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Title: memoir on crop yield and nutrient uptake
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    Title Page
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    Abstract
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    Main
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        Page 15
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    List of Figures
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    Figures
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Full Text






A MEMOIR ON CROP YIELD AND NUTRIENT UPTAKE

Comparison of Polynomial and Logistic Models





Allen R. Overman





Agricultural and Biological Engineering

University of Florida


Copyright 2009 Allen R. Overman










A MEMOIR ON CROP YIELD AND NUTRIENT UPTAKE


Comparison of Polynomial and Logistic Models

Allen R. Overman

Agricultural and Biological Engineering Department
University of Florida, Gainesville, FL 32611-0540

ABSTRACT

The extended logistic model has been developed to describe crop response to applied nutrients.
Control variables include levels of applied nutrients (N, P, K), harvest interval (for perennial
grasses), and water availability. Response variables include biomass yield (Y), plant nutrient
uptake (Nu), and plant nutrient concentration (Nc). The model also describes phase relations (Y
and Nc vs. N,). Analysis of data from the literature has established application of the model to
various crops (perennial grasses, grain crops, and vegetables) grown on different soils and under
a wide range of environmental conditions. Model parameters can be estimated by one of three
methods: (1) graphical, (2) linearization, and (3) nonlinear regression. Method (3) provides the
most rigorous procedure. Linearization represents a rearrangement of the logistic equation so that
the exponential coefficients can be estimated by linear regression. Some have suggested that a
simple polynomial model may fit the data more accurately, so why not use this approach? For
example, with five measurements a fourth order polynomial may fit all the data points exactly.
After all, isn't that the goal of a mathematical model? In this memoir data from a field
experiment using a perennial grass with five levels of applied nitrogen, five harvest intervals, and
two levels of water availability are analyzed by a fourth order polynomial and by the extended
logistic model. It is shown that the polynomial model fits the data for biomass yield and plant
nitrogen uptake exactly for each harvest interval and each year. Analysis generates 100 model
parameters! However, the polynomials exhibit strange behavior just outside the domain of the
measured nitrogen levels. In fact, yield estimates become negative. Yield measurements at even
higher levels in other studies show no sign of downturn in yield response. This is where
knowledge of the physics of the system becomes important for interpretation purposes. It is
shown that the extended logistic model can be used to describe data for this complex experiment
rather well without strange behavior. In fact the equations are well-behaved and bounded. In
addition, the model generates well-behaved phase relations for the system. It is concluded that
the extended logistic model is far superior to the polynomial model for understanding crop
response to applied nitrogen. This memoir contains 25 pages, including 48 equations, 13
references, 6 tables, and 8 figures.








INTRODUCTION


Application of the logistic model for response of various crops to applied nutrients has been
described by (Overman and Scholtz, 2002). In this memoir both the logistic and fourth order
polynomial models are used to analyze data from a field experiment with coastal bermudagrass
[Cynodon dactylon (L.) Pers.] grown on Tifton loamy sand (fine-loamy, kaolinitic, thermic
Plinthic Kandiudult) at Tifton, GA by Prine and Burton (1956). The experiment included five
levels of applied nitrogen (0, 112, 336, 672, and 1008 kg N ha-'), five harvest intervals (2, 3, 4, 6,
and 8 wk), and for two years (1953 and 1954). In 1953 rainfall was quite adequate, while 1954
showed the worst drought in more than 30 years. Response variables included biomass yield,
plant nitrogen uptake, and plant nitrogen concentration. Data from the experiment are
summarized in Table 1 (1953) and Table 2 (1954).
A comparison of polynomial and logistic models for these data is now discussed.

