A MEMOIR ON MODEL RESPONSE OF FORAGE GRASS TO
FERTILIZER AND BROILER LITTER
Allen R. Overman
Agricultural and Biological Engineering
University of Florida
Copyright 2007 Allen R. Overman
MEMOIR ON MODEL RESPONSE OF FORAGE GRASS TO
FERTILIZER AND BROILER LITTER
Allen R. Overman
Agricultural and Biological Engineering Department, University of Florida,
Gainesville, FL 326110570
ABSTRACT
The extended logistic model has proven very useful for describing response of crops to applied
nutrients. It conforms to the requirement of transferability in space (location to location) and
time (year to year) for a variety of crops, soils, and environmental conditions. Equations of the
model are bounded and monotone increasing functions. In this memoir the model is applied to a
field study with bermudagrass and tall fescue at Fayetteville, Arkansas, USA. Response to
nitrogen from fertilizer and' broiler litter was measured over several years. Input (control)
variables consisted of nitrogen rate (N) and irrigation, while output (response) variables
consisted of biomass yield (Y), plant nitrogen uptake (Nu), and plant nitrogen concentration (NQ).
Model variables were bounded by 0 < Y< 21.33, 12.33 Mg ha1; 0 < Nu < 495, 340 kg ha'; and
13.2, 17.9 < Nc < 23.2, 27.6 g kg1 for bermudagrass and tall fescue, respectively, for response to
nitrogen fertilizer. The nitrogen response coefficient Cn = 0.0078 ha kg1 was common for both
grasses grown on the same soil. This parameter is believed to be characteristic of the soil, and is
a measure of soil availability of the applied nutrient. The phase plots of Nc vs. Nu followed a
linear relationship as predicted by the model. The intercepts of these plots provide estimates of
the minimum plant nitrogen concentration (Nct) at reduced (depleted) soil N. Effect of water
availability on yield and plant N uptake was accounted for in the linear parameters Ay and An for
maximum yield and maximum plant N uptake, respectively, at high N. Applied N and efficiency
of nitrogen recovery (Np, Ep) for optimum efficiency of plant utilization was (390 kg ha1, 78%)
for bermudagrass and (300 kg ha1, 58%) for tall fescue. Nitrogen application from broiler litter
varied among years due to variation in litter composition. Once the plots became acclimated to
litter application, parameter c, assumed the same value for bermudagrass as response to
fertilizer. The same was true for Nct. However, maximum plant N concentration (Ncm) was
somewhat lower for litter than fertilizer as was Ay and An. This is believed due to the impact of
solids on the grass. The model provided excellent description of response of both grasses to
fertilizer nitrogen and for bermudagrass response to broiler litter. Phase plots behaved as
predicted for all cases. The model is based on two relatively simple equations, which are easy to
use in practice. Extended logistic and expanded growth models are both used to analyze response
of bermudagrass to fertilizer and broiler litter at Watkinsville, GA, USA. The nitrogen response
coefficient was found to be Cn = 0.0072 ha kg1. Model variables were bounded by 0 < Y< 17.2
Mg ha ; 0 < Nu < 550 kg ha1; and 17.6 < Nc < 32.0 g kg1 for bermudagrass response to nitrogen
from fertilizer and broiler litter. The phase plot of Nc vs. Nu followed a linear relationship as
predicted by the model. Peak nitrogen utilization at Watkinsville, GA was 64% compared to 78%
at Fayetteville, AR.
Key words: Model, nitrogen, fertilizer, broiler litter, bermudagrass, tall fescue.
INTRODUCTION
The logistic model has been used to describe crop response of biomass yield to applied
nitrogen by perennial grasses (Overman et al., 1990). The model was later extended to include
plant nitrogen uptake in response to applied nitrogen (Overman et al., 1994a). It was shown that
the extended logistic model also described response of corn to applied nitrogen (Overman et al.,
1994b). The extended logistic model has been applied to a variety of crops on different soils, and
to applied phosphorus and potassium as well as nitrogen (Overman and Scholtz, 2002). The
expanded growth model (Overman, 1998) describes accumulation of biomass with time through
an analytical function which incorporates effects of energy input, partitioning of biomass
between lightgathering and structural components, and aging as the plant grows. These ideas
are now demonstrated by analysis of a field study with forage grass response to fertilizer and
broiler litter.
MODEL DESCRIPTION
The extended logistic model of crop response to applied nutrients can be developed from two
postulates given by the two equations
N = A, (1)
1 + exp(b, c.N)
Y= YmNu (2)
K, +N,
where N is level of applied nutrient (N, P, or K), kg ha'; Nu is plant nutrient uptake (N, P, or K),
kg ha1; Yis biomass yield, Mg ha'; An is maximum plant nutrient uptake at high N, kg ha'1; b, is
the intercept parameter for plant nutrient uptake; cn is the nutrient response coefficient, ha kg1;
Ym is potential biomass yield, Mg ha1; and Kn is nutrient uptake coefficient, kg ha'1. Note that the
units on cn are the reciprocal of those on N. Equation (2) predicts a linear relationship between
plant nutrient concentration, Nc, and plant nutrient uptake given by
Nc = N+ K+ N (3)
Y Y Y
r rm rm
Equation (3) can be tested directly from data. In fact, linear regression of Nc vs. Nu leads to
estimates of Ym and K,.
Substitution of Eq. (1) into Eq. (2) leads, after rearrangement, to a second logistic equation
Y = (4)
1+ exp(by c,N)
where Ay is maximum biomass yield at high N, Mg ha'1; and by is the intercept parameter for
yield. The yield parameters can be calculated from
Ab= b b, = In + A (5)
A A.f Y (6)
A A, + K,,'"
Dependence of plant nutrient concentration on applied nutrient results from Eqs. (1) and (4)
N 1 + exp(by c,N)
N, = Ncm (7)
Y 1 + exp(b, c, N)
where Ncm = A,/Ay is maximum plant nutrient concentration at high N, g kg1. This completes
development of the model, which revolves around the two postulates given by Eqs. (1) and (2).
The alternative is to develop the model from Eqs. (1) and (4), as discussed by Overman and
Scholtz (2002).
The expanded growth model for biomass accumulation (Y) with calendar time (t) from Jan. 1
for a perennial grass is described by
AY, = AAQ, (8)
where AYj is biomass yield for the i th growth interval, Mg ha'; AQ; is growth quantifier for the
i th growth interval; and A is the yield factor, Mg ha1'. The growth quantifier is defined by
AQ, = {( kx,)[erf x erf x,] [exp( x2) exp( x? )] exp(2ocx,) (9)
with dimensionless time, x, defined in terms of calendar time, t, by
t 2oc (10)
V2cr 2
where model parameters are defined by: 1u is time to the mean of the energy distribution, wk; o
is time spread of the energy distribution, wk; c is the aging coefficient, wk'1; and k is the partition
coefficient between lightgathering and structural components of the plant. Note that xi
corresponds to the time of initiation of growth, ti. The error function, erf, in Eq. (13) is defined
by
erf x = exp(u2)du (11)
0
where u is the variable of integration. Values of the error function can be obtained from
mathematical tables (Abramowitz and Stegun, 1965). Cumulative growth quantifier, Q, and
biomass, Y, are then given by
Q = AQ (12)
i
Y=XAY. (13)
i
It follows from Eq. (8) that cumulative yield and growth quantifier are coupled by the linear
relationship
Y= AY. = A AQ, = AQ (14)
i i
Data from two field studies are now used to illustrate application of the models.
Field Study from Arkansas
DATA ANALYSIS
Data for this analysis are adapted from a field study at Fayetteville, AR (Huneycutt et al.,
1988) on response of forage grasses to applied nitrogen. The grasses were the warm season
'Tifton 44' bermudagrass (Cynodon dactylon (L.) Pers.) and the cool season 'Kenhy' tall fescue
(Festuca arundinacea Schreb.). Both grasses were harvested on a 4 to 5 week interval. The soil
was Captina silt loam (finesilty, siliceous, active, mesic Typic Fragiudult). Fertilizer was
applied as ammonium nitrate at annual rates of 0, 112, 224, 336, 448, 560, and 672 kg N ha .
Broiler litter was applied at rates of 0, 4.5, 9.0, and 13.5 Mg ha1 wet weight and solids content of
approximately 30%, with nitrogen content varying from year to year. Treatments included
irrigated and nonirrigated plots. This analysis includes response of bermudagrass and fescue to
fertilizer and of bermudagrass to broiler litter. Bermudagrass measurements were taken in 1981
through 1985, while fescue data were for 19811982, 19821983, 19831984, and 19841985.
Data for the first season are ignored for both grasses, since this represented acclimation of the
plots to the treatments.
Response of Bermudagrass to Fertilizer
Results are given in Table 1 for yield (Y), plant N concentration (Nc), and plant N uptake (Nu)
averaged over 1982 through 1985 for both irrigated and nonirrigated plots. Irrigation treatments
are then averaged to obtain overall response to applied nitrogen. Entries for each nitrogen level
represent the average of eight values (4 yr x 2 irrigation treatments). Response curves are shown
in Figure 1. Equation (1) can be linearized to the form
n495
Z, = ln4 1 = b, +c, N= 2.06+0.00781N r= 0.9923 (15)
where A, = 495 kg ha1 has been chosen to optimize the correlation coefficient, r. Linear
regression of Z, vs. N then leads to Eq. (15). The response equation for Nu vs. N becomes
A An 495
1+exp(b, cnN) 1+exp(2.060.0078N) (16)
The phase plots (Y and Nc vs. Nu) are shown in Figure 2, where the line and curve are drawn
from
K 1
N ++N =13.23+0.02018N, r=0.9964 (17)
Y Y.
m m
SYN = 49.55N, (18)
K +N. 655+N.
It follows from Eqs. (5) and (6) that
Ab=bnby =ln 1+ = In + 495= 0.56= 2.06b b by 1.50 (19)
A ,= A Y1m 495= 5549.55 = 21.33 Mg ha (20)
The yield response and nitrogen concentration equations become
= A = 21.33 (21)
1 + exp(by c,N) 1+exp(1.500.0078N)
N = N 1+ exp(b cnN) 23 1+exp(1.50 0.0078N)
N = NcmI+ =23.2 (22)
Y 1 + exp(b, cN)_ 1 + exp(2.06 0.0078N) (
Response curves in Figure 1 are drawn from Eqs. (16), (21), and (22). The model describes
response of bermudagrass to fertilizer nitrogen rather well.
