• TABLE OF CONTENTS
HIDE
 Title Page
 Table of Contents
 Preface
 Introduction
 Empirical model of crop growth
 Phenomenological model of crop...
 Expanded model of crop growth
 Simulation of leaf area with...
 Coupling among applied, soil, and...
 Application of the expanded growth...
 Coupling of crops on the same...
 Crop response to plant populat...
 Plant nutrient uptake and...
 Coupling of biomass and plant nutrient...
 Modeling crop response to applied...
 Coupling of growth model and seasonal...
 Mathematical characteristics of...
 Summary and conclusions






Group Title: Memoir on crop growth: accumulation of biomass and mineral elements
Title: A memoir on crop growth
CITATION PAGE IMAGE ZOOMABLE PAGE TEXT
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00072283/00001
 Material Information
Title: A memoir on crop growth accumulation of biomass and mineral elements
Physical Description: 1 v. (unpaged, ca. 400 leaves) : ;
Language: English
Creator: Overman, Allen R., 1937-
Publication Date: c2006
 Subjects
Subject: Crops -- Growth   ( lcsh )
Plant biomass   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Bibliography: Includes bibliographical references.
Statement of Responsibility: Allen R. Overman.
Funding: Florida Historical Agriculture and Rural Life
 Record Information
Bibliographic ID: UF00072283
Volume ID: VID00001
Source Institution: Marston Science Library, George A. Smathers Libraries, University of Florida
Holding Location: Florida Agricultural Experiment Station, Florida Cooperative Extension Service, Florida Department of Agriculture and Consumer Services, and the Engineering and Industrial Experiment Station; Institute for Food and Agricultural Services (IFAS), University of Florida
Rights Management: All rights reserved, Board of Trustees of the University of Florida
Resource Identifier: aleph - 003592859
oclc - 70833695

Table of Contents
    Title Page
        Title Page
    Table of Contents
        Table of Contents 1
        Table of Contents 2
        Table of Contents 3
    Preface
        Preface 1
        Preface 2
    Introduction
        Page 1
        Mathematical models
            Page 1
        Theory of crop growth and nutrient uptake
            Page 1
        Crops described in this memoir
            Page 2
        References
            Page 3
    Empirical model of crop growth
        Page 4
        Background
            Page 4
        Accumulation of plant nitrogen with time
            Page 4
            Page 5
        Accumulation of plant nitrogen with time
            Page 6
        Summary
            Page 7
        References
            Page 7
        Figures
            Page 8
            Page 9
            Page 10
            Page 11
    Phenomenological model of crop growth
        Page 12
        Background
            Page 12
        Environmental driving function
            Page 12
        Intrinsic growth function
            Page 12
        Net growth function
            Page 12
            Page 13
        Dependence of seasonal yield on harvest interval
            Page 14
        Summary
            Page 15
        References
            Page 15
        Figures
            Page 16
            Page 17
    Expanded model of crop growth
        Page 18
        Background
            Page 18
        Expanded growth function
            Page 19
        Coupling of biomass and plant nitrogen for perennial grass
            Page 20
        Confirmation of the expanded growth model for perennial grass
            Page 21
            Page 22
        Confirmation of the expanded growth model for annual crop
            Page 23
            Study with corn at Florence, South Carolina
                Page 23
                Page 24
            Study with corn at Tallahassee, Florida
                Page 25
                Page 26
                Page 27
            Study with peanut at Lewiston, North Carolina
                Page 28
                Page 29
                Page 30
                Page 31
        Partitioning of biomass
            Page 32
            Study with elephantgrass at Gainesville, Florida
                Page 33
            Study with soybean at Ames, Iowa
                Page 34
                Page 35
                Page 36
            Study with soybean at Clayton, North Carolina
                Page 37
                Page 38
                Page 39
            Study with potato at Old Town, Maine
                Page 40
                Page 41
                Page 42
                Page 43
                Page 44
                Page 45
            Study with alfalfa at Guelph, Ontario, Canada
                Page 46
                Page 47
                Page 48
                Page 49
            Study with corn at Tallahassee, Florida
                Page 50
                Page 51
                Page 52
        Harvest interval and plant digestibility
            Page 53
            Study with Bermudagrass at Tifton, Georgia
                Page 54
                Page 55
            Study with perennial peanut at Gainesville, Florida
                Page 56
                Page 57
                Page 58
        Response of plant digestibility to applied nitrogen
            Page 59
            Page 60
        Summary
            Page 61
        References
            Page 62
            Page 63
        Figures
            Page 64
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    Simulation of leaf area with time
        Page 140
        Background
            Page 140
        Response of leaf area to carbon dioxide concentration
            Page 140
        Accumulation of leaf area with time
            Page 140
            Simulation with the empirical model
                Page 141
            Simulation with the expanded growth model
                Page 142
                Page 143
        Summary
            Page 144
        References
            Page 144
        Figures
            Page 145
            Page 146
            Page 147
            Page 148
            Page 149
    Coupling among applied, soil, and plant nutrients
        Page 150
        Background
            Page 150
            Page 151
        Model application
            Page 152
            Response of Bermudagrass to applied nitrogen
                Page 152
                Page 153
            Response of Bermudagrass to applied phosphorus
                Page 154
            Response of Bermudagrass to applied potassium
                Page 155
                Page 156
            Response of cotton to applied phosphorus and potassium
                Page 157
                Page 158
                Page 159
                Page 160
                Page 161
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            Response of millet to applied nitrogen, phosphorus and potassium
                Page 164
                Page 165
                Page 166
        Summary
            Page 167
        References
            Page 167
            Page 168
        Figures
            Page 169
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            Page 189
    Application of the expanded growth model to cotton growth and nutrient uptake
        Page 190
        Background
            Page 190
        Application of the expanded growth model to cotton
            Page 190
            Study with cotton at San Joaquin valley, California
                Page 190
                Page 191
                Page 192
                Page 193
            Study with cotton in northern Alabama
                Page 194
        Summary
            Page 195
        References
            Page 196
        Figures
            Page 197
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    Coupling of crops on the same soil
        Page 207
        Background
            Page 207
        Response of three crops to applied nitrogen
            Page 207
            Page 208
            Page 209
        Summary
            Page 210
        References
            Page 211
        Figures
            Page 212
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    Crop response to plant population
        Page 219
        Background
            Page 219
        Exponential model of crop response to plant population
            Page 219
        Response of corn to plant population
            Page 220
            Study with corn at Aurora, New York
                Page 220
                Page 221
                Page 222
            Study with corn in northern and southern, Wisconsin
                Page 223
                Page 224
                Page 225
            Study with corn at Deerfield, Massachusetts
                Page 226
                Page 227
                Page 228
                Page 229
            Study with corn at Gainesville, Florida
                Page 230
            Study with corn at Quincy, Florida
                Page 231
                Page 232
                Page 233
                Page 234
        Response of a broadleaf crop to plant population
            Page 235
            Study with Maryland tobacco at upper Marlboro, Maryland
                Page 235
            Study with bright leaf tobacco in North Carolina
                Page 236
        Response of cotton to plant population
            Page 237
            Page 238
        Summary
            Page 239
        References
            Page 240
        Figures
            Page 241
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    Plant nutrient uptake and recovery
        Page 264
        Background
            Page 264
        Model description
            Page 264
            Page 265
        Estimates of plant nutrient uptake and recovery
            Page 266
            Application to coastal bermudagrass
                Page 266
            Application to corn
                Page 267
                Page 268
        Summary
            Page 269
        References
            Page 269
        Figures
            Page 270
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    Coupling of biomass and plant nutrient accumulation with time for tops and roots
        Page 274
        Background
            Page 274
        Data analysis
            Page 274
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        Summary
            Page 278
        References
            Page 279
        Figures
            Page 280
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    Modeling crop response to applied nitrogen, phosphorus, and potassium
        Page 295
        Background
            Page 295
        Model description
            Page 295
        Data analysis
            Page 296
            Study with oats at Tifton, Georgia
                Page 296
                Page 297
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            Study with orchardgrass in Howard County, Maryland
                Page 300
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        Summary
            Page 306 (MULTIPLE)
        Figures
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    Coupling of growth model and seasonal response model
        Page 321
        Background
            Page 321
        Model description
            Page 321
            Page 322
        Data analysis
            Page 323
            Study with canola at Greenethorpe, New South Wales, Australia
                Page 324
                Page 325
                Page 326
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            Study with bromegrass at Lincoln, Nebraska
                Page 330
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        Summary
            Page 335
        References
            Page 335
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        Figures
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    Mathematical characteristics of the crop models
        Page 357
        Background
            Page 357
        Growth model development
            Page 358
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        Mathematical characteristics of the expanded growth model
            Page 361
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        Mathematical characteristics of the extended logistic model
            Page 365
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        Summary
            Page 368
        References
            Page 368
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        Figures
            Page 370
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    Summary and conclusions
        Page 377
        Discussions
            Page 377
            Page 378
            Page 379
        References
            Page 380
Full Text







A MEMOIR ON CROP GROWTH

Accumulation ofBiomass and Mineral Elements



Allen R. Overman





Agricultural and Biological Engineering

University of Florida


Copyright 2006 Allen R. Overman









Table of Contents


Preface

1. Introduction
1.1 Mathematical Models
1.2 Theory of Crop Growth and Nutrient Uptake
1.3 Crops Described in this Memoir
1.4 References
2. Empirical Model of Crop Growth
2.1 Background
2.2 Accumulation of Biomass with Time
2.3 Accumulation of Plant Nitrogen with Time
2.4 Summary
2.5 References
3. Phenomenological Model of Crop Growth
3.1 Background
3.2 Environmental Driving Function
3.3 Intrinsic Growth Function
3.4 Net Growth Function
3.5 Dependence of Seasonal Yield on Harvest Interval
3.6 Summary
3.7 References
4. Expanded Model of Crop Growth
4.1 Background
4.2 Expanded Growth Function
4.3 Coupling of Biomass and Plant Nitrogen for Perennial Grass
4.4 Confirmation of the Expanded Growth Model for Perennial Grass
4.5 Confirmation of the Expanded Growth Model for Annual Crop
4.5.1 Study with Corn at Florence, South Carolina
4.5.2 Study with Corn at Tallahassee, Florida
4.5.3 Study with Peanut at Lewiston, North Carolina
4.6 Partitioning of Biomass
4.6.1 Study with Elephantgrass at Gainesville, Florida
4.6.2 Study with Soybean at Ames, Iowa
4.6.3 Study with Soybean at Clayton, North Carolina
4.6.4 Study with Potato at Old Town, Maine
4.6.5 Study with Alfalfa at Guelph, Ontario, Canada
4.6.6 Study with Corn at Tallahassee, Florida
4.7 Harvest Interval and Plant Digestibility
4.7.1 Study with Bermudagrass at Tifton, Georgia
4.7.2 Study with Perennial Peanut at Gainesville, Florida
4.8 Response of Plant Digestibility to Applied Nitrogen
4.9 Summary
4.10 References









5. Simulation of Leaf Area with Time
5.1 Background
5.2 Response of Leaf Area to Carbon Dioxide Concentration
5.3 Accumulation of Leaf Area with Time
5.3.1 Simulation with the Empirical Model
5.3.2 Simulation with the Expanded Growth Model
5.4 Summary
5.5 References
6. Coupling Among Applied, Soil, and Plant Nutrients
6.1 Background
6.2 Model Description
6.3 Model Application
6.3.1 Response of Bermudagrass to Applied Nitrogen
6.3.2 Response of Bermudagrass to Applied Phosphorus
6.3.3 Response of Bermudagrass to Applied Potassium
6.3.4 Response of Cotton to Applied Phosphorus and Potassium
6.3.5 Response of Millet to Applied Nitrogen, Phosphorus and Potassium
6.4 Summary
6.5 References
7. Application of the Expanded Growth Model to Cotton Growth and Nutrient Uptake
7.1 Background
7.2 Application of the Expanded Growth Model to Cotton
7.2.1 Study with Cotton at San Joaquin Valley, California
7.2.2 Study with Cotton in Northern Alabama
7.3 Summary
7.4 References
8. Coupling of Crops on the Same Soil
8.1 Background
8.2 Response of Three Crops to Applied Nitrogen
8.3 Summary
8.4 References
9. Crop Response to Plant Population
9.1 Background
9.2 Exponential Model of Crop Response to Plant Population
9.3 Response of Corn to Plant Population
9.3.1 Study with Corn at Aurora, New York
9.3.2 Study with Corn in Northern and Southern, Wisconsin
9.3.3 Study with Corn at Deerfield, Massachusetts
9.3.4 Study with Corn at Gainesville, Florida
9.3.5 Study with Corn at Quincy, Florida
9.4 Response of a Broadleaf Crop to Plant Population
9.4.1 Study with Maryland Tobacco at Upper Marlboro, Maryland
9.4.2 Study with Bright Leaf Tobacco in North Carolina
9.5 Response of Cotton to Plant Population
9.6 Summary
9.7 References









10. Plant Nutrient Uptake and Recovery
10.1 Background
10.2 Model Description
10.3 Estimates of Plant Nutrient Uptake and Recovery
10.3.1 Application to Coastal Bermudagrass
10.3.2 Application to Corn
10.4 Summary
10.5 References
11. Coupling of Biomass and Plant Nutrient Accumulation with Time for Tops and Roots
11.1 Background
11.2 Data analysis
11.3 Summary
11.4 References
12. Modeling Crop Response to Applied Nitrogen, Phosphorus, and Potassium
12.1 Background
12.2 Model Description
12.3 Data Analysis
12.3.1 Study with Oats at Tifton, Georgia
12.3.2 Study with Orchardgrass in Howard County, Maryland
12.4 Summary
12.5 References
13. Coupling of Growth Model and Seasonal Response Model
13.1 Background
13.2 Model Description
13.3 Data Analysis
13.3.1 Study with Canola at Greenethorpe, New South Wales, Austalia
13.3.2 Study with Bromegrass at Lincloln, Nebraska
13.4 Summary
13.5 References
14. Mathematical Characteristics of the Crop Models
14.1 Background
14.2 Growth Model Development
14.3 Mathematical Characteristics of the Expanded Growth Model
14.4 Mathematical Characteristics of the Extended Logistic Model
14.5 Summary
14.6 References
15. Summary and Conclusions
15.1 Discussion
15.2 References









Preface


During my childhood on a farm in eastern North Carolina I came up with many questions:
What was the origin of the seasons? How did farmers know when to plant crops? What was the
role of fertilizer and rainfall in crop production? What was the composition of plants? What
made plants (such as corn) at different locations and from year to year look essentially the same?
My parents advised me to go to school and learn about science to find some of these answers.
And so I did, some 22 years of formal education. Beginning with chemistry (my favorite subject
in high school), followed by college courses in math, chemistry, physics, engineering, and
agriculture. All of the courses seemed logical and useful until I encountered quantum mechanics.
What minds invented this stuff? Later I was to read that the eminent theoretical physicist John A.
Wheeler found himself really up a tree with this subject. Richard Feynman remarked in his
famous lecture notes on physics that nobody understands quantum mechanics! And these are two
major contributors to the subject. Over the years I have learned to accept the mysteries of nature
and to dine frequently on "humble pie." Albert Einstein has written that one who claims to hold
the truth is destined to be laughed at by the gods. And so I have continued a life long search for
answers to my childhood questions. The search can continue for as long as the mind remains
open to curiosity and is not closed by dogma or suppressed by authority.
The benchmark for the modern environmental era might be 1962 with the publication of
Rachael Carson's book Silent Spring, which called attention to negative impacts of some
chemicals in our environment and ecosystem. I was just beginning my PhD program in
Agricultural Engineering at North Carolina State University at the time. So I decided to work
with a soil physical chemist on some fundamental processes related to chemical transport. This
work continued when he and I joined the Agronomy Department at the University of Illinois. I
have pursued this subject in greater depth during my career in Agricultural and Biological
Engineering at the University of Florida. The work has also led me into soil chemistry and soil
fertility as related to crop growth and yield.
This Memoir is an attempt to bring together my work in crop growth and nutrient uptake. A
central theme of science for at least 400 years has been unification terrestrial and celestial
mechanics, electricity and magnetism, matter and motion, space and time. In my case this centers
on accumulation of biomass by photosynthesis (carbon fixation from the atmosphere) and of
mineral elements from soil (rhizosphere). This necessarily leads to mathematical models to
describe and connect things together in a comprehensive framework. This work is full of
equations because the language of modern science is mathematics. The end goal is a
comprehensive description of the physical system. References have been included in each
chapter to provide some documentation for this work and to guide readers to further information.
During my career I have been guided by the search for five characteristics: (1) patterns in
data, (2) mathematical relationships consistent with the patterns, (3) connections among various
components of the system, (4) consistency among various studies, and (5) mathematical beauty
in the models. This process has necessarily involved many failed attempts, but has been
sufficiently successful to sustain my continued search.
I am indebted to a number of people for inspiration and guidance. These include: W.P.
Hollowell (chemistry teacher), W.S. Lamm (4-H advisor), George Blum (advisor), F.J. Hassler
(advisor), J. van Schilfgaarde (MS advisor), R.J. Miller (soil physical chemist, PhD advisor), T.P
Smith (engineer), W.G. Leseman (chemistry), L.C. Hammond (soil physics), C.C. Hortenstein
(soil chemistry), W.G. Blue (soil fertility), F.M. Rhoads (soil chemistry), R.L. Stanley (crop









science), O.C. Ruelke (crop science), F.G. Martin (statistics), E.J. Kamprath (soil chemistry),
G.W. Evers (crop science), and D.L. Robinson (crop science). I have drawn heavily from work
by W.E. Adams (soil fertility, USDA-ARS) and G.W. Burton (plant genetics, USDA-ARS).
I am particularly indebted to Stanley R. Wilkinson (soil fertility, USDA-ARS) with whom I
conducted cooperative research for more than 10 years, and to Richard V. Scholtz III (University
of Florida) who has been my cooperator beginning as a student in our program.
Finally, I thank my wife, Deanye, for many years of support and patience while I struggled
with ideas and data analysis. Our children have come to appreciate her loving support and
encouragement.

Allen R. Overman
July, 2006









1. Introduction


1.1 Mathematical Models

Mathematical modeling covers a vast domain in the field of science and engineering. Models
include physical, chemical, and biological systems. It is common practice in engineering to
construct scale models of larger objects such as buildings, planes, ships, and other devices. In the
medical field animals may be used to evaluate chemicals and surgical procedures. Mathematical
models include both geometric and arithmetic. The focus here is on arithmetic models.
Arithmetic models include everything from regression models to ones derived from fundamental
principles. All of the models used in this work will be empirical in the sense that they arise out of
measurements and observations rather than from abstract postulates.
There is much controversy over the definition and application of mathematical models. The
interested reader is referred to articles in the inaugural issue of the journal Mathematical
Modeling (Aris and Penn, 1980; Spanier, 1980) for some thoughts on the subject. In science new
ideas are sometimes greeted with intense hostility, as has occurred in the field of chemistry
(Jaffe, 1976). I consider myself a modeler in the sense of the eminent physical chemist Henry
Eyring (Eyring, 1977; Eyring and Caldwell, 1980). Models are developed to describe and explain
observations and measurements, to couple various phenomena, and to make predictions. While
no model has proven totally complete, high value is placed on consistency among various data
sets at different times, at different places, and by different observers (Pagels, 1982).
This memoir represents efforts over a period of nearly 50 years, from graduate student days
until the present. The flavor of my approach has been described in the book by Overman and
Scholtz (2002). The common theme has been unification, or what Chandrasekhar (1987) has
termed systemization. Some of the models described here are based on reductionism, while
others are of the more holistic type (Casti, 1997; Laughlin, 2006). Solutions to differential
equations involve the analytical type.

1.2 Theory of Crop Growth and Nutrient Uptake

Evolution of a scientific theory can be described by the sequence: idea -* observation --
data -* information analysis --+ theory -- knowledge -- understanding. The sequence is
written as if it moves forward in a simple, logical manner. In practice, there are usually many
steps forward and backward as experience provides new insights. I have listed the starting point
as an 'idea', hypothesis, or guess. Of course there are always many antecedent factors leading up
to the idea. Then a set of observations are made, frequently from a controlled experiment. This
produces a set of 'data' for analysis. Included in this process is 'information' about the details
including relevant factors or conditions which may influence the outcome. 'Analysis' often
constitutes the most intensive step in the process. Included are such things as graphing the data,
adopting a mathematical model, performing statistical analysis, comparing results to other work,
revising the model, etc. This involves operating between bias and a blank mind. There is always
a risk of being wrong a risk which is essential to progress. Formulating a 'theory' is a bold step
and should follow extensive analysis of numerous cases. In my experience a theory is always
tentative when viewed in the larger span of time, and represents the best insight available at the
time. It is subject to revision down the road. Out of this process arises a body of 'knowledge'









1. Introduction


1.1 Mathematical Models

Mathematical modeling covers a vast domain in the field of science and engineering. Models
include physical, chemical, and biological systems. It is common practice in engineering to
construct scale models of larger objects such as buildings, planes, ships, and other devices. In the
medical field animals may be used to evaluate chemicals and surgical procedures. Mathematical
models include both geometric and arithmetic. The focus here is on arithmetic models.
Arithmetic models include everything from regression models to ones derived from fundamental
principles. All of the models used in this work will be empirical in the sense that they arise out of
measurements and observations rather than from abstract postulates.
There is much controversy over the definition and application of mathematical models. The
interested reader is referred to articles in the inaugural issue of the journal Mathematical
Modeling (Aris and Penn, 1980; Spanier, 1980) for some thoughts on the subject. In science new
ideas are sometimes greeted with intense hostility, as has occurred in the field of chemistry
(Jaffe, 1976). I consider myself a modeler in the sense of the eminent physical chemist Henry
Eyring (Eyring, 1977; Eyring and Caldwell, 1980). Models are developed to describe and explain
observations and measurements, to couple various phenomena, and to make predictions. While
no model has proven totally complete, high value is placed on consistency among various data
sets at different times, at different places, and by different observers (Pagels, 1982).
This memoir represents efforts over a period of nearly 50 years, from graduate student days
until the present. The flavor of my approach has been described in the book by Overman and
Scholtz (2002). The common theme has been unification, or what Chandrasekhar (1987) has
termed systemization. Some of the models described here are based on reductionism, while
others are of the more holistic type (Casti, 1997; Laughlin, 2006). Solutions to differential
equations involve the analytical type.