POLYNOMIAL MODEL

The fourth order polynomial response equations to applied nitrogen are given by

Y= ao + aN + aN2 + aN3 +a4N4 (1)

= a'o +a[N + a2N + aN3 + aN4 (2)

where N is applied nitrogen, kg ha'; Y is biomass yield, Mg ha'1; Nu is plant nitrogen uptake, kg
ha-'; ai are coefficients for yield response; and a' are coefficients for plant N uptake. It follows
from Eqs. (1) and (2) that plant nitrogen concentration (Nc = Nu/Y) is described by

N_ .N a ',+aN+a N2 + a'3 +a N4
a 2 a3N +a4N (3)
fc ao + aN + aN2 + a3,N + aN4

A procedure is needed to optimize values of the coefficients for the model from the data. The
least squares criterion is adopted for this procedure (Draper and Smith, 1981). Error sum of
squares (Ey) of the difference between measured yield (Y) and estimated yield (Y) is defined by


E, = ( )2 (4)
i=1

where n is the number of observations. The goal is to choose model coefficients to minimize Eq.
(4). So Eq. (4) can be viewed as a function Ey(ai) of the parameters ai. At the optimum values of
the coefficients ai the differential in the error (dEy) at minimum Ey is given by


dE = da + da, + da2 + + da4 = 0 (5)
S ao a, a2) 9a4 )







This condition requires that all partial derivatives vanish simultaneously. For the first coefficient
ao the partial derivative becomes

aE= a [Y-Y = (-1 Y-]=- Y+ (ao +a 2N+a2N2 +aN3 +aN4)=0 (6)
aaO aaO

which leads to the expanded form

na0 +( Na, + ( _N2)a2 + ( N3)3 +(I N44 = Y (7)

Performance of partial derivatives on the other coefficients leads to the system of five equations
in five unknowns

nao + (N)aI +(ZaN2)2 +(ZNE 3 +(AI N44 =INY
(IN)ao +(I N21)a, + N 3 )a2 + ( N4 )a3 N54)a, =INY

N(E )a + ( N4 + 5 6 ( 7 3
_N40)o + ,_NS)a, +~N62 +( 7)a3 +_N. 4 = N4
which constitutes a system of linear equations in the coefficients since all the sums can be
calculated from the set of observations. In fact, Eq. (8) can be written in matrix form as

[A].[a]= [B] (9)

where the square (5x5) coefficient matrix [A] is defined by

n ZN ZN2 ZN3 ZN4
ZN ZN2 2N3 N4 ZN5
[A]= IN2 ZN3 IN4 ZN5 ZN6 (10)
ZN3 ZN4 ZN5 ZN6 ZN7
ZN4 JN5 IN6 IN7 IN8
Also the [By] and [a] vectors (1x5) in Eq. (9) are defined by

Y ao-
SYN al
[By]= ZYN2 and [a]= a (11)
SYN3 a3
SYN4 a4










The challenge is to determine the unknowns a, from the observations.
Error sum of squares (E,) between observed plant N uptake (N,) and estimated plant N
uptake ( N) is given by


E = (N, u.)2 (12)
i=1

In a similar manner the coefficients a' for Eq. (2) form the matrix representation

[A]. [a']= [B,] (13)

where the coefficient matrix is the same as Eq. (10) and the vectors [B,] and [a'] are defined by

E N2 a,
SNN a:
[B,]= E .N 2 and [a']= aa; (14)
E NN3 a3
ENUN4 a,

Again the coefficients a' are determined from the observations.
The coefficients are estimated from the data in Table 1 with an HP 50g calculator using the
numerical solver and matrix writer, and are listed in Table 3 for the year 1953. In order to limit
the size of the numbers in the matrix procedure applied N has been scaled to N/1000 as input. It
is important to carry all digits to avoid roundoff errors in the procedure. A similar procedure is
used for the data in Table 2 for 1954 to obtain the coefficients in Table 4.
Results of this analysis are now examined in detail for the data of 1953 (Tables 1 and 3).
Attention is first focused on yield response to applied nitrogen and harvest interval. Since yields
are all positive (by definition), the ao coefficients are all positive. Since yields show an increase
with increase in N, it follows that the linear coefficients al are all positive. And since the rate of
increase of yield with increase in N shows a decrease, the quadratic coefficients a2 are all
negative. It is observed from Table 2 that the cubic coefficients a3 are all positive and the quartic
coefficients a4 are all negative. There is a clear pattern in the yield coefficients for 1953, and
calculations show that Equation (1) fits the data perfectly with the coefficients in Table 2. It can
be concluded that the 4th order polynomial provides the 'best fit' of any model available for the
five data points (five observations and five coefficients). Response of plant N uptake to applied
N is less clear cut, but the model again provides perfect fit to the data. This is illustrated in
Figure 1 for the data from 1953 for a harvest interval of 8 wk. The estimator equations are given
by


