, g/kg
132
0 50 100 150 200 250 300 350 400 450
Nitrogen Removal, kg/ha
Figure 4-5 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for dallisgrass grown at Baton Rouge, LA. Data from Robinson
etal. (1988); curves drawn from Eq. [4.8] and [4.9],

Dry Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
133
0.0 0.2 0.4 0.6 0.8 1.0 1.2
N Removal/Estimated Maximum
Figure 4-6 Dimensionless plot of dry matter and plant N concentration as a function of
plant N removal for dallisgrass grown at Baton Rouge, LA. Original data
from Robinson el al. (1988); curves drawn from Eq. [4.10] and [4.11],

134
Figure 4-7 Scatter plot of dry matter yield for dallisgrass grown at Baton Rouge, LA.
Original data from Robinson el al. (1988).

135
Figure 4-8 Scatter plot of plant N removal for dallisgrass grown at Baton Rouge, LA.
Original data from Robinson et al. (1988).

Residual Dry Matter Yield, Mg/ha
136
1.5
1.0
0.5
0.0
-0.5
O
o
0)
o
o
o
-1.0 -
-1.5
5 10 15
Predicted Dry' Matter Yield, Mg/ha
20
Figure 4-9 Residual plot of dry matter yield for dallisgrass grown at Baton Rouge,
LA. Original data from Robinson el al. (1988). Solid line is mean and
dashed lines are 2 standard errors.

137
Figure 4-10 Residual plot of plant N removal for dallisgrass grown at Baton Rouge,
LA. Original data from Robinson el at. (1988). Solid line is mean and
dashed lines are 2 standard errors.

Dry Matter Yield. Mg/ha N Removal, kg/ha N Concentration, g/kg
138
Figure 4-11 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bermudagrass grown at Thorsby, AL and
cut at two harvest intervals. Data from Doss et a/. (1966); curves drawn
from Eq. [4.12] through [4.17],

139
Figure 4-12 Seasonal dry matter yield and plant N concentration as a function of N
removal for bermudagrass grown at Thorsby, AL and cut at two harvest
intervals. Data from Doss et a/. (1966); curves drawn from Eq. [4.18]
through [4.21],

Dry Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
140
Figure 4-13 Dimensionless plot of dry matter and plant N concentration as a function of
plant N removal for bermudagrass grown at Thorsby, AL and cut at two
harvest intervals. Original data from Doss et al. (1966); curves drawn
from Eq. [4.22] and [4.23],

141
Figure 4-14 Scatter plot of dry matter yield for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals. Original data from Doss et al. (1966).

142
Figure 4-15 Scatter plot of plant N removal for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals. Original data from Doss et al. (1966).

143
1
2
2
"o
£

g
B
C3
J3
- -1

o
0 -
-2 -
G

3.0 weeks
4.5 weeks
O


TJ
-GO:
@


--O-
10 15
Predicted Dry Matter Yield, Mg/ha
20
25
Figure 4-16 Residual plot of dry matter yield for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals. Original data from Doss et al. (1966).
Solid line is mean and dashed lines are 2 standard errors.

144
T3
>
o
a
g
od
C3
3
O
100
75
50
25
S -25
-50
-75
-100
O

3.0 weeks
4.5 weeks

O
o

o
o
CD


O
100 200 300 400
Predicted N Removal, kg/lia
500
600
Figure 4-17 Residual plot of plant N removal for bermudagrass grown at Thorsby, AL
and cut at two harvest intervals. Original data from Doss et al. (1966).
Solid line is mean and dashed lines are 2 standard errors.

Dry Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
145
Figure 4-18 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bermudagrass grown at Maryland and cut at
five harvest intervals. Data from Decker et al. (1971); curves drawn from
Eq. [4.24] through [4.38],

Dry Matter Yield, Mg/ha Nitrogen Concentration, g/kg
146
0 100 200 300 400 500
Nitrogen Removal, kg/ha
Figure 4-19 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bermudagrass grown at Maryland and cut at five harvest
intervals. Data from Decker et al. (1971); curves drawn from Eq. [4.39]
through [4.48].

Dry Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
147
0.0 0.2 0.4 0.6 0.8 1.0 1.2
N Removal/Estinialed Maximum
Figure 4-20 Dimensionless plot of dry matter and plant N concentration as a function of
plant N removal for bermudagrass grown at Maryland and cut at five
harvest intervals. Original data from Decker el al. (1971); curves drawn
from Eq. [4.49] and [4.50],

148
Figure 4-21 Scatter plot of dry matter yield for bermudagrass grown at Maryland and
cut at five harvest intervals. Original data from Decker et al. (1971).

149
O 100 200 300 400 500
Measured N Removal, kg/ha
Figure 4-22 Scatter plot of plant N removal for bermudagrass grown at Maryland and
cut at five harvest intervals. Original data from Decker et al. (1971).

150
i
c
Q
15
P
rs
CO

12 1
13
*>-
Vh
y
t A
C3 U
-1
-2
-3
-4
0
3.2 weeks

3.6 weeks
A
4.3 weeks
V
5.5 weeks
<0
7.7 weeks /S
V
V
.A
V
0
%
O
->zr
O
&
A

v
O
5 10 15
Predicted Dry Matter Yield, Mg/lia
20
25
Figure 4-23 Residual plot of dry matter yield for bermudagrass grown at Maryland and
cut at five harvest intervals. Original data from Decker et al. (1971). Solid
line is mean and dashed lines are 2 standard errors.

151
C3
§
on
A
-a
>
o
S
s
Q
13
3
IS
w
0
125
G
1
3.2 weeks
r
i 1
100

A
3.6 weeks
4.3 weeks
-
V
5.5 weeks
75
- O
7.7 weeks
50
-
<>
25
V
G _
0
G
o A
o
)
&
£>
G
-25
IS
O Ov
-
-50
-
-75
-
-100
-125
100 200 300
Predicted N Removal, kg/ha
400
500
Figure 4-24 Residual plot of plant N removal for bermudagrass grown at Maryland and
cut at five harvest intervals. Original data from Decker et al. (1971). Solid
line is mean and dashed lines are 2 standard errors.

