A MEMOIR ON MATHEMATICAL MODELS OF CROP GROWTH
AND YIELD
Effect of Geographic Location
Allen R. Overman
Agricultural and Biological Engineering
University of Florida
Copyright 2008 Allen R. Overman
A Memoir on Mathematical Models of Crop Growth and Yield
Effect of Geographic Location
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 light-gathering 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 field studies from several
geographic locations and planting times to clarify values of model parameters. The phase
relations for the growth model imply that biomass accumulation by photosynthesis is the rate
limiting process in the field studies which have been analyzed, and that accumulation of mineral
elements by the plant proceeds in virtual equilibrium. The document contains 80 pages,
including 101 equations, 27 tables, 40 figures, and 13 references.
Key Words: Mathematical models, biomass accumulation, nutrient accumulation, response plots,
phase plots, perennial grasses, annual crops.
Table of Contents
1. Model Description
1.1 Accumulation of Biomass with Calendar Time
1.2 Accumulation of Plant Nutrients with Calendar Time
1.3 Summary
2. Field Study at Lincoln, New Zealand
2.1 Introduction
2.2 Accumulation of Biomass and Plant P with Calendar Time
2.3 Response of Seasonal Biomass and Plant P to Applied P
2.4 Conclusions
2.5 Tables
2.6 Figures
3. Field Study at R6duit, Mauritius
3.1 Introduction
3.2 Accumulation of Biomass and Plant N with Calendar Time
3.3 Conclusions
3.4 Tables
3.5 Figures
4. Field Study at Paramaribo, Suriname
4.1 Introduction
4.2 Response of Sorghum Growth with Calendar Time
4.3 Response of Groundnut Growth with Calendar Time
4.4 Response of Soybean Growth with Calendar Time
4.5 Conclusions
4.6 Tables
4.7 Figures
5. References
6. Acknowledgment
1. Model Description
1.1 Accumulation of Biomass with Calendar Time
A mathematical model of crop growth has been presented by Overman and Scholtz (2002).
The model describes accumulation ofbiomass with calendar time due to photosynthesis. It
contains three major functions: (1) an energy driving function, (2) a partition function between
light-gathering and structural components of the plant, and (3) an aging function. The rate of
accumulation ofbiomass with time, dY/dt, is written as an ordinary first order linear differential
equation
dY = constant. {[a + b(t t)]exp[- c(t t)]} exp (1.1)
dt cr
in which Y is biomass, Mg ha-'; t is calendar time since Jan. 1, wk; ti is reference time since Jan.
1 for the growth interval, wk; a an intercept parameter, Mg ha'; b is a slope parameter, Mg ha'"
wk-'; c is the aging coefficient, wk''; is time to the mean of the energy distribution, wk; and o
is the time spread of the energy distribution, wk. In Eq. (1.1) the first bracket term represents the
linear partition function, the second bracket term represents the exponential aging function, and
the third bracket term represents the Gaussian energy distribution function. The first two terms
together constitute the intrinsic growth function.
Integration of Eq. (1.1) leads to the simple linear relation
Y = AQ (1.2)
where A is the yield factor for the particular crop, Mg ha-'; and the growth quantifier, Q, is
defined by
Q ={(1- kx)erfx erf x,]-- [exp(- x2 )}exp((-ocx,) (1.3)
in which k = -Job / a is the partition coefficient; and dimensionless time, x, is defined by
t -,u Jo-c
x = +-- (1.4)
0-J 2
Note that x1 corresponds to ti in Eq. (1.4). In Eq. (1.3) the error function is defined by
(Abramowitz and Stegun, 1965)
erf x =
erf x = exp(-u')du (1.5)
0i.F 0
where u is the variable of integration.
1.2 Accumulation of Plant Nutrients with Calendar Time
The next step is to describe accumulation of mineral elements (such as nitrogen, phosphorus,
and potassium) with time. Examination of extensive crop data led Overman and Scholtz (2002)
to assume a hyperbolic phase relation between plant nutrient (N,) and biomass (Y) given by
N Y
N. "- N(1.6)
SK +Y
in which Num is potential maximum plant nutrient uptake, kg ha&'; and Ky is the yield coefficient,
Mg ha-'. Equation (1.6) is easily rearranged to the linear form
Y K, 1
+ Y (1.7)
N, N.,, N..
Data can be used to test the validity of Eq. (1.7) directly. The significance of the phase relation
will be discussed later.
1.3 Summary
The present memoir is part of a continuing series on mathematical models of crop growth
and yield (Overman, 2006a, b, c; 2007a, b, c). My goal has been to present an integrated theory
which brings together fundamental processes, but from an operational perspective. The focus has
been on analytical functions in the classical tradition. While the mathematics may seem
unfamiliar to some readers, it is necessary as the language of scientific analysis and
communication. Data have been drawn extensively from the literature, which constitutes a very
large data base of 150 years of agricultural research from across the world. I have taken care to
reference the source of each set of data, and I am indebted to these many authors for their
outstanding research programs.
1
2. Field Study at Lincoln, New Zealand
2.1 Introduction
Nguyen and Goh (1992) conducted a field study with the growth of perennial ryegrass
(Lolium perenne L.) on Lismore stony silt loam (Udic Ustochrept), a well drained soil, at
Lincoln, New Zealand (S4339' E17229'). Fertilizer treatments included annual applied
phosphorus levels of 0, 17.5, and 35.0 kg P ha-'. Each treatment was replicated four times.
Grazed fields were divided into camp and non-camp areas. Plant samples were collected monthly
between September and May of the 1983-1984 season, and analyzed for biomass (dry matter)
and plant P uptake. This work is part of a long-term study initiated in 1952.
2.2 Accumulation of Biomass and Plant P with Calendar Time
Results are given in Tables 2.1 through 2.6. Sampling times (t) are taken as the end of each
month. Negative and positive numbers indicate times before and after Jan. 1, respectively. Model
parameters are chosen as: u = 0.0 wk, 1ao- = 12.0 wk, c = 0.15 wk-', k = 5. Dimensionless time
(x) and growth quantifier (AQi) are given by
t-,u J c t-0 t+10.8
x + -i- = + 0.90 = 8 (2.1)
V2o 2 12.0 12.0
AQ = {(1-kx)[erfx-erf x ]- -[exp (- x )]} exp( cx ) (2.2)
= {- 5x, )[erf x erfx, ]- 2.82[exp- x2- p- x exp()-exp(-.80x,)
Note that t; is reset at the beginning of each growth interval. Measurements include biomass (A Y)
and plant P uptake (AP,) for each harvest. Values for cumulative growth quantifier (Q),
cumulative yields (Y), and cumulative plant P uptake (Pu) are included in each table. Since A Yi
and AQi for a given growth interval are assumed to be related by
AY, = AAQ~ (2.3)
it follows that cumulative yield (1) and cumulative growth quantifier (Q) are related by
Y = AYi = A AQ, = AQ (2.4)
i i
Linear regression of Y vs. Q leads to an estimate of parameter A. For data in Tables 2.1 through
2.3 this leads to
Camp area P = 0: = 0.021+ 0.838Q r = 0.9965 (2.5)
P= 17.5 kgha': =0.764+2.13Q r= 0.9938 (2.6)
P = 35.0 kg ha-: 1 =0.962 + 2.48Q r =0.9934 (2.7)
with correlation coefficients of r = 0.9965, 0.9938, and 0.9934, respectively, as shown in Figure
2.1. Since plant samples were collected on approximately fixed time intervals (At w 4.9 wk) it
turns out that cumulative plant P uptake and cumulative biomass follows a straight line
Camp area P = 0: P, = -0.050 + 2.50Y r = 0.99965 (2.8)
P= 17.5 kg ha': P = 1.52+3.64Y r = 0.9988 (2.9)
P = 35.0 kg ha': = 2.00 + 4.41Y r = 0.9992 (2.10)
as shown in Figure 2.2. Corresponding values from Tables 2.4 through 2.6 for the non-camp area
leads to
Non-camp area P= 0: = 0.020 + 0.540Q r= 0.9958 (2.11)
P= 17.5 kg ha-: = 0.685 +1.47Q r = 0.9926 (2.12)
P= 35.0 kg ha': = 0.723+1.66Q r = 0.9942 (2.13)
P= 0: P= =0.012+2.36Y r= 0.9996 (2.14)
P= 17.5 kgha': P = 0.79 + 3.50Y r= 0.9991 (2.15)
P = 35.0 kg ha-: P, = 0.73 + 4.55Y r = 0.9995 (2.16)
as shown in Figures 2.3 and 2.4.
Overman and Scholtz (2002) present a theorem which shows that for perennial grass
harvested on a fixed time interval (At) cumulative biomass can be related to calendar time by the
probability equation
S=Y,.F = Y, [ +erf (tI )]
2 IV2-)J
(2.17)
where Y, is total cumulative biomass for the entire growing season. Results are shown in Figure
2.5 for the camp area, where the curves are drawn from
Y, {I1+erft)]} (2.18)
2 12.0
with Yt 5.09, 13.10, and 15.54 Mg ha' for P = 0, 17.5, and 35.0 kg ha', respectively.
2.3 Response of Seasonal Biomass and Plant P to Applied P
The final step is to relate seasonal total yield to applied phosphorus P. Results are given in
Table 2.7. Overman and Scholtz (2002) have shown that response can be described by the
logistic model
Y= A(2.19)
1 + exp(by cP)
P. = (2.20)
1+exp(b cP) (2.20)
P l+ exp(b c ) P)
Pc P = Pcm I P) (2.21)
=Y 1+ exp(b c P)
where Pc=Pu/Y is plant P concentration, g kg-'; Ay is maximum yield at high P, Mg ha'; Ap is
maximum plant P uptake at high P, kg ha'; Pcm = A/Ay is maximum plant P concentration at
high P, g kg-'; by is intercept parameter for yield; bp is intercept parameter for plant P uptake; and
c, is response coefficient for applied P, ha kg''. Note that the units on cp must be the reciprocal of
those on P. It can be shown that the phase relations (Yand Pc vs. Pu) are a consequence of Eqs.
(2.19) and (2.20), and are given by
Y_ P.
Y += (2.22)
KP + P,
p P K+ 1
Pc -- +-P, (2.23)
Y Y,, Y m
where Ym is potential maximum biomass yield, Mg ha'1; and Kp is the response coefficient for
plant P uptake, kg ha1. The validity of Eq. (2.23) can be tested directly from data. Coupling of
hyperbolic and logistic parameters is given by
A
, = A (2.24)
1- exp(-Ab)
A
K = AP
Sexp(Ab)-l
with the shift in intercept parameters defined by
Ab= bp -by
These relations are now used to describe data in Table 2.7. Resulting equations are
Camp area
SAy 16.00
1 + exp(b, cP) 1 + exp(0.70- 0.125P)
Ap 75.0
1 + exp(b, cP) 1+ exp(1.60 0.125P)
P. 1+exp(b -c,P) +exp(0.70-0.125P)
c cm exp(1.60- .12.6
Y 1+ exp(bp cpP) = 1+ exp(1.60 0.125P)
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
S_ YPu
K, +P,
Non-camp area
27.0P,
51.4 + P
S=- _= + 1 P =1.90+0.0370P,
Y Y Y.
A _____=10.95
1+ exp(b, coP) 1 + exp(0.70 0.125P)
SAp 51.1
1 + exp(b, cP) l+exp(l.60 0.125P)
p P1 + exp(b -cP)-
Pc -" =Pcm ep -
S -Y 1+ exp(b, cP)
1+ exp(0.70 0.125P)
4.67 exp.60 0.125P)
1_+ exp(1.60 -0.125P)
18.5P,
35.0 + P
Pf =- =-- + -P = 1.89 + 0.0541P,
r rm r.
(2.30)
(2.31)
(2.32)
(2.33)
(2.34)
(2.35)
(2.36)
KP + P,
Response plots are shown in Figure 2.6, where the curves are drawn from Eqs. (2.27) through
(2.29) and (2.32) through (2.34). The corresponding phase plots are shown in Figure 2.7, where
the curves are drawn from Eqs. (2.30) and (2.35) while the lines are drawn from Eqs. (2.31) and
(2.36).
2.4 Conclusions
The growth model describes the data from Lincoln, New Zealand for perennial ryegrass
rather well. It was necessary to choose p = 0.0 for the extreme southern hemisphere, which
compares to the value / = 26.0 wk for the southern USA. The linear relationship predicted
between biomass accumulation (Y) and the growth quantifier (Q) was shown to hold (Figures 2.1
and 2.3). The relationship between plant P uptake (Pu) and biomass accumulation (Y) was found
to be linear as well (Figures 2.2 and 2.4). Slopes of the lines in Figures 2.2 and 2.4 represent
average plant P concentrations, which is clearly dependent on applied P. It was also shown that
for a perennial grass harvested on a fixed time interval (At) growth curves can be described by
the simple empirical model (Eq. (2.18)).