1.2 Theory of Crop Growth and Nutrient Uptake

Evolution of a scientific theory can be described by the sequence: idea -* observation --
data -* information analysis --+ theory -- knowledge -- understanding. The sequence is
written as if it moves forward in a simple, logical manner. In practice, there are usually many
steps forward and backward as experience provides new insights. I have listed the starting point
as an 'idea', hypothesis, or guess. Of course there are always many antecedent factors leading up
to the idea. Then a set of observations are made, frequently from a controlled experiment. This
produces a set of 'data' for analysis. Included in this process is 'information' about the details
including relevant factors or conditions which may influence the outcome. 'Analysis' often
constitutes the most intensive step in the process. Included are such things as graphing the data,
adopting a mathematical model, performing statistical analysis, comparing results to other work,
revising the model, etc. This involves operating between bias and a blank mind. There is always
a risk of being wrong a risk which is essential to progress. Formulating a 'theory' is a bold step
and should follow extensive analysis of numerous cases. In my experience a theory is always
tentative when viewed in the larger span of time, and represents the best insight available at the
time. It is subject to revision down the road. Out of this process arises a body of 'knowledge'









1. Introduction


1.1 Mathematical Models

Mathematical modeling covers a vast domain in the field of science and engineering. Models
include physical, chemical, and biological systems. It is common practice in engineering to
construct scale models of larger objects such as buildings, planes, ships, and other devices. In the
medical field animals may be used to evaluate chemicals and surgical procedures. Mathematical
models include both geometric and arithmetic. The focus here is on arithmetic models.
Arithmetic models include everything from regression models to ones derived from fundamental
principles. All of the models used in this work will be empirical in the sense that they arise out of
measurements and observations rather than from abstract postulates.
There is much controversy over the definition and application of mathematical models. The
interested reader is referred to articles in the inaugural issue of the journal Mathematical
Modeling (Aris and Penn, 1980; Spanier, 1980) for some thoughts on the subject. In science new
ideas are sometimes greeted with intense hostility, as has occurred in the field of chemistry
(Jaffe, 1976). I consider myself a modeler in the sense of the eminent physical chemist Henry
Eyring (Eyring, 1977; Eyring and Caldwell, 1980). Models are developed to describe and explain
observations and measurements, to couple various phenomena, and to make predictions. While
no model has proven totally complete, high value is placed on consistency among various data
sets at different times, at different places, and by different observers (Pagels, 1982).
This memoir represents efforts over a period of nearly 50 years, from graduate student days
until the present. The flavor of my approach has been described in the book by Overman and
Scholtz (2002). The common theme has been unification, or what Chandrasekhar (1987) has
termed systemization. Some of the models described here are based on reductionism, while
others are of the more holistic type (Casti, 1997; Laughlin, 2006). Solutions to differential
equations involve the analytical type.

1.2 Theory of Crop Growth and Nutrient Uptake

Evolution of a scientific theory can be described by the sequence: idea -* observation --
data -* information analysis --+ theory -- knowledge -- understanding. The sequence is
written as if it moves forward in a simple, logical manner. In practice, there are usually many
steps forward and backward as experience provides new insights. I have listed the starting point
as an 'idea', hypothesis, or guess. Of course there are always many antecedent factors leading up
to the idea. Then a set of observations are made, frequently from a controlled experiment. This
produces a set of 'data' for analysis. Included in this process is 'information' about the details
including relevant factors or conditions which may influence the outcome. 'Analysis' often
constitutes the most intensive step in the process. Included are such things as graphing the data,
adopting a mathematical model, performing statistical analysis, comparing results to other work,
revising the model, etc. This involves operating between bias and a blank mind. There is always
a risk of being wrong a risk which is essential to progress. Formulating a 'theory' is a bold step
and should follow extensive analysis of numerous cases. In my experience a theory is always
tentative when viewed in the larger span of time, and represents the best insight available at the
time. It is subject to revision down the road. Out of this process arises a body of 'knowledge'









which allows further speculation, estimates, and predictions. Finally, I have listed
'understanding', our best efforts to understand how Nature really works.
This work is part of the triology of publications:

Mathematical Models of Crop Growth and Yield
A. R. Overman and R. V. Scholtz III (2002)
A Memoir on Chemical Transport: Applications to Soils and Crops
A. R. Overman (2006)
A Memoir on Crop Growth: Accumulation ofBiomass and Mineral Elements
A. R. Overman (2006)

This memoir contains 757 equations, 84 tables, and 184 figures. Actually, there are only a hand
full of basic equations, with the large number representing applications to various crops, soils,
and experimental conditions. Data are provided in the tables to provide documentation and for
use by the curious reader who may wish to either verify results or try an alternative approach.
Key results are shown in the figures, where trends and scatter of the data become more apparent.
I have followed the practice of providing data in both tabular and graphical format (a practice
generally not allowed in journals) to make the analysis more transparent. Each chapter is written
to be somewhat self-contained.
I do feel comfortable to call this work a 'theory', since it has now been shown to be
transferable in space (location), time (year-to-year and over long periods), and over a wide range
of environmental conditions. It constitutes a large body of knowledge which provides some
understanding of how processes in soils and plants couple together to affect crop production. It is
offered as a contribution to the intellectual enterprise of curiosity and inquiry.

1.3 Crops Described in this Memoir


Common Name
Alfalfa
Bahiagrass
Bermudagrass
Bromegrass
Canola
Corn
Cotton
Elephantgrass
Oats
Orchardgrass
Peanut
Pearlmillet
Perennial Peanut
Potato
Soybean
Squash
Tobacco
Tomato


Scientific Name
Medicago sativa L.
Paspalum notatum Flilgge
Cynodon dactylon (L.) Pers.
Bromus inermis Leyss
Brassica napus L.
Zea mays L.
Gossypium hirsutum L.
Pennisetum purpureum Schumach.
Avena L.
Dactylis glomerata L.
Arachis hypogaea L.
Pennisetum americanum (L.) Leeke
Arachis globrata Benth.
Solanum tuberosum L.
Glycine max (L.) Merr.
Cucurbita piop L.
Nicotiana tabaccum L.
Lycopersicon esculentum Mill










1.4 References

Aris, R. and M. Penn. 1980. The mere notion of a model. Mathematical Modeling 1:1-12.
Casti, J.L. 1997. Would-be Worlds: How Simulation is Changing the Frontiers of Science. John
Wiley & Sons. New York, NY.
Chandrasekhar, S. 1987. Truth and Beauty: Aesthetics and Motivations in Science. University of
Chicago Press. Chicago, IL.
Eyring, H. 1977. Men, mines, and molecules. Annual Reviews of Physical Chemistry 28:1-13.
Eyring, H. and D. Caldwell. 1980. Inductive and deductive science. Mathematical Modeling
1:33-40.
Jaffe, B. 1976. Crucibles: The Story of Chemistry. Dover Publications. New York, NY.
Laughlin, R.B. 2005. A Different Universe: Reinventing Physics from the Bottom Down. Basic
Books. New York, NY.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. Philadelphia, PA.
Pagels, H.R. 1982. The Cosmic Code: Quatum Physics as the Language of Nature. Simon &
Schuster. New York, NY.
Spanier, J. Thoughts about the essentials of mathematical modeling. Mathematical Modeling
1:99-108.









2. Empirical Model of Crop Growth


2.1 Background

This chapter will focus on a simple mathematical model of biomass accumulation with time
for a warm-season perennial grass. Results from this simple approach offer a clue toward
development of a more sophisticated model, which is pursued in greater detail in chapters 3 and
4. As a model becomes more comprehensive, mathematical complexity tends to increase.

2.2 Accumulation of Biomass with Time

Overman (1984) used the probability equation to describe accumulation of biomass with time
for a perennial grass harvested on a fixed time interval


Y = A + erf t (2.1)
2 -

where Y is accumulated biomass, Mg ha-1; t is calendar time since Jan. 1, wk; A is maximum
accumulated biomass, Mg ha-1, p is time to the mean of the biomass distribution, wk; a is time
spread of the biomass distribution, wk. The error function in Eq. (2.1) is defined by

2x
erf x = exp(-u2)du (2.2)


where u is simply the variable of integration. The error function can be evaluated from
mathematical tables (Abramowitz and Stegun, 1965). It should be noted that erf (0) = 0, erf (oo) =
1, and that erf (-x) = -erf (+x). Equation (2.1) can be written in normalized form


F =- =1[l+erf(tP (2.3)
A 2 -2o-aJ

so that normalized yield, Fy, is bounded by 0 < Fy < 1. A plot of Fy vs. t on probability paper
produces a straight line. It should be noted that p serves as a reference time for the system.
Equation (2.1) is simply an empirical model chosen to describe a pattern similar to that of
data, and not derived from any basic principles. The model contains three parameters (A, p, a) to
be evaluated for a given set of data. Parameters can be evaluated by either of three methods: (1)
graphical, (2) regression of the linearized form, or (3) nonlinear regression. The graphical
method is carried out by plotting Y vs. t on linear graph paper, from which a visual estimate of
parameter A (the upper plateau) is made. Values of Fy are then calculated for each time by
dividing values of Yby A. A graph of Fy vs. t is then plotted on probability paper. A straight line
is constructed by visual inspection of the probability plot and used to estimate times for Fy =
0.50, Fy = 0.16, and Fy = 0.84. Parameter p is then estimated as p = t(F = 0.50). Parameter a is
estimated from a = [t(Fy = 0.84) t(Fy = 0.16)]/2. The curve of Y vs. t can then be constructed









2. Empirical Model of Crop Growth


2.1 Background

This chapter will focus on a simple mathematical model of biomass accumulation with time
for a warm-season perennial grass. Results from this simple approach offer a clue toward
development of a more sophisticated model, which is pursued in greater detail in chapters 3 and
4. As a model becomes more comprehensive, mathematical complexity tends to increase.

2.2 Accumulation of Biomass with Time

Overman (1984) used the probability equation to describe accumulation of biomass with time
for a perennial grass harvested on a fixed time interval


Y = A + erf t (2.1)
2 -

where Y is accumulated biomass, Mg ha-1; t is calendar time since Jan. 1, wk; A is maximum
accumulated biomass, Mg ha-1, p is time to the mean of the biomass distribution, wk; a is time
spread of the biomass distribution, wk. The error function in Eq. (2.1) is defined by

2x
erf x = exp(-u2)du (2.2)


where u is simply the variable of integration. The error function can be evaluated from
mathematical tables (Abramowitz and Stegun, 1965). It should be noted that erf (0) = 0, erf (oo) =
1, and that erf (-x) = -erf (+x). Equation (2.1) can be written in normalized form


F =- =1[l+erf(tP (2.3)
A 2 -2o-aJ

so that normalized yield, Fy, is bounded by 0 < Fy < 1. A plot of Fy vs. t on probability paper
produces a straight line. It should be noted that p serves as a reference time for the system.
Equation (2.1) is simply an empirical model chosen to describe a pattern similar to that of
data, and not derived from any basic principles. The model contains three parameters (A, p, a) to
be evaluated for a given set of data. Parameters can be evaluated by either of three methods: (1)
graphical, (2) regression of the linearized form, or (3) nonlinear regression. The graphical
method is carried out by plotting Y vs. t on linear graph paper, from which a visual estimate of
parameter A (the upper plateau) is made. Values of Fy are then calculated for each time by
dividing values of Yby A. A graph of Fy vs. t is then plotted on probability paper. A straight line
is constructed by visual inspection of the probability plot and used to estimate times for Fy =
0.50, Fy = 0.16, and Fy = 0.84. Parameter p is then estimated as p = t(F = 0.50). Parameter a is
estimated from a = [t(Fy = 0.84) t(Fy = 0.16)]/2. The curve of Y vs. t can then be constructed









2. Empirical Model of Crop Growth


2.1 Background

This chapter will focus on a simple mathematical model of biomass accumulation with time
for a warm-season perennial grass. Results from this simple approach offer a clue toward
development of a more sophisticated model, which is pursued in greater detail in chapters 3 and
4. As a model becomes more comprehensive, mathematical complexity tends to increase.

2.2 Accumulation of Biomass with Time

Overman (1984) used the probability equation to describe accumulation of biomass with time
for a perennial grass harvested on a fixed time interval


Y = A + erf t (2.1)
2 -

where Y is accumulated biomass, Mg ha-1; t is calendar time since Jan. 1, wk; A is maximum
accumulated biomass, Mg ha-1, p is time to the mean of the biomass distribution, wk; a is time
spread of the biomass distribution, wk. The error function in Eq. (2.1) is defined by

2x
erf x = exp(-u2)du (2.2)


where u is simply the variable of integration. The error function can be evaluated from
mathematical tables (Abramowitz and Stegun, 1965). It should be noted that erf (0) = 0, erf (oo) =
1, and that erf (-x) = -erf (+x). Equation (2.1) can be written in normalized form


F =- =1[l+erf(tP (2.3)
A 2 -2o-aJ

so that normalized yield, Fy, is bounded by 0 < Fy < 1. A plot of Fy vs. t on probability paper
produces a straight line. It should be noted that p serves as a reference time for the system.
Equation (2.1) is simply an empirical model chosen to describe a pattern similar to that of
data, and not derived from any basic principles. The model contains three parameters (A, p, a) to
be evaluated for a given set of data. Parameters can be evaluated by either of three methods: (1)
graphical, (2) regression of the linearized form, or (3) nonlinear regression. The graphical
method is carried out by plotting Y vs. t on linear graph paper, from which a visual estimate of
parameter A (the upper plateau) is made. Values of Fy are then calculated for each time by
dividing values of Yby A. A graph of Fy vs. t is then plotted on probability paper. A straight line
is constructed by visual inspection of the probability plot and used to estimate times for Fy =
0.50, Fy = 0.16, and Fy = 0.84. Parameter p is then estimated as p = t(F = 0.50). Parameter a is
estimated from a = [t(Fy = 0.84) t(Fy = 0.16)]/2. The curve of Y vs. t can then be constructed









on linear graph paper from parameter estimates by obtaining estimates of erf from mathematical
tables. This should produce a sigmoid curve.
The second method of parameter estimation is now outlined. Again the linear graph of Y vs. t
is constructed, from which parameter A is estimated by visual inspection. Values of Fy are then
calculated for each time. At this point it may be noted that Eq. (2.3) can be rearranged and
written in the linearized form


Z = erf'(2F -1)= 1 t (2.4)


where erf -' designates the inverse error function, which is also read from mathematical tables for
the error function. Linear regression is then performed on Zy vs. t to obtain the slope and
intercept, and hence values for p and a. This method is generally superior to the graphical
method.
The method of nonlinear regression has been outlined by Overman and Scholtz (2002). It is
the most objective approach, but also the most tedious of the three. Method 2 represents a
balanced approach which is generally acceptable. All of these methods assume the data exhibit
the sigmoid distribution characterized by the probability equation.
Application of the empirical model is now illustrated with field data for bahiagrass
(Paspalum notatum Fltigge) on a sandy soil at Gainesville, Florida (Leukel et al., 1934). Data for
applied nitrogen of N = 295 kg ha'1 and with irrigation are given in Table 2.1. Accumulation of

Table 2.1 Accumulation of biomass (Y) and plant nitrogen (N,) with calendar time (t) for
bahiagrass on sandy soil with irrigation and N = 295 kg ha'1 at Gainesville, FL.a
t Y Fy Zy Nu Fn Zn
wk Mg ha-1 kg ha'-1
--- 0.000 0.000 --- 0.00 0.000 ---
20.3 0.061 0.029 -1.340 0.90 0.024 -1.398
22.3 0.126 0.060 -1.100 1.95 0.052 -1.150
24.4 0.437 0.208 -0.578 9.13 0.244 -0.888
27.3 0.851 0.405 -0.170 15.91 0.426 -0.138
29.4 1.045 0.498 -0.004 18.95 0.507 0.013
31.4 1.272 0.606 0.190 22.93 0.613 0.203
34.3 1.446 0.689 0.349 26.16 0.699 0.369
36.7 1.777 0.846 0.722 31.32 0.838 0.696
39.6 1.962 0.934 1.065 34.82 0.932 1.055
43.7 2.100 1.000 ------- 37.38 1.000 -------
aData for biomass and plant nitrogen accumulation adapted from Leukel et al. (1934, table 1).

biomass and plant nitrogen with calendar time are shown in Figure 2.1. For this case normalized
yield and plant N are calculated from Fy = Y/2.100 and Fn = Nu/37.38. Values of Zy are
calculated from Eq. (2.4), which leads to the regression equation


Z= erf(2F-1)=- -- +--t =-3.625+0.1195t r = 0.987 (2.5)
-,.12a 2i










with a correlation coefficient of r = 0.987. It follows from Eq. (2.5) that Vi2o = 8.36 wk and p =
30.3 wk. The plot of Zy vs. t is shown in Figure 2.2, where the line is drawn from Eq. (2.5). The
estimation equation for biomass accumulation is given by


Y = [1 + erf 2.0 [1 + erf (t3-.3
2 2o-J 2 8.36

The curve for Y vs. t in Figure 2.1 is drawn from Eq. (2.6).
Estimation of Y for various t is illustrated in Table 2.2, where variable x is defined by

t-p, _t-30.3
x =72c, 8.36


(2.6)






(2.7)


Table 2.2 Estimation of biomass (Y) with calendar time (t) for the data of Leukel et al. (1934).
t x erfx Y


-1.471
-1.232
-0.993
-0.754
-0.514
-0.275
-0.036
0.203
0.443
0.682
0.921
1.160
1.400
1.639
1.878


-0.962
-0.918
-0.840
-0.714
-0.533
-0.302
-0.041
0.226
0.469
0.665
0.807
0.899
0.952
0.980
0.9920


Mg ha1
0.04
0.09
0.17
0.30
0.49
0.73
1.01
1.29
1.54
1.75
1.90
1.99
2.05
2.08
2.09


Values of the error function erf x are obtained from mathematical tables (Abramowitz and
Stegun, 1965). It should be noted that the curve in Figure 2.1 does not represent biomass at any
time t, but rather represents cumulative yield for other harvest times following the same harvest
interval, which for this case averaged 2.60 wk. The effect of harvest interval on yield is
discussed in subsequent sections.

2.3 Accumulation of Plant Nitrogen with Time

Description of plant nitrogen accumulation with time is the next question to be considered. A
plot of normalized plant nitrogen (Fn) vs. normalized biomass (Fy) is shown in Figure 2.3. Linear
regression leads to the correlation equation


.1 --


,v









F, = 0.010 + 0.9928Fy r = 0.9990


with a very high correlation coefficient of r = 0.9990. The line in Figure 2.3 is drawn from Eq.
(2.8). This result suggests that plant N concentration at each harvest is approximately the same
(for constant harvest interval). In fact, average plant N concentration is Nc = 37.38/2.100 = 17.8 g
kg-' over the entire season. The curves in Figure 2.1 is drawn from Eqs. (2.6) and (2.8).
These results suggest that accumulation of plant N uptake can also be described with the
probability model. Linear regression of Zn vs. t from Table 2.1 gives


Z = erf -(2F -1)=- + t = -3.857 +0.1260t r= 0.985 (2.9)


which leads to -2cr = 7.94 wk and p = 30.6 wk, which are very similar to those for yields.

2.4 Summary

Biomass accumulation by a warm-season perennial grass harvested on a fixed time interval
appears to be described rather well by the simple probability equation. Plant N and biomass
accumulation appear to be related through a simple linear relationship with plant N concentration
remaining constant for each harvest over the season. Overman and Scholtz (2002) demonstrated
that water availability and harvest interval are accounted for in the linear parameter A. It appears
that the effect of applied nutrients occurs in this parameter as well. It will be shown later that this
simple model does not describe response of annual crops, where plant nutrient concentration is
decreasing as the plant ages.
The success of the probability model suggests the presence of a Gaussian component in the
growth process. This insight will prove useful in developing more detailed models.

2.5 References

Abramowitz, M. and I.A. Stegun. 1965. Handbook of Mathematical Functions. Dover
Publications. New York, NY.
Leukel, W.A., J.P. Camp, and J.M. Coleman. 1934. Effect of frequent cutting and nitrate
fertilization on the growth behavior and relative composition of pasture grasses. Florida
Agricultural Experiment Station Bulletin 269. University of Florida. Gainesville, FL.
Overman, A.R. 1984. Estimating crop growth rate with land treatment. J. Environmental
Engineering Division, ASCE 110:1009-1012.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. Philadelphia, PA.


(2.8)









F, = 0.010 + 0.9928Fy r = 0.9990


with a very high correlation coefficient of r = 0.9990. The line in Figure 2.3 is drawn from Eq.
(2.8). This result suggests that plant N concentration at each harvest is approximately the same
(for constant harvest interval). In fact, average plant N concentration is Nc = 37.38/2.100 = 17.8 g
kg-' over the entire season. The curves in Figure 2.1 is drawn from Eqs. (2.6) and (2.8).
These results suggest that accumulation of plant N uptake can also be described with the
probability model. Linear regression of Zn vs. t from Table 2.1 gives


Z = erf -(2F -1)=- + t = -3.857 +0.1260t r= 0.985 (2.9)


which leads to -2cr = 7.94 wk and p = 30.6 wk, which are very similar to those for yields.

2.4 Summary

Biomass accumulation by a warm-season perennial grass harvested on a fixed time interval
appears to be described rather well by the simple probability equation. Plant N and biomass
accumulation appear to be related through a simple linear relationship with plant N concentration
remaining constant for each harvest over the season. Overman and Scholtz (2002) demonstrated
that water availability and harvest interval are accounted for in the linear parameter A. It appears
that the effect of applied nutrients occurs in this parameter as well. It will be shown later that this
simple model does not describe response of annual crops, where plant nutrient concentration is
decreasing as the plant ages.
The success of the probability model suggests the presence of a Gaussian component in the
growth process. This insight will prove useful in developing more detailed models.

2.5 References

Abramowitz, M. and I.A. Stegun. 1965. Handbook of Mathematical Functions. Dover
Publications. New York, NY.
Leukel, W.A., J.P. Camp, and J.M. Coleman. 1934. Effect of frequent cutting and nitrate
fertilization on the growth behavior and relative composition of pasture grasses. Florida
Agricultural Experiment Station Bulletin 269. University of Florida. Gainesville, FL.
Overman, A.R. 1984. Estimating crop growth rate with land treatment. J. Environmental
Engineering Division, ASCE 110:1009-1012.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. Philadelphia, PA.


(2.8)









List of Figures

Figure 2.1 Accumulation of biomass yield (Y) and plant N uptake (N,) with calendar time (t) for
bahiagrass (with irrigation and N = 295 kg ha-') at Gainesville, FL. Data adapted from Leukel et
al. (1934). Curves drawn from Eqs. (2.6) and (2.8).

Figure 2.2 Dependence of reduced yield (Zy) on calendar time (t) for bahiagrass (with irrigation
and N = 295 kg ha-') at Gainesville, FL. Data adapted from Leukel et al. (1934). Line drawn
from Eq. (2.5).

Figure 2.3 Correlation of fractional plant N uptake (Fn) with fractional yield (Fy) for bahiagrass
(with irrigation and N = 295 kg ha-') at Gainesville, FL. Data adapted from Leukel et al. (1934).
Line drawn from Eq. (2.8).






























>2


/0 ~Lc~ I~Q
twA.
































/ / I 1
4/0


tv~


Fia


I----- T -7--' /--1 /


t, tWA



































/ //







/

0<


/

0~ ~ //
I




9 OtL /,cJ









3. Phenomenological Model of Crop Growth


3.1 Background

In Chapter 2 we dealt with the simple probability model for accumulation of biomass with
calendar time by a perennial grass harvested on a fixed time interval. While the model appeared
to describe data rather well, it gave no insight into processes which govern plant growth with
time. The model represented an effort to begin at an elementary level of analysis. This is a
common approach in mathematical modeling of physical systems. Now a more comprehensive
model is developed to describe the phenomenon of crop growth.

3.2 Environmental Driving Function

It was noted in Chapter 2 that the empirical model involved the Gaussian function. The
success of this model suggests that a Gaussian component could be incorporated into a more
detailed model. Since biomass accumulation is the result of photosynthesis driven by solar
energy, it seems quite logical to assume that the energy driving function, E, can be described by a
Gaussian distribution of the form


E = constant, exp t U1 (3.1)


where t is time since Jan. 1, wk; / is time to the mean of the energy distribution, wk; a is the time
spread of the energy distribution, wk. Equation (3.1) is proposed as a reasonable approximation
of the energy distribution. The exact distribution varies from day to day, depending on cloud
cover and other variables. At this point these variations are treated as noise in the general trend.