Y = 5.6400 + 87.2401N -155.970N2 +153.1468N3 60.6880N4


(15)









N, = 62.000 +1183.432N 875.141N2 + 315.468N3 62.767N4


N, 62.000 +1183.432N 875.141N2 + 315.468N3 62.767N4 (17)
Y 5.6400 + 87.2401N -155.970N2 +153.1468N3 60.6880N4

Perfect fit of the model to the data points is apparent.
Now let us examine the results in greater detail to see the implications of the polynomial
model. First, there is evidence that coastal bermudagrass (Creel, 1957; Doss et al., 1966) and
other forage grasses (Little et al., 1959) show continued increase in yield up to 2000 kg N ha-' in
contrast to the downward curvature exhibited in Figure 1. Second, the yield curve becomes
negative at negative (reduced) nitrogen levels, which is not meaningful under the definition of
positive yields. The curve for plant N uptake also becomes negative at reduced applied N levels.
Third, plant N concentration response to applied N is described by Eq. (17). This curve exhibits
strange behavior as shown in Figure 1. Since the yield curve passes through Y= 0 at N= -58 and
1470 kg ha-', the concentration curve exhibits discontinuities at these points. This of course is
meaningless from a physical point of view. So, while the polynomial model provides perfect fit
of the data points, it makes no sense from a physical point of view. The phase plots among the
response variables (Y and Nc vs. Nu) are shown in Figure 2, where the curves are calculated from
Eqs. (15) through (17). While the curves pass through the data points, strange behavior is
apparent just outside the data. Phase plots will be discussed in more detail for the case of the
logistic model.
Deficiencies of a truncated power series have been illustrated by Rudy Rucker (Rucker,
1987, p. 144) for the sine function. The power series representation for sin x is given by the
infinite series

X X3 X5 X7 Xn
y= sinx= -- + - +- + (18)
1! 3! 5! 7! n!

Various orders of approximation of this series are given by the truncated series

1st order: y, = x (19)
3
x
3rd order: Y3 = x -- (20)
6
X3 X5
5th order: y = x --+- (21)
6 120
3 5 7
7th order: y = x - + (22)
6 120 5040

Curves in Figure 3 are drawn from Eqs. (18) through (22). Note that in these cases x is given in
radians (versus degrees). In the domain 0.50 < x < +0.50 Eq. (19) is within a 5% error of the
true value. Equation (20) expands the domain to- 1.50 < x < +1.50. While Eq. (18) is bounded
by -1 < y < +1 for all values of x, all of the truncated power series are unbounded iny. This
highlights the deficiency in Eqs. (15) through (17) for describing crop response to applied N.


(16)









EXTENDED LOGISTIC MODEL

The extended logistic model of crop response to applied nitrogen is described by

Y= -A (23)
1+ exp(by cN)


N. = A (24)
1 + exp(b, c N)


SN N 1+ exp(by cN) (25)
fc YI + exp(b, -c, N)

where N is applied nitrogen, kg ha-'; Y is biomass yield, Mg ha '; Nu is plant N uptake, kg ha1';
Nc is plant N concentration, g kg-'; Ay is maximum yield at high N, Mg ha-'; An is maximum
plant N uptake at high N; kg ha-1; by is intercept parameter for yield; b, is intercept parameter for
plant N uptake; and c, is response coefficient for applied N, ha kg-'; and Ncm = A/Ay = maximum
plant N concentration at high N, g kg'. Equations (23) and (24) can be combined to derive the
hyperbolic phase relation described by

S= Y, N. (26)
K,, + N,

where Ym is maximum potential yield, Mg ha'; and Kn is response coefficient for plant N uptake,
kg ha'. Equation (26) can be rearranged to the linear form

SN K 1
N,- = -- + -N, (27)
rY Y. Y

Hyperbolic and logistic parameters are related by

A,
Y. = (28)
1 exp(-Ab)


K, = A (29)
exp(Ab) -1

where shift in the intercept parameter is defined by


Ab = b, by


(30)