Response of seasonal total biomass, plant P uptake, and plant P concentration to applied P
was described rather well by the extended logistic model (Figure 2.6). It can be proven that
minimum plant P concentration (Pct) for depleted soil P (negative P) is given by
Pc, = P,, exp(-Ab) = 4.68 exp(-0.90) = 1.90 g kg"' (2.37)
in agreement with Figure 2.7. This conclusion can be deduced from Eq. (2.31) as well. This
analysis explains the dependence of parameter A in the growth model on applied P. It appears
that parameters Pc and Pem are characteristic of plant species. Parameter cp is related to soil
characteristics. Parameters Ay and Ap are dependent upon water availability (Overman and
Scholtz, 2002).
2.5 Tables
Table 2.1. Biomass (Y) and plant phosphorus (Pu) accumulation with calendar time (t) for
perennial ryegrass at P = 0 in the camp area at Lincoln, New Zealand.'
t x erfx exp(-x2) AQ, Q AY Y AP, Pu
wk Mg ha'- Mg ha' kg ha' kg ha-
-17.3 -0.5417 -0.5564 0.7457 0.000 0.00 0.00
0.261 0.15 0.31
-13.0 -0.1833 -0.2012 0.9669 0.261 0.15 0.31
0.596 0.60 1.50
- 8.4 0.2000 0.2227 0.9608 0.857 0.75 1.81
0.870 0.78 1.87
- 4.3 0.5417 0.5564 0.7457 1.727 1.53 3.68
1.160 0.97 2.63
0.0 0.9000 0.7969 0.4449 2.887 2.50 6.31
1.186 1.02 2.44
4.4 1.2667 0.9264 0.2010 4.073 3.52 8.75
1.426 0.76 2.14
13.0 1.9833 0.9950 0.0196 5.499 4.28 10.89
0.264 0.63 1.25
17.3 2.3417 0.99905 0.00416 5.763 4.91 12.14
0.076 0.18 0.38
21.7 2.7083 0.99987 0.00065 5.839 5.09 12.52
IYield data adapted from Nguyen and Goh (1992, table 6).
Table 2.2. Biomass (Y) and plant phosphorus (Pu) accumulation with calendar time (t) for
perennial ryegrass at P = 17.5 kg ha"' in the camp area at Lincoln, New Zealand.'
t x erfx exp(-x2) AQ Q AY Y AP, P,
wk Mg ha'' Mgha-' kg ha'- kg ha
-17.3 -0.5417 -0.5564 0.7457 0.000 0.00 0.00
0.261 0.60 2.20
-13.0 -0.1833 -0.2012 0.9669 0.261 0.60 2.20
0.596 2.36 10.60
- 8.4 0.2000 0.2227 0.9608 0.857 2.96 12.80
0.870 1.73 6.59
- 4.3 0.5417 0.5564 0.7457 1.727 4.69 19.39
1.160 2.08 7.71
0.0 0.9000 0.7969 0.4449 2.887 6.77 27.10
1.186 3.61 12.62
4.4 1.2667 0.9264 0.2010 4.073 10.38 39.72
1.426 1.47 5.16
13.0 1.9833 0.9950 0.0196 5.499 11.85 44.88
0.264 1.11 3.22
17.3 2.3417 0.99905 0.00416 5.763 12.96 48.10
0.076 0.14 0.42
21.7 2.7083 0.99987 0.00065 5.839 13.10 48.52
'Yield data adapted from Nguyen and Goh (1992, table 6).
Table 2.3. Biomass (Y) and plant phosphorus (P,) accumulation with calendar time (t) for
perennial ryegrass at P = 35.0 kg ha'- in the camp area at Lincoln, New Zealand.'
t x erfx exp(-x2) AQi Q AY Y AP, P,
wk Mgha' Mg ha' kg ha' kg ha'
-17.3 -0.5417 -0.5564 0.7457 0.000 0.00 0.00
0.261 0.68 3.42
-13.0 -0.1833 -0.2012 0.9669 0.261 0.68 3.42
0.596 3.15 15.10
- 8.4 0.2000 0.2227 0.9608 0.857 3.83 18.52
0.870 1.75 9.28
- 4.3 0.5417 0.5564 0.7457 1.727 5.58 27.80
1.160 2.18 10.25
0.0 0.9000 0.7969 0.4449 2.887 7.76 38.05
1.186 4.14 15.73
4.4 1.2667 0.9264 0.2010 4.073 11.90 53.78
1.426 1.96 8.80
13.0 1.9833 0.9950 0.0196 5.499 13.86 62.58
0.264 1.46 6.88
17.3 2.3417 0.99905 0.00416 5.763 15.32 69.46
0.076 0.22 0.97
21.7 2.7083 0.99987 0.00065 5.839 15.54 70.43
'Yield data adapted from Nguyen and Goh (1992, table 6).
Table 2.4. Biomass (Y) and plant phosphorus (Pu) accumulation with calendar time (t) for
perennial ryegrass at P = 0 in the non-camp area at Lincoln, New Zealand.'
t x erfx exp(-x2) AQi Q AY Y AP, P
wk Mg ha-' Mgha1' kg ha-' kg ha-
-17.3 -0.5417 -0.5564 0.7457 0.000 0.00 0.00
0.261 0.07 0.14
-13.0 -0.1833 -0.2012 0.9669 0.261 0.07 0.14
0.596 0.40 0.93
- 8.4 0.2000 0.2227 0.9608 0.857 0.47 1.07
0.870 0.52 1.35
- 4.3 0.5417 0.5564 0.7457 1.727 0.99 2.42
1.160 0.70 1.67
0.0 0.9000 0.7969 0.4449 2.887 1.69 4.09
1.186 0.62 1.30
4.4 1.2667 0.9264 0.2010 4.073 2.31 5.39
1.426 0.45 1.21
13.0 1.9833 0.9950 0.0196 5.499 2.76 6.60
0.264 0.38 0.83
17.3 2.3417 0.99905 0.00416 5.763 3.14 7.43
0.076 0.11 0.18
21.7 2.7083 0.99987 0.00065 5.839 3.25 7.61
'Yield data adapted from Nguyen and Goh (1992, table 6).
Table 2.5. Biomass (Y) and plant phosphorus (P,) accumulation with calendar time (t) for
perennial ryegrass at P = 17.5 kg ha' in the non-camp area at Lincoln, New Zealand.'
t x erfx exp(-x2) AQ1 Q AY Y AP, P
wk Mg ha-1 Mg ha' kg ha kg ha-
-17.3 -0.5417 -0.5564 0.7457 0.000 0.00 0.00
0.261 0.38 1.34
-13.0 -0.1833 -0.2012 0.9669 0.261 0.38 1.34
0.596 1.88 7.35
- 8.4 0.2000 0.2227 0.9608 0.857 2.26 8.69
0.870 1.28 5.14
- 4.3 0.5417 0.5564 0.7457 1.727 3.54 13.83
1.160 1.49 5.21
0.0 0.9000 0.7969 0.4449 2.887 5.03 19.04
1.186 2.06 6.80
4.4 1.2667 0.9264 0.2010 4.073 7.09 25.84
1.426 1.13 3.74
13.0 1.9833 0.9950 0.0196 5.499 8.22 29.58
0.264 0.93 2.87
17.3 2.3417 0.99905 0.00416 5.763 9.15 32.45
0.076 0.11 0.34
21.7 2.7083 0.99987 0.00065 5.839 9.26 32.79
'Yield data adapted from Nguyen and Goh (1992, table 6).
Table 2.6. Biomass (Y) and plant phosphorus (P,) accumulation with calendar time (t) for
perennial ryegrass at P = 35.0 kg ha-' in the non-camp area at Lincoln, New Zealand.l
t x erfx exp(-x2) AQi Q AY Y APu Pu
wk Mgha-' Mghal' kg ha-' kg ha
-17.3 -0.5417 -0.5564 0.7457 0.000 0.00 0.00
0.261 0.49 2.20
-13.0 -0.1833 -0.2012 0.9669 0.261 0.49 2.20
0.596 2.03 9.95
- 8.4 0.2000 0.2227 0.9608 0.857 2.52 12.15
0.870 1.40 7.00
- 4.3 0.5417 0.5564 0.7457 1.727 3.92 19.15
1.160 1.62 7.75
0.0 0.9000 0.7969 0.4449 2.887 5.54 26.90
1.186 2.32 9.53
4.4 1.2667 0.9264 0.2010 4.073 7.85 36.43
1.426 1.45 6.66
13.0 1.9833 0.9950 0.0196 5.499 9.30 43.09
0.264 1.04 4.49
17.3 2.3417 0.99905 0.00416 5.763 10.34 47.58
0.076 0.18 0.70
21.7 2.7083 0.99987 0.00065 5.839 10.52 48.28
'Crop data adapted from Nguyen and Goh (1992, table 6).
Table 2.7. Dependence of seasonal total biomass yield (Y), plant P uptake (Pu), and plant P
concentration (Pc) on applied phosphorus (P) for perennial ryegrass at Lincoln, New Zealand.'
Area P Y Pu Pc
kg ha-' Mg ha-' kg ha'' g kg'-
Camp 0.0 5.09 12.5 2.46
17.5 13.10 48.5 3.70
35.0 15.54 70.4 4.53
Non-camp 0.0 3.25 7.61 2.34
17.5 9.26 32.8 3.54
35.0 10.52 48.3 4.59
'Data adapted from Nguyen and Goh (1992).
2.6 Figures
Figure 2.1. Linear correlation between cumulative biomass (Y) and cumulative growth quantifier
(Q) as related to applied phosphorus (P) for perennial ryegrass grown in the camp area of the
pasture study at Lincoln, New Zealand by Nguyen and Goh (1992). Data from Tables 2.1
through 2.3. Lines drawn from Eqs. (2.5) through (2.7).
Figure 2.2. Phase plot between cumulative plant P uptake (Pu) and cumulative biomass (Y) as
related to applied phosphorus (P) for perennial ryegrass grown in the camp area of the pasture
study at Lincoln, New Zealand by Nguyen and Goh (1992). Data from Tables 2.1 through 2.3.
Lines drawn from Eqs. (2.8) through (2.10).
Figure 2.3. Linear correlation between cumulative biomass (Y) and cumulative growth quantifier
(Q) as related to applied phosphorus (P) for perennial ryegrass grown in the non-camp area of
the pasture study at Lincoln, New Zealand by Nguyen and Goh (1992). Data from Tables 2.4
through 2.6. Lines drawn from Eqs. (2.11) through (2.13).
Figure 2.4. Phase plot between cumulative plant P uptake (Pu) and cumulative biomass (Y) as
related to applied phosphorus (P) for perennial ryegrass grown in the non-camp area of the
pasture study at Lincoln, New Zealand by Nguyen and Goh (1992). Data from Tables 2.4
through 2.6. Lines drawn from Eqs. (2.14) through (2.16).
Figure 2.5. Response of cumulative biomass (Y) with calendar time as related to applied
phosphorus (P) for perennial ryegrass grown in the camp area of the pasture study at Lincoln,
New Zealand by Nguyen and Goh (1992). Data from Tables 2.1 through 2.3. Curves drawn from
Eq. (2.18) with Y, = 5.09, 13.10, and 15.54 Mg ha1 forP = 0, 17.5, and 35.0 kg ha1,
respectively.
Figure 2.6. Response of seasonal total biomass (Y), plant P uptake (Pu), and plant P concentration
(Pc) to applied phosphorus (P) for camp and non-camp areas for perennial ryegrass from the
pasture study at Lincoln, New Zealand by Nguyen and Goh (1992). Data from Table 2.7. Curves
drawn from Eqs. (2.27) through (2.29) and (2.32) through (2.34).
Figure 2.7. Phase plots between seasonal total biomass yield (Y) and plant P concentration (Pc)
vs. seasonal total plant P uptake (Pu) for camp and non-camp areas for perennial ryegrass grown
in the pasture study at Lincoln, New Zealand by Nguyen and Goh (1992). Data from Table 2.7.
Curves drawn from Eqs. (2.30) and (2.35); lines from Eqs. (2.31) and (2.36).
12-2442
I'
I,
~1-~
- -'-F-
.1
I..
177-
F
t-. I
.1.
20 Squares to tbe Inch
1/
/ I-,
i i "
7. _..-1 -
;L i" ;:: ::
7, li-nt4
I"
4 -
-7f777
7 I-7, :_ 71 __________
I ~,
jI
1;1 i M
iITBT
7. I I..
71 4<
I,,;
1'~
-V
I,,
LLI
17I
>'5$ K.;
~1'
'I
- -- -- i 2 4
I--- .
*1 f
T 4
1
-[J7
4 -
77, .
Tl
I.
~1~~
-- I-.
'I
.4.-
-I."
411.1
I:
*. F-'
1~
7-1-77
~1
I.
----I
I.