3.3 Intrinsic Growth Function

The next step is to incorporate a function which describes the essential characteristics of the
plant, which is labeled the intrinsic growth function. While there are numerous possibilities for
this function, we choose a simple linear form (Overman et al., 1989)

dY'
a + b(t t,) (3.2)
dt

where dY'/dt is intrinsic growth rate under constant environmental conditions, Mg ha-1 wk-'; t, is
reference time, wk; a is the intercept parameter, Mg ha-1 wk-'; b is the slope parameter, Mg ha-1
wk-2. Note that ti constitutes a second reference time.

3.4 Net Growth Function

Equations (3.1) and (3.2) must be combined to write a differential equation for the overall
growth rate, dY/dt. The simplest approach is to assume a product form such that









3. Phenomenological Model of Crop Growth


3.1 Background

In Chapter 2 we dealt with the simple probability model for accumulation of biomass with
calendar time by a perennial grass harvested on a fixed time interval. While the model appeared
to describe data rather well, it gave no insight into processes which govern plant growth with
time. The model represented an effort to begin at an elementary level of analysis. This is a
common approach in mathematical modeling of physical systems. Now a more comprehensive
model is developed to describe the phenomenon of crop growth.

3.2 Environmental Driving Function

It was noted in Chapter 2 that the empirical model involved the Gaussian function. The
success of this model suggests that a Gaussian component could be incorporated into a more
detailed model. Since biomass accumulation is the result of photosynthesis driven by solar
energy, it seems quite logical to assume that the energy driving function, E, can be described by a
Gaussian distribution of the form


E = constant, exp t U1 (3.1)


where t is time since Jan. 1, wk; / is time to the mean of the energy distribution, wk; a is the time
spread of the energy distribution, wk. Equation (3.1) is proposed as a reasonable approximation
of the energy distribution. The exact distribution varies from day to day, depending on cloud
cover and other variables. At this point these variations are treated as noise in the general trend.

3.3 Intrinsic Growth Function

The next step is to incorporate a function which describes the essential characteristics of the
plant, which is labeled the intrinsic growth function. While there are numerous possibilities for
this function, we choose a simple linear form (Overman et al., 1989)

dY'
a + b(t t,) (3.2)
dt

where dY'/dt is intrinsic growth rate under constant environmental conditions, Mg ha-1 wk-'; t, is
reference time, wk; a is the intercept parameter, Mg ha-1 wk-'; b is the slope parameter, Mg ha-1
wk-2. Note that ti constitutes a second reference time.

3.4 Net Growth Function

Equations (3.1) and (3.2) must be combined to write a differential equation for the overall
growth rate, dY/dt. The simplest approach is to assume a product form such that









3. Phenomenological Model of Crop Growth


3.1 Background

In Chapter 2 we dealt with the simple probability model for accumulation of biomass with
calendar time by a perennial grass harvested on a fixed time interval. While the model appeared
to describe data rather well, it gave no insight into processes which govern plant growth with
time. The model represented an effort to begin at an elementary level of analysis. This is a
common approach in mathematical modeling of physical systems. Now a more comprehensive
model is developed to describe the phenomenon of crop growth.

3.2 Environmental Driving Function

It was noted in Chapter 2 that the empirical model involved the Gaussian function. The
success of this model suggests that a Gaussian component could be incorporated into a more
detailed model. Since biomass accumulation is the result of photosynthesis driven by solar
energy, it seems quite logical to assume that the energy driving function, E, can be described by a
Gaussian distribution of the form


E = constant, exp t U1 (3.1)


where t is time since Jan. 1, wk; / is time to the mean of the energy distribution, wk; a is the time
spread of the energy distribution, wk. Equation (3.1) is proposed as a reasonable approximation
of the energy distribution. The exact distribution varies from day to day, depending on cloud
cover and other variables. At this point these variations are treated as noise in the general trend.

3.3 Intrinsic Growth Function

The next step is to incorporate a function which describes the essential characteristics of the
plant, which is labeled the intrinsic growth function. While there are numerous possibilities for
this function, we choose a simple linear form (Overman et al., 1989)

dY'
a + b(t t,) (3.2)
dt

where dY'/dt is intrinsic growth rate under constant environmental conditions, Mg ha-1 wk-'; t, is
reference time, wk; a is the intercept parameter, Mg ha-1 wk-'; b is the slope parameter, Mg ha-1
wk-2. Note that ti constitutes a second reference time.

3.4 Net Growth Function

Equations (3.1) and (3.2) must be combined to write a differential equation for the overall
growth rate, dY/dt. The simplest approach is to assume a product form such that









3. Phenomenological Model of Crop Growth


3.1 Background

In Chapter 2 we dealt with the simple probability model for accumulation of biomass with
calendar time by a perennial grass harvested on a fixed time interval. While the model appeared
to describe data rather well, it gave no insight into processes which govern plant growth with
time. The model represented an effort to begin at an elementary level of analysis. This is a
common approach in mathematical modeling of physical systems. Now a more comprehensive
model is developed to describe the phenomenon of crop growth.

3.2 Environmental Driving Function

It was noted in Chapter 2 that the empirical model involved the Gaussian function. The
success of this model suggests that a Gaussian component could be incorporated into a more
detailed model. Since biomass accumulation is the result of photosynthesis driven by solar
energy, it seems quite logical to assume that the energy driving function, E, can be described by a
Gaussian distribution of the form


E = constant, exp t U1 (3.1)


where t is time since Jan. 1, wk; / is time to the mean of the energy distribution, wk; a is the time
spread of the energy distribution, wk. Equation (3.1) is proposed as a reasonable approximation
of the energy distribution. The exact distribution varies from day to day, depending on cloud
cover and other variables. At this point these variations are treated as noise in the general trend.

3.3 Intrinsic Growth Function

The next step is to incorporate a function which describes the essential characteristics of the
plant, which is labeled the intrinsic growth function. While there are numerous possibilities for
this function, we choose a simple linear form (Overman et al., 1989)

dY'
a + b(t t,) (3.2)
dt

where dY'/dt is intrinsic growth rate under constant environmental conditions, Mg ha-1 wk-'; t, is
reference time, wk; a is the intercept parameter, Mg ha-1 wk-'; b is the slope parameter, Mg ha-1
wk-2. Note that ti constitutes a second reference time.

3.4 Net Growth Function

Equations (3.1) and (3.2) must be combined to write a differential equation for the overall
growth rate, dY/dt. The simplest approach is to assume a product form such that









3. Phenomenological Model of Crop Growth


3.1 Background

In Chapter 2 we dealt with the simple probability model for accumulation of biomass with
calendar time by a perennial grass harvested on a fixed time interval. While the model appeared
to describe data rather well, it gave no insight into processes which govern plant growth with
time. The model represented an effort to begin at an elementary level of analysis. This is a
common approach in mathematical modeling of physical systems. Now a more comprehensive
model is developed to describe the phenomenon of crop growth.

3.2 Environmental Driving Function

It was noted in Chapter 2 that the empirical model involved the Gaussian function. The
success of this model suggests that a Gaussian component could be incorporated into a more
detailed model. Since biomass accumulation is the result of photosynthesis driven by solar
energy, it seems quite logical to assume that the energy driving function, E, can be described by a
Gaussian distribution of the form


E = constant, exp t U1 (3.1)


where t is time since Jan. 1, wk; / is time to the mean of the energy distribution, wk; a is the time
spread of the energy distribution, wk. Equation (3.1) is proposed as a reasonable approximation
of the energy distribution. The exact distribution varies from day to day, depending on cloud
cover and other variables. At this point these variations are treated as noise in the general trend.

3.3 Intrinsic Growth Function

The next step is to incorporate a function which describes the essential characteristics of the
plant, which is labeled the intrinsic growth function. While there are numerous possibilities for
this function, we choose a simple linear form (Overman et al., 1989)

dY'
a + b(t t,) (3.2)
dt

where dY'/dt is intrinsic growth rate under constant environmental conditions, Mg ha-1 wk-'; t, is
reference time, wk; a is the intercept parameter, Mg ha-1 wk-'; b is the slope parameter, Mg ha-1
wk-2. Note that ti constitutes a second reference time.

3.4 Net Growth Function

Equations (3.1) and (3.2) must be combined to write a differential equation for the overall
growth rate, dY/dt. The simplest approach is to assume a product form such that









dY constant- [a+ b(t-t,)]- exp- t 2 (3.3)
dt -

Equation (3.3) constitutes a linear differential equation which can be solved analytically. In order
to solve Eq. (3.3) it is convenient to define a dimensionless time variable, x, as

x t (3.4)
V72c

It follows that the reference time becomes

t-
x ti (3.5)


Equation (3.4) can now be written as

dY 2 + 2k(x -x) exp x2) (3.6)
dx -.7 F7C

where the new parameters are defined by

constant- a Vo-/ (37
A = constan. = yield factor (3.7)
2


k = -b = partition factor (3.8)
a

The next step is to integrate Eq. (3.6). From mathematical tables (Abramowitz and Stegun, 1965)
we find that


2 xexp(-u2)du = 2 exp(-u2)du --2 exp(-u2)du = erf x- erf x, (3.9)
2 exp(_)u = e-o f71 o'
xf 0 0

X X Xi
f2u exp(-u2 )du = J2u exp(-u2 du 2uexp(-u2 u =-[exp(-x2) exp(-x)] (3.10)
xi 0 0

where u is the variable of integration. It follows that the yield increment, A Yi, for the ith growth
interval is given by


AY. = A (1 kx)[erf x erf x,]- [exp(-x2)-exp(-x? )]} (3.11)
k 2~ J










It is now convenient to define the growth quantifier, AQ,, by


AQi = (1- kx)[erf x erf x ]- [exp(-x') exp(-x )I (3.12)


This definition leads to a linear relationship between A Yi and AQi given by

AY. = AAQ, (3.13)

Cumulative yield for I harvests of a perennial grass, Y;, is given by the sum

I I
Y, = AY, = A AQ, (3.14)
i=1 i=1

If Y, is the total biomass yield for all harvests, then normalized yield, FI, is given by

F, = (3.15)
Y,

It has been shown (Overman et al., 1990) that for constant harvest interval, At, a plot of F vs. t
produces a straight line on probability paper. It has also been shown (Overman et al., 1989) that
seasonal total biomass follows a linear relationship with harvest interval

Y, = ay + flyAt (3.16)

where ay and fly are regression coefficients.

3.5 Dependence of Seasonal Yield on Harvest Interval

A practical test of the phenomenological model can be obtained by comparison of
dependence of seasonal yield vs. harvest interval to Eq (3.16). Data for this test are taken from a
field study with coastal bermudagrass (Cynodon dactylon L.) at Tifton, Georgia by Prine and
Burton (1956). The study was conducted over a two year period in which 1953 was a relatively
wet year and 1954 experienced a severe drought. Data are presented in Table 3.1 for applied
nitrogen of 672 kg ha-'. Response is also shown in Figure 3.1, where the lines are drawn from.

1953: Y, = ac + fy At = 11.34+ 2.74At r = 0.9956 (3.17)

1954: Y, = ay + flyAt = 4.41+1.60At r= 0.9836 (3.18)

for 0 < At < 6 wk. Two conclusions are evident from Figure 3.1: (1) yields are considerably
lower for 1954 than for 1953, and (2) yields for At = 8 wk lie below the lines. Conclusion (1)









indicates that response is sensitive to seasonal rainfall, which is not surprising. Conclusion (2)
means that this model is deficient for harvest intervals beyond 6 wk.

Table 3.1 Dependence of seasonal dry matter (Y,) on harvest interval (At1) for coastal
bermudagrass at Tifton, GA.a
Ati Yt
wk Mg ha-1
1953 1954
1 14.00 5.33
2 17.43 7.84
3 19.24 9.90
4 21.68 11.13
6 28.11 13.57
8 27.93 15.86
aData adapted from Prine and Burton (1956).


3.6 Summary

At this point the physical significance of the terms in Eq. (3.2) should be clarified. Overman
and Wilkinson (1989) analyzed data from the literature on partitioning of dry matter between leaf
and stem for coastal bermudagrass to suggest that terms in the intrinsic growth function relate to
partitioning of biomass. The first term on the right of Eq. (3.2) corresponds to the rate of growth
of the light-gathering component (leaf) while the second term corresponds to the rate of growth
of the structural component (stem).
There does seem to be a deficiency in the net growth function, Eq. (3.3). It suggests that if
the seasonal input of radiant energy could be made constant (say with artificial lighting) then
plant growth would continue unbounded. Remember the story of Jack and the bean stalk? We
know that this simply does not happen. Plants actually age and mature. A function related to
aging of the plant needs to be incorporated into the model. This subject will be explored in the
next chapter. At least the phenomenological model appears to be a step in the right direction.

3.7 References

Abramowitz, M. and I.A. Stegun. 1965. Handbook of Mathematical Functions. Dover
Publications. New York, NY.
Overman, A.R. and S.R. Wilkinson. 1989. Partitioning of dry matter between leaf and stem in
coastal bermudagrass. Agricultural Systems 30:35-47.
Overman, A.R., E.A. Angley, and S.R. Wilkinson. 1989. A phenomenological model of coastal
bermudagrass production. Agricultural Systems 29:137-148.
Overman, A.R., E.A. Angley, and S.R. Wilkinson. 1990. Evaluation of phenomenological model
of coastal bermudagrass production. Trans. American Society ofAgricultural Engineers
33:443-450.
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.]. Agronomy J 48:296-301.









indicates that response is sensitive to seasonal rainfall, which is not surprising. Conclusion (2)
means that this model is deficient for harvest intervals beyond 6 wk.

Table 3.1 Dependence of seasonal dry matter (Y,) on harvest interval (At1) for coastal
bermudagrass at Tifton, GA.a
Ati Yt
wk Mg ha-1
1953 1954
1 14.00 5.33
2 17.43 7.84
3 19.24 9.90
4 21.68 11.13
6 28.11 13.57
8 27.93 15.86
aData adapted from Prine and Burton (1956).


3.6 Summary

At this point the physical significance of the terms in Eq. (3.2) should be clarified. Overman
and Wilkinson (1989) analyzed data from the literature on partitioning of dry matter between leaf
and stem for coastal bermudagrass to suggest that terms in the intrinsic growth function relate to
partitioning of biomass. The first term on the right of Eq. (3.2) corresponds to the rate of growth
of the light-gathering component (leaf) while the second term corresponds to the rate of growth
of the structural component (stem).
There does seem to be a deficiency in the net growth function, Eq. (3.3). It suggests that if
the seasonal input of radiant energy could be made constant (say with artificial lighting) then
plant growth would continue unbounded. Remember the story of Jack and the bean stalk? We
know that this simply does not happen. Plants actually age and mature. A function related to
aging of the plant needs to be incorporated into the model. This subject will be explored in the
next chapter. At least the phenomenological model appears to be a step in the right direction.

3.7 References

Abramowitz, M. and I.A. Stegun. 1965. Handbook of Mathematical Functions. Dover
Publications. New York, NY.
Overman, A.R. and S.R. Wilkinson. 1989. Partitioning of dry matter between leaf and stem in
coastal bermudagrass. Agricultural Systems 30:35-47.
Overman, A.R., E.A. Angley, and S.R. Wilkinson. 1989. A phenomenological model of coastal
bermudagrass production. Agricultural Systems 29:137-148.
Overman, A.R., E.A. Angley, and S.R. Wilkinson. 1990. Evaluation of phenomenological model
of coastal bermudagrass production. Trans. American Society ofAgricultural Engineers
33:443-450.
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.]. Agronomy J 48:296-301.










List of Figures

Figure 3.1 Dependence of seasonal total yield (YI) on harvest interval (At) for two years for
coastal bermudagrass at Tifton, GA. Data adapted from Prine and Burton (1956). Lines drawn
from Eqs. (3.17) and (3.18).



















































20 Squares to the Inch


fijg 3,J1









4. Expanded Model of Crop Growth


4.1 Background

It was suggested in Chapter 3 that the intrinsic growth function should be modified to include
a factor related to aging in the plant. What form should this function take? Perhaps it would be
helpful to examine additional data on response of seasonal yield to harvest interval. That will be
the next step in the process.
Burton et al. (1963) conducted additional studies on this question with coastal bermudagrass
at Tifton, Georgia. Harvest intervals included At = 3, 4, 5, 6, 8, 12, and 24 wk. Applied nitrogen
was 672 kg ha-' in all cases. Data for seasonal biomass yield (Y,), plant nitrogen uptake (Nu), and
plant nitrogen concentration (Nc) are given in Table 4.1. Biomass yield vs. harvest interval is

Table 4.1 Dependence of seasonal biomass yield (Ye), plant N uptake (Nu), and plant N
concentration (Nc) on harvest interval (At) for coastal bermudagrass at Tifton, GA.a

At Y, Nu Nc Yt exp(0.077At) N, exp(0.077At)
wk Mg ha1' kg ha-1 g kg-'
3 15.2 438 28.8 19.1 552
4 16.2 415 25.6 22.0 565
5 17.8 417 23.4 26.2 613
6 19.9 411 20.6 31.6 652
8 19.9 340 17.1 36.8 630
12 20.1 289 14.4 50.6 728
24 14.6 198 13.6 92.7 1257
aCrop response data adapted from Burton et al. (1963).

shown in Figure 4.1, which further confirms that yields are approximately linear up to about 6
wk. Beyond this point yield appears to peak and then decline. The total growing season for
coastal bermudagrass at Tifton, GA is approximately 26 wk. An empirical relationship is now
assumed between Yj and At of the form

Y, = (a, + y At) exp(-yAt) (4.1)

where ay, fiy, y are to be determined from analysis of data. Note that for 7 = 0 Eq. (4.1) reduces
to Eq. (3.16) as expected. Now Eq. (4.1) can be linearized by defining standardized yield as
Y,* = Y, exp(TAt). A trial-and-error procedure is used to obtain the best straight line for Y,* vs.
At, as shown in Figure 4.1 for y = 0.077 wk-1. This definition leads to the regression equation

Y,* = Y, exp(0.077At) = ay + fy At = 8.91 + 3.49At r = 0.99950 (4.2)

with a correlation coefficient of r = 0.99950. The prediction equation now becomes


Y, = (ay + yAt) exp(-yAt) = (8.91 +3.49At)exp(-0.077At)


(4.3)









4. Expanded Model of Crop Growth


4.1 Background

It was suggested in Chapter 3 that the intrinsic growth function should be modified to include
a factor related to aging in the plant. What form should this function take? Perhaps it would be
helpful to examine additional data on response of seasonal yield to harvest interval. That will be
the next step in the process.
Burton et al. (1963) conducted additional studies on this question with coastal bermudagrass
at Tifton, Georgia. Harvest intervals included At = 3, 4, 5, 6, 8, 12, and 24 wk. Applied nitrogen
was 672 kg ha-' in all cases. Data for seasonal biomass yield (Y,), plant nitrogen uptake (Nu), and
plant nitrogen concentration (Nc) are given in Table 4.1. Biomass yield vs. harvest interval is

Table 4.1 Dependence of seasonal biomass yield (Ye), plant N uptake (Nu), and plant N
concentration (Nc) on harvest interval (At) for coastal bermudagrass at Tifton, GA.a

At Y, Nu Nc Yt exp(0.077At) N, exp(0.077At)
wk Mg ha1' kg ha-1 g kg-'
3 15.2 438 28.8 19.1 552
4 16.2 415 25.6 22.0 565
5 17.8 417 23.4 26.2 613
6 19.9 411 20.6 31.6 652
8 19.9 340 17.1 36.8 630
12 20.1 289 14.4 50.6 728
24 14.6 198 13.6 92.7 1257
aCrop response data adapted from Burton et al. (1963).

shown in Figure 4.1, which further confirms that yields are approximately linear up to about 6
wk. Beyond this point yield appears to peak and then decline. The total growing season for
coastal bermudagrass at Tifton, GA is approximately 26 wk. An empirical relationship is now
assumed between Yj and At of the form

Y, = (a, + y At) exp(-yAt) (4.1)

where ay, fiy, y are to be determined from analysis of data. Note that for 7 = 0 Eq. (4.1) reduces
to Eq. (3.16) as expected. Now Eq. (4.1) can be linearized by defining standardized yield as
Y,* = Y, exp(TAt). A trial-and-error procedure is used to obtain the best straight line for Y,* vs.
At, as shown in Figure 4.1 for y = 0.077 wk-1. This definition leads to the regression equation

Y,* = Y, exp(0.077At) = ay + fy At = 8.91 + 3.49At r = 0.99950 (4.2)

with a correlation coefficient of r = 0.99950. The prediction equation now becomes


Y, = (ay + yAt) exp(-yAt) = (8.91 +3.49At)exp(-0.077At)


(4.3)









The line and curve in Figure 4.1 are drawn from Eqs. (4.2) and (4.3), respectively. This model
describes the data very closely. It can be shown from calculus that the peak of the curve occurs at
1 y 1 8.91
At --- 10.4 wk (4.4)
7 fly 0.077 3.49

which corresponds to a peak yield of Y, = 20.3 Mg ha-1. This can be confirmed from Figure 4.1.

4.2 Expanded Growth Function

Success of the exponential term in Eq. (4.1) provides a clue as to modification of Eq. (3.2).
Perhaps the aging function can be written in exponential form, so that the intrinsic growth
function is now expanded to

dY'
-d = [a + b(t t,)] exp[-c(t t,)] (4.5)
dt

where c is the coefficient of aging. Equation (4.5) can be combined with the energy driving
function, Eq. (3.1), to write the equation for net growth rate

dY = constant- {[a + b(t t, )]exp[- c(t t,)]}. exp t J- 2 (4.6)
dt 2 -

The solution to Eq. (4.6) can be written (Overman, 1998; Overman and Scholtz, 2002a)

AY, = A (1 kx, )[erf x erf x, ] --k [exp(-x2) exp(-x )]}. exp(V2oucx,) (4.7)

which constitutes the expanded growth model. It is now convenient to define the growth
quantifier, AQi, by

AQ = (1- kx,)[erf x erf x,]-- [exp(-x2) exp(-x2 ) }.exp(F2ocx,) (4.8)


so that Eq. (4.7) can be simplified to the linear form

AY. = AAQ, (4.9)


where dimensionless time, x, is now defined as









xt- -Frov (4.10)



xi = C +- (4.11)


with the new parameters defined as


A = constant-. Fa exp C2 (4.12)



k = (4.13)
a

Note that for c = 0, this model reduces to the phenomenological model of Chapter 3.
The next challenge is to link the expanded growth model to seasonal yield for a perennial
grass harvested on a fixed time interval. What is required is proof of a mathematical theorem
(Overman, 2001; Overman and Scholtz, 2002a), which is no small task. After much
mathematical manipulation the result is obtained

( kA' c
Y,= 2A'+ -- At exp --At = (a + fAt)exp(-yAt) (4.14)


F = 1+ erf (4.15)


where


A' = constant 12oan-j (4.16)


Note from Eq. (4.14) that7 = c/2. This result justifies Eq. (4.1) and provides a rationale for the
probability model as well. This theorem provides a valuable link between the growth model and
seasonal production.

