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Title: Memoir on model analysis of switchgrass response to applied nitrogen and calendar time
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Permanent Link: http://ufdc.ufl.edu/UF00075467/00001
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Title: Memoir on model analysis of switchgrass response to applied nitrogen and calendar time
Physical Description: Book
Creator: Overman, Allen R.
Affiliation: University of Florida -- Gainesville -- College of Agricultural and Life Sciences(CALS -- Agricultural and Biological Engineering
Publication Date: 2007
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Subject: Congongrass   ( lcsh )
Switchgrass
Panicum virgatum
Forage plants
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Bibliographic ID: UF00075467
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Table of Contents
    Title Page
        Page 1
    Abstract
        Page 2
    Model response of switchgrass to applied nitrogen and calendar time
        Page 3
        Introduction and model description
            Page 3
            Page 4
            Page 5
        Data analysis
            Page 6
            Page 5
            Page 7
        Discussion
            Page 8
            Page 7
            Page 9
            Page 10
            Page 11
            Page 12
            Page 13
        Conclusion
            Page 13
        References
            Page 14
            Page 13
        Tables
            Page 15
            Page 16
            Page 17
        List of Figures
            Page 18
        Figures
            Page 19
            Page 20
            Page 21
            Page 22
            Page 23
            Page 24
            Page 25
            Page 26
            Page 27
    Model response of switchgrass to applied nitrogen and harvest interval
        Page 28
        Introduction and model description
            Page 28
            Page 29
            Page 30
        Data analysis
            Page 31
            Page 32
            Page 33
            Page 34
            Page 35
        Discussion
            Page 36
            Page 37
            Page 38
            Page 35
            Page 39
        Conclusions and references
            Page 40
            Page 39
        Tables
            Page 41
            Page 42
            Page 43
            Page 44
            Page 45
            Page 46
            Page 47
            Page 48
            Page 49
            Page 50
            Page 51
            Page 52
            Page 53
            Page 54
        List of Figures
            Page 55
            Page 56
        Figures
            Page 57
            Page 58
            Page 59
            Page 60
            Page 61
            Page 62
            Page 63
            Page 64
            Page 65
            Page 66
            Page 67
            Page 68
            Page 69
            Page 70
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            Page 74
            Page 75
            Page 76
            Page 77
Full Text







A MEMOIR ON MODEL ANALYSIS OF SWITCHGRASS
RESPONSE TO APPLIED NITROGEN AND CALENDAR TIME






Allen R. Overman





Agricultural and Biological Engineering

University of Florida


Copyright 2007 Allen R. Overman









A MEMOIR ON MODEL ANALYSIS OF SWITCHGRASS RESPONSE TO APPLIED
NITROGEN AND CALENDAR TIME

ALLEN R. OVERMAN

Agricultural and Biological Engineering Department, University ofFlorida,
Gainesville, FL 32611-0570

ABSTRACT

Mathematical models are discussed which describe crop response to applied nutrients and
accumulation ofbiomass with calendar time. The extended logistic model describes crop
response (biomass yield and plant nutrient uptake) to applied nutrients (N, P, and K) through two
logistic equations. This leads to a hyperbolic phase relation between biomass yield (Y) and plant
nutrient uptake (Nu). It follows that plant nutrient concentration (Nc) is a linear function of plant
nutrient uptake. The variables are bounded by 0 < Y < Ay, 0 < Nu < An, and Nci < N < Ncm,
where Ay is maximum biomass yield at high nutrient application, An is maximum plant nutrient
uptake at high nutrient application, Nc, is lower limit on plant nutrient concentration for depleted
soil nutrient, and Ncm = An/Ay is maximum plant nutrient concentration at high nutrient
application. According to this model system variables are bounded. The logistic equations are
well-behaved monotone increasing functions. Peak nutrient application (Np) to achieve
maximum efficiency of plant nutrient utilization (Ep) can be estimated from the model. The
expanded model describes crop growth (biomass accumulation) with calendar time by
photosynthesis. The model contains components for an energy driving function, a function for
partitioning ofbiomass into light-gathering and structural components, and an aging function as
the plant grows. This results in a linear first order differential equation which can be integrated to
obtain an analytical solution. A linear relationship is established between biomass accumulation
(AY) and a growth quantifier (AQ). A hyperbolic phase relationship between plant nutrient
accumulation (AN,) and biomass accumulation (A Y) is assumed. This leads to a linear
relationship between yield/plant nutrient ratio and biomass. For perennial grasses seasonal
biomass yield (Y) can be linked to a fixed harvest interval (At) through a linear-exponential
equation. Peak harvest interval (Atp) for maximum biomass yield can be calculated from the
model. Cumulative biomass yield (Y) vs. calendar time (t) is described by the probability
equation, which contains a Gaussian distribution. The growth model explains the decrease in
plant nutrient concentration with time as the plant ages, and biomass shifts from domination by
the light-gathering component (higher plant nutrient concentration) to the structural component
(lower plant nutrient concentration). In this document the mathematical models are used to
describe response of switchgrass (Panicum virgatum L.) to applied nitrogen for two field studies
- one at Ames, Iowa, USA and the other at Montreal, Quebec, Canada. Detailed procedures for
model parameter estimation are discussed. Inferences of the models are confirmed with the
analysis. The models can be applied to annuals as well as perennials.


Keywords: Models, nitrogen, switchgrass, forage grass.









MODEL RESPONSE OF SWITCHGRASS TO APPLIED NITROGEN AND
CALENDAR TIME

ALLEN R. OVERMAN

Agricultural and Biological Engineering Department, University ofFlorida,
Gainesville, FL 32611-0570

INTRODUCTION

Mathematical models are useful for describing output (response variables) in response to
input (control variables). Extensive field studies have been conducted in various parts of the
world over the past 150 years on crop response to applied nutrients and other factors, such as
crop species, soil type, harvest interval for perennial grasses, environmental conditions (such as
water availability), and plant population. The present analysis will focus on two particular
models. The extended logistic model (Overman et al., 1994) describes coupling of seasonal
biomass yield and plant nutrient uptake with applied nutrients (such as nitrogen) through logistic
equations. Biomass yield is then linked to plant nutrient uptake through a hyperbolic phase
relation, which predicts a linear relationship between plant nutrient concentration and plant
nutrient uptake. The expanded growth model (Overman, 1998) describes accumulation of
biomass with time through an analytical function which incorporates effects of energy input,
partitioning of biomass between light-gathering and structural components, and aging as the
plant grows. Plant nutrient and plant biomass accumulation are coupled through a hyperbolic
phase relation. Field data have been used to confirm this model for coastal bermudagrass
(Cynodon dactylon L.) grown in Georgia (Overman and Brock, 2003). The growth model
predicts a linear-exponential dependence of seasonal yield with a fixed harvest interval for
perennial grasses. All of the results have been shown to be consistent with field data for
numerous crops, soils, and environmental conditions (Overman and Scholtz, 2002).
This analysis will focus on response of the warm-season perennial switchgrass (Panicum
virgatum L.) to applied nitrogen and with calendar time.

MODEL DESCRIPTION

Response of biomass yield and plant nitrogen uptake to applied nitrogen can be described by
the extended logistic model given by


Y = (1)
1+exp(b, cN)

N, = (2)
1+ exp(b, cN)
N = N 1 + exp(b c,N) (
Y + exp(b, cN)









MODEL RESPONSE OF SWITCHGRASS TO APPLIED NITROGEN AND
CALENDAR TIME

ALLEN R. OVERMAN

Agricultural and Biological Engineering Department, University ofFlorida,
Gainesville, FL 32611-0570

INTRODUCTION

Mathematical models are useful for describing output (response variables) in response to
input (control variables). Extensive field studies have been conducted in various parts of the
world over the past 150 years on crop response to applied nutrients and other factors, such as
crop species, soil type, harvest interval for perennial grasses, environmental conditions (such as
water availability), and plant population. The present analysis will focus on two particular
models. The extended logistic model (Overman et al., 1994) describes coupling of seasonal
biomass yield and plant nutrient uptake with applied nutrients (such as nitrogen) through logistic
equations. Biomass yield is then linked to plant nutrient uptake through a hyperbolic phase
relation, which predicts a linear relationship between plant nutrient concentration and plant
nutrient uptake. The expanded growth model (Overman, 1998) describes accumulation of
biomass with time through an analytical function which incorporates effects of energy input,
partitioning of biomass between light-gathering and structural components, and aging as the
plant grows. Plant nutrient and plant biomass accumulation are coupled through a hyperbolic
phase relation. Field data have been used to confirm this model for coastal bermudagrass
(Cynodon dactylon L.) grown in Georgia (Overman and Brock, 2003). The growth model
predicts a linear-exponential dependence of seasonal yield with a fixed harvest interval for
perennial grasses. All of the results have been shown to be consistent with field data for
numerous crops, soils, and environmental conditions (Overman and Scholtz, 2002).
This analysis will focus on response of the warm-season perennial switchgrass (Panicum
virgatum L.) to applied nitrogen and with calendar time.