152
O 2 4 6 8 10
Harvest Interval, weeks
Figure 4-25 Estimated maximum dry matter yield and estimated maximum plant N
removal as a function of harvest interval for bermudagrass in Maryland.
Lines drawn from Eq. [4 51] and [4.52]

Dry Matter Yield, Mg/ha N Removak kg/ha N Concentration, g/kg
153
Figure 4-26 Seasonal dry matter yield, plant N removal, and plant N concentration for a
two week clipping interval over two years for bermudagrass grown at
Tifln, GA. Data from Prine and Burton (1956); curves drawn from Eq.
[4.53] through [4.56], [4.73], and [4.74],

Diy Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
154
Figure 4-27 Seasonal dry matter yield, plant N removal, and plant N concentration for a
three week clipping interval over two years for bermudagrass grown at
Tifton, GA. Data from Prine and Burton (1956); curves drawn from Eq.
[4.57] through [4.60], [4.75], and [4.76],

Dry Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
155
Figure 4-28 Seasonal dry matter yield, plant N removal, and plant N concentration for a
four week clipping interval over two years for bermudagrass grown at
Tifton, GA. Data from Prine and Burton (1956); curves drawn from Eq.
[4.61] through [4.64], [4.77], and [4.78],

Dry Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
156
Figure 4-29 Seasonal dry matter yield, plant N removal, and plant N concentration for a
six week clipping interval over two years for bermudagrass grown at
Tifln, GA. Data from Prine and Burton (1956); curves drawn from Eq.
[4.65] through [4.68], [4.79], and [4.80]

Dry Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
157
Figure 4-30 Seasonal dry matter yield, plant N removal, and plant N concentration for a
eight week clipping interval over two years for bermudagrass grown at
Tifln, GA. Data from Prine and Burton (1956); curves drawn from Eq.
[4.69] through [4.72], [4.81], and [4.82],

158
Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a two week clipping interval over two years for
bermudagrass grown at Tifton, GA. Data from Prine and Burton (1956);
curves drawn from Eq. [4.83] through [4.86],
Figure 4-31

159
O 100 200 300 400 500 600 700
Nitrogen Removal, kg/ha
Figure 4-32 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a three week clipping interval over two years for
bermudagrass grown at Tifln, GA. Data from Prine and Burton (1956);
curves drawn from Eq. [4.87] through [4.90],

160
Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a four week clipping interval over two years for
bermudagrass grown at Tifton, GA. Data from Prine and Burton (1956);
curves drawn from Eq. [4.91] through [4.94],
Figure 4-33

161
O 75 150 225 300 375 450 525 600 675 750
Nitrogen Removal, kg/ha
Figure 4-34 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a six week clipping interval over two years for bermudagrass
grown at Tifton, GA. Data from Prine and Burton (1956); curves drawn
from Eq. [4.95] through [4.98],

162
Figure 4-35 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for a eight week clipping interval over two years for
bermudagrass grown at Tifln, GA. Data from Prine and Burton (1956);
curves drawn from Eq. [4.99] through [4.102],

Drv Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
163
N Removal/Estimaled Maximum
Figure 4-36 Dimensionless plot of seasonal dry matter yield and seasonal plant N
concentration versus seasonal plant N removal. Original data from Prine
and Burton (1956); curves drawn from Eq. [4.103] and [4.104],

1000
800
600
400
200
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40
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164
1953
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R = 0.9986
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Harvest Interval, weeks
10
Estimated mximums of seasonal dry matter yield and plant N removal as a
function of harvest interval for two years of bermudagrass grown at Tifton,
GA. Lines drawn from Eq. [4.105] through [4.108],

165
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Measured Dry Matter Yield, Mg/ha
Figure 4-38 Scatter plot of seasonal dry matter yield for bermudagrass grown over two
years at Tifln, GA, and cut at five different harvest intervals. Original
data from Prine and Burton (1956).

166
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Measured N Removal, kg/lia
600
700
Figure 4-39 Scatter plot of seasonal plant N removal for bermudagrass grown over two
years at Tifton, GA, and cut at five different harvest intervals. Original
data from Prine and Burton (1956).

167
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Predicted Dry Matter Yields, Mg/ha
25
30
Figure 4-40 Residual plot of seasonal dry matter yield for bermudagrass grown over
two years at Tifton, GA, and cut at five different harvest intervals. Original
data from Prine and Burton (1956). Solid line is mean and dashed lines are
2 standard errors.

168
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200
150
100
50
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200 400
Estimated N Removal, kg/lia
600
800
Figure 4-41 Residual plot of seasonal plant N removal for bermudagrass grown over
two years at Tifton, GA, and cut at five different harvest intervals. Original
data from Prine and Burton (1956). Solid line is mean and dashed lines are
2 standard errors.

Dry Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
169
Figure 4-42 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for three different number of cuttings over the
season for ryegrass grown at England. Data from Reid (1978); curves
drawn from Eq. [4 109] through [4.117],

170
Figure 4-43 Seasonal dry matter yield and plant N removal as a function of plant N
concentration for three different number of cuttings over the season for
ryegrass grown at England. Data from Reid (1978); curves drawn from
Eq. [4.118] through [4.123],

Dry Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
171
0.0 0.2 0.4 0.6 0.8 1.0 1.2
N Removal/Estimatcd Maximum
Figure 4-44 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for three different number of cuttings over
the season for ryegrass grown at England. Original data from Reid (1978);
curves drawn from Eq. [4.124] through [4.129],

172
Figure 4-45 Scatter plot of seasonal dry matter yield for ryegrass grown at England and
cut different times over the season. Data from Reid (1978).

