A MEMOIR ON MATHEMATICAL MODELS OF CROP GROWTH
AND YIELD
Miscellaneous Applications
Allen R. Overman
Agricultural and Biological Engineering
University of Florida
Copyright 2007 Allen R. Overman
A Memoir on Mathematical Models of Crop Growth and Yields
Miscellaneous Applications
Allen R. Overman
Agricultural and Biological Engineering, University of Florida, Gainesville, FL 32611
Abstract: Two mathematical models have been developed to couple plant biomass and mineral
elements (N, P, and K). Both models use analytical functions (in contrast to numerical
procedures). The growth model describes accumulation of biomass with calendar time due to
photosynthesis. It contains a linear partition function between lightgathering and structural plant
components, an exponential aging function, and a Gaussian energy driving function.
Accumulation of plant nutrients is coupled to biomass through a hyperbolic phase relation.
Accumulation of biomass appears to be the rate limiting process in the system. The seasonal
model assumes logistic dependence of plant nutrient accumulation on applied nutrient. Biomass
is coupled to plant nutrient through a hyperbolic relation. The model has been extended to cover
response to multiple levels of N, P, and K. Both models have been shown to apply to annuals and
perennial grasses. In this document the models are applied to a variety of examples to further
confirm the general applicability. Data from the literature are used extensively. The document
contains 211 pages, including 65 tables, 88 figures, and 40 references.
Key Words: Mathematical models, biomass accumulation, nutrient accumulation, response plots,
phase plots, perennial grasses, annual crops.
General references:
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. New York, NY. 328 p.
Overman, A. R. 2006a. A Memoir on Chemical Transport: Application to Soils and Crops.
University of Florida. Gainesville, FL. 364 p. (13 Tables and 34 Figures).
http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00072282&v=00001
Overman, A. R. 2006b. A Memoir on Crop Growth: Accumulation ofBiomass and Mineral
Elements. University of Florida. Gainesville, FL. 386 p. (84 Tables and 184 Figures).
http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00072283&v=00001
Overman, A. R. 2006c. A Memoir on Crop Yield and Nutrient Uptake. University of Florida.
Gainesville, FL. 116 p. (46 Tables and 62 Figures).
http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00072010&v=00001
Overman, A. R. 2007a. A Memoir on Model Response of Forage Grass to Fertilizer and Broiler
Litter. University of Florida. Gainesville, FL. 53 p. (12 Tables and 22 Figures).
http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00075466&v=00001
Overman, A. R. 2007b. A Memoir on Model Analysis ofSwitchgrass Response to Applied
Nitrogen and Calendar Time. University of Florida. Gainesville, FL. 75 p. (23 Tables and 30
Figures).
http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00075467&v=00001
Table of Contents
1. Response of Bahiagrass to Applied Nitrogen and Harvest Interval
2. Response of Wheat Yield and Nitrogen Uptake with Calendar Time
3. Response of Cotton to Carbon Dioxide Levels
4. Response of Rice to Carbon Dioxide Levels
5. Response of Wheat to Applied Nitrogen.
6. Response of Forage Grasses to Applied Nitrogen.
7. Response of Barley to Applied Nitrogen and Legume
8. Response of Biomass Yield to Applied K for Potato.
9. Coupling of Plant P Uptake and Extractable Soil P with Applied P.
10. Coupling of Alfalfa Yield and Extractable Soil K with Applied K
11. Response of Bahiagrass to Applied Nitrogen, Phosphorus, and Potassium
12. Response of Corn to Applied Nitrogen, Phosphorus, and Potassium
13. Dependence of Growth Quantifier on Harvest Interval for the Expanded Growth Model
14. Dependence of Growth Quantifier on Time of Initiation for the Expanded Growth Model
15. Response of a Forage Grass to Organic Waste
16. Coupling of Applied K, Extractable Soil K, Plant K Uptake, and Yield
17. Discussion
1. Response of Bahiagrass to Applied Nitrogen and Harvest Interval
References:
Beaty, E.R., J.D. Powell, R.H. Brown, and W.J. Ethredge. 1963. Effect of nitrogen rate and
clipping frequency on yield of Pensacola bahiagrass. Agronomy Journal 55:34.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. New York, NY.
Mathematical models: Models for this analysis are discussed in Overman and Scholtz (2002).
Response ofbiomass yield to applied nitrogen can be described by the logistic equation
Y= (1.1)
1+exp(b, cN)
where N is applied nitrogen, kg ha'; Y is biomass yield, Mg ha'1; Ay is maximum biomass yield
at high N, Mg ha1; by is the intercept parameter for yield; and Cn is response coefficient for
applied N, ha kg'. Note that the units on c, are the reciprocal of those on N. Equation (1.1) is a
wellbehaved, monotone increasing function, bounded by 0 < Y < Ay.
Dependence of yield on harvest interval for a perennial grass harvested on a fixed time
interval can be described by the linearexponential equation
A, = (a, + P, At). exp( y At) (1.2)
where At is harvest interval, wk; a, is the intercept parameter for yield, Mg ha'1; fly is the slope
parameter for yield, Mg ha"1 wk'; and y is the aging parameter, wk1.
Data Analysis: Data for this analysis are adapted from Beaty et al. (1963). The crop was the
warmseason perennial Pensacola bahiagrass (Paspalum notatum Fligge), grown at Americus,
GA, USA during the period 19571958. Soil type was Red Bay sandy loam (fineloamy,
kaolinitic, thermic Rhodic Kandiudult). Nitrogen was applied as NH4NO3 in four equal amounts
during the growing season. Annual applications were N= 0, 56, 112, and 224 kg ha1. Harvest
intervals were At = 1, 2, 3, 4, and 6 wk. The experiment was a complete factorial of 4 x 5 = 20
combinations, with three replications of each treatment. Data are given in Table 1.1.
The challenge is how to analyze this large array of data. Our approach is to segment the
analysis into two parts one on response to N and the other on response to At. Response of yield
averaged over the 5 harvest intervals is given in Table 1.1. Is this averaging statistically
justified? We assume so at this point and return to the question later. Results are shown in Figure
1.1, where an increase in yield with applied N can be noted. The logistic model assumes that
response will approach a maximum (Ay). Now Eq. (1.1) can be linearized to the form
Z, =In 1 = b, +c,N (1.3)
Calculation of Zy for each Y requires an estimate of Ay. One approach is to try values of Ay to find
the one which maximizes the correlation coefficient, r. Several tries are listed in Table 1.2. For
simplicity, Ay = 8.50 Mg ha' is chosen so that Eq. (1.3) becomes
z =ln 1 = 1.62 + 0.0103N r =0.99942 (1.4)
ZyY
as shown in Figure 1.2, where the line is drawn from Eq. (1.4). The response equation becomes
S8.50 (1.5)
1 + exp(1.62 0.0103N)
The curve in Figure 1.1 is drawn from Eq. (1.5), which provides excellent agreement with data.
It remains to determine dependence of yield on harvest interval. It is assumed that all the
dependence can be assigned to the linear parameter Ay, with common by and cn. It can be shown
from regression theory that the best estimate of the linear parameter for each harvest interval is
obtained from
4
Ay = '(1.6)
F,2
i=1
where the weighting factor, F, is defined in this case by
F = (1.7)
1 + exp(1.62 0.0103N)
From the values in Table 1.3 we obtain
4.119
At = 1 wk: Ay = =6.00 Mg ha (1.8)
0.6864
At =2 wk: A, 5.073 = 7.39 Mg ha (1.9)
0.6864
At= 3 wk: Ay 6.299 9.18 Mg ha' (1.10)
0.6864
6.385
At= 4 wk: A 6.385 = 9.30 Mgha&' (1.11)
0.6864
At = 6 wk: A =7.305 10.64 Mg ha' (1.12)
S0.6864
Values of Ay are summarized in Table 1.4 and plotted in Figure 1.3. The aging parameter in Eq.
(1.2) is chosen as y = 0.075wk' from Overman and Scholtz (2002), so that standardized Ay
(Ay) for each harvest interval can be calculated from
A; = A, exp(0.075At)= a, + fAt (1.13)
Values of A; are given in Table 1.4. Linear regression of Ay vs. At leads to
A; = 4.69 + 2.022 At r = 0.9942 (1.14)
with a correlation coefficient of r = 0.9942. The estimation equation for A, becomes
Ay = (4.69 + 2.022 At)exp(0.075At) (1.15)
The line and curve in Figure 1.3 are drawn from Eqs. (1.14) and (1.15), respectively.
It can be shown from calculus that the harvest interval (Atp) for peak value of Ay in Figure 1.3
is given by
1 a, 1 4.69
Atp =  =11.0 wk (1.16)
A y p, 0.075 2.022
with corresponding yield parameter of Ay = 11.8 Mg ha''. This can be confirmed from Figure 1.3.
The models can now be used to estimate biomass yield as related to applied nitrogen and
harvest interval. Results are given in Table 1.5 where yield estimates are calculated from
= Ay (1.17)
1+ exp(1.62 0.0103N)
with values of parameter A, made from Eq. (1.15) as listed in Table 1.5. Correlation of with Y
is shown in Figure 1.4, where the line is drawn from
Y = 0.03 + 0.9897Y r = 0.9953 (1.18)
The correlation appears to be relatively free of bias and reflects scatter in the distribution.
It appears that the procedure of segmenting analysis into response to applied N and
dependence on harvest interval is justified. Averaging yields over harvest intervals appears to be
justified in this case as well.
Table 1.1. Response ofbiomass yield (Y) to applied nitrogen (N) and harvest interval (At) for
Pensacola bahiagrass grown at Americus, GA, USA (19571958).'
N Y
kg ha' Mg ha'
1 wk 2 wk 3 wk 4 wk 6 wk Avg
0 1.08 1.36 1.50 1.47 1.57 1.39
56 1.77 2.17 2.33 2.42 2.75 2.29
112 2.26 2.86 3.44 3.46 4.01 3.21
224 3.92 4.78 6.19 6.28 7.19 5.67
'Data adapted from Beaty et al. (1963).
Table 1.2. Correlation of linearized yield (Zy) with applied nitrogen (N) for Pensacola bahiagrass
grown at Americus, GA, USA.1
N Y Zy
kg ha"' Mg ha'
0 1.39 1.559 1.632 1.700 1.674
56 2.29 0.914 0.998 1.075 1.045
112 3.21 0.400 0.500 0.590 0.555
224 5.67 0.889 0.695 0.532 0.594
Ay, Mg ha 8.00 8.50 9.00 8.80
bn 1.56 1.62 1.67 1.65
c,, ha kg1 0.0108 0.0103 0.00986 0.0100
r 0.999247 0.999422 0.999384 0.999415
'Yield data adapted from Beaty et al. (1963).
Table 1.3. Response ofbiomass yield (Y) and the weighting factor (F) to applied nitrogen (N)
and harvest interval (At) for Pensacola bahiagrass grown at Americus, GA, USA (19571958).1
N Y F
kg ha'' Mg ha
1wk 2wk 3wk 4wk 6wk
0 1.08 1.36 1.50 1.47 1.57 0.1652
56 1.77 2.17 2.33 2.42 2.75 0.2605
112 2.26 2.86 3.44 3.46 4.01 0.3855
224 3.92 4.78 6.19 6.28 7.19 0.6653
'Yield data adapted from Beaty et al. (1963).
Table 1.4. Dependence of yield parameter (Ay) and standardized parameter (Ay*) on harvest
interval (At) for Pensacola bahiagrass grown at Americus, GA, USA.
At Ay Ay* Ay
wk Mg ha' Mg ha'' Mg ha'1
1 6.00 6.47 6.23
2 7.39 8.59 7.52
3 9.18 11.50 8.59
4 9.30 12.55 9.47
6 10.64 16.69 10.73
Table 1.5. Response of biomass yield (Y) and estimated yield ( ) to applied nitrogen (N) and
harvest interval (At) for Pensacola bahiagrass grown at Americus, GA, USA (19571958).1
N Y Y
kg ha'1 Mg ha' Mg ha1
1 wk 2wk 3 wk 4 wk 6wk lwk 2wk 3 wk 4wk 6wk
0 1.08 1.36 1.50 1.47 1.57 1.03 1.24 1.42 1.56 1.77
56 1.77 2.17 2.33 2.42 2.75 1.62 1.96 2.24 2.47 2.80
112 2.26 2.86 3.44 3.46 4.01 2.40 2.90 3.31 3.65 4.14
224 3.92 4.78 6.19 6.28 7.19 4.15 5.00 5.72 6.30 7.14
Af,,Mgha1     6.23 7.52 8.59 9.47 10.73
'Yield data adapted from Beaty et al. (1963).
List of Figures
Figure 1.1. Response ofbiomass yield (Y) to applied nitrogen (N) for Pensacola bahiagrass
grown at Americus, GA, USA. Yields average over five harvest intervals. Data adapted from
Beaty et al. (1963). Curve drawn from Eq. (1.5).
Figure 1.2. Correlation of linearized biomass yield (Zy) with applied nitrogen (N) for Pensacola
bahiagrass grown at Americus, GA, USA. Data from Table 2 for Ay = 8.80 Mg ha'. Line drawn
from Eq. (1.4).
Figure 1.3. Dependence of maximum yield parameter (Ay) and standardized yield parameter
(Ay*) on harvest interval (At) for Pensacola bahiagrass grown at Americus, GA, USA. Parameter
values from Table 1.4. Line and curve drawn from Eqs. (1.14) and (1.15), respectively.
Figure 1.4. Correlation between measure yield (Y) and estimated yield (Y) for Pensacola
bahiagrass grown at Americus, GA, USA. Line drawn from Eq. (1.18).
 Ij
Ii I
I I'
.I I
K
1. i
.s Th
F~
20 Sq ilIres I to li Inch
i
1
i r
i : I
"
" i
i Ir

j
i :
Sr I 
FYI tI N
i I : I :
1 I
il;: ~ i~ 1F r: I Ti
I ~ / .
i ( I i. 1.
, \It.
i : I;'
Ni
1:
Ke' '
i i .
i : i
j '
: I i
: i I
'
; i :; i
j 'I : I i i : : ';
1 I: i : :
:
; i i
.1 I,
I 
'H .
4."
2.~.
II '
1 1 I' ,
4010 pq i i~ i
TWOI: t .I.i
20 ISt'rer t the Iv1ch
U
~.Z' ~1
~ v~
9: i
j
.f
re3~
i
;i
'
; ; ;
i :
i
Ld.
To, OT Il i
ANA I
''
N.c~
': I .
i
'I I
An. i
ad
i__ ;_i___ I.:.ii.
i _i
: i :
I ;
..1.1.!_1._._1 1 ._. i
:: i : I
i
'
1 j i /
I
;
: i
i : :
: : i
..
20 Squrcs ti> 1h1 Inch
: : : '
I
...h
2. Response of Wheat Yield and Nitrogen Uptake with Calendar Time
References:
Abramowitz, M. and I.A. Stegun. 1965. Handbook of Mathematical Functions. Dover
Publications. New York, NY.
Knowles, F. and J.E. Watkins. 1931. The assimilation and translocation of plant nutrients in
wheat during growth. J. Agricultural Science 21:612637.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. New York, NY.