MODEL DESCRIPTION

Response of biomass yield and plant nitrogen uptake to applied nitrogen can be described by
the extended logistic model given by


Y = (1)
1+exp(b, cN)

N, = (2)
1+ exp(b, cN)
N = N 1 + exp(b c,N) (
Y + exp(b, cN)









where N is applied nitrogen, kg ha-'; Y is seasonal total biomass yield, Mg hal'; Nu is seasonal
total plant nitrogen uptake, kg ha'1; Nc is plant nitrogen concentration, g kg"1; Ay is maximum
yield at high N, Mg ha'; An is maximum plant nitrogen uptake at high N, kg ha ; Ncm = An/Ay is
maximum plant nitrogen concentration at high N, g kg-'; by is intercept parameter for plant yield;
bn is intercept parameter for plant nitrogen uptake; and Cn is response coefficient for applied
nitrogen, ha kg-1. Note that the units on c, are the reciprocal of those for N. Equations (1) through
(3) are well-behaved monotone increasing functions. Since variables N, Y, and Nu are defined as
positive, parameters Ay, An, and cn must be positive as well, while parameters by and bn can be
either positive, zero, or negative. Equations (1) and (2) with common c, can be combined to give
the hyperbolic phase relation between Y and Nu

SY N. (4)
K, + N,

where hyperbolic and logistic parameters are coupled through


Y. = (5)
1- exp(-Ab)

K. (6)
exp(Ab)-l1

with the shift in intercept parameters defined by

Ab = b,, b (7)

For Y, and K, to be positive requires that Ab > 0. Equation (4) can be rearranged to the linear
form

N K, 1
Nc = N.- = -K. + N, (8)
Y Y, Y,

which predicts a linear relationship between Nc and Nu. This prediction is easily tested from data.
The expanded growth model for biomass accumulation (Y) with calendar time (t) from Jan. 1
is described by

Y= AQ (9)

where Y is accumulated biomass, Mg ha-'; Q is accumulated growth quantifier; and A is the yield
factor, Mg ha-'. The growth quantifier is defined by


Q = {( -kx,)[erfx erfx, ]-- exp(-x )-exp(- x,)]. exp(locx,) (10)
k 2Vex 2









with dimensionless time, x, defined in terms of calendar time, t, by


t p 2o'c
x-= +o- (11)
Jj- 2

where model parameters are defined by: p is time to the mean of the energy distribution, wk; ac
is time spread of the energy distribution, wk; c is the aging coefficient, wk-1; and k is the partition
coefficient between light-gathering and structural components of the plant. Note that xi
corresponds to the time of initiation of growth, ti. The error function, erf, in Eq. (10) is defined
by

2 x
erf x = exp(-u (12)
0

where u is the variable of integration. Values of the error function can be obtained from
mathematical tables (Abramowitz and Stegun, 1965).

DATA ANALYSIS

Data for this analysis are adapted from field studies with switchgrass (cv. 'Cave-in-Rock') at
Meade, NE and Ames, IA by Vogel et al. (2002). Applied nitrogen levels were 0, 60, 120, 180,
240, and 300 kg ha-1. Measurements included biomass (dry matter) and plant nitrogen uptake.
Samples were also collected periodically to measure growth from the beginning of the season.
This analysis will focus on the study at Ames, IA where the soil was a Webster-Nicollet complex
(fine-loamy, mixed, mesic Typic Haplaquoll-Aquic Hapludoll). No supplemental irrigation was
provided.
Nitrogen response is given in Table 1 and shown in Figure 1 for the first cutting only. The
increase of all three measurements with increase in applied N may be noted. The procedure for
evaluation of model parameters is now discussed. Equations (1) and (2) can be linearized to the
forms


Zy= -ln ( 1)=-by +cN= 0.39+0.0130N r= 0.9982 (13)


Z, =-n 152-1 = -b, +cN= -0.89+0.0131N r=0.9971 (14)


where Ay = 11.60 Mg ha-1 and An = 152 kg ha- have been chosen to give the same value of cn.
Note the high correlation coefficients (r > 0.99). The response equations now become

11.60
Y =11.60 (15)
1 + exp(-0.39 0.0130N)
152
N, = (16)
1 + exp(0.89 0.0130N)









N =131 l+exp(-0.39 0.0130N) (17)
Y L 1 +exp(0.89 0.0130N)

Curves in Figure 1 are drawn from Eqs. (15) through (17). Hyperbolic phase parameters can be
estimated from

Y 11.60 16.07 Mgha-1 (18)
S1-exp(-Ab) 1-exp(-1.28)
K, A,, 152- =58.5 kg ha1 (19)
exp(Ab)-l exp(1.28)-1

which leads to the phase relations

Y YN, 16.07N. (20)
K, + N, 58.5 + N,

N, =K ,+- N =3.64+0.0622N, (21)
m m

Phase plots are shown in Figure 2, where the curve and line are drawn from Eqs. (20) and (21),
respectively. The model describes the seasonal data rather well.
Growth data are given in Table 2 and shown in Figure 3. The first step in the growth analysis
is to calculate the growth quantifier (Q) as a function of calendar time (t) from Jan. 1. Model
parameters are assumed to be: p = 26.0 wk, 1/2o- = 8.00 wk, c = 0.15 wk-', k = 5. Model
parameters are the same as those used for coastal bermudagrass (Cynodon dactylon L.) by
Overman and Scholtz (2002). Time of initiation is assumed to be ti = 21.8 wk. This leads to the
dimensionless time (x) and growth quantifier equations

t-u -crc t 26.0 0.6 t-21.2
x= + =-- +0.600=, xi = 0.075 (22)
/2o- 2 8.00 8.00

Q = l kx)[erf x erf x, [exp(- x2)- exp(- x,)]} exp(v-ccx) (23)

= {0.625[erf x 0.085]- 2.821[exp(- x2)- 0.9944]- 1.094

Values of model variables in Table 2 are calculated from Eqs. (22) and (23). Correlation of
biomass (Y) with the growth quantifier is shown in Figure 4, where the line is drawn from

Y = -0.069 + 3.97Q r = 0.980 (24)

with an intercept of essentially zero, in accordance with the growth model. In fact ti = 21.8 wk
has been chosen to make the intercept zero. The phase relations for growth are given by









with dimensionless time, x, defined in terms of calendar time, t, by


t p 2o'c
x-= +o- (11)
Jj- 2

where model parameters are defined by: p is time to the mean of the energy distribution, wk; ac
is time spread of the energy distribution, wk; c is the aging coefficient, wk-1; and k is the partition
coefficient between light-gathering and structural components of the plant. Note that xi
corresponds to the time of initiation of growth, ti. The error function, erf, in Eq. (10) is defined
by

2 x
erf x = exp(-u (12)
0

where u is the variable of integration. Values of the error function can be obtained from
mathematical tables (Abramowitz and Stegun, 1965).

DATA ANALYSIS

Data for this analysis are adapted from field studies with switchgrass (cv. 'Cave-in-Rock') at
Meade, NE and Ames, IA by Vogel et al. (2002). Applied nitrogen levels were 0, 60, 120, 180,
240, and 300 kg ha-1. Measurements included biomass (dry matter) and plant nitrogen uptake.
Samples were also collected periodically to measure growth from the beginning of the season.
This analysis will focus on the study at Ames, IA where the soil was a Webster-Nicollet complex
(fine-loamy, mixed, mesic Typic Haplaquoll-Aquic Hapludoll). No supplemental irrigation was
provided.
Nitrogen response is given in Table 1 and shown in Figure 1 for the first cutting only. The
increase of all three measurements with increase in applied N may be noted. The procedure for
evaluation of model parameters is now discussed. Equations (1) and (2) can be linearized to the
forms


Zy= -ln ( 1)=-by +cN= 0.39+0.0130N r= 0.9982 (13)


Z, =-n 152-1 = -b, +cN= -0.89+0.0131N r=0.9971 (14)


where Ay = 11.60 Mg ha-1 and An = 152 kg ha- have been chosen to give the same value of cn.
Note the high correlation coefficients (r > 0.99). The response equations now become

11.60
Y =11.60 (15)
1 + exp(-0.39 0.0130N)
152
N, = (16)
1 + exp(0.89 0.0130N)









Y KY 1
+ Y = 0.0208 + 0.00679Y r = 0.957 (25)
Nu N,, N..
NumY 147Y (
SK + Y 3.06 + Y

Results are shown in Figure 5, where the line and curve are drawn from Eqs. (25) and (26),
respectively. It follows from Eq. (26) that plant N concentration is given by

N, 147
SN 147 (27)
Y 3.06 + Y

The growth response curves in Figure 3 are drawn from Eqs. (22) through (27), with model
estimates listed in Table 3. Maximum plant N concentration at t = 21.8 wk where Y-- 0 is Nc =
147/3.06 = 48.0 g kg"1. The model describes results reasonably well.