It is now possible to estimate parameters Ay and An separately for irrigated and nonirrigated
treatments by assuming that by, b,, and c, are common for the two. Standardized yield (Y*) and
standardized plant N uptake (Nu*) can be calculated from
Y = Y[l + exp(1.50 0.0078N)] = Ay (23)
N = N, [1+ exp(2.06 0.0078N)] = A, (24)
Values are listed as the last two columns of Table 1. On average there is approximately 25%
increase in yield and plant N uptake with irrigation over natural rainfall. These results suggest
that effect of water availability can be assigned to the linear model parameters Ay and An.
Q = AQ (12)
i
Y=XAY. (13)
i
It follows from Eq. (8) that cumulative yield and growth quantifier are coupled by the linear
relationship
Y= AY. = A AQ, = AQ (14)
i i
Data from two field studies are now used to illustrate application of the models.
Field Study from Arkansas
DATA ANALYSIS
Data for this analysis are adapted from a field study at Fayetteville, AR (Huneycutt et al.,
1988) on response of forage grasses to applied nitrogen. The grasses were the warm season
'Tifton 44' bermudagrass (Cynodon dactylon (L.) Pers.) and the cool season 'Kenhy' tall fescue
(Festuca arundinacea Schreb.). Both grasses were harvested on a 4 to 5 week interval. The soil
was Captina silt loam (finesilty, siliceous, active, mesic Typic Fragiudult). Fertilizer was
applied as ammonium nitrate at annual rates of 0, 112, 224, 336, 448, 560, and 672 kg N ha .
Broiler litter was applied at rates of 0, 4.5, 9.0, and 13.5 Mg ha1 wet weight and solids content of
approximately 30%, with nitrogen content varying from year to year. Treatments included
irrigated and nonirrigated plots. This analysis includes response of bermudagrass and fescue to
fertilizer and of bermudagrass to broiler litter. Bermudagrass measurements were taken in 1981
through 1985, while fescue data were for 19811982, 19821983, 19831984, and 19841985.
Data for the first season are ignored for both grasses, since this represented acclimation of the
plots to the treatments.
Response of Bermudagrass to Fertilizer
Results are given in Table 1 for yield (Y), plant N concentration (Nc), and plant N uptake (Nu)
averaged over 1982 through 1985 for both irrigated and nonirrigated plots. Irrigation treatments
are then averaged to obtain overall response to applied nitrogen. Entries for each nitrogen level
represent the average of eight values (4 yr x 2 irrigation treatments). Response curves are shown
in Figure 1. Equation (1) can be linearized to the form
n495
Z, = ln4 1 = b, +c, N= 2.06+0.00781N r= 0.9923 (15)
where A, = 495 kg ha1 has been chosen to optimize the correlation coefficient, r. Linear
regression of Z, vs. N then leads to Eq. (15). The response equation for Nu vs. N becomes
Response of Tall Fescue to Fertilizer
Results are given in Table 2 for yield (Y), plant N concentration (Nc), and plant N uptake (Nu)
averaged over the growing seasons 19821983, 19831984, 19841985 for both irrigated and
nonirrigated plots. Irrigation treatments are then averaged to obtain overall response to applied
nitrogen. Response curves are shown in Figure 3. Equation (1) can be linearized to the form
Z, In 340 = b,, + c,N = 1.54 + 0.00777N r= 0.9963 (25)
( N, )
where A, = 340 kg ha'1 has been chosen to optimize the correlation coefficient, r. Linear
regression of Z, vs. N then leads to Eq. (25). The response equation for Nu vs. Becomes
N = A, = 340 (26)
1 + exp(b, cN) 1+ exp(1.54 0.0078N)
The phase plots (Y and Nc vs. Nu) are shown in Figure 4, where the line and curve are drawn
from
K 1
c "+ N, = 17.86+ 0.02862N. r = 0.9750 (27)
S Y.N 34.95N (28)
K, +N, 624+N,
It follows from Eqs. (5) and (6) that
Ab =b,by = In 1+A =ln 1+ = 0.43 =1.54b > b = 1.11 (29)
yK,), 624)
A, = Y = 340 34.95 =12.33 Mg ha' (30)
An + K 340 + 624)
The yield response and nitrogen concentration equations become
= Ay = 12.33 (31)
l+exp(by cN) 1+exp(1.110.0078N)
N = = N exp(by c,,N) 27 1 +exp(1.11 0.0078N)
Y 1 + exp(b, c,,N) 1 +exp(l.54 0.0078N)
Response curves in Figure 3 are drawn from Eqs. (26), (31), and (32). The model describes
response of tall fescue to fertilizer nitrogen rather well.
It is now possible to estimate parameters Ay and An separately for irrigated and nonirrigated
treatments by assuming that by, bn, and Cn are common for the two. Standardized yield (Y*) and
standardized plant N uptake (Nu*) can be calculated from
Y* =Y[l+ exp(1.11 0.0078N)]= AY (33)
N = N, [1 + exp(1.54 0.0078N)] = A, (34)
Values are listed as the last two columns of Table 2. On average there is approximately 65%
increase in yield and plant N uptake with irrigation over natural rainfall. These results suggest
that effect of water availability can be assigned to the linear model parameters Ay and An.
Response of Bermudagrass to Broiler Litter
Results are given in Tables 3 through 6 for yield (Y), plant N concentration (Nc), and plant N
uptake (Nu) for individual years for both irrigated and nonirrigated plots. Irrigation treatments are
then averaged to obtain overall response to applied nitrogen, as done with the fertilizer
treatments. Response plots are shown in Figures 5 through 8 for the individual years. Model
parameters for plant N uptake are determined from
1982: Z, = ln 300 =b, + c,N = 1.41+ 0.00546N r = 0.9960 (35)
An 300
N 00 (36)
1 + exp(b, c,N) 1 + exp(l.41 0.0055N)
1983: Z, =ln 1 = b, +c,N = 1.92+ 0.00781N r= 0.9964 (37)
N.
An, 355
1+ exp(b, c,N) 1+ exp(1.92 0.0078N)
285
198: Z, = In N 1) = b +c,N = 1.65 + 0.00779N r = 0.9982 (39)
A,, 285
N 1 (40)
1+ exp(bn c,,N) 1 + exp(1.65 0.0078N)
1985: Z, = ln 2N lJ = bn + cN = 2.06 + 0.00785N r = 0.9970 (41)
A, 220
1+ exp(b, c,N) 1+ exp(2.06 0.0078N) (42)
where values of An have been chosen to optimize the correlation coefficients, r. Phase plots (Y
and Nc vs. N,) are shown in Figures 9 through 12 for 1982 through 1985, where the lines and
curves are drawn from
1982:
K 1
N, "+Nm = 14.01+0.01267N,
Y.N.= 78.91N,
K, +N, 1105+N.
c_ = +_ N. = 13.51+0.01885N.
Y_ mN 53.05N.
Kn +N. 717+N.
1983:
1984:
KY
1
N. =12.95+0.02810N.
 l
YmNu 35.58Nu
K +N 461+ N
N =+N =12.84+0.02501N,
.m Ym
SYmN 39.99N
K +N, 513+N.
r= 0.9351
r= 0.9950
r= 0.9649
r= 0.9958
Estimates of Ay, by, and Ncm are obtained from
1982: Ab =ln1 +"L = In(+ '0' =0.24=b, b =1.41by > b =1.17
Kn) 1105)
A YJA. m )=300+ 10578.91= 16.85 Mgha'
Ncm. = 300 = 17.8 gkg1
Ay 16.85
1983: Ab=ln 1l+ A =ln l+355 = 0.40 = bn by = 1.92 by > by =1.52
A=( An )m
An +K,
1984:
= 355 3.05= 17.57 Mgha"
S355+717)
Ncm = A 355 = 20.2 gkg'
AY 17.57
Ab =ln 1+ = ln 1+ 28 =0.48=b by =1.65by + by =1.17
(A Y ( 285 35.58 =13.59 Mg ha'1
An +K)k 285+461)
1985:
(43)
(44)
(45)
(46)
(47)
(48)
(49)
(50)
(51)
(52)
(53)
(54)
(55)
(56)
(57)
(58)
A 285
Ncme,, = 285 21.0 gkg (59)
Ay 13.59
1985: Ab=ln 1+A =ln( 1+ 1 =0.36=b,, b =2.06b b =1.70 (60)
K,, ) 513) '
A= "x = = 9.99 = 12.00 Mg ha (61)
A,, +K, 220+513J
N A 220 18.3 g kg (62)
Ay 12.00
Response equations for individual years are then given by
1982: Y = Ay 16.8563)
1+ exp(by c,N) 1+ exp(1.17 0.0055N) (63)
N =N 1+exp(b c,N) 17.8+exp(1.170.0055N) (64)
c l+exp(b, c,N) 1l+exp(1.410.0055N)
1983: Y = A _=17.57 (65)
1+exp(b c,,N) l+exp(l.520.0078N)
[1 + exp(b cN) 1 + exp(.520.0078N)(6)
1 + exp(b, c,N) 1 + exp(1.92 0.0078N)
1984: Y A =_13.9 (67)
1+exp(by c,N) 1+exp(1.17 0.0078N) (67
S= N [1+ *exp(b cN) 0 1+ exp(1.17 0.0078N) (68)
[1+ exp(b, c,N)J l 1+ exp(1.65 0.0078N)
1985: Y =Ay 12.0069)
1+ exp(by c,N) 1+ exp(1.70 0.0078N)
= N + exp(b c,N) 18.31 + exp(1.70 0.0078N)
1+ exp(b, c,N) 1 +exp(2.06 0.0078N)(70)
Response curves in Figures 5 through 8 are drawn from Eqs. (36), (38), (40), (42), and (63)
through (70).