Consider the data of Knowles and Watkins (1931) for wheat (Triticum aestivum L.) discussed
on p. 13 of Overman and Scholtz (2002). We focus on accumulation ofbiomass (Y) and plant
nitrogen uptake (Nu) with calendar time (t) referenced to Jan. 1. The expanded growth model is
used to simulate growth. Assume the parameters:
pu = 26.0wk, N2cr = 8.OOwk, c = 0.20wk', k = 5. Assume the time for beginning of significant
growth to be ti = 16.0 wk. Dimensionless time (x) can be related to calendar time (t) by
t,u a2c t 26.0 t19.6
x + = +0.800= xi =0.450 (2.1)
2oa 2 8.00 8.00
It follows that the growth quantifier (Q) is given by
Q= {( kx,)[erf xerf x,] [exp( x2 )exp( x )] exp(2oAcx,) (2.2)
= {3.25[erf x + 0.4755] 2.821[exp( x2) 0.8167} 0.4868
Accumulation ofbiomass (Y) with calendar time is assumed to follow the linear relationship
Y = AQ (2.3)
where A is the yield factor.
Values ofx, erfx, exp(x2), and Q are listed in Table 2.1 for each sampling time (t), along
with biomass yield (Y) and plant N accumulation (N,). Values of erfx are obtained from
Abramowitz and Stegun (1965).
1. Confirm the calculations in Table 2.1 using tables for the error function.
2. Plot Yvs. Q on linear graph paper.
3. Perform linear regression for Yvs. Q (excluding points at t = 30.3 and 31.3 wk).
4. Plot Yvs. t on linear graph paper. Draw the curve from part (3) on the same graph.
5. Plot N, and Y/Nu vs. Y on linear graph paper.
6. Perform linear regression on Y/N, vs. Y for the data in Table 2.1.
7. Write the equation for Nu as a function of Y.
8. Plot the line and curve for parts (6) and (7) on linear graph paper.
9. Discuss your results. How well does the model describe the data? What are the maximum
values for Y and Nu? Do these values seem reasonable?
Table 2.1. Accumulation of biomass (Y) and plant N (N,) with calendar time (t) for wheat grown
at Essex, England.1
t x erfx exp(x2) Q Y Nu Y/Nu
wk kg g kg g'
16.0 0.4500 0.4755 0.8167 0.000   
17.3 0.2875 0.3175 0.9207 0.107 0.77 27 0.028
20.3 0.0875 0.098 0.9924 0.666 2.97 55 0.054
22.3 0.3375 0.367 0.8923 1.229 6.19 80 0.077
24.3 0.5875 0.594 0.7081 1.841 9.51 90 0.106
26.3 0.8375 0.764 0.4959 2.401 12.27 96 0.128
28.3 1.0875 0.876 0.3065 2.839 14.35 110 0.130
29.3 1.2125 0.9136 0.2299 3.003 15.06 109 0.138
30.3 1.3375 0.9415 0.1671 14.73 109 0.135
31.3 1.4625 0.9615 0.1178 14.21 109 0.130
00 1.0000 0.0000 3.456  
'Crop data adapted from Knowles and Watkins (1931).
Table 2.2. Simulation ofbiomass and plant N with calendar time for wheat at Essex, England.
t x erfx exp(2) Q Y N, Nc
wk kg g g kg'
16.0 0.450 0.4755 0.8167 0.000 0.00 0.0 33.9
17 0.325 0.355 0.900 0.076 0.38 11.7 31.0
18 0.200 0.223 0.9608 0.201 1.02 27.4 27.1
19 0.075 0.085 0.9944 0.374 1.89 43.5 23.1
20 0.050 0.056 0.9975 0.592 3.00 58.3 19.5
21 0.175 0.195 0.9698 0.850 4.30 70.6 16.4
22 0.300 0.329 0.9139 1.139 5.76 80.5 14.0
23 0.425 0.452 0.835 1.442 7.30 88.1 12.1
24 0.550 0.563 0.739 1.750 8.86 94.1 10.6
25 0.675 0.660 0.634 2.047 10.36 98.5 9.52
26 0.800 0.742 0.527 2.324 11.76 102.0 8.67
27 0.925 0.808 0.425 2.568 12.99 104.4 8.04
28 1.050 0.862 0.332 2.781 14.07 106.4 7.56
29 1.175 0.9032 0.251 2.958 14.97 107.9 7.21
30 1.300 0.9340 0.185 3.097 15.67 108.9 6.95
31 1.425 0.9560 0.131 3.206 16.22 109.7 6.76
32 1.550 0.9716 0.0905 3.286 16.63 110.2 6.63
33 1.675 0.9821 0.0605 3.344 16.92 110.6 6.54
34 1.800 0.9891 0.0392 3.385 17.13 110.9 6.47
35 1.925 0.9935 0.0246 3.412 17.26 111.0 6.43
oo 1.0000 0.0000 3.456 17.49 111.3 6.36
Solutions:
f = 0.0009 + 5.06Q r = 0.99924 (2.4)
The intercept is essentially zero as assumed for the model. This value depends on the choice of ti.
Y K 1
 +  Y = 0.0296 + 0.00732Y r= 0.9912 (2.5)
N, N,, N,,
NumY 137Y (2.6)
SK + Y 4.04+Y
N 137 (2.7)
Y 4.04 + Y
The maximum value for Q is 3.456, which leads to a maximum projected value of Yof 17.5 kg
from Eq. (4) and N, of 111 g from Eq. (2.6).
List of Figures
Figure 2.1. Correlation of cumulative biomass (Y) with the growth quantifier (Q) for wheat
grown at Essex, England for the study of Knowles and Watkins (1931). Line drawn from Eq.
(2.4).
Figure 2.2. Response of biomass (Y) with calendar time (t) for wheat grown at Essex, England.
Data adapted from Knowles and Watkins (1931). Curve drawn from Eqs. (2.1), (2.2), and (2.4).
Figure 2.3. Phase plot of plant N uptake (N,) and yield/plant N ratio (Y/N,) vs. biomass yield (Y)
for wheat grown at Essex, England. Data adapted from Knowles and Watkins (1931). Line
drawn from Eq. (2.5); curve drawn from Eq. (2.6).
Figure 2.4. Response of biomass (Y), plant N uptake (N,), and plant N concentration (Nc) with
calendar time from Jan. 1 for wheat grown at Essex, England. Data adapted from Knowles and
Watkins (1931). Curves drawn from Eqs. (2.1), (2.2), (2.4), (2.6), and (2.7) as compiled in Table
2.2.
i .. >
I I,
I I r .i i .
I I.
.1
: i i ~~I. ...
1~ .
i lI
____::I' ii
i ; I
; iI_~lil '' i '
i '
I i .
i : i
j
i i
!
_~..i.' ..: I.I.L_.. i LI
i. ~.l.._i..
i /1' i :i I Iii:: I i
:i ~1
I ,
i .
i : / i .
i
i I :
ii ii ''
jl
:1 i
:' : ': i
'
i. 1 :i:
i '
,. :._.i,.. _;i i /
i
i I,
i I
i i I i i i
,.. ..Lri
i..
jL;1:. ti :LLI.LLI_:/ i j
20 Squa'es to tI e Inch
i
i .
i
r_ i i i
:i: : : i:ii::
Ii I...,.,. i
i
_~i.~.L:. __II_.L_..1 :.._~
i
:j
i '
'i r : i i _.._,.. ... ij :
:Ir ;
I
i I
i : i
i
I : I / ~ i i .
i : : : : iT_: i /
i : : ~ :I ~r
i_
i r
i'
i :
~_ ,_ ~_~__:
i::!i,
'' .'
''~:'
'
i I~lj
i
i
I :
i ;:t i j
i '
i i
i : 1
i : 1
j: j :
I '
.;i
i I / ~ i 1
_
1.
1 t
1 J
i I
I ,
*1
*
i i I
i:
i. i:
i i i j : ii : I:
1.._: i .~.I I :
; 
i;:i ''
I
/ i
i :
1~ i j: i
1
.. i _:...._ .... j
l ^.~
29'I 'S,I krr, (o the, Inch
I.i
I I.. :1.: .:
I i
:i.I.
I I .
ii:
j *j
KI j... : I
.1 II
I i I
I .
j j: :_:. :i I L[..
I
.1 :
LI
I ,i
I.
ii
*1,I 1
i ,
~. I i 1
I V..i : i 1111. ;I i ''
1,'
I'::
Li
LI
L I
[IN
20 Squnrrs I1o tlhe Inch
"""K'i0 L
1?3~ p I0
0K Y
>cD
1?
/0
Ut^k
r':i r_i
i .... 1; i :"il i
3. Response of Cotton to Carbon Dioxide Levels
References:
Kimball, B.A. and J.R. Mauney. 1993. Response of cotton to varying CO2, irrigation, and
nitrogen: Yield and growth. Agronomy J. 85:706712.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. New York, NY.
Mathematical model: The logistic model for this analysis is discussed in Overman and Scholtz
(2002). Response ofbiomass yield to levels of atmospheric carbon dioxide is assumed to follow
Y Y= (3.1)
1+ exp(b, ccC)
where C is carbon dioxide concentration, pmol mol1; Y is biomass yield, g nm2; Ay is maximum
biomass yield at high C, g m'2; by is the intercept parameter for yield; and cc is response
coefficient for carbon dioxide, mol Imol'. Note that the units on cc are the reciprocal of those on
C. Equation (3.1) is a wellbehaved, monotone increasing function, bounded by 0 < Y < Ay.
Data analysis: Data for this analysis are adapted from a controlledenvironment study by Kimball
and Mauney (1993) with cotton (Gossypium hirsutum L.) at Phoenix, AZ, USA. The soil was
Avondale clay loam (fineloamy, mixed (calcareous), hyperthermic Anthropic Torrifuvent).
Carbon dioxide levels were ambient (about 350 pmol mol'), 500, and 650 Rmol mol"1. Two
water treatments were included, listed as 'dry' and 'wet'. Measurements included whole plant
and seed cotton. Data are given in Table 3.1 and shown in Figure 3.1.
The challenge now is to obtain values for the model parameters. It should be noted that Eq.
(3.1) can be linearized to the form
Z, =In 1 j=by + cc (3.2)
We choose to average yields for the whole plant and estimate Ay = 2200 g m2. Linear regression
then leads to
Z, =n(22001 =b +cC = 2.04 +0.00572C r= 0.999923 (3.3)
with a correlation coefficient ofr = 0.999923. Results are shown in Figure 3.2, where the line is
drawn from Eq. (3.3). This leads to the response equation
S= Ay = Ay (3.4)
1+exp(b, cC) 1+ exp(2.040.00572C)
with appropriate values of Ay for the individual cases. Visual inspection leads to the equations
1800
Whole plant, dry: =1800 (3.5)
1 + exp(2.04 0.00572C)
2600
Whole plant, wet: =2600 (3.6)
1+ exp(2.04 0.00572C)
750
Seed cotton, dry: =750 (3.7)
1+ exp(2.04 0.00572C)
Seed cotton, wet: = 000 (3.8)
1+ exp(2.04 0.00572C)
The curves in Figure 3.1 are drawn from Eqs. (3.5) through (3.8).
It is appropriate to examine whether or not there is significant difference between Eqs. (3.5)
and (3.6) vs. Eq. (3.4) with average Ay = 2200 tmol mol Analysis of variance is used for this
purpose, as shown in Table 3.2. Residual sum of squares (RSS) of deviations between measured
and estimated yields is defined by
RSS= (Y J (3.9)
i=l
where n is the number of points in the data set. In Mode (1) Eqs. (3.5) and (3.6) are used to
estimate values of yields for dry and wet conditions individually. This leads to a residual sum of
squares of deviations between observed yields (Y,) and estimated yields ( Y) of RSS = 3,267.
Mean sum of squares (MSS) is then given by MSS = RSS/df = 1,634. In Mode (2) the
corresponding values are RSS = 475,608 and MSS = 158,536. The variance ratio for the
comparison is F = 472,341/1,634 = 289. The critical value for the comparison is F(1, 2, 95%) =
18.5. It follows that Mode (1) with individual Ay is significantly better than the simple average. A
similar analysis between Eqs. (3.7) and (3.8) for seed cotton leads to results in Table 3.3. Since F
= 61 exceeds the critical value of 18.5, it also follows that Mode (1) is preferable for seed cotton
as well.
Discussion: The logistic model describes response of cotton to concentrations of CO2 rather well.
There is a significant difference between 'dry' and 'wet' conditions, i.e. water availability does
have a significant impact on cotton yield. It can be shown that the level of CO2 required to reach
50% of maximum yield is given by
b 2.04
C,/2  2. = 357 imol mol1 (3.10)
c, 0.00572
which can be confirmed from Figure 3.1.
Equation (3.1) can be rewritten in dimensionless form
(3.11)
1+ exp(Y)
by defining the dimensionless variables
y = ccC by = 0.00572C 2.04
Y
A,
(3.12)
(3.13)
Results of this procedure are given in Table 3.4. Average values of y, vs. y are shown in
Figure 3.3, where the curve is drawn from Eq. (3.11). This confirms the generalized form of the
logistic model.
Table 3.1. Response ofbiomass yields (Y) to carbon dioxide concentrations (C) for cotton grown
at Phoenix, AZ, USA.1
Water C Y
Whole plant
918
1219
1515
1800
1253
1828
2199
2600
Sm2
g m .
Seed cotton
356
493
654
750
482
697
842
1000
'Yield data adapted from Kimball and Mauney (1993) Table 3.
Table 3.2. Analysis of variance for model estimates for whole plant yield response (Y) to carbon
dioxide concentration (C) for cotton grown at Phoenix, AZ, USA for study of Kimball and
Mauney (1993).
Mode p df RSS MSS F
(1) Individual Ay 4 2 3,267 1,634 
Common by, Cc
(2) Common, Ay, by, cc 3 3 475,608 158,536 
(2) (1)  1 472,341 472,341 289
p = model parameters df = degrees of freedom
RSS = residual sum of squares MSS = mean sum of squares
F = variance ratio
Table 3.3. Analysis of variance for model estimates for seed cotton yield response (Y) to carbon
dioxide concentration (C) for cotton grown at Phoenix, AZ, USA for study of Kimball and
Mauney (1993).
Mode p df RSS MSS F
(1) Individual Ay 4 2 1,503 751 
Common by, cc
(2) Common, Ay, by, cc 3 3 47,148 15,716 
(2)(1)  1 45,645 45,645 61
p = model parameters
RSS = residual sum of squares
F = variance ratio
df = degrees of freedom
MSS = mean sum of squares
inmnl monl
Dry
Ay, gm2
Wet
Ay, g m2
350
500
650
350
500
650
"Mol mol I
Table 3.4. Response of dimensionless yield (y) to dimensionless carbon dioxide concentration
( y) for cotton grown at Phoenix, AZ, USA.'