I-.
ii-'
1-
I r ,
I
4..... ;;
-;^ i ,,
41 -
H-
.7
* ~jJ +
12 282
20 Squares to the Inch
IKI K1
- 4-.-
'K
1*
-~ 7
'-I
T-F t -
7,
-,-.-T-
1_77
77-71
j i !- at
iiin iii^ ill
T7
7-77.-7
A' i'I I ; ~ ~ '' a
a'~~~ a-- ,,Ia I
af j f(~ t" I
7K
-i 7
1".l
L.- -
T ,: 1 1 1 1 1f I II iJ1iIr ITd I
.,1
V_
K Ii
^_
i
--I
I
L I i I E m J I
I
I
.'17
L- "'I-'--
A L:
LK
t t
L!I L LIJt=
4-
12-282
TT
1 7~
4-
:..
1I
20 Squares to the Inch
1
4 14
I
7-,.
VA777 h I
177
7
7I I I
7 -7.~
7~ 1- -7T j7 -
'2 T.^:
LII
I]
h!3 ~3
1:: tH 1 ni i
- o'
14
Ii-
1-
'il
I.
F'
*1
7,,,
I-,
"7
T I
-7-IL
i 12T7 ~
1 4 .
7I
H
*1-
-4
'.1
i 14
LL.-
I '
I`7
'1'
I
1
i
12 282
--4-
F''
* ''--I-:
'-K4~i~
1'
~~1~
41
<1
20) Square, to the Inch
-~-1~
''I
7.
)
'F~q Q~4
;J f I F. .
'F .7K i ,F',', ~ 7A
F I ':.................... ,.
77 I
''i' ,
T~~~' .1
F F
7 7 F
'FF.~ ,FL
]'F~~~~ 4'* 7 i .,T F
I 4F
4.,I F.. F F.. 17
4 {7
. F-_.. F ........ .
,4 l .
.1
I I
ff LTI 4 4' l zLIhi 4J 1 -1--F
17-f-
---_77,-
"J
4--
20 Squares to tile Inch
'I
I -
4i 1 11 1
i7i
TIT
.-=2
*1VIF ni T
KIM
MINAl
IT !
.4F71T
H I- M i
KIN
7TT
LIJ
]~ L~
1
* I4-
AN
777-L ~ T I. T
Ell
.4, "
T~Ik
1:)
wk;
7T
j-iT
ITT
71,.
N LJi
* *~iJ.
'ji
[4t~
I-^^
IL
--1-
* Ii
.. ii,^i~~i~
10itt ^-^
t. I -- -i ; -lT'P
- -'h j-t
I i
1-7
vii
CP
I
I
I
12-282
Thi
7-
-:7-7
'n::
LL L-L, LL
II
I-
-'v-i
fi-
--u
Iv]
-"7--i-
WHO
P hi
777
i;
A2
J __ _
1- i -` .r
f / I,
+2,-
f- I
'1
-~ IF I
- 111.11
j j I --
--I.,- 4 .~ -
NOW: n
;Jf
i1.2 HI
'-47I
-7-
17 7-7-
-fit -
!1- I! H
20 Sq uares to thle Inrch
- 1 nI
LW'1''"I.' .
;: :- : .-- "_ ( --- _:
-I ",t
IS Ij 41
Li
. if "
L_-- I
1-4
_. _-p -^
I 1-
I.-FT
f~'
7F:77
-<1
1K (
oilbp
I
-I.
E
t
L
I
t
B
KI I Ib I f -
I 4 1 1 ] i I t
f7 0-~
2~2~
V
K'
(z2k2
r
-1
H'
'1
:0-'
0. ~
201 Squares to t ite I nch
--I
$r7 t
I *r~i
.1
i ,I-
r ';
*
T-1
i t
H 4,
if:T
ite
I-
1:
____ [ "J
i I
I:
J i-
'.2K *
7477 *
T
h
I -
71i21
AI f
1**
I-.
K
:7
7rjti~~7
+
t i t 1 !
. :Tj.I 4 ; L i
F;7 TK;
K I
It i
77
g i
I
I
i
|
3. Field Study at R6duit, Mauritius
3.1 Introduction
Ng Kee Kwong and Deville (1994) conducted a field study with the growth of sugarcane
(Saccharum hybrid spp.) on silty clay soil (Humic Nitosol) at R6duit, Mauritius (S20*13'
E57028'). Nitrogen was applied as ammonium sulphate in a single application of 100 kg N ha-' in
September, 1983. Each treatment was replicated four times. Samples were collected monthly
between September and July of the 1983-1984 season, and analyzed for biomass (dry matter) and
plant N uptake.
3.2 Accumulation of Biomass and Plant N with Calendar Time
Results are given in Table 3.1. Sampling times (t) are taken as the end of each month.
Negative and positive numbers indicate times before and after Jan. 1, respectively. Model
parameters are chosen as: p = 18.4 wk, V2o- = 12.0 wk, c = 0.20 wk~', k = 5. Time of initiation
of significant growth (t;) is assumed to be -8.0 wk. Dimensionless time (x) and growth quantifier
(Q) are given by
t-/u /2orc t-18.4 t-4.0
x t +- = 18.4 +1.20=- 4.0 xi= -1.000 (3.1)
V2 2 12.0 12.0
Q = (1- kx, )[erf x erf x ]- k [ x)-exp((x2 ex(- x )] exp(Vfo-cx)(
I -I J (3.2)
= {(l 5x, )[erf x erf x, ]- 2.82 l[exp(- x2 )- exp(- x2 )} exp(2.40x,)
Correlation of Y with Q in Table 3.1 leads to
f = 0.056 + 34.04Q r = 0.99937 (3.3)
with a correlation coefficient of r = 0.99937, which is shown in Figure 3.1.
The next step is to couple plant N uptake (N,) with accumulated biomass (Y). Following Eq.
(1.7) we obtain
y Ky 1
+ Y = 0.0524 + 0.00725Y r = 0.9972 (3.4)
N NN N,,
Equation (3.4) leads to the hyperbolic phase equation
SNY (3.5 138Y)
K +Y 7.22+Y
Phase plots are shown in Figure 3.2, where the line and curve are drawn from Eqs. (3.4) and
(3.5), respectively. It follows that plant N concentration (N,) can be estimated from
N N u m 138
Y Ky+Y 7.22+Y (3.6)
Response plots are shown in Figure 3.3, where the curves are drawn from Eqs. (3.3), (3.5),
and (3.6). Simulation calculations are given in Table 3.2.
3.3 Conclusions
The mean time of the energy distribution was assumed to be / = 18.4 wk, reflecting the
location south of the equator. Results confirmed the linear relationship between biomass
accumulation (Y) and the growth quantifier (Q) as seen in Figure 3.1. Accumulation of plant N
(N,) with accumulation of biomass (Y) followed the hyperbolic relationship (Figure 3.2). As a
result the growth model described dependence of biomass, plant N uptake, and plant N
concentration with calendar time rather well.
Note from Table 3.2 that maximum growth quantifier has the value Q"" = 1.907. It is easily
shown that for x, = 0.000 maximum growth quantifier shifts to the value Qm = 3.821. This
corresponds to t4 = 4.0 wk. According to this analysis, shifting ti by 12 wk could virtually double
capture of solar energy by the crop.
3.4 Tables
Table 3.1. Accumulation of biomass (Y), plant N uptake (N,), and plant N concentration (Nc)
with calendar time (t) for a single nitrogen application by sugarcane at R6duit, Mauritius (1983-
1984).'
t x erfx exp(-x2) Q Y Nu Nc
wk Mg ha-' kg hal' g kg-'
-8.0 -1.000 -0.8427 0.3679 0.000 ----- ---- -----
-4.4 -0.700 -0.6778 0.6126 0.027 1.5 28 18.9
0.0 -0.333 -0.3620 0.8948 0.127 4.0 50 12.5
4.4 0.033 0.0341 0.9989 0.316 10.0 76 7.6
8.4 0.367 0.3960 0.8742 0.545 19.3 96 5.0
12.9 0.742 0.7060 0.5769 0.789 27.3 106 3.9
17.1 1.092 0.8777 0.3037 0.953 32.0 110 3.4
21.6 1.467 0.9619 0.1164 1.047 36.0 116 3.2
25.9 1.825 0.9901 0.0358 1.083 36.7 118 3.2
30.0 2.167 0.9978 0.00915 1.094 37.3 120 3.2
00 1.0000 0.0000 1.097 ----- ---- -----
'Data adapted from Ng Kee Kwong and Deville (1994, figure 1).
Table 3.2. Simulation of yield (Y), plant N uptake (N,a), and plant N concentration ( N^) with
calendar time (t) for sugarcane at R6duit, Mauritius.
t x erfx exp(-x2) Q N, N,
wk Mg ha-' kg ha'' g kg-'
-8.0 -1.000 -0.8427 0.3679 0.000 0.06 1.2 19.11
-5.0 -0.750 -0.7112 0.5698 0.020 0.74 12.8 17.34
-2.0 -0.500 -0.5205 0.7788 0.070 2.44 34.9 14.29
1.0 -0.250 -0.2763 0.9394 0.162 5.57 60.1 10.79
4.0 0.000 0.0000 1.0000 0.297 10.17 80.7 7.94
7.0 0.250 0.2763 0.9394 0.463 15.82 94.8 5.99
10.0 0.500 0.5205 0.7788 0.637 21.74 103.6 4.77
13.0 0.750 0.7112 0.5698 0.794 27.08 109.0 4.02
16.0 1.000 0.8427 0.3679 0.917 31.27 112.1 3.59
19.0 1.250 0.9229 0.2096 1.002 34.16 113.9 3.33
22.0 1.500 0.9661 0.1054 1.052 35.87 114.9 3.20
25.0 1.750 0.9867 0.0468 1.078 36.75 115.3 3.14
28.0 2.000 0.9953 0.0183 1.090 37.15 115.5 3.11
31.0 2.250 0.9985 0.00633 1.095 37.33 115.6 3.10
34.0 2.500 0.99959 0.00193 1.096 37.37 115.7 3.09
00oo 1.00000 0.00000 1.097 37.40 115.7 3.09
3.5 Figures
Figure 3.1. Correlation of accumulated biomass (Y) with the growth quantifier (Q) for sugarcane
grown at R6duit, Mauritius (1983-1984). Data adapted from Ng Kee Kwong and Deville (1994).
Line drawn from Eq. (3.3).
Figure 3.2. Phase plots of biomass/plant N uptake ratio (Y/N,) and plant N uptake (Nu) vs.
biomass (Y) for sugarcane grown at R6duit, Mauritius (1983-1984). Data adapted from Ng Kee
Kwong and Deville (1994). Line and curve drawn from Eqs. (3.4) and (3.5), respectively.
Figure 3.3. Response of biomass (Y), plant N uptake (Nu), and plant N concentration (Nc) with
calendar time (t) for sugarcane grown at R6duit, Mauritius (1983-1984). Data adapted from Ng
Kee Kwong and Deville (1994). Curves drawn from Eqs. (3.3), (3.5), and (3.6).
Ir
12 0213
-t
.1!
f
F'
.1~7~~
55
17
-0- t -
I.
'7, --1..-.
______________ *1.
2,I
17 -
.7
fI'
!N :.
4F- 7 7
'I
'I'41
SS7
-7-f-7-,-7
-; 5,
<47
747' :I
i t 7~
7
/A^
, TI
t I
I Y4()
K'T
t I
1 T
'5
LL!
7i ;
fI
555..
1~
'.1
5. I
I'
is I
S I
6~t
I'
14 1 ~Ii~
7:7FT- T-75
I
.9
A
-77-1'
1, .. 'A
,- ; ;i
4I
4+~i
/)
517
i 77
4-
201 SquarSIes to( the I nch
I
i I':!
'i-:-I--
I
'V
N- -T7
Milli
J
I
L;
I
I
I I I tI : ;
7T
4-
-:7 :
20 Squares to the Inchl
pI3
1E~
0
.4
I ]
)
-~oI
'V
7 .' \'
* F.
I
'
* F'
- -
I r
.7~~t;Iji
7: i
>nI" il '17
Fj>FI ;2+.
:T'
t, I
4 ______ -
I 1
7 47 77 T
t,
f
'I'
0'
ii
A1~ A~7'
'I
I-.
'I,
'.4
~1'
I'
pL ?,p
-4-^
. -_ ;
I'''
~j~if,
t
ti
7: I
4 .. ...
F'
I:,
'I'
--1-
I.
T1.
t t
,7=
V
7' .1:
'1
-. '~*1
JFi
t
i
4,
1
--4
''I till jl l:
I-.-
'7
J2~
'T.
'1 i
12-2E32
--171
20 -Sq iares to (fieC Inchi
0
/PO
F
0
;L1~)
Q ~
I
~1
I t
313
! : I E ': i ._i
1:^111-111
1
It
--:77T
Ii: :
2-Hii~
il-I,
I I
. . 2
! : ^ | I : ':
l '; ; i : l ;
M
1-. ^ i \ '-'
j
I'
'('~'~
1 4
I. i I
l.'i
)1I
-I-.-.. -. *~ -
7 71
ill.
.-iH
II
-i
It
1L42,
I.
II
II
I I I I I I I
I.
7,-
il
II
''I
,1.