The model describes response of bermudagrass to broiler litter rather well for the individual
years. Model parameters are summarized in Table 7. Several things become apparent from these
results. After 1982, parameter c, = 0.0078 hg kg'1, which is the same value as for bermudagrass
response to fertilizer. Apparently two years (1981 and 1982) were required for plots to acclimate
to broiler litter application. For the four years Nct = 13.3 g kg1, which is the same as for
bermudagrass response to fertilizer. However, Ncm = 19.3 g kg1 is less than the value of 23.2 g
kg1 for fertilizer. This leads to Ab = 0.37, which is less than 0.56 for fertilizer treatments. There
was a decline in productivity between 1983 and 1985 as measured by parameters Ay and An as
shown in Figure 13. The dashed curves in Figure 13 are drawn from
Ay = 10.50 + 7.00 exp[ 0.775(Y 1983)] (71)
A, = 185 +185exp[ 0.775(Y1983)] (72)
where Y > 1983 is the year. Projections are shown for two years beyond 1985. Equations (71)
and (72) suggests that values would stabilize at Ay = 10.50 Mg ha1, An = 185 kg ha', and Ncm =
17.6 g kg'1. These values are highly speculative, of course. Suppression of yields and plant N
uptake by applications of broiler litter probably result from accumulation of solids on the
vegetation which could reduce the rate of photosynthesis.
DISCUSSION
The extended logistic model of crop response to applied nutrients describes response of
bermudagrass and tall fescue to fertilizer and broiler litter rather well (Figures 1 through 12). The
predicted linear relationship between plant N concentration (Nc) and plant N uptake (Nu) is
confirmed. The nitrogen response coefficient (cn) is the same for bermudagrass and fescue grown
on the same soil, which lends support to the hypothesis that Cn is a measure of nutrient
availability in the soil (Overman, 2006a, b). Since minimum and maximum limits on plant N
concentration (Nci and Ncm) are believed to be characteristics of the plant species, it follows from
Ab = ln I N I (73)
that the shift in intercept parameters (Ab) is characteristic of the plant species as well. Effect of
water availability occurs in the linear model parameters Ay and An.
The value of applied N (N1/2) to achieve 50% of maximum yield can be estimated from
Bermudagrass: N1/2 = by / c, = 1.50/0.0078 = 190 kg ha' (74)
Tall fescue: N11/2 = by /c, = 1.11/0.0078 = 140 kg ha' (75)
Similarly, the value of applied N (N,'/2) to achieve 50% of maximum plant N uptake can be
estimated from
Bermudagrass: N/12 = b,/c, = 2.06/0.0078= 260 kg ha1 (76)
Tall fescue: N'12 = b,, / c, = 1.54/0.0078 = 200 kg ha1 (77)
The model can also be used to estimate peak efficiency of plant N recovery (Overman,
2006b) given by (N = Np, E = Ep)
b
N=Np = 1.5N'/2 = 1.5 bn
E =Ep=N.Nu)=(AA (c 4 1 1 (78)
S Np 4 1Nl.5n 1l+exp(0.5b,) 1+exp(b,,)
Equation (78) represents the point of maximum overall efficiency of recovery. For the Arkansas
study these become
Bermudagrass: Np = 390 kg ha, Ep = (0.965)(1.295)(0.737 0.113) = 0.78 = 78%
Tall fescue: Np = 300 kg ha Ep = (0.663)(1.732)(0.684 0.177) = 0.58 = 58%
It is apparent that bermudagrass produces a higher level of nitrogen utilization than tall fescue
(78% vs. 58%).
The logistic model can be generalized by writing the equations in dimensionless format.
Input variables are defined by
y =cNby (79)
, = cNb b (80)
Dimensionless response equations for yield, plant N uptake, and plant N concentration can be
written as
Y 1
S==  (81)
Ay 1 + exp() (81)
0 N 1 (82)
A, 1 + exp(n,)
N= N, Nl 1 (83)
Y Nc Nl Nc_ 1+ exp(4n)
Dimensionless phase equations can be written as
Y N, /K (84)
 (84)
Nm 1+N.
c 1 (85)
Net K,
Dimensionless variables are given in Table 8 for average data in Tables 1 and 2 for
bermudagrass and tall fescue, respectively. Appropriate values of model parameters have been
used for each grass. Response results are shown in Figure 14, where the curves are drawn from
Eqs. (81) through (83). Phase results are shown in Figure 15, where the curve and line are drawn
from Eqs. (84) and (85), respectively.
Dimensionless plots demonstrate that various sets of data can be plotted on the same graph
when generalized variables are employed. This is important if the model is to have broad
applicability over various crops, soils, and environmental conditions. In physics this is referred to
as invariance of the model with respect to changes in input conditions, and it has proven to be a
guiding principle for some 400 years.
Quadratic Model of Crop Response to Applied Nitrogen
Some investigators utilize a quadratic model of crop response. Procedures for this model are
now discussed for the forage grass study. Assume yield and plant N uptake response to applied
nitrogen are described by the quadratic equations
Y = ao + aN + a2N2 (86)
N, = ao +aN+a'N2 (87)
N= N = a, +a'N + a2N (88)
Y ao + aN+aN2
Equation (88) is defined automatically from Eqs. (86) and (87). The next step is to evaluate
model parameters by regression analysis for bermudagrass response to applied nitrogen (Table
1).
Regression analysis by the least squares criterion leads to the three simultaneous equations
for yield
ZY=nao + (N)a + (ZN2)a2
ZNY =(N)o + (N2 )a, + (I )a2 (89)
SNzY=( 2 N2)a" + (I 3)ai + (N4)a2
where n is the number of observations and the sums are over the seven observations. Since the
equations are linear in the parameters (ao, a,, a2) the procedure for evaluation is called linear
regression. The system of equations can be written in matrix form as
n N N 2 I ao1 [
ZN XN2 N3 a, = a NY (90)
N 2 IN'3 Z 4 a2 2
This represents three simultaneous equations in three unknowns (ao, a1, a2). Note that elements
of the coefficient matrix and the righthand vector can be calculated directly from the data. The
system can be solved provided that the determinant of the coefficient matrix does not vanish. For
the present set of data Eq. (90) becomes (with scaling of N/100) for bermudagrass response to
fertilizer (1982 through 1985 average from Table 1)
7 23.52 114.1504 "ao 100.29
23.52 114.1504 619.573248 a1 = 428.0528 (91)
114.1504 619.573248 3579.75654400 _a2i 2194.2341120
We can solve for parameters ao, a,. and a2 by Cramer's rule using determinants (Ayers, 1962)
n ZN ZN2 7 23.52 114.1504
D= ZN EN2 N3 = 23.52 114.1504 619.73248 =32,497 (92)
N2 N3 ZN4 114.1504 619.73248 3579.756544
ZY EN N2 100.29 23.52 114.1504
DaE = Y N2 ZN3 = 428.0528 114.1504 619.73248 = 100,315 (93)
ZN2Y ZN3 ZN4 2194.2341120 619.73248 3579.756544
n ZY N2 7 100.29 114.1504
Da, = Z N ZNY N3 = 23.52 428.0528 619.73248 = 172,267 (94)
Z N2 ZN2y ZN4 114.1504 2194.2341120 3579.756544
n ZN ZY 7 23.52 100.29
Da2 ZN ZN2 ZNY = 23.52 114.1504 428.0528 =13,095 (95)
ZN2 ZN3 ZN2Y 114.1504 619.73248 2194.2341120
Parameters are now estimated from
Da. 100,315
ao D 3,9 = 3.087 (96)
a, = D 10 172,267102 = 0.0530 (97)
D 32,497
D 13,095
a2 = Da, 104 13,095104 = 0.0000403 (98)
D 32,497
The regression equation for yield becomes
= 3.087 + 0.0530N 0.0000403N2 (99)
Regression analysis leads to the three simultaneous equations for plant N uptake
N. = nao + (E Na;+(N2a
Z NN = N) + XN )a + N3 )a (100)
NN, = + + ( 4 2
which can be written in matrix form as
[n N ZN2 YFa N
ZN ZN 2 NJ3 [a = NN.
ZN2 N3 4ZN L 2
For the present set of data Eq. (101) becomes (with scaling of N/100 and Nu/1O0)
23.52
114.1504
619.573248
114.1504 a'o 20.4540
619.573248 a' = 91.828800
3579.75654400 a' 481.626880
We can solve for a', a', and a' by Cramer's rule using determinants
n
D= IN
IN2
IN
N 2
IN3
IN
Da = ZNNu
SN2NU
n
Da, = ZN
ZN2
n
Da; = N
N2
ZN
ZN
ZAW.
ZNN
SN2N
ZN
SN2
ZN2 7 23.52 114.1504
N 3 = 23.52 114.1504 619.573248 = 32,497
IZN4 114.1504 619.573248 3579.756544
I N2 20.4540
2 ZN3 = 91.828800
3 ZN4 481.62688
ZN2 7
N3 = 23.52
I N4 114.1504
ZNu 7
SNN, = 23.52
Z N2Nu 114.1504
23.52
114.1504
619.57324!
20.4540
91.828800
481.62688
23.52
114.1504
619.573248
114.1504
619.573248 = 12,069
8 3579.756544
114.1504
619.573248 = 33,231
3579.756544
20.4540
91.828800 = 1,764
481.62688
Parameters are now estimated from
D, 12,069102=37.14
D 32,497
a Dbl 10 2 33,231 =1.023
D 32,497
a = c102 104 764102 = 0.000543
D 32,497
The regression equation for plant N uptake becomes
7
23.52
114.1504
(101)
(102)
(103)
(104)
(105)
(106)
(107)
(108)
(109)
N, = 37.14 +1.023N 0.000543N2
It follows that plant N concentration is described by
 Nu 37.14+1.023N 0.000543N2
Y 3.087 + 0.0530N 0.0000403N2
Estimates of bermudagrass response to fertilizer nitrogen by the quadratic model are given in
Table 9. Response of yield (Y), plant N uptake (Nu), and plant N concentration (Nc) are shown in
Figure 16, where the curves are drawn from Eqs. (99), (110), and (111). Note that Y vs. N
reaches a peak at N = 660 kg ha1, and then declines with increase in applied N. Similarly, Nu vs.
N reaches a peak at N= 940 kg ha"1, and then declines with increase in applied N. The curve for
Nc vs. N exhibits strange behavior below N = 100 kg ha1 and above N= 700 kg ha1. There is
evidence that yield, plant N uptake, and plant N concentration all approach plateaus at high
applied nitrogen (N _> 1000 kg ha'") as discussed by Overman and Scholtz (2002). While the
quadratic model provides excellent fit of the data points in this study, it fails to describe general
response to applied N. Phase plots (Y and Nc vs. N,) are shown in Figure 17, where the curves are
drawn from values in Table 9. While the curves fit the data points reasonably well, they exhibit
strange behavior at low and high ends. The fallacy in using the quadratic model for crop
response to applied nitrogen now becomes apparent.