173
Figure 4-46 Scatter plot of seasonal plant N removal for ryegrass grown at England and
cut different times over the season. Data from Reid (1978).

174
Predicted Dry Matter Yield, Mg/ha
Figure 4-47 Residual plot of seasonal dry matter yield for ryegrass grown at England
and cut different times over the season. Original data from Reid (1978).
Solid line is mean and dashed lines are 2 standard errors.

175
Predicted N Removal, kg/ha
Figure 4-48 Residual plot of seasonal plant N removal for ryegrass grown at England
and cut different times over the season. Original data from Reid (1978).
Solid line is mean and dashed lines are 2 standard errors.

176
0123456789 10
Averaged Harvest Interval, weeks
Figure 4-49 Estimated mximums of seasonal dry matter yield and plant N removal as a
function of average harvest interval for ryegrass grown at England. Lines
drawn from Eq. [4.130] and [4.131],

Div Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
177
Figure 4-50 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bermudagrass grown over three years at
Fayetteville, AR, with and without irrigation. Data from Huneycutt et al.
(1988); curves drawn from Eq. [4.132] through [4.149],

178
Figure 4-51 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bermudagrass grown over three years at Fayetteville, AR,
with and without irrigation Data from Huneycutt et al. (1988); curves
drawn from Eq.[4.150] through [4.161],

Drv Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
179
0.0 0.2 0.4 0.6 0.8 1.0 1.2
N Removal/Estimaled Maximum
Figure 4-52 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for bermudagrass grown over three years
at Fayetteville, AR, with and without irrigation. Original data from
Huneycutt el a!. (1988); curves drawn from Eq.[4.162] and [4.163],

180
Measured Dry Matter Yield. Mg/ha
Figure 4-53
Scatter plot of seasonal dry matter yield for bermudagrass grown over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt el al. (1988).

181
600
500 -
03
in
~
>
o
6
o
0
-o
o
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8
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400
300
200
100
O Non-lrrigatcd 1983
Non-lrrigatcd 1984
A Non-lrrigatcd 1985
V Irrigated 1983
O Irrigated 1984
O Irrigated 1985
A
i j

A
f)
100 200
Measured N Removal, kg/lia
Figure 4-54 Scatter plot of seasonal plant N removal for bermudagrass grown over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt el al. (1988).

182
Figure 4-55 Residual plot of seasonal dry matter yield for bermudagrass grown over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt el al. (1988) Solid line is mean and dashed lines are 2
standard errors.

183
Figure 4-56 Residual plot of seasonal plant N removal for bermudagrass grown over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt el al. (1988). Solid line is mean and dashed lines are 2
standard errors.

y Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
184
Figure 4-57 Response of seasonal dry matter yield, plant N removal and plant N
concentration for bermudagrass averaged over three years at Fayetteville,
AR, with and without irrigation. Data from Huneycutt el al. (1988);
curves drawn from Eq. [4.164] through [4 169],

Dry Matter Yield, Mg/ha N Concentration, g/kg
185
Figure 4-58 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bermudagrass averaged over three years at Fayetteville, AR,
with and without irrigation Data from Huneycutt et aL (1988); curves
drawn from Eq. [4.170] through [4.173],

Drv Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
186
N Removal/Estimated Maximum
Figure 4-59 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for bermudagrass averaged over three
years at Fayetteville, AR, with and without irrigation. Original data from
Huneycutt el a!. (1988); curves drawn from Eq. [4.174] and [4.175],

187
0
0

10 15
Measured Dry Matter Yield. Mg/ha
Figure 4-60 Scatter plot of seasonal dry matter yield for bermudagrass averaged over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt el al. (1988).

188
Figure 4-61 Scatter plot of seasonal plant N removal for bermudagrass averaged over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt el al. (1988).

189
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S
12
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£
V-I

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Non-lrrigatcd
Irrigated
o
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-
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-

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10 15 20
Predicted Dry Matter Yield. Mg/ha
25
30
Figure 4-62 Residual plot of seasonal dry matter yield for bermudagrass averaged over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt et al. (1988). Solid line is mean and dashed lines are 2
standard errors.

190
Figure 4-63 Residual plot of seasonal plant N removal for bermudagrass averaged over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt el af. (1988). Solid line is mean and dashed lines are 2
standard errors.

Dry Matter Yield. Mg/ha N Removal, kg/ha N Concentration, g/kg
191
Figure 4-64 Response of seasonal dry matter yield, plant N removal and plant N
concentration for tall fescue grown over three years at Fayetteville, AR,
with and without irrigation. Data from Huneycutt el al (1988); curves
drawn from Eq [4.176] through [4 193],

Dry Matter Yield, Mg/ha Nitrogen Concentration, g/kg
192
Figure 4-65 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for tall fescue grown over three years at Fayetteville, AR, with
and without irrigation. Data from Huneycutt e1 at. (1988); curves drawn
from Eq. [4.194] through [4.205],

Drv Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
193
4:i *
(;i Non-lrrigatcd 1981-2
Non-Irrigated 1982-3
A Non-lrrigatcd 1983-4
Irrigated 1981-2
Irrigated 1982-3
m Irrigated 1983-4
Q
i:)
0.8
1.0
1.2
Nitrogen Removal/Estimated Maximum
Figure 4-66 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for tall fescue grown over three years at
Fayetteville, AR, with and without irrigation. Original data from
Huneycutt et al. (1988); curves drawn from Eq. [4.206] and [4.207],

194
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Non-Irri gated 1982-3
A
Non-lrrigatcd 1983-4
V
Irrigated 1981-2
16
-
o
Irrigated 1982-3
0
Irrigated 1983-4
14
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12
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10
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8
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w
6
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Measured Dry Matter Yield. Mg/ha
Figure 4-67 Scatter plot of seasonal dry matter yield for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation Original data from
Huneycutt et al. (1988).