L, ;
... 1 ... .. J .. ... .. .. ... ^ ^ __ _
- 1, -4 -., .
i:
; ,w
L
l
I
I7t~
.1 i-'
II
* I
* I
F
II
* I
K:
iv K
f T`-
4 ip
t
4. Field Study at Paramaribo, Suriname
4.1 Introduction
The growth model is now used to describe results from field studies by Everaarts (1992a,
1992b, 1993) with sorghum (Sorghum bicolor (L.) Moench), groundnut (Arachis hypogaea L.),
and soybean (Glycine max (L.) Merr.) at Paramaribo, Suriname (5 20' N, 55 30' W). Soils were
a sandy loam (first year) and loamy sand (second year). Measurements included accumulated
biomass (Y), plant nitrogen uptake (Nu), and plant nitrogen concentration (Nc), plant phosphorus
uptake (Pu), plant phosphorus concentration (Pc), plant potassium uptake (Ku), and plant
potassium concentration (Kc) for the six sampling times, labeled as days after planting (DAP).
Data from weed-free plots are used for this analysis. Model parameters are chosen as:
p = 32 wk, 2io- = 8.00 wk, c = 0.20 wk-, k = 5 for the growth model. This leads to
dimensionless time (x) and growth quantifier (Q)
t -u or-c t 32 0.80t 25.6 (4.1)
x -+- = -- +0.80=- (4.1)
V2o- 2 8.00 8.00
Q = {(1 kx, )erf x erf x, ]- [exp(- x2 )- exp(x )] exp(ocx,) (4.2)
= {(1 5x,)[erf x erf x, ]- 2.82 l[exp(- x2 )-exp(x,2 )]-exp(.60x,)
Equations (4.1) and (4.2) apply for both crop species.
4.2 Response of Sorghum Growth with Calendar Time
Data are given in Table 4.1 for 1982 and Table 4.2 for 1983 for sorghum. Planting times were
1982 July 16 and 1983 June 14. Using the dates of planting times, sampling data can be
converted to calendar time (t) as given in Tables 4.3 and 4.4. For 1982 the reference time for
significant growth after planting is chosen as t, = 30.9 wk, which leads to the values listed in
Table 4.3. The first test is for linearity between Y and Q as assumed in the model. Linear
regression leads to
f =-0.028 +3.111Q r = 0.9941 (4.3)
as shown in Figure 4.1. Note that the intercept is essentially zero, which means that the choice of
ti is correct. The second test is for linearity between Y/N, and Y for the phase relation. Linear
regression leads to
Y Kv 1
Ky + Y = 0.0252 + 0.00762Y r = 0.9903 (4.4)
N NN N,
i-
NuY 131Y (45)
K+Y 3.37+Y
as shown in Figure 4.2. Note that plant N concentration Nc is described by
131
Y 3.37 + Y
Response plots are shown in Figure 4.3, where the curves are drawn from Eqs. (4.3), (4.5), and
(4.6). Phase relations are now established for phosphorus and potassium. For 1982 these relations
are
y K, 1
- = -- + Y = 0.252 + 0.0335Y r = 0.9734 (4.7)
P. P, Pu.
P Y =" 29.8Y (4.8)
"Ky+Y 7.52+Y
y K, 1
y-=- K + Y = 0.0147 +0.00537Y r = 0.9716 (4.9)
Ku Kum Kum
S Kum = 186 (4.10)
K +Y 2.74+Y
The plots for phosphorus are shown in Figure 4.4, where the line and curve are drawn from Eqs.
(4.7) and (4.8), respectively. Plots for potassium are shown in Figure 4.5, where the line and
curve are drawn from Eqs. (4.9) and (4.10), respectively.
For 1983 the reference time is chosen as ti = 26.3 wk, which leads to the values in Table 4.4.
Linear correlation of Y with Q is shown in Figure 4.6, where the line is drawn from
f= -0.032+3.361Q r = 0.9983 (4.11)
with an intercept of essentially zero. Linear regression of Y/Nu vs. Y leads to
Y K 1y
-y = + Y = 0.0207 + 0.00799Y r = 0.9895 (4.12)
N NN N,
which leads to the hyperbolic relation
N, Y 125Y
-NuY 125Y (4.13)
KY+Y 2.59+ Y
as shown in Figure 4.7. Plant N concentration (Nc) is described by
-N, 125
N Y -2.59+Y (4.14)
Y 2.59 + Y
Response curves are shown in Figure 4.8, where the curves are drawn from Eqs. (4.11), (4.13),
and (4.14). Phase relations are now established for phosphorus and potassium. For 1983 these
relations are
y K 1
-=-- + -- Y =0.198+0.0213Y r 0.9492 (4.15)
P. P. u P.
P_ P _= 47.Y (4.16)
K + Y 9.31+Y
y Ky 1
+IY = 0.0163+0.00539Y r= 0.9880 (4.17)
Ku Kur Kum
yk KumY_ 186Y
Ku~ (418)
KY+Y 3.02+Y
The plots for phosphorus are shown in Figure 4.9, where the line and curve are drawn from Eqs.
(4.15) and (4.16), respectively. Plots for potassium are shown in Figure 4.10, where the line and
curve are drawn from Eqs. (4.17) and (4.18), respectively.
4.3 Response of Groundnut Growth with Calendar Time
Data are given in Table 4.7 for 1982 and Table 4.8 for 1982-1983 for groundnut. Planting times
were 1982 June 26 and 1982 December 15. Using the dates of planting times, sampling data can
be converted to calendar time (t) as given in Tables 4.9 and 4.10. For 1982 the reference time for
significant growth after planting is chosen as ti = 27.4 wk, which leads to the values listed in
Table 4.9. The first test is for linearity between Y and Q as assumed in the model. Linear
regression leads to
Y = 0.035+2.113Q r = 0.9739 (4.19)
as shown in Figure 4.11. Note that the intercept is essentially zero, which means that the choice
of ti is correct. The second test is for linearity between Y/Nu and Y for the phase relation. Linear
regression leads to
Y K + 1
- = + -- Y = 0.0235 + 0.00221Y r = 0.9575 (4.20)
N. NN N ,m
N,- NY 452Y (4.21)
Ky + Y 10.62+Y ( )
as shown in Figure 4.12. Note that plant N concentration Nc is described by
N 452 (4.22)
Y 10.62+ Y
Response plots are shown in Figure 4.13, where the curves are drawn from Eqs. (4.19), (4.21),
and (4.22). Phase relations are now established for phosphorus and potassium. For 1982 these
relations are
Y K, 1
-=y -- +- Y = 0.239+0.0404Y r= 0.9609 (4.23)
P. P. Pu
_ P Y 24.7Y
K +K Y 5.91+Y
Y Ky l1=
-= +- Y = 0.0282 + 0.00528Y r = 0.9729 (4.25)
Ku Kum Kum
I= KumY = 189Y (4.26)
K +Y 5.34+Y
The plots for phosphorus are shown in Figure 4.14, where the line and curve are drawn from Eqs.
(4.23) and (4.24), respectively. Plots for potassium are shown in Figure 4.15, where the line and
curve are drawn from Eqs. (4.25) and (4.26), respectively.
For 1982-1983 the reference time is chosen as ti = -0.4 wk and i = 6.0 wk, which leads to
dimensionless time of
S- t.0 +0.80 = 0 xi= 0.000 (4.27)
2o- 2 8.00 8.00
and which leads to the values in Table 4.10. Linear correlation of Y with Q is shown in Figure
4.16, where the line is drawn from
f = 0.038 + 2.254Q r = 0.9978 (4.28)
with an intercept of essentially zero. Linear regression of Y/N, vs. Y leads to
- = -K- +-- = 0.0221+0.00162Y r= 0.7468 (4.29)
N. Num N,,
which leads to the hyperbolic relation
N u N 617Y (4.30)
Ky + Y 13.62+Y
as shown in Figure 4.17. Plant N concentration (Nc) is described by
N 617 (4.31)
Y 13.62 + Y
Response curves are shown in Figure 4.18, where the curves are drawn from Eqs. (4.28), (4.30),
and (4.31). Phase relations are now established for phosphorus and potassium. For 1982-1983
these relations are
y K 1
--- + Y = 0.241+0.0261Y r= 0.9156 (4.32)
P. P. P,
SP, Y 38.3Y (4.33)
SK + Y 9.24+Y
-= +- Y = 0.0247 +0.00416Y r= 0.9858 (4.34)
Ku Kum Kum
SKum 240Y (435)
SKy + Y 5.93+Y
The plots for phosphorus are shown in Figure 4.19, where the line and curve are drawn from Eqs.
(4.32) and (4.33), respectively. Plots for potassium are shown in Figure 4.20, where the line and
curve are drawn from Eqs. (4.34) and (4.35), respectively.
4.4 Response of Soybean Growth with Calendar Time
Data are given in Table 4.13 for 1982 and Table 4.14 for 1982-1983 for soybean. Planting
times were 1982 June 26 and 1982 December 15. Using the dates of planting times, sampling
data can be converted to calendar time (t) as given in Tables 4.15 and 4.16. For 1982 the
reference time for significant growth after planting is chosen as ti = 28.9 wk, which leads to the
values listed in Table 4.15. The first test is for linearity between Y and Q as assumed in the
model. Linear regression leads to
S= 0.063+1.895Q r= 0.9951
(4.36)
as shown in Figure 4.21. The point at t = 36.9 wk has been omitted from regression. Note that the
intercept is essentially zero, which means that the choice of t is correct. The second test is for
linearity between Y/Nu and Y for the phase relation. Linear regression leads to
=-- K + 1Y = 0.0280 + 0.00222Y r= 0.9298 (4.37)
N, Nu. N..
NY 450 (4.38)
Ky + Y 12.59+Y
as shown in Figure 4.22. Note that plant N concentration Nc is described by
N = 450 (4.39)
Y 12.59 +Y
Response plots are shown in Figure 4.23, where the curves are drawn from Eqs. (4.36), (4.38),
and (4.39). Phase relations are now established for phosphorus and potassium. For 1982 these
relations are
-y + Y =0.325+0.0145Y r= 0.8893 (4.40)
Pu Purn urn
PurmY 69.1Y (4.41)
Kx +Y -22.47 + Y
y Ky 1
-- + Y = 0.0308 + 0.00446Y r = 0.9215 (4.42)
Ku Kum Kum
k_ KumY 224Y
Ky+Y 6.90+Y
The plots for phosphorus are shown in Figure 4.24, where the line and curve are drawn from Eqs.
(4.40) and (4.41), respectively. Plots for potassium are shown in Figure 4.25, where the line and
curve are drawn from Eqs. (4.42) and (4.43), respectively.
For 1982-1983 the reference time is chosen as t1 = 1.1 wk and p = 6.0 wk, which leads to
dimensionless time of
t- i -Lov t 6.0 t + 0.4
x -6+- +0.80= 0 x= 0.1875 (4.44)
Vo- 2 8.00 8.00
and which leads to the values in Table 4.16. Linear correlation of Y with Q is shown in Figure
4.26, where the line is drawn from
Y = -0.056 +1.972Q r = 0.9977 (4.45)
with an intercept of essentially zero. The value at t = 13.7 wk has been omitted from regression
Linear regression of Y/N,, vs. Y leads to
Y K
-= +- Y = 0.0253+0.00432Y r= 0.7699 (4.46)
N. Num Num
which leads to the hyperbolic relation
N_ NmY 232Y (4.47)
Ky+Y 5.86+Y
as shown in Figure 4.27. Plant N concentration (Nc) is described by
_N 232
Y 5.86 +Y
Response curves are shown in Figure 4.28, where the curves are drawn from Eqs. (4.45), (4.47),
and (4.48). Phase relations are now established for phosphorus and potassium. For 1982-1983
these relations are
-y Ky + -- y=0.212 +0.0394Y r= 0.9545 (4.49)
PT P.. P.
P Y 25.4Y
-. = Y 25.4Y (4.50)
Ky+Y 5.39+Y
Y K 1l
= -- + Y = 0.0265 + 0.00419Y r = 0.9731 (4.51)
Ku Kum Kum
k KumY 239Y (4.52)
Ky +Y 6.32+Y
The plots for phosphorus are shown in Figure 4.29, where the line and curve are drawn from Eqs.
(4.49) and (4.50), respectively. Plots for potassium are shown in Figure 4.30, where the line and
curve are drawn from Eqs. (4.51) and (4.52), respectively.
4.5 Conclusions
This analysis confirms the growth model for sorghum, groundnut, and soybean. Linear
correlation of biomass accumulation with the growth quantifier is demonstrated. The hyperbolic
phase relation between plant accumulation of mineral elements (N, P, and K) and biomass is
established for all three crop species.
It is also shown that the value of model parameter u is dependent on time of planting
(midyear or December) as well as location near the equator. For plantings in June or July p = 32
wk was assumed, while for December planting p = 6.0 wk was assumed. The difference of 26 wk
appears reasonable.
Data for soybean shows a decrease in biomass for the last sampling time. This is due to
senescence in the plant following maturity.