Mitscherlich Model of Crop Response to Applied Nitrogen
Another alternative model for crop response to applied nitrogen is one proposed by
Mitscherlich (Overman and Scholtz, 2002). Assume that response of biomass yield (Y), plant N
uptake (Nu), and plant N concentration (Nc) to applied nitrogen (N) is given by
Y = Yo + (Y. Yo)[1 exp(cyN)] (112)
N. =Nuo + (Nu Nu0)[1 exp(cN)] (113)
N, N N. +(Num No0)[1exp(cN)] (114)
c Y Yo +(Y.Yo)[1exp(cN)]
where 0 and m subscripts designate intercept and plateau values, and cy and cn are response
coefficients for yield and plant N uptake, respectively. This model is now applied to
bermudagrass response to applied nitrogen (Table 1). Equations (112) and (113) can be
linearized to the forms
Z = ln(Y, Y)= ln(22.25 Y)= ln(Ym Yo)cyN = 3.013 0.00380N r =0.9975 (115)
Z, =ln(Num N,,)=ln(650Nu)=ln(Nm No) c,N = 6.433 0.00191N r = 0.9987(116)
where Ym = 22.25 Mg ha1' and Num = 650 kg ha1 have been chosen to optimize correlation
coefficients, r. It follows from Eqs. (115) and (116) that the response functions are given by
(110)
was a decline in productivity between 1983 and 1985 as measured by parameters Ay and An as
shown in Figure 13. The dashed curves in Figure 13 are drawn from
Ay = 10.50 + 7.00 exp[ 0.775(Y 1983)] (71)
A, = 185 +185exp[ 0.775(Y1983)] (72)
where Y > 1983 is the year. Projections are shown for two years beyond 1985. Equations (71)
and (72) suggests that values would stabilize at Ay = 10.50 Mg ha1, An = 185 kg ha', and Ncm =
17.6 g kg'1. These values are highly speculative, of course. Suppression of yields and plant N
uptake by applications of broiler litter probably result from accumulation of solids on the
vegetation which could reduce the rate of photosynthesis.
DISCUSSION
The extended logistic model of crop response to applied nutrients describes response of
bermudagrass and tall fescue to fertilizer and broiler litter rather well (Figures 1 through 12). The
predicted linear relationship between plant N concentration (Nc) and plant N uptake (Nu) is
confirmed. The nitrogen response coefficient (cn) is the same for bermudagrass and fescue grown
on the same soil, which lends support to the hypothesis that Cn is a measure of nutrient
availability in the soil (Overman, 2006a, b). Since minimum and maximum limits on plant N
concentration (Nci and Ncm) are believed to be characteristics of the plant species, it follows from
Ab = ln I N I (73)
that the shift in intercept parameters (Ab) is characteristic of the plant species as well. Effect of
water availability occurs in the linear model parameters Ay and An.
The value of applied N (N1/2) to achieve 50% of maximum yield can be estimated from
Bermudagrass: N1/2 = by / c, = 1.50/0.0078 = 190 kg ha' (74)
Tall fescue: N11/2 = by /c, = 1.11/0.0078 = 140 kg ha' (75)
Similarly, the value of applied N (N,'/2) to achieve 50% of maximum plant N uptake can be
estimated from
Bermudagrass: N/12 = b,/c, = 2.06/0.0078= 260 kg ha1 (76)
Tall fescue: N'12 = b,, / c, = 1.54/0.0078 = 200 kg ha1 (77)
The model can also be used to estimate peak efficiency of plant N recovery (Overman,
2006b) given by (N = Np, E = Ep)
Y = Yo + (Y )[1 exp(cN)]= 1.90 + 20.35[1 exp(0.00380N)]
S= No +(N,, N)1l exp(c,,N)]= 28+ 622[1exp(0.00191N)] (118)
N N, 28+622[1exp(0.00191N)]
SY 1.90 + 20.35[1 exp[(0.00380N)]
Estimates are given in Table 10. Response of bermudagrass to applied nitrogen is shown in
Figure 18, where the curves are drawn from Eqs. (117) through (119). Phase plots are shown in
Figure 19, where the curves are drawn from values given in Table 10. Since the Mitscherlich
model appears more sensible than the quadratic model, we now explore it a little further.
Response of tall fescue to fertilizer nitrogen is taken from Table 2 and given in Table 11. For
this analysis it is assumed that ey = 0.00380 ha kg"1 and c,, = 0.00191 ha kg1 as with
bermudagrass response to fertilizer nitrogen. It becomes appropriate to form Mitscherlich
variables as given in columns three and five in Table 11. Linear regression then leads to the
response equations
S= Y, + (Y,, Y0 )[1 exp( 0.00380N)] (120)
= 2.65 +10.22[1 exp( 0.00380N)] r = 0.9957, Y. = 12.87 Mgha'
N= No +(N,, N0o)[l exp(0.00191N)] (121)
= 53.5 + 401.2[1exp( 0.00191N)] r = 0.9932, Nu, = 454.7 kgha'
c = N, 53.5 + 401.2[1 exp(0.00191N)]
c Y 2.65+10.22[1exp(0.00380N)]
The linear correlations are shown in Figure 20, where the lines are drawn from Eqs. (120) and
(121). Response curves are shown in Figure 21, where the curves are drawn from Eqs. (120)
through (122). Phase plots are shown in Figure 22, where the curves are drawn from values in
Table 12. The plot of Nc vs. Nu shows strong upward curvature for Nu < 50 kg ha1, which seems
inconsistent with actual response.
This analysis suggests for experiments on the same soil that c, = cy/2. Whether this is
generally true or is an artifact of these particular data has not been established. It does appear that
the Mitscherlich model represents a better model for crop response to applied nitrogen that the
quadratic model. However, the extended logistic model is a more rational approach than either of
these two.
Field Study from Georgia
DATA ANALYSIS
Data for this analysis are adapted from a field study conducted at the USDA Southern
Piedmont Conservation Research Center at Watkinsville, GA during the period 1972 through
1979. Details of the study can be found in Carreker et al. (1977) and Dudzinsky et al. (1983).
Data were provided by Wilkinson (1988). Coastal bermudagrass was grown on Cecil sandy loam
(117)
clayeyy, kaolinitic, thermic Typic Hapludult). Depth of the topsoil was approximately 25 cm and
field slope averaged 7%. Data for seasonal response are shown in Table 13 and Figure 23.
Parameters for yield response are estimated by nonlinear regression to obtain
A Ay17.2
Y = r = 0.99960 (123)
1+ exp(by c,N) 1+ exp(2.70 0.0072N)
Other parameters are selected by inspection to obtain uptake response
An 550
N = 55(124)
+ exp(b, cN) 1+ exp(3.30 0.0072N)
from which it follows that plant N concentration is given by
SN cm[ 1+ exp(by c,,N) 1 + exp(2.70 0.0072N) (125)
N = N = 32.o0(OO (125))
Y 1 + exp(b, c,N) + exp(3.30 0.0072N)
Curves in Figure 23 are drawn from Eqs. (123) through (125). Phase parameters are calculated
from
Ay 17.2
Y. Y 72 38.1 Mg ha (126)
m exp(Ab) 1 exp(0.60)
K, = = 550 = 669 kg ha1 (127)
exp(Ab)1 exp(0.60) 1
which leads to the phase relations
YYNu 38.1N,
Y=N 38. 1N (128)
K, + N, 669+ N,
N = _K + Nu =17.6+0.0263N (129)
Y Y. Y.r
Phase plots are shown in Figure 24, where the curve and line are drawn from Eqs. (128) and
(129), respectively.
The next step is to simulate the growth data for the study. Average harvest interval was 4.3
wk. Growth parameters are assumed to be p = 26.0 wk, V2o = 8.00 wk, c = 0.15 wk, k = 5.
These values lead to the equations for dimensionless time
t p r2oc t 26.0
,2 2 8.00
and growth quantifier
AQi = (1 kxi)[erf x erf xi ] [exp(x2 exp( X .exp(2ocx)
= {(1 5x1 )[erf x erf x, ] 2.821 [exp( x2 ) exp( x )}. exp(l.20xi)
Results are given in Table 14. Note that reference time x; is reset for each growth interval to
calculate AQi. Correlation of cumulative yield with cumulative growth quantifier is shown in
Figure 25, where the lines are drawn from
N= 0: =0.076+0.410Q r = 0.9865 (132)
N= 700 kg ha1: = 0.29 + 3.381Q r = 0.9949 (133)
N= 1400 kg ha'1: =0.35+ 3.491Q r = 0.9938 (134)
It may be noted from Table 14 that the maximum value of the cumulative growth quantifier is Q
= 4.50. This value can be substituted into Eqs. (132) through (134) to estimate maximum
cumulative yields, which leads to values of the yield factors of A(N= 0) = 0.43, A(N= 700) =
3.32, and A(N= 1400) = 3.57 Mg ha"1. Recall from Eq. (116) that A = 17.2 Mg ha, so that the
maximum value of the yield factor is Ao = 17.2/4.50 = 3.82 Mg ha Dependence of A on applied
N is shown in Figure 26, where the curve is drawn from
= 382 (135)
1+ exp(2.70 0.0072N)
Cumulative biomass (1) with calendar time (t) can be estimated from the probability equation
(for constant harvest interval)
f= [1+erft 260 (136)
2 8.00 )J (
where A. is estimated from Eqs. (132) through (134) with Qoo = 4.50. Results are shown in
Figure 27, where the curves are drawn from Eq. (136) with appropriate values of Aoo.
The final step is to couple cumulative plant N uptake with cumulative biomass. Results are
shown in Figure 28, where the lines are drawn from
N= 0: N = 2.33+15.4Y r= 0.9977 (137)
N= 700 kg ha': = 5.91+ 27.9Y r = 0.99974 (138)
N= 1400 kg ha'1 N =12.3+31.6Y r= 0.99995 (139)
It should be noted that the slopes in Eqs. (136) through (138) represent estimates of average plant
N concentration over the season.