In order for Ym and K, to be positive requires that Ab > 0. It can be shown from Eqs. (27) through
(29) that the lower limit on plant N concentration (NeI) can be related to model parameters by

NeA = Nc exp(- Ab) (31)

Response data averaged over all harvest intervals are given in Table 5 and shown in Figure 4.
The response curves in Figure 4 are drawn from (Overman and Scholtz, 2002, p. 111)

S= Ay (32)
1 + exp(1.38 0.0077N)

S= An, (33)
1 + exp(2.10 0.0077N)

= N, = 1 + exp(1.38 0.0077N) (34)
c Y l1+ exp(2.10 0.0077N)J

where the parameters are given by Ay (1953) = 23.87 Mg ha-' and Ay (1954) = 12.05 Mg ha'; A,
(1953) = 650 kg ha1 and A, (1954) = 340 kg ha-; Ncm (1953) = 27.2 g kg-' and Ncm (1954) = 28.2
g kg'1. Corresponding phase plots are shown in Figure 5, where the hyperbolic curves are drawn
from


1953: = 46.50N (35)
616 + N


1954: Y- 23.47N. (36)
322 + N

and the lines are drawn from

1953: =13.2 + 0.0215N (37)

1954: =13.7 + 0.0426N. (38)

A summary of parameters for the extended logistic model is given in Table 6. Estimates of
parameters were obtained by nonlinear regression (Overman and Scholtz, 2002, p. 108). It was
shown by analysis of variance that parameters by, bn, and c, were common for all of the data.
Parameter A, was common over harvest intervals for each year, while parameter Ay varied with
harvest interval and year. Correlation of model parameter Ay with harvest interval ( At) is shown
in Figure 6, where the lines are drawn from









1953: A, =11.40+2.96At =11.40(1+0.260At)


1954: Ay = 5.44+ 1.49At = 5.44(1 + 0.274At) (40)

using data through 6 wk harvest interval. Equations (39) and (40) can be combined to form the
overall equation

Both: Ay = A ( + 0.267At) (41)

where Ao depends on water availability (rainfall). Optimum values ofAo for the two years can be
estimated from


1(l+0.267At)Ay
1953: A, = =18706 11.22 Mg ha- (42)
S(1+0.267At)2 16.6715
1=1

( ( + 0.267At)Ay 91.7872
1954: A0 = =5.51 Mgha- (43)
(1+0.267At)2 16.6715
i=1

POWER SERIES ESTIMATE OF THE LOGISTIC MODEL

Perhaps it would be possible to replace the logistic model with a power series representation
to convert the model into linear algebra. Equation (23) can be written in dimensionless form

1
0 = (44)
1+ exp(-)

with the definitions = Y / Ay and = cN by. Recalling that the exponential function can be
expanded in Taylor series (Abramowitz and Stegun, 1965)

e 2 43 (4 _- _
24 nr(45)
exp(-)=1-- + + +--+ +... (45)
1! 2! 3! 4! 5! n!

with a domain of convergence of oo < 4 < +oo, it follows that Eq. (44) can be written as


= 2 + a 4 a5=a2 + a3 + a24 4+ + an, + (46)
1+1 + + 3 4 +
1! 2! 3! 4! 5!


(39)









where it has been assumed that 0 can be written as a power series in . Cross multiplication and
collection of like terms leads to the series

1 1 1 1 1 1 1 1
0 1+ 1 1 3+1 5 1 7 + 1 9 11 + 13 .1+..
24+ +48 _80_ + __--
2 4 48 480 4743.5 46,823.2 462,133.7 4,561,084.6
(47)
1 3 9 )11" + 13
2 (4.000 3.634 3.438 3.351 3.303 3.273 3.253

to as many terms as desired. The coefficients in Eq. (47) are not arbitrary but are calculated by
the sequence of recurrence relations