4.6 Tables
Table 4.1. Accumulation of biomass yield (Y), plant N uptake (AN), plant N concentration (NA),
plant P uptake (Pu), plant P concentration (Pc), plant K uptake (Ku), and plant K concentration
(Kc) with days after planting (DAP) for sorghum (with weed control) at Paramaribo, Suriname
(1982).'
DAP Y Nu Nc P, Pc Ku Kc
Mg ha- kg ha' g kg-' kg ha'' g kg'' kg ha'' g kg-1
19 0.1 6 45.7 0.55 4.3 6 50.4
33 1.9 41 21.9 5.5 2.9 88 46.8
54 6.2 85 13.6 13.7 2.2 153 24.5
68 8.0 92 11.4 14.7 1.8 140 17.4
82 8.1 90 11.2 15.5 1.9 128 15.9
89 8.5 99 11.6 16.8 2.0 137 16.1
'Data adapted from Everaarts (1993, tables 4 and 5).
Table 4.2. Accumulation of biomass yield (Y), plant N uptake (N,), plant N concentration (N,),
plant P uptake (Pu), plant P concentration (Pc), plant K uptake (Ku), and plant K concentration
(Kc) with days after planting (DAP) for sorghum (with weed control) at Paramaribo, Suriname
(1983).'
DAP Y Nu Nc Pu Pc K& Kc
Mg ha-1 kg ha-' g kg-' kg ha-' g kg' kg ha'1 g kg-1
15 0.02 1 48.3 0.1 6.5 1 50.9
29 0.86 30 35.0 3.5 4.1 39 44.5
43 3.29 77 23.6 12.2 3.7 110 33.4
56 6.32 85 13.3 17.3 2.7 141 22.3
77 9.44 92 9.7 23.2 2.5 135 14.3
91 10.80 108 10.0 27.4 2.5 141 13.1
'Data adapted from Everaarts (1993, tables 4 and 5).
Table 4.3. Accumulation of biomass yield (Y), plant N uptake (N,), and plant N concentration
(N) with calendar time (t) for sorghum (with weed control) at Paramaribo, Suriname (1982).'
t x erfx exp(-x2) Q Y NA Nc
wk Mg ha-' kg ha g kg-'
28.1 planting
30.9 0.6625 0.6602 0.6447 0.000 0.1 6 45.7
32.9 0.9125 0.8030 0.4349 0.755 1.9 41 21.9
35.9 1.2875 0.9314 0.1906 1.887 6.2 85 13.6
37.9 1.5375 0.9702 0.0940 2.415 8.0 92 11.4
39.9 1.7875 0.9885 0.0410 2.724 8.1 90 11.2
40.9 1.9125 0.9939 0.0258 2.812 8.5 99 11.6
00 1.0000 0.0000 2.981 --- --- -----
'Data adapted from Everaarts (1993, tables 4 and 5).
Table 4.4. Accumulation of biomass yield (Y), plant N uptake (Nu), and plant N concentration
(Nc) with calendar time (t) for sorghum (with weed control) at Paramaribo, Suriname (1983).'
t x erfx exp(-x2) Q Y N, Nc
wk Mg ha-' kg ha'' g kg-'
23.6 planting
26.3 0.0875 0.0980 0.9924 0.000 ---
27.7 0.2625 0.2895 0.9334 0.315 0.86 30 35.0
29.7 0.5125 0.5314 0.7690 1.005 3.29 77 23.6
31.6 0.7500 0.7112 0.5698 1.768 6.32 85 13.3
34.6 1.1250 0.8884 0.2821 2.816 9.44 92 9.7
36.6 1.3750 0.9481 0.1510 3.280 10.80 103 10.0
00oo 1.0000 0.0000 3.804 ------
IData adapted from Everaarts (1993, tables 4 and 5).
Table 4.5. Simulation of biomass yield (f), plant N uptake (.N), and plant N concentration
( c) with calendar time (t) for sorghum (with weed control) at Paramaribo, Suriname (1982).
t x erfx exp(-x2) Q Y NR N,
wk Mg ha' kg ha' g kg'
28.1 planting
30.9 0.6625 0.6602 0.6447 0.000 0.00 0 38.9
32.0 0.8000 0.7421 0.5273 0.409 1.24 35.2 28.4
33.0 0.9250 0.8092 0.4250 0.794 2.44 55.0 22.5
34.0 1.0500 0.8624 0.3320 1.197 3.70 68.6 18.5
35.0 1.1750 0.9032 0.2514 1.580 4.89 77.6 15.9
36.0 1.3000 0.9340 0.1845 1.920 5.95 83.6 14.1
37.0 1.4250 0.9560 0.1313 2.206 6.84 87.8 12.8
38.0 1.5500 0.9716 0.0905 2.434 7.55 90.6 12.0
39.0 1.6750 0.9814 0.0605 2.613 8.10 92.5 11.4
40.0 1.8000 0.9891 0.0392 2.735 8.48 93.7 11.1
41.0 1.9250 0.9935 0.0246 2.824 8.76 94.6 10.8
42.0 2.0500 0.9961 0.0150 2.885 8.95 95.2 10.6
44.0 2.3000 0.99886 0.0050 2.948 9.15 95.7 10.5'
00oo 1.0000 0.0000 2.981 9.25 96.0 10.4
Table 4.6. Simulation of biomass yield (Y), plant N uptake (^N), and plant N concentration
(N,) with calendar time (t) for sorghum (with weed control) at Paramaribo, Suriname (1983).'
t x erfx exp(-x2) Q Y N, Nc
wk Mg ha'' kg ha'' g kg-'
26.3 0.0875 0.0980 0.9924 0.000 0.000 0 48.3
27.0 0.175 0.1950 0.9698 0.136 0.425 17.6 41.5
28.0 0.300 0.3286 0.9139 0.404 1.33 42.4 31.9
29.0 0.425 0.4520 0.8347 0.741 2.46 60.9 24.8
30.0 0.550 0.5633 0.7390 1.123 3.74 73.9 19.7
31.0 0.675 0.6602 0.6340 1.527 5.10 82.9 16.3
32.0 0.800 0.7421 0.5273 1.926 6.44 89.1 13.8
33.0 0.925 0.8092 0.4250 2.301 7.70 93.5 12.1
34.0 1.050 0.8624 0.3320 2.638 8.83 96.7 10.9
35.0 1.175 0.9032 0.2514 2.925 9.80 98.9 10.1
36.0 1.300 0.9340 0.1845 3.162 10.60 100.5 9.48
37.0 1.425 0.9560 0.1313 3.349 11.22 101.6 9.05
38.0 1.550 0.9716 0.0905 3.492 11.70 102.3 8.75
39.0 1.675 0.9822 0.0605 3.596 12.05 102.9 8.54
40.0 1.800 0.9891 0.0392 3.670 12.30 103.3 8.39
41.0 1.925 0.9935 0.0246 3.720 12.47 103.5 8.30
42.0 2.050 0.9962 0.0150 3.753 12.58 103.7 8.24
43.0 2.175 0.9979 0.0088 3.774 12.65 103.8 8.20
44.0 2.300 0.9989 0.0050 3.787 12.70 103.8 8.18
00oo 1.0000 0.0000 3.804 12.75 103.9 8.15
I
Table 4.7. Accumulation of biomass yield (Y), plant N uptake (Nu), plant N concentration (Ny),
plant P uptake (Pu), plant P concentration (Pc), plant K uptake (Ku), and plant K concentration
(Kc) with days after planting (DAP) for groundnut (with weed control) at Paramaribo, Suriname
(1982).'
DAP Y Nu Nc Pu Pc Ku Kc
Mg ha-' kg ha' g kg-' kg ha-1 g kg-' kg hal' g kg-'
17 0.16 7 38.1 0.8 4.4 5 30.4
31 0.96 37 38.4 3.0 3.2 33 34.2
53 4.68 131 28.0 10.3 2.2 83 17.7
66 6.67 180 27.0 13.3 2.0 107 16.0
80 6.28 157 25.0 12.3 2.0 110 17.8
95 6.68 187 27.0 14.0 2.1 100 14.9
'Data adapted from Everaarts (1992a, tables 4 and 5).
Table 4.8. Accumulation of biomass yield (1), plant N uptake (N,,), plant N concentration (Nc),
plant P uptake (P,), plant P concentration (Pc), plant K uptake (Ku), and plant K concentration
(Kc) with days after planting (DAP) for groundnut (with weed control) at Paramaribo, Suriname
(1982-1983).'
DAP Y Nu Nc Pu Pc K. Kc
Mg ha-' kg ha'' g kg-' kg ha'1 g kg-' kg ha'1 g kg'
13 0.09 5 54.8 0.4 4.8 3 35.8
27 0.64 28 43.8 2.4 3.7 25 39.2
49 3.81 106 27.8 10.3 2.7 98 25.8
69 5.92 173 29.2 13.7 2.3 120 20.4
83 7.05 213 30.2 16.7 2.3 137 19.1
92 7.58 260 34.3 20.0 2.6 127 16.9
'Data adapted from Everaarts (1992a, tables 4 and 5).
Table 4.9. Accumulation of biomass yield (Y), plant N uptake (Nu), plant N concentration (Nc)
with calendar time (t) for groundnut (with weed control) at Paramaribo, Suriname (1982).'
t x erfx exp(-x2) Q Y N, Nc
wk Mg ha'' kg ha-' g kg''
25.3 planting
27.4 0.2250 0.2500 0.9506 0.000 ----- ---- ----
27.7 0.2625 0.2895 0.9334 0.062 0.16 7 38.1
29.7 0.5125 0.5314 0.7690 0.684 0.96 37 38.4
32.9 0.9125 0.8031 0.4349 1.986 4.68 131 28.0
34.7 1.1375 0.8924 0.2742 2.620 6.67 180 27.0
36.7 1.3875 0.9503 0.1459 3.128 6.28 157 25.0
38.9 1.6625 0.9813 0.0630 3.458 6.68 187 28.0
00 1.0000 0.0000 3.709 --- ---- ---
'Data adapted from Everaarts (1992a, tables 4 and 5).
Table 4.10. Accumulation of biomass yield (1), plant N uptake (N,), and plant N concentration
(Nc) with calendar time (t) for groundnut (with weed control) at Paramaribo, Suriname (1982-
1983).'
t x erfx exp(-x2) Q Y N, Nc
wk Mg ha-' kg ha1' g kg-'
-2.3 planting
-0.4 0.0000 0.0000 1.0000 0.000 0.09 5 54.8
1.6 0.2500 0.2227 0.9394 0.394 0.64 28 43.8
4.7 0.6375 0.5939 0.6660 1.536 3.87 106 27.8
7.6 1.0000 0.8209 0.3679 2.604 5.92 173 29.2
9.6 1.2500 0.9103 0.2096 3.140 7.05 213 30.2
10.9 1.4125 0.9460 0.1360 3.383 7.58 260 34.3
00 1.0000 0.0000 3.821 -- ---- -----
'Data adapted from Everaarts (1992a, tables 4 and 5).
Table 4.11. Simulation of biomass yield (Y), plant N uptake ( N), plant N concentration (N )
with time after planting (t) for groundnut (with weed control) at Paramaribo, Suriname (1982).
t x erfx exp(-x2) Q fY N
wk Mg ha'1 kg ha'1 g kg'1
25.3 planting
27.4 0.225 0.2500 0.9506 0.000 0.00 0.0 42.6
29.0 0.425 0.4522 0.8347 0.432 0.95 37.1 39.1
30.0 0.550 0.5633 0.7390 0.799 1.72 63.0 36.6
31.0 0.675 0.6602 0.6341 1.206 2.58 88.3 34.2
32.0 0.800 0.7421 0.5273 1.623 3.46 111.1 32.1
33.0 0.925 0.8092 0.4250 2.025 4.31 130.5 30.3
34.0 1.050 0.8624 0.3320 2.392 5.09 146.4 28.8
35.0 1.175 0.9034 0.2514 2.710 5.76 158.9 27.6
36.0 1.300 0.9340 0.1845 2.975 6.32 168.6 26.7
37.0 1.425 0.9560 0.1313 3.186 6.77 176.0 26.0
38.0 1.550 0.9716 0.0905 3.354 7.12 181.4 25.5
39.0 1.675 0.9822 0.0605 3.468 7.36 185.0 25.1
40.0 1.800 0.9891 0.0392 3.553 7.54 187.7 24.9
41.0 1.925 0.9935 0.0246 3.611 7.66 189.4 24.7
42.0 2.050 0.9961 0.0150 3.649 7.75 190.7 24.6
43.0 2.175 0.9979 0.00882 3.674 7.80 191.4 24.5
44.0 2.300 0.99886 0.00504 3.689 7.83 191.8 24.5
45.0 2.425 0.99940 0.00279 3.698 7.85 192.1 24.5
00 1.0000 0.0000 3.709 7.87 192.4 24.4
Table 4.12. Simulation of biomass yield (1), plant N uptake (A ), and plant N concentration
(AN ) with calendar time (t) for groundnut (with weed control) at Paramaribo, Suriname (1982-
1983).