DISCUSSION
Analysis of data from Watkinsville, GA confirm that the nitrogen response coefficient, Cn, is
common for fertilizer and broiler litter. The shift in the intercept parameters, Ab = 0.60, means
that plant N concentration is bounded by 17.6 < Nc < 32.0 g kg1. Response data are described
rather well by the extended logistic model.
Growth data are described quite well by the expanded growth model, using parameters
common to other studies from various locations. The linear relationship between cumulative
biomass and cumulative growth quantifier is confirmed. A linear relation between cumulative
plant N and cumulative biomass is also confirmed for constant harvest interval.
Overall efficiency of plant N utilization can be estimated from
NP =1.5 =1.5 330 =688 kgha1
p c 0.0072
N. N.uo Ae 4 1 1
E=E N= 1 (n 1 1e = 1 0.642 = 64.2%
I NP 4 1.5b Ll+exp(0.5b,) 1+exp(b,)
This value is somewhat less than obtained at Fayetteville, AR.
Another important question was examined in the study at Watkinsville, GA. What is the
residual effect of soil nitrogen from either fertilizer or broiler litter with time if application is
discontinued. In this case nitrogen was applied during years 1 and 2, and none during years 3, 4,
5, 6, and 7. We first examine data for fertilizer N= 1344 kg ha'1 during years 1 and 2. Data are
given in Table 15 and shown in Figure 29. Decrease in Y and Nu appears essentially exponential
with elapsed time. It is assumed that Nc is bounded by 17.6 < Nc < 32.0 g kg1 as noted above. It
is further assumed that Yin Figure 29 is bounded by 20.0 > Y> 2.0 Mg ha1. By inspection the
equation for Y vs. t is found to be
Y = 2.0 +18.0 exp( 0.50t) (141)
as shown in Figure 29. A similar equation is assumed for Nu vs. t given by
N, = 35+605exp(0.50t) (142)
as also shown in Figure 29. Equations (141) and (142) lead to the equation for Nc vs. t
N, 35+605exp(0.50t)
N= (143)
N Y 2.0+18.0exp(0.50t) (143)
The values 35 and 605 have been chosen to satisfy the constraints on Nc. While there is
considerable scatter in the data, Eqs. (141) through (143) appear to describe the results
reasonably well.
We next examine data for broiler litter N= 5100 kg ha"1 during years 1 and 2. Data are given
in Table 16 and shown in Figure 30. Decrease in Y and Nu appears essentially exponential with
elapsed time. It is again assumed that Nc is bounded by 17.6 < Nc < 32.0 g kg' as noted above. It
Y = Yo + (Y )[1 exp(cN)]= 1.90 + 20.35[1 exp(0.00380N)]
S= No +(N,, N)1l exp(c,,N)]= 28+ 622[1exp(0.00191N)] (118)
N N, 28+622[1exp(0.00191N)]
SY 1.90 + 20.35[1 exp[(0.00380N)]
Estimates are given in Table 10. Response of bermudagrass to applied nitrogen is shown in
Figure 18, where the curves are drawn from Eqs. (117) through (119). Phase plots are shown in
Figure 19, where the curves are drawn from values given in Table 10. Since the Mitscherlich
model appears more sensible than the quadratic model, we now explore it a little further.
Response of tall fescue to fertilizer nitrogen is taken from Table 2 and given in Table 11. For
this analysis it is assumed that ey = 0.00380 ha kg"1 and c,, = 0.00191 ha kg1 as with
bermudagrass response to fertilizer nitrogen. It becomes appropriate to form Mitscherlich
variables as given in columns three and five in Table 11. Linear regression then leads to the
response equations
S= Y, + (Y,, Y0 )[1 exp( 0.00380N)] (120)
= 2.65 +10.22[1 exp( 0.00380N)] r = 0.9957, Y. = 12.87 Mgha'
N= No +(N,, N0o)[l exp(0.00191N)] (121)
= 53.5 + 401.2[1exp( 0.00191N)] r = 0.9932, Nu, = 454.7 kgha'
c = N, 53.5 + 401.2[1 exp(0.00191N)]
c Y 2.65+10.22[1exp(0.00380N)]
The linear correlations are shown in Figure 20, where the lines are drawn from Eqs. (120) and
(121). Response curves are shown in Figure 21, where the curves are drawn from Eqs. (120)
through (122). Phase plots are shown in Figure 22, where the curves are drawn from values in
Table 12. The plot of Nc vs. Nu shows strong upward curvature for Nu < 50 kg ha1, which seems
inconsistent with actual response.
This analysis suggests for experiments on the same soil that c, = cy/2. Whether this is
generally true or is an artifact of these particular data has not been established. It does appear that
the Mitscherlich model represents a better model for crop response to applied nitrogen that the
quadratic model. However, the extended logistic model is a more rational approach than either of
these two.
Field Study from Georgia
DATA ANALYSIS
Data for this analysis are adapted from a field study conducted at the USDA Southern
Piedmont Conservation Research Center at Watkinsville, GA during the period 1972 through
1979. Details of the study can be found in Carreker et al. (1977) and Dudzinsky et al. (1983).
Data were provided by Wilkinson (1988). Coastal bermudagrass was grown on Cecil sandy loam
(117)
is further assumed that Yin Figure 29 is bounded by 20.0 > Y> 2.0 Mg ha . By inspection the
equation for Y vs. t is found to be
Y = 5.0 +15.0 exp( 0.25t) (144)
as shown in Figure 30. A similar equation is assumed for Nu vs. t given by
N, = 87 + 555exp( 0.25t) (145)
as also shown in Figure 30. Equations (144) and (145) lead to the equation for Nc vs. t
SN,, 87+555exp(0.25t)
c Y 5.0 +15.0exp(0.25t)
The values 87 and 555 have been chosen to satisfy the constraints on Nc. While there is
considerable scatter in the data, Eqs. (144) through (146) appear to describe the results
reasonably well.
These results indicate that plants can recover residual nitrogen from the soil during
succeeding years. For the case of fertilizer applications the decay coefficient is 0.50 yr1, which
leads to a characteristic time of tc = 1/0.50 = 2 years. This means that for t = 2 yr, maximum
yield drops to 2.0 + 18.0 exp (1) = 8.6 Mg ha1. For the case of broiler litter applications tc =
1/0.25 = 4 years, so that yield drops to 5.0 + 15.0 exp (1) = 10.5 Mg ha'1. Broiler litter acts as a
slow release nitrogen source with a longer release time. Reck and Overman (2006) analyzed
response of corn (Zea mays L.) to residual nitrogen from plowdown of grass sod on this same
soil at Watkinsville, GA. The characteristic time was also 2 yr.
CONCLUSIONS
This analysis has shown that the extended logistic model describes response curves and
phase relations for bermudagrass and tall fescue with fertilizer and broiler litter. The model
provides a unified treatment of biomass and plant nitrogen uptake for both grasses. Detailed
procedures for model calibration have been discussed. It has been shown elsewhere that the
model applies to both perennials and annuals (Overman and Scholtz, 2002; Overman et al.,
1994b). A more detailed analysis of the data from Arkansas by nonlinear regression and analysis
of variance has been performed by Brock (2004).
The expanded growth model provides reasonable simulation of accumulation of biomass
with calendar time with model parameters (,a, 2o, c, k) which have been used for various
locations (Overman, 2006b).
A guide to composting of poultry litter has been published (Brake, 1992).
DISCUSSION
Analysis of data from Watkinsville, GA confirm that the nitrogen response coefficient, Cn, is
common for fertilizer and broiler litter. The shift in the intercept parameters, Ab = 0.60, means
that plant N concentration is bounded by 17.6 < Nc < 32.0 g kg1. Response data are described
rather well by the extended logistic model.
Growth data are described quite well by the expanded growth model, using parameters
common to other studies from various locations. The linear relationship between cumulative
biomass and cumulative growth quantifier is confirmed. A linear relation between cumulative
plant N and cumulative biomass is also confirmed for constant harvest interval.
Overall efficiency of plant N utilization can be estimated from
NP =1.5 =1.5 330 =688 kgha1
p c 0.0072
N. N.uo Ae 4 1 1
E=E N= 1 (n 1 1e = 1 0.642 = 64.2%
I NP 4 1.5b Ll+exp(0.5b,) 1+exp(b,)
This value is somewhat less than obtained at Fayetteville, AR.
Another important question was examined in the study at Watkinsville, GA. What is the
residual effect of soil nitrogen from either fertilizer or broiler litter with time if application is
discontinued. In this case nitrogen was applied during years 1 and 2, and none during years 3, 4,
5, 6, and 7. We first examine data for fertilizer N= 1344 kg ha'1 during years 1 and 2. Data are
given in Table 15 and shown in Figure 29. Decrease in Y and Nu appears essentially exponential
with elapsed time. It is assumed that Nc is bounded by 17.6 < Nc < 32.0 g kg1 as noted above. It
is further assumed that Yin Figure 29 is bounded by 20.0 > Y> 2.0 Mg ha1. By inspection the
equation for Y vs. t is found to be
Y = 2.0 +18.0 exp( 0.50t) (141)
as shown in Figure 29. A similar equation is assumed for Nu vs. t given by
N, = 35+605exp(0.50t) (142)
as also shown in Figure 29. Equations (141) and (142) lead to the equation for Nc vs. t
N, 35+605exp(0.50t)
N= (143)
N Y 2.0+18.0exp(0.50t) (143)
The values 35 and 605 have been chosen to satisfy the constraints on Nc. While there is
considerable scatter in the data, Eqs. (141) through (143) appear to describe the results
reasonably well.
We next examine data for broiler litter N= 5100 kg ha"1 during years 1 and 2. Data are given
in Table 16 and shown in Figure 30. Decrease in Y and Nu appears essentially exponential with
elapsed time. It is again assumed that Nc is bounded by 17.6 < Nc < 32.0 g kg' as noted above. It
is further assumed that Yin Figure 29 is bounded by 20.0 > Y> 2.0 Mg ha . By inspection the
equation for Y vs. t is found to be
Y = 5.0 +15.0 exp( 0.25t) (144)
as shown in Figure 30. A similar equation is assumed for Nu vs. t given by
N, = 87 + 555exp( 0.25t) (145)
as also shown in Figure 30. Equations (144) and (145) lead to the equation for Nc vs. t
SN,, 87+555exp(0.25t)
c Y 5.0 +15.0exp(0.25t)
The values 87 and 555 have been chosen to satisfy the constraints on Nc. While there is
considerable scatter in the data, Eqs. (144) through (146) appear to describe the results
reasonably well.