DISCUSSION

The extended logistic model appears to give reasonable correlation of yield, plant N uptake,
and plant N concentration with applied N (Figure 1) for switchgrass grown at Ames, IA. This
leads to excellent phase plots of yield vs. plant N uptake (Figure 2). Prediction of a linear
relationship between plant N concentration and plant N uptake is confirmed. Dependence of
yield, plant N uptake, and plant N concentration on calendar time is described adequately by the
expanded growth model (Figure 3). The linear relationship between the growth quantifier and
biomass accumulation with time is confirmed (Figure 4). The linear relationship between
yield/plant N uptake ratio and biomass accumulation predicted by the model is also confirmed
(Figure 5).
Several inferences follow from this analysis. The point of maximum slope of the yield
response curve is determined by

N -39 kg ha, 2 -= 5.8 Mg ha' (28)
c, 0.0130 2

The negative value indicates that more than enough nitrogen is already present in the soil to
reach 50% of maximum yield, which may be noted in Figure 1. The point of maximum slope of
the plant N response curve is determined by


N = N/2_ b 0.89 = 68 kg ha-1, (N,),,2 A= = 76 kg ha1 (29)
cn 0.0130 2

Differential response of plant N uptake at this point is given by

SdN. Ac (152)(0.0130)
dN 0.49 (30)










This means that maximum efficiency of nitrogen recovery is 49%. Actual efficiency of plant N
recovery, E, is defined by

E=N-N.o (31)
N

where Nuo is plant N uptake at N = 0. Peak nitrogen recovery, Ep, can be estimated from
(Overman, 2006a)

b A. 1 4 1 1
NP = 1.5 b E = (32)
N a p 4 [1.5b, 1+ exp(-0.5bn) 1+exp(b,)

For the Ames, IA study, these values become Np = 102 kg ha-1 and Ep = 0.47 = 47%.
The logistic model can be used to establish bounds on the response variables. According to
Eq. (17) maximum plant N concentration is Ncm = 13.1 g kg-'. Now the lower limit, NcI, can be
estimated from Eqs. (5), (6), and (8) as


N = = N,,, exp(-Ab)= 13.1exp(-1.28)= 3.64 g kg- (33)


which is also the intercept in Eq. (21). It follows that response variables are bounded by 0 < Y<
11.6 Mg ha-1, 0 The expanded growth model can be used to establish bounds on the growth variables.
According to Eq. (23) the upper bound on the growth quantifier is Q 3.695. From Eqs. (24)
and (26) the upper bounds on biomass and plant N uptake are Y -- 14.60 Mg ha- and Nu -+ 122
kg ha-1. The lower bound on plant N concentration is Nc = N/Y- 8.3 g kg-, compared to an
upper bound of Nc -- 147/3.06 = 48.0 g kg-1. It follows that the growth variables are bounded by
0 < Y< 14.60 Mg ha1, 0 < Nu < 122 kg ha', and 48.0 > Nc > 8.3 g kg-1. This behavior may be
noted in Figure 3. The decline in plant N concentration with age is due to a shift from dominance
by the light-gathering component (higher plant N concentration) to that of the structural
component (lower plant N concentration) as the plant ages.
Since Njc and Ncm are characteristics of the plant, it follows from Eq. (33) that Ab is
characteristic of the plant. There is some indication that parameter c may relate to availability of
nutrients in the soil (Overman, 2006b). Parameters A, and An have been shown to relate to water
availability and to plant population. Furthermore, it appears from the definition Ncm = A,/Ay that
this ratio is a characteristic of the plant.
Some investigators use the quadratic model for data analysis. This is due in part because it
represents a linear model in the regression sense (linear in the model parameters). Data from the
Ames, IA study are now analyzed by this approach. Assume yield and plant N uptake response to
applied nitrogen are described by the quadratic equations.

Y = ao + aN + a2N2 (34)
N, = a'+a'N+a N2 (35)









Y KY 1
+ Y = 0.0208 + 0.00679Y r = 0.957 (25)
Nu N,, N..
NumY 147Y (
SK + Y 3.06 + Y

Results are shown in Figure 5, where the line and curve are drawn from Eqs. (25) and (26),
respectively. It follows from Eq. (26) that plant N concentration is given by

N, 147
SN 147 (27)
Y 3.06 + Y

The growth response curves in Figure 3 are drawn from Eqs. (22) through (27), with model
estimates listed in Table 3. Maximum plant N concentration at t = 21.8 wk where Y-- 0 is Nc =
147/3.06 = 48.0 g kg"1. The model describes results reasonably well.

DISCUSSION

The extended logistic model appears to give reasonable correlation of yield, plant N uptake,
and plant N concentration with applied N (Figure 1) for switchgrass grown at Ames, IA. This
leads to excellent phase plots of yield vs. plant N uptake (Figure 2). Prediction of a linear
relationship between plant N concentration and plant N uptake is confirmed. Dependence of
yield, plant N uptake, and plant N concentration on calendar time is described adequately by the
expanded growth model (Figure 3). The linear relationship between the growth quantifier and
biomass accumulation with time is confirmed (Figure 4). The linear relationship between
yield/plant N uptake ratio and biomass accumulation predicted by the model is also confirmed
(Figure 5).
Several inferences follow from this analysis. The point of maximum slope of the yield
response curve is determined by

N -39 kg ha, 2 -= 5.8 Mg ha' (28)
c, 0.0130 2

The negative value indicates that more than enough nitrogen is already present in the soil to
reach 50% of maximum yield, which may be noted in Figure 1. The point of maximum slope of
the plant N response curve is determined by


N = N/2_ b 0.89 = 68 kg ha-1, (N,),,2 A= = 76 kg ha1 (29)
cn 0.0130 2

Differential response of plant N uptake at this point is given by

SdN. Ac (152)(0.0130)
dN 0.49 (30)








N a' +aN+a'N2
N, -= 0 (36)
Y ao + aN +a2N2

Equation (36) is defined automatically from Eqs. (34) and (35). The next step is to evaluate
model parameters by regression analysis.

Yield response

Regression analysis by the least squares criterion leads to the three simultaneous equations
for yield

Y = nao + N)a, + NN2)a2
CNY= (N)ao + (N2 )a + (N3) 2 (37)
SN2Y= ( N2 )a, +( N 3 )a +(N)a

where n is the number of observations and the sums are over the six observations. Since the
equations are linear in the parameters (a0, al, a2) the procedure for evaluation is called linear
regression. The system of equations can be written in matrix form as

n IN IN2 0
ZN ZN2 1N3 al = [ NY (38)
IN N N3 N4 la 2 YN 2y

This represents three simultaneous equations in three unknowns (a0, a,, a2). Note that elements
of the coefficient matrix and the right-hand vector can be calculated directly from the data. The
system can be solved provided that the determinant of the coefficient matrix does not vanish. For
the present set of data Eq. (38) becomes (with scaling of N/100)

6 9.00 19.8 Iao 59.48
9.00 19.8 48.6 a1 = 98.604 (39)
19.8 48.6 126.8784 ia2 220.968

We can solve for parameters ao, a and a2 by Cramer's rule using determinants (Ayers, 1962)

n IN YN2 6 9.00 19.8
D= N N2 N3 = 9.00 19.8 48.6 =182.89 (40)
SN2 N3 ZN4 19.8 48.6 126.8784









Y ZN
Do = NY N2
ZN2Y ZN3
n EY
D = ZN ZNY
ZN2 ZN2Y
n ZN
Da = ZN ZN2
N2 ZN3


ZN2 59.48
ZN3 = 98.604
ZN4 220.968
ZN2 6 5
ZN3 = 9.00 9
ZIN4 19.8 22
EY 6 9.
ZNY = 9.00 41
ZN2Y 19.8 48


9.(
19
48
;9.48
8.60
00.9
00
>.8
.6


00 19.8
'.8 48.6 =1247.12
.6 126.8784
8 19.8
4 48.6 =665.69
68 126.8784
59.48
98.604 = -131.09
220.968


Parameters are now estimated from

D. 1247.12
ao_ = 6.82
D 182.89
a, = 10-2 665.6910-2 = 0.0364
D 182.89
a2 = D"10-4 -131.0910-4 =-0.0000717
D 182.89

The regression equation for yield becomes

Y= 6.82 + 0.0364N 0.0000717N2

Plant N uptake response

Regression analysis leads to the three simultaneous equations for plant N uptake

Z N = nao +(2 N)a + (IN2 )a'
ZNN, = (N)a + N22 +(N3 )
Nw2Nh =n N2)b + N3wn)a + N a a

which can be written in matrix form as


n
SZN
ZN2


ZN
IN2
ZN3


NN2 [a0
N'3 a;
N 4 a


For the present set of data Eq. (49) becomes (with scaling of N/100 and Nu/100)


(41)



(42)



(43)


(44)

(45)

(46)




(47)


(48)


(49)









6 9.00 19.8 Fa 6.235
9.00 19.8 48.6 a' = 11.494
19.8 48.6 126.8784 aJ 26.6245

We can solve for a'o, a', and a2 by Cramer's rule using determinants


n I
D= N I
YN2



N2Y
n

Da>; =N N


N2 :
n

ZN2 ,


N
N2
N3




-4
EN

IN


2

ZN
NN2
NN3


-N:


5NN


9.00
19.8
48.6


V NN2 6.2:
r2 -N3 = 11.4
r3 N4 26.6

N2 6
IN' = 9.00
u -N4 19.8
EN 6
SNN, = 9.00
SN2Nu 19.8


19.8
48.6 =182.89
126.8784


35
94


9.(
19
48


6.235
11.49
26.62e
9.00
19.8
48.6


00 19.8
.8 48.6 = 79.769
.6 126.8784
5 19.8
4 48.6 =104.75
45 126.8784
6.235
11.494 = -14.194
26.6245