t x erfx exp(-x2) Q c N ,
wk Mg ha'- kg ha-' g kg'
-2.3 planting
-0.4 0.000 0.0000 1.0000 0.000 0.04 1.81 45.3
0.0 0.050 0.0564 0.9975 0.063 0.18 8.05 44.7
1.0 0.175 0.1950 0.9698 0.280 0.67 28.9 43.2
2.0 0.300 0.3286 0.9139 0.571 1.32 54.5 41.3
3.0 0.425 0.4522 0.8348 0.918 2.11 82.8 39.2
4.0 0.550 0.5633 0.7390 1.300 2.97 110.5 37.2
5.0 0.675 0.6602 0.6340 1.693 3.85 136.0 35.3
6.0 0.800 0.7421 0.5273 2.076 4.72 158.8 33.6
7.0 0.925 0.8092 0.4250 2.431 5.52 177.9 32.2
8.0 1.050 0.8624 0.3320 2.747 6.23 193.6 31.1
9.0 1.175 0.9034 0.2514 3.015 6.83 206.1 30.2
10.0 1.300 0.9340 0.1845 3.235 7.33 215.9 29.5
11.0 1.425 0.9562 0.1312 3.407 7.72 223.2 28.9
12.0 1.550 0.9716 0.0905 3.537 8.01 228.5 28.5
13.0 1.675 0.9822 0.0605 3.633 8.23 232.4 28.2
14.0 1.800 0.9891 0.0392 3.700 8.38 235.0 28.0
15.0 1.925 0.9935 0.0246 3.745 8.48 236.7 27.9
16.0 2.050 0.9961 0.0150 3.775 8.55 237.9 27.8
17.0 2.175 0.9979 0.00882 3.794 8.59 238.6 27.8
18.0 2.300 0.99886 0.00504 3.806 8.62 239.1 27.7
19.0 2.425 0.99938 0.00279 3.813 8.63 239.3 27.7
20.0 2.550 0.99967 0.00150 3.816 8.64 239.5 27.7
00 1.0000 0.0000 3.821 8.65 239.7 27.7
Table 4.13. Accumulation of biomass yield (Y), plant N uptake (Nu), plant N concentration (Nc),
plant P uptake (Pu), plant P concentration (Pc), plant K uptake (Ku), and plant K concentration
(Kc) with days after planting (DAP) for soybean (with weed control) at Paramaribo, Suriname
(1982).1
DAP Y Nu Nc Pu Pc Ku Kc
Mg ha-1 kg ha-' g kg-1 kg ha-' g kg-1 kg ha'' g kg"'
25 0.2 7 41.3 0.6 3.5 6 36.0
46 1.8 58 32.7 5.2 3.0 50 28.7
60 3.7 102 27.4 9.7 2.6 85 22.6
67 4.4 122 27.7 10.9 2.5 86 19.5
81 3.7 96 26.1 10.3 2.8 72 19.6
'Data adapted from Everaarts (1992b, tables 4 and 5).
Table 4.14. Accumulation of biomass yield (Y), plant N uptake (Nu), plant N concentration (Nc),
plant P uptake (Pu), plant P concentration (Pc), plant K uptake (Ku), and plant K concentration
(Kc) with days after planting (DAP) for soybean (with weed control) at Paramaribo, Suriname
(1982-1983).'
DAP Y Nu Nc Pu Pc Ku Kc
Mg hal kg ha'' g kg-' kg ha'' g kg-' kg hal g kg'1
14 0.05 3 60.5 0.2 5.6 2 38.5
28 0.36 16 44.0 1.4 3.8 12 32.4
48 2.00 44 22.0 7.1 3.2 57 28.5
56 2.94 62 21.1 9.1 3.2 83 28.0
70 4.73 104 22.0 11.1 2.7 106 22.5
90 6.22 140 22.5 15.0 2.1 113 18.1
112 4.71 139 29.5 15.0 3.0 82 17.3
'Data adapted from Everaarts (1992b, tables 4 and 5).
Table 4.15. Accumulation of biomass yield (Y), plant N uptake (Nu), plant N concentration (Nc)
with calendar time (t) for soybean (with weed control) at Paramaribo, Suriname (1982).'
t x erfx exp(-x2) Q Y N. Nc
wk Mg ha'' kg ha'' g kg'l
25.3 planting
28.9 0.4125 0.4400 0.8435 0.000 0.2 7 41.3
31.9 0.7875 0.7345 0.5379 1.063 1.8 58 32.7
33.9 1.0375 0.8575 0.3408 1.886 3.7 102 27.4
34.9 1.1625 0.8998 0.2589 2.246 4.4 122 27.7
36.9 1.4125 0.9542 0.1360 2.805 3.7 96 26.1
0o 1.0000 0.0000 3.453 --- ---- -----
IData adapted from Everaarts (1992b, tables 4 and 5).
Table 4.16. Accumulation of biomass yield (Y), plant N uptake (Nu), and plant N concentration
(Nc) with calendar time (t) for soybean (with weed control) at Paramaribo, Suriname (1982-
1983).'
t x erfx exp(-x2) Q Y Nu Nc
wk Mg ha-' kg ha'' g kg-
-2.3 planting
-0.3 -0.0500 0.05 3
1.1 0.1875 0.2084 0.9655 0.000 ---
1.7 0.2625 0.2894 0.9334 0.129 0.36 16 44.0
4.6 0.6250 0.6234 0.6766 1.135 2.00 44 22.0
5.7 0.7625 0.7192 0.5591 1.591 2.94 62 21.0
7.7 1.0125 0.8478 0.3587 2.365 4.73 104 22.0
10.6 1.3750 0.9481 0.1510 3.164 6.22 140 22.5
13.7 1.7625 0.9873 0.0448 3.572 4.71 139 29.5
00 1.0000 0.0000 3.743 ---- -----
'Data adapted from Everaarts (1992b, tables 4 and 5).
Table 4.17. Simulation of biomass yield (Y), plant N uptake (N ), plant N concentration ( Nc)
with calendar time (t) for soybean (with weed control) at Paramaribo, Suriname (1982).
t x erfx exp(-x2) Q 1Y N Nc
wk Mg ha'' kg ha-' g kg-'
25.3 planting
28.9 0.4125 0.4400 0.8435 0.000 0.00 0 35.7
30.0 0.550 0.5633 0.7390 0.317 0.66 22.4 34.0
31.0 0.675 0.6602 0.6341 0.690 1.37 44.2 32.2
32.0 0.800 0.7421 0.5273 1.105 2.15 65.6 30.5
33.0 0.925 0.8092 0.4250 1.525 2.95 85.4 29.0
34.0 1.050 0.8624 0.3320 1.923 3.70 102.2 27.6
35.0 1.175 0.9034 0.2514 2.279 4.38 116.1 26.5
36.0 1.300 0.9340 0.1845 2.581 4.95 127.0 25.7
37.0 1.425 0.9561 0.1313 2.826 5.42 135.4 25.0
38.0 1.550 0.9716 0.0905 3.017 5.78 141.6 24.5
39.0 1.675 0.9822 0.0605 3.159 6.05 146.1 24.1
40.0 1.800 0.9891 0.0392 3.261 6.24 149.1 23.9
41.0 1.925 0.9935 0.0246 3.332 6.37 151.2 23.7
42.0 2.050 0.9961 0.0150 3.379 6.46 152.6 23.6
00 1.0000 0.0000 3.453 6.60 154.8 23.4
Table 4.18. Simulation of biomass yield (1 ), plant N uptake (Nu), and plant N concentration
(Ne,) with calendar time (t) for soybean (with weed control) at Paramaribo, Suriname (1982-
1983).
t x erfx exp(-x2) Q Y Rc
wk Mg ha-' kg ha-1 g kg'
-2.3 planting
1.1 0.1875 0.2084 0.9655 0.000 0.00 0.0 39.6
2.0 0.300 0.3286 0.9139 0.207 0.351 13.1 37.4
3.0 0.425 0.4522 0.8348 0.518 0.966 32.8 34.0
4.0 0.550 0.5633 0.7390 0.892 1.70 52.2 30.7
5.0 0.675 0.6602 0.6340 1.300 2.51 69.6 27.7
6.0 0.800 0.7421 0.5273 1.714 3.32 83.9 25.3
7.0 0.925 0.8092 0.4250 2.109 4.10 95.5 23.3
8.0 1.050 0.8624 0.3320 2.468 4.81 104.6 21.7
9.0 1.175 0.9034 0.2514 2.778 5.42 111.5 20.6
10.0 1.300 0.9340 0.1845 3.035 5.93 116.7 19.7
11.0 1.425 0.9561 0.1313 3.240 6.33 120.5 19.0
12.0 1.550 0.9716 0.0905 3.396 6.64 123.2 18.6
13.0 1.675 0.9822 0.0605 3.511 6.87 125.2 18.2
14.0 1.800 0.9891 0.0392 3.593 7.03 126.5 18.0
15.0 1.925 0.9935 0.0246 3.649 7.14 127.4 17.8
16.0 2.050 0.9961 0.0150 3.686 7.21 128.0 17.8
00 1.0000 0.0000 3.743 7.33 128.9 17.6
4.7 Figures
Figure 4.1. Correlation ofbiomass accumulation (Y) with the growth quantifier (Q) for sorghum
at Paramaribo, Suriname in 1982. Biomass data adapted from Everaarts (1993). Line drawn from
Eq. (4.3).
Figure 4.2. Phase plots between plant N uptake (Nu) and yield/plant N uptake ratio (Y/Nu) with
biomass yield (Y) for sorghum at Paramaribo, Suriname in 1982. Data adapted from Everaarts
(1993). Line and curve drawn from Eqs. (4.4) and (4.5), respectively.
Figure 4.3. Response plots for biomass yield (Y), plant N uptake (Nu), and plant N concentration
(Nc) vs. calendar time (t) for sorghum at Paramaribo, Suriname in 1982. Data adapted from
Everaarts (1993). Curves drawn from Eqs. (4.3), (4.5), and (4.6).
Figure 4.4. Phase plots between plant P uptake (P,) and yield/plant P uptake ratio (Y/Pu) with
biomass yield (Y) for sorghum at Paramaribo, Suriname in 1982. Data adapted from Everaarts
(1993). Line and curve drawn from Eqs. (4.7) and (4.8), respectively.
Figure 4.5. Phase plots between plant K uptake (K,,) and yield/plant K uptake ratio (Y/Ku) with
biomass yield (Y) for sorghum at Paramaribo, Suriname in 1982. Data adapted from Everaarts
(1993). Line and curve drawn from Eqs. (4.9) and (4.10), respectively.
Figure 4.6. Correlation of biomass accumulation (Y) with the growth quantifier (Q) for sorghum
at Paramaribo, Suriname in 1983. Biomass data adapted from Everaarts (1993). Line drawn from
Eq. (4.11).
Figure 4.7. Phase plots between plant N uptake (Nu) and yield/plant N uptake ratio (Y/N,) with
biomass yield (Y) for sorghum at Paramaribo, Suriname in 1983. Data adapted from Everaarts
(1993). Line and curve drawn from Eqs. (4.12) and (4.13), respectively.
Figure 4.8. Response plots for biomass yield (Y), plant N uptake (Nu), and plant N concentration
(Nc) vs. calendar time (t) for sorghum at Paramaribo, Suriname in 1983. Data adapted from
Everaarts (1993). Curves drawn from Eqs. (4.11), (4.13), and (4.14).
Figure 4.9. Phase plots between plant P uptake (Pu) and yield/plant P uptake ratio (Y/P,) with
biomass yield (Y) for sorghum at Paramaribo, Suriname in 1983. Data adapted from Everaarts
(1993). Line and curve drawn from Eqs. (4.15) and (4.16), respectively.
Figure 4.10. Phase plots between plant K uptake (Ku) and yield/plant K uptake ratio (Y/K,) with
biomass yield (Y) for sorghum at Paramaribo, Suriname in 1983. Data adapted from Everaarts
(1993). Line and curve drawn from Eqs. (4.17) and (4.18), respectively.
Figure 4.11. Correlation of biomass accumulation (Y) with the growth quantifier (Q) for
groundnut at Paramaribo, Suriname in 1982. Biomass data adapted from Everaarts (1992a). Line
drawn from Eq. (4.19).
Figure 4.12. Phase plots between plant N uptake (N,) and yield/plant N uptake ratio (Y/NJ) with
biomass yield (Y) for groundnut at Paramaribo, Suriname in 1982. Data adapted from Everaarts
(1992a). Line and curve drawn from Eqs. (4.20) and (4.21), respectively.
Figure 4.13. Response plots for biomass yield (Y), plant N uptake (N,), and plant N concentration
(Nc) vs. calendar time (t) for groundnut at Paramaribo, Suriname in 1982. Data adapted from
Everaarts (1992a). Curves drawn from Eqs. (4.19), (4.21), and (4.22).
Figure 4.14. Phase plots between plant P uptake (P,) and yield/plant P uptake ratio (Y/P,) with
biomass yield (Y) for groundnut at Paramaribo, Suriname in 1982. Data adapted from Everaarts
(1992a). Line and curve drawn from Eqs. (4.23) and (4.24), respectively.