These results indicate that plants can recover residual nitrogen from the soil during
succeeding years. For the case of fertilizer applications the decay coefficient is 0.50 yr1, which
leads to a characteristic time of tc = 1/0.50 = 2 years. This means that for t = 2 yr, maximum
yield drops to 2.0 + 18.0 exp (1) = 8.6 Mg ha1. For the case of broiler litter applications tc =
1/0.25 = 4 years, so that yield drops to 5.0 + 15.0 exp (1) = 10.5 Mg ha'1. Broiler litter acts as a
slow release nitrogen source with a longer release time. Reck and Overman (2006) analyzed
response of corn (Zea mays L.) to residual nitrogen from plowdown of grass sod on this same
soil at Watkinsville, GA. The characteristic time was also 2 yr.
CONCLUSIONS
This analysis has shown that the extended logistic model describes response curves and
phase relations for bermudagrass and tall fescue with fertilizer and broiler litter. The model
provides a unified treatment of biomass and plant nitrogen uptake for both grasses. Detailed
procedures for model calibration have been discussed. It has been shown elsewhere that the
model applies to both perennials and annuals (Overman and Scholtz, 2002; Overman et al.,
1994b). A more detailed analysis of the data from Arkansas by nonlinear regression and analysis
of variance has been performed by Brock (2004).
The expanded growth model provides reasonable simulation of accumulation of biomass
with calendar time with model parameters (,a, 2o, c, k) which have been used for various
locations (Overman, 2006b).
A guide to composting of poultry litter has been published (Brake, 1992).
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Publications. New York, NY. 1046 p.
Ayers, F. 1962. Theory and Problems of Matrices. McGrawHill. New York, NY. 219 p.
Brake, J.D. 1992. A Practical Guide for Composting Poultry Litter. Mississippi Agricultural &
Forestry Experiment Station Bulletin 981. Mississippi State, MS. 8 p.
Brock, K.H. 2004. Extended logistic model of crop response to applied nutrients. PhD
Dissertation. University of Florida. Gainesville, FL. 163 p.
Carreker, J.R., S.R. Wilkinson, A.P. Barnett, and J.E. Box, Jr. 1977. Soil and Water Management
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Office, Washington, DC. 76 p.
Dudzinsky, M.L., S.R. Wilkinson, R.N. Dawson, and A.P. Barnett. 1983. Fate of Nitrogen from
NH4NO3 and Broiler Litter Applied to Coastal Bermudagrass. Pp. 373388. Nutrient Cycling
in Agricultural Ecosystems. R. Todd, L. Asmusen, and R. Leonard (eds). University of
Georgia Agricultural Experiment Stations Special Publication 23. Athens, GA.
Huneycutt, H.J., C.P. West, and J.M. Phillips. 1988. Responses of Bermudagrass, Tall Fescue,
and Tall FescueClover to Broiler Litter and Commercial Fertilizer. Bulletin 913. Arkansas
Agricultural Experiment Station, University of Arkansas, Fayetteville, Arkansas. 20 p.
Overman, A.R. 1998. An expanded growth model for grasses. Communications in Soil Science
and Plant Analysis 29:6785.
Overman, A.R. 2006a. A Memoir on Chemical Transport: Application to Soils and Crops.
University of Florida. Gainesville, FL. 364 p.
Overman, A.R. 2006b. A Memoir on Crop Growth: Accumulation ofBiomass and Mineral
Elements. University of Florida. Gainesville, FL. 386 p.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. Philadelphia, PA. 328 p.
Overman, A.R., F.G. Martin, and S.R. Wilkinson. 1990. A logistic equation for yield response of
forage grass to nitrogen. Commun. Soil Science and Plant Analysis 21:595609.
Overman, A.R., S.R. Wilkinson, and D.M. Wilson. 1994a. An extended model of forage grass
response to applied nitrogen. Agronomy J. 86:617620.
Overman, A.R., D.M. Wilson, and E.J. Kamprath. 1994b. Estimation of yield and nitrogen
removal by corn. Agronomy J. 86:10121016.
Reck, W.R. and A.R. Overman. 2006. Modeling effect of residual nitrogen on response of corn
to applied nitrogen. Commun. Soil Science and Plant Analysis 37:16511662.
Wilkinson, S.R. 1988. Personal communication.
Table 1. Response ofbiomass yield (Y), plant nitrogen uptake (N,), and plant nitrogen
concentration (Nc) to applied nitrogen (N) from fertilizer for bermudagrass grown at Fayetteville,
AR. Data are averages for 1982 through 1985.1
Irrigation N Y Nc Nu Y* N*
kg ha" Mg ha g kg' kg ha' Mg ha kg ha
Yes
Avg (SD)
No
Avg (SD)
0
112
224
336
448
560
672
0
112
224
336
448
560
672
Avg 0
112
224
336
448
560
672
Avg (SD) 
3.58
9.80
14.01
18.55
20.35
21.86
22.85
2.42
7.80
11.09
14.59
17.47
17.79
18.43
3.00
8.80
12.55
16.57
18.91
19.82
20.64
14.7
16.2
18.2
19.3
20.0
21.4
22.4
13.3
15.5
18.5
19.8
22.0
22.6
24.1
14.1
15.9
18.3
19.6
20.9
21.9
23.2
52.6
159
255
358
407
468
512
32.2
121
205
289
384
402
444
42.4
140
230
324
396
435
478
19.62
28.13
24.95
24.60
23.12
23.10
23.39
23.84 (2.56)
13.27
22.39
19.75
19.35
19.85
18.80
18.87
18.90 (2.76)
16.45
25.26
22.35
21.97
21.48
20.95
21.13
21.37 (2.61)
'Crop data adapted from Huneycutt et al. (1988).
465
680
604
562
504
515
533
552 (72)
285
517
485
454
475
442
462
446 (75)
375
599
544
509
490
478
498
499 (68)
Table 2. Response ofbiomass yield (Y), plant nitrogen uptake (Nu), and plant nitrogen
concentration (Nc) to applied nitrogen (N) from fertilizer for fescue grown at Fayetteville, AR.
Data are averages for 19821983, 19831984, 19841985.'
Irrigation
Yes
Avg (SD)
N
kg ha1
0
112
224
336
448
560
672
0
112
224
336
448
560
672
Y
Mg ha
3.53
7.37
9.55
13.21
14.49
15.30
15.46
2.09
4.82
6.35
7.42
8.29
7.89
8.37
Nc
g kg'
20.9
20.2
21.4
24.5
25.8
26.2
27.2
20.2
21.5
23.1
25.7
27.2
29.0
29.4
Avg (SD)
Avg 0
112
224
336
448
560
672
Ave (SD) 
2.81
6.10
7.95
10.32
11.39
11.60
11.92
'Crop data adapted from Huneycutt et al.
20.6
20.7
22.1
25.0
26.3
27.2
27.9
(1988).


Nu
kg ha"1
73.8
149
204
324
374
401
420
42.2
104
147
191
225
229
246
58.0
126
176
258
300
315
333
Mg ha1
14.24
16.71
14.60
16.13
15.83
15.89
15.71
15.59 (0.87)
8.43
10.93
9.71
9.06
9.05
8.19
8.50
9.12 (0.94)
11.34
13.83
12.15
12.60
12.54
12.05
12.11
12.37 (0.76)
Nu *
kg ha1
418
439
370
434
427
425
430
420 (23)
239
307
266
256
257
243
252
260 (23)
329
371
319
346
342
334
341
340 (16)
Table 3. Response of biomass yield (Y), plant nitrogen uptake (Nu), and plant nitrogen
concentration (Nc) to applied nitrogen (N) from broiler litter for bermudagrass grown at
Fayetteville, AR (1982).1
Irrigation N Y Nc Nu Y* Nu*
kg ha1 Mg ha' g kg' kg ha1 Mg ha1 kg ha'
Yes 0 3.88 15.4 59.8 16.38 305
200 10.01 15.2 152 20.75 359
400 13.82 16.3 225 18.75 327
600 15.72 17.1 269 17.59 310
Avg (SD)     18.37 (1.86) 325(24)
No 0 3.70 14.7 54.4 15.62 277
200 8.04 15.2 122 16.66 288
400 10.10 16.6 168 13.71 244
600 13.98 18.1 253 15.64 291
Avg (SD)     15.41 (1.23) 275(22)
Avg 0 3.79 15.1 57.1 16.00 291
200 9.02 15.2 137 18.69 324
400 11.96 16.4 196 16.23 285
600 14.85 17.6 261 16.61 300
Avg (SD)    16.88 (1.23) 300(17)
1Data adapted from Huneycutt et al. (1988).
Table 4. Response of biomass yield (Y), plant nitrogen uptake (Nu), and plant nitrogen
concentration (Nc) to applied nitrogen (N) from broiler litter for bermudagrass grown at
Fayetteville, AR (1983).1
Irrigation N Y Nc Nu Y* Nu*
kg ha' Mg ha1 g kg'' kg ha1 Mg ha' kg ha1
Yes 0 4.28 14.7 62.9 23.85 492
165 9.45 15.8 149 21.38 430
330 14.87 17.9 266 20.05 404
495 18.19 19.7 358 19.94 409
Avg (SD)    21.30 (1.82) 434 (40)
No 0 1.95 13.8 26.9 10.87 210
165 6.94 16.5 115 15.70 332
330 10.37 16.3 169 13.98 257
495 13.89 19.4 269 15.23 308
Avg (SD)     13.95 (2.17) 277 (54)
Avg 0 3.12 14.4 44.9 17.39 351
165 8.20 16.1 132 18.55 381
330 12.62 17.3 218 17.02 331
495 16.04 19.6 314 17.58 359
Avg (SD)     17.64 (0.65) 356 (21)
'Data adapted from Huneycutt et al. (1988).