Parameters are now estimated from

D, 79.769
a0 =- 1 = 43.6
D 182.89
a, =b 102 .10-2 = 104.75 = 0.573
D 182.89
a = 102 10-4 -= -1419410-2 = -0.000776
D 182.89

The regression equation for plant N uptake becomes

, = 43.6 + 0.573N 0.000776N2

It follows that plant N concentration is described by


N.
Y


43.6 + 0.573N 0.000776N2
6.82 + 0.0364N 0.0000717N2


(50)


(51)



(52)



(53)



(54)


(55)

(56)

(57)




(58)


(59)








Response plots are shown in Figure 6, where the curves are drawn from Eqs. (47), (58), and
(59). The correlation coefficients for yield (Y), plant N uptake (Nu), and plant N concentration
(Nc) are r = 0.9912, 0.99971, and 0.9936, respectively. It is obvious that the quadratic model fits
the data very well. However, for extrapolations above N= 300 kg ha'1 the curves exhibit patterns
which are not consistent with data from other studies at higher applied N values, where the
curves all tend to approach plateaus (Overman, 2006a). For N< 0 (reduced soil N) the curves go
negative, which is incompatible with the definitions of positive response variables. In fact values
go negative at Y= 0 N= -145, 650 kg ha- and Nu = 0 -- N= -70, 810 kg ha1.
The phase plots (Y and Nc vs. N,) are shown in Figure 7, where the curves are again drawn
from Eqs. (47), (58), and (59). While the curves pass through the data points rather well, these
exhibit very strange behavior outside the range of applied N used in the experiments.
The expanded growth model can be used to estimate partitioning ofbiomass into light-
gathering and structural components. For the data from Ames, IA the growth quantifier is given
by Eq. (23). It follows that the light-gathering growth quantifier (QL) and structural growth
quantifier (Qs) are defined by

QL = {[erf x erf x ]} -exp(J o-cxi )= {[erf x 0.085]} 1.094 (60)

Qs = kxi [erf x erf x, ]- -= [exp(-x )- exp(- x )] exp(V2o-c(1)
I,7 t J(61)
= {- 0.375[erf x 0.085]- 2.821[exp(- x2)- 0.9944D. 1.094

The light-gathering fraction (fL) is then defined by

fL = L (62)
Q

Values are listed in Table 4 and shown in Figure 8. This allows simulation ofbiomass for light-
gathering (YL) and structural ( 7s) components from

L =fL (63)
is =(1-fL)- (64)

as given in Table 5. In order to simulate plant nitrogen partitioning it is necessary to make some
assumption about plant N concentration. The choice is made to assume that plant N
concentration in the structural component (N9s) remains constant at 6.0 g kg' during growth.
Plant N uptake in the structural (N^s) and light-gathering (N,,) components are estimated by

Ns = Ns .Y = 6.0Ys (65)
NuL = N N.s (66)









as given in Table 5. Estimates of plant N concentration in the light-gathering component ( N )
are made from


Nc=L (67)
L

which is listed as the last column in Table 5. Results are shown in Figure 9. It should be noted
that NcL declines as the plant ages. The last step should be considered as somewhat speculative at
this point.

CONCLUSIONS

The extended logistic model of seasonal response of biomass yield, plant N uptake, and plant
N concentration to applied N has been shown to describe field data of switchgrass grown at
Ames, IA rather well. The model predicts a linear relationship between plant N concentration
and plant N uptake, which has been confirmed from the data. While a quadratic model can be
used to describe response data, it leads to inconsistencies with patterns observed from studies
with other grasses. A proper model should describe connections among variables in a given
study, but should be consistent among locations, times (years), and environmental conditions.
Model parameters may change with physical conditions, but the form of the model should
remain unchanged. This has been a guiding principle in physics for over four hundred years.
The expanded growth model has been shown to describe accumulation ofbiomass with
calendar time by the process of photosynthesis. A linear relationship between biomass
accumulation and the growth quantifier defined in the model has been confirmed. Parameter
values appear to be the same as those for a different grass grown at Tifton, GA. The model
assumes a linear relationship between biomass/plant N ratio and biomass accumulation with
calendar time, which has been confirmed from the data. This phase relation implies that biomass
accumulation by photosynthesis is the rate limiting step in plant growth, and that plant nitrogen
accumulates in virtual equilibrium with biomass. This is consistent with results for other crops as
well.
It should be noted that this analysis has utilized analytical functions to describe the data. This
offers distinct advantages over finite difference procedures in that the models are well behaved
and avoid problems with numerical stability in simulations. Rucker (1987) has pointed out the
limitations of a truncated power series. It has been shown previously that effects due to water
availability (rainfall or irrigation) are accounted for in the linear model parameters (Overman and
Scholtz, 2002). This greatly simplifies model application.
Peak efficiency of plant N utilization can be estimated from Eq. (32) where values of model
parameters A,, bn, and c, are available. Results are dependent on plant species, soil
characteristics, and water availability. Typical values range between 50% and 80%.









as given in Table 5. Estimates of plant N concentration in the light-gathering component ( N )
are made from


Nc=L (67)
L

which is listed as the last column in Table 5. Results are shown in Figure 9. It should be noted
that NcL declines as the plant ages. The last step should be considered as somewhat speculative at
this point.

CONCLUSIONS

The extended logistic model of seasonal response of biomass yield, plant N uptake, and plant
N concentration to applied N has been shown to describe field data of switchgrass grown at
Ames, IA rather well. The model predicts a linear relationship between plant N concentration
and plant N uptake, which has been confirmed from the data. While a quadratic model can be
used to describe response data, it leads to inconsistencies with patterns observed from studies
with other grasses. A proper model should describe connections among variables in a given
study, but should be consistent among locations, times (years), and environmental conditions.
Model parameters may change with physical conditions, but the form of the model should
remain unchanged. This has been a guiding principle in physics for over four hundred years.
The expanded growth model has been shown to describe accumulation ofbiomass with
calendar time by the process of photosynthesis. A linear relationship between biomass
accumulation and the growth quantifier defined in the model has been confirmed. Parameter
values appear to be the same as those for a different grass grown at Tifton, GA. The model
assumes a linear relationship between biomass/plant N ratio and biomass accumulation with
calendar time, which has been confirmed from the data. This phase relation implies that biomass
accumulation by photosynthesis is the rate limiting step in plant growth, and that plant nitrogen
accumulates in virtual equilibrium with biomass. This is consistent with results for other crops as
well.
It should be noted that this analysis has utilized analytical functions to describe the data. This
offers distinct advantages over finite difference procedures in that the models are well behaved
and avoid problems with numerical stability in simulations. Rucker (1987) has pointed out the
limitations of a truncated power series. It has been shown previously that effects due to water
availability (rainfall or irrigation) are accounted for in the linear model parameters (Overman and
Scholtz, 2002). This greatly simplifies model application.
Peak efficiency of plant N utilization can be estimated from Eq. (32) where values of model
parameters A,, bn, and c, are available. Results are dependent on plant species, soil
characteristics, and water availability. Typical values range between 50% and 80%.









REFERENCES


Abramowitz, M. and I.A. Stegun. 1965. Handbook of Mathematical Functions. Dover
Publications. New York, NY. 1046 p.
Ayers, F. 1962. Theory and Problems ofMatrices. McGraw-Hill. New York, NY. 219 p.
Overman, A.R. 1998. An expanded growth model for grasses. Communications in Soil Science
and Plant Analysis 29:67-85.
Overman, A. R. 2006a. A Memoir on Crop Yield and Nutrient Uptake. University of Florida.
Gainesville, FL. 116 p. (46 Tables and 62 Figures).
http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00072010&v=00001
Overman, A. R. 2006b. A Memoir on Chemical Transport: Application to Soils and Crops.
University of Florida. Gainesville, FL. 364 p. (13 Tables and 34 Figures).
http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00072282&v=00001
Overman, A.R. and K.H. Brock. 2003. Confirmation of the expanded growth model for a warm-
season perennial grass. Communications in Soil Science and Plant Analysis 34:1105-1117.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. Philadelphia, PA. 328 p.
Overman, A.R., S.R. Wilkinson, and D.M. Wilson. 1994. An extended model of forage grass
response to applied nitrogen. Agronomy J. 86:617-620.
Rucker, R. 1987. Mind Tools: The Five Levels of Mathematical Reality. Houghton Mifflin Co.
Boston, MA. 328 p.
Vogel, K.P., J.J. Brejda, D.T. Walters, and D.R. Buxton. 2002. Switchgrass biomass production
in the Midwest USA: Harvest and nitrogen management. Agronomy J. 94:413-420.









as given in Table 5. Estimates of plant N concentration in the light-gathering component ( N )
are made from


Nc=L (67)
L

which is listed as the last column in Table 5. Results are shown in Figure 9. It should be noted
that NcL declines as the plant ages. The last step should be considered as somewhat speculative at
this point.