Figure 4.15. Phase plots between plant K uptake (Ku) and yield/plant K uptake ratio (Y/Ku) with
biomass yield (Y) for groundnut at Paramaribo, Suriname in 1982. Data adapted from Everaarts
(1992a). Line and curve drawn from Eqs. (4.25) and (4.26), respectively.
Figure 4.16. Correlation of biomass accumulation (Y) with the growth quantifier (Q) for
groundnut at Paramaribo, Suriname in 1982-1983. Biomass data adapted from Everaarts (1992a).
Line drawn from Eq. (4.28).
Figure 4.17. Phase plots between plant N uptake (N,) and yield/plant N uptake ratio (Y/N,) with
biomass yield (1) for groundnut at Paramaribo, Suriname in 1982-1983. Data adapted from
Everaarts (1992a). Line and curve drawn from Eqs. (4.29) and (4.30), respectively.
Figure 4.18. Response plots for biomass yield (Y), plant N uptake (N,), and plant N concentration
(Nc) vs. calendar time (t) for groundnut at Paramaribo, Suriname in 1982-1983. Data adapted
from Everaarts (1992a). Curves drawn from Eqs. (4.28), (4.30), and (4.31).
Figure 4.19. Phase plots between plant P uptake (P,) and yield/plant P uptake ratio (Y/P,) with
biomass yield (Y) for groundnut at Paramaribo, Suriname in 1982-1983. Data adapted from
Everaarts (1992a). Line and curve drawn from Eqs. (4.32) and (4.33), respectively.
Figure 4.20. Phase plots between plant K uptake (Ku) and yield/plant K uptake ratio (Y/K,) with
biomass yield (1) for groundnut at Paramaribo, Suriname in 1982-1983. Data adapted from
Everaarts (1992a). Line and curve drawn from Eqs. (4.34) and (4.35), respectively.
Figure 4.21. Correlation of biomass accumulation (Y) with the growth quantifier (Q) for soybean
at Paramaribo, Suriname in 1982. Biomass data adapted from Everaarts (1992b). Line drawn
from Eq. (4.36).
Figure 4.22. Phase plots between plant N uptake (Nu) and yield/plant N uptake ratio (Y/N,) with
biomass yield (Y) for soybean at Paramaribo, Suriname in 1982. Data adapted from Everaarts
(1992b). Line and curve drawn from Eqs. (4.37) and (4.38), respectively.
I
Figure 4.23. Response plots for biomass yield (Y), plant N uptake (Nu), and plant N concentration
(Nc) vs. calendar time (t) for soybean at Paramaribo, Suriname in 1982. Data adapted from
Everaarts (1992b). Curves drawn from Eqs. (4.36), (4.38), and (4.39).
Figure 4.24. Phase plots between plant P uptake (P,) and yield/plant P uptake ratio (Y/P,) with
biomass yield (Y) for soybean at Paramaribo, Suriname in 1982. Data adapted from Everaarts
(1992b). Line and curve drawn from Eqs. (4.40) and (4.41), respectively.
Figure 4.25. Phase plots between plant K uptake (Ku) and yield/plant K uptake ratio (Y/K,) with
biomass yield (Y) for soybean at Paramaribo, Suriname in 1982. Data adapted from Everaarts
(1992b). Line and curve drawn from Eqs. (4.42) and (4.43), respectively.
Figure 4.26. Correlation of biomass accumulation (Y) with the growth quantifier (Q) for soybean
at Paramaribo, Suriname in 1982-1983. Biomass data adapted from Everaarts (1992b). Line
drawn from Eq. (4.45).
Figure 4.27. Phase plots between plant N uptake (Nu) and yield/plant N uptake ratio (Y/N,) with
biomass yield (Y) for soybean at Paramaribo, Suriname in 1982-1983. Data adapted from
Everaarts (1992b). Line and curve drawn from Eqs. (4.46) and (4.47), respectively.
Figure 4.28. Response plots for biomass yield (Y), plant N uptake (N,), and plant N concentration
(Nc) vs. calendar time (t) for soybean at Paramaribo, Suriname in 1982-1983. Data adapted from
Everaarts (1992b). Curves drawn from Eqs. (4.45), (4.47), and (4.48).
Figure 4.29. Phase plots between plant P uptake (Pu) and yield/plant P uptake ratio (Y/Pu) with
biomass yield (Y) for soybean at Paramaribo, Suriname in 1982-1983. Data adapted from
Everaarts (1992b). Line and curve drawn from Eqs. (4.49) and (4.50), respectively.
Figure 4.30. Phase plots between plant K uptake (Ku) and yield/plant K uptake ratio (Y/K,) with
biomass yield (Y) for soybean at Paramaribo, Suriname in 1982-1983. Data adapted from
Everaarts (1992b). Line and curve drawn from Eqs. (4.51) and (4.52), respectively.
97-
I.22e
. : : : :* F
77 1-
-I-
\_J _
~
* 2
J:^
! : ;
~~.1..
- C;)
: 1
4,
"I-i 1222:
ft
I
SThf~I1!~J / J(')d1
-9
7777.
2 F
h--i
I.. -~
'-1~----)
-F -
72
Ti
4-
F.
;:.j* 2
I :
S. ,J ; ,
' ) '
": ''*~^~
F'' 'I
- t r2 4
17* 2
IL
}7t
L 14'
F
20 Sq uares to) t he InchI
-I---
-L U
77
. . . .
i
-771
p
7
122E2 ~ fl27
-H---'-
-7-
`01 Squhares to the Incwh
4r-
& I
T
-----
. .. .... ...
V 7 7-
-0-~'"^7
.'i4^_'_
* I ;
TT>
0 -j
/ K
>1
II
t 7-
I L
I,
7
/
A
:I---,
-K--\
-4-1
F
T,
77 -T
j; --
.1
I,
:2; I
*1~~~~
t:
Lij'
771-L
~~2~~
*1~~~
-I }
-4i
?
I, 1 4
r I
i
12 2 32
-t
I
^JT
l : t
- -F
-s
~1~
*'1 'I~'
20 Sq uares to the Inch
0
/00
-1
-^--I
'- til
; ': ;i
..... r *
7-747 7
~QJI
~~ri
I -~I -;
-1:.-
-7 1.
I- -
..
-j
I -
I.- -
H-',
-I'
-I
7-77
. i ; ,i i . ..' l
1!- I
I -- I'
M^i
*~1~
I:
:7f4,j
-I,..:-.
4- 4 -
lit-
7 1+
4-
-I--
I IL I
I-
I-
I-.
* I
I.
[1
[I
4 V
ii
Ki
4 -i .-----t-4 ---r
LLIIi.2 *jjj77ij t ~.'k I I ..
~{v> / I __ [ ~71 :;Ij~~:.
ffi _
-~ th3
* ~'-* I:.
-I
Ii-
<--1-i
I.
~d.
ILI*
4+
1T1
, LL
I
12 282
f~ 1-h11
17
4-4
to
i -T
F. i
F! t i
77-
, 9..
I.,
-4
I,
~1
.r.
I,:
V' 71 1
I. >1)
I. .
77
I.
-I.
t 1
'il 1
rivt l
iT-7
7-
i 7
44, -7
+ I T7
4-H
77T-, F7
! t
T : ^ 1. ; :"
'-JM^
K
<1 ~i
I,
1.
1-
20 Squanres to h Ic
Tt
-4
17
7- 7.
7
- T
tile inch
, I
-4
LI--.L
+4-@- +- FH-,b-- i
1 ,L- -.--H------- !-
I
, i
.... LLZZ .
I
I
12 232
I -
--1-----'-
0) 0']
- I--- -
i1t~o-
- -j I'-
-'I----'
-0
F
F
2 'C
'F F
1-FH -
hL 1 2.HPhM I
7__ i
F -
.1;,
I I
1 :: __________
2
* F
I.-,
1'^i
: 'i l li
HUT:
Vj7K~7f~~F~*
'1-
7: -
- ; -,
FiH F l
- !! 7 ;/K -K 4!-i,
F: F F: i
iii: I ii
- F : :
- -^ 1^-.
i
OT 1K i 41K[ :T
2
> ~
I -
F-Ir-7
17~"'
--I-
-j
I -'-------- I ,
>2
* Ii*. -
- I-
ypI
I !JJL1 F
F I
I
2W Sq uare' to the Inoch
I
I
7i
-I-4`
21
I
t
I
I
I
l
S09 P 11 rh I 6 1-
12- ,9
I-7
:1 t I
7: _7
-T
- I--.
1, -~
.1~;-.
*1~~
I -
.4L
*LL.
F'
- I
--4,.
.1:
- ---I----
'-Ti---
I,'
I),
I'
-1'
* TT
'.4
I',
'.1..
717k
:1
I-
I f
-F-L I -t- 7
-LLL~~~~~~r~y' TL LL
FC2 tj
-C,
-/l-
I'
ri.
- -.
-17
k J
--7
F.7
-i-.
20 Squares to the Inch
-77 --'
7F
77
1-28
1, 1,
I- '
7Kii
I .
4--7--7
77 :. -
I' LL
5 *'~ *L
V 17T 7{7~~i4 1 ~ K I~4,
til II
ft t l
-7;
1+~
.1
'I
it
I.
L I'
~- ~
-v
-~ A
.:j.
I I
- ~o
K
4 tLK L LI
20 SjIta1.res to the Incli
r
*1
~1*~* t-.
----iii
r
0
It!K-,:
tt
. ; i : .
*1
K
71 7
FiT7
lit
T j
.i
' i:-
-1, 77.
500Aa.'m 110
k
I '
I J I ,
F:
v-17
.11:
12-282
QN
v ~ ~~~ 71-:" rn
7
T^ji
Zt
^-7--
2]
/ 1
0-o
I '.'
2-L^
0;J :
1-7
( -
11"iir; ;
- i f *
I,
1 T
I
-7-7'?:
K
t~7II~> 71 V
I
- _________ _________ T I I -
-I -
I,
I
* 1
1~
I -*
2---
4
I: I I i I
* it
jP,4
F
A..
'T:It .
1~
.1
7-/ :-: I !
LP'~
] ;
. . t
.' : i :
7-I-.
-
': f > .I i .
'1j ,; 'iK
41
''' '1 -."
i iT .. .-i
I: I'~ ,*~ j, i ..
:'7L
[V
1I~j
1-
4~. 2::
11 21:14
ITh
20 Squares to t lie Inch
-7
1,
2: -
t~-,mi .iii iii i.. i. f ....- ~ iin il.- n 1 1, t .t..,i,,it,,1 iin ~ imi .in i i.i., ,iJ -.. i ..^ii ll ~~'iJ F-.t.--.---^ t ^*t -
] : 1 1 1 1 1 L I : I I 1 ; I I [ I I .
l
I
I
!I
'' : ^ '. .'
'.- i ;:
,i.~:12:;
I-
1221II
F.
12-22 8 2T F
5044VJ1171 EI.
Em ',
Q0
71-~~
; Z
-7
201 Squares to I le I fch
TV
'I
>1 F
2 2 F
I'
T-,7 7
,,7
J'
t t
q
.4 II-
T'77
* F, '
.1' Ii
'.2
I L
+ T i
4i
-' 1 7 -
F 1 F
2 I'
hi 'ILl
IF'
4
...2.'
'-'1-' -', 7
t t
t t
7t,
-17= m 77.V
1.2T
It __r____ qI''.
I L'
>77
I'
F -
m
t
_71
.......
:T ::
I' .7
gr20 I, I -
4,
71
Ot~
c~
a
'-C' '~
7T277 -
7 7
j -~
I,
I
I 7
I-.
12
I.
I,
7-777
-~ 4IL
-vi
)
''I
7-:
7
VT
1'1
1~.
-I..
-'-I iK;i
:41
I.
I:
+7
7-
tI
I H
+1 -.(;
.1 4.1
I 11 ('.
I I
I, j I
~ I-
I j L~Ii
A T
LL1 KhiL~i 4A~
-4-
Ell~ <1-4
-fl
7 : f 41
i t t l 1I
20 SquLares to the Inch
-V
- -j t .,
', .' .. I. \ '
I
m I
I
[
i
4L-,
i .....
12 22
1it
774
77,.
1
VK
J. fr-\ ,\','
-p '4-
/ f/ (' V
I i
6
1~-~
/
7 -7
-p-- --7-. -p T V
i^ .,!,. ,.,
*+-.4- **i-i-i-i-1-i-i I f \'i \
; : : : : ; :^ ; : m : t
., .,,^- ^ n.,
j^; H jilf i ir
'1
- ,-' ; "- ,- T rT r"
T 7 .. : ..:
it
7'-7-
7I 1 77
II
*-1-~
.1
.1
20 Squares to the Inch
I~ t+m
7iFI' TJi
-7F 77
-7-
7 7
II
I-T-f
I
I
1
'
1
, ,
, -- r
t
I
I[llll ,liIL
I
[
I
I.