4.3 Coupling of Biomass and Plant Nitrogen for Perennial Grass

We now return to the data in Table 4.1 for coastal bermudagrass. Following a similar
assumption to Eq. (4.1) for seasonal plant N accumulation (Nu), it seems reasonable to write


N,, = (a,, + 8,At) exp(-7At)


(4.17)










where a,, and ,,, are to be determined from analysis of data. Parameters is assumed the same as
for biomass. It follows from Eqs. (4.1) and (4.17) that plant N concentration (Nc) is related to
harvest interval by

N, a,, + At (4.18)
S Y, a, +PAt

Standardized plant N uptake is defined as N* = N,, exp(yAt), which is also listed in Table
4.1. This leads to the regression equation

N = N,, exp(0.077At) = a,, + ,At = 422 + 33.0At r = 0.9780 (4.19)

which leads to the prediction equation

N, = (a,, + f3,At) exp(-yAt) = (422 + 33.0At) exp(-0.077At) (4.20)

Dependence of plant N concentration on harvest interval is now described by

N a,, + 3,At 422+33.0At
N = -= = (4.21)
c Y, ay +f3yAt 8.91+3.49At

Response of biomass, plant N uptake, and plant N concentration to harvest interval is shown in
Figure 4.2, where the curves are drawn from Eqs. (4.3), (4.20), and (4.21). It may be noted that
peak plant N uptake occurs at

1 a, 1 422
At,, -" ----=0.2 wk (4.22)
y /3,, 0.077 33.0

According to Eq. (4.21) plant N concentration is bounded by 47.4 > Nc > 9.5 g kg-'. This
result indicates that as the plants age (longer harvest interval) plant N concentration decreases.
This happens as plants shift from dominance by leaves (higher N concentration) to dominance by
stems (lower N concentration) as plants age, which is consistent with the expanded growth
model.

4.4 Confirmation of the Expanded Growth Model for Perennial Grass

Analysis of data in the previous section lends strong support for the expanded growth model.
It would still be useful to obtain direct confirmation of Eq. (4.9). Overman and Brock (2003)
have outlined such a procedure using field data of Burton and Hart (1961) for coastal
bermudagrass at Tifton, Georgia. The analysis led to the values p = 25.0 wk and o- = 8.5 wk.
From Section 4.1 we obtained y = 0.077 wk'1, which leads to the aging coefficient of









c = 2Y = 0.15 wk-'. From previous work (Overman and Wilson, 1999) we assume k = 5.
Dimensionless time can be calculated from

t f o-c t 25.0 t 14.2
x = + + 0.90= xi = 0.0333 (4.23)
o- 2 12.0 12.0

It follows from Eq. (4.8) that the growth quantifier can be estimated from


AQ = {(1 kx, )[erf x erf x, ]- [exp(-x2) exp(-x2 )]} *exp(Vorcx,)

= {0.834[erf x 0.038]- 2.821[exp(-x2) 0.9989]} 1.062

Data for the first cutting of each harvest interval of the study at Tifton, GA are listed in Table
4.2. Values ofx and AQ, as calculated from Eqs. (4.23) and (4.24), are included as well. A
reference time ofti = 14.6 wk has been estimated for this case. Linear regression of AY vs. AQ
gives

Table 4.2. Accumulation of biomass yield (A Y), plant N uptake (AN,,), and plant N concentration
(Nc) with calendar time (t) for coastal bermudagrass at Tifton, GA.a

t AY AN, Nc AY/AN, x AQ
wk Mg ha-1 kg ha'' g kg-1 kg g''
14.6 -- -- -- --- 0.033 0.000
17.6 2.37 61.8 26.1 0.0384 0.283 0.469
18.6 3.49 84.0 24.1 0.0415 0.367 0.691
19.6 4.26 92.6 21.7 0.0460 0.450 0.933
20.6 6.83 138 20.2 0.0495 0.533 1.190
22.6 8.78 126 14.4 0.0697 0.700 1.724
26.6 12.54 148 11.8 0.0847 1.033 2.686
38.6 16.96 230 13.6 0.0737 2.033 3.793
0o 3.845
aCrop data adapted from Burton and Hart (1961).


AY = 0.70 + 4.38AQ r = 0.9948 (4.25)

as shown in Figure 4.3 with a correlation coefficient ofr = 0.9948, which lends support to the
growth model. The fact that the intercept is not zero means that the reference time of 14.6 wk is
not quite the optimum choice. Results are shown in Figure 4.3, where the line is drawn from Eq.
(4.25).
The next question relates to simulation of plant N accumulation with time. Overman and
Scholtz (1999) showed that a hyperbolic phase relation between biomass (AY) and plant nutrient
accumulation (AN,) worked very well for corn (Zea mays L.). This approach leads to









AAN NmAY (4.26)
K +AY


where ANum is potential maximum plant N uptake and Ky is the response coefficient for yield.
Equation (4.26) can be rearranged to the linear form and applied to data for coastal bermudagrass

AY K + 1- AY = 0.0338+0.00302AY r= 0.885 (4.27)
AN. AN,,, ANAu

The prediction equation becomes


AN = AN AY 330AY(4.28)
+AY 11.2+AY

Results are shown in Figure 4.4, where the line is drawn from Eq. (4.27) and the curve from Eq.
(4.28). Scatter in the data resulted primarily from variability in kjeldahl nitrogen determinations
of plant N concentration. Plant N concentration can then be calculated from Eq. (4.28) as

N AN 330 (4.29)
AY 11.2 + AY

Accumulation of biomass, plant N uptake, and plant N concentration is shown in Figure 4.5,
where the curves are drawn from Eqs. (4.25), (4.28), and (4.29). It is apparent that the model
provides excellent simulation of biomass accumulation with time. Results for plant N show
considerably more scatter and uncertainty in the simulation. Equation (4.29) illustrates the strong
sensitivity of plant N concentration to plant age.

4.5 Confirmation of the Expanded Growth Model for Annual Crop

The expanded growth model is now applied to a warm-season annual. Results from field
studies at Florence, SC and Tallahassee, FL with corn and Lewiston, NC with peanut are
analyzed to illustrate procedures for the expanded growth model. Analysis includes accumulation
of both biomass and mineral elements.

4.5.1 Study with Corn at Florence, South Carolina

Data are taken from a field study by Karlen et al. (1987) with 'Pioneer 3382' corn on Norfolk
loamy fine sand (fine-loamy, siliceous, thermic Typic Paleudult) at Florence, South Carolina.
Data for 1982 and plant population density of 7 plants m-2 are used for this analysis. Planting
date was April 2 (t = 13.3 wk). Fertilizer application was N-P-K = 268-36-224 kg ha"-. Data are
given in Table 4.3. The delta symbol has been dropped for convenience. The following
parameters are chosen for this analysis: p = 26.0 wk, o- = 5.66 wk, k = 5, and c = 0.206 wk-1. This
leads to the equations









AAN NmAY (4.26)
K +AY


where ANum is potential maximum plant N uptake and Ky is the response coefficient for yield.
Equation (4.26) can be rearranged to the linear form and applied to data for coastal bermudagrass

AY K + 1- AY = 0.0338+0.00302AY r= 0.885 (4.27)
AN. AN,,, ANAu

The prediction equation becomes


AN = AN AY 330AY(4.28)
+AY 11.2+AY

Results are shown in Figure 4.4, where the line is drawn from Eq. (4.27) and the curve from Eq.
(4.28). Scatter in the data resulted primarily from variability in kjeldahl nitrogen determinations
of plant N concentration. Plant N concentration can then be calculated from Eq. (4.28) as

N AN 330 (4.29)
AY 11.2 + AY

Accumulation of biomass, plant N uptake, and plant N concentration is shown in Figure 4.5,
where the curves are drawn from Eqs. (4.25), (4.28), and (4.29). It is apparent that the model
provides excellent simulation of biomass accumulation with time. Results for plant N show
considerably more scatter and uncertainty in the simulation. Equation (4.29) illustrates the strong
sensitivity of plant N concentration to plant age.

4.5 Confirmation of the Expanded Growth Model for Annual Crop

The expanded growth model is now applied to a warm-season annual. Results from field
studies at Florence, SC and Tallahassee, FL with corn and Lewiston, NC with peanut are
analyzed to illustrate procedures for the expanded growth model. Analysis includes accumulation
of both biomass and mineral elements.

4.5.1 Study with Corn at Florence, South Carolina

Data are taken from a field study by Karlen et al. (1987) with 'Pioneer 3382' corn on Norfolk
loamy fine sand (fine-loamy, siliceous, thermic Typic Paleudult) at Florence, South Carolina.
Data for 1982 and plant population density of 7 plants m-2 are used for this analysis. Planting
date was April 2 (t = 13.3 wk). Fertilizer application was N-P-K = 268-36-224 kg ha"-. Data are
given in Table 4.3. The delta symbol has been dropped for convenience. The following
parameters are chosen for this analysis: p = 26.0 wk, o- = 5.66 wk, k = 5, and c = 0.206 wk-1. This
leads to the equations









t- up Jo-c = t-26.0 +0 t-19.4
x= +- = + 0.824=- xi = 0.00
=82o- 2 8.00 8.00


Q = (1- kx,)[erf x erf x ]- [exp(-x2) exp(-x )]}. exp(2 ocx,)

= {l.000[erf x 0.000]- 2.82 l[exp(-x2) 1.000]}. 1.000


(4.30)



(4.31)


Parameter c was chosen to make xi = 0. Values of x and Q listed in Table 4.3 were calculated
from Eqs. (4.30) and (4.31).

Table 4.3. Accumulation of biomass yield (Y), plant N uptake (N,), and plant N concentration
(N,) with calendar time (t) for corn at Florence, SC (1982).a

t Y Nu Nc Y/N. x Q
wk Mg ha-' kg ha-1 g kg'1 kg g-'
19.4 -- -- -- --- 0.0000 0.000
19.5 0.41 13 32.0 0.0315 0.0125 0.014
21.0 1.83 60 32.8 0.0305 0.200 0.333
22.0 4.21 110 26.1 0.0383 0.325 0.636
24.0 10.2 171 16.8 0.0596 0.575 1.380
26.8 18.8 198 10.5 0.0949 0.925 2.431
28.8 21.3 204 9.6 0.104 1.175 3.016
29.5 22.5 207 9.2 0.109 1.262 3.174
co 3.821
aCrop data adapted from Karlen et al. (1987).


The first step is to calibrate the yield equation. Correlation ofbiomass
quantifier (Q) is shown in Figure 4.6, where the line is drawn from


(Y) with the growth


Y = -0.048 + 7.25Q r = 0.9976


(4.32)


which provides very high correlation. The intercept is essentially zero in accordance with the
theory. The second step is to couple plant N and biomass accumulation. Following the same
procedure as in Section 4.4 leads to the phase relations


y Ky 1
S-= + -- + Y = 0.0251+0.00368Y r= 0.9972
N NN N,,.

The prediction equation for plant N accumulation becomes

N,,Y 272Y
Ky +7Y 6.82+Y


(4.33)


(4.34)









Phase relations are shown in Figure 4.7, where the line and curve are drawn from Eqs. (4.33) and
(4.34), respectively. It follows from Eq. (4.34) that plant N concentration is given by


N Num 272
N KY 6.82 "
c Y Ky+Y 6.82+Y


(4.35)


Response of biomass, plant N uptake, and plant N concentration with calendar time is shown in
Figure 4.8, where the curves are drawn from Eqs. (4.32), (4.34), and (4.35). The model describes
results very well.

4.5.2 Study with Corn at Tallahassee, Florida

Data for this analysis are taken from a field study at the Southeast Farm at Tallahassee, FL
with corn (cv 'DeKalb T1 100') on Kershaw fine sand (fine-sand, siliceous, thermic
Quartzipsamment) as reported by Overman and Scholtz (2004). Plant population was
approximately 8.5 plants m-2, with a planting date of 6 March 1990 (t = 9.3 wk). Measurements
were made of biomass and mineral elements (N, P, K, Ca, Mg) on a weekly basis between 15
April 1990 and 8 July 1990. During the 19 wk growing season rainfall was 26 cm and irrigation
with municipal reclaimed water was 134 cm through a center pivot unit. Nutrient applications
through reclaimed water and supplemental fertilizer were: N-P-K-Ca-Mg = 143-13-262-470-150
kg ha-1. Results for total plant (stalks, leaves, ears) are given in Table 4.4 for biomass (Y), plant
N uptake (Nu), and plant N concentration (Nc) with calendar time (t). Model parameters are
chosen the same as for Florence, SC: = 26.0 wk, u = 5.66 wk, k = 5, and c = 0.206 wk-l. This
leads to the equations for x and Q of

Table 4.4. Accumulation of biomass yield (Y), plant N uptake (Nu), and plant N concentration
(Nc) with calendar time (t) for corn at Tallahassee, FL (1990).a

t Y Nu Nc YI/N x Q
wk Mg ha-1 kg ha"' g kg-' kg g-'
13.8 -- -- -- --- -0.700 0.000
15.3 0.45 12.5 27.8 0.0360 -0.512 0.070
16.3 1.03 26.9 26.1 0.0383 -0.388 0.150
17.3 1.98 40.5 20.5 0.0488 -0.262 0.266
18.3 4.17 61.2 14.7 0.0680 -0.138 0.416
19.3 5.96 112 18.8 0.0532 -0.012 0.599
20.3 7.40 137 18.5 0.0541 0.112 0.807
21.3 9.00 125 13.9 0.0719 0.238 1.040
22.3 10.82 137 12.7 0.0787 0.362 1.281
23.3 13.03 127 9.8 0.103 0.488 1.528
24.3 16.09 139 8.6 0.116 0.612 1.764
25.3 17.22 153 8.9 0.113 0.738 1.987
26.3 16.29 160 9.8 0.102 0.862 2.185
27.3 17.24 138 8.0 0.125 0.988 2.359
00 oo2.972
aCrop data adapted from Overman and Scholtz (2004).









t p Eoac t 26.0 t 19.4
= +- =- +0.824 xi=-0.700 (4.36)
2o- 2 8.00 8.00

Q = (1- kx,)[erf x erf x ]- k [exp(_x2) exp(--x ) exp( -cx,)
Q N J 1(4.37)
= {4.500[erf x + 0.678]- 2.821[exp(-x2) 0.613]}. 0.315

Values ofx and Q listed in Table 4.4 were calculated from Eqs. (4.36) and (4.37). Correlation of
biomass (Y) with the growth quantifier (Q) is shown in Figure 4.9, where the line is drawn from

Y = 0.034 + 8.77Q r = 0.9973 (4.38)

Data points at 26.3 and 27.3 wk were omitted from regression. The intercept is essentially zero in
accordance with the theory. The second step is to couple plant N and biomass accumulation.
Following the same procedure as in Section 4.4 leads to the phase relations

y Ky 1
-y + --Y = 0.0336 + 0.00472Y r = 0.9619 (4.39)
N Num Num

The prediction equation for plant N accumulation becomes

NY 212Y
Nu -= N (4.40)
Ky +Y 7.12+Y

Phase relations are shown in Figure 4.10, where the line and curve are drawn from Eqs. (4.39)
and (4.40), respectively. It follows from Eq. (4.40) that plant N concentration is given by

N Num 212
N= =- (4.41)
Y Ky +Y 7.12+Y

Response of biomass, plant N uptake, and plant N concentration with calendar time is shown in
Figure 4.11, where the curves are drawn from Eqs. (4.38), (4.40), and (4.41). The model appears
to give reasonable results.
Accumulation of plant phosphorus (P,), potassium (K,), calcium (Ca,), and magnesium
(Mg,,) with calendar time (t) is given in Table 4.5. Hyperbolic phase relations with biomass (Y) is
assumed for each mineral element. Phase plots for phosphorus are shown in Figure 4.12, where
the line and curve are drawn from

- K + Y = 0.263 + 0.00769Y r = 0.9388 (4.42)
Pu um P Pur









p = P Y_ 130Y
Ky+Y 34.2+Y


(4.43)


Table 4.5. Accumulation of biomass yield (Y), plant P uptake (Pu), and plant K uptake (Ku), plant
Ca uptake (Cau), and plant Mg uptake (Mgu) with calendar time (t) for corn at Tallahassee, FL
(1990).a
t Y Pu Y/PI Ku Y/K. Can Y/Cau Mgn Y/Mgu
wk Mg ha'1 kg ha-' kg g-' kg ha-' kg g-1 kg ha-' kg g'1 kg ha'' kg g-1
13.8 -- -- -- --- --- -- -- -- --
15.3 0.45 1.65 0.272 9.5 0.0474 3.0 0.149 1.4 0.322
16.3 1.03 4.07 0.253 22.8 0.0452 7.3 0.141 3.5 0.294
17.3 1.98 7.50 0.264 50.5 0.0392 11.4 0.174 5.9 0.336
18.3 4.17 12.7 0.328 97.6 0.0427 17.2 0.243 9.2 0.452
19.3 5.96 18.6 0.321 112 0.0532 23.1 0.254 11.9 0.500
20.3 7.40 23.3 0.317 120 0.0617 21.7 0.341 12.3 0.602
21.3 9.00 27.4 0.329 116 0.0775 21.9 0.412 13.3 0.676
22.3 10.82 32.2 0.336 117 0.0926 25.0 0.433 14.6 0.741
23.3 13.03 36.5 0.357 124 0.105 26.2 0.498 15.7 0.833
24.3 16.09 45.5 0.353 130 0.124 26.6 0.606 18.3 0.877
25.3 17.22 42.4 0.406 128 0.226 25.1 0.685 16.0 1.08
26.3 16.29 39.4 0.413 101 0.161 21.7 0.752 14.4 1.14
27.3 17.24 43.4 0.397 120 0.144 20.4 0.847 14.6 1.18
aCrop data adapted from Overman and Scholtz (2004).


Phase plots for potassium are shown in Figure 4.13, where the line and curve are drawn from


y Ky 1
- = +- +-Y = 0.0260 + 0.00658Y r = 0.9503
K,, Kum Kum

S= KY 152Y
"Ky+Y 3.95+Y


(4.44)



(4.45)


The point at 25.3 wk is omitted from regression. Phase plots for calcium are shown in Figure
4.14, where the line and curve are drawn from


y K 1
-= -'-+- Y = 0.109+0.0314Y r= 0.9930
SC, Cau Caum


a CauY 31.8Y
K + Y 3.47 + Y


(4.46)



(4.47)









The points at 26.3 and 27.3 wk are omitted from regression. Phase plots for magnesium are
shown in Figure 4.15, where the line and curve are drawn from

Y Ky 1
-K- + -- Y = 0.268 + 0.0431Y r = 0.9884 (4.48)
Mg, Mg9m Mg.,

MgmY 23.2Y
Mg MgY 23.2Y (4.49)
SK + Y 6.21+Y

The points at 26.3 and 27.3 wk are omitted from regression.
The hyperbolic phase relation appears to work well for the elements N, P, K, Ca, and Mg.
The phase relation has been assumed, and not derived from basic principles. This analysis
suggests that biomass accumulation by photosynthesis is the rate limiting step in the growth
process, and that accumulation of mineral elements proceeds in virtual equilibrium with
bioimass.

4.5.3 Study with Peanut at Lewiston, North Carolina

Data for this analysis are adapted from a field study at Lewiston, NC with peanut (Arachis
hypogaea L.) by Nicholaides (1968). Average data for the cultivars 'NC2' and 'NC5' are used
from the study (figures 1, 3, 5, 7, and 9). The soil was Goldsboro sandy loam (fine-loamy,
siliceous, subactive, thermic Aquic Paleudult). Plots were planted on 6 May 1966 (t = 17.9), with
plant samples collected on 13 July, 13 August, 24 August, 14 September, and 5 October 1966.
Samples were partitioned into tops, shells, and kernels as growth allowed. Measurements were
made of biomass and plant nitrogen, phosphorus, potassium, and magnesium. Results for
biomass and plant N are given in Table 4.6. Model parameters are chosen p = 26.0 wk, o- = 5.66
wk, k = 5, and c = 0.050 wk'1. Time of initiation is chosen as t, = 26.6 wk. This leads to
dimensionless time and growth quantifier of

Table 4.6. Accumulation of biomass yield (Y), plant N uptake (N,), and plant N concentration
(Nc) with calendar time (t) for peanut (whole plant) at Lewiston, NC (1966).a

t Y Nu Nc Y/N, x Q
wk Mg ha'' kg ha'' g kg-' kg g-'
26.6 -- -- -- --- 0.275 0.000
27.7 1.58 51.0 32.3 0.0310 0.412 0.204
30.7 2.85 73.8 25.9 0.0386 0.788 1.043
33.7 5.88 146 24.8 0.0403 1.162 1.853
36.7 9.02 194 21.5 0.0465 1.538 2.343
39.7 9.94 217 21.8 0.0458 1.912 2.548
00 2.626


aCrop data adapted from Nicholaides (1968).

t P -c t 26.0 t 24.4
a2 2 8.00 8.00 x=0.275


(4.50)










Q = {(1- kx,)[erf x erf x,]- [exp(-x2) exp(-x ) exp( cx) (4.51)
1 7- i 1(4.51)
= {- 0.375[erf x 0.302]- 2.821[exp(-x2) 0.927]} 1.116

Values in Table 4.6 are calculated from Eqs. (4.50) and (4.51). Correlation of Y vs. Q is shown in
Figure 4.16, where the line is drawn from

Y = -0.021 + 3.68Q r = 0.972 (4.52)

Coupling of plant nutrients with biomass is examined next. Results for plant N are shown in
Figure 4.17, where the line and curve are drawn from

y K _1
+ Y = 0.0310+ 0.00162Y r = 0.942 (4.53)
N NN N.


N.,, Y 618Y (4.54)
Ky + Y 19.2+Y

It follows from Eq. (4.54) that plant N concentration is given by

N 618
Nc (4.55)
Y 19.2+ Y

Accumulation of biomass, plant N uptake, and plant N concentration with calendar time are
shown in Figure 4.18, where the curves are drawn from Eqs. (4.52), (4.54), and (4.55).
Accumulation of plant P, K, and Mg with calendar time is given in Table 4.7. The phase
equations for these elements are given by

Table 4.7. Accumulation of biomass yield (Y), plant P uptake (Pu), plant K uptake (Ku), and plant
Mg uptake (Mgu) with calendar time (t) for peanut (whole plant) at Lewiston, NC (1966).a

t Y P. Y/Pu K& Y/K. Mgu Y/Mg-
wk Mg ha-1 kg ha-1 kg g-1 kg ha'' kg g-1 kg ha-' kg g-1
26.6 -- -- -- -- --- -- --
27.7 1.58 4.31 0.367 48.0 0.0329 8.97 0.176
30.7 2.85 5.66 0.504 65.6 0.0435 17.5 0.163
33.7 5.88 12.7 0.463 159 0.0370 22.2 0.265
36.7 9.02 14.8 0.609 178 0.0507 30.7 0.294
39.7 9.94 15.3 0.650 174 0.0571 32.0 0.311
aCrop data adapted from Nicholaides (1968).









- = -- + Y = 0.353 + 0.0282Y r = 0.908 (4.56)
P. P.. P..

S-un (4.57)
Ky +Y 12.5+Y

y Ky 1
+ Y = 0.0308 + 0.00230Y r = 0.854 (4.58)
Ku Kum Kum

KumY 436Y
K= (4.59)
K + Y 13.4+Y

Y K 1
y +-- Y = 0.136+0.0180Y r= 0.965 (4.60)
Mg, Mg,, Mgm

mg -mgumY 55.5Y (4.61)
K + Y 7.55+Y

Results are shown in Figures 4.19 through 4.21 for phosphorus, potassium, and magnesium,
respectively. Lines are drawn from Eqs. (4.56), (4.58), and (4.60); curves are drawn from Eqs.
(4.57), (4.59), and (4.61).
Now Eq. (4.51) can be partitioned into light-gathering and structural components of the plant

QL = {[erf x erf x ]exp(/2crcx,) = {[erf x 0.302]} 1.116 (4.62)

Qs = {- kx, [erf x erf x, ]- -- [exp(- x2 )- exp(- x )] ) exp(/lo-cx,)
I J (4.63)
= -1.375[erf x 0.302]- 2.82 l[exp(- x2 )- 0.927}. 1.116

The biomass fraction for the light-gathering component,fL, can be calculated from

fL QL (4.64)
QL + QS

Estimates for biomass partitioning are listed in Table 4.8. Dependence of partitioning with
calendar time is shown in Figure 4.22, where the curve has been drawn from values in Table 4.8.
According to this analysis the light-gathering component drops from approximately 100% at the
beginning of growth to 30% at t = 40 wk. It should be noted that in this analysis kernels + shells
have been considered as part of the structural component.