195
500
400
a
5
o
S

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£
300
200
100
O Non-lrrigated 1981-2
Non-Irrigatcd 1982-3
A Non-Irrigatcd 1983-4
V Irrigated 1981-2
<> Irrigated 1982-3
O Irrigated 1983-4
/
7)

i -M
I 11
A

A
A
100
200 300
Measured N Removal, kg/lia
400
500
Figure 4-68 Scatter plot of seasonal plant N removal for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation. Original data from
Huneycutt el al. (1988).

196
Figure 4-69 Residual plot of seasonal dry matter yield for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation. Original data from
Huneycutt el al. (1988). Solid line is mean and dashed lines are 2 standard
errors.

Residual N Removal, kg/ha
197
Predicted N Removal, kg/lia
Figure 4-70 Residual plot of seasonal plant N removal for tall fescue grown over three
years at Fayetteville, AR, with and without irrigation. Original data from
Huneycutt el at. (1988). Solid line is mean and dashed lines are 2 standard
errors.

Dry Matter Yield. Mg/ha N Removal, kg/ha N Concentration, g/kg
198
Figure 4-71 Response of seasonal dry matter yield, plant N removal and plant N
concentration for tall fescue averaged over three years at Fayetteville, AR,
with and without irrigation Data from Huneycutt el al. (1988); curves
drawn from Eq. [4.208] through [4.213],

Dry Matter Yield. Mg/ha N Concentration, g/kg
199
Figure 4-72 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for tall fescue averaged over three years at Fayetteville, AR,
with and without irrigation. Data from Huneycutt el al. (1988); curves
drawn from Eq. [4.214] through [4.217],

Drv Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
200
1.2 1 1 i
0.4
O Non-Irrigated
Irrigated
0.0 1 1 1 J
0.0 0.2 0.4 0.6 0.8 1.0 1.2
N Removal/Estimatcd Maximum
Figure 4-73 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for tall fescue averaged over three years at
Fayetteville, AR, with and without irrigation. Original data from
Huneycutt etol. (1988); curves drawn from Eq. [4.218] and [4.219],

201
20
18
16
t 14
s
H 12
o
S io
s
! *
B
o
-5
S 6
O Non-lrrigatcd
Irrigated
o
&

f)
x
6
rn
8
10
12
Measured Dr\ Matter Yield. Mg/ha
Figure 4-74 Scatter plot of seasonal dry matter yield for tall fescue averaged over three
years at Fayetteville, AR, with and without irrigation. Original data from
Huneycutt el al. (1988).

202
C3
OI)
T3
>
o
S
o
o
-o
o
-a
1
200 300
Measured N Removal, kg/lia
Figure 4-75 Scatter plot of seasonal plant N removal for tall fescue averaged over three
years at Fayetteville, AR, with and without irrigation. Original data from
Huneycutt el a!. (1988).

203
Figure 4-76 Residual plot of seasonal dry matter yield for tall fescue averaged over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt el al. (1988). Solid line is mean and dashed lines are 2
standard errors.

204
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00
a
>
o
a
u
o£
ed
3
ts
to
o
Oi
75
50
25
-25
-50
-75
O

Non-Irrigatcd
Irrigated
G

TT-O-
(1
()

jgJU
()
o
n
100
200 300
Predicted N Removal, kgdia
400
500
Figure 4-77 Residual plot of seasonal plant N removal for tall fescue averaged over
three years at Fayetteville, AR, with and without irrigation. Original data
from Huneycutt et al. (1988). Solid line is mean and dashed lines are 2
standard errors.

Dry Matter Yield. Mg/ha N Removal, kg/ha N Concentration, g/kg
205
40
30
20
10
0
250
200
150
100
50
0
15
10
5
0
0 50 100 150 200 250 300 350 400
Applied Nitrogen, kg/ha
n
o
(>

rv
FT
Figure 4-78 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bermudagrass grown over two years at
Eagle Lake, TX. Data from Evers (1984); curves drawn from Eq. [4.220]
through [4.223], [4.228] and [4 229]

Dry Matter Yield. Mg/ha N Removal, kg/ha N Concentration, g/kg
206
Applied Nitrogen, kg/lia
Figure 4-79 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bahiagrass grown over two years at Eagle
Lake, TX. Data from Evers (1984), curves drawn from Eq. [4.224]
through [4.227], [4 230], and [4 231 ]

Dry Matter Yield, Mg/ha Nitrogen Concentration, g/kg
207
Nitrogen Removal, kg/ha
Figure 4-80 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bermudagrass grown over two years at Eagle Lake, TX.
Data from Evers (1984); curves drawn from Eq [4.232] through [4.235],

208
/A
A
A
A
VA'
100
Nitrogen Removal, kg/lia
Figure 4-81 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bahiagrass grown over two years at Eagle Lake, TX. Data
from Evers (1984); curves drawn from Eq. [4.236] through [4.239],

Drv Matter/Estimated Maximum N concentration/Estimated Maximum
209
N Removal/Estimated Maximum
Figure 4-82 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for bahiagrass and bermudagrass grown
over two years at Eagle Lake, TX. Original data from Evers (1984);
curves drawn from Eq. [4.240] and [4.241],

Predicted Dry Matter Yield, Mg/ha
210
15
12
O Bemiudagrass 1979
Bemiudagrass 1980
A Bahiagrass 1979
V Bahiagrass 1980
n A
id
0 V
yH
A
&
V
-A
i >
A
i
6
j
9
"71
/
/O
o
12
15
Measured Dry Matter Yield. Mg/ha
Figure 4-83 Scatter plot of seasonal dry matter yield for bahiagrass and bermudagrass
grown over two years at Eagle Lake, TX. Original data from Evers
(1984).