Water C ,y Y Oy Y Oy
Umol mol' g m2 gm'2
Whole plant Seed cotton
Dry 350 0.04 918 0.510 356 0.475
500 0.82 1219 0.677 493 0.657
650 1.68 1515 0.842 654 0.872
Ay, gm2   1800  750 
Wet 350 0.04 1253 0.482 482 0.482
500 0.82 1828 0.703 697 0.697
650 1.68 2199 0.846 842 0.842
Ay, g m2   2600  1000 
'Yield data adapted from Kimball and Mauney (1993).
List of Figures
Figure 3.1. Response ofbiomass yield (Y) to carbon dioxide concentration (C) for cotton (whole
plant and seed cotton) for dry and wet water supply at Phoenix, AZ, USA. Data adapted from
Kimball and Mauney (1993). Curves drawn from Eqs. (3.5) through (3.8).
Figure 3.2. Correlation of linearized yield (Zy) vs. carbon dioxide concentration (C) for whole
plant of cotton grown at Phoenix, AZ, USA. Data adapted from Kimball and Mauney (1993).
Line drawn from Eq. (3.3).
Figure 3.3. Response of dimensionless yield (r,) to dimensionless carbon dioxide concentration
(4,) for cotton grown at Phoenix, AZ, USA. Data averaged for values in Table 3.2, based on the
study of Kimball and Mauney (1993). Curve drawn from Eq. (3.11).
p ~
'i I
: i ';  : iI
r
: I i :
. i : j
i' ' ': iI ~Do1 i
AL
.1l
*'1
i t
i
V:
Ukr
* : K:~
'i
:" ''
ii:; t i
iii. rL:iiri~
i i ,,,,
i ;il~LI
'
ii i~iit~
Ti i ii
T1:7
<2Yl i""
~~/Ffl1~ vriI :"
: i
i :
; i
4 .* iv ;
'I i
i I
r
I1.
.11
I
:I .1 'I
Li.r L .. *
113 i ; ii' ': '' i'~
i
..
_:Ot
~ ~
I;
i :_,:
1r
i ::
'~''
rrriii.t i
i
I :
I I
A
~itii
J .. .
20 Squares to the Inch
i
ii
,
1 Ci 1 i i Ij (:irr'I;ii: I
I,
~1.1 j_
ili~~' 'i f /
i
ii
:, II i i i i iii' '
::: :;:1
J" :i!:
''9 "~
:
~~~`
I i i
I
' ~~~ 
i
t
iI
,,, ~. I...
I ~ i iii. i
I
:I , 1
'
i
i
i
I
I I I
i l
* .. I 
ii
:II
i
'~''~''
i
I
i
i
i
: :
LL ..i
i i: I Iii
"" ;
i

Ililiilj : r
'1; '''`
i
I
'
I i
i :
~
i
i
i i i
'
: : : i
.. ~I ..~.L.
i
i: i.
.:::.:I
:i'
: 1 ; i
:~:! I
i :j
Y
': i
''
'` ''~'
t_ ,t i..
: : ~ : I
"t
i : : :
I I 1 i
i
j
I
1
i 441
m4f
*j,,
jI :1~ V'. .~ A
i I ti i i i 1.:; i L ..1iI~i I ;
I' If 1i+
it
4
4: 4
n H
b ~~
4 I 1
I ~ j.._.L_ .jii
Ii .K.. I
I .. i
ii
I... 
4, I
iKit
rir 1 i
i...!..;.
i :
1
2.u
.1
i 4 I
20 Squares to the Inch
lir; I", IIi I
i
i i i
i .
~ :
..I... _,.. i:
:i
'.
''.1 : ; JI i
il Ia:! i!it
.1
4 
I :
''
1
 1
: ~:i:
i ; : i '
~+'~T
: : : i ,
i '
!
Cii1 i i :
'''`
''
1
I1~1~
 i ~
: : '. ': .
<^:
S: :ili
i:
___'I___!
 L I
ii f~
:jil : i i
I
I ~1
I
I
I    
i:
: :. 1.
i :
ri
i :
I
i:; i
i i .
i `
i i : i i i ii
1 i :i ~ t r Ii:
'' ~'~
i '
i :
i.. ... :
'
:
i T7
L J
i
i I.
I 
i ,..i.r/l.
i ~:
'
:1
i 
'
: i
: : : 1: /I
i.
;
I j. III
* I i*I I..... :i :
I I _' iLIi i :
Ii *I*
10
or 
I: I
1:
*' I,_.: ..
I .1i
ij
i:
t I i i
i
~~.j.l.. ii
i
I
_.._t.
i : I :
i T1 : j
i
'
i I
J K
KIL
''
i '
Ai
4. Response of Rice to Carbon Dioxide Levels
References:
Abramowitz, M. and I.A. Stegun. 1965. Handbook of Mathematical Functions. Dover
Publications. New York, NY.
Baker, J.T., L.H. Allen, and K.J. Boote. 1990. Growth and yield responses of rice to carbon
dioxide concentration. J. ofAgricultural Science 115:313320.
Marschner, H. 1986. Mineral Nutrition of Higher Plants. Academic Press. Orlando, FL.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. New York, NY.
Mathematical models: The logistic model for this analysis is discussed in Overman and Scholtz
(2002). Response of biomass yield to levels of carbon dioxide can be described by
y= (4.1)
1 + exp(by ccC)
where C is carbon dioxide concentration, umol mol'; Y is biomass yield, g; Ay is maximum
biomass yield at high C, g; by is the intercept parameter for yield; and cc is response coefficient
for carbon dioxide, mol umol". Note that the units on cc are the reciprocal of those on C.
Equation (4.1) is a wellbehaved, monotone increasing function, bounded by 0 < Y< Ay.
Accumulation ofbiomass with time can be described by the expanded growth model.
Dependence of Ay in Eq. (4.1) is related to time, t, by
A, = AyQ (4.2)
where Aym is the yield factor, g; and Q is the growth quantifier defined by
Q = {(l kx, erf x erf x, ] [exp( x2 )exp(x ]}.exp(a2o'cx,) (4.3)
where dimensionless time (x) is related to real time (t) by
xt + (4.4)
V2o 2
with model parameters defined by: j is mean time of the energy distribution, wk; o is time
spread of the energy distribution, wk; c is the aging coefficient, wk'; k is the partition coefficient
between lightgathering and structural components of the plant; and xi is the reference value
corresponding to time of initiation of significant growth, ti. The error function, erfx, in Eq. (4.3)
is defined by
2
erf x exp( u)u (4.5)
I0
where u is the variable of integration. Values of the error function can be obtained from
mathematical tables (Abramowitz and Stegun, 1965).
Data analysis: Data for this analysis are adapted from a study with rice (Oryza sativa L.) grown
in growth chambers exposed to natural sunlight at Gainesville, FL, USA (Baker et al., 1990).
Concentrations of CO2 were 160, 250, 330, 500, 600, and 900 gmol mol'. Plant samples were
collected at 30, 58, and 110 days after sowing of plants on 23 June 1987. Measurements were
made of top and root biomass. Data are given in Table 4.1 and shown in Figure 4.1 for yields of
tops (Yt).
Data for t = 8.3 wk are used to estimate parameters At, bt, and cc for response of top biomass
to CO2 concentration. Since Eq. (4.1) can be linearized to the form
Z, =n( 1 =bt + ccC (4.6)
Linear regression of Zt vs. C for A = 52 g leads to
Z, =In = 1.33+0.00534C r=0.984 (4.7)
with a correlation coefficient ofr = 0.984. Results are shown in Figure 4.2, where the line is
drawn from Eq. (4.7). The response equation becomes
t = 8.3 wk: = 52 (4.8)
1 + exp(1.33 0.00534C)
The middle curve in Figure 4.1 is drawn from Eq. (4.8).
The next challenge is to estimate parameters for t = 4.3 and 15.7 wk. It is assumed that
parameters bt and cc are common to all three sampling times, and that all the variation in yield is
accounted for in parameter At. It is therefore possible to standardize yield ( YI ) over C with the
equation
Y' = Y, [1+ exp(1.33 0.00534C)] (4.9)
as shown in Table 4.2. This leads to the response equations
t = 4.3 wk: =15 (4.10)
1+ exp(1.33 0.00534C)
t = 15.7 wk: Y = 1 (4.11)
1 + exp(1.33 0.00534C)
The response curves in Figure 4.1 are drawn from Eqs. (4.8), (4.10), and (4.11).
Response of grain yield (Yg) to carbon dioxide concentration is given in Table 4.3 and shown
in Figure 4.3. It is now assumed that parameters bg = 1.33 and cc = 0.00534 mol amol1 as for top
yields. Standardized grain yield (Yg) can be defined by
Yg = Yg[1 + exp(1.33 0.00534C)] (4.12)
as listed in Table 4.3. This results in a value ofAg = 3.28 0.43 g. It follows that the yield
response equation for grain becomes
., 3.28 0.43
Grain: 3.28 0.43 (4.13)
S1+ exp(1.33 0.00534C)
The curves in Figure 4.3 are drawn from Eq. (4.13), with the dashed curves representing one
standard deviation above and below the average. This provides a measure of sensitivity of
response to parameter Ag.
The next step is to couple yield with sampling time. This is equivalent to coupling At to t.
Growth model parameters are assumed to be: p = 8 wk, 2o" = 8.0 wk, c = 0.20 wk1, and k = 5.
Time of initiation of significant growth is assumed to be ti = 2.1 wk. This leads to dimensionless
time (x) and growth quantifier (Q) of
t,u oc t8 t1.6
S + + 0.800 = xi = 0.0625 (4.14)
/2oa 2 8.00 8.00
Q = (l kx,)[erf x erf x, ] 2 [exp( x2 ) exp(x exp(o ) (4.15)
In Vi ]I. ) (4.15)
= {.6875[erf x 0.070] 2.821[exp( x )0.9961. 1.105
Results are given in Table 4.4. Linear regression ofAt vs. Q leads to
A, =0.14+ 25.76Q r = 0.99940 (4.16)
The correlation is shown in Figure 4.4, where the line is drawn from Eq. (4.16). These results
confirm Eq. (4.2) for the growth model. Dependence of At on time is shown in Figure 4.5 based
on estimates given in Table 4.5. Since the maximum value of the growth quantifier is Qm =
3.815, maximum yield is estimated to be Y, = 98 g.
Discussion: The mathematical models used for this analysis appear to describe results for
response of rice to carbon dioxide concentrations and with elapsed time reasonably well. This
agrees with results obtained with crop response to mineral elements (N, P, K, Ca, Mg).
According to Marschner (1986, p. 115) more than 90% of plant dry matter consists of organic
compounds, which is related to photosynthesis. It therefore seems reasonable to treat CO2 as a
nutrient, derived from the atmosphere.
It can be shown that CO2 to reach 50% of maximum yield is given by
C /2 =b 1.33 250 [imol mo1 (4.17)
cc 0.00534
which also happens to be the point of maximum differential response of yield to CO2 according
to this mathematical model.
Table 4.1. Response of yields of plant tops (Yt) to carbon dioxide concentration (C) with
sampling time (t) for rice grown at Gainesville, FL, USA.1
C Yt
xmol mol' g
t = 4.3 wk t = 8.3 wk t = 15.7 wk
160 6.2 15 35
250 8.5 31 43
330 9.1 33 65
500 12 41 89
660 14 47 84
900 14 51 76
'Data adapted from Baker et al. (1990) Table 1.
Table 4.2. Response of standardized yield ( Y,) of tops to carbon dioxide concentration (C) with
sampling time (t) for rice grown at Gainesville, FL, USA.'
C Y, ry Y, r,' Y, Yr,
jtmol mol' gg g g
t = 4.3 wk t = 8.3 wk t =15.7wk
160 6.2 16 15 39 35 91
250 8.5 17 31 62 43 86
330 9.1 15 33 54 65 107
500 12 15 41 52 89 112
660 14 16 47 52 84 93
900 14 14 51 53 76 78
avg  15  52  95
'Yield data adapted from Baker et al. (1990) Table 1.
Table 4.3. Response of grain yield (Yg) and standardized grain yield (Y) to carbon dioxide
concentration (C) for rice grown at Gainesville, FL, USA.'
C Y, Y;
pmol mol'1 g
160 1.4 3.65
250 1.3 2.59
330 1.9 3.13
500 3.0 3.79
660 2.8 3.11
900 3.3 3.40
A  3.28 (0.43)
'Yield data adapted from Baker et al. (1990) Table 3. The number in parenthesis represents
standard deviation.
Table 4.4. Coupling of yield parameter At with growth quantifier (Q) and sampling time (t) for
rice grown at Gainesville, FL, USA.
t x erfx exp(x2) Q At
wk g
2.1 0.0625 0.070 0.9961 0.000 
4.3 0.3375 0.375 0.892 0.556 15
8.3 0.8375 0.764 0.496 2.086 52
15.7 1.7625 0.9873 0.0448 3.662 95
oo 1 0 3.812 
Table 4.5. Estimated values of yield parameter At with growth quantifier (Q) and sampling time
(t) for rice grown at Gainesville, FL, USA.
t x erfx exp(x2) Q A,
wk g
2.1 0.0625 0.070 0.9961 0.000
3 0.175 0.195 0.970 0.176 4.5
4 0.300 0.329 0.914 0.453 11.7
5 0.425 0.452 0.835 0.792 20.4
6 0.550 0.563 0.739 1.176 30.3
7 0.675 0.660 0.634 1.577 40.6
8 0.800 0.742 0.527 1.973 50.8
9 0.925 0.810 0.425 2.342 60.3
10 1.050 0.862 0.332 2.672 68.8
12 1.300 0.934 0.185 3.185 82.0
14 1.550 0.9716 0.0905 3.508 90.4
16 1.800 0.9891 0.0392 3.681 94.8
18 2.050 0.9961 0.150 3.762 96.9
20 2.300 0.9989 0.00504 3.795 97.8
00 1 0 3.812 98.2
List of Figures
Figure 4.1. Response of biomass yields of plant tops (Yt) to carbon dioxide concentrations (C) for
three sampling times (t) for rice grown at Gainesville, FL, USA. Data adapted from Baker et al.
(1990). Curves drawn from Eqs. (4.8), (4.10), and (4.11).
Figure 4.2. Correlation of linearized yield of plant tops (Zt) with carbon dioxide concentration
(C) for sampling time oft = 8.3 wk for rice grown at Gainesville, FL, USA. Based on study of
Baker et al. (1990). Line drawn from Eq. (4.7).
Figure 4.3. Response of biomass yields of grain (Yg) to carbon dioxide concentrations (C) for rice
grown at Gainesville, FL, USA. Data adapted from Baker et al. (1990). Curves drawn from Eq.
(4.13). Dashed curves represent one standard deviation above and below the average.
Figure 4.4. Correlation of yield parameter (At) with growth quantifier (Q) for the expanded
growth model for rice grown at Gainesville, FL, USA. Line drawn from Eq. (4.16).
Figure 4.5. Response of yield parameter (At) with elapsed time (t) for rice grown at Gainesville,
FL, USA. Curve drawn from values given in Table 4.5.