Table 5. Response of biomass yield (Y), plant nitrogen uptake (Nu), and plant nitrogen
concentration (Nc) to applied nitrogen (N) from broiler litter for bermudagrass grown at
Fayetteville, AR (1984).1
Irrigation N Y Nc N, Y* Nu*
kg ha1 Mg ha1 g kg' kg ha1 Mg ha1 kg ha
Yes 0 4.17 14.7 61.3 17.61 380
150 7.66 17.4 133 15.32 348
300 12.34 19.4 239 16.17 359
450 12.81 19.8 254 14.04 294
Avg(SD)     15.78 (1.50) 345(37)
No 0 2.35 12.2 28.7 9.92 178
150 5.49 14.6 80.2 10.98 210
300 8.20 19.0 156 10.75 234
450 12.68 18.4 233 13.90 269
Avg (SD)     11.39 (1.74) 223(38)
Avg 0 3.26 13.8 45.0 13.76 279
150 6.58 16.3 107 13.16 280
300 10.27 19.3 198 13.46 297
450 12.75 19.1 244 13.98 282
Avg (SD)    13.59 (0.36) 285 (8.4)
'Data adapted from Huneycutt et al. (1988).
Table 6. Response ofbiomass yield (Y), plant nitrogen uptake (Nu), and plant nitrogen
concentration (Nc) to applied nitrogen (N) from broiler litter for bermudagrass grown at
S... __* A T [1 I'n \ 1
rayetteville, AK I198).
Irrigation N
kg ha'
Yes 0
125
250
375
Avg (SD) 
y
Mg ha'
1.75
5.96
8.15
11.51
Nc
g kg'
14.1
14.9
16.8
17.3
12.5
13.0
13.8
15.4
13.4
14.3
15.6
16.6
Avg (SD)  
1Data adapted from Huneycutt et al. (1988).
Nu
kg ha'
24.7
88.8
137
199
21.2
35.0
69.8
108
23.0
61.9
103
154
Y*
Mg ha1
11.33
18.27
14.50
14.89
14.75 (2.84)
11.00
8.24
9.00
9.07
9.33(1.18)
11.14
13.24
11.74
11.98
12.02 (0.88)
Table 7. Summary of logistic model parameters for bermudagrass response to broiler litter for
Fayetteville, AR study.
Year Ay An by bn Ab cn Nc Ncm
Mg ha1 kg ha1 ha kg1 g kg1 g kg'1
1982 16.85 300 1.17 1.41 0.24 0.0055 14.0 17.8
1983 17.57 355 1.52 1.92 0.40 0.0078 13.5 20.2
1984 13.59 285 1.17 1.65 0.48 0.0078 13.0 21.0
1985 12.00 220 1.70 2.06 0.36 0.0078 12.8 18.3
Avg 15.00 290 1.37 1.76 0.37  13.3 19.3
SD 2.65 56 0.29 0.29 0.10  0.54 1.52
Rel Error 0.18 0.19 0.21 0.16 0.27  0.04 0.08
Nu*
kg ha1
218
352
290
283
286 (55)
188
139
148
153
157 (21)
203
245
218
219
221 (17)
1.70
2.69
5.06
7.01
1.72
4.32
6.60
9.26
Avg (SD)
Avg
0
125
250
375
0
125
250
375
Table 8. Dimensionless formulation of the extended logistic model for response of bermudagrass
and tall fescue to applied nitrogen at Fayetteville, AR.1
Grass N y Y, c Y/Ym Nu/Kn N/Nct
kg ha'
Bermuda 0 1.50 2.06 0.141
112 0.63 1.19 0.413
224 0.25 0.31 0.588
336 1.12 0.56 0.777
448 1.99 1.43 0.887
560 2.87 2.31 0.929
672 3.74 3.18 0.968
Fescue 0 1.11 1.54 0.228
112 0.24 0.67 0.495
224 0.64 0.21 0.645
336 1.51 1.08 0.837
448 2.38 1.95 0.924
560 3.26 2.83 0.941
672 4.13 3.70 0.967
'Crop data taken from Tables 1 and 2 averages.
0.086
0.283
0.465
0.655
0.800
0.879
0.966
0.171
0.371
0.518
0.759
0.882
0.926
0.979
0.090
0.270
0.510
0.640
0.770
0.870
1.000
0.278
0.289
0.433
0.732
0.866
0.959
1.031
0.061 0.065 1.066
0.178 0.214 1.202
0.253 0.351 1.383
0.334 0.495 1.481
0.382 0.605 1.580
0.400 0.664 1.655
0.417 0.730 1.754
0.080 0.093 1.153
0.175 0.202 1.159
0.227 0.282 1.237
0.295 0.413 1.400
0.326 0.481 1.473
0.332 0.505 1.523
0.341 0.534 1.562
Table 9. Estimates of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) by
the quadratic model for bermudagrass response to applied nitrogen (N) at Fayetteville, AR.
N Y N, N,
kg ha'1 Mg ha' kg ha'1 g kg"1
50 0.34 15.4 45.7
35 1.18 0.67 0.57
25 1.74 11.23 6.46
15 2.28 21.67 9.49
0 3.09 37.14 12.0
25 4.39 62.38 14.2
50 5.64 86.93 15.4
75 6.84 110.8 16.2
100 7.98 134.0 16.8
125 9.08 156.5 17.2
150 10.13 178.4 17.6
175 11.13 199.5 17.9
200 12.08 220.0 18.2
250 13.82 259.0 18.7
300 15.36 295.2 19.2
350 16.70 328.7 19.7
400 17.84 359.5 20.2
450 18.78 387.5 20.6
500 19.51 412.9 21.2
550 20.05 435.5 21.7
600 20.38 455.5 22.4
650 20.51 472.7 23.0
700 20.44 487.2 23.8
750 20.17 499.0 24.7
800 19.70 508.0 25.8
850 19.02 514.4 27.0
900 18.14 518.0 28.6
950 17.07 518.9 30.4
1000 15.79 517.1 32.8
1050 14.31 512.6 35.8
1100 12.62 505.4 40.0
1200 8.66 482.8 55.8
1300 3.88 449.4 115.8
Table 10. Estimates of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) by
the Mitscherlich model for bermudagrass response to applied nitrogen (N) at Fayetteville, AR.
N Y N, N,
kg ha' Mg ha' kg ha'' g kg1
40 1.44 21.4 14.84
30 0.557 8.68 15.58
20 0.293 3.78 12.90
10 1.11 16.0 14.40
0 1.90 28.0 14.74
25 3.74 57.0 15.22
50 5.42 84.7 15.61
75 6.95 111.0 15.98
100 8.33 136.1 16.34
125 9.59 160.1 16.69
150 10.74 182.9 17.03
175 11.78 204.7 17.37
200 12.73 225.5 17.71
250 14.38 264.2 18.37
300 15.74 299.3 19.01
350 16.87 331.2 19.64
400 17.80 360.3 20.24
450 18.57 386.7 20.82
500 19.21 410.6 21.38
550 1973 432.4 21.91
600 20.17 452.3 22.42
650 20.53 470.3 22.91
700 20.83 486.6 23.37
750 20.07 501.5 23.80
800 21.28 515.0 24.21
850 21.45 527.3 24.59
900 21.58 538.5 24.95
950 21.70 548.7 25.28
1000 21.79 557.9 25.60
1050 21.87 566.3 25.89
1100 21.94 573.9 26.16
1200 22.04 587.1 26.64
1300 22.10 598.1 27.06
Table 11. Response of biomass yield (Y) and plant N uptake (Nu) to applied nitrogen (N) for tall
fescue at Fayetteville, AR.1
N Y [1 exp(0.00380N)] Nu [1 exp(0.00191N)]
kg ha1 Mg ha' kg ha1
0 2.81 0.000 58.0 0.000
112 6.10 0.347 126 0.193
224 7.95 0.573 176 0.348
336 10.32 0.721 258 0.474
448 11.39 0.818 300 0.575
560 11.60 0.881 315 0.657
672 11.92 0.922 333 0.723
'Crop data adapted from Huneycutt et al. (1988).
Table 12. Estimates of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) by
the Mitscherlich model for tall fescue response to applied nitrogen (N) at Fayetteville, AR.
N Y NR Nc
kg ha1 Mg ha1 kg ha'1 g kg1
60 0.0328 4.78 145.8
50 0.511 13.3 26.00
40 0.972 21.6 22.26
30 1.42 29.8 21.07
20 1.84 37.9 20.55
10 2.25 45.8 20.30
0 2.65 53.5 20.19
25 3.58 72.2 20.19
50 4.42 90.0 20.38
75 5.18 107.0 20.65
100 5.88 123.3 20.96
125 6.51 138.7 21.29
150 7.09 153.4 21.64
175 7.61 167.5 22.00
200 8.09 180.9 22.36
250 8.92 205.8 23.08
300 9.60 228.5 23.80
350 10.17 249.1 24.50
400 10.63 267.8 25.18
450 11.02 284.8 25.84
500 11.34 300.3 26.48
550 11.60 314.4 27.09
600 11.82 327.2 27.67
650 12.00 338.8 28.22
700 12.16 349.3 28.74
750 12.28 358.9 29.23
800 12.38 367.7 29.69
850 12.47 375.6 30.13
900 12.54 382.8 30.54
950 12.59 389.3 30.92
1000 12.64 395.3 31.27
1050 12.68 400.7 31.60
1100 12.71 405.6 31.90
1200 12.76 414.2 32.45
1300 12.80 421.2 32.91
Table 13. Response ofbiomass yield (Y), plant nitrogen uptake (Nu), and plant nitrogen
concentration (Nc) to applied nitrogen (N) from fertilizer and broiler litter for bermudagrass
grown at Watkinsville, GA.1
Source N Y Nu Nc
kg ha1 Mg ha1 kg ha' a gkg'
Check 0 1.16 19.7 17.0
Fertilizer 448 10.84 288 26.6
1344 17.45 568 32.5
Broiler Litter 700 15.79 432 27.4
1400 17.02 526 30.9
'Data adapted from Wilkinson (1988).