Predicted N Removal, kg/ha
211
300
250
200
150
100
50
O Bcrmudagrass 1979
Bemuidagrass 1980
A Bahiagrass 1979
V Bahiagrass 1980
/
A
O e
-V

0
$0
V
V
50
100
Measured N Removal, kg/lia
Figure 4-84 Scatter plot of seasonal plant N removal for bahiagrass and bermudagrass
grown over two years at Eagle Lake, TX. Original data from Evers
(1984).

212
1
2
13
£
a
i o
£
c3
3
3 .i
o
Q
-2
-3
O Bcrmudagrass 1979
Bcmmdagrass 1980
A Baliiagrass 1979
V Baliiagrass 1980
D
V Dv v
A
( )
O
.A-
AQ
A
-<)--
i >
J I L I
6 8 10 12
Predicted Dry Matter Yield, Mg/ha
14
16
Figure 4-85 Residual plot of seasonal dry matter yield for bahiagrass and berntudagrass
grown over two years at Eagle Lake, TX. Original data from Evers
(1984). Solid line is mean and dashed lines are 2 standard errors.

213
50
O Bermudagrass 1979
Bermudagrass 1980
A Bahiagrass 1979
V Bahiagrass 1980
100 150 200
Predicted N Removal, kg/ha
250
300
Figure 4-86 Residual plot of seasonal plant N removal for bahiagrass and bermudagrass
grown over two years at Eagle Lake, TX. Original data from Evers
(1984). Solid line is mean and dashed lines are 2 standard errors.

Dry Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
214
Figure 4-87 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for grain and total plant of corn grown at
Clayton and Kinston, NC, on Dothan and Goldsboro soils, respectively.
Data from Kamprath (1986); curves drawn from Eq. [4.242] through
[4.253],

Dry Matter Yield, Mg/ha Nitrogen Concentration, g/kg
215
Figure 4-88 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for grain and total plant of com grown at Clayton and Kinston,
NC, on Dothan and Goldsboro soils, respectively. Data from Kamprath
(1986); curves drawn from Eq. [4.254] through [4.261],

Dry Matter Yield/Estimated Maximum Nitrogen Concentration/Estimated Maximum
216
Nitrogen Removal/Estimatcd Maximum
Figure 4-89 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for grain and total plant of corn grown at
Clayton and Kinston, NC, on Dothan and Goldsboro soils, respectively.
Original data from Kamprath (1986); curves drawn from Eq. [4.262] and
[4.263],

Predicted Dry Matter Yield, Mg/ha
217
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1 ~T
Dothan Grain
j
1

Dothan Total
A
Goldsboro Grain
V
Goldsboro Total



j
V
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-
V/
A
Q
-
a/
i i
i
i
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5 10
15
20
25
Dry Matter Yield. Mg/ha
Figure 4-90 Scatter plot of seasonal dry matter yield for grain and total plant of com
grown at Clayton and Kinston, NC, on Dothan and Goldsboro soils,
respectively. Original data from Kamprath (1986).

218
Figure 4-91 Scatter plot of seasonal plant N removal for grain and total plant of corn
grown at Clayton and Kinston, NC, on Dothan and Goldsboro soils,
respectively. Original data from Kamprath (1986).

219
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Predicted Dry Matter Yield. Mg/ha
20
25
Figure 4-92 Residual plot of seasonal dry matter yield for grain and total plant of com
grown at Clayton and Kinston, NC, on Dothan and Goldsboro soils,
respectively. Original data from Kamprath (1986). Solid line is mean and
dashed lines are 2 standard errors.

220
Figure 4-93 Residual plot of seasonal plant N removal for grain and total plant of corn
grown at Clayton and Kinston, NC, on Dothan and Goldsboro soils,
respectively. Original data from Kamprath (1986). Solid line is mean and
dashed lines are 2 standard errors.

Dry Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
221
Figure 4-94 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for grain and total plant of corn grown at
Plymouth, NC. Data from Kamprath (1986); curves drawn from Eq.
[4.264] through [4.269]

222
Nitrogen Removal, kg/ha
Figure 4-95 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for grain and total plant of corn grown at Plymouth, NC. Data
from Kamprath (1986); curves drawn from Eq [4.270] through [4.273],

Dry Matter Yield/Estimated Maximum Nitrogen Concentration/Estimated Maximum
223
Nitrogen Removal/Estinialed Maximum
Figure 4-96 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for grain and total plant of corn grown at
Plymouth, NC. Original data from Kamprath (1986); curves drawn from
Eq. [4.274] and [4 275]

224
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Measured Dry Matter Yield. Mg/lia
16
18 20
Figure 4-97 Scatter plot of seasonal dry matter yield for grain and total plant of com
grown at Plymouth, NC. Original data from Kamprath (1986).

Predicted N Removal, kg/ha
225
C
n
90
Measured N Removal, kg/ha
Figure 4-98 Scatter plot of seasonal plant N removal for grain and total plant of corn
grown at Plymouth, NC. Original data from Kamprath (1986).

226
a
Figure 4-99 Residual plot of seasonal dry matter yield for grain and total plant of corn
grown at Plymouth, NC. Original data from Kamprath (1986). Solid line
is mean and dashed lines are 2 standard errors.

227
Predicted N Removal, kg/ha
Figure 4-100 Residual plot of seasonal plant N removal for grain and total plant of corn
grown at Plymouth, NC. Original data from Kamprath (1986). Solid line is
mean and dashed lines are 2 standard errors.