2ao = 1 -> ao =
2
1
2a, = ao a = +-
4
2a2 a1 a0 2 =0
1! 2!
a2 a, 1
2a3 =- + -> a3=
1! 2! 3! 48
2a4 a3 a2 a 0 a 4
2a =- ---+---- a =0
1! 2! 3! 4!
a4 a3 a2 a1 o 1
2a, = + + -= -> a+ = +--
1! 2! 3! 4! 5! 480
2a6a5 a4 +3 -2 ao =0
2a, = + + a6 =0
1! 2! 3! 4! 5! 6!
a a a 4 a3 a2 a1 a0 1 (
2a=-- -+ --- --a -- (48)
1! 2! 3! 4! 5! 6! 7! 4743.5
2a a7 a6 a4 a3 a a2 a 0
2a- = + + -- -, a, =0
1! 2! 3! 4! 5! 6! 7! 8!
2a9 a a a6 a, a a, a2 a, ao 1
2a, =---+---+ 4 -+ ---+ --) a, =+-
1! 2! 3! 4! 5! 6! 7! 8! 9! 46,823.2
Sa a8 a7 a6 a5 a4 a3 a2 a1 ao
2a1o = + __ a_ __+ a a a a, 0=0
1! 2! 3! 4! 5! 6! 7! 8! 9! 10!
a,,1 a a a a8 a7 a a5 a+ a4 a o aa a1 1
2a, = +---+---+- -- +-- a
1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 11! 462,133.7
al ao a9 a, a0 a6 a5 4 a3 a2 al ao
2a =22a + + -+ + --a12 =0
1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 11! 12!
a12 a,, + ao_ a9 -t aa7 a6 a, 4 a3 -2 a a, + a 3 1
2a13 = 3 4 5 6_ ___- + -> a,3 =1 ,
1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 11! 12! 13! 4,561,085










Note that the series contains only odd powers, i.e. the even order coefficients (a2, a4, .. ) all
vanish. A comparison of the truncated series at the 13th order to the true solution is shown in
Figure 7. It appears that we have succeeded in replacing the logistic equation with a power
series. However, there is the sticky question of the domain of convergence for Eq. (47). It turns
out that the denominator of the series is advancing by approximately nt2 and that the second line
is converging slowly toward i = 3.1415 -, as shown in Figure 8. It has been shown (Lanczos,
1988, p. 441) that the domain of convergence for the series is 7t < < + t, which should be
apparent from the series (Eq. (47)). In fact, outside this domain adding more terms to the series
increases divergence! So, this procedure does not turn out to be very productive.

SUMMARY AND CONCLUSIONS

This exercise should have taught us a powerful lesson beware of truncated power series for
model analysis. Analytical functions (such as the logistic model) are much to be preferred where
possible. In this particular case, the logistic equation represents a well-behaved, bounded,
monotone increasing function which describes response data rather well. Furthermore, it has
allowed us to couple biomass yield and plant nitrogen uptake in a rational way. An added bonus
is that the effects of harvest interval and water availability have been accounted for in the linear
model parameters. The logistic model describes the data rather well. It is well to remember that
the goal of statistical analysis is to draw inferences about the physical system from the limited
data available.
Numerous applications of the model have been presented by Overman and Scholtz (2002). A
rational basis for the logistic model has been given by Overman (1995). Comparison of the
logistic model to quadratic and linear-plateau models has been presented by Overman and
Scholtz (2003).
A survey of applications of mathematics to physics has been given by Devlin (1998), and is
highly recommended to the reader interested in mathematical modeling. A survey of physics has
been published by Roger Newton (Newton, 2007).