CONCLUSIONS

The extended logistic model of seasonal response of biomass yield, plant N uptake, and plant
N concentration to applied N has been shown to describe field data of switchgrass grown at
Ames, IA rather well. The model predicts a linear relationship between plant N concentration
and plant N uptake, which has been confirmed from the data. While a quadratic model can be
used to describe response data, it leads to inconsistencies with patterns observed from studies
with other grasses. A proper model should describe connections among variables in a given
study, but should be consistent among locations, times (years), and environmental conditions.
Model parameters may change with physical conditions, but the form of the model should
remain unchanged. This has been a guiding principle in physics for over four hundred years.
The expanded growth model has been shown to describe accumulation ofbiomass with
calendar time by the process of photosynthesis. A linear relationship between biomass
accumulation and the growth quantifier defined in the model has been confirmed. Parameter
values appear to be the same as those for a different grass grown at Tifton, GA. The model
assumes a linear relationship between biomass/plant N ratio and biomass accumulation with
calendar time, which has been confirmed from the data. This phase relation implies that biomass
accumulation by photosynthesis is the rate limiting step in plant growth, and that plant nitrogen
accumulates in virtual equilibrium with biomass. This is consistent with results for other crops as
well.
It should be noted that this analysis has utilized analytical functions to describe the data. This
offers distinct advantages over finite difference procedures in that the models are well behaved
and avoid problems with numerical stability in simulations. Rucker (1987) has pointed out the
limitations of a truncated power series. It has been shown previously that effects due to water
availability (rainfall or irrigation) are accounted for in the linear model parameters (Overman and
Scholtz, 2002). This greatly simplifies model application.
Peak efficiency of plant N utilization can be estimated from Eq. (32) where values of model
parameters A,, bn, and c, are available. Results are dependent on plant species, soil
characteristics, and water availability. Typical values range between 50% and 80%.









Table 1. Response of biomass yield (Y), plant nitrogen uptake (Nu), and plant nitrogen
concentration (Nc) to applied nitrogen (N) by switchgrass grown at Ames, IA. Averaged over
1994 and 1995.1
N Y Nu Nc


kg ha' Mg ha1' kg ha-'
0 6.59 43.8
60 9.15 74.7
120 10.16 101
180 10.89 123
240 11.25 135
300 11.44 146
IData adapted from Vogel et al. (2002).


g kg-1
6.65
8.16
9.94
11.3
12.0
12.8


Table 2. Accumulation ofbiomass (Y), plant nitrogen (Nu), and plant nitrogen concentration (Nc)
with calendar time (t) from Jan. 1 by switchgrass grown at Ames, IA. Averaged over 1994 and
1995 and applied nitrogen.1
t x erfx exp(-x2) Q Y Nu Nc
wk Mg ha' kg ha' g kg'
21.8 0.0750 0.085 0.9944 0.000 ----- ---- -----
25.8 0.5750 0.584 0.718 1.194 4.6 85 18.5
27.7 0.8125 0.750 0.517 1.928 6.7 97 14.5
28.8 0.9500 0.821 0.406 2.320 9.9 120 12.1
29.8 1.0750 0.872 0.315 2.635 11.1 129 11.6
30.8 1.2000 0.910 0.237 2.902 11.7 117 10.0
32.3 1.3875 0.9503 0.146 3.210 12.6 118 9.37
33.7 1.5625 0.9728 0.0870 3.408 12.8 108 8.44
'Crop data adapted from Vogel et al. (2002).









(Y), plant nitrogen (Nu), and plant nitrogen concentration (Nc)


with calendar time (t) from Jan. 1 by switchgrass grown at Ames, IA.
t x erfx exp(-x2) Q
wk Mg ha- kg ha' g kg'1
21.8 0.0750 0.085 0.9944 0.000 0.00 0.0 48.0
23 0.225 0.250 0.9506 0.248 0.916 33.9 37.0
24 0.350 0.379 0.8847 0.540 2.08 59.5 28.6
25 0.475 0.499 0.798 0.889 3.46 78.0 22.5
26 0.600 0.604 0.698 1.270 4.97 91.0 18.3
27 0.725 0.695 0.591 1.662 6.53 100.1 15.3
28 0.850 0.771 0.486 2.038 8.02 106.4 13.3
29 0.975 0.832 0.386 2.389 9.42 111.0 11.8
30 1.100 0.880 0.298 2.693 10.62 114.1 10.7
31 1.225 0.9173 0.223 2.950 11.64 116.4 10.0
32 1.350 0.9438 0.162 3.157 12.46 118.0 9.47
33 1.475 0.9630 0.114 3.318 13.10 119.2 9.10
34 1.600 0.9763 0.0773 3.440 13.59 120.0 8.83
35 1.725 0.9853 0.0510 3.528 13.94 120.5 8.65
36 1.850 0.9911 0.0326 3.588 14.18 120.9 8.53
38 2.100 0.9970 0.0122 3.655 14.44 121.3 8.40
40 2.350 0.9991 0.0040 3.682 14.55 121.46 8.35
00 1. 0. 3.695 14.60 121.53 8.32

Table 4. Simulation of growth quantifier (Q), light-gathering component (QL), structural
component (Qs), and ligt-gathering fraction (fL) with calendar time (t) from Jan. 1 by switchgrass
grown at Ames, IA.
t x erfx exp(-x2) Q QL Qs fL
wk
21.8 0.0750 0.085 0.9944 0.000 0.000 0.000
23 0.225 0.250 0.9506 0.248 0.181 0.067 0.730
24 0.350 0.379 0.8847 0.540 0.322 0.218 0.596
25 0.475 0.499 0.798 0.889 0.453 0.436 0.510
26 0.600 0.604 0.698 1.270 0.568 0.702 0.447
27 0.725 0.695 0.591 1.662 0.667 0.995 0.401
28 0.850 0.771 0.486 2.038 0.750 1.288 0.368
29 0.975 0.832 0.386 2.389 0.817 1.572 0.342
30 1.100 0.880 0.298 2.693 0.870 1.823 0.323
31 1.225 0.9173 0.223 2.950 0.910 2.040 0.308
32 1.350 0.9438 0.162 3.157 0.940 2.217 0.298
33 1.475 0.9630 0.114 3.318 0.961 2.357 0.290
34 1.600 0.9763 0.0773 3.440 0.975 2.465 0.283
35 1.725 0.9853 0.0510 3.528 0.985 2.543 0.279
36 1.850 0.9911 0.0326 3.588 0.991 2.597 0.276
38 2.100 0.9970 0.0122 3.655 0.998 2.657 0.273
40 2.350 0.9991 0.0040 3.682 1.000 2.682 0.272
00 1. 0. 3.695 1.001 2.694 0.271


Table 3. Simulation ofbiomass









Table 5. Simulation of biomass (fL and 1s), plant N uptake (NL and s,), and plant N
concentration ( NR and s ) with calendar time (t) from Jan. 1 for light-gathering and structural
components, respectively.
t fL AL Ys R NuS NuL cL
wk Mg ha- Mg ha1 Mg ha- kg ha kg ha-1 kg ha-' g kg1
21.8 ------ 0.00 0.00 0.00 0.0 0.0 0.0
23 0.730 0.916 0.669 0.247 33.9 1.5 32.4 48.4
24 0.596 2.08 1.24 0.84 59.5 5.0 54.5 43.9
25 0.510 3.46 1.76 1.70 78.0 10.2 67.8 38.5
26 0.447 4.97 2.22 2.75 91.0 16.5 74.5 33.6
27 0.401 6.53 2.62 3.91 100.1 23.5 76.6 29.2
28 0.368 8.02 2.95 5.07 106.4 30.4 76.0 25.8
29 0.342 9.42 3.22 6.20 111.0 37.2 73.8 22.9
30 0.323 10.62 3.43 7.19 114.1 43.1 71.0 20.7
31 0.308 11.64 3.59 8.05 116.4 48.3 68.1 19.0
32 0.298 12.46 3.71 8.75 118.0 52.5 65.5 17.7
33 0.290 13.10 3.80 9.30 119.2 55.8 63.4 16.7
34 0.283 13.59 3.85 9.74 120.0 58.4 61.6 16.0
35 0.279 13.94 3.89 10.05 120.5 60.3 60.2 15.5
36 0.276 14.18 3.91 10.27 120.9 61.6 59.3 15.2
38 0.273 14.44 3.94 10.50 121.3 63.0 58.3 14.8
40 0.272 14.55 3.96 10.59 121.46 63.5 58.0 14.6
co 0.271 14.60 3.96 10.64 121.53 63.8 57.7 14.6









List of Figures


Figure 1. Response of biomass yield (Y), plant nitrogen uptake (Nu), and plant nitrogen
concentration (Nc) to applied nitrogen (N) for switchgrass grown at Ames, IA. Data adapted from
Vogel et al. (2002). Curves drawn from Eqs. (15) through (17).

Figure 2. Phase plots between biomass yield (Y) and plant nitrogen concentration (Nc) and plant
nitrogen uptake (Nu) for switchgrass grown at Ames, IA. Data adapted from Vogel et al. (2002).
Curve and line drawn from Eqs. (20) and (21), respectively.

Figure 3. Accumulation ofbiomass yield (Y), plant nitrogen (Nu), and plant nitrogen
concentration (Nc) with calendar time (t) for switchgrass grown at Ames, IA. Data adapted from
Vogel et al. (2002). Curves drawn from Eqs. (22) through (27).

Figure 4. Correlation ofbiomass (Y) with growth quantifier (Q) for switchgrass grown at Ames,
IA. Yield data adapted from Vogel et al. (2002). Line drawn from Eq. (24).