~.. I j
; I i ~
f
:I
i
/ I i r: i
:i I .i I
~31 WI Ji I~ii 
,__.1
^i
L ii *i I[''
~ r)
! i I:
i. C
I`
I ^ 
20 Squrre fl I h' Inch
1
i
: I
II
I
I
:j
: :
i /
i
i
I I i
iii
I 1;
I i
i :
j
i .
: i
r!
:
'
i
ii
i
 1 i: ii
1
i:
K' j y\
I .
I *
ir1.
* I
~ I. I
w.1
1*1
I1 n
II L....
'1 4:i 1;I::
i :
I ~ i
I
/
Iii
1
F
i
I :
i ~ i
I
II
jrY~C
~I
1:~....,
:; S r>s to (L Iin
I *
i *
I .
~~ it>. r
SI j. 1' jlj
,i .
I I
k *j
*1 ;
I"
'
'
I
[. ,
i .
i
;i ''
r.:.. i. I ii i::
i 1 i:
i r ~ ; i
I 1 j
i ;:i i I
i
i ~ '
I : : i
i;.
I '! i
I, I
jij: .I K iL
.4 '1
J
20 Squires t th, 1i1 h
i;
I I
I :
i
:
I
: : : i :
1
i `
ji crr~ :
i
ii: ii
J
i :
;
~ ~
i
I: :
i I
P1()
i 7I
21 S
I,;
* I
j ii
:::
I i
II!
j: : i :j i I i
I
12 1
:::~
I
i
!
1
i I i ,
j
; i ,
i I I 1 I i :
j I
'
'
i
i ,
ii :I ~/i:T ;
i : : : : j
(7'n
If,,, inch
' 1 ;
i '''
i :
 ' '
i
':'"
t '
i I
;I
:i
;  '
r :lliil i
i ;
i: 1
'
:i
i' i
i:
i .I:
''
: i i
1
I '.. i
: i j
j :
I
,.,.
i ,
; ~
$ "
i' i '
~ I I I I
..__1___ .._,
i :
: ...

1'
:I:: i
LI ii
;; ~
; : I
: i ,: i
i
I .
: i :
i : ,
I i '
I '
I i '
: i
i
i '
i
i :
''
i::j
I '
5. Response of Wheat to Applied Nitrogen.
Definition of the system: Field experiments were conducted at Keiser, Arkansas, USA on
response of wheat (Triticum aestivum L.) to applied nitrogen. The soil was Sharkey clay (very
fine, smectitic, thermic Chromic Epiaquert). The experiment was conducted over the period 1987
through 1989. Applied nitrogen rates were 0, 50, 100, and 150 kg N ha1. Irrigation was provided
to maintain a soil moisture deficit of 1.3 cm. Measurements included biomass yields for total
plant (above ground portion) and grain, with plant N uptake measured for total plant.
References:
Mascagni, H.J. Jr. and W.E. Sabbe. 1990. Nitrogen Fertilization of Wheat Grown on Raised,
Wide Beds. Arkansas Agricultural Experiment Station Report Series 317. University of
Arkansas. Fayetteville, AR. 16 p.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. New York, NY.
Mathematical Model: The logistic model will be used to analyze yield response to applied
nitrogen (Overman and Scholtz, 2002)
Y = y (5.1)
1+ exp(b, cN)
where Nis applied nitrogen, kg ha'; Yis biomass yield, Mg ha'1; Ay is maximum yield at high N,
Mg ha; by is the intercept parameter for yield; and c, is the response coefficient for applied N,
ha kg'. Note that the units on c, are the reciprocal of those for N. Equation (5.1) is a well
behaved monotone increasing function bounded by 0 < Y < Ay.
Plant N uptake can be related to applied nitrogen through a second logistic equation
N, = A. (5.2)
1 + exp(b, c N)
where Nu is plant N uptake, kg ha'; A, is maximum plant N uptake at high N, kg ha"', and b, is
the intercept parameter for plant N uptake. It follows directly from Eqs. (5.1) and (5.2) that plant
N concentration is described by
Nc N, =Ncm1 + exp(by cN) (5.3)
N Non y (5.3)
Y 1l+exp(b, cN)
where Nc is plant N concentration, g kg'; and Ncm = An/Ay is maximum plant N concentration at
high N, g kg'.
A phase relation between Y and N, can now be developed. Combination of Eqs. (5.1) and
(5.2) with common Cn leads to the hyperbolic equation
Y= Y, N (5.4)
K + N,
where the hyperbolic and logistic parameters are related by
A(
Y. = (5.5)
1 exp(Ab)
K,, (5.6)
exp(Ab)1
with the shift in intercept parameters defined by
Ab = b, by (5.7)
It follows immediately from Eq. (5.4) that Nc and Nu are related by the linear equation
Nc =  = + N (5.8)
Y Y, Y.
Equation (5.8) can be tested directly from data. In fact, linear regression of Nc vs. Nu provides
estimates of Ym and K,.
Data Analysis: We first examine yield data among individual years for total and grain portions.
Data are given in Table 5.1. Averages of Y vs. N are shown in Figure 5.1. Several points may be
noted from the graph. The control variable (N) appears to have been chosen very well for the
experiment. Relative errors (Std Dev/Avg) appear very reasonable. Yield appears to be
increasing toward a plateau in the neighborhood of 14 and 4 Mg ha' for total and grain portions,
respectively. A procedure is needed for estimating the three model parameters Ay, by, and c,.
Nonlinear regression is used for this purpose. Results are shown in Table 5.2 for various modes
of analysis. The most accurate mode is to fit the model to each year and portion individually.
Parameters are given in Table 5.2. In mode (2) common parameters are assumed for both total
and grain. Common parameters do not work due to large differences in yields between total and
grain portions. Individual Ay with common by and cn (mode 3) does appear reasonable. Analysis
of variance (ANOVA) can be used to test for significance, as shown in Table 5.3. Mode (3) is
shown to be acceptable at the 95% level of confidence, where the variance ratio is 2.55 compared
to a critical value ofF(10, 6, 95%) = 4.08. This leads to low standard errors of the estimates and
an overall correlation coefficient ofr = 0.9970. Average yield response equations are given by
Total: 13.8 (5.9)
1+ exp(0.87 0.0386N)
4.18
Grain: = .18(5.10)
1+ exp(0.87 0.0386N)
The yield curves in Figure 5.1 are drawn from Eqs. (5.9) and (5.10).
Phase plots are shown in Figure 5.2 using data from Table 5.4. Equation (5.8) leads to
+ +N, =5.56+0.0252N, r=0.867 (5.11)
rY Y. rY.
with the corresponding hyperbolic phase relation
= Y,=N, 39.7N, (5.12)
K + N 221+N,
The line and curve in Figure 5.2 are drawn from Eqs. (5.11) and (5.12), respectively. Equations
(5.5) and (5.6) lead to
Ay 13.81
Y = 39.7  > Ab = 0.43 > b = by + Ab= 0.87 +0.43 =1.30
1 exp(Ab) 1 exp(Ab)
K= 221 A A, =1.88A A, =118 kgha
exp(Ab)l exp(0.43)1
It follows from Eqs. (5.2) and (5.3) that dependence of Nu and Nc on N are given by
= A, 118 (5.13)
S1+exp(b, cN) 1+exp(1.300.0386N)
S=N= 1 + exp(by cN)] = 54I1+ exp(0.87 0.0386N) (5.14)
c Y "l+exp(b, c,N) + exp(l.30 0.0386N)
Curves in Figure 5.1 are drawn from Eqs. (5.9), (5.10), (5.13), and (5.14).
Interpretation: Analysis of variance shows that the logistic model describes response ofbiomass
yield to applied N rather well with common parameters by and Cn, and individual Ay for total and
grain portions. The extended logistic model includes response of plant N uptake to applied N.
Linear coupling of plant N concentration and plant N uptake appears to be confirmed. Output
variables for total plant are bounded by 0 < Y< 13.81 Mg ha1, 0 < N, < 118 kg ha', and 5.56 <
Nc < 8.54 gkg1.
Estimates of yields for total plant and grain are given in Table 5.5. Correlations between
estimated yields (Y) and measured yields (Y) are shown in Figure 5.3 (total plant) and Figure 5.4
(grain). Linear regression leads to
Total plant: Y = 0.092 + 0.9896Y
r = 0.9921
(5.15)
Grain: Y =0.15 + 0.9558Y r = 0.9959 (5.16)
The correlations appear to be relatively free of bias.
Shift in intercept parameters is given by
Ab = In N = 54= 0.429 (5.17)
Naj 5.56
Since the concentrations are functions of the crop, Ab is assumed to be characteristic of the crop
as well.
The point of maximum overall efficiency of plant utilization of applied N (Np, Ep) is defined
by
b (1.30
N =1.5 1.5 0 = 50.5 kg ha' (5.18)
c 0.0386
E Ac, 4 ) 1 1
"E 4 J1.5b, 1+exp(0.5b,) 1+exp(b,)
(5.19)
= (1.139){2.05[0.657 0.214} = 1.03 = 103%
Table 5.1. Response ofbiomass yield (Y) to applied nitrogen (N) for wheat grown at Keiser, AR,
USA (19871989).'
N Y
kg ha' Mg ha'
Total Grain
1987 1988 1989 Avg (Std Dev) 1987 1988 1989 Avg (Std Dev)
0 4.54 4.40 3.35 4.10 (0.65) 1.39 1.22 0.90 1.17 (0.25)
50 11.12 10.17 9.87 10.39 (0.65) 3.43 3.10 2.54 3.02 (0.45)
100 14.07 11.92 13.07 13.02 (1.08) 4.60 3.80 3.66 4.02 (0.51)
150 16.06 11.94 13.26 13.75 (2.10) 4.76 3.96 3.92 4.21 (0.47)
'Data adapted from Mascagni and Sabbe (1990).
Table 5.2. Estimates of model parameters for different modes for wheat grown at Keiser, AR,
USA.
Mode p df Year Ay by Cn r
Mg ha' ha kg1
(1) IndividualAy, by, c, 18 6 1987 16.03 (0.63) 0.88(0.15) 0.0322 (0.0050) 0.9965
1988 12.04 (0.08) 0.55(0.03) 0.0452 (0.0016) 0.9998
1989 13.44 (0.13) 1.11(0.06) 0.0430 (0.0017) 0.9997
1987 4.84 (0.06) 0.92 (0.05) 0.0366 (0.0014) 0.9997
1988 3.96 (0.02) 0.81 (0.03) 0.0415 (0.0010) 0.9999
1989 3.99 (0.01) 1.24(0.01) 0.0361 (0.0004) 1.0000
(2) Common Ay, by, c 3 21 all 9.01(1.14) 0.88(0.64) 0.0387 (0.0213) 0.5393
(3) Individual A 8 16 1987 15.38 (0.24) 0.87(0.06) 0.0386 (0.0020) 0.9970
Common by, c, 1988 12.66 (0.22) 0.87(0.06) 0.0386 (0.0020) 0.9970
1989 13.40 (0.23) 0.87(0.06) 0.0386 (0.0020) 0.9970
1987 4.76 (0.20) 0.87(0.06) 0.0386 (0.0020) 0.9970
1988 4.03 (0.20) 0.87(0.06) 0.0386 (0.0020) 0.9970
1989 3.76 (0.20) 0.87(0.06) 0.0386 (0.0020) 0.9970
p = number of parameters estimated
df= degrees of freedom
r = correlation coefficient
numbers in parentheses are standard errors of the estimates
Table 5.3. Analysis of variance for model with different modes for wheat grown at Keiser, AR,
USA.
Mode p df RSS MS F
(1) Individual Ay, by, c, 18 6 0.598 0.0997 
(2) Common Ay, by, c 3 21 366.6 17.5 
(2)(1) 15 366.0 24.4 245
(3) Individual A, 8 16 3.14 0.196 
Common by, c,
(3) (1) 10 2.54 0.254 2.55
p = number of parameters estimated
df= degrees of freedom
RSS = residual sum of squares
MSS = RSS/df = mean sum of squares
Table 5.4. Response of average biomass yield (Y), plant N uptake (Nu), and plant N concentration
(N,) to applied nitrogen (N) for total plant of wheat grown at Keiser, AR, USA.'
N Y Nu Nc
kg ha'' Mg ha'' kg ha'1 g kg
0 4.10 27.7 6.75
50 10.39 69.7 6.70
100 13.02 98.7 7.58
150 13.75 129 9.38
IData adapted from Mascagni and Sabbe (1990).
Table 5.5. Response of estimated yield (Y) to applied nitrogen (N) for wheat grown at Keiser,
AR, USA.
N
kg ha' Mg ha'
Total
1988 1
3.74
9.40
12.05 1
12.57 1
1989
3.96
9.95
12.76
13.30
Avg (Std Dev)
4.08 (0.41)
10.26 (1.04)
13.15 (1.34)
13.71 (1.40)
1987
1.41
3.54
4.53
4.73
Grain
1988 1989
1.19 1.11
2.99 2.79
3.84 3.58
4.00 3.73
Avg (Std Dev)
1.24 (0.16)
3.11 (0.39)
3.98 (0.49)
4.15 (0.52)
13.81 (1.41)
0.87
0.0386
4.76 4.03 3.76 4.18 (0.52)
0.87 0.87 0.87 0.87
0.0386 0.0386 0.0386 0.0386
1987
4.54
11.42
14.64
15.27
Ay, Mg ha'
by
cn, ha kg'
15.38
0.87
0.0386
12.66
0.87
0.0386
13.40
0.87
0.0386
List of Figures
Figure 5.1. Response ofbiomass yield (Y) to applied nitrogen (N) for total plant and grain of
wheat grown at Keiser, AR. Data adapted from Mascagni and Sabbe (1990). Curves drawn from
Eqs. (5.9), (5.10), (5.13), and (5.14).
Figure 5.2. Phase plots of biomass yield (Y) and plant N concentration (Nc) vs. plant N uptake
(N,) for total plant of wheat grown at Keiser, AR. Data adapted from Mascagni and Sabbe
(1990). Line and curve drawn from Eqs. (5.11) and (5.12), respectively.
Figure 5.3. Correlation of estimated yield (Y) with measured yield (Y) for total plant of wheat
for the experiment of Mascagni and Sabbe (1990) at Keiser, AR.
Figure 5.4. Correlation of estimated yield (Y) with measured yield (Y) for wheat grain for the
experiment of Mascagni and Sabbe (1990) at Keiser, AR.
/S
1
171
I I
I '
I
/41)r
'
i
;
1'!''
' ; i I Bb
i:
1,
;s
J7~TI I
I/v
5,l
:i
~ ''
r,
3
1,     ~  I I_
 r I'
I pll L ~ ~~LI
I
t
.' "" AfV
I
i
:li;7~"1~
i i it: : ;
AU
....... ,,
j '
I
i
I.....~. i
i
I
.i
: i :

i 
i : .
:i i:'
i
i '
i
i ;i
i. ../i
i.. i..
; i:
I I
i
i
I i i : i
i : j
(I
'
I i
1 i /i
i
1
ji :1 i
1 '
:.. i : 1
i
Bi
i1ii
IT:
ID
I i
20 Squ:r< ; to whe Inch
j ii
:.i
1
i.. .~
: i
i ...._. i :.
i i
i
i
'
i i:J
i .:._:i
: 1
j:r! iii
!