Table 4.8. Estimated partitioning of biomass between light-gathering and structural components
of peanut at Lewiston, NC (1966).
t x erfx exp(-x2) QL Qs Q fL
wk
26.6 0.275 0.302 0.927 0.000 0.0000 0.000 1.000
27 0.325 0.354 0.900 0.058 0.0052 0.063 0.918
28 0.450 0.475 0.817 0.193 0.081 0.274 0.704
30 0.700 0.678 0.613 0.420 0.443 0.863 0.487
32 0.950 0.821 0.406 0.579 0.884 1.423 0.407
34 1.200 0.910 0.237 0.679 1.239 1.918 0.354
36 1.450 0.960 0.122 0.734 1.525 2.259 0.325
38 1.700 0.984 0.0556 0.761 1.697 2.458 0.310
40 1.950 0.9942 0.0223 0.772 1.787 2.559 0.302
42 2.200 0.9981 0.00791 0.777 1.825 2.602 0.299
44 2.450 0.9995 0.00247 0.778 1.841 2.619 0.297
co 1.0000 0.00000 0.779 1.847 2.626 0.297


Finally, we examine accumulation of biomass of plant nuts (kernels + shells) with time.
These results are given in Table 4.9. Time of initiation of growth is chosen as ti = 33.0 wk,

Table 4.9 Accumulation of biomass (Y), plant N uptake (Nu), plant P uptake (Pu), plant K uptake
(Ku), and plant Mg uptake (Mgu) with calendar time (t) for nuts (kernels + shells) for peanut
grown at Lewiston, NC (1966).a
t Y N. P" K. Mg. x Q
wk Mg ha-1 kg ha-1 kg ha'' kg ha'' kg ha-'
33.0 ----- ---- ---- --- ---- 1.075 0.000
33.7 0.60 18 1.5 7 0.8 1.162 0.055
36.7 3.17 73 6.0 23 4.0 1.538 0.299
39.7 4.48 120 8.4 27 4.0 1.912 0.440
00o 0.505
aCrop data adapted from Nicholaides (1968).


which leads to equations for dimensionless time and growth quantifier of

t -+ P-c t- 26.0 t 24.4
x= + = + 0.200=- x,=1.075
2- 2 8.00 8.00


Q = (1- kx,)[erf x erf x, ]- [exp(-x2) exp(-x ) exp(.-x bcx)

={- 4.375[erf x 0.872]- 2.821[exp(-x2) -0.315]J. 1.537


(4.65)



(4.66)


Values in Table 4.9 are calculated from Eqs. (4.65) and (4.66). Correlation of biomass with Q for
nuts is shown in Figure 4.23, where the line is drawn from









Y = 0.069 +10.1Q r = 0.99948


Correlation of plant N, P, and K with biomass is shown in Figure 4.24. Since the trends are
essentially linear, the lines are drawn from



S= y = 779.8 =25.6Y (4.68)
I 2 L30.481
L i=1


[57.55] 1.855
P.= Y = Y = 1.89Y (4.69)
Y>2 >30.48
F=l

3
E YiK,,1
K, = i Y= 198.1 Y = 6.50Y (4.70)
y2 30.48J
i=1

in which the lines have been constrained to pass through (0, 0) intercepts on intuitive grounds.
The expanded growth model appears to provide reasonable description of biomass and plant
nutrients with time. Partitioning of biomass between light-gathering and structural components
accounts for the drop in plant nutrient concentration as plants age.

4.6 Partitioning of Biomass

The intrinsic growth function of the expanded growth model incorporates partitioning of
biomass into light-gathering and structural components. In this section data are analyzed which
lend support to this assumption. Because of the linear form of the intrinsic growth function it is
possible to partition the growth quantifier into a light-gathering component, QL, and a structural
component, Qs, which can be written as

QL = {[erf x erf x, ]exp(A2 orx,) (4.71)


Qs= kx, [erfx erf x]- [exp(- x2) exp(- xj)] exp(--cx,) (4.72)


It follows that the growth quantifier for total biomass, Q, can be written as the algebraic sum


Q=QL +QS


(4.67)


(4.73)








4.6.1 Study with Elephantgrass at Gainesville, Florida


Data for this analysis are taken from a field study with elephantgrass (Pennisetum purpureum
Schum.) by Woodard et al. (1993) as discussed by Overman and Woodard (2006). Field
experiments were conducted on Arredondo fine sand (loamy, siliceous, hyperthermic
Grossarenic Paleudult) at Gainesville, Florida. Plots were mowed on 28 March 1989 (t = 12.6
wk) to promote stand uniformity, and then sampled every 4 wk beginning on 16 May 1989 (t =
19.6 wk) and until 28 November (t = 47.6 wk). Fertilizer was applied at N-P-K = 200-22-83 kg
ha''. Plants were partitioned into stems and leaves. Results are given in Table 4.10 for stem
biomass (Ys), leaf biomass (YL), and leaf fraction (fj) with calendar time (t).
Model parameters are chosen from Overman and Woodard (2006) as: p = 26.0 wk, o- = 13.6
wk, k = 5.7, and c = 0.050 wk-1. This leads to the dimensionless time equation of


t p 12crc t 26.0 0 t 16.7
x = --- -- +0.4825 -- x = 0.00
72a 2 19.3 19.3


The growth quantifiers for stem biomass and leaf biomass are given by (for xi = 0)


(4.74)


Table 4.10 Accumulation of stem biomass (Ys), leaf biomass (YL),
nelac dar time (t) by ele hanterass at
a


and leaf fraction (/J) with


t Ys YL fL X Qs QL
wk Mg ha-' Mg ha'
16.7 -- -- -- 0.000 0.000 0.000
19.6 0.91 4.55 0.83 0.150 0.071 0.168
23.6 4.64 6.05 0.57 0.358 0.386 0.387
27.6 10.12 9.05 0.47 0.565 0.878 0.576
31.6 17.16 10.48 0.38 0.772 1.44 0.725
35.6 21.18 12.00 0.36 0.979 1.98 0.834
39.6 25.90 11.86 0.31 1.187 2.43 0.907
43.6 31.31 13.43 0.30 1.394 2.76 0.951
47.6 35.46 13.13 0.27 1.601 2.97 0.976
00 3.22 1.000
aYield data are adapted from Woodard et al. (1993).


Qs = --[exp-exp- x2) exp(- x )] exp(4 Vcocx,)=
-77r


{- 3.216[exp(- x2 )-1.000. 1.000


QL = {[erf x erf x, J~ exp(V-2ocx,) = {1.000[erf x OD -1.000


(4.75)


(4.76)


Values for Qs and QL listed in Table 4.6 are calculated from Eqs. (4.75) and (4.76). Correlation
of biomass with growth quantifier is shown in Figure 4.25, where the lines are drawn from


Ys = 0.055 +11.3Qs


r= 0.9962


(4.77)









(4.78)


YL = 2.37 +11.2QL r = 0.9914


The intercept in Eq. (4.78) reflects accumulation of leaf biomass between mowing (t = 12.6 wk)
and time of initiation of significant growth (ti = 16.7 wk). The choice of k = 5.7 was made to
make the slopes in Eqs. (4.77) and (4.78) the same. It follows from Eqs. (4.77) and (4.78) that
leaf fraction can be calculated from


fL = L (4.79)
YL S

Accumulation of biomass by stem and leaf fractions are shown in Figure 4.26, where the curves
are drawn from Eqs. (4.77) through (4.79) with inputs from Table 4.10.
The theory can also be used to describe total biomass with calendar time. System growth
quantifier, Q, can be written as

Q = (1 kx)[erf x erf x,] [exp(-x2) exp(-x )] exp(-2orcx,)

= {l1.000[erf x- 0]- 3.216[exp(-x2) -1.000]}. 1.000 (4.80)
= 0, + QS

By definition, total biomass is the sum of stem and leaf fractions, so

Y = YL + Y (4.81)

Linear regression of Y vs. Q leads to

Y = 2.37 +11.3Q r = 0.9969 (4.82)

which is used to draw the line in Figure 4.27. Accumulation of total biomass with calendar time
is shown in Figure 4.28, where the curve is drawn from Eqs. (4.80) and (4.82).
This analysis lends support to the expanded model of crop growth and to partitioning of
biomass into light-gathering and structural components. The long growing season of
elephantgrass is reflected in the large value of a and the low value of c. The grass ages at a
slower rate than many crops and captures solar energy over a longer period of time. These two
parameters couple together so as to make xi = 0 for the experiment at Gainesville, FL. It can also
produce large amounts of biomass, with yields of 80 Mg ha-' reported in Puerto Rico (Vicente-
Chandler et al., 1959).

4.6.2 Study with Soybean at Ames, Iowa

Data for this analysis are taken from a field study with soybean (Glycine max L. Merr., cv
'Richland') by Hammond et al. (1951) at Ames, IA. The soils were Webster silt loam (fine-
loamy, mixed, superactive, mesic Typic Endoaquoll) and Clarion loam (fine-loamy, mixed,
superactive, mesic Typic Hapludoll). The former was flat, but well-drained and uniform; the









latter area was located on a small knoll where the slope varied from 1 to 4%, with variable
surface soil. Both soils had a pH of approximately 7. However, the Webster soil showed higher
fertility than the Clarion as evidenced by cation exchange capacity, adsorbed phosphorus, and
inorganic nitrogen. Planting date was 24 May 1946 (t = 20.4 wk). Beginning on 10 August (t =
31.9 wk) plant samples were divided into 'vegetative' and 'fruit' (seeds + pods) components. No
fertilizers were applied during the experiment.
Results are given in Table 4.11. Model parameters are chosen as: p = 26.0 wk, a = 5.66 wk, c
= 0.050 wk-', and k = 5. Dimensionless time and growth quantifier are given by

Table 4.11. Accumulation of biomass (Y) with calendar time (t) by soybean vegetation on two
soils at Ames, IA (1946).a
t x erfx exp(-x2) Q Y
wk Mg ha-
Webster Clarion
20.4 --- --- --- -- -- ---
23.6 --- --- --- -- 0.113 0.110
24.9 --- --- --- -- 0.226 0.228
25.3 0.112 0.126 0.987 0.000 -- --
26.3 0.238 0.264 0.945 0.187 0.614 0.535
26.9 0.312 0.341 0.907 0.335 0.785 0.737
27.9 0.438 0.464 0.826 0.630 1.41 1.05
28.9 0.562 0.573 0.729 0.966 1.93 1.40
29.9 0.688 0.670 0.623 1.323 2.63 2.00
30.9 0.812 0.749 0.517 1.672 3.44 2.82
31.9 0.938 0.815 0.415 2.004 4.69 3.40
32.9 1.062 0.867 0.323 2.300 5.43 3.88
33.9 1.188 0.907 0.244 2.550 6.25 4.21
34.9 1.312 0.9364 0.179 2.756 6.93 5.58
35.9 1.438 0.9580 0.127 2.919 6.69 4.24
36.9 1.562 0.9728 0.0870 3.044 5.69 3.62
37.9 1.688 0.9830 0.0580 3.134 5.51 3.39
38.9 1.812 0.9896 0.0374 3.198 5.67
39.7 1.912 0.9931 0.0258 3.234 5.34
00 1.0000 0.0000 3.313 ----- -----
aCrop data adapted from Hammond et al. (1951).


t-P, F2gc t t-26.0 t-24.4
x=,-+--=- +0.200=- ;x=0.112
x 2o- 2 8.00 8.00


Q = {(1- kx,)[erf x erf x,] [exp(-2) exp(-x )]}. exp(V2o-cx,)

= {0.438[erfx-0.126]-2.821[exp(-x2)-0.987]}. 1.046

Values ofx and Q in Table 4.11 are calculated from Eqs. (4.83) and (4.84), respectively.
Correlation of Y on Q is shown in Figure 4.29, where the lines are drawn from


(4.83)



(4.84)










Webster soil: Y =-0.14+ 2.37Q r = 0.9913

Clarion soil: Y = 0.08 +1.61Q r = 0.9949


(4.85)

(4.86)


where data for 26.3 < t < 33.9 wk have been used for calibration. Biomass accumulation for the
vegetative component is shown in Figure 4.30, where the curves are drawn from Eqs. (4.83)
through (4.86).
The next step is to analyze data for fruit (seeds + pods). Results are given in Table 4.12.
Time of initiation is assumed to be ti = 31.5 wk with other parameters the same as above.
Dimensionless time and growth quantifier are given by

Table 4.12. Accumulation of biomass (Y) with calendar time (t) by soybean fruit (seeds + pods)
on two soils at Ames, IA (1946).a
t x erfx exp(-x2) Q Y
wk Mg ha-1
Webster Clarion
31.5 0.8875 0.791 0.455 0.000
31.9 0.938 0.815 0.415 0.043 0.208 0.286
32.9 1.062 0.867 0.323 0.158 0.759 0.500
33.9 1.188 0.907 0.244 0.280 1.48 0.932
34.9 1.312 0.9364 0.179 0.395 1.80 1.68
35.9 1.438 0.9580 0.127 0.505 2.26 2.00
36.9 1.562 0.9728 0.0870 0.589 2.76 2.21
37.9 1.688 0.9830 0.0580 0.656 3.20 2.16
38.9 1.812 0.9896 0.0374 0.706 3.41
39.7 1.912 0.9931 0.0258 0.736 3.50
co 1.0000 0.0000 0.806 ----- -----
aCrop data adapted from Hammond et al. (1951).


t -/ P o-c t 26.0 t 24.4
x J- +-= +0.200 = ,
2a 2 8.00 8.00


xi = 0.8875


(4.87)


Q = (1 kx,)[erf x erf x, ] k- [exp(-x2) exp(-x? )] exp(. 2-cx6)

= 3.438[erf x 0.791]- 2.82 Iexp(-x2) 0.4551-1.426


(4.88)


Values ofx and Q in Table 4.12 are calculated from Eqs. (4.87) and (4.88). Correlation of Y with
Q is shown in Figure 4.31, where the lines are drawn from


Webster soil: Y = 0.011+ 4.74Q


r= 0.9974


(4.89)

(4.90)


Clarion soil: Y = 0.082 + 3.50Q r = 0.9796









Accumulation of fruit biomass with calendar time is shown in Figure 4.32, where the curves are
drawn from Eqs. (4.87) through (4.90).
The expanded growth model provides reasonable description of biomass accumulation with
calendar time for both vegetative and fruit components on the two soils. Linear correlation of
biomass with the growth quantifier is confirmed for both plant components. Production was
higher for Webster compared to Clarion soil. Final grain yields were 2.70 Mg ha-1 for Webster
compared to 1.62 Mg ha-1 for Clarion soil. Extractable soil P was 9.2 mg kg-' for Webster
compared to 2.9 mg kg-1 for Clarion. Furthermore, total cation exchange capacity was 427 meq
kg-' vs. 160 meq kg-1 for Webster compared to Clarion soils. Based on these factors, production
would be expected to be higher for the Webster compared to the Clarion soil. Overman and
Wilkinson (2003) have shown that soil erosion on hillsides can reduce soil fertility for corn.

4.6.3 Study with Soybean at Clayton, North Carolina

Data for this analysis are taken from a field study with soybean (cv 'Lee') during the period
1966 and 1967 by Henderson and Kamprath (1970) at Clayton, NC. The soil was Norfolk loamy
sand (fine-loamy, kaolinitic, thermic Typic Kandiudult). Planting date was approximately 10
May (t = 18.7 wk). Plant samples were collected every 10 days from June through September.
Starting 110 d after planting, plants were divided into fractions labeled 'vegetative' and 'seeds &
pods'. For the sake of brevity the latter is referred to as 'fruit'.
Results for the vegetative component are given in Table 4.13. Time of initiation is chosen as
ti = 23.6 wk. Model parameters are chosen as: p = 26.0 wk, u = 5.66 wk, c = 0.050 wk'1, and k =
5. Dimensionless time and growth quantifier become

Table 4.13. Accumulation of biomass (Y) with calendar time (t) by soybean vegetation for two
years at Clayton, NC.a
t x erfx exp(-x2) Q Y
wk Mg ha-1
1966 1967
18.7 --- --- --- -- -- ---
23.6 -0.100 -0.112 0.9900 0.000 -- --
24.4 0.000 0.000 1.000 0.134 0.445 0.457
25.9 0.188 0.210 0.965 0.532 0.943 1.72
27.3 0.362 0.391 0.877 1.03 1.55 2.59
28.7 0.538 0.553 0.749 1.61 2.52 5.19
30.1 0.712 0.686 0.602 2.20 3.25 6.50
31.6 0.900 0.797 0.445 2.79 4.27 7.09
33.0 1.075 0.872 0.315 3.25 5.15 8.99
34.4 1.250 0.923 0.210 3.61 6.24 10.89
35.9 1.438 0.9580 0.127 3.88 6.31 8.37
37.4 1.625 0.9784 0.0713 4.06 5.69 7.02
38.7 1.788 0.9884 0.0410 4.16 4.75 5.33
00 1.0000 0.0000 4.29 ----- -----
aCrop data adapted from Henderson and Kamprath (1970).









t 1 2o-c t 26.0 t 24.4
x =+ -- -+ 0.200 = --, x= -0.100
cr 2 8.00 8.00


Q = (1- kx,)[erf x erf x, ]- [exp(-x2) exp(-x ) }. exp(-2ccx.)

= {1.500[erf x + 0.112]- 2.821[exp(-x2) 0.9900 -. 0.9608


(4.91)



(4.92)


Values of x and Q in Table 4.13 are calculated from Eqs. (4.91) and (4.92), respectively.
Correlation of Y vs. Q for 24.4 < t < 34.4 wk is shown in Figure 4.33, where the lines are drawn
from


1966: Y = 0.008 +1.60Q


r = 0.9933


(4.93)

(4.94)


1967: Y = 0.082 + 2.82Q r = 0.9906


Accumulation of vegetative biomass with calendar time is shown in Figure 4.34, where the
curves are drawn from Eqs. (4.91) through (4.94). The drop in biomass at the later sampling
times reflect shedding of leaves by the plants.
The next step is to analyze data for fruit (seeds + pods). Results are given in Table 4.14.
Time of initiation is assumed to be ti = 33.3 wk with other parameters the same as above.
Dimensionless time and growth quantifier are given by

Table 4.14. Accumulation of biomass (Y) with calendar time (t) by soybean fruit (seeds + pods)
for two years at Clayton, NC.a
t x erfx exp(-x2) Q Y
wk Mg ha-1
1966 1967
33.3 1.112 0.884 0.290 0.000
34.4 1.250 0.923 0.210 0.075 1.09 1.90
35.9 1.438 0.9580 0.127 0.191 2.09 4.43
37.4 1.625 0.9784 0.0713 0.291 3.13 6.50
38.7 1.788 0.9884 0.0410 0.353 3.80 8.55
co 1.0000 0.0000 0.451 ----- -----
aCrop data adapted from Henderson and Kamprath (1970).


t r2cc t 26.0 t 24.4
x 2o- 2 8.00 8.00

Q = {(1 kx, )[erf x erf x, ]- k- [exp(-x2) exp(-x2 ) ]} exp('F2o-cx1)

={- 4.562[erfx 0.884]- 2.82 l[exp(-x2) -0.290o}. 1.560


(4.95)



(4.96)








Values ofx and Q in Table 4.14 are calculated from Eqs. (4.95) and (4.96). Correlation of Y with
Q is shown in Figure 4.35, where the lines are drawn from

1966: Y = 0.307 +9.76Q r= 0.9986 (4.97)

1967: Y = 0.042 + 23.3Q r = 0.9959 (4.98)

Accumulation of fruit biomass with calendar time is shown in Figure 4.36, where the curves are
drawn from Eqs. (4.95) through (4.98).
The question naturally occurs as to the basis for the difference in response of fruit with time
between the two years. One possibility is difference in water availability between the two years.
Seasonal rainfall was 34.4 cm for 1966 and 55.6 cm for 1967. Overman and Scholtz (2002b)
assumed an exponential relationship between water availability and yield for corn of the form


Y = Y 1- exp R- o] (4.99)


where R is seasonal rainfall, Y is biomass yield, Ym is maximum yield at high R, Ro is intercept
value of R for Y= 0, and R'is characteristic rainfall. Soybean fruit yields are estimated as 4.15
Mg ha-1 (1966) and 9.22 Mg ha-' (1967) at calendar time t = 40 wk from Figure 4.36. This result
is shown in Figure 4.37. We estimate Ro = 25 cm, R'= 25 cm, and Ym = 13.0 Mg ha-i for this
case. The curve in Figure 4.37 is drawn from

Y = 13.0 1- exp R 5) (4.100)


While the estimates used here are very approximate, this result does emphasize the sensitivity of
crop production to water availability. The results of this study indicate that the effect of water
availability can be accounted for in the yield factor A with the basic model parameters remaining
constant. This is illustrated in Figures 4.34 and 4.36. This of course simplifies model application.
Estimates of partitioning of biomass between light-gathering and structural components can
be made from the expanded growth model. Following the same procedure as in 4.5.3 for peanut,
we obtain for the growth quantifiers

t-p V2-c t-26.0 t-24.4
x- 8.+0 0.200- xi=-0.100 (4.101)


QL = {[erf x erf x, ]} exp(,cxj) = {[erf x + 0.112]} 0.9608 (4.102)


Qs = kx, [erf x erf x, ]-- [exp(- x2 )- exp(- x?)] -l exp(V-4ccx,)
L x + 0 1 2J ) (4.103)
= {0.500[erfx + 0.112]- 2.82 exp(- x2)- 0.9900-. 0.9608









The light-gathering fraction is then given by


(4.104)


fL = QL
QL + QS


Values are listed in Table 4.15 and shown in Figure 4.38. Note thatfL drops to approximately
25% at t = 40 wk. This occurs as the plants age and the structural component becomes more
dominant.