Dry Matter Yield, Mg/ha N Removal, kg/ha N Concentration, g/kg
228
Applied Nitrogen, kg/ha
Figure 4-101 Response of seasonal dry matter yield, plant N removal, and plant N
concentration to applied N for bahiagrass grown on two soils in Florida.
Data from Blue (1987); curves drawn from Eq. [4.276] through [4.281],

229
Nitrogen Removal, kg/ha
Figure 4-102 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for bahiagrass grown on two soils in Florida. Data from Blue
(1987); curves drawn from Eq. [4 282] through [4.285],

Drv Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
230
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-e"
EL
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0.4
0.2
O Entisol
Spodosol
0
El
£7
0.0
0.0
J I
0.2 0.4
N Removal/Estimated Maximum
Figure 4-103 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for bahiagrass grown on two soils in
Florida. Original data from Blue (1987); curves drawn from Eq. [4.286]
and [4.287]

231
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Measured Dry Matter Yield, Mg/ha
Figure 4-104 Scatter plot of seasonal dry matter yield for bahiagrass grown on two soils
in Florida. Original data from Blue (1987).

Predicted N Removal, kg/ha
232
300
250
200
150
100
50
O Enlisol
Spodosol
/
/
0
i
50
Q
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100
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250
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Measured N Removal, kg/lia
Figure 4-105 Scatter plot of seasonal plant N removal for bahiagrass grown on two soils
in Florida. Original data from Blue (1987).

233
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Predicted Dry Matter Yield. Mg/ha
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16 18 20
Figure 4-106 Residual plot of seasonal dry matter yield for bahiagrass grown on two
soils in Florida. Original data from Blue (1987). Solid line is mean and
dashed lines are 2 standard errors.

234
1
p
%
a
p4
O 50 100 150 200 250 300 350
Predicted N Removal, kg/ha
Figure 4-107 Residual plot of seasonal plant N removal for bahiagrass grown on two
soils in Florida. Original data from Blue (1987). Solid line is mean and
dashed lines are 2 standard errors.

235
1.50
1.25
1.00
< 0.75
0.50
0.25
0.00
50
40
<
30
20
10
0 5 10 15 20 25 30
Site
Figure 4-108 Plot of the mean and 2 standard errors of An/A and Ab for twenty sites in
England.

Drv Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
236
N Removal/Esiimated Maximum
Figure 4-109 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a function of plant N removal for twenty sites in England. Original data
from Morrison el al. (1980); curves drawn from Eq. [4.288] and [4.289],

Drv Matter Yield/Estimated Maximum N Concentration/Estimated Maximum
237
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
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r 4 0.9793
0.4 0.6 0.8
N Rcmoval/Estimatcd Maximum
1.2
Figure 4-110 Dimensionless plot of seasonal dry matter yield and plant N concentration
as a (unction of plant N removal for twenty sites in England. Original data
from Morrison et al. (1980); curves drawn from Eq. [4.290] and [4.291]
with Ab=0.76.

Nitrogen Removal, kg/ha N Concentration, g/kg
238
Figure 4-111 Response of seasonal dry matter, plant N removal, and plant N
concentration to applied N for rye grown at Tifton, GA and fixed
application rates of 40 and 74 kg/ha of P and K, respectively. Data from
Walker and Morey (1962). Curves drawn by Eq. [4.292], [4.293] and
[4.296],

Dry Matter Yield, Mg/ha P Remov al, kg/ha P Concentration, g/kg
239
Figure 4-112 Response of seasonal dry matter, plant P removal, and plant P
concentration to applied P for rye grown at Tifton, GA and fixed
application rates of 135 and 74 kg/ha of N and K, respectively. Data from
Walker and Morey (1962). Curves drawn by Eq. [4.292], [4.294], and
[4.297],

240
Figure 4-113 Response of seasonal dry matter, plant K removal, and plant K
concentration to applied K for rye grown at Tifton, GA and fixed
application rates of 135 and 40 kg/ha of N and P, respectively. Data from
Walker and Morey (1962). Curves drawn by Eq. [4.292], [4.295], and
[4 298],

241
Nitrogen Removal, kg/ha
Figure 4-114 Seasonal dry matter yield and plant N concentration as a function of plant
N removal for rye grown at Tifln, GA. Data from Walker and Morey
(1962). Curves drawn by Eq. [4 299] and [4.300],

Dry Matter Yield, Mg/ha Phosphorus Concentration, g/kg
242
Phosphorus Removal, kg/ha
Figure 4-115 Seasonal dry matter yield and plant P concentration as a function of plant P
removal for rye grown at Tifton, GA. Data from Walker and Morey
(1962). Curves drawn by Eq. [4.301 ] and [4.302],

243
Potassium Removal, kg/ha
Figure 4-116 Seasonal dry matter yield and plant K concentration as a function of plant
K removal for rye grown at Tifton, GA. Data from Walker and Morey
(1962). Curves drawn by Eq. [4.303] and [4.304],

Drv Matter Yield/Estimated Maximum Nutrient Concentration/Estimated Maximum
244
Nutrient Removal/Estimalcd Maximum
Figure 4-117 Dimensionless plot of seasonal dry matter yield and plant nutrient
concentration as a function of plant nutrient removal for rye grown at
Tifton, GA. Original data from Walker and Morey (1962). Curves drawn
by Eq. [4.305] through [4.310],

245
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2 .
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()
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00
Measured Dry Matter Yield, Mg/lia
Figure 4-118 Scatter plot of dry matter yield for rye grown at Tifln, GA. Original data
from Walker and Morey (1962).

Estimated N Removal, kg/ha
246
250
100 150
Measured N Removal, kg/lia
Figure 4-119 Scatter plot of plant N removal for rye grown at Tifton, GA. Original data
from Walker and Morey (1962).

247
()
O
9' o
<~> C >
o
(I
<)
20 25 30
Measured P Removal, kg/ha
35 40
Figure 4-120 Scatter plot of plant P removal for rye grown at Tifton, G A. Original data
from Walker and Morey (1962).