: i i
i
.I.._i~__ ;___/
t
i j/
~~~i_......i
r i i :
: 1:
;
i
; i ..
I I
.
jr
Si,.
oii i.
1/ i
I // I i i
iii 1_
:i. t. ~ ''
I
i^._i _I.i
:
20 qu'itres to t1e IIn<11
i :
i
I
i
j
I
I
jl
j I 1 I I i
i j '
i: '
1:
i
i :I
i.
: i .
; i : 1
;:
i .I:j
'1 f
:2 '1
6. Response of Forage Grasses to Applied Nitrogen.
Definition of the system: Field experiments were conducted in Mongalia County, West Virginia,
USA on response of Kentucky bluegrass (Poa pratensis L.), tall fescue (Festuca arundinacea
Schreb), orchardgrass (Dactylis glomerata L.), reed canarygrass (Phalaris arundinacea L.),
smooth bromegrass (Bromus inermis Leyss.), and timothy (Phleum pretense L.) to applied
nitrogen by Colyer et al. (1977). The soil was a combination of Wharton silt loam clayeyy,
mixed, mesic Aquic Hapludult) and Cookport loam (fineloamy, mixed, Aquic Fragiudult) with
average slope of 10%. The experiment was conducted over the period 1968 through 1970.
Applied nitrogen rates were 0, 112, 224, and 448 kg N ha1. Treatments were replicated three
times. Seasonal biomass yields were reported.
Characteristics of these forage grasses are discussed by Barnes et al. (1995).
References:
Barnes, R.F., D.A. Miller, and C.J. Nelson. 1995. Forages. Volume 1: An Introduction to
Grassland Agriculture. 5 th Edition. Iowa State University Press. Ames, IA.
Colyer, D, F.L. Alt, J.A. Balasko, P.R. Henderlong, G.A. Jung, and V. Thang. 1977. Economic
optima and price sensivity of N fertilization for six perennial grasses. Agronomy J. 69:514517.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. New York, NY.
Mathematical Model: The logistic model will be used to analyze yield response to applied
nitrogen (Overman and Scholtz, 2002)
Y = y (6.1)
1+exp(b, c,N)
where Nis applied nitrogen, kg ha'; Yis biomass yield, Mg ha'1; Ay is maximum yield at high N,
Mg ha'; by is the intercept parameter for yield; and c, is the response coefficient for applied N,
ha kg'. Note that the units on c, are the reciprocal of those for N. Equation (6.1) is a well
behaved monotone increasing function bounded by 0 < Y < Ay.
Data Analysis: We first examine data among individual years for one of the grasses, viz. tall
fescue. Data are given in Table 6.1. Averages and standard deviations of Y vs. N are shown in
Figure 6.1. Several points may be noted from the graph. The control variable (N) appears to have
been chosen very well, using a geometrical distribution. Relative errors (Std Dev/Avg) appear
very reasonable. Yield appears to be increasing toward a plateau in the neighborhood of 12 Mg
ha'. A procedure is needed for estimating the three model parameters Ay, by, and c,. The most
rigorous approach is nonlinear regression. Perhaps it is possible to linearize the model and to use
linear regression. Note that Eq. (6.1) can be rearranged to the form
Z, =lny  1 =by +cN (6.2)
which does linearize the model with a positive slope cn. This approach requires an estimate of Ay
in order to calculate Zy for each N. Some criterion is needed for selecting Ay. The criterion chosen
is to maximize the linear correlation coefficient for Eq. (6.2). A few tries leads to the result
Zy=ln(1200 = 0.296 + 0.00955N r= 0.9962 (6.3)
with a correlation coefficient ofr = 0.9962. This leads to the response equation
f = Ay 12.00 (6.4)
1+ exp(by c,N) 1+ exp(0.296 0.00955N)
The curve in Figure 6.1 is drawn from Eq. (6.4). The model appears to give a reasonable
description of the yield data.
Several questions now occur from this analysis. Is averaging of the data a legitimate step? Is
the assumption of common Ay, by, and c justified on statistical grounds? The best fit of the
model to the data is to fit Eq. (6.1) to each year separately. Is some compromise procedure
acceptable? These questions are pursued in more detail. Results of this analysis are given in
Table 6.2. In Mode (1) each year of data is analyzed separately, so the number of parameter
estimates required is p = 9, with degrees of freedom of df = 12 9 = 3. In Mode (2) all three
years are analyzed together, with p = 3 and df= 9. Mode (3) simplifies the analysis by assuming
individual Ay with common by and cn. Model parameters have been optimized by the least
squares criteria to minimize the sum of squares of deviations between observed and estimated
yields, which leads to a residual sum of squares (RSS) defined by
RSS = (Y, (6.5)
i=1
where Yi is measured value of yield for the i th observation, Yi is estimated value of yield from
the model (Eq. (6.1)) for the i th observation, and n = 12 is the number of observations. A
procedure for comparison of the different modes can now be developed as shown in Table 6.3.
The Fisher Ftest for analysis of variance (ANOVA) is used for statistical evaluation. For
example, comparison of Modes (1) and (2) leads to a variance ratio ofF = 0.415/0.387 = 1.07.
The critical value ofF for a 95% confidence level can be obtained from a statistics text or
handbook as F(6, 3, 95%) = 8.94. Since our value F = 1.07 < 8.94 it is concluded that Mode (2)
with common Ay, by, and c, is acceptable for these data. For Mode (3) analysis shows that F =
0.088/0.387 = 0.23 << 9.12 = F(4, 3, 95%). So, Mode (3) improves the 'goodness of fit'
considerably over Mode (2). It should be noted that values for parameters in Table 6.2 for Mode
(2) are quite close to those in Eq. (6.4) obtained by linearization of the model.
Yield response data are given in Table 6.4 for the six grasses, averaged over the three years.
The first step might be to calculate averages and standard deviations among the six grasses, as
shown in Table 6.4. The second step is to plot these data on linearlinear graph paper. A
procedure is needed for estimating the three model parameters Ay, by, and Cn. From Figure 6.2 a
reasonable estimate appears to be Ay = 13.50 Mg ha'1. Linear regression of Zy vs. N leads to
Z = ln130 1 = by+c,N =0.267+ 0.00664N r= 0.9974 (6.6)
with a correlation coefficient ofr = 0.9974. Actually, the correlation coefficient can be increased
slightly by choosing Ay = 13.40 to obtain
Zy= n(13A 1= by + c,N = 0.269 +0.00690N r = 0.9975 (6.7)
which leads to the response equation shown in Figure 6.2
Y= A = 13.40 (6.8)
1 + exp(by c,N) 1+ exp(0.269 0.00690N)
Estimates of model parameters are given in Table 6.5 by different modes. Nonlinear regression
of the 24 data points (6 grasses x 4 nitrogen levels) with common Ay, by, and c, leads to
___= Ay 13.23
= Y = r= 0.9153 (6.9)
1+ exp(by c,N) 1+ exp(0.25 0.0072N)
with estimates and standard errors of Ay = 13.23 0.66 Mg ha', by = 0.25 + 0.11,
c, = 0.0072 0.0013 ha kg', and overall correlation coefficient ofr = 0.9153. The smallest
relative error occurs with parameter Ay, with the greatest uncertainty in by. Comparison of Modes
(1) and (2) in Table 6.6 leads to F = 3.83 compared to the critical value ofF(16, 6, 95%) = 3.93.
Critical F for Modes (3) and (4) is F(10, 6, 95%) = 4.08, so that either can be reasonably
justified. Mode (4) gives slightly better fit, where the response equation is given by
= Ay = 12.88= 0.9627 (6.10)
1+exp(by c,N) 1+exp(by 0.0080N)
with appropriate values of by taken from Table 6.5. It may be noted from Table 6.5 that the
outlier for by is for timothy grass. If this grass is ignored, then it appears reasonable to use Eq.
(6.9) for the other five grasses. This goes back to the high background yield (Y= 8.48 Mg ha')
for timothy in Table 6.4 and leads to the high uncertainty in by. The contrast between timothy
and brome is illustrated in Figure 6.3.
An alternative method for estimating Ay and by for each grass is now described, using the
linearization procedure for Eq. (6.8). Since it has been established that a common c, for the six
grasses is appropriate, we choose cn = 0.0069 ha kg'. Values of Ay are then selected to produce
this result for each grass. Estimated yields (Y) are given in Table 6.7 using these parameter
values. A scatter diagram of Y vs. Yis shown in Figure 6.4, where the line is drawn from
Y = 0.26 +0.9800Y r= 0.9858 (6.11)
This result indicates that the correlation is relatively free of bias.
Interpretation: Analysis of yields of six grasses grown on the same soil type indicate that use of a
common response coefficient c, = 0.0080 ha kg' is appropriate. This is consistent with the
premise that c, is a characteristic of the soil. Intercept parameter by does vary with grass,
particularly for timothy. The reason for this difference is not known.
Table 6.1. Response ofbiomass yield (Y) to applied nitrogen (N) for tall fescue grown at
Mongalia County, WV, USA (19681970).1
N Y
kg ha' Mg ha'
1968 1969 1970 Avg (Std Dev)
0 5.59 4.77 4.56 4.97 (0.54)
112 9.00 8.86 8.21 8.69 (0.42)
224 10.82 10.09 9.32 10.08 (0.75)
448 11.90 12.33 11.14 11.79 (0.60)
'Data adapted from Colyer et al. (1977).
Table 6.2. Estimates of model parameters for different modes for tall fescue grown at Mongalia
County, WV, USA.
Mode p df Year Ay by cn r
Mg ha' ha kg"
(1) Individual A, by, c 9 3 1968 11.98 (0.07) 0.13(0.02) 0.0108 (0.0003) 0.9998
1969 12.31 (0.79) 0.38(0.19) 0.0100 (0.0027) 0.9884
1970 11.09 (0.61) 0.30(0.18) 0.0103(0.0027) 0.9900
(2) Common Ay, by, c, 3 9 all 11.78 (0.32) 0.27(0.09) 0.0104 (0.0013) 0.9764
(3) Individual Ay 5 7 1968 12.28 (0.29) 0.27(0.07) 0.0104 (0.0010) 0.9903
Common by, cn 1969 12.01 (0.29) 0.27(0.07) 0.0104 (0.0010) 0.9903
1970 11.03 (0.28) 0.27(0.07) 0.0104(0.0010) 0.9903
p = number of parameters estimated
df= degrees of freedom
r = correlation coefficient
numbers in parentheses are standard errors of the estimates
Table 6.3. Analysis of variance for model with different modes for tall fescue grown at Mongalia
County, WV, USA.
Mode p df RSS MS F
(1) IndividualAy, by, cq 9 3 1.16 0.387 
(2) Common Ay, by, cn 3 9 3.65 0.406 
(2)(1) 6 2.49 0.415 1.07
(3) Individual Ay 5 7 1.51 0.216 
Common by, cn
(3)(1) 4 0.35 0.088 0.23
p = number of parameters estimated
df = degrees of freedom
RSS = residual sum of squares
MSS = RSS/df= mean sum of squares
Table 6.4. Response of biomass yield (Y) to applied nitrogen (N) for six forage grasses at
Mongalia County, WV, USA (19681970).1
N Y
kg ha' Mg ha1
Grass
Blue Fescue Orchard Canary Brome Timothy Avg (Std Dev)
0 5.33 4.97 5.37 5.00 4.83 8.48 5.66 (1.40)
112 7.31 8.69 7.50 9.33 8.58 10.97 8.73 (1.34)
224 9.59 10.08 9.73 11.22 9.21 11.68 10.25 (0.98)
448 11.93 11.79 11.72 13.95 13.70 12.86 12.66 (1.00)
'Data adapted from Colyer et al. (1977).
Table 6.5. Estimates of model parameters for different modes for six forage grasses grown at
Mongalia County, WV, USA.
Mode p df Grass Ay by Cn r
Mg ha' ha kg
(1) Individual A, by, cn 18 6 Blue 13.15 (0.39) 0.41 (0.05) 0.0060 (0.0005) 0.9992
Fescue 11.78 (0.46) 0.27 (0.13) 0.0104 (0.0019) 0.9944
Orchard 12.54 (0.28) 0.31 (0.04) 0.0067 (0.0005) 0.9992
Canary 14.16 (0.74) 0.53 (0.15) 0.0093 (0.0018) 0.9937
Brome 16.47 (5.18) 0.74 (0.39) 0.0051 (0.0029) 0.9775
Timothy 12.92 (0.38) 0.67 (0.11) 0.0083 (0.0019) 0.9929
(2) Common Ay, by, cn 3 21 all 13.23 (0.66) 0.25 (0.11) 0.0072 (0.0013) 0.9153
(3) Individual A 8 16 Blue 12.11 (0.65) 0.23 (0.09) 0.0072 (0.0011) 0.9592
Common by, Cn Fescue 12.56 (0.65) 0.23 (0.09) 0.0072 (0.0011) 0.9592
Orchard 12.13 (0.65) 0.23 (0.09) 0.0072 (0.0011) 0.9592
Canary 14.14 (0.70) 0.23 (0.09) 0.0072 (0.0011) 0.9592
Brome 13.02 (0.68) 0.23 (0.09) 0.0072 (0.0011) 0.9592
Timothy 15.03 (0.72) 0.23 (0.09) 0.0072 (0.0011) 0.9592
(4) Individual by 8 16 Blue 12.88 (0.64) 0.32 (0.09) 0.0080 (0.0011) 0.9627
Common Ay, cq Fescue 12.88 (0.64) 0.34 (0.09) 0.0080 (0.0011) 0.9627
Orchard 12.88 (0.64) 0.25 (0.09) 0.0080 (0.0011) 0.9627
Canary 12.88 (0.64) 0.37 (0.09) 0.0080 (0.0011) 0.9627
Brome 12.88 (0.64) 0.46 (0.09) 0.0080 (0.0011) 0.9627
Timothy 12.88 (0.64) 0.68 (0.09) 0.0080 (0.0011) 0.9627
p = number of parameters estimated
df= degrees of freedom
r = correlation coefficient
numbers in parentheses are standard errors of the estimates
Table 6.6. Analysis of variance for model with different modes for six forage grasses grown at
Mongalia County, WV, USA.
(1) Individual Ay, by, c,
(2) Common Ay, by, c,
(2)(1)
(3) Individual Ay
Common by, c,
(3)(1)
(4) Individual by
Common Ay, Cn
(4) (1
3 21
 15
8 16
 10
8 16
 10
RSS
2.80
29.66
26.86
14.60
11.80
13.39
10.59
Fp
0.467
1.41
1.79
0.912
1.18
0.837
1.06
3.83
2.53
2.27
p = number of parameters estimated
df= degrees of freedom
RSS = residual sum of squares
MSS = RSS/df = mean sum of squares
Table 6.7. Estimated yield (Y) vs. applied nitrogen (N) for six forage grasses grown at Mongalia
County, WV, USA (19681970).