References


Abramowitz, M. and I. Stegun. 1965. Handbook ofMathematical Functions. Dover Publications.
New York, NY.
Creel, J.M. Jr. 1957. The effect of continuous high nitrogen fertilization on coastal
bermudagrass. PhD Dissertation. University of Florida. Gainesville, FL.
Devlin, K. 1998. The Language ofMathematics: Making the Invisible Visible. W.H. Freeman &
Co. New York, NY.
Doss, B.D., D.A.Ashley, O.L. Bennet, and R.M. Patterson. 1966. Interaction of soil moisture,
nitrogen, and clipping frequency on yield and nitrogen content of coastal bermudagrass.
Agronomy J. 58:510-512.
Draper, N.R. and H. Smith. 1981. Applied Regression Analysis. John Wiley & Sons. New York,
NY.
Lanczos, C. 1988. Applied Analysis. Dover Publications. New York, NY.
Little, S., J. Vicente, and F. Abruna. 1959. Yield and protein content of irrigated napiergrass,
guineagrass, and pangolagrass as affected by nitrogen fertilization. Agronomy J. 51:111-113.
Newton, R.G. 2007. From Clockwork to Crapshoot: A History of Physics. Harvard University
Press. Cambridge, MA.
Overman, A.R. 1995. Rational basis for the logistic model for forage grasses. J. Plant Nutrition
18:995-1012.
Overman, A.R. and R.V. Scholtz. 2002. Mathematical Models of Crop Growth and Yield. Taylor
& Francis. New York, NY.
Overman, A. R. and R. V. Scholtz. 2003. In defense of the extended logistic model of crop
production. Commun. Soil Sci. and Plant Anal. 34(5&6):851-864.
Prine, G.M. and G.W. Burton. 1956. The effect of nitrogen rate and clipping frequency upon
yield, protein content and certain morphological characteristics of coastal bermudagrass
[Cynodon dactylon (L.) Pers.]. Agronomy J 48:296-301.
Rucker, R. 1987. Mind Tools: The Five Levels of Mathematical Reality. Houghton Mifflin Co.
Boston, MA.









Table 1. Dependence ofbiomass yield (Y), plant nitrogen uptake (N,), and plant nitrogen
concentration (N,) on applied nitrogen (N) and harvest interval (At) for coastal bermudagrass
grown at Tifton. GA (1953).1
At N, kg ha'
wk 0 112 336 672 1008


Y Mg ha-1


2.33 5.96 11.76 17.43 19.71
3.33 8.91 13.64 19.24 20.47
2.71 9.86 17.65 21.68 23.61
4.35 12.77 21.79 28.11 30.11
5.64 13.66 22.38 27.93 29.30

N,, kg ha-1
37 130 327 582 721
51 184 363 579 682
40 176 431 590 739
53 158 392 621 738
62 184 372 545 624

AN, g kg-1


16.0
15.4
14.8
12.1
11.0


21.8
20.6
17.9
12.4
13.5


27.8
26.6
24.4
18.0
16.6


33.4
30.1
27.2
22.1
19.5


36.6
33.3
31.3
24.5
21.3


'Data adapted from Prine and Burton (1956).


__









Table 2. Dependence ofbiomass yield
concentration (Nc) on applied nitrogen
grown at Tiflon, GA (1954).'
At
wk 0 112


(Y), plant nitrogen uptake (Nu), and plant nitrogen
(N) and harvest interval (At) for coastal bermudagrass


N, kg ha'
336


672


1008


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

N,, kg ha1
2 17 70 224 299 320
3 15 75 208 294 364
4 19 80 233 323 344
6 21 92 261 293 390
8 26 100 223 325 417

N., g kg-1


22.6
16.0
17.5
16.4
13.5


26.0
20.5
17.5
14.9
15.5


32.8
28.1
24.7
22.5
18.2


38.2
29.7
29.0
21.6
20.5


37.1
36.4
29.0
27.6
25.7


'Data adapted from Prine and Burton (1956).









Table 3. Estimates of the polynomial coefficients for data in Table 1 for 1953.
At Coefficent
wk


2.3300 34.8604 22.8340 9.0005 3.6581
3.3300 66.9428 -180.1202 257.5893 -127.0316
2.7100 76.9222 -128.165 104.9869 32.8759
4.3500 92.4004 -173.225 182.2483 75.5811
5.6400 87.2401 -155.970 153.1468 60.6880

a, a, a2 a3 a4


37.00
51.00
40.00
53.00
62.00


788.62
1405.03
1117.77
807.23
1183.432


455.84
-2262.13
1325.31
1504.56
- 875.141


- 765.16
3029.62
-4440.21
-3227.74
315.468


203.01
-1539.84
2686.31
1596.71
- 62.767


Table 4. Estimates of the polynomial coefficients for data in Table 2 for 1954.
At Coefficent
wk
ao a1 a2 a3 a4