Figure 5. Phase plots of plant nitrogen uptake (Nu) and yield/plant N uptake ratio (Y/Nu) vs.
biomass (Y) for switchgrass grown at Ames, IA. Data adapted from Vogel et al. (2002). Line and
curve drawn from Eqs. (25) and (26), respectively.

Figure 6. Response ofbiomass yield (Y), plant nitrogen uptake (Nu), and plant nitrogen
concentration (Nc) to applied nitrogen (N) for switchgrass grown at Ames, IA. Data adapted from
Vogel et al. (2002). Curves drawn from Eqs. (47), (58), and (59) for the quadratic model.

Figure 7. Phase plots between biomass yield (Y) and plant nitrogen concentration (Nc) and plant
nitrogen uptake (Nu) for switchgrass grown at Ames, IA. Data adapted from Vogel et al. (2002).
Curves drawn from Eqs. (47), (58), and (59) for the quadratic model.

Figure 8. Simulation of the growth quantifier for light-gathering (QL) and structural (Qs)
components, and light-gathering fraction (fL) with calendar time (t) from Jan. 1 for switchgrass
grown at Ames, IA. Curves are drawn from values in Table 4.

Figure 9. Simulation of biomass (YL and s ), plant N uptake (NuL and Nus), and plant N
concentration ( 9 and Ns ) with calendar time (t) from Jan. 1 for light-gathering and structural
components, respectively. Curves are drawn from Table 5.







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MODEL RESPONSE OF SWITCHGRASS TO APPLIED NITROGEN AND HARVEST
INTERVAL

ALLEN R. OVERMAN

Agricultural and Biological Engineering Department, University of Florida,
Gainesville, FL 32611-0570


INTRODUCTION

Mathematical models are useful for describing output (response variables) in response to
input (control variables). Extensive field studies have been conducted in various parts of the
world over the past 150 years on crop response to applied nutrients and other factors, such as
crop species, soil type, harvest interval for perennial grasses, environmental conditions (such as
water availability), and plant population. The present analysis will focus on two particular
models. The extended logistic model (Overman et al., 1994) describes coupling of seasonal
biomass yield and plant nitrogen uptake with applied nutrients (such as nitrogen) through logistic
equations. Biomass yield is then linked to plant nitrogen uptake through a hyperbolic phase
relation, which predicts a linear relationship between plant nitrogen concentration and plant
nitrogen uptake. The expanded growth model (Overman, 1998) describes accumulation of
biomass with time through an analytical function which incorporates effects of energy input,
partitioning of biomass between light-gathering and structural components, and aging as the
plant grows. Plant nutrient and biomass accumulation are coupled through a hyperbolic phase
relation. Field data have been used to confirm this model for coastal bermudagrass (Cynodon
dactylon L.) grown in Georgia (Overman and Brock, 2003). The growth model predicts a linear-
exponential dependence of seasonal yield with a fixed harvest interval for perennial grasses. All
of the results have been shown to be consistent with field data for numerous crops, soils, and
environmental conditions (Overman and Scholtz, 2002).
This analysis will focus on response of the warm-season perennial switchgrass (Panicum
virgatum L.) to applied nitrogen at various harvest intervals.

MODEL DESCRIPTION

Response of biomass yield and plant nitrogen uptake to applied nitrogen can be described by
the extended logistic model given by


Y= Ay (1)
1+ exp(by c.N)

N. = A,(2)
1+exp(b.-cN)
1 + exp(b exp( -c.N)N)
N [1 + exp(by N) (3)
Y 1l +exp(b, c(N)









MODEL RESPONSE OF SWITCHGRASS TO APPLIED NITROGEN AND HARVEST
INTERVAL

ALLEN R. OVERMAN

Agricultural and Biological Engineering Department, University of Florida,
Gainesville, FL 32611-0570


INTRODUCTION

Mathematical models are useful for describing output (response variables) in response to
input (control variables). Extensive field studies have been conducted in various parts of the
world over the past 150 years on crop response to applied nutrients and other factors, such as
crop species, soil type, harvest interval for perennial grasses, environmental conditions (such as
water availability), and plant population. The present analysis will focus on two particular
models. The extended logistic model (Overman et al., 1994) describes coupling of seasonal
biomass yield and plant nitrogen uptake with applied nutrients (such as nitrogen) through logistic
equations. Biomass yield is then linked to plant nitrogen uptake through a hyperbolic phase
relation, which predicts a linear relationship between plant nitrogen concentration and plant
nitrogen uptake. The expanded growth model (Overman, 1998) describes accumulation of
biomass with time through an analytical function which incorporates effects of energy input,
partitioning of biomass between light-gathering and structural components, and aging as the
plant grows. Plant nutrient and biomass accumulation are coupled through a hyperbolic phase
relation. Field data have been used to confirm this model for coastal bermudagrass (Cynodon
dactylon L.) grown in Georgia (Overman and Brock, 2003). The growth model predicts a linear-
exponential dependence of seasonal yield with a fixed harvest interval for perennial grasses. All
of the results have been shown to be consistent with field data for numerous crops, soils, and
environmental conditions (Overman and Scholtz, 2002).
This analysis will focus on response of the warm-season perennial switchgrass (Panicum
virgatum L.) to applied nitrogen at various harvest intervals.

MODEL DESCRIPTION

Response of biomass yield and plant nitrogen uptake to applied nitrogen can be described by
the extended logistic model given by


Y= Ay (1)
1+ exp(by c.N)

N. = A,(2)
1+exp(b.-cN)
1 + exp(b exp( -c.N)N)
N [1 + exp(by N) (3)
Y 1l +exp(b, c(N)









where N is applied nitrogen, kg ha-1; Y is seasonal total biomass yield, Mg ha-1; Nu is seasonal
total plant nitrogen uptake, kg ha-1; Nc is plant nitrogen concentration, g kg''; Ay is maximum
yield at high N, Mg ha-'; A, is maximum plant nitrogen uptake at high N, kg ha'1; Ncm = An/Ay is
maximum plant nitrogen concentration at high N, g kg-'; by is intercept parameter for plant yield;
bn is intercept parameter for plant nitrogen uptake; and Cn is response coefficient for applied
nitrogen, ha kg-'. Note that the units on Cn are the reciprocal of those for N. Equations (1) through
(3) are well-behaved monotone increasing functions. Since variables N, Y, and Nu are defined as
positive, parameters Ay, An, and cn must be positive as well, while parameters by and bn can be
either positive, zero, or negative. Equations (1) and (2) with common cn can be combined to give
the hyperbolic phase relation between Y and Nu

Y = Ymu (4)
K, + N,

where the hyperbolic and logistic parameters are coupled through

Ay
Y.= A (5)
1 exp(-Ab)

K, A (6)
exp(Ab) -1

with the shift in intercept parameters defined by

Ab = b, by (7)

For Ym and Kn to be positive requires that Ab > 0. Equation (4) can be rearranged to the linear
form

NU N- _.K + 1 N (8)
Y Y. Y.

which predicts a linear relationship between Nc and Nu.
Dependence of biomass yield and plant nitrogen uptake on harvest interval for a perennial
grass can be described by

Y = (ay + flyAt)exp(- 7 At) (9)
N, =(a,, + IAt)exp(- rAt) (10)

c Y ay + IAt









where At is a fixed harvest interval, wk; and a and / are intercept and response parameters,
respectively, for yield and plant nitrogen uptake. Parameter r is a plant aging factor and is
assumed common for yield and plant nitrogen uptake.
The expanded growth model for biomass accumulation (Y) with calendar time (t) from Jan. 1
for a perennial grass is described by

AY = AAQj (12)

where A YI is biomass yield for the i th growth interval, Mg ha-'; AQj is growth quantifier for the
i th growth interval; and A is the yield factor, Mg ha"1. The growth quantifier is defined by


AQJ = {- kx,)[erf x erf x ]- exp(- x2 )- exp(- x )]}. exp(,o-cx,) (13)


with dimensionless time, x, defined in terms of calendar time, t, by


x + (14)
j2r+ 2

where model parameters are defined by: p. is time to the mean of the energy distribution, wk; o-
is time spread of the energy distribution, wk; c is the aging coefficient, wk'1; and k is the partition
coefficient between light-gathering and structural components of the plant. Note that x,
corresponds to the time of initiation of growth, ti. The error function, erf, in Eq. (13) is defined
by

2x
erf x = exp(-u 2)du (15)


where u is the variable of integration. Values of the error function can be obtained from
mathematical tables (Abramowitz and Stegun, 1965). Cumulative growth quantifier, Q, and
biomass, Y, are then given by

Q = AQ, (16)
i
Y = AY, (17)
i

It follows from Eq. (12) that cumulative yield and growth quantifier are coupled by the linear
relationship