248
200
175
150
125
a
2 100
J 75
50
25
0
0 25 50 75 100 125 150 175 200
Measured K Removal, kg/ha
Figure 4-121 Scatter plot of plant K removal for rye grown at Tifton, GA. Original data
from Walker and Morey (1962).
4

Residual Dry Matter Yield, Mg/ha
249
0 12 3 4 5
Estimated Dry Matter Yield. Mg/ha
Figure 4-122 Residual plot of dry matter yield for rye grown at Tifton, GA. Original
data from Walker and Morey (1962). Solid line is mean and dashed lines
are 2 standard errors.

250
100
75
50
25
o
S
2 0
CO
3
-25
C/J
-50 -
-75 -
-100
O
ri
co
O o O
o
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50 100 150
Estimated N Removal, kg/lia
200
250
Figure 4-123 Residual plot of plant N removal for rye grown at Tifln, GA. Original
data from Walker and Morey (1962). Solid line is mean and dashed lines
are 2 standard errors.

Residual P Removal, kg/ha
251
0 5 10 15 20 25 30
Estimated P Removal, kg/lia
Figure 4-124 Residual plot of plant P removal for rye grown at Tifton, GA. Original
data from Walker and Morey (1962). Solid line is mean and dashed lines
are 2 standard errors.

Residual K Removal, kg/ha
252
0 25 50 75 100 125 150 175 200
Estimated K Removal, kg/lia
Figure 4-125 Residual plot of plant K removal for rye grown at Tifton, GA. Original
data from Walker and Morey (1962). Solid line is mean and dashed lines
are 2 standard errors.

CHAPTER 5
SUMMARY AND CONCLUSIONS
Lately, environmental issues have become a popular topic in public discussion.
People are interested in controlling pollution in soil, water and air. Land application of
treated wastes and effluent is being used to help control contamination of surface and
groundwater of Florida A simple method is needed to aid engineers, farmers, and
managers in obtaining estimates of nutrient removal by various forage crops. This
research project has focused on modeling the seasonal production of seven different
forage grasses, at different locations and under different factors, such as water availability,
harvest interval and plant partitioning.
The simple logistic model, Equation [2.8], the extended logistic model, Equations
[2.8] and [2.9], and the extended triple logistic (NPK) model, Equations [2.17] through
[2.20], were found to adequately describe grass response to applied nutrients (R > 0.99).
Data from various studies were used to relate management parameters of water
availability, harvest interval, soil type, and crop species to the parameters of the equation.
Second-order Newton-Raphson method for nonlinear regression was used to optimize the
fit of the model to the data. Analysis of variance (ANOVA) was used to search for
simplification of the model in the form of common parameter values. The conclusions
from the analysis of the data from the various studies are presented below.
1. The logistic model is relatively simple to use. Once the parameters are known,
estimates can be computed on a basic hand calculator. The logistic model involves
analytical functions rather than finite difference (numerical techniques).
253

254
2. The logistic model is well behaved. The response is positive for the entire
domain of applied nitrogen The slope is greater than zero (dY/dN > 0) for all values of
N, i.e., it is monotone increasing. It asymptotically approaches a maximum at high N and
zero at low N.
3. As shown with various data sets, harvest interval, water availability and plant
partitioning can be accounted for in the linear parameter, A. This means that once the
relationship has been identified, the effects of different levels and combinations of these
factors on dry matter yield or nutrient removal response to applied nutrients can be
estimated. Had these factors been accounted for only by the nonlinear parameters b
and/or c, the task of simplification would have been greatly limited Furthermore, since
the effects are linear, the dependence upon yield and N removal can be factored into
product terms:
A = Aw Ah Ap [5.1]
where A = maximum yield or N removal parameter for Eq [2.8] or [2.9],
Aw = water availability coefficient.
Ah = harvest interval coefficient,
Ap = plant partitioning coefficient.
Because these effects fall into products, averaging over years only affects the A parameter.
Why was the A parameter linearly related to harvest interval? The answer lies in how the
plant grows between harvests. Overman et a/. (1989) have shown that between harvests
the plant follows a quadratic intrinsic growth function When these sawtooth growths for
multiple harvests are summed, the result is linear dependence of seasonal yield on harvest
interval. Further, if the harvest interval is too long, the quadratic trend falls off and this
effect appears in the sum as well as in the slow loss of linearity.
4. Dimensionless plots are a valuable tool in evaluating the form of a model
(Segr, 1984, p. 168). All data are normalized to the same scale, thereby removing the

255
units. After normalizing, the curve and line drawn are only dependent upon the Ab
value of the logistic model. In cases such as the Prine and Burton (1956) study
(bermudagrass grown at Tifton, GA), the model could be simplified since all the different
combinations of factors had common Ab and c parameters as evidenced by the
dimensionless plot (Figure 4-36). If after normalizing the data there had been no distinct
relationship, it would be difficult to separate or identify the effects of different factors. In
many different aspects of science and engineering, dimensionless plots are valuable tools in
examining the underlying relationship between two variables and in collapsing data with
different ranges onto the same scale.
5. The logistic equation exhibits symmetry suggesting that something in the
system is conserved. In the case of these models, it is the total capacity of the system, A,
the estimated maximum. But how exactly does this occur? Recall the rumor model. The
total capacity of the system is analogous to the number of people in the room. The
number of people in the room remains constant as the rumor is spread throughout the
room. Furthermore, the transfer rate of the rumor is a product of two entities, the people
who know the rumor and those who still need to be told. Similarly to the logistic crop
growth system, the total capacity of the system is the sum of the dry matter that has been
produced (y) and that which still can be produced (A-y). Response to increased input (N)
is a product of two things: that which has already been produced and that which can be
produced. As shown with Equation [2.3], the differential form of the logistic is directly
proportional to the amount present and to the amount left to be produced.
6. The extended model was developed assuming the logistic equation fits both dry
matter and plant N removal and further that the nitrogen response coefficient, c, was
common for both. The hyperbolic relationship between dry matter yield and plant N
removal resulted from these assumptions. One could ask: What combination of these
assumptions is more fundamental? The original set of assumptions resulted from the ease