Table 4.15. Estimated partitioning of biomass
for soybean at Clavton. NC (1970V-


between light-gathering and structural components


t x erfx exp(-x2) QL Qs Q fL
wk
23.6 -0.100 -0.112 0.9900 0.000 0.000 0.000 1.000
25 0.075 0.085 0.9944 0.189 0.083 0.272 0.695
26 0.200 0.223 0.961 0.322 0.240 0.562 0.573
27 0.325 0.354 0.900 0.448 0.468 0.916 0.489
28 0.450 0.475 0.817 0.564 0.751 1.315 0.429
30 0.700 0.678 0.613 0.759 1.401 2.160 0.351
32 0.950 0.821 0.406 0.896 2.031 2.927 0.306
34 1.200 0.910 0.237 0.982 2.532 3.514 0.279
36 1.450 0.960 0.122 1.030 2.868 3.898 0.264
38 1.700 0.984 0.0556 1.053 3.059 4.112 0.256
40 1.950 0.9942 0.0223 1.063 3.154 4.217 0.252
42 2.200 0.9981 0.00791 1.067 3.195 4.262 0.250
44 2.450 0.9995 0.00247 1.068 3.211 4.279 0.250
00oo 1.0000 0.00000 1.068 3.218 4.286 0.249


4.6.4 Study with Potato at Old Town, Maine

Data for this analysis are taken from a field study with potato (Solanum tuberosum L., cv
'Kennebec') by Carpenter (1963) at Old Town, ME for the period 1957-1959. Only data for
1959 are used in this analysis. Soil type was unspecified. Planting date was not specified and is
assumed to be late May (t = 20 wk). Nitrogen application rates for the study were 0, 50, 67, 84,
and 134 kg ha-1. Phosphorus and potassium rates were each 270 kg ha-i. All treatments were
replicated four times. Measurements were made of biomass and plant nitrogen for vegetative
(tops) and tuber components of the plant. Data are given in Table 4.16.
The first challenge is to calibrate the growth model with time. Model parameters are assumed
to be p = 26.0 wk, Vo- = 8.0 wk, c = 0.10 wk-', k = 5. Biomass and plant N accumulation for
the vegetative component at applied N of 134 kg ha-' are shown in Figure 4.39. Time of
initiation is assumed to be ti = 25.2 wk. This leads to dimensionless time, x, and growth
quantifier, Qv, of








Table 4.16 Accumulation of vegetative (Yv) and tuber (Y,) biomass and plant nitrogen for
vegetative (Nv,) and tuber (Nut) with calendar time (t) for different applied nitrogen (N) for potato
at Old Town, ME (1959)a
N t, wk 25.6 26.4 27.7 28.6 29.6 30.6 31.6 32.6
kg ha-1
0 Y,, Mg ha'' 0.083 0.120 0.208 0.262 0.476 0.657 0.642 0.644
Y,, Mg ha-1 --- --- 0.055 0.052 0.171 0.353 0.504 0.834
N,,, kg ha-' 2.65 3.51 6.13 7.65 15.5 20.0 18.3 17.8
Nu,, kg ha' -- -- 0.75 0.71 2.64 5.30 8.49 13.2

50 Yv, Mg ha-' 0.230 0.445 0.715 0.898 1.13 1.42 1.54 1.54
Y,, Mg ha1 --- --- 0.199 0.448 0.600 1.47 2.00 2.93
N., kg ha-1 10.5 17.9 25.1 29.6 35.6 41.3 42.6 39.5
Nut, kg ha-1 -- -- 3.38 6.91 9.11 21.4 32.1 44.7

67 Y, Mg ha' 0.215 0.561 0.810 1.24 1.74 2.58 2.38 2.36
Y,, Mg ha --- ------- 0.152 0.484 0.878 1.66 2.59 3.58
N, kg ha- 9.28 26.3 30.0 42.9 59.6 77.8 66.5 55.4
Nt, kg ha -- -- 2.93 8.06 14.4 26.3 38.1 53.9

84 Y,, Mg ha-1 0.256 0.489 0.989 1.47 1.95 2.23 2.03 2.53
Y,, Mg ha' --- --- 0.181 0.540 1.09 2.58 2.41 4.75
N,,, kg ha-1 12.4 22.8 39.0 51.7 61.2 65.7 54.2 58.8
Nut, kg ha -- -- 3.45 8.74 17.9 28.1 36.1 74.8

134 Y,, Mg ha-' 0.233 0.618 1.20 1.78 2.46 3.09 3.65 3.60
Yt, Mg ha' ------- ------- 0.181 0.484 1.06 2.04 4.16 4.82
N,, kg ha-' 12.4 31.0 47.4 67.5 85.7 104 103 82.0
N., kg ha-' -- -- 3.81 8.77 20.3 35.3 54.7 80.9
aData adapted from Carpenter (1963).

t- p -ocr t-26 t- 22.8
x = +2- t-26 +0.4= -22.8 xi=0.300 (4.105)
2 8 8


Q, = (1- kx,)[erf x erf x, F7[ [exp(- x2 exp(- x ) exp0c6cx,)
SJ (4.106)
= {-0.500[erf x 0.329]- 2.821[exp(- x2 )-0.914]}. 1.271

Values are listed in Table 4.17 along with the crop data. Correlation of biomass, Yv, with the
growth quantifier is shown in Figure 4.40, where the line is drawn from


Y, = 0.069 + 1.91Q, r = 0.9992


(4.107)









with a correlation coefficient ofr = 0.9992. Note that the intercept (0.069) is essentially zero, in
accordance with the model. The phase plots (Nuv and Y/Nuv vs. Y,) is shown in Figure 4.41, where
the line and curve are drawn from

Table 4.17 Calibration of the growth model for accumulation of vegetative biomass (Y,) and plant
nitrogen (Nuv) with calendar time (t) at applied nitrogen (N) of 134 kg ha-1 for potato grown at Old
Town, ME (1959)a
t x erfx exp(-x2) Q Yv N.v Nc,
wk Mg ha'1 kg ha'' g kg-1
25.2 0.300 0.329 0.914 0
25.6 0.350 0.379 0.885 0.072 0.233 12.4 53.2
26.4 0.450 0.475 0.817 0.255 0.618 31.0 50.2
27.7 0.612 0.613 0.687 0.633 1.20 47.4 39.5
28.6 0.725 0.695 0.591 0.926 1.78 67.5 37.9
29.6 0.850 0.771 0.486 1.254 2.46 85.7 34.8
30.6 0.975 0.832 0.386 1.573 3.09 104 33.7
31.6 1.100 0.880 0.298 1.858 3.65 103 28.2
32.6 1.225 0.917 0.223 2.104 3.60 82.0 22.8
00 1.000 0.000 2.851 -- --
aCrop data adapted from Carpenter (1963).


- = 0.0181 + 0.00440Y, r = 0.9734
Nuv

N 227Y,
4.11 + Y,

It follows that plant N concentration, Ncv, is given by

N ,= 227
Yv 4.11+Y,


(4.108)


(4.109)


(4.110)


Biomass and plant N accumulation by tubers at applied N of 134 kg ha1- are given in Table
4.18 and shown in Figure 4.42. Time of initiation is assumed to be t4 = 28.3 wk. This leads to
dimensionless time, x, and growth quantifier, Q,, of


t -/. 2 ocr t 26 t 22.8
x -+ --= -+0.4= --
o- 2 8 8


xi = 0.6875


(4.111)


Q, = {(- kx )[erf x erf x, ]-- [exp(- x2)-exp(-x )}. exp(2-cxj)

= {- 2.438[erf x 0.670]- 2.821[exp(- x2)- 0.623]}. 1.733


(4.112)










Table 4.18 Calibration of the growth model for accumulation of tuber biomass (Y,) and tuber
nitrogen (Nu) with calendar time (t) at applied nitrogen (N) of 134 kg ha-1 for potato grown at
Old Town, ME (1959)a
t x erfx exp(-x2) Q Yt Nut Nc
wk Mg ha'-1 kg ha'-1 g kg-1
28.3 0.688 0.670 0.623 0.000 -- -- 19.3
28.6 0.725 0.695 0.591 0.051 0.484 8.77 18.1
29.6 0.850 0.771 0.486 0.243 1.06 20.3 19.2
30.6 0.975 0.832 0.386 0.474 2.04 35.3 17.3
31.6 1.100 0.880 0.298 0.702 4.16 54.7 13.1
32.6 1.225 0.917 0.223 0.912 4.82 80.9 16.8
00oo 1.000 0.000 1.651 ----- -- ---
aCrop data adapted from Carpenter (1963).

Correlation of biomass, Y,, with the growth quantifier is shown in Figure 4.41, where the line is
drawn from

Y, = -0.069 + 5.42Q, r = 0.9824 (4.113)

with a correlation coefficient of r = 0.9824. Note that the intercept (-0.069) is essentially zero, in
accordance with the model. The phase plot (Nut and Y/Nut vs. Y,) is shown in Figure 4.43, where
the line and curve are drawn from

-- = 0.0519 + 0.00328Y, r = 0.6727 (4.114)
Nut

305Y,
N,, 305Y, (4.115)
15.8+ YI

It follows that plant N concentration, Nc,, is given by

N,, 305
N, = = 305 (4.116)
Y, 15.8+ Y

The second challenge is to calibrate the logistic equations. Response of vegetative and tuber
components to applied N at sampling time t = 32.6 wk is shown in Figure 4.44. While there is
significant scatter in the data, several things can be noted. Plant N concentration appears to be
essentially independent of applied N. This requires that parameters by and b, be equal. Plant N
uptake appears to be equal for vegetative and tuber components, which requires that parameter
A, be equal for the two components. By inspection of Figure 4.44, plant N uptake for vegetation
and tubers, Nuv and Nu, is given by









Nuv, = Nu, = (4.117)
1 + exp(1.65 0.0278N)

Now average plant N concentrations are Ncv = 23.8 g kg-1 and Nc, = 15.8 g kg-1 for vegetation and
tubers, respectively. It follows that biomass yields are given by


vegetation: Y, = 4.00 (4.118)
1+ exp(1.65 0.0278N)


tubers:


6.00
1 + exp(1.65 0.0278N)


Plant N concentration follows from the definition


N, -N -95 23.8
Y,, 4.00

N 95
Nt = 15.8
Y, 6.00


(4.119)


(4.120)



(4.121)


Curves in Fig. 4.44 are drawn from Eqs. (4.117) through (4.119), while the lines are drawn from
Eqs. (4.120) and (4.121).
The final challenge is to couple biomass production with applied N and time of growth. This
can be accomplished by writing for N = 134 kg ha-1


vegetation:


Y4 = A.. Q, = Q = 1.91Q, -> A = 2.15 Mg ha-'(4.122)
1+ exp(1.65 0.0278N) 1.126


A
Yr = e- Q. -
1 + exp(1.65 0.0278N)


A Q, = 5.42Q, -- A, = 6.10 Mg ha-' (4.123)
1.126


The yield equations can now be written in the final form


2.15 Q
1 + exp(1.65 0.0278N)A

6.10 1
1 + exp(1.65 0.0278N)Q


(4.124)



(4.125)


It is also necessary to generalize the parameters in the phase relations. For N = 134 kg ha-1 these
become


tubers:


vegetation:


tubers:


Y, =










vegetation: N = uMNum =" 227 -> N,, = 255 kg ha-' (4.126)
i 1+ exp(1.65 -0.0278N) 1.126 2


K, = K= = 4.11 -> K = 4.63 Mg ha- (4.127)
1+ exp(1.65-0.0278N) 1.126


tubers: Num7 = -AN,,m Nu = 305 -+ N = 343 kg ha-1 (4.128)
1+ exp(1.65-0.0278N) 1.126


K, = K, = K' = 15.8 -> K = 17.8 Mg ha-1 (4.129)
1+ exp(1.65-0.0278N) 1.126 '"

The phase parameter equations can now be written as

255
vegetation: NA., = (4.130)
1 + exp(1.65 0.0278N)


4.63
K, = 4.63 (4.131)
1 + exp(1.65 0.0278N)

343
tubers: Nu., = (4.132)
1 + exp(1.65 0.0278N)

17.8
K, = 17.8 (4.133)
1 + exp(1.65 0.0278N)

The prediction equations for plant N uptake and plant N concentration still retain the form


N =
K +Y


Nc =- Num (4.135)
Y Ky+Y

A summary of growth model parameters (A, Num, and Ky) is given in Table 4.19. Estimates of
A are from Eqs. (4.124) and (4.125), while those for Num and Ky are from Eqs. (4.130) through
(4.133). Estimates for other N can be made as well. Appropriate estimates of growth quantifier
(Q) can be made from Eqs. (4.106) and (4.112) for vegetative and tuber components,
respectively.
This analysis has demonstrated several characteristics of the expanded growth model for
potato. Biomass accumulation follows linear dependence on the growth quantifier for both
vegetative and tuber components of the plant. The hyperbolic phase relation between plant N and









biomass accumulation has been confirmed for both plant components. Dependence of model
parameters on applied N has also been established.

Table 4.19 Estimates of growth model parameters yield factor (A), potential maximum plant N
uptake (Num), and yield coefficient (Ky) as related to applied nitrogen (N) for vegetative and tuber
components of potato grown at Old Town, ME.
N A Num Ky A Num Ky
kg ha-1 Mg ha'1 kg ha-1 Mg ha'' Mg ha-1 kg ha-' Mg ha'1
vegetation tubers
0 0.35 41 0.75 0.98 55 2.9
50 0.94 111 2.02 2.66 149 7.7
67 1.19 141 2.56 3.37 190 9.8
84 1.43 170 3.08 4.06 228 11.8
134 1.91 227 4.11 5.42 305 15.8


4.6.5 Study with Alfalfa at Guelph, Ontario, Canada

Data for this analysis are taken from a study with alfalfa (Medicago sativa L., cv 'Vernal') at
Guelph, Ontario, Canada by Fulkerson (1983). Plant samples were collected at different stages of
growth beginning on May 22. Results for dry matter for leaves (YL) and stems (Ys) are given in
Table 4.20. For these data it is assumed that t1 = 17.3 wk, and that p = 26 wk and a = 5.66 wk
from previous analysis (Overman and Scholtz, 2006). To facilitate data analysis it is further
assumed that x, = 0 (following Overman and Woodard, 2006), so it follows that

Table 4.20 Accumulation of stem (Ys) and leaf (YL) biomass with time (t) for alfalfa grown at
Guelph, Ontario, Canadaa
t x erfx exp(-x2) QL Qs YL Ys
wk Mg ha'' Mg ha-1
17.3 0.000 0.000 1.000 0.000 0.000
20.3 0.375 0.404 0.869 0.404 0.259 1.14 0.61
21.1 0.475 0.499 0.798 0.499 0.399 1.62 1.12
22.1 0.600 0.604 0.698 0.604 0.596 1.67 1.73
23.1 0.725 0.695 0.591 0.695 0.808 2.02 2.43
24.1 0.850 0.771 0.486 0.771 1.015 2.22 2.93
25.1 0.975 0.832 0.386 0.832 1.213 2.44 3.46
26.3 1.125 0.888 0.282 0.888 1.418 2.49 3.89
27.1 1.225 0.917 0.223 0.917 1.535 2.72 4.40
28.1 1.350 0.944 0.162 0.944 1.655 2.36 4.87
29.1 1.475 0.963 0.114 0.963 1.750 2.14 4.68
00 1.000 0.000 1.000 1.975 ----- ---
aYield data adapted from Fulkerson (1983).


ti -/ p 2o-c 2
x=- = +---= 0 -2 cO2 =


p-ti, =26-17.3=8.7wk -> c=0.27wk-'


(4.136)










Dimensionless time (x) can now be related to calendar time (t) by

t P -crc t-26 t-17.3
x=- +- =- +1.080- xi;=0
o- 2 8 8

The growth quantifiers for leaves (QL) and stems (Qs) can be written as

QL = erf x

Qs= -1.975exp( x2)-]

by assuming k = 3.5. Linear regression of data in Table 4.20 leads to

YL = 0.059 + 2.83QL r = 0.9879

is = 0.017 + 2.81Qs r = 0.9959


(4.137)


(4.138)

(4.139)



(4.140)

(4.141)


where the intercepts in Eqs. (4.140) and (4.141) are essentially zero. The value of parameter k
has been chosen to make the slopes of Eqs. (4.140) and (4.141) equal. Correlation between
biomass and growth quantifier is shown in Figure 4.45, where the lines are drawn from Eqs.
(4.140) and (4.141).
Attention is now turned to biomass and plant nitrogen accumulation for the whole (above
ground) plant. Data are given in Table 4.21. The hyperbolic relation between plant N uptake and


Table 4.21 Accumulation of biomass (Y), plant N uptake (Nu), and plant N
with time (t) for alfalfa grown at Gue aa


concentration (Nc)


t x erfx exp(-x2) Q Y N. Nc Y/Nu
wk Mg ha-' kg ha-1 g kg-' kg g-
17.3 0.000 0.000 1.000 0.000 -- -- -- ---
20.3 0.375 0.404 0.869 0.663 1.75 -- -- ---
21.1 0.475 0.499 0.798 0.898 2.74 --- -- ---
22.1 0.600 0.604 0.698 1.200 3.40 124 36.5 0.0274
23.1 0.725 0.695 0.591 1.503 4.45 144 32.4 0.0309
24.1 0.850 0.771 0.486 1.786 5.15 153 29.7 0.0337
25.1 0.975 0.832 0.386 2.045 5.90 160 27.3 0.0366
26.3 1.125 0.888 0.282 2.306 6.38 162 25.2 0.0397
27.1 1.225 0.917 0.223 2.452 7.12 171 24.0 0.0417
28.1 1.350 0.944 0.162 2.600 7.23 162 22.4 0.0446
29.1 1.475 0.963 0.114 2.713 6.82 151 22.1 0.0452
co 1.000 0.000 2.975-- ---- --- --------
aPlant data adapted from Fulkerson (1983).









yield can be written as


N,, Y
N = u" (4.142)
K +Y

where N,, is plant N accumulation, kg ha-'; Num is potential maximum plant N uptake, kg ha'1;
and Ky is yield response coefficient, Mg ha-1. Equation (4.142) can rearranged to the linear form

y Ky 1
-= -+-Y (4.143)
N N,, N ,,n

The growth quantifier for the whole plant becomes

Q = {.000[erf x 0]-1.975[exp(- x2)-1. 1.000 (4.144)

with values listed in Table 4.21. Linear regression ofbiomass (Y) vs. growth quantifier (Q) leads
to

Y = 0.086 + 2.81Q r = 0.9973 (4.145)

where the correlation is shown in Figure 4.46. The phase relations now become

Y K, 1
-= + Y = 0.0121+0.00428Y r= 0.9896 (4.146)
SN.N Nm

NY 2347
N,, un- = ---- (4.147)
Ky +Y 2.82+Y

as shown in Figure 4.47, where the line is drawn from Eq. (4.146) and the curve from Eq.
(4.147). Plant nitrogen concentration can be estimated from

N N, 234
Nc =--y-=2 (4.148)
Y 2.82 + Y

Results are shown in Figure 4.48, where the curves are drawn from Eqs. (4.145), (4.147), and
(4.148).
The final step is to couple plant N and biomass accumulation for leaves and stems. It is
assumed that Eqs. (4.142) and (4.143) apply for leaves and stems separately. Data are given in
Table 4.22 for each plant component at each sampling time. Regression analysis of the data leads
to the equations









Table 4.22 Growth response of dry matter (Y), plant N concentration (Nc), and plant N uptake
(N,) with calendar time (t) for alfalfa leaves and stems growth at Guelph, Ontario, Canadaa
t Y Nc Nu Y/Nu Y Nc N, Y/N.
wk Mg ha'' g kg-' kg ha-' kg g-1 Mg ha-' g kg-'1 kg ha-1 kg g-'
Leaves Stems
20.3 1.14 -- -- --- 0.61 -- -- ---
21.1 1.62 -- -- --- 1.12 -- -- ---
22.1 1.67 50.9 85.0 0.0196 1.73 22.4 38.8 0.0446
23.1 2.02 48.5 98.0 0.0206 2.43 19.0 46.2 0.0526
24.1 2.22 46.6 103 0.0215 2.93 17.0 49.8 0.0588
25.1 2.44 43.5 106 0.0230 3.46 15.7 54.3 0.0637
26.3 2.49 41.6 104 0.0240 3.89 14.9 58.0 0.0671
27.1 2.72 39.2 107 0.0255 4.40 14.6 64.2 0.0685
28.1 2.36 38.4 90.6 0.0260 4.87 14.7 71.6 0.0680
29.1 2.14 37.0 79.2 0.0270 4.68 15.4 72.1 0.0649
aData adapted from Fulkerson (1983).


Stems: Ys K + Ys = 0.0301+ 0.00928Ys r = 0.984 (4.149)
N N N
uS umS umS
NN sYs 108Ys
Nus S S (4.150)
Ks +Ys 3.24 + Ys

Ns 108 (4.151)
Ys 3.24+ Ys

Leaves: -L_ + L = 0.00946 + 0.00571YL r = 0.9700 (4.152)
NuL NumL NumL
NA Y 175YL
N.LA UMLL (4.153)
KL + YL 1.66 + YL
N L 175
NcL -u- (4.154)
YL 1.66 + YL

The correlations are shown in Figure 4.49, where the lines are drawn from Eqs. (4.149) and
(4.152) and with curves drawn from Eqs. (4.150), (4.151), (4.153), and (4.154). Biomass and
plant N accumulation curves are shown in Figure 4.50 for stem and leaf components, where the
curves are drawn from Eqs. (4.140), (4.141), (4.150), (4.151), (4.153), and (4.154). Simulations
describe data rather well up to t = 28 wk, when leaf mass begins to decline.
This analysis has confirmed the linear relationship between biomass accumulation and the
growth quantifier of the expanded growth model (Figures 4.45 and 4.46). Hyperbolic coupling of
plant nitrogen and biomass accumulation has also been confirmed (Figures 4.47 and 4.49). The
model describes accumulation of plant biomass and nitrogen with time rather well for whole
plant (Figure 4.48) and for leaves and stems (Figure 4.50). The phase relations suggest that the
rate limiting process in plant growth is biomass accumulation by photosynthesis, and that plant N
accumulation exhibits virtual equilibrium with biomass accumulation. This same issue occurs in
the study of chemical rate processes, where equilibrium or steady state processes often have a









dynamic basis (Laidler 1950). An example is the Michaelis-Menten approximation of the Briggs-
Haldane model of enzyme kinetics.
Most field experiments measure and only report biomass and plant nutrient accumulation for
the whole plant. The postulates of the expanded growth model have been confirmed for this case
as well, so that the model can be used to describe results for such cases.
It may be noted that these results are self-consistent by examining the upper limits of the
values. These are given by

QL =lim QL = 1.000 YL = lim YL = 2.89 Mg ha- NuL, = lim NL = 111 kg ha-
X--co x-).co j----o

Qs =limQs = 1.975 Ys. =limYs =5.57 Mg ha -- Ns = lim Ns =68 kg ha-
S.-.o X-)--oo X.-oo

Q. = lim Q= 2.975 -- Y. = lim Y= 8.45 Mgha-1 N = lim N =175 kgha-1
X''+W X"-*a X--.o

Note that the sums of lines 1 and 2 are equal to or very close to line 3.