N
kg ha' Mg ha'1
Grass
Blue Fescue Orchard Canary Brome Timothy Avg (Std Dev)
0 5.13 5.80 5.28 5.63 4.88 8.73 5.91 (1.42)
112 7.55 8.35 7.64 8.50 7.63 10.68 8.39 (1.19)
224 9.66 10.47 9.63 11.11 10.32 11.90 10.52 (0.87)
448 11.90 12.65 11.68 14.03 13.55 12.90 12.79 (0.91)
Ay 12.70 13.40 12.40 15.10 14.80 13.20 13.60 (1.11)
by 0.39 0.27 0.30 0.52 0.71 0.67 0.25 (0.48)
c, ha kg 0.0069 0.0069 0.0069 0.0069 0.0069 0.0069 
M.(Z 'R
I*IVUV C~ ul
MS

List of Figures
Figure 6.1. Response ofbiomass yield (Y) to applied nitrogen (N) for tall fescue grown at
Montgalia County, WV, USA (averaged over 1968 through 1970). Data adapted from Colyer et
al. (1977). Curve drawn from Eq. (6.4).
Figure 6.2. Response ofbiomass yield (Y) to applied nitrogen (N) for six grasses grown at
Montgalia County, WV, USA (averaged over six grasses and three years). Data adapted from
Colyer et al. (1977). Curve drawn from Eq. (6.8).
Figure 6.3. Response ofbiomass yield (Y) to applied nitrogen (N) for brome and timothy grown
at Montgalia County, WV, USA (averaged over three years). Data adapted from Colyer et al.
(1977). Curves drawn from Eq. (6.10) with by = 0.46 and 0.68 for brome and timothy,
respectively.
Figure 6.4. Correlation of estimated yield (Y) with measured yield (Y) for six forage grasses
grown at Montgalia County, WV, USA (averaged over three years). Yield data adapted from
Colyer et al. (1977). Line drawn from Eq. (6.11).
I / , 
~ Ka
IxI.
I 4.1 4 .. i.
',I,.41
: Ph
I' I I
I. I
Ii I' I
:~ISI.I IhIII I I i A _ii' /:1j
I,
4'
>2
I 
0
A jia
1^ ^ l .
iA^A~J~j
i : '
ir;ql~a:
ri
rS
I;]
. Ui l I l t" itn h
AI% II ~r
0rr jot~ 5t:J() / 'o :
i; '1 i: i i
i....
I .
V
.1
I:
* 4
1~*'
1'~
wJ
K
~
20 Squares to the Inch
 r
..._... U
I L7
:::: /
'
./
'
i
;n
: 1
1*
*1L
77
.1
:1 j *
Kill'
Li
~FiW4,i_
41
h__. 1
/
''
I
'
;IT
1
 'I.
I.
SI. I I
A
' I
i : r :
'
Lli ilii
: : : '
i i ; i I i'
i
:I '''
'''
""~'
''"'
'
i, lr,!
i :I i.i
I
i ''
I t I
''I ':~
i it;'
.j+ .:v T, 1
;I fii::
i
i
I
i
i'
it..
i 
 '
:' i i
AL.
* ..~ ti c
;ij, i:
iJ
,
/i
7.i
.1i
I i
I. '
*j I
I
I
i
! '
I _
::j:
:I i ::
~,i
:ti~l i.
ill; Iii
::::I;
i
ill
1
7. Response of Barley to Applied Nitrogen and Legume
Definition of the system: Field experiments were conducted at Wobum Agricultural Experiment
Station at Harpenden, Herts, England on response of barley (Hordeum vulgare L.) to applied
nitrogen. The soil was classified as Cottenham series (sand to loamy sand). The experiment was
conducted over the period 1965 and 1966. Applied nitrogen rates were 0, 38, 75, and 113 kg N
ha'. Biomass yields for grain were reported. Note that the cover crops ryegrass (Lolium perenne
L.) and trefoil (Lotus corniculatus L.) were plowed under before planting of barley.
References:
Dyke, G.V., H.D. Patterson, and T.W. Barnes. 1977. The Woburn longterm experiment on
green manuring, 193667; results with barley. P 119149. Rothamsted Experimental Station
Report for 1976. Part 2. Lawes Agricultural Trust.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. New York, NY.
Mathematical Model: The logistic model will be used to analyze yield response to applied
nitrogen (Overman and Scholtz, 2002)
Y= (7.1)
1 + exp(b, c N)
where Nis applied nitrogen, kg ha'1; Y is biomass yield, Mg ha'; Ay is maximum yield at high N,
Mg ha'1; by is the intercept parameter for yield; and c, is the response coefficient for applied N,
ha kg'. Note that the units on c. are the reciprocal of those for N. Equation (7.1) is a well
behaved monotone increasing function bounded by 0 < Y< Ay.
Data Analysis: Data for response of grain yield to applied nitrogen with different rotations are
given in Table 7.1 and shown in Figure 7.1. Yields for Trefoil rotation are considerably higher
than the others because this is a legume which fixes nitrogen from the atmosphere. Since yields
for columns 2 and 3 are very similar, these values are averaged for analysis. Now Eq. (7.1) can
be linearized to the form
Zy =In 1 =b, + cN (7.2)
This procedure can be applied to average data from Table 7.1 to give
Z = ln(4 1 = by +cN = 0.98 + 0.0390N r= 0.9925 (7.3)
where Ay = 4.80 Mg ha'1 has been chosen to optimize the linear correlation coefficient to r =
0.9925. Correlation of the linear yield variable (Zy) vs. applied nitrogen (N) is shown in Figure
7.2, where the line is drawn from Eq. (7.3). This leads to the response equation
A 4.80 (.
None & Ryegrass: k = (7.4)
1+ exp(by c,N) 1 + exp(0.98 0.0390N)
The lower curve in Figure 7.1 is drawn from Eq. (7.4). It appears that yield data for the trefoil
rotation approach the same upper limit, so it is reasonable to assume the same value of Ay. On the
premise that c, is characteristic of the soil, this value is kept the same as well. Since by reflects
background level of soil N, this parameter is adjusted for the legume to by = 1.00 to give the
response equation
4.80
Trefoil: 4 = (7.5)
1+ exp(1.00 0.0390N)
The upper curve in Figure 7.1 is drawn from Eq. (7.5).
Interpretation: The logistic model appears to describe yield response to applied nitrogen
reasonably well (Figures 7.1 and 7.2). Including Trefoil (a legume) in the rotation provides
additional nitrogen to production of barley. The model can be used to estimate the quantity of
nitrogen from fixation by the legume. Nitrogen required to reach 50% of maximum yield is given
by
by 0.98
None & Ryegrass: N,,2 = = 25 kg ha1 (7.6)
c, 0.0390
Trefoil: N2 1.00 = 25 kg ha1 (7.7)
c, 0.0390
The negative sign in Eq. (7.7) indicates that more than enough nitrogen is provided from the soil
to achieve 50% of maximum yield. It follows that trefoil provides the nitrogen equivalent of
AiN12 = 25 (25) = 50 kg ha1 (7.8)
This conclusion can be confirmed from Figure 7.1 by the shift between the lower and upper yield
response curves.
Table 7.1. Response of grain yield (Y) to applied nitrogen (N) for barley with different rotations
at Wobum Experiment Station at Harpenden, Herts, England.
N Y
kg ha' Mg ha"'
Rotation
None Ryegrass Trefoil
0 1.44 1.51 3.76
38 2.54 2.67 4.10
75 4.29 4.27 4.64
113 4.54 4.76 4.83
'Data adapted from Dyke et al. (1977) Table 12.
List of Figures
Figure 7.1. Response of grain yield (Y) to applied nitrogen (N) for barley grown with and without
legume at Wobum Experiment Station at Harpenden, Herts, England. Data adapted from Dyke et
al. (1977). Lower and upper curves drawn from Eqs. (7.4) and (7.5), respectively.
Figure 7.2. Correlation of linear yield variable (Zy) with applied nitrogen (N) for barley grown
without legume at Woburn Experiment Station at Harpenden, Herts, England. Line drawn from
Eq. (7.3).
iii
I /F :
I i ]
iRZ
0C
t! Ii,.
Fi 71
I
j I
I
TT:1TI
14r
Yb 6~: :::4
Al An
K:~: i i : i
II Ir~i
.il ii i
9 t 0
F^~ir T iR ^
I 2
8. Response of Biomass Yield to Applied K for Potato.
Definition of the system: Field experiments were conducted at Rothamsted Agricultural
Experiment Station at Harpenden, Herts, England on response of various crops to applied
potassium. The soil was classified as Stackyard series (sandy loam). The experiment was
conducted over the period 1957 and 1958. Applied K levels were 0, 31, 62, and 124 kg K ha'.
This analysis will focus on yield of potato tubers. One set of plots contained high residue of K.
References:
Johnston, A.E., R.G. Warren, and A. Penny. 1970. The value of residues from longterm
manuring at Rothamsted and Woburn. The value to arable crops of residues accumulated
from potassium fertilizers. P 6990. Rothamsted Experimental Station Report for 1969. Part
2. Lawes Agricultural Trust.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. New York, NY.
Mathematical Model: The logistic model will be used to analyze response ofbiomass yield to
applied potassium (Overman and Scholtz, 2002)
A
Y = (8.1)
1+ exp(by cK)
where K is applied potassium, kg ha'l; Y is biomass yield, Mg ha'; Ay is maximum biomass yield
at high K, Mg ha'; by is the intercept parameter for yield; and ck is the response coefficient for
applied K, ha kg'1. Note that the units on ck are the reciprocal of those for K. Equation (8.1) is a
wellbehaved monotone increasing function bounded by 0 < Y < Ay.
Data Analysis: Data for this analysis are adapted from Johnston et al. (1976) as listed in Table
8.1. Response of biomass yield (Y) to applied potassium (K) is shown in Figure 8.1 for averages
from 1957 and 1958. Model parameters are now estimated by inspection. It appears that yield are
approaching a plateau which is estimated to be Ay = 32.0 Mg ha"1. Intercepts are then estimated at
K= 0 to be
No residue: by = In30 1 0.20
14.5
With residue: bye In(3201 1.20
S24.5
Further inspection leads to ck = 0.025 ha kg'. The response equations become
No K residue: y 32.0 (8.2)
1+ exp(by ckK) 1+ exp(0.20 0.0250K)
With K residue: Y
Ar 32.0
1 + exp(b, ckK) 1+ exp(1.20 0.0250K)
(8.3)
Curves in Figure 8.1 are drawn from Eqs. (8.2) and (8.3).
The model can be used to describe data for individual years by assuming that by and ck are
common between years. Standardized yield (Y*) can be defined by
(8.4)
Y* = Y[ + exp(by ckK)]
Equation (8.4) becomes for the two cases
No K residue:
With K residue:
Y* = Y[1 + exp(0.20 0.0250K)] = Ay
Y' = Y[1 + exp(1.20 0.0250K)]= Ay
Values are given in Table 8.2 as calculated from Eqs. (8.5) and (8.6). Note that average values
are close to 32.0 Mg ha1 for both cases. Response for individual years is shown in Figure 8.2,
where the curves are drawn from
No K residue:
With K residue:
= A AY
1+ exp(b, ckK) 1+ exp(0.20 0.0250K)
= AY A
1+ exp(b, ckK) 1+ exp(1.20 0.0250K)
with appropriate values of Ay selected from Table 8.2.
Interpretation: The logistic model describes response of potato to applied K reasonably well. The
effect of potassium residue in the soil is accounted for in the intercept parameter by, as would be
expected. Variation in yield between years is accounted for in the yield parameter Ay. Yields
were consistently higher in 1958 than 1957, which probably reflected difference in water
availability.
(8.5)
(8.6)
(8.7)
(8.8)
Table 8.1. Response ofbiomass yield (Y) to applied potassium (K) with and without potassium
residue for potato tubers grown at Rothamsted Agricultural Experiment Station at Harpenden,
Herts, England.1
K Residue K Y
kg ha Mg ha'
1957 1958 Avg
Without 0 10.5 19.7 15.2
31 17.7 21.7 19.7
62 25.8 27.1 26.4
124 24.4 31.1 27.8
With 0 22.4 27.1 24.6
31 25.5 30.9 28.2
62 31.4 33.4 32.4
124 31.6 34.0 32.7
'Data adapted from Johnston et al. (1970) Table A2.
Table 8.2. Dependence of standardized yield (Y*) on applied potassium (K) with and without
potassium residue for potato tubers grown at Rothamsted Agricultural Experiment Station at
Harpenden, Herts, England.
K Residue K Y*
kp ha' Mg ha
Without
1957
23.3
27.7
32.5
25.7
27.3 (3.9)
With 0 29.1
31 29.0
62 33.4
124 32.0
avg 30.9 (2.2)
Numbers in parentheses are standard errors.
1958
43.8
33.9
34.1
32.8
36.2 (5.1)
35.3
35.2
35.5
34.5
35.1 (0.43)
Avg
31.8
33.0
List of Figures
Figure 8.1. Response ofbiomass yield (Y) to applied potassium (K) with and without K residue
for potato grown at Wobum Agricultural Experiment Station at Harpenden, Herts, England. Data
adapted from Johnston et al. (1970) with averages for 1957 and 1958. Curves drawn from Eqs.
(8.2) and (8.3).
Figure 8.2. Response ofbiomass yield (Y) to applied potassium (K) with and without K residue
for potato grown at Wobum Agricultural Experiment Station at Harpenden, Herts, England. Data
adapted from Johnston et al. (1970) for 1957 and 1958. Curves drawn from Eqs. (8.7) and (8.8)
with parameter A, selected from Table 8.2.
How much potassium does the residue provide?
No residue:
With residue:
Difference:
b 0.20
K112 =c 0.20= 8 kg ha
ck 0.025
by 1.20
K,,2 = 48 kg ha
ck 0.025
Aby 1.40
AK/2 = 56 kg ha
ck 0.025
This is the equivalent K provided by the soil residue.
I I
EQ
3 NHo Pe; eI
a k racith
.i i. i .
___m :1:
1<
p
1'
B ,I
T) :, u e o Tl i, i n" 1h
I 1
dA '
*^
Fi~ Bi
SRO
N.ri
/ .j I'~
A A iI_
I ( ~ i ~ I
I 1
I I
L .. ...
**I .S r i K i, 1 1 n; !
j
'F;1Ii~ala;
9. Coupling of Plant P Uptake and Extractable Soil P with Applied P.
Definition of the system: Field experiments were conducted at Wobum Agricultural Experiment
Station at Harpenden, Herts, England on response of various crops to applied phosphorus. The
soil was classified as Stackyard series (sandy loam). The experiment was conducted over the
period 1968 through 1970. Applied P levels were 0, 82, 164, 328, and 492 kg P ha'. Soil P was
measured with 0.5 M NaHCO3 as extracting solution.
References:
Johnston, A.E., G.E.G Mattingly, and P.R. Poulton. 1976. Effect of phosphate residues on soil P
values and crop yields. I. Experiments on barley, potatoes and sugar beet on sandy loam
soils. P 535. Rothamsted Experimental Station Report for 1975. Part 2. Lawes Agricultural
Trust.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. New York, NY.