256
of regressing the two responses to a logistic with a common c. An alternative approach
would be to fit the logistic to the dry matter yield response to applied N and the hyperbolic
to the dry matter yield response to plant N removal This is a more fundamental approach,
since it involves fitting a relationship between two extensive variables: dry matter and
plant N removal. In this relationship, the effect of the independent variable, applied N, has
been removed and only the measured variables are included. This procedure would be
more difficult to compute since it would involve regressing data to two different functions
simultaneously. It is important to remember that the hyperbolic relationship is based upon
two measurable quantities, i.e. the independent variable (applied nitrogen) has been
removed from the picture From data set to data set, the same hyperbolic trend appears.
What physically in the system causes this relationship? Recall the dimensionless parameter
for the hyperbolic, Ym/A, represented by Equation [2.14] This describes the maximum
dry matter attained at high levels of plant N removal For the studies investigated in this
work, the value of this ratio ranges from 1 7 to 2.7, suggesting that approximately half of
the potential capacity of the system is being observed in the various experiments. For
Yni/A = 2, it follows from Equation [2.14] that Ab = 0.69. One possible explanation for
this lies in a paper coupling various plant components for forage production by Overman
(1995b). He has shown that the response of leaf area to C02 concentration is hyperbolic.
Assuming a current C02 concentration equal to the level in the atmosphere, 300
pmol/mol, it appears from Figure 13 in the article that for days after planting (DAP)
greater than 30, only a little more than half of the potential maximum leaf area is attained.
Although this has not directly linked dry matter production to C02 concentration,
production is directly related to leaf area. This suggests that at higher levels of
atmospheric C02 concentration would result in higher leaf areas, which would then result
in higher dry matter production

257
7. The England study (Morrison et a/., 1980) suggests that Ab and Ncm may be
constant for a particular grass. A compilation of Ab values for the various studies
examined in this work is summarized in Table 5-1. The mean, standard deviation, and
relative error of Ab is also shown in the table It appears from the table that the two
grasses grown at Fayetteville, AR have the same Ab. Also the two grasses grown at Eagle
Lake, TX have the same Ab. Corn grown at two sites with similar soils (Clayton and
Kinston, NC) has the same Ab as well. Finally, bahiagrass grown on two different soils in
Florida (Entisol and Spodosol) has the same Ab. What does this mean? Based upon the
data presented in the table, it appears that Ab may be associated with specific grasses.
Recall that the b parameter is a measure of the amount of available nitrogen in the soil
before nutrient application (response at N = 0). Although no clear connection between the
b parameters and types of grasses or measurable soil properties exists at this point, further
investigation might reveal the nature of the relationship
8. A compilation of c and N' parameter values for the various studies examined in
this work is summarized in Table 5-2. Recall that N' = 1/c. It appears from the table that
the two grasses grown at Fayetteville, AR probably have the same c. Also the two grasses
grown at Eagle Lake, TX have the same c. Corn grown at two sites with similar soils
(Clayton and Kinston, NC) has the same c as well. Bahiagrass grown on two different
soils in Florida (Entisol and Spodosol) has the same c. Finally, for the twenty sites in
England where ryegrass was grown, they all had the same c. The c values for
bermudagrass grown at different sites are not the same; however, they all lie within a
relatively small range (0.0067 0.0112). The c values compare well with those reported
in the literature for other data sets (Overman and Wilkinson, 1992, 1995). What does this
mean? It is not clear what effects are incorporated in the c parameter, but the table
suggests that this parameter might be related to type of grass. One possible clue is that c
is the same for both dry matter response to applied nitrogen and plant N removal response

258
to applied N. What in the system causes this phenomenon? At this point, it is not clear
what physiological process might explain this behavior. Additional investigation might
reveal a connection between this parameter and various effects: type of grass, water, soil
characteristics, etc. It also appears from the table that rye and corn have higher c values
than the other grasses. One possible explanation is that these grasses are annuals, while
the others are grown as perennials.
9. This analysis has focused on seasonal quantities. Although time is not
explicitly contained within the model, it is implicit within the A parameter. The maximum
parameter contains within it the accumulation of dry matter over time, as evidenced by its
relationship to harvest interval (Maryland and Tifton, GA, studies).
10. The models developed and analyzed in this study adequately describe seasonal
relationships. The focus has been to determine, by dimensionless plots, if the form of the
logistic adequately describes the behavior, not to determine parameter values. No attempt
has been made to develop a "cookbook" model for the general public, so that site specific
information, such as weather and soil properties, can be entered and automatically produce
estimates. Rather the approach as been to determine if the form is sufficient in describing
the relationship. If so, then the model can be used universally with little input.
Although this study has not developed a table of parameter values, reasonable
estimates can be obtained. Of the parameters, b is the least significant since it is the
response at N = 0. Typically for sandy soils the b estimate is high (low intercept) as a
result of low residual nitrogen in the soil. The A parameter can be estimated relatively
easily since it is an estimate of the maximum Very often in practice, high application rates
are not used since the efficiency of the system to remove N decreases as the maximum is
approached. The normal operating range is concentrated around the Nt/2 value. Recall
that this parameter is the amount of nitrogen applied to achieve half of the maximum yield.
This is also the point of maximum slope (incremental increase of yield with N). For

259
estimation purposes, the key is narrowing down the value for c. As mentioned before,
this is believed to be the most critical parameter. As with many engineering situations,
safety factors are always included. If the use of this model in a specific situation is to
estimate N removal for a field receiving municipal wastewater, it would be better to
underestima