4.6.6 Study with Corn at Tallahassee, Florida

Data for this analysis are taken from a field study at the Southeast Farm at Tallahassee, FL
with corn (cv 'DeKalb T 1100') on Kershaw fine sand (fine-sand, siliceous, thermic
Quartzipsamment) by Overman and Scholtz (2004). Plant population was approximately 8.5
plants m-2, with a planting date of 6 March 1990 (t = 9.3 wk). Measurements were made of
biomass and mineral elements (N, P, K, Ca, Mg) on a weekly basis between 15 April 1990 and 8
July 1990. During the 19 wk growing season rainfall was 26 cm and irrigation with municipal
reclaimed water was 134 cm through a center pivot unit. Nutrient applications through reclaimed
water and supplemental fertilizer were: N-P-K-Ca-Mg = 143-13-262-470-150 kg ha-1. Results for
the ear component of the plant are given in Table 4.23 and shown in Figure 4.51 for biomass
(YE), plant N uptake (NuE), and plant N concentration (NcE) with calendar time (t). Model
parameters are chosen the same as in Section 4.5.2: p = 26.0 wk, a = 5.66 wk, k = 5, and c =
0.206 wk'-. This leads to the equations for x and Q of

Table 4.23 Accumulation of biomass and plant nitrogen by corn ears at Tallahassee, FL (1990)a
t x erfx exp(-x2) QE YE NuE NcE
wk Mg ha-' kg ha'-1 g kg'
19.4 0.000 0.000 1.000 0.000 --- --
21.3 0.238 0.264 0.945 0.419 2.46 33.5 13.6
22.3 0.362 0.391 0.877 0.738 3.80 39.2 10.3
23.3 0.488 0.510 0.788 1.108 5.84 51.4 8.80
24.3 0.612 0.613 0.687 1.496 8.80 80.6 9.16
25.3 0.738 0.703 0.580 1.888 10.89 101 9.27
26.3 0.862 0.777 0.475 2.258 10.97 123 11.2
27.3 0.988 0.838 0.377 2.595 11.85 113 9.54
00 1.000 0.000 3.821 ----- ------- ------
aCrop data adapted from Overman and Scholtz (2004).








t 2 o-c t 26.0 08 t 19.4
x = + -- ----8.0+ 0.825=- x- =0 (4.155)
2o 2 8.00 8.00


QE= {(l kxi )[erf x erf x,] -= [exp(- x2 exp(- x? )] exp(ox,)(4.156)

= {1.000[erf x 0]- 2.82 l[exp(-x2 )-1]}. 1.000

Values of QE vs. t in Table 4.23 are calculated from Eqs. (4.155) and (4.156). Correlation of ear
biomass, YE, with QE is shown in Figure 4.52, where the line is drawn from

YE = -0.30 + 5.91QE r = 0.9964 (4.157)

with the linear correlation coefficient r = 0.9964.
The phase plots for ears (NuE and YE/NuE vs. YE) are shown in Figure 4.53, where the line and
curve are drawn from

YE = 0.0840 + 0.00239YE r = 0.634 (4.158)
N.uE

419Y
NuE E (4.159)
35.2+ YE

It follows from Eq. (4.159) that plant N concentration is given by

N E 419
NcE -352Y (4.160)
E 35.2 + YE

The curves in Figure 4.51 are drawn from Eqs. (4.157), (4.159), and (4.160).
The final step is to partition ear data (comprised of shucks, cobs, and grain) for biomass into
light-gathering and structural components. To do this the growth quantifier, given by Eq. (4.156),
is partitioned into light-gathering (QLE) and structural (QsE) components given by

QLE = {1.000[erf x O]} -1.000 = erf x (4.161)

QSE = -2.821exp(- x2)- j1.000 (4.162)

Biomass for the two components is then estimated from

YLE = 5.91QLE (4.163)

YsE = 5.91QsE (4.164)









to be consistent with Eq. (4.157) with YE = YLE + YSE and QE = QLE + QSE. The fraction of light-
gathering component is then estimated from


fLE = YE- QLE (4.165)
LE + SE QLE + QSE

Results are shown in Figure 4.54, where the curves are drawn from Eqs. (4.163) through (4.165).
This analysis suggests that part of the shuck biomass functions as a light-gathering component of
the plant, which seems reasonable since the outer shuck contains chlorophyll and conducts
photosynthesis. The fraction decreases with time, as would be expected as the plant ages.
In the Florida study, biomass was partitioned into stalks, leaves, and ears. The analysis
confirms the linear relation between Yand Q for ears (Figure 4.52), using the same parameters as
for the whole plant.Data for corn ears exhibit more scatter than for the whole plant.
The model is then used to partition ear data into light-gathering and structural components. It
is concluded that the fraction of biomass in the light-gathering component shifts from
approximately 100% at the beginning of ear formation (t = 19.4 wk) to 30% at the hard dent
stage (t = 28 wk). This implies that photosynthesis occurs in the outer shuck, which seems
reasonable due to the green color. Independent data are needed to verify this conclusion.
Data are given in Table 4.24 for mineral composition of corn ears for sampling times listed.

Table 4.24 Accumulation of biomass (YE), plant phosphorus (PuE), potassium (Ku,), calcium
(Ca,E), and magnesium (MgE) with calendar time (t) by corn ears at Tallahassee, FL (1990)a
t YE PuE YE/PuE KuE YE/KuE CauE YE/CauE MguE YE/MgE
wk Mg ha-1 kg ha"' kg g-" kg ha-' kg g-1 kg ha-1 kg g-1 kg ha-1 kg g-1
19.4 ------- ------ ----- ----- ------ ---- ...---- ---...
21.3 2.46 6.9 0.357 18.6 0.132 1.5 1.6 3.2 0.77
22.3 3.80 10.3 0.369 22.3 0.170 2.0 1.9 3.0 1.27
23.3 5.84 14.6 0.400 29.0 0.201 2.1 2.8 4.0 1.46
24.3 8.80 23.4 0.376 38.2 0.230 2.3 3.8 6.6 1.33
25.3 10.89 26.2 0.416 36.7 0.297 1.6 6.8 7.0 1.56
26.3 10.97 29.1 0.377 36.4 0.301 2.7 4.1 7.9 1.39
27.3 11.85 32.0 0.370 40.0 0.296 2.8 4.2 8.8 1.35
aCrop data adapted from Overman and Scholtz (2004).

Phase plots for plant P, K, Ca, and Mg are shown in Figures 4.55 through 4.58, where the lines
and curves are drawn from

Y 186Y
Plant P: -- =0.349+0.00538YE Pu = 186Y (4.166)
PuE 64.9 + YE


Y 56.2Y4
Plant K: -E =0.0936+0.0178 YE KuE 562Y (4.167)
KuE 5.26 + YE









YE =0.944+0.291YE
Ca uE


Plant Mg: -E 0.894 + 0.0565YE
Mgu.E


- CaE 3.43Y
3.24 + YE

17.7YE
15.8 +YE


Data for the corn ears exhibit considerably more scatter than for the whole plant. The data
generally follow the hyperbolic phase relations.
The question naturally occurs as to connections of accumulation among the various mineral
elements by the plant as growth occurs. Perhaps the logical choice as a base for comparison is
plant N concentration. Equations (4.159) and (4.166) though (4.169) provide the framework for
the calculations of concentration ratios for corn ears. The prediction equations are


P = 35.2 + YE
Plant P/N; cE = 0.444
NcE 64.9 + YE


(4.170)


Plant K/N;



Plant Ca/N;



Plant Mg/N;


KcE -0.134[35.2+ Y,
NcE 5.26+YE]

CaE 0.0081935.2 + YE
NcE 3.24 + YE


Mg 0.0422 35.2 +YE
NcE L 15.8 + YE


Results are shown in Figures 4.59 through 4.62, where the data are taken from Tables 4.23 and
4.24. While there is considerable scatter in the data, the general trends are apparent. The ratio
PcE/NcE increases slightly as the plants grow. The ratios for the other elements all show
appreciable decline with plant growth.

4.7 Harvest Interval and Plant Digestibility

It was established in Section 4.1 that biomass yield of a warm-season perennial is dependent
on time interval between harvests. The mathematical relationship for a constant harvest interval
is given by


Y = (a, + f, At)exp(- yAt)


(4.174)


where Y is seasonal total biomass and At is harvest interval. Parameters a, fl, y are to be
estimated from data analysis. It was shown in Section 4.2 that Eq. (4.174) follows from the
expanded growth model. It is known that digestibility of forage grasses declines as the plants


Plant Ca:


(4.168)


(4.169)


(4.171)



(4.172)



(4.173)








age. In this section we hypothesize that dependence of digestible biomass, D, is related to harvest
interval by

D = (ad + /dAt)exp(- yAt) (4.175)

with parameters ad and f/d to be determined from data analysis. It follows from Eqs. (4.174)
and (4.175) that dependence of digestible fraction,fd, on harvest interval is given by

D ad+I dAt
fd = D d +dAt (4.176)
Y ay +,6yAt

The question is how to estimate parameters from data. If it assumed that parameter 7 is
known, then Eqs. (4.174) and (4.175) can be written in the standardized forms

y* = Yexp(yAt)= ay +/yAt (4.177)

D* = Dexp(yAt)=ad +, fdAt (4.178)

Regression of Y* and D* on At leads to estimates of the linear parameters. Once model
parameters are known, estimates of harvest intervals for peak production, Atpy and Atpd, for yield
and digestible biomass can be estimated from

1 a
At py- (4.179)
fly

Atpd ad (4.180)


Data from two separate field studies are now used to illustrate the procedures.

4.7.1 Study with Bermudagrass at Tifton, Georgia

Data for this analysis are taken from a field study with coastal bermudagrass at Tifton, GA
by Burton et al. (1963). Plots were replicated seven times on Tifton loamy sand (fine-loamy,
kaolinitic, thermic Plinthic Kandiudult). Applied nitrogen was 672 kg ha''. Harvest intervals
included 3, 4, 5, 6, 8, 12, and 24 wk. Seasonal total yields and in vitro digestible dry matter are
given in Table 4.25. It was found in Section 4.1 that y = 0.077 wk-'. Regression analysis then
gives

Y* = Yexp(0.077At) = ac + fAt = 8.91 + 3.49At r = 0.99950 (4.181)


D* = D exp(0.077At) = ad +d dAt = 9.93 + 1.29At r = 0.9929


(4.182)










Table 4.25. Dependence of total seasonal biomass (Y) and digestible biomass (D) on harvest
interval (At) for coastal bermudagrass at Tifton, GA.a
At Y D/Y D Y* D*
wk Mg ha-1 Mg ha-1 Mg ha-1 Mg ha1
3 15.2 0.652 9.91 19.1 12.5
4 16.2 0.637 10.3 22.0 14.0
5 17.8 -- -- 26.2
6 19.9 0.597 11.9 31.6 18.9
8 19.9 0.566 11.3 36.8 20.9
12 20.1 0.525 10.6 50.6 26.7
24 14.6 0.432 6.31 92.7 40.0
aData adapted from Burton et al. (1963).


The response equations become


Y = (a, + 8y At)exp(- 7At) = (8.91 + 3.49At)exp(- 0.077At)

D = (a,+ ,dAt)exp(- yAt) = (9.93 +1.29At)exp(- 0.077At)

It follows immediately that digestible fraction is described by


D ad +jdAt 9.93+1.29At
Y ay +j3,At 8.91+3.49At


(4.183)

(4.184)


(4.185)


Correlation of Y* and D* with At is shown in Figure 4.63, where the lines are drawn from Eqs.
(4.181) and (4.182). The corresponding response of Y, D, andfd with harvest interval is shown in
Figure 4.60, where the curves are drawn from Eqs. (4.183) through (4.185). Results for digestible
dry matter exhibit more scatter than for total dry matter. This probably accounts for the intercept
of fd= 9.93/8.91 = 1.11, which should not exceed 1.00. In fact it can be shown that the 95%
confidence intervals for the intercepts in Eqs (4.183) and (4.184) are ay = 8.91 1.41 Mg ha1
and ad = 9.93 2.52 Mg ha'1, which clearly overlap. Therefore, it seems quite reasonable to
assume an intercept of 1.00 in Figure 4.64.
Additional inferences can be drawn from this analysis. Peak harvest intervals to maximize
total dry matter and digestible dry matter are estimated as


1 ay 1 8.91
Aty 10.4 wk
7y /,y 0.077 3.49

1 ad 1 9.93
Atd -- d -- = 5.3 wk
7"Y ,d 0.077 1.29


(4.186)


(4.187)









which can be confirmed from Figure 4.65. Since the expanded growth model incorporates
partitioning of biomass into light-gathering and structural components, Eqs. (4.183) and (4.184)
can be partitioned into

Y= YL +Ys = 8.91 exp(- 0.077At)+ 3.49At exp(- 0.077At) (4.188)

D = DL +Ds = 9.93exp(- 0.077At)+1.29Atexp(-0.077At) (4.189)

It follows that the fractions of digestible dry matter contained in light-gathering and structural
components,fdL andfds, are given by

9.93exp(-0.077At)
fdL= = 1.11 (4.190)
8.91 exp(-0.077At)

1.29Atexp(-0.077At) 0.37 (4.191)
3.49At exp(-0.077At)

from which it may be concluded that digestibility of the light-gathering and structural
components are approximately 100 and 37%, respectively. According to Eq. (4.185) digestibility
of the whole plant drops to 61% for a harvest interval of At = 5.3 wk.

4.7.2 Study with Perennial Peanut at Gainesville, Florida

Data for this analysis are taken from a field study with perennial peanut (Arachis glabrata
Benth., cv 'Florigraze') by Beltranena (1980) and Beltranena et al. (1981) at Gainesville, FL.
Treatments were replicated four times on Arredondo loamy fine sand (loamy, siliceous,
semiactive, hyperthermic Grossarenic Paleudult). Plant samples were collected in 1979 on 2, 4,
6, 8, 10, and 12 wk intervals to measure dry matter yields, plant N accumulation, and in vitro
organic matter digestibility. Data are given in Table 4.26. It will be assumed that parameter c =
0.077 wk-' is the same as for coastal bermudagrass. This leads to the standardized equations

Table 4.26. Dependence of seasonal total dry matter yield (Y), plant N uptake (N,), plant N
concentration (Nc), and digestible dry matter (D) on harvest interval (At) for perennial peanut at
Gainesville, FL (1979).a
At Y N. Nc D/Y D Y' N' D*
wk Mg ha-' kg ha-1 g kg-1 Mg ha' Mg ha-' kg ha-1 Mg ha-1
2 8.0 280 35.0 0.711 5.7 9.3 327 6.6
4 9.0 300 33.3 0.699 6.3 12.2 408 8.6
6 11.4 310 27.2 0.646 7.4 18.1 492 11.7
8 12.4 300 24.2 0.616 7.6 23.0 555 14.1
10 11.6 270 23.3 0.562 6.5 25.1 583 14.0
12 12.0 270 22.5 0.561 6.7 30.2 680 16.9
aCrop data adapted from Beltranena et al. (1981).









Y*= Yexp(0.077At)= ay + /,yAt = 4.84+ 2.12At r = 0.9938 (4.192)

N, = N, exp(0.077At)= a, +J, ,At = 272+33.6At r= 0.9916 (4.193)

D* = Dexp(0.077At)= ad +dAt = 4.97 +1.OOAt r= 0.9791 (4.194)

Standardized results are shown in Figure 4.65, where the lines are drawn from Eqs. (4.192)
through (4.194). The response equations become

Y = (ay + ,y At)exp(-yAt)= (4.84 +2.12At)exp(-0.077At) (4.195)

N, = (a,, +, ,, At)exp(- yAt)= (272 + 33.6At)exp(- 0.077At) (4.196)

D = (ad + /JdAt)exp(- yAt) = (4.97 +1 .OOAt)exp(- 0.077At) (4.197)

which are shown in Figure 4.66. Plant N concentration, Nc, and digestible dry matter fraction,fd,
are related to harvest interval by

SN, 272+33.6At
Y 4.84 + 2.12At

D 4.97+1.00At
Y 4.84+2.12At

which are shown in Figure 4.67. Both measures exhibit a rapid drop near At = 0. The intercept
forfd is 4.97/4.84 = 1.03 & 1.00.
The peak harvest intervals for total dry matter, plant N uptake, and digestible dry matter are
given, respectively, by

1 a 1 4.84
Atpy =10.7 wk (4.200)
y 8fy 0.077 2.12

1 a, 1 272
At =4.9 wk (4.201)
y 3,, 0.077 33.6

1 ad 1 4.97
Atpd =- -- =8.0 wk (4.202)
y /fd 0.077 1.00

which can be confirmed from Figure 4.67. It appears that a reasonable harvest interval in practice
would be about 6 wk.









An interpretation of Eqs. (4.195) through (4.197) is now proposed. Assume the above-ground
plant is composed of two components, viz. light-gathering (L) and structural (S). Furthermore,
assume that biomass for the two is described by

YL = ay exp(-yAt) = 4.84 exp(-0.077At) (4.203)

Ys = fly At exp(-yAt) = 2.12At exp(-0.077At) (4.204)

Likewise, assume that plant N uptake is described by

N,L = a, exp(-yAt) = 272 exp(-0.077At) (4.205)

Nas = 6, At exp(-yAt) = 33.6At exp(-0.077At) (4.206)

And finally, that digestible dry matter is described by

DL = ad exp(-yAt) = 4.97 exp(-0.077At) (4.207)

Ds = dAt exp(-yAt) = 1.OOAt exp(-0.077At) (4.208)

Plant N concentrations in the light-gathering and structural components are estimated as

N 272
NCL =_ N- 56.2 g kg (4.209)
YL 4.84


Ns =_ Ns 33.6 15.8 g kg' (4.210)
Ys 2.21

which indicates that the light-gathering component contains considerably higher plant N than the
structural component. The fractions of digestible dry matter in L and S components are estimated
as

4.97
fdL 4.84 =1.027 1.00 (4.211)

1.00
fds= -1 =0.47 = 0.50 (4.212)
2.12

By this analysisfd is bounded by 1 >fd > 0.50 for perennial peanut forage. Note that these values
are in the same range as for coastal bermudagrass in the previous section. Forage quality (protein
concentration and digestibility) declines as plants age due to a shift from dominance by the light-
gathering fraction to the structural fraction.









4.8 Response of Plant Digestibility to Applied Nitrogen


The affect of applied nitrogen on digestibility of plant material is now examined. Response
of biomass yield to applied nitrogen is described by the logistic equation

A
Y = Ay (4.213)
1 + exp(by c,,N)

where N is applied nitrogen, kg ha-'; Y is biomass yield, Mg ha-'; Ay is maximum yield at high N,
Mg ha-'; by is intercept parameter for yield; and c, is response coefficient for applied nitrogen, ha
kg''. It seems logical to assume that digestible dry matter would follow the same type equation


D = Ad (4.214)
1 + exp(bd c,,N)

where D is yield of digestible dry matter, Mg ha-'; Ad is maximum digestible dry matter at high
N, Mg ha-'; and bd is intercept parameter for digestible dry matter. It follows from Eqs. (4.213)
and (4.214) that the fraction of digestible dry matter,fd, is given by

D [+exp(b -c.N)
fd = dm + exp(by (4.215)
Y 1+ exp(bd -cN)

wherefdm = Ad/Ay is maximum fraction at high N.
Equations (4.213) and (4.214) can be linearized to the form


ZY =-In AY1 =cN-by (4.216)



Zd =-ln AdI cN bd (4.217)


Parameters Ay and Ad can be estimated visually from plots of Y vs. N and D vs. N, respectively.
The exponential parameters can then be estimated by linear regression.
Data for this analysis are taken from a field study by Cummins (1972) at Experiment, GA
with corn silage for the period 1966-1969. The soil was Cecil sandy loam (fine, kaolinitic,
thermic Typic Kanhapludult). Nitrogen was applied at rates of 0, 56, 112, 168, and 224 kg ha'.
All plots received 56 and 168 kg ha'' of phosphorus and potassium, respectively. Treatments
were replicated four times. Corn was harvested at the dent (dough) stage. Plant population was
7.16 plants m-2. No irrigation was provided. In vitro dry matter digestibility (IVDMD) was
measured on the samples.
Average data for the four year period are given in Table 4.27 and shown in Figure 4.68. By
visual inspection, Ay = 14.0 Mg ha'1 and Ad = 9.3 Mg ha-' are chosen. Linear regression then
leads to










Table 4.27. Response of biomass (Y) and digestible dry matter (D) to applied nitrogen (N) for
corn silage at Experiment, GA (1966-1969).a
N Y Dry matter D D/Y Dry matter fraction
kg ha-' Mg ha-' % Mg ha'' Leaf Stalk Ear
green dry
0 23.7 7.8 32.9 4.8 0.62 0.20 0.34 0.46
56 32.5 11.0 33.8 7.1 0.65 0.18 0.26 0.56
112 35.4 11.6 32.8 7.5 0.65 0.17 0.28 0.55
168 38.1 12.8 33.6 8.3 0.65 0.21 0.27 0.52
224 40.5 13.4 33.1 8.9 0.66 0.18 0.26 0.56
avg 0.188 0.282 0.530
std dev. 0.016 0.033 0.042
aData adapted from Cummins (1972).


Zy = -ln( y _-1) = 0.0122N +0.35


Zd =-ln 9-1 = 0.0125N + 0.17
(D )=


(4.218)


r= 0.9889


r= 0.9842


Equations (4.218) and (4.219) lead immediately to the estimation equations

14.0
1 + exp(-0.35 0.0125N)


9.3
D-=
1 + exp(-0.17 0.0125N)

D =06641 + exp(-0.35 0.0125N)
d Y i+ exp(-0.17-0.0125N)


(4.219)


(4.220)


(4.221)


(4.222)


where a common value c, = 0.0125 ha kg-' has been used. Curves in Figure 4.68 are drawn from
Eqs. (4.220) through (4.222).
Yield response to applied N for individual years is given in Table 4.28 and shown in Figure
4.69. A procedure is now described for estimating response curves for individual years. It is
assumed that parameters by and cn are common among years, and that all variation can be
assigned to parameter Ay. Now it can be shown from least squares analysis that optimum Ay for
given by and cn can be calculated from









Table 4.28 Response of biomass yield (Y) to applied nitrogen (N) by year for corn silage at
Experiment, GA (1966-1969).a
N Y 1
1 + exp(-0.35 0.0125N)
kg ha-1 Mg ha-'
1966 1967 1968 1969
0 9.9 7.2 8.7 5.6 0.587
56 12.5 11.4 11.4 8.1 0.741
112 14.1 14.1 10.8 7.6 0.852
168 14.1 14.3 13.9 8.5 0.921
224 15.7 15.5 13.9 8.7 0.959
aData adapted from Cummins (1972).


5 yr
1 + exp(by c, N)
A4 = x (4.223)

1+ +exp(by c.N,)

These computations lead to Ay = 16.3, 15.6, 14.4, and 9.4 Mg ha-' for 1966, 1967, 1968, and
1969, respectively. Curves in Figure 4.69 are drawn from Eq. (4.220) with appropriate Ay
parameters.
The logistic model describes response of biomass and digestible dry matter to applied N
rather well (Figure 4.68). Negative values of by and bd indicate that the soil contains more than
enough nitrogen to reach 50% of maximum yield. Digestibility is relatively insensitive to applied
N. It can be shown from Eq. (4.222) that the lower limit, fdl, for highly depleted soil N is related
to the upper limit,fd, by

fd/ = Jfn exp(- Ab) (4.224)

where Ab = bd by = 0.18. It follows from Eq. (4.224) that fd = 0.555. This means that digestible
dry matter is bounded by 0.555 Yields varied among years (Figure 4.69), with the variation being accounted for in the linear
parameter Ay. The value for 1969 was considerably below the other years. This was attributed to
dry weather (Cummins, 1972). Dependence of corn yields on water availability has been shown
to follow an exponential function (Overman and Scholtz, 2002b).
Distribution of dry matter among leaf, stalk, and ear fraction was relatively insensitive to
applied N (Table 4.27), with approximately 50% of dry matter occurring in the ears. This agrees
with results from a field study in North Carolina (Overman et al., 1994).

4.9 Summary

Results presented in this chapter lend strong support for the expanded model for crop growth
and nutrient accumulation. Applications included both perennial and annual crops. Accumulation









of biomass with time is defined by the linear Eq. (4.9). The growth quantifier is defined by Eq.
(4.8), while dimensionless time is defined by Eq. (4.10). The growth model incorporates a
Gaussian energy driving function, a linear function for partitioning of biomass between light-
gathering and structural components, and an exponential aging function. No attempt has been
made to incorporate the daily solar energy cycle. Accumulation of mineral elements and biomass
has been described by a hyperbolic phase relation, Eq. (4.26).
The structure of the growth model allows a procedure for partitioning of biomass between
light-gathering and structural components of plants. This procedure has been applied for both
perennial and annual crops.