Mathematical Model: The logistic model will be used to analyze response of extractable soil P to
applied phosphorus (Overman and Scholtz, 2002)
S= A (9.1)
e 1+ exp(be cP)
where P is applied phosphorus, kg ha'; Pe is extractable soil P, mg P kg'; Ae is maximum
extractable soil P at high P, mg P kg"1; be is the intercept parameter for extractable soil P; and c,
is the response coefficient for applied P, ha kg'. Note that the units on c, are the reciprocal of
those for P. Equation (9.1) is a wellbehaved monotone increasing function bounded by 0 < Px <
Ae. It is next assumed that plant P uptake is related to extractable soil P through the hyperbolic
phase equation
P = P,, P (9.2)
K + P,
where Pu is plant P uptake, kg ha'1; Pum is potential maximum plant P uptake, kg ha'; and Ke is
response coefficient for extractable soil P, mg P kg'1. Substitution of Eq. (9.1) into Eq. (9.2)
leads, after simplification, to a second logistic equation
p. = (9.3)
1+ exp(b, cP)
where Ap is maximum plant P uptake at high P, kg ha'; and bp is the intercept parameter for
plant P uptake. Parameters for plant P uptake and extractable soil P are related by
AP = Ae P (9.4)
bp = be + In K (9.5)
(,Ke +Ae,
The model can be extended further to couple biomass yield with extractable soil P through
Y= Y PI (9.6)
K + Pe
where Yis biomass yield, Mg ha'; Y, is potential maximum yield, Mg ha'; and K; is response
coefficient for extractable soil P, mg P kg'. Substitution of Eq. (9.1) into Eq. (9.6) leads, after
simplification, to a third logistic equation
A,
Y = y (9.7)
1+ exp(b, cP)
where Ay is maximum biomass yield at high P, Mg ha'; and by is the intercept parameter for
yield. Parameters for plant biomass yield and extractable soil P are related by
A = AAe K1 (9.8)
by = be K + InK (9.9)
The model is now used to estimate response to phosphorus from a field study.
Data Analysis: Data for this analysis are adapted from Johnston et al. (1976) as listed in Table
9.1 and shown in Figure 9.1. Equation (9.2) can be written in the linearized form
P Kv 1
P K + (9.10)
SP P. P "
Linear regression of data from Table 9.1 leads to
L = 0.324 + 0.0276P r =0.9980 (9.11)
P
with a correlation coefficient ofr = 0.9980. Equation (9.11) leads immediately to
p .= P_ 37.4Pe (9.12)
K, ++P, 12.1+P,
Results are shown in Figure 9.2, where the line and curve are drawn from Eqs. (9.11) and (9.12).
Now Eq. (9.1) can be written in the linearized form
Z =n e = be + cP (9.13)
( ex )
Linear regression of results in Table 9.1 leads to
Z = In 6 1=1.18 + 0.00450P r= 0.99914 (9.14)
with a correlation coefficient ofr = 0.99914. Parameter Ae = 65.0 mg P kg' has been chosen to
maximize the correlation coefficient. Linearized extractable soil P (Zx) vs. applied phosphorus
(P) is shown in Figure 9.3, where the line is drawn from Eq. (9.14). Equation (9.14) leads to the
response equation
Ae, 65.0 (9.15)
1+ exp(b, cP) 1+ exp(1.18 0.00450P)
Equations (9.4) and (9.5) can be used to estimate parameters for plant P uptake
A, = Ae P.,, ,= 65.0+137.4 = 31.5 kg ha1 (9.16)
__ A,_ + K 65.0 +12.1gh
bp =be +ln( =1.18+ln 2 =1.181.85 =0.67 (9.17)
SKe +Ae) ,12.1+ 65.0)
Equation (9.3) now becomes
= A, 31.5
^ AP 315 (9.18)
1+ exp(b, cPP) 1+ exp(0.67 0.00450P)
It follows from Eqs. (9.15) and (9.18) that the ratio Pex/Pu is related to applied P by
P A 1+ exp(b, cP) 061+ exp(0.67 0.00450P)
S+exp(bcP) L + exp( 0.00450P)(9.19)
P" AP 1+ exp(b cP) I+ exp(1. 18 0.00450P)
Curves in Figure 9.1 are drawn from Eqs. (9.15), (9.18), and (9.19).
Response of extractable soil P (P,) and biomass yield (Y) for barley (grain, adjusted to 85%
DM), potato (tubers), sugar beet (sugar) to applied phosphorus (P) is examined in detail. Results
are given in Table 9.2. Dependence of yield and extractable soil P/yield ratio on extractable soil
P for barley is shown in Figure 9.4, where the line and curve are drawn, respectively, from
Barley:
Pe. K' 1
 =  + P = 1.07 + 0.222P,
Y Y' Y "
r= 0.9989
y= Y'Pe = 4.50P,
K' + P 4.82+Pe
Similar results are shown in Figure 9.5 for potato, where the line and curve are drawn,
respectively, from
Potato:
Pe K' 1
" = + P = 0.131+0.0222P
Y Ym
r= 0.9888
Y, YP 45.0Pe
K' + Pe 5.90 + P
(9.20)
(9.21)
(9.22)
(9.23)
Results are shown in Figure 9.6 for sugar beet, where the line and curve are drawn, respectively,
from
Sugar beet:
P, K; 1
e =  + P = 0.0990 + 0.160P,
Y Y Y ex
m m
r= 0.9943
(9.24)
(9.25)
Y_ Y'Px 6.25P,
K'+P, 0.619+P,
These results confirm the hyperbolic relationship between Y and Px.
The next step is to examine dependence of extractable soil P on applied P. Response of P, to
P is shown in Figure 9.7. Note that Eq. (9.1) can be rearranged to the linear form
(9.26)
Parameter Ae is chosen to produce c, = 0.00450 ha kg'1. Analysis by linear regression leads to
Ze = In465 1 = 0.65 + 0.00450P
(Pe )
Z, = In Pl =b, +,P
\ ex )=
Barley:
r= 0.99950
(9.27)
P 46.5 (9.28)
1+ exp(0.65 0.00450P)
Potato: Z = ln 45 = 0.65 + 0.00450P r= 0.99911 (9.29)
45.0
P 45.0 (9.30)
1+ exp(0.65 0.00450P) (9
Sugar beet: Zi = In 56.0 = 1.13+0.00450P r= 0.9961 (9.31)
56.0
S56.0 (9.32)
e 1+ exp(1.13 0.00450P) (9
Linearized plots are shown in Figure 9.8, where the lines are drawn from Eqs. (9.27), (9.29), and
(9.31). Curves in Figure 9.7 are drawn from Eqs. (9.28), (9.30), and (9.32).
The final step is to examine response ofbiomass yield to applied P for the three crops.
Results are shown in Figure 9.9. Equations (9.7) through (9.9) are used for this purpose.
A 46.5
Barley: A A, e. 4 4.84 4.50=4.08 Mg ha (9.33)
_K' 4.84
by =be + An = 0.65 +ln 4.84 0.65 2.36 = 1.71 (9.34)
(K' + A, 4.84 + 46.5
Ay 4.08
Y= A = 4.08 (9.35)
1+ exp(b, cP) 1+ exp(1.71 0.00450P)
(A = ( 45.0
Potato: A = A Y= 45.0 45.0 = 39.8 Mgha" (9.36)
Ae + Ke ) 45.0+5.90
KK', 5.90
S, b+( = 0.65 + I(5.90 0= 0.65 2.15= 1.50 (9.37)
b K K + Ae ) 5.90 + 45.0)
= AY 39.8 (9.38)
1+ exp(b, cP) 1+ exp(1.50 0.00450P)
Sugar be =t A, ) 56.0 1
Sugar bet: A, Ae = 56.0 6.25 = 6.18 Mg ha (9.39)
K' 0.62
b =be ( 1.13+n 0.62= 1.134.51=3.38 (9.40)
(K + Ae )0.62 + 56.0)
iY Ay 6.18 (9.41)
1+ exp(b, cP) 1+ exp(3.38 0.00450P)
Curves in Figure 9.9 are drawn from Eqs. (9.35), (9.38), and (9.41).
Interpretation: The logistic model describes results rather well. Logistic dependence of
extractable soil P on applied P is confirmed (Figures 9.7 and 9.8). A common response parameter
cp = 0.00450 ha kg' applies throughout. Intercept parameters are in the order be > bp > by, as is
required by the model. Response variables Pe, Pu, and Y are all bounded. Hyperbolic
relationships are confirmed by Eqs. (9.12), (9.21), (9.23), and (9.25) as illustrated in Figures 9.2,
9.4, 9.5, and 9.6. Biomass yields of barley, potato, and sugar beet are relatively unresponsive to
applied P, as reflected in the negative by values for the three crops.
The analysis can be extended even further. Assume the phase relation between plant P uptake
(Pu) and extractable soil P (Px)
p = P". (9.42)
SKe+Px
Then assume another phase relation between biomass yield (Y) and plant P uptake (Pu)
Y= YmP" (9.43)
K, +P"
where Ym is potential maximum yield, Mg ha''; and Kp is response coefficient for plant P uptake,
kg ha'. Substitution of Eq. (9.42) into Eq. (9.43) leads to the third phase relation between yield
(Y) and extractable soil P (P,)
Y= Y P (9.44)
K' +P,
where Y, and K" are defined by
y' =( P Y (9.45)
Y" Kp +P )m
^p um
K; K(9.46)
( P,,,, + KP )
Equations (9.42) through (9.44) describe clear coupling between extractable soil P, plant P
uptake, and biomass yield. These hyperbolic phase relations are compatible with the three
logistic response equations for extractable soil P, plant P uptake, and biomass yield
A,
P = Ae(9.47)
S1+ exp(be cP)
A
P,= p (9.48)
1 + exp(b, cP)
A)
Y = A (9.49)
1+ exp(b, cP)
Several characteristics of the model should be noted. The response coefficient for applied P
(c) is common to all three logistic equations, and thus it can be considered a property of the soil.
This premise appears to be supported by this analysis. Hyperbolic and logistic parameters are
related by
Y = (9.50)
1 exp(Ab)
A
K = (9.51)
K exp(Ab)l
with Ab = b by,
Y' = (9.52)
1 exp(Ab') (
K' A (9.53)
exp(Ab') 1
with Ab'= be by,
P. A" (9.54)
1 exp(Ab")
K, = A (9.55)
exp(Ab") 1
with Ab" = be bp In order for the hyperbolic parameters to be positive, the theory requires that
be > bp > by. This analysis appears to support this premise as well.
The weak response of yields to applied P for the three crops may be noted from Figure 9.9, as
reflected in the negative values of by. This would make direct calibration of the model for yield
response to applied P rather difficult. Data for response of extractable soil P to applied P made
possible estimates of by.
Table 9.1. Response of plant P uptake (Pu) and extractable soil P (Px) to applied phosphorus (P)
at Wobum Agricultural Experiment Station at Harpenden, Herts, England.'
P Pu Pex Pex/Pu Zex
kg ha'' kg ha' mg kg'
0 21.4 15.4 0.72 1.17
82 23.4 20.6 0.88 0.77
164 24.4 24.6 1.01 0.50
328 28.9 37.2 1.29 0.29
492 29.6 48.1 1.62 1.05
Ae, mg kg1    65.0
be    1.18
c,, ha kg    0.00450
r  0.99914
'Data adapted from Johnston et al. (1976) Table 7.
Table 9.2. Response of extractable soil phosphorus (Pex) and biomass yield (Y) to applied
phosphorus (P) at Woburn Agricultural Experiment Station at Harpenden, Herts. England.1
P Pe Y Px/Y Pex Y Pe/Y Pe Y Pex/Y
kg ha' mgkg'1 Mg ha' mg kg'1 Mg ha' mg kg'' Mg ha'
Barley (grain) Potato (tuber) Sugar Beet (sugar)
0 16.0 3.40 4.71 15.6 31.6 0.494 13.8 5.78 2.39
82 20.4 3.69 5.53 19.6 34.4 0.570 18.8 5.76 3.26
164 23.8 3.74 6.36 22.7 36.4 0.624 21.6 6.33 3.41
328 32.5 3.98 8.17 31.4 40.0 0.785 31.6 6.46 4.89
492 38.5 3.96 9.72 37.2 37.5 0.992 42.5 6.00 7.08
'Data adapted from Johnston et al. (1976).
List of Figures
Figure 9.1. Response of plant P uptake (Pu), extractable soil P (Px), and extractble P/plant P ratio
(P,/P,) to applied phosphorus (P) at Wobum Agricultural Experiment Station at Harpenden,
Herts, England. Data adapted from Johnston et al. (1976). Curves drawn from Eqs. (9.15), (9.18),
and (9.19).
Figure 9.2. Dependence of plant P uptake (Pu) and extractable soil P/plant P ratio (Pe/Pu) on
extractable soil P (P,) at Wobum Agricultural Experiment Station at Harpenden, Herts, England.
Data adapted from Johnston et al. (1976). Line and curve drawn from Eqs. (9.11) and (9.12),
respectively.
Figure 9.3. Dependence of linearized extractable soil P (Ze) on applied phosphorus (P) at
Wobum Agricultural Experiment Station at Harpenden, Herts, England. Data from Table 9.1.
Line drawn from Eq. (9.14).
Figure 9.4. Dependence ofbiomass yield (Y) and extractable soil P/yield ratio (Pe/Y) on
extractable soil P (Px) for barley grown at Wobum Agricultural Experiment Station at
Harpenden, Herts, England. Data adapted from Johnston et al. (1976). Line and curve drawn
from Eqs. (9.20) and (9.21), respectively.
Figure 9.5. Dependence ofbiomass yield (Y) and extractable soil P/yield ratio (Px/Y) on
extractable soil P (Pe) for potato grown at Woburn Agricultural Experiment Station at
Harpenden, Herts, England. Data adapted from Johnston et al. (1976). Line and curve drawn
from Eqs. (9.22) and (9.23), respectively.
Figure 9.6. Dependence ofbiomass yield (Y) and extractable soil P/yield ratio (P/Y) on
extractable soil P (Px) for sugar beet grown at Wobum Agricultural Experiment Station at
Harpenden, Herts, England. Data adapted from Johnston et al. (1976). Line and curve drawn
from Eqs. (9.24) and (9.25), respectively.
Figure 9.7. Dependence of linearized extractable soil P (Ze) on applied phosphorus (P) for plots
with barley, potato, and sugar beet grown at Wobum Agricultural Experiment Station at
Harpenden, Herts, England. Data from Table 9.2. Lines drawn from Eqs. (9.27), (9.29), and
(9.31).
Figure 9.8. Dependence of extractable soil P (Pe) on applied phosphorus (P) for barley, potato,
and sugar beet grown at Woburn Agricultural Experiment Station at Harpenden, Herts, England.
Data adapted from Johnston et al. (1976). Curves drawn from Eqs. (9.28), (9.30), and (9.32).