2 0.7600 13.1431 50.2871 -131.5195 75.7645
3 0.9400 27.0810 -26.9835 11.2272 2.2563
4 1.0800 33.0550 -14.4620 39.6362 31.3643
6 1.3000 51.2531 -74.8376 42.2513 5.8765
8 1.9300 46.4068 -57.9854 36.6706 -10.7798

a, a, a2 a3 a4

2 17.00 248.34 2570.42 -5339.89 2818.75
3 15.00 429.03 1277.68 -3097.90 1734.99
4 19.00 378.65 1920.89 -4163.64 2185.17
6 21.00 409.35 2700.21 -6642.22 3889.74
8 26.00 676.96 60.33 822.74 593.35


---- --









Table 5. Response ofbiomass yield (Y), plant N uptake (N,), and plant N concentration (Ne) to
applied nitrogen (N) averaged over harvest intervals (At) for coastal bermudagrass at Tifton,
GA.'
Year N Y Nu Nc
kg ha-' Mg ha'' kg ha' g kg-1
1953 0 3.67 49 13.4
112 10.23 166 16.2
336 17.44 377 21.6
672 22.88 583 25.5
1008 24.64 700 28.4

1954 0 1.20 20 16.7
112 4.70 83 17.7
336 9.50 230 24.2
672 11.66 307 26.3
1008 12.09 367 30.4
'Data adapted from Prine and Burton (1956).


Table 6. Summary of parameters for the extended logistic model for coastal bermudagrass
response to applied nitrogen and harvest interval at Tifton, GA.1
Year At Ay A, by b, Cn Ym Kn
wk Mg ha' kg ha1' ha kg' Mg ha- kg ha-
1953 2 17.81 650 1.38 2.10 0.0077 34.69 616
3 19.75 38.47
4 23.00 44.80
6 29.38 57.23
8 29.40 57.27
avg. 23.87 46.50

1954 2 8.34 340 1.38 2.10 0.0077 16.24 322
3 9.88 19.25
4 11.59 22.58
6 14.28 27.82
8 16.14 31.44
avg. 12.05 23.47
'Values adapted from Overman and Scholtz (2002, p. 108).









List of Figures


Figure 1. Response of biomass yield (Y), plant nitrogen uptake (Nu), and plant nitrogen
concentration (Nc) to applied nitrogen (N) for harvest interval (At) of 8 wk and year 1953 for
coastal bermudagrass grown at Tifton, GA. Data adapted from Prine and Burton (1956). Curves
drawn from Eqs. (15) through (17) for the polynomial model.

Figure 2. Phase plots ofbiomass yield (Y) and plant nitrogen concentration (Nc) vs. plant
nitrogen uptake (Nu) for harvest interval (At) of 8 wk and year 1953 for coastal bermudagrass
grown at Tifton, GA. Data adapted from Prine and Burton (1956). Curves drawn from Eqs. (15)
through (17) for the polynomial model.

Figure 3. Representations of the function sin x as given by various approximations. Curves
drawn from Eqs. (18) through (22).

Figure 4. Response ofbiomass yield (Y), plant nitrogen uptake (Nu), and plant nitrogen
concentration (Nc) to applied nitrogen (N) averaged over harvest intervals (At) for years 1953
and 1954 for coastal bermudagrass grown at Tifton, GA. Data adapted from Prine and Burton
(1956). Curves drawn from Eqs. (32) through (34) for the extended logistic model.

Figure 5. Phase plots ofbiomass yield (Y) and plant nitrogen concentration (Nc) vs. plant
nitrogen uptake (Nu) averaged over harvest intervals (At) for years 1953 and 1954 for coastal
bermudagrass grown at Tifton, GA. Data adapted from Prine and Burton (1956). Curves drawn
from Eqs. (35) and (36), lines from Eqs. (37) and (38) for the extended logistic model.

Figure 6. Dependence of logistic model parameter for biomass yield (Ay) on harvest interval (At)
for years 1953 and 1954 for coastal bermudagrass grown at Tifton, GA. Lines drawn from Eqs.
(39) and (40) for the extended logistic model.

Figure 7. Comparison of response of dimensionless yield ( ) to dimensionless applied nitrogen
() for the true solution and the 13th order approximation for the logistic model. Solid curve
drawn from Eq. (44); dashed curve drawn from Eq. (47).


Figure 8. Convergence of the logistic coefficients toward 7n.







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