4.10 References

Abramowitz, M. and I.A. Stegun. 1965. Handbook of Mathematical Functions. Dover
Publications. New York, NY.
Beltranena, R. 1980. Yield, growth, and quality of Florigraxe rhizome peanut (Arachis glabrata
Benth.) as affected by cutting height and frequency. M.S. Thesis. University of Florida.
Gainesville, FL. 72 p.
Beltranena, R., J. Breman, and G.M. Prine. 1981. Yield and quality of Florigraze rhizome peanut
(Arachis glabrata Benth.) as affected by cutting height and frequency. Proc. Soil and Crop
Science Society of Florida 40:153-156.
Burton, G.W. and R.H. Hart. 1961. Grass and Turf Investigations; Annual Report Georgia
Coastal Plain Experiment Station: Tifton, GA. p. 231, 236.
Burton, G.W., J.E. Jackson, and R.H. Hart. 1963. Effects of cutting frequency and nitrogen on
yield, in vitro digestibility, and protein, fiber, and carotene content of coastal bermudagrass.
Agronomy J. 55:500-502.
Carpenter, P.N. 1963. Mineral Accumulation in Potato Plants as Affected by Fertilizer
Application and Potato Variety. Maine Agricultural Experiment Station Bulletin 610.
University of Maine. Orono, ME.
Cummins, D.G. 1972. Yield and Quality of Corn Silage Grown Under Fertilizer Regimes.
Bulletin 105. Georgia Agricultural Experiment Station. University of Georgia. Experiment,
GA.
Fulkerson, R.S. 1983. Research Review of Forage Production; Crop Science Department.
University of Guelph. Ontario, Canada.
Hammond, L.C., C.A. Black, and A.G. Norman. 1951. Nutrient Uptake by Soybeans on Two
Soils. Research Bulletin No. 384. Iowa State Agricultural Experiment Station. Ames, IA. p.
461-512.
Henderson, J.B. and E.J. Kamprath. 1970. Nutrient and Dry Matter Accumulation by Soybeans.
Technical Bulletin No. 197. North Carolina Agricultural Experiment Station. Raleigh, NC.
27 p.
Karlen, D.L., E.J. Sadler, and C.R. Camp. 1987. Dry matter, nitrogen, phosphorus, and
potassium accumulation rates by corn on Norfolk loamy sand. Agronomy J. 79:649-656.
Laidler, K.J. 1950. Chemical Kinetics. McGraw-Hill. New York, NY.
Overman, A.R. 1998. An expanded growth model for grasses. Commun. Soil Science and Plant
Analysis 67-85.
Overman, A.R. 2001. A mathematical theorem to relate seasonal dry matter to harvest interval
for the expanded growth model. Commun. Soil Science and Plant Analysis 32:389-399.









Overman, A.R. and K.H. Brock. 2003. Confirmation of the expanded growth model for a warm-
season perennial grass. Commun. Soil Science and Plant Analysis 34:1105-1117.
Overman, A.R. and R.V. Scholtz III. 1999. Model for accumulation of dry matter and plant
nutrients by corn. Commun. Soil Science and Plant Analysis 30:2059-2081.
Overman, A.R. and R.V. Scholtz III. 2002a. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. Philadelphia, PA.
Overman, A.R. and R.V. Scholtz III. 2002b. Corn response to irrigation and applied nitrogen.
Commun. Soil Science and Plant Analysis 33:3609-3619.
Overman, A.R. and R.V. Scholtz III. 2004. Model analysis for growth response of corn. J. Plant
Nutrition 27:883-904.
Overman, A.R. and S.R. Wilkinson. 2003. Extended logistic model of forage grass response to
applied nitrogen as affected by soil erosion. Trans. American Society ofAgricultural
Engineers 46:1375-1380.
Overman, A.R. and D.M. Wilson. 1999. Physiological control of forage grass yield and growth.
In Crop Yield: Physiology and Processes; Smith D.L. and C. Hamel (eds). Springer-Verlag.
New York, NY. p. 443-473.
Overman, A.R. and K.R. Woodard. 2006. Simulation of biomass partitioning and production in
elephantgrass. Commun. Soil Science and Plant Analysis 37:1999-2010.
Overman, A.R., D.M. Wilson, and E.J. Kamprath. 1994. Estimation of yield and nitrogen
removal by corn. Agronomy J. 86:1012-1016.
Vicente-Chandler, J., S. Silva, and J. Figarella. 1959. The effect of nitrogen fertilization and
frequency of cutting on the yield and composition of three tropical grasses. Agronomy J.
51:202-206.
Woodard, K.R., G.M. Prine, and S. Bacherin. 1993. Solar energy recovery by elephantgrass,
energycane, and elephantmillet canopies. Crop Science 33:824-830.









List of Figures


Figure 4.1 Dependence of seasonal total yield (Y1) and standardized seasonal total yield (Y,) on
harvest interval (At) for coastal bermudagrass at Tifton, GA. Data adapted from Burton et al.
(1963). Line drawn from Eq. (4.2); curve from Eq. (4.3).

Figure 4.2 Dependence of seasonal total yield (Y,), seasonal plant N uptake (Nu), and plant N
concentration (Nc) on harvest interval (At) for coastal bermudagrass at Tifton, GA. Data adapted
from Burton et al. (1963). Curves drawn from Eqs. (4.3), (4.20), and (4.21).

Figure 4.3 Correlation of accumulated biomass yield (AY) with the growth quantifier (AQ) for
coastal bermudagrass at Tifton, GA. Data adapted from Overman and Brock (2003) for the study
of Burton and Hart (1961). Line drawn from Eq. (4.25).

Figure 4.4 Phase plots of plant N uptake (ANu) and yield/plant N uptake ratio (A Y/AN,,) vs. yield
(AY) for coastal bermudagrass at Tifton, GA. Data adapted from Overman and Brock (2003) for
the study of Burton and Hart (1961). Line drawn from Eq. (4.27); curve from Eq. (4.28).

Figure 4.5 Accumulation of biomass yield (A Y), plant N uptake (AN,,), and plant N concentration
(N,) with calendar time (t) for coastal bermudagrass at Tifton, GA. Data adapted from Overman
and Brock (2003) for the study of Burton and Hart (1961). Curves drawn from Eqs. (4.25),
(4.28), and (4.29).

Figure 4.6 Correlation of biomass yield (Y) with the growth quantifier (Q) for corn at Florence,
SC. Data adapted from Karlen et al. (1987). Line drawn from Eq. (4.32).

Figure 4.7 Phase plots of plant N uptake (N,) and yield/plant N uptake ratio (Y/Nu,) vs. yield (Y)
for corn at Florence, SC. Data adapted from Karlen et al. (1987). Line drawn from Eq. (4.33);
curve from Eq. (4.34).

Figure 4.8 Accumulation of biomass yield (Y), plant N uptake (N,), and plant N concentration
(N,) with calendar time (t) for corn at Florence, SC. Data adapted from Karlen et al. (1987).
Curves drawn from Eqs. (4.32), (4.34), and (4.35).

Figure 4.9 Correlation of biomass yield (Y) with the growth quantifier (Q) for corn at
Tallahassee, FL. Data from Overman and Scholtz (2004). Line drawn from Eq. (4.38).

Figure 4.10 Phase plots of plant N uptake (N,) and yield/plant N uptake ratio (Y/NA,) vs. yield (Y)
for corn at Tallahassee, FL. Data adapted from Overman and Scholtz (2004). Line drawn from
Eq. (4.39); curve from Eq. (4.40).

Figure 4.11 Accumulation of biomass yield (1), plant N uptake (N,), and plant N concentration
(Nc) with calendar time (t) for corn at Tallahassee, FL. Data adapted from Overman and Scholtz
(2004). Curves drawn from Eqs. (4.38), (4.40), and (4.41).









Figure 4.12 Phase plots of plant P uptake (P,) and yield/plant P uptake ratio (Y/P,) vs. yield (Y)
for corn at Tallahassee, FL. Data adapted from Overman and Scholtz (2004). Line drawn from
Eq. (4.42); curve from Eq. (4.43).

Figure 4.13 Phase plots of plant K uptake (Ku) and yield/plant K uptake ratio (Y/K,) vs. yield (Y)
for corn at Tallahassee, FL. Data adapted from Overman and Scholtz (2004). Line drawn from
Eq. (4.44); curve from Eq. (4.45).

Figure 4.14 Phase plots of plant Ca uptake (Can) and yield/plant Ca uptake ratio (Y/Cau) vs. yield
(Y) for corn at Tallahassee, FL. Data adapted from Overman and Scholtz (2004). Line drawn
from Eq. (4.46); curve from Eq. (4.47).

Figure 4.15 Phase plots of plant Mg uptake (Mg,) and yield/plant Mg uptake ratio (Y/Mg,) vs.
yield (Y) for corn at Tallahassee, FL. Data adapted from Overman and Scholtz (2004). Line
drawn from Eq. (4.48); curve from Eq. (4.49).

Figure 4.16 Correlation of biomass yield (Y) with the growth quantifier (Q) for peanut (total
plant) at Lewiston, NC. Data from Nicholaides (1968). Line drawn from Eq. (4.52).

Figure 4.17 Phase plots of plant N uptake (N,) and yield/plant N uptake ratio (Y/NV,) vs. yield (Y)
for peanut (total plant) at Lewiston, NC. Data adapted from Nicholaides (1968). Line drawn
from Eq. (4.53); curve from Eq. (4.54).

Figure 4.18 Accumulation of biomass yield (Y), plant N uptake (N,), and plant N concentration
(N,) with calendar time (t) for peanut (total plant) at Lewiston, NC. Data adapted from
Nicholaides (1968). Curves drawn from Eqs. (4.52), (4.54), and (4.58).

Figure 4.19 Phase plots of plant P uptake (P,) and yield/plant P uptake ratio (Y/P,) vs. yield (Y)
for peanut (total plant) at Lewiston, NC. Data adapted from Nicholaides (1968). Line drawn
from Eq. (4.56); curve from Eq. (4.57).

Figure 4.20 Phase plots of plant K uptake (Ku) and yield/plant K uptake ratio (Y/K,) vs. yield (Y)
for peanut (total plant) at Lewiston, NC. Data adapted from Nicholaides (1968). Line drawn
from Eq. (4.58); curve from Eq. (4.59).

Figure 4.21 Phase plots of plant Mg uptake (Mg,) and yield/plant Mg uptake ratio (Y/Mg,) vs.
yield (Y) for peanut (total plant) at Lewiston, NC. Data adapted from Nicholaides (1968). Line
drawn from Eq. (4.60); curve from Eq. (4.61).

Figure 4.22 Dependence of leaf fraction (fL) on calendar time (t) for peanut (total plant) at
Lewiston, NC. Curve drawn from Table 4.8.

Figure 4.23 Correlation of biomass yield (Y) with the growth quantifier (Q) for peanut (kernels +
shells) at Lewiston, NC. Data from Nicholaides (1968). Line drawn from Eq. (4.67).








Figure 4.24 Correlation of plant N uptake (N,), plant P uptake (Pu), and plant K uptake (K,) with
biomass yield (Y) for peanut (kernels + shells) at Lewiston, NC. Data from Nicholaides (1968).
Lines drawn from Eqs. (4.68) through (4.70).

Figure 4.25 Correlation of stem (Ys) and leaf (YL) yields with the growth quantifier (Q) for
elephantgrass at Gainesville, FL. Data from Overman and Woodard (2006) based on study of
Woodard et al. (1993). Lines drawn from Eqs. (4.77) and (4.78).

Figure 4.26 Accumulation of stem (Ys) and leaf (YL) biomass yields, and leaf fraction (fL) with
calendar time (t) for elephantgrass at Gainesville, FL. Data from Overman and Woodard (2006)
based on study of Woodard et al. (1993). Curves drawn from Eqs. (4.77) through (4.79).

Figure 4.27 Correlation of biomass yield (Y) with the growth quantifier (Q) for elephantgrass
(total plant) at Gainesville, FL. Data from Overman and Woodard (2006) based on study of
Woodard et al. (1993). Line drawn from Eq. (4.82).

Figure 4.28 Accumulation of biomass yield (Y) with calendar time (t) for elephantgrass (whole
plant) at Gainesville, FL. Data from Overman and Woodard (2006) based on study of Woodard
et al. (1993). Curve drawn Eqs. (4.80) through (4.82).

Figure 4.29 Correlation of biomass yield (Y) with the growth quantifier (Q) for soybean
(vegetation) on two soils at Ames, IA. Data adapted from Hammond et al. (1951). Lines drawn
from Eqs. (4.85) and (4.86).

Figure 4.30 Accumulation of biomass yield (Y) with calendar time (t) for soybean (vegetation)
on two soils at Ames, IA. Data adapted from Hammond et al. (1951). Curves drawn from Eqs.
(4.83) through (4.86).

Figure 4.31 Correlation of biomass yield (Y) with the growth quantifier (Q) for soybean (seeds +
pods) on two soils at Ames, IA. Data adapted from Hammond et al. (1951). Lines drawn from
Eqs. (4.89) and (4.90).

Figure 4.32 Accumulation of biomass yield (Y) with calendar time (t) for soybean (seeds + pods)
on two soils at Ames, IA. Data adapted from Hammond et al. (1951). Curves drawn from Eqs.
(4.87) through (4.90).

Figure 4.33 Correlation of biomass yield (Y) with the growth quantifier (Q) for soybean
(vegetation) for two years at Clayton, NC. Data adapted from Henderson and Kamprath (1970).
Lines drawn from Eqs. (4.93) and (4.94).

Figure 4.34 Accumulation of biomass yield (Y) with calendar time (t) for soybean (vegetation)
for two years at Clayton, NC. Data adapted from Henderson and Kamprath (1970). Curves
drawn from Eqs. (4.91) through (4.94).








Figure 4.35 Correlation of biomass yield (Y) with the growth quantifier (Q) for soybean (seeds +
pods) for two years at Clayton, NC. Data adapted from Henderson and Kamprath (1970). Lines
drawn from Eqs. (4.97) and (4.98).

Figure 4.36 Accumulation of biomass yield (Y) with calendar time (t) for soybean (seeds + pods)
for two years at Clayton, NC. Data adapted from Henderson and Kamprath (1970). Curves
drawn from Eqs. (4.95) through (4.98).

Figure 4.37 Dependence of biomass yield (Y) on seasonal rainfall (R) for soybean (seeds + pods)
at Clayton, NC. Data adapted from Henderson and Kamprath (1970). Curve drawn from Eq.
(4.100).

Figure 4.38 Estimated dependence of leaf fraction (fL) on calendar time (t) for soybean at
Clayton, NC. Curve drawn from Table 4.15.

Figure 4.39 Accumulation of biomass yield (Yv), plant N uptake (Nuv), and plant N concentration
(Ncv) with calendar time (t) for potato (vegetation) at Old Town, ME. Data from Carpenter
(1963). Curves drawn from Eqs. (4.105) through (4.107) and (4.109) through (4.110).

Figure 4.40 Correlation ofbiomass yield (Y) with the growth quantifier (Q) for potato
(vegetation and tubers) at Old Town, ME. Data from Carpenter (1963). Lines drawn from Eqs.
(4.107) and (4.113).

Figure 4.41 Phase plots of plant N uptake (Nuv) and yield/plant N uptake ratio (YI/Nv) vs. yield
(Y,) for potato (vegetation) at Old Town, ME. Data from Carpenter (1963). Line drawn from Eq.
(4.108); curve from Eq. (4.109).

Figure 4.42 Accumulation of biomass yield (Y,), plant N uptake (Nu), and plant N concentration
(N,,) with calendar time (t) for potato (tubers) at Old Town, ME. Data from Carpenter (1963).
Curves drawn from Eqs. (4.111) through (4.113), (4.115) and (4.116).

Figure 4.43 Phase plots of plant N uptake (Nut) and yield/plant N uptake ratio (YIvNut) vs. yield
(Yt) for potato (tubers) at Old Town, ME. Data from Carpenter (1963). Line drawn from Eq.
(4.114); curve from Eq. (4.115).

Figure 4.44 Response of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc)
to applied nitrogen (N) for potato (vegetation and tubers) at Old Town, ME. Data from Carpenter
(1963). Curves drawn from Eqs. (4.117) through (4.119); lines from (4.120) and (4.121).

Figure 4.45 Correlation of leaf (YL) and stem (Ys) biomass with growth quantifier (Q) for alfalfa
grown at Guelph, Ontario, Canada. Yield data adapted from Fulkerson (1983). Lines drawn from
Eqs. (4.140) and (4.141).

Figure 4.46 Correlation of plant biomass (Y) with growth quantifier (Q) for alfalfa grown at
Guelph, Ontario, Canada. Yield data adapted from Fulkerson (1983). Line drawn from Eq.
(4.145).










Figure 4.47 Phase plots of plant N uptake (N,) and plant biomass/plant N (Y/N,) vs. biomass (Y)
for alfalfa grown at Guelph, Ontario, Canada. Plant data adapted from Fulkerson (1983). Line
drawn from Eq. (4.146); curve drawn from Eq. (4.147).

Figure 4.48 Accumulation of plant biomass (Y) and plant nitrogen (N,) with calendar time (t) for
alfalfa grown at Guelph, Ontario, Canada. Plant data adapted from Fulkerson (1983). Curves
drawn from Eqs. (4.145), (4.147), and (4.148).

Figure 4.49 Phase plots of plant N uptake (N,) and plant biomass/plant N (Y/N,) vs. biomass (Y)
of leaves and stems for alfalfa grown at Guelph, Ontario, Canada. Plant data adapted from
Fulkerson (1983). Lines drawn from Eqs. (4.149) and (4.152); curves drawn from Eqs. (4.150),
(4.151), (4.153), and (4.154).

Figure 4.50 Accumulation of plant biomass (Y) and plant nitrogen (N,) with calendar time (t) of
leaves and stems for alfalfa grown at Guelph, Ontario, Canada. Plant data adapted from
Fulkerson (1983). Curves drawn from Eqs. (4.140), (4.141), (4.150), (4.151), (4.153), and
(4.154).

Figure 4.51 Accumulation of biomass (YE), plant nitrogen uptake (NuE), and plant nitrogen
concentration (NcE) with calendar time (t) for corn ears at Tallahassee, FL. Data adapted from
Overman and Scholtz (2004). Curves drawn from Eqs. (4.157), (4.159), and (4.160).

Figure 4.52 Correlation of biomass accumulation (YE) with the growth quantifier (QE) for corn
ears at Tallahassee, FL. Yield data adapted from Overman and Scholtz (2004). Line drawn from
Eq. (4.157).

Figure 4.53 Phase plots of plant N uptake (NuE) and plant biomass to plant N ratio (YE/NUE) vs.
biomass (YE) for corn ears at Tallahassee, FL. Data adapted from Overman and Scholtz (2004).
Line drawn from Eq. (4.158); curve from Eq. (4.159).

Figure 4.54 Estimated accumulation of light-gathering (YLE) and structural (YsE) biomass, and
fraction of light-gathering component (fLE) with calendar time (t) for corn ears at Tallahassee,
FL. Curves drawn from Eqs. (4.163) through (4.165).

Figure 4.55 Phase plots of plant P uptake (PuE) and plant biomass to plant P ratio (YE/PuE) vs.
biomass (YE) for corn ears at Tallahassee, FL. Data adapted from Overman and Scholtz (2004).
Line and curve drawn from Eq. (4.166).

Figure 4.56 Phase plots of plant K uptake (KuE) and plant biomass to plant K ratio (YE/KuE) vs.
biomass (YE) for corn ears at Tallahassee, FL. Data adapted from Overman and Scholtz (2004).
Line and curve drawn from Eq. (4.167).

Figure 4.57 Phase plots of plant Ca uptake (CaE) and plant biomass to plant Ca ratio (YE/CaE)
vs. biomass (YE) for corn ears at Tallahassee, FL. Data adapted from Overman and Scholtz
(2004). Line and curve drawn from Eq. (4.168).










Figure 4.58 Phase plots of plant Mg uptake (MguE) and plant biomass to plant Mg ratio (YE/MguE)
vs. biomass (YE) for corn ears at Tallahassee, FL. Data adapted from Overman and Scholtz
(2004). Line and curve drawn from Eq. (4.169).

Figure 4.59 Dependence of phosphorus/nitrogen concentration ratio (PcENcE) on plant biomass
(YE) for corn ears at Tallahassee, FL. Data adapted from Overman and Scholtz (2004). Curve
drawn from Eq. (4.170).

Figure 4.60 Dependence of potassium/nitrogen concentration ratio (KcE/NcE) on plant biomass
(YE) for corn ears at Tallahassee, FL. Data adapted from Overman and Scholtz (2004). Curve
drawn from Eq. (4.171).

Figure 4.61 Dependence of calcium/nitrogen concentration ratio (CacE/NcE) on plant biomass
(YE) for corn ears at Tallahassee, FL. Data adapted from Overman and Scholtz (2004). Curve
drawn from Eq. (4.172).

Figure 4.62 Dependence of magnesium/nitrogen concentration ratio (MgcE/NcE) on plant biomass
(YE) for corn ears at Tallahassee, FL. Data adapted from Overman and Scholtz (2004). Curve
drawn from Eq. (4.173).

Figure 4.63 Correlation of standardized biomass yield (Y*) and standardized digestible dry
matter (D*) with harvest interval (At) for coastal bermudagrass at Tifton, GA. Data adapted
from Burton et al. (1963). Lines drawn from Eqs. (4.181) and (4.182).

Figure 4.64 Dependence of biomass yield (Y), digestible dry matter (D), and digestible fraction
(fd) on harvest interval (At) for coastal bermudagrass at Tifton, GA. Data adapted from Burton et
al. (1963). Curves drawn from Eqs. (4.183) through (4.185).

Figure 4.65 Correlation of standardized biomass yield (Y*), standardized plant N uptake ( N ),
and standardized digestible dry matter (D*) with harvest interval (At) for perennial peanut at
Gainesville, FL. Data adapted from Beltranena (1980) and Beltranena et al. (1981). Lines drawn
from Eqs. (4.192) through (4.194).

Figure 4.66 Dependence of biomass yield (Y), plant N uptake (Nu), and digestible dry matter (D)
on harvest interval (At) for perennial peanut at Gainesville, FL. Data adapted from Beltranena
(1980) and Beltranena et al. (1981). Curves drawn from Eqs. (4.195) through (4.197).

Figure 4.67 Dependence of plant N concentration (Nc) and digestible dry matter fraction (fd) on
harvest interval (At) for perennial peanut at Gainesville, FL. Data adapted from Beltranena
(1980) and Beltranena et al. (1981). Curves drawn from Eqs. (4.198) and (4.199).

Figure 4.68 Dependence of biomass yield (Y), digestible dry matter (D), and digestible fraction
(fd) on applied nitrogen (N) for corn silage grown at Experiment, GA. Data adapted from
Cummins (1972). Data averaged over four years. Curves drawn from Eqs. (4.220) through
(4.222).










Figure 4.69 Dependence of biomass yield (Y) on applied nitrogen (N) for individual years for
corn silage grown at Experiment, GA. Data adapted from Cummins (1972). Curves drawn from
Eq. (4.220) withAy = 16.3, 15.6, 14.4, and 9.4 Mg ha-1 for 1966, 1967, 1968, and 1969,
respectively.


















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