Table 14. Response of cumulative biomass (Y) and plant N uptake (Nu) with calendar time (t)
from Jan. 1 in response to applied nitrogen (N) from broiler litter for coastal bermudagrass at
Watkinsville, GA.'
t x erfx exp(x2) AQ Q Y Nu
wk Mg ha1' kg ha1
N, kg ha' N, kg ha1
0 700 1400 0 700 1400
14.7 0.8125 0.750 0.517  0.000 0.00 0.00 0.00 0.0 0 0
19.0 0.2750 0.302 0.9272 0.419 0.419 0.32 1.47 2.30 6.9 35 58
23.3 0.2625 0.290 0.9334 0.998 1.417 0.64 4.47 5.14 12.2 122 152
27.1 0.7375 0.703 0.5805 1.187 2.604 1.07 8.06 8.89 18.7 216 271
31.4 1.2750 0.9286 0.1968 1.154 3.758 1.47 11.75 12.83 25.8 316 392
35.7 1.8125 0.9896 0.0374 0.562 4.320 1.83 14.18 15.24 31.2 393 469
40.6 2.4250 0.99938 0.00279 0.165 4.485 2.09 15.78 17.04 33.4 435 526
44.9 2.9625 0.999971 0.00015 0.016 4.501      
1Crop data adapted from Wilkinson (1988).
Table 15. Effect of residual fertilizer nitrogen of 1344 kg N ha'1 applied during years 1 and 2 on
biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) with elapsed time (t) for
coastal bermudagrass grown at Watkinsville. GA.'
Year t Y N. Nc
vr Mg ha'' kg ha' g kg'
1 0 18.39 593 32.2
2 1 16.50 542 32.8
3 2 11.94 361 30.2
4 3 6.71 127 18.9
5 4 2.07 38.4 18.6
6 5 4.32 62.0 14.4
7 6 2.70 55.7 20.6
'Data adapted from Wilkinson (1988).
Table 16. Effect of residual broiler litter nitrogen of 5100 kg N ha' applied during years 1 and 2
on biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) with elapsed time (t)
for coastal bermudagrass grown at Watkinsville, GA.'
Year t Y N, Nc
vr Mg ha' kg ha' g kg1
1 0 18.94 663 35.0
2 1 15.04 513 34.1
3 2 14.37 473 32.9
4 3 17.59 445 25.3
5 4 10.28 248 24.1
6 5 11.19 287 25.6
7 6 8.91 218 24.5
'Data adapted from Wilkinson (1988).
List of Figures
Figure 1. Response of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) to
fertilizer nitrogen (N) for bermudagrass at Fayetteville, AR averaged over four years (1982
through 1985). Data adapted from Huneycutt et al. (1988). Curves drawn from Eqs. (16), (21),
and (22).
Figure 2. Phase plots of biomass yield (Y) and plant N concentration (Nc) vs. plant N uptake (Nu)
for bermudagrass at Fayetteville, AR averaged over four years (1982 through 1985). Data
adapted from Huneycutt et al. (1988). Line and curve drawn from Eqs. (17) and (18),
respectively.
Figure 3. Response of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) to
fertilizer nitrogen (N) for tall fescue at Fayetteville, AR averaged over three seasons (19821983,
19831984, and 19841985). Data adapted from Huneycutt et al. (1988). Curves drawn from Eqs.
(26), (31), and (32).
Figure 4. Phase plots of biomass yield (Y) and plant N concentration (Ne) vs. plant N uptake (Nu)
for tall fescue at Fayetteville, AR averaged over three seasons (19821983, 19831984, 1984
1985). Data adapted from Huneycutt et al. (1988). Line and curve drawn from Eqs. (27) and
(28), respectively.
Figure 5. Response of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) to
broiler litter nitrogen (N) for bermudagrass at Fayetteville, AR (1982). Data adapted from
Huneycutt et al. (1988). Curves drawn from Eqs. (36), (63), and (64).
Figure 6. Response of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) to
broiler litter nitrogen (N) for bermudagrass at Fayetteville, AR (1983). Data adapted from
Huneycutt et al. (1988). Curves drawn from Eqs. (38), (65), and (66).
Figure 7. Response of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) to
broiler litter nitrogen (N) for bermudagrass at Fayetteville, AR (1984). Data adapted from
Huneycutt et al. (1988). Curves drawn from Eqs. (40), (67), and (68).
Figure 8. Response of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) to
broiler litter nitrogen (N) for bermudagrass at Fayetteville, AR (1985). Data adapted from
Huneycutt et al. (1988). Curves drawn from Eqs. (42), (69), and (70).
Figure 9. Phase plots of biomass yield (Y) and plant N concentration (Nc) vs. plant N uptake (Nu)
for bermudagrass at Fayetteville, AR (1982). Data adapted from Huneycutt et al. (1988). Line
and curve drawn from Eqs. (43) and (44), respectively.
Figure 10. Phase plots of biomass yield (Y) and plant N concentration (Nc) vs. plant N uptake
(Nu) for bermudagrass at Fayetteville, AR (1983). Data adapted from Huneycutt et al. (1988).
Line and curve drawn from Eqs. (45) and (46), respectively.
Figure 11. Phase plots of biomass yield (Y) and plant N concentration (Nc) vs. plant N uptake
(N,) for bermudagrass at Fayetteville, AR (1984). Data adapted from Huneycutt et al. (1988).
Line and curve drawn from Eqs. (47) and (48), respectively.
Figure 12. Phase plots of biomass yield (1) and plant N concentration (Nc) vs. plant N uptake
(Nu) for bermudagrass at Fayetteville, AR (1985). Data adapted from Huneycutt et al. (1988).
Line and curve drawn from Eqs. (49) and (50), respectively.
Figure 13. Dependence of model parameters maximum yield (Ay), maximum plant N uptake
(Nu), and maximum plant N concentration (Ncm) on year for response of bermudagrass to broiler
litter at Fayetteville, AR. Data from Table 7. Curves drawn from Eqs. (71) and (72).
Figure 14. Response of dimensionless yield (50y), plant N uptake (0,), and plant N concentration
(9c) to dimensionless nitrogen input ( , or ,, ) for bermudagrass and tall fescue grown at
Fayetteville, AR. Data from Table 8. Curves drawn from Eqs. (79) through (83).
Figure 15. Phase plots for dimensionless yield (Y/Y,,) and plant N concentration (Nc/Nei) vs.
dimensionless plant N uptake (Nu/K,) for bermudagrass and tall fescue grown at Fayetteville,
AR. Data from Table 8. Curve and line drawn from Eqs. (84) and (85), respectively.
Figure 16. Response of biomass yield (Y), plant N uptake (N,), and plant N concentration (Nc) to
fertilizer nitrogen (N) for bermudagrass at Fayetteville, AR averaged over four years (1982
through 1985). Data adapted from Huneycutt et al. (1988). Curves drawn from Eqs. (99), (110),
and (111) for the quadratic model.
Figure 17. Phase plots of biomass yield (Y) and plant N concentration (Nc) vs. plant N uptake
(Nu) for bermudagrass at Fayetteville, AR averaged over four years (1982 through 1985). Data
adapted from Huneycutt et al. (1988). Curves drawn from values given in Table 8 for the
quadratic model.
Figure 18. Response of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) to
fertilizer nitrogen (N) for bermudagrass at Fayetteville, AR averaged over four years (1982
through 1985). Data adapted from Huneycutt et al. (1988). Curves drawn from Eqs. (117)
through (119) for the Mitscherlich model.
Figure 19. Phase plots of biomass yield (Y) and plant N concentration (Nc) vs. plant N uptake
(N,) for bermudagrass at Fayetteville, AR averaged over four years (1982 through 1985). Data
adapted from Huneycutt et al. (1988). Curves drawn from values given in Table 9 for the
Mitscherlich model.
Figure 20. Dependence of biomass yield (Y) and plant N uptake (Nu) on the Mitscherlich
variables for tall fescue at Fayetteville, AR average over three seasons (19821983, 19831984,
19841985). Crop data adapted from Huneycutt et al. (1988). Lines drawn from Eqs. (120) and
(121) for the Mitscherlich model.
Figure 21. Response of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) to
fertilizer nitrogen (N) for tall fescue at Fayetteville, AR averaged over three seasons (19821983,
19831984, 1841985). Data adapted from Huneycutt et al. (1988). Curves drawn from Eqs.
(120) through (122) for the Mitscherlich model.
Figure 22. Phase plots of biomass yield (Y) and plant N concentration (Nc) vs. plant N uptake
(Nu) for tall fescue at Fayetteville, AR averaged over three seasons (19821983, 19831984,
19841985). Data adapted from Huneycutt et al. (1988). Curves drawn from values given in
Table 11 for the Mitscherlich model.
Figure 23. Response of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) to
fertilizer and broiler litter nitrogen (N) for bermudagrass at Watkinsville, GA. Data adapted from
Wilkinson (1988). Curves drawn from Eqs. (123) through (125).
Figure 24. Phase plots of biomass yield (Y) and plant N concentration (Nc) vs. plant N uptake
(Nu) for bermudagrass at Watkinsville, GA. Data adapted from Wilkinson (1988). Curve and line
drawn from Eqs. (128) and (129), respectively.
Figure 25. Correlation of cumulative biomass yield (Y) with cumulative growth quantifier (Q) as
a function of applied nitrogen (N) for bermudagrass at Watkinsville, GA. Data from Table 14.
Lines drawn from Eqs. (132) through (134).
Figure 26. Dependence of crop yield factor (A) on applied nitrogen (N) for bermudagrass at
Watkinsville, GA. Curve drawn from Eq. (135).
Figure 27. Cumulative biomass (Y) vs. calendar time (t) as related to applied nitrogen (N) from
broiler litter for bermudagrass grown at Watkinsville, GA. Yield data from Table 14. Curves
drawn from Eq. (136) with Yoo(N= 0) = 1.92, Yo(N= 700) = 14.92, and Yo,(N= 1400) = 16.06
Mg ha .
Figure 28. Correlation of cumulative plant N uptake (Nu) with cumulative biomass yield (Y) as a
function of applied nitrogen (N) for bermudagrass at Watkinsville, GA. Crop data from Table 14.
Lines drawn from Eqs. (137) through (139).
Figure 29. Effect of residual soil nitrogen from fertilizer on biomass yield (Y), plant N uptake
(N,), and plant N concentration (Nc) with elapsed time for bermudagrass at Watkinsville, GA.
Data adapted from Wilkinson, (1988). Curves drawn Eqs. (141) through (143).
Figure 30. Effect of residual soil nitrogen from broiler litter on biomass yield (Y), plant N uptake
(Nu), and plant N concentration (Nc) with elapsed time for bermudagrass at Watkinsville, GA.
Data adapted from Wilkinson, (1988). Curves drawn Eqs. (144) through (146).
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