Y = AY, = A AQ, = AQ (18)
i i









DATA ANALYSIS


Data for this analysis are adapted from a field study by Madakadze et al. (1999) at McGill
University in Montreal, Quebec, Canada. Switchgrass (cv. 'Cave-in-Rock', 'Pathfinder', and
'Sunburst') was grown on St. Bernard sandy clay loam (Typic Hapludalf), a free draining soil.
Nitrogen treatments were 0, 75, and 150 kg ha-1 yr'" applied as ammonium nitrate. Harvest
intervals were 4, 6, and 15 wk. Measurements included dry matter yields and plant nitrogen
uptake. Surface plant residue was removed at the beginning of each season.
Results are summarized in Table 1 and shown in Figure 1 for the 1996 season. The challenge
is estimation of parameters in Eqs. (1) through (3) from the data. Results for 4- and 6-wk harvest
intervals are averaged for this purpose. Equations (1) and (2) can be linearized to the forms

Zy =-In12 -1 = -by -c,N= -0.22 + 0.00975N r = 0.9971

-0.22 + 0.0098N

Z = -In250 -1 = -b -c,N = -0.98+0.00981N r = 0.99917
.N, ) (20)
-0.98 + 0.0098N

where Ay = 12.00 Mg ha-1 and An = 250 kg ha"' have been chosen to give equal values of c,. The
next step is to estimate Ay and An for each harvest interval. On the assumption that by, bn, and cn
are common for all harvest intervals, standardized yields (Y*) and standardized plant N uptake
(Nu*) can be calculated, respectively, from

Y* = Y[i + exp(0.22 0.0098N)] = Ay (21)
Nu* = N. [1+ exp(0.98 0.0098N)]= A, (22)

for each nitrogen level and each harvest interval as shown in Table 2. This leads to estimates of
Ay(4 wk) = 11.04, Ay(6 wk) = 12.95, and Ay(15 wk) = 16.18 Mg ha-1; An(4 wk) = 250, A,(6 wk) =
250, and An(15 wk) = 127 kg ha- ; and Ncm(4 wk) = 22.6, Ncm(6 wk) = 19.3, and Ncm(15 wk) =
7.85 g kg-'. The yield and plant N response equations now become

A
Y=- = (23)
1+ exp(0.22 0.0098N)

N, = A (24)
1 + exp(0.98 0.0098N)
N = N = 1+exp(0.22-0.0098N) (25)
Yc cm1 + exp(0.98 0.0098N)

Curves in Figure 1 are drawn from Eqs. (23) through (25) with appropriate values for Ay, An, and
Ncm,
The next step is to estimate hyperbolic parameters for the phase relations. These are given by





At= 4 wk: Y. = 11.04 =20.74 Mgha-1, K = 250 = 220kgha-' (2
1 exp(-0.76) exp(0.76) -1
Y =20.74N,
Y- 20"74N. (27
220 + N
N, =10.61+ 0.0482N,, (28
12.95 250
At =6wk: Y,= 12.95 =24.33Mgha-', K, =- 250 =220kgha-' (29
exp(-0.76) exp(0.76)-1
y 24.33N, (3(
220 + N
Nc =9.04 + 0.0411N, (31

At= 15 wk: Y, = 16.18 =30.39Mgha-1, K, = 127 112kgha-' (32
1 exp(-0.76) exp(0.76) -1
Y- 30."39N (33
112+N.
Nc = 3.69 + 0.0329N. (34

Phase plots are shown in Figure 2, where curves are drawn from Eqs. (27), (30), and (33) and
lines from Eqs. (28), (31), and (34).
The second challenge is to determine response to harvest interval. This is equivalent to
determining dependence of A. and An on harvest interval. It is assumed that these relations are
given by

AY = (ac + 6 yAt)exp(- 7 At) (35
A, =(a, + ,, At)exp(- At) (3(

Following analysis by Overman and Scholtz (2002) for coastal bermudagrass, it is assumed that
y = 0.075 wk"1. It follows that standardized Ay, (Ay*), and An, (An*), can be calculated from

A; = A, exp(0.075At) = y + fyAt (3;
A = A exp(0.075At) = a, +fAt (38

for each harvest interval. Results are given in Table 3. This leads to the standardized equations

Ay =1.61+3.21At r=0.99963 (3S
A = 267 + 20.0At (4(

and to dependence of Ay, A,, and Ncm on harvest interval of

Ay = (1.61+ 3.21At)exp(- 0.075 At) (41


7)


1)


I.)


5)
')


)
))









A, = (267 + 20.0 At)exp(- 0.075 At)
A., 267 + 20.0 At
cm A 1.61+3.21At


(42)

(43)


Results are shown in Figure 3, where the curves are drawn from Eqs. (41) through (43). It can be
shown that peak harvest interval for yield (Atpy) and plant N uptake (Atp,) are determined by

1 a_ 1 1.61
Atpy 1 a 1 1.61 =12.8wk (44)
ry7 8, 0.075 3.21
1 a,, 1 267
Atp -- = 0.0 wk (45)
SY7 ,n 0.075 20.0

Values of a,, and/i,, have been chosen to make Atpn = 0, consistent with the data.
Accumulation of plant nitrogen and biomass can be coupled. Data are given in Table 4 and
shown in Figure 4. The data appear to follow straight lines for the constant harvest intervals.
Since the lines should exhibit intercepts of (Y, Nu) = (0, 0) on intuitive grounds, simple moment
analysis leads to the lines shown in Figure 5 given by


At =4 wk:


At = 6 wk:


At = 15 wk:


N=0:


N= 75 kg ha-:

N= 150 kg ha-1:

N= 0:

N= 75 kg ha4:

N= 150 kg ha'1:

N=0:

N= 75 kg ha-:

N= 150 kg ha' :


SYNU j(662.6y46
N. = )=, = 14.6Y
[Y 45.38)
(1389y=.
N. =1 Y = 16.OY
,86.57)
r2511y
N. = 2511Y = 18.0y
S139.8)
(576.2
N, =(. Y =11.3Y
S51.02)
(1100
N, = Y = 13.6Y
80.73)
(2105Y
N, ,136.0J
(350.3
N.Y = 356.9 Y=5.08Y
68.9)

Nu = Y= 5.39Y
K 93.3)
(752.6
N," 126.6) 5.95Y


It may be noted that the slopes represent plant N concentrations (Nc). This is illustrated in Figure
5, where the curves are drawn from Eq. (25) with Ncm(4 wk) = 22.6, Ncm(6 wk) = 19.3, and


(46)


(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)








Ncm(15 wk) = 7.85 g kg-' as estimated previously. Results by two different approaches are
consistent.
Finally, biomass accumulation with time can be described by the expanded growth model.
Assume the parameters: p = 26.0 wk, Fi-a = 8.00 wk, c = 0.15 wk-1, k = 5. These values lead to
dimensionless time (x) and growth quantifier (AQi) given by


t-+t o-c t-26.0 0.600= t-21.2
x=- +- --= +0.600=- --
-2o- 2 8.00 8.00

AQi = (1 kx )[erf x erf x; ]- [exp(- x2) exp(- x ) exp(i2o-cx

= {(1- 5x, )[erf x erf x, ]- 2.821[exp(- x2 )- x )} exp(l .20x,)


Estimates of the cumulative growth quantifier (Q) with calendar time (t) are given in Table 5 for
the three harvest intervals. Note that time of initiation (t4) must be reset for each growth interval.
The growing season is assumed to begin at t = 23.1 wk. The next step is to couple cumulative
biomass yield (Y) with cumulative growth quantifier (Q) as given in Table 6. Results are also
shown in Figure 6. Due to the limited number of points, slopes of the lines in Figure 6 are
estimated by simple moment analysis to be


At=4wk N=0:

N= 75 kg ha':

N= 150 kg ha&-:

At = 6 wk N= 0:

N= 75 kg ha':

N= 150 kg ha':

At= 15 wk N= 0:

N= 75 kg ha>':

N= 150 kg ha':


Y 2Q= 6.64 Q1.70Q
(36.86) =

15.71
6.86 = 2.98Q
15.71)
46. = Q = 1.93Q
13.76)
33.29 = .Q
3. 2 17 = 2.42Q
13.76)
(43.23)Q = 3.14Q
13.76
(27.56) = 2.50Q
11.02
32.0 = = 2.91Q
11.02
(37.35)
S= 11 = 3.39Q
111.02)


(55)



(56)


(57)

(58)

(59)

(60)

(61)

(62)

(63)

(64)

(65)








Correlation of yield factor (A) with applied nitrogen (N) is shown in Figure 7. It seems logical
that this correlation should be described by the logistic model

A = A- (66)
1+ exp(by -c,,N) 1+ exp(0.22- 0.0098N)

where Am is maximum A at high N, Mg ha"'. It follows from Eq. (66) that standardized A, (A*),
can be calculated for each nitrogen level and harvest interval from

A* = A[l + exp(0.22 0.0098N)] = A (67)

Results are given in Table 7. It follows from Table 7 that response of parameter A to applied
nitrogen is given by Eq. (66) with Aom(4 wk) = 3.80, Am(6 wk) = 4.08, A.o(15 wk) = 4.88 Mg ha'1.
The curves in Figure 7 are drawn from Eq. (66) with appropriate values of Am. Results are very
similar to those for Y vs. N in Figure 1.