Figure 9.9. Response ofbiomass yield (Y) to applied phosphorus (P) for barley, potato, and sugar
beet grown at Wobum Agricultural Experiment Station at Harpenden, Herts, England. Data from
Table 9.2. Curves drawn from Eqs. (9.35), (9.38), and (9.41).
i 
'1
i
i
r .'~ J
~
i
~z~.a
I
~ i
i :
f
"I$~~
i
1 : I; i I
i '
"`''''
~ ' '
' '
~
i :
i ; i
f Ito In l
f) / L)U :.
t' t' 1\
/
/I
I
U11" L~ti
I : i i FC1I i i ; i i i
i ~1
:D i : iii I 1
r i7
i;
SI _______ii
/ :7
Ki I
I V .
4. i.
I I
1!' I);. j* 
SI I' 1.1.. 
I II *: iiI I 1 Ii, :l~~
Ii I
I .I
''Ii
: i :: .I : 1 I ,. I
: 1I1
I' I ,ii
I I
77 .Qf 
I:'.
mU ~;
 ;j'I
7I: 1 i I
I
I. .i i.
,~ I.
Ir\F ..l1 V'
.C
0i:
20 n'utr,'.
)

I
i ~ i ~ ~
i
1
[ : '
* I '
^'l
;
L
,i i
. I.:
, ~I
; i
. ." j '. '. \ .
'*^ ;Hi<.:1I
l ii~ll
; i: : : :
p '
: : 1
I..
IA
Kbi
V i I
V $
iiiC)
F' ; / i
K I .
{21
JO
Ki
I,
F.
4:
2 *. i .l
:II,
:I~~~~4. :::
f f:
* F
i..I F I_ 
F
I j rp
7
F F
1:'
.4. ::
SI
1 I t rr.
* .i 
* F _
I j : j I J
i
ii; iji I : I : :
i
i''''
;i ::1 :
'I
i II i.
I''
:I:I
:i
~ f
i
i i
I;
; ~ ; '
I '
~'~'~~''
___.. _i._j
1;1:
_..1..
t rl~
....
,....
1**'
j~ __
11<:
I '
20 IS r l h e Inch
I .~ I~ .~I ~_
~IIT  ^  I C  ~i ~ ~^~~ I
E
b
i j
i
I 1 i
j ii i iI :
: I
i:
: i : i
1
I 
j
I
: i ;
i
i
i
i
i i
1
: i .
i
'
;
~ ; '
:
''
.,
:
'
i '
L ; i
i
i :
1~ i. :
1;_
i '
i i : i '
: : r : ; : 1::
 i::i
'''
.._ ..,
"~C~
: : : : : I:
1,
,,
j I:
; 1 / I i .
: : : j
: : i..
.. ~.
I I. 1. .1 i
I I iij
.11
~I I '.1:. *1
I:11
.: I:
ti .i I,1
tt I* '
j~~K 'r .I r
1 i I.:. il .1.I.
*11
I i *I :. :.
I I I4
1 I
I; ~ I .1...: I ir I' .',:~ I
~Th 
K I'
Ij~ I
I*
I. 1 j/:
I II
~~.
1 I
: I!
I
'
1
 : I~
;;i i;
: : i : I
'
'
aI
i : I
;I!
j i :
'' '
: : : f
jijl
20I t I,
,sI~
 J
i. '
i 
L
h ci
I.
II3~
/1i
i
i: i i'i
 i
I i I i ; I i i
j
i :i i
ilij;
` ~ '
:I
1 .'~
i I :
i :
ir
~ 1 :
5Y1
2 ;(itIIrst to If Inch
ii:l
p~.
j)
ctC~
~s~
1'
: i
I
; i
i i i: i i
i ;
:!::;
~. : 
i : : :
it91~I
I I
I I
*y it j 1
~ I: ~i
i i
*i
: t : I
.; I i ,
i 1
i + 'i i
iiv
'.**i ,
i .
i;^ i..
[ *
T>
I i
 1
i
t:
..n ; .
01
: i I
i+ I
I i i '.
. \ \ ", +
 I+I~
I I.
1'
I II
4 iii
4
.1 i i
iv
liiil
Wii
I,
I ..i
SI ,
.1`
/I4 r;4 ix '
* :i iii
.' 
I,.
2i 1
I :: ,i *cJ :;
i
'
I
i .
_i_..: I:......
i 1 : i .
: ... ~ i ;
I
i : : i; i : ii :
i
I:
i
I ~
i i ~ ; i
i
J
I
I~*Y~r~rrlr;lr~11~sll
I
R~YLUInllYI3LP(L~P(Pj
1: ii
.I ..
i..
i i
I
i .
i I ;
...~:
i iii.l
'1
I I ~ ;
'' i
~~;
~i_
i i i I i 1 i i i
: i I : : :
:i i
i
:I
20 Sq'u,1 res t< f;I Inch
I.
j 1.1. i'.
Wi
:.. 1 .I
v
i
i I i
!
i ~
*1
.1
~YPotiirto~
i~
J
/00
: .
I I
^^*M'^i::'
I .\ '
LII__
w w
i
x
'
1
i
I ~
i
 ?
8:
i : : I
: i .
,e
1 :
S i ii Ti I
A ii !1 2 I i U
nu
1 Bd~vle
J
l0oo
9i00
W.
/7, kq Ihi~
* I
I I
: 1
j .
I ,
(~
&tLQ
_~_I_
 ~ ~ ~ ~~~~~Y~""~YI~^ULU
t:l+I xc~~
,r  11~1~1.~~~11~
~ ~~ 1. I
'
11%"1vic~:w.v~
: i
'
r SIe
fo
: I^
3 zo
A;
iIg 9
it
0~  I'
~t (L4' )
iQ Uo
10. Coupling of Alfalfa Yield and Extractable Soil K with Applied K
References:
Markus, D.K. and W.R. Battle. 1965. Soil and plant responses to longterm fertilization of alfalfa
(Medicago sativa L.). Agronomy Journal 57:613616.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. New York, NY.
Mathematical model: The logistic model for this analysis is discussed in Overman and Scholtz
(2002). Response of biomass yield to applied potassium can be described by
Y= Ay (10.1)
1+exp(b ckK)
where K is applied potassium, kg ha'1; Y is biomass yield, Mg ha1 ; Ay is maximum biomass yield
at high K, Mg ha'1; by is the intercept parameter for yield; and ck is response coefficient for
applied K, ha kg'". Note that the units on Ck are the reciprocal of those on K. Equation (10.1) is a
wellbehaved, monotone increasing function, bounded by 0 < Y < Ay.
Response of extractable potassium (Kx) to applied K can be described by
K = Ae (10.2)
1 + exp(be ckK)
where K is applied potassium, kg ha'1; Ke is exchangeable potassium, kg ha&; Ae is maximum
exchangeable potassium at high K, Mg ha'; be is the intercept parameter for exchangeable K;
and ck is response coefficient for applied K, ha kg'. Note that the units on ck are the reciprocal of
those on K. Equation (10.2) is a wellbehaved, monotone increasing function, bounded by 0 <
Ke < Ae. Equation (10.2) can be rearranged to the linear form
Z, = In A 1 = be +kK (10.3)
.K
For a particular value ofAe, Eq. (10.3) can be used to obtain estimates of parameters be and ck
from linear regression of Z, vs. K.
Equations (10.1) and (10.2) can be combined to give the hyperbolic phase relation
Y= YmKex (10.4)
K' +K,
where Y' is potential maximum yield, Mg ha''; and K' is response coefficient for exchangeable
K, kg ha"1. Equation (10.4) can be rearranged to the linear form
K K' 1
 =K +Ke (10.5)
Y Y' Y'
Equation (10.5) is easily tested by linear regression.
A set of data is now used to illustrate procedures for parameter estimation.
Data analysis: Data for this analysis are adapted from a field study with alfalfa (Medicago sativa
L.) on Nixon loam (fineloamy, mixed, semiactive, mesic Typic Hapludult) at New Brunswick,
NJ, USA. Applied potassium included rates of 0, 46.5, 93, 186, 280, and 372 kg ha. Soil
samples were collected from the upper 30 cm. Soil extracts were obtained by electrodialysis.
Treatments were replicated five times.
Data for response to applied K are given in Table 10.1 and shown in Figure 10.1. The
corresponding phase plots are shown in Figure 10.2. Linear regression of K,/Y vs. K, from
Table 10.1 leads to
K K' 1
K =  + K = 11.36 + 0.0838K, r= 0.984 (10.6)
Y Y' Y'
m m
which leads immediately to the hyperbolic phase equation
Y'K 11.94K (10.7)
Y (10.7)
K +K, 136+ K
The line and curve in Figure 10.2 are drawn from Eqs. (10.6) and (10.7), respectively.
The second step is to estimate parameters for Eq. (10.2). Linear regression with Eq. (10.3)
leads to
Z= =8 1 =be + ckK = 2.09+0.00879K r= 0.982 (10.8)
Ke )
where Ae = 850 kg ha"1 has been chosen to maximize the correlation coefficient (r). Results are
shown in Figure 10.3, where the line is drawn from Eq. (10.8). This leads to the response
equation
A, 850
K = A (10.9)
1+ exp(be ck K) 1+ exp(2.09 0.00879K)
It is now possible to estimate parameters for yield vs. applied K in Eq. (10.1). Logistic and
hyperbolic parameters are related by
A b' A\ A) 850 =01 (1.
K=ex A Ab'= n + =In1+ j=1.98=2.09by by=O.11 (10.10)
K exp(Ab') 1 Kb 136) 5
Y exA Ay = Y [l exp(Ab')]= 11.94[l exp(1.98)]= 10.28 Mg ha (10.11)
1 exp(Ab')
which leads to the yield response equation
Y= Ay 10.28 (10.12)
1+ exp(b, ckK) 1 + exp(0.11 0.00879K)
Equations (10.9) and (10.12) can now be combined to give
K,_ Ae [1+exp(b ckK) 1+ exp(0.110.00879K) (
Y A, 1 + ex c exp(2.09 0.00879K)
The curves in Figure 10.1 are drawn from Eqs. (10.9), (10.12), and (10.13).
Interpretation: The model provides reasonable description of the data. Figure 10.1 appears to
confirm the common Ck = 0.00879 ha kg1 for response of yield and exchangeable soil K to
applied K. The analysis also confirms that be > by as required by the theory.
The analysis can be extended even further. Assume the phase relation between plant K
uptake (Ku) and exchangeable soil K (K,)
K, =K K, (10.14)
K, + K,
Then assume another phase relation between biomass yield (Y) and plant K uptake (Ku)
Y = YaKu (10.15)
Kk +K,
where Ym is potential maximum yield, Mg ha'; and Kk is response coefficient for plant K uptake,
kg ha'. Substitution of Eq. (10.14) into Eq. (10.15) leads to the third phase relation between
yield (Y) and exchangeable soil K (K,)
Y= Y'K (10.16)
K + K,
where Y, and K' are defined by
Y K + Km Y (10.17)
S K' k Ke (10.18)
e = um + Kk
Equations (10.14) through (10.16) describe clear coupling between exchangeable soil K, plant K
uptake, and biomass yield. These hyperbolic phase relations are compatible with the three
logistic response equations for exchangeable soil K, plant K uptake, and biomass yield
K Ae (10.19)
S+ exp(be ckK)
K Ak (10.20)
1+ exp(bk ckK)
A,
Y= y (10.21)
1+exp(b ckK)
It can be shown that be > bu > by must hold for internal consistency.
Table 10.1. Response ofbiomass yield (Y) and exchangeable soil potassium (Ke) to applied
potassium (K) for alfalfa grown at New Brunswick, NJ. USA.'
K Y Kex K/Y
kg ha'' Mg ha'' kg ha' g kg'
0 4.10 105 25.6
46.5 5.98 141 23.6
93 7.73 185 23.9
186 8.92 240 26.9
280 9.45 545 57.7
372 9.77 660 67.6
IData adapted from Markus and Battle (1965).
Table 10.2. Response of exchangeable potassium (Ke) to applied potassium (K) for alfalfa grown
at New Brunswick, NJ, USA (1961).1
K Ke Ze
kg ha1' kg ha'1
0 105 1.89 1.82 1.86 1.92 1.96
46.5 141 1.54 1.46 1.51 1.57 1.62
93 185 1.20 1.12 1.17 1.23 1.28
186 240 0.85 0.75 0.81 0.88 0.93
280 545 0.76 0.98 0.84 0.68 0.58
372 660 1.55 1.99 1.70 1.42 1.25
Ae, kg ha'  800 750 780 820 850
be  2.05 2.04 2.04 2.07 2.09
ck, ha kg  0.00937 0.0103 0.00967 0.00911 0.00879
r 0.9799 0.9774 0.9792 0.98108 0.98198
'Data adapted from Markus and Battle (1965).
List of Figures
Figure 10.1. Response ofbiomass yield (Y) of alfalfa, exchangeable potassium (Ke), and
exchangeable K/yield ratio (Kx/Y) to applied potassium (K) at New Brunswick, NJ,USA. Data
adapted from Markus and Battle (1965). Curves drawn from Eqs. (10.9), (10.12), and (10.13).
Figure 10.2. Phase plots ofbiomass yield (Y) for alfalfa and exchangeable K/yield ratio (K/Y)
vs. exchangeable potassium (Ke) at New Brunswick, NJ,USA. Data adapted from Markus and
Battle (1965). Line and curve drawn from Eqs. (10.6) and (10.7), respectively.
Figure 10.3. Dependence of linearized exchangeable potassium (Kx) on applied potassium (K)
for alfalfa grown at New Brunswick, NJ, USA. Line drawn from Eq. (10.8).
i !
iL
n'i
\ 0vj!
/
<
C
S00
0
y
>2
0 ~r '~pe100
Pz: 1O hI
'7IT777
rn
I:1 ri,, iII
i
: I
F'( iea~'
12 ?'
/j0
;
i
j
'i
~I::::; i
i ~ i
j : : j :
i :
Qh
X
tl
`7
^r^
,:;
1
:i :iii
i.... i :i::
i .
i
:r:i
i
i
oA ll
<,^^
'
: : j
'
I
I '
''
I `
'
~ ~ '
I
'
* I
* .
j 
i
1 i I: i' '
i.: _.
I : 1
.I;
.11
.11 : I
i 1 *.*' iii:i
I I~'
I 
!I I. '~i
I ,* I ~ i~' I i
i I 1II
II *I I :
I~ II
/
I ii.
i:"
B ha:
i '
i ;;
i~i7
~illli Ii I: I: i `
*1
.. .. I t i.
i i : i .. .
: i i : ; : : i : i: i l l
: i i !
. .. .... ._ .^i . 
: .
i"~~ i
i
i, j..
>1]
i
; i.
1Y.$l
i
SI'
F j
rl 1
: : : i .
:!:; i,
I:
' '
'
i... ...~. ... i
i
'
i:
'
'~'''
i::: i::l ii;ii;li: ; 1
i
;li
j~ .....J... :  i
I i : i
I:i i I i 'liii' ~ '
I : i
I : I~ ,.
i : : ; 1
::iiriii1 ;
I:r:t i i :.: !:_i.. :_r ..i : .....: :I
i" ii 7tj tjl._j_~
: : I 1 I
'
I.....:.
ri
__I__L._IL~/
~'~
~ i
!:;;
...
''''~
i
i ! 1
~ i
I j~j : 
i
 i : I
20 SI'i r, s to i, in" inchI