--I
1~
4-
----K'
* 1*
t
=7
fii
77-
7--
210 Squares to t he Inoch
M--
4,4
/ .']
I --
I,..
^
I
T.
I...
K
Ii
I---
.1
K
- 7
2
*1
I, 4
/ T;
1 -
V
II.
I.
11
1T7
V
* ; ; i ;
4 44
.'1
11.1
-I-~ L
F
KA+H
:j 1
I.'
'-tim
-4
1- 1
7t77:
L
J L i J
4 ............... L ...... ..... ..
l
i
I
t
l ....
.4Th
'-^ i
*F,"
,-4
i-i
12,202
C
r 'I1-rr^
1~7~
IF-.
T <
-17
----1 7
LL -
1:.7
* I L~~I
I* I
iiL
1<
* w
IF I I::
.1
1.
t -
t ; : ;: : 1 : ; '
HI,.t.f1Ii. I
':1 I:..,
-I -
.2
tII '*'
I
A.I
I~i'
I, I
K
-I
F..
'I.
:7
I;
V
I
K
K.].
* I
:1.
i11ii~
.1
20 Sq~iires to the~ Inch,-
12; 2j'
--4
:1 2
vi
A-2 ~0
t
7T7
I..
I! 15&
0
r~1
1-7
'I
* ..J L
I A
L r
7FTTT
-4-----
, t i
I Ill
h
Hill ]
I
r
I
!ILL
12 2442
'j!.4 -.';
I,
H .' :;:.
717.7
* I,
r
-~ 14JLJ
20 Squares to the Inch
4>
P,4~ -
Ir~ tJ
I I,
I ~
I -
- 'TI
, 7 !-,, -
i-:
f I
77 j -
i 1 i i -\'' -'
iLJ: Li!
Ii.r
'-I
7 T'T
t
1. *'
77K;~I'
'I
rv-.
'1
7777I
-4 1
7 l Ti7
.-LhIL:-
.
' ; I i >
I '* I
- 1
:v.
1
H
-VI
I1'.-
* I
I
/ fill
.i7 [I~
* '
7
* 4, 4..
PIT
* (I
/
-0--
7 .: TI
* I
- .1.
'-I
1 1
-I-
I
I',
- I
i i ii i I I i 1 i ll i
m
' .' I I. I- .,I I .1 I 1 1- -- .-' --- -1 -
1 [ J r J i l l ; d 4 . . . . . . .
I
I
- 7 T -7.
7-
12 202
-- ~F7,
Ott
201 Sq uares to the InochI
~tL7
'-I
I' -:
Ul) --
1~
TIT
F4\I~
K--; 42
''I
,~-"
I-
''I
~~~~~1
I''
7t7'i
.1
I '
it-i
1-i-, 1~-~
~ f>77.7)
'7~
&
4 I
I !:}F .'
-l
I-'
4t''
' l---
'F i FjjL
r I'
~Th
'Ii'
7-V
I,;
All
... t I
I I77
. .... ...
77'
'I
;-~i 7-. i -7i~
/-77 T -7T" -
1<
(if. K
-. I
IL t::: If~7~
[
.1
]I
'7 74
- -,F-, 7 j
F'A
fi'77
iri
I I 11,I
41
W- IL
fit i-
I, >. ,! lw
F"** F'i; r 4'4
*.:'t^ t~t ^ ^ I
1 1 I I I
. i---
r
Il
LIN
AL
ly
i 2 2 rl
f/WI" / /~ -~ '3
>2< ~
K C)
3 27/
I. **~ I
L
110 I ,Ia:uie t t lI' liI h
I
7
:1+
111
/ /f 1M -'
I I
'/
pit
Al2,
'1 1qt >Ic'l Kh Ic
Fi, qt
I I
':1
*~1
~ >1
'1
L ~-~J
20 t tI ,- linch
>~
x
K
C
-~2~ /90
K
Ii&z
i.o~
'9
I I
I I
C I I I
I I I
I I
I I I I I I
I I I
I I
~
F I..
I
~ I
I I
I I I
Wi I. K _
.1 lj I I*
.1 ~'I .1 -
I I A.
I I I
r-i~
II
I 1
2
21) ~I~AI.AfA~ tA, ~ I ~..1'J:1IIIi2.IiA L I~1
tilt II), II JJ*. i-I II
F>- r1nq
-I
I.
-I '7,
1~
-7
I,
T,
I i, :
f
IL,'
it
-~
L L t:"I ~ 1'I;i u
A-1 III'!
.. 1 ;, i .... .... . i ; i
12~
20 SqItrI s to t t' I Inf h
12-2B2
t1
4 TU1
24 [4
/
)
K
? ~j77~
ip i i.
f~gLJ~
- - -
121.1>
-V
2'1
-I>
.1
'I-I
~jI I
I /1.
i~J
f V.
J- I
4- i-
J-1I
177
1 T -
-4' 1~~*
4-t _1
I -4
I,,L
7 47 -
H '
i ti _____ i
I
1I
20 Sqttares otheIc
o the Inch
a .
1.
a as
r -- IJl! lil L
I
i
"y\ ~..
p I
282 F7
,I
4 I
1' __ _
I'
Kr :;j7TIij~
II
4f! f i1
t
'I f
>'
.4- -I.
I
I I 1~~~
1L~
'''I I I
-. t ---I 4
.1,
I J 4
-. I
I I..:
--I
-~ I
Li
IL I ~fILffiI1'~~IITI4UIi
20 Squares to the Inch
I 1~r~
Lfl
14
I
~~1
(T
I
till,
, ,la,
i
14HH1 ifflll t11W
LL !:A
j
K
11
14
~~1 -- I
12.22
20 Squares to the Inch
12 202
~1 7
20 Squares to the Inch
4
;T:
'U
I'
*1'
-b t---"~
43<
I.
Ii
~3 LJ~L~
I4
- L
1 :L
.4
17
4
-4--
t : t it
T I
i 4
i~7
lit
4;
*~ ~ M. *
4T t
iit4
All L!
4: H, f
It <
<2
1.LI v
7i4
'.1
~ ~1-*
t1I~ ~r9
T- Y
41
lo~
7-77~~
n J
I
|
I I--
I
1
I r I I
L-
-Lil
ti
12-2132
ri-i
I.
/
IL-I1
7 F) U
-1-6d,
I JI
*--.1
4-
I I
1.1 I
V
7 iT77
r 11-1
If~
Li 4%
K ~
-~ I
2~zj
- .111
~
N
~-~-1
I I
Slo:t
M _
i '1 *i -
Hi
~; 1i1K i'~i. ~ j
IL-
101,
* y1~*1ij -
- --4-
I I
AI I
20Sqars o he It I
Pat
U P
I
:1 ___ 1'
HV+-
AM 14
a qn0 r
PIh AM; nap
i K
______________ *-.--*---*---'-r -
II"
ALL
r Y
1 him,
T 7111
---- -----
I
I
i,
L
I
I
JI'l
4
minallnum
1A
12082~
wil
7T !7ii7
rii!K
1-2 11v
VV'T
.! < < .1 .
''I
F' 'I i
WH IM F
O 'F 0
7 j
''1 -
K'
L
V~ j K ]'
P Q;:,1
ii.limit
H IT
11 ph
I W51
n'
Fi F'[;:
TOF
Mud1"R
liI IJ -!i 1 l
''F "I
F" "I "'f' ''
7'F
A l., --
1 F
WANK'WhI
'4,q q 1.
; is I'I
lill
P I N IT
I 1F
, IP[ ;A4 '''''i -h
q Hap'
WHY hFl T
'F' An'
S" '
.. ,- I : .. .
.. .. I) ... .. .
[ i
-----------,------- L
M AP ,..' h
"Msl"t
. I I 121 4 n
LH
p'
20 Squares to the Inch
! ip
i ili i;[i
p I I : I I )-~
12 241 81
2IZ
20 Squares to the Inch
7'7v ~
7-7, T
r.t
m ^Sn"
4i7)7
.I f
hI 1 L-itI
F,,,,
<0
ii
.~
.1
I
I
(I
I..
-A
.77)
1-
I:
I:
I
-I.
~ E~J7 II
4- t -
*4-- i i
7,,
7
V -
I-.
* .1
4:
.4...
4Ftt t
'V.
1-
.4
f' t
r
7.7
'I:
I -
II:
I -
Ii:
7
Ii
12-202
/ ;ikh~
Ii,
____ I- -
*ht
I T2
if:
+-- 7--77
L ___________ -4-4----
I
I,
V-7
41 f4--
-- I,
Pr)
II,
I~IO
-V
I,,
-I
41 4.
__ I-I
Ii
--1,
I444~
77,
-- I t1t4
I i ', i l l i
f
r,. j
12
C,,
~~1~
4'-
I-~
4'
K
.444f
iF
V 44,,
-I
/4
* 4.4.
4,444
41
'I
'I;
"--1-
I::
20 Squares to the Inch
-71
I,
-1~
nf
7:.7
L
-t -
I-
* -7.--
7jT
7
I I I -
I
t
. t
!,i
4
it
12 21C2
I-'
;L_ : \ i _
I'^
F F
'F F
': : 4
F 7T
'I,
''F i : L
F'' F''
-7-rH
V "
- --
F''''
'I
11: I::'',
J-:~ I,~&~!
i !_L-.!ill
F .' F' :
71 i
4i
'F`
'F,
: ,: . '
I ;. I.
F' '~
3 '4'
''I
--I
F .1
':4,;
r IF
\K.,
: .
F -
1'.^
... ....
11
7'
* F
* I''
'i\tz)
ii
F'
F F
''I
* I I
K ~'j'
.
''I
I''
--7"
'I-
F~F'v'
* I'
F' 7;
f .-.- .
'F
,,.,F.,
F H. i ;FFI
tF
h i
F' i :-.:: -,
77, i tl i-r! -.
T I7
: :j 4,
1 ^ L ;
; 1i -
t. _i-j- _*
4 ti2
I;
I
YI
20 Squares to (ihe Inch
l l
i
- --t-t-I
L7K
-i7 7-7
2 202
~:.
'-I
O~
'~ I
* ~'~~-1
* .1
1
[00
- F 2,
I.
12
777
* 'I
ill,,
I -
~T7~
:1~
.77 7221
777 : V .'177 -T
'--i^ '- 1 -"- '
I I 2ii t
aii i!!!:i __
IAL > _I_ J!!?!!
i I
'-I-'--
AD
.1~
1141:2
'I
-I'
I-
'7-
41
->1 ~
I'
1''~
4
- I
V
-777
4' 2
2'
-I
* 2
* K- -~
2 :11
* 2
- I'
<11 Ii
I--
7Tii7-
- .
- -4
- '-I
-I
7I 1
-F
V
77"V
20 Squares to the Inch
I- I
:1
i I
I
I I
i
I
t
5. References
Abramowitz, M. and I. Stegun. 1965. Handbook of Mathematical Functions. Dover. New York,
NY.
Everaarts, A.P. 1992a. Effects of competition with weeds on the growth, development, and yield
of groundnuts. Netherlands J. ofAgricultural Science 40:73-90.
Everaarts, A.P. 1992b. Effects of competition with weeds on the growth, development, and yield
of soybeans. Netherlands J. ofAgricultural Science 40:91-107.
Everaarts, A.P. 1993. Effects of competition with weeds on the growth, development, and yield
of sorghum. J. Agricultural Science, Cambridge 120:187-196.
Ng Kee Kwong, K.F. and J. Deville. 1994. The course of fertilizer nitrogen uptake by rainfed
sugarcane in Mauritius. J. Agrcultural Science, Cambridge 122:385-391.
Nguyen, M.L. and K.M. Goh. 1992. Nutrient cycling and losses based on a mass-balance model
in grazed pastures receiving long-term superphosphate in New Zealand. 1. Phosphorus. J
Agrcultural Science, Cambridge 199:89-106.
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). S585.9 093
http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00072282&v=00001
Overman, A. R. 2006b. A Memoir on Crop Growth: Accumulation of Biomass and Mineral
Elements. University of Florida. Gainesville, FL. 386 p. (84 Tables and 184 Figures).
SB112.5 095
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). SB112.5 096
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). SB201.B35
095
http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00075466&v=00001
Overman, A. R. 2007b. A Memoir on Model Analysis of Switchgrass Response to Applied
Nitrogen and Calendar Time. University of Florida. Gainesville, FL. 75 p. (23 Tables and 30
Figures). SB201.P2 095
http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00075467&v=00001
Overman, A. R. 2007c. A Memoir on Mathematical Models of Crop Growth and Yield:
Miscellaneous Applications. University of Florida. Gainesville, FL. 211 p. (65 Tables and 88
Figures). SB112.5.0965
http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00075468&v=00001
Overman, A.R. and R.V. Scholtz. 2002. Mathematical Models of Crop Growth and Yield. Taylor
& Francis. Philadelphia, PA.
6. Acknowledgment
The author is indebted to Professor Arij Everaarts of Wageningen University and Researchcentre
for supplying the planting dates for the study at Paramaribo, Suriname.
|