DISCUSSION

The extended logistic model appears to give reasonable correlation of yield, plant N uptake,
and plant N concentration with applied N (Figure 1) for switchgrass grown in Canada. This leads
to excellent phase plots of yield vs. plant N uptake (Figure 2). Prediction of a linear relationship
between plant N concentration and plant N uptake is confirmed. Dependence of the phase plots
on harvest interval is illustrated. Dependence of yield, plant N uptake, and plant N concentration
parameters on harvest interval is described adequately by the expanded growth model (Figure 3).
Several inferences follow from this analysis. The point of maximum slope of the yield
response curves is determined by
by_ 0.22 A(68)
N=N1 =y = 22 22 kg ha-1, Y/2 = (68)
c,, 0.0100 2

with the corresponding value of the yield depending on harvest interval. The point of maximum
slope of the plant N response curves is determined by
N=N'2 0.98 98 kg ha-, (NA)l/2 = A (69)

c, 0.0100 2

with the corresponding value of plant N uptake depending on harvest interval. Differential
response of plant N uptake at this point is given by

dN Ac,, (248)(0.0100)=0.62 (70)
dN 4 4

for harvest intervals of 4 and 6 wk. This means that maximum efficiency of nitrogen recovery is
62% for this harvest interval. Actual efficiency of plant N recovery, E, is defined by










E-= N -N. (71)
N

where Nuo is plant N uptake at N = 0. Peak nitrogen recovery, Ep, can be estimated from
(Overman, 2006a)


N = 1.5 E = A, 1x 4 (72)
S Cn' 4 )[1.5b, +exp(-0.5b,) l+exp(b,)J

For harvest intervals of 4 and 6 wk these values become Np = 147 kg ha-' and Ep = 0.585 =
58.5%.
A dramatic decrease in maximum plant N concentration (Ncm) with increased harvest interval
(At) may be noted from Figure 3. This occurs because as the plant ages the fraction of light-
gathering component (higher N concentration) decreases in relation to the structural component
(lower N concentration) of the plant. The expanded growth model describes this phenomenon
quite well.
The lower limit of plant N concentration, Nct, at reduced soil nitrogen (N <<0) can be
estimated by combining Eqs. (5), (6), and (8) to obtain

Net = '- Non exp(-Ab) (73)


For a harvest interval of 4 wk, this value is 10.6 g kg-1. At this harvest interval the variables are
bounded by 0 < Y< 11.04 Mg ha"1, 0 < Nu < 250 kg ha-1, and 10.6 < Nc < 22.6 g kg-'. Actual
values of the variables depend on the level of applied N.
Attention is now focused in greater detail on the growth curves, particularly between
harvests. Calculations are given in Table 8 for a harvest interval of At = 4 wk. Equations (55) and
(56) are used for this purpose. Calculations cover the entire calendar year for completeness. The
slight jump in time step at t = 35.4 wk is made to conform to sampling times. Because of the
linear relationship between biomass accumulation (A Y) and growth quantifier (AQi) given by Eq.
(12), it is sufficient to focus on dependence ofAQi on calendar time (t). Resulting curves are
shown in Figure 8. Note that the first point in each growth increment represents the reference
point at which the growth quantifier is set to zero. It should also be noted that for t < 26 wk
growth curves bend upward, whereas for t > 26 wk the curves bend downward. This is due to the
Gaussian distribution of the energy driving function in the model. Overman and Scholtz (2002)
have shown that the terminus points in Figure 8 follow a Gaussian distribution with calendar
time. These values are listed in Table 9. Cumulative growth quantifier (Q) is given as the third
column in Table 9. Column 4 lists the normalized values (F), which is obtained by dividing each
value of Q by the maximum value of 4.586. Now if the distribution of AQi is Gaussian, then it
follows that F vs. t is described by the probability function


F = -~ + erf (tj (74)
2 12o-










To test this hypothesis it is convenient to write Eq. (74) in linearized form


Z= erf-'(2F-l)=- + 1 (75)


where erf designates the inverse of the error function. It should be noted that erf (0) = 0, erf (oo)
= 1, and erf (-x) = -erf (+x). Values of Z are listed in Table 9. Linear regression of Zvs. t leads
to

At = 4 wk: Z = -3.267 + 0.1267t r = 0.999928 (76)

with a correlation coefficient of r = 0.999928. The distribution of Z vs. t is shown in Figure 9,
where the line is drawn from Eq. (76). It follows from Eqs. (75) and (76) that
N2cr = 7.894 = 8.00 wk and p/ = 25.79 = 26.0 wk as assumed for the expanded growth model.
Estimates of Q are then obtained from

S4.586 rf(t-26.0 (77)
At= 4 wk: Q=- 1+ erf (77)
2 8.00

as given by the last column in Table 9. The distribution of Q vs. t is shown in Figure 10, where
the curve is drawn from Eq. (77).
Other harvest intervals are now examined. Results for At = 6 wk are given in Tables 10 and
11, and shown in Figures 11 and 12. The curves in Figure 11 are drawn from Table 10, while the
line in Figure 12 is drawn from

At= 6 wk: Z = -3.200 + 0.1235t r = 0.99973 (78)

which leads to F2cr = 8.096 8.00 wk and p/ = 25.91= 26.0wk. It follows that Q vs. t is
described by


At= 6 wk: 5.28 1+erf t 26.0 (79)
2 1 8.00

Results for At = 8 wk are given in Tables 12 and 13, and shown in Figures 13 through 15. The
curves in Figure 13 are drawn from Table 12, while the line in Figure 14 is drawn from

At = 8 wk: Z = -3.268 + 0.1267t r = 0.9999954 (80)

which leads to ,o- = 7.891 8.00 wk and ,u = 25.79 = 26.0 wk.


It follows that Q vs. t is described by








5.024 t(-26.0V,
At= 8 wk: 24 1 + erf 26 (81)
2 1 8.00 .]

as shown in Figure 15. Results for At = 12 wk are given in Tables 14 and 15, and shown in
Figures 16 through 18. The line in Figure 17 is drawn from

At= 12 wk: Z= -3.343+0.1285t r= 1 (82)

which leads to Vio- = 7.779 = 8.00 wk and p = 26.01 = 26.0wk. Distribution of Q vs. t is
described by

4.868 26.0] (83)
At= 12 wk: Q-= 1 +erf ( .00 3)

as shown in Figure 18. Results for At = 16 wk are given in Table 16, without any graphs since
this would be redundant for one point. Finally, results are given in Table 17 for At = 2 wk, and
shown in Figures 19 and 20. The line in Figure 19 is drawn from

At = 2 wk: Z = -3.214 + 0.1236t r = 0.99924 (84)

which leads to V-ia = 8.093 =- 8.00 wk and/p = 26.01 = 26.0wk. Distribution of Q vs. t is
described by

4.0001 t- 26.0
At=2wk: 4.000 1+erf t-26.0 (85)
2 8.00

as shown in Figure 20.
The question now occurs as to dependence of seasonal growth quantifier (Q) on harvest
interval (At). A summary of values is listed in Table 18 from the various simulations. It can be
shown from the expanded growth model (Overman and Scholtz, 2002) that

Q = (a + ,At)exp(- yAt) (86)

where a and f, are to be determined by regression analysis. It was shown that
y = c / 2 = 0.15 / 2 = 0.075 wk'1. It follows that Eq. (86) can be linearized to the form

Q*= Qexp(0.075At) = a + ,At = 3.40 + 0.709At r = 0.99920 (87)

The prediction equation then becomes


0 = (a + fAt)exp(- yAt)= (3.40 + 0.709At)exp(- 0.075At)


(88)








Correlation of yield factor (A) with applied nitrogen (N) is shown in Figure 7. It seems logical
that this correlation should be described by the logistic model

A = A- (66)
1+ exp(by -c,,N) 1+ exp(0.22- 0.0098N)

where Am is maximum A at high N, Mg ha"'. It follows from Eq. (66) that standardized A, (A*),
can be calculated for each nitrogen level and harvest interval from

A* = A[l + exp(0.22 0.0098N)] = A (67)

Results are given in Table 7. It follows from Table 7 that response of parameter A to applied
nitrogen is given by Eq. (66) with Aom(4 wk) = 3.80, Am(6 wk) = 4.08, A.o(15 wk) = 4.88 Mg ha'1.
The curves in Figure 7 are drawn from Eq. (66) with appropriate values of Am. Results are very
similar to those for Y vs. N in Figure 1.

DISCUSSION

The extended logistic model appears to give reasonable correlation of yield, plant N uptake,
and plant N concentration with applied N (Figure 1) for switchgrass grown in Canada. This leads
to excellent phase plots of yield vs. plant N uptake (Figure 2). Prediction of a linear relationship
between plant N concentration and plant N uptake is confirmed. Dependence of the phase plots
on harvest interval is illustrated. Dependence of yield, plant N uptake, and plant N concentration
parameters on harvest interval is described adequately by the expanded growth model (Figure 3).
Several inferences follow from this analysis. The point of maximum slope of the yield
response curves is determined by
by_ 0.22 A(68)
N=N1 =y = 22 22 kg ha-1, Y/2 = (68)
c,, 0.0100 2

with the corresponding value of the yield depending on harvest interval. The point of maximum
slope of the plant N response curves is determined by
N=N'2 0.98 98 kg ha-, (NA)l/2 = A (69)

c, 0.0100 2

with the corresponding value of plant N uptake depending on harvest interval. Differential
response of plant N uptake at this point is given by

dN Ac,, (248)(0.0100)=0.62 (70)
dN 4 4

for harvest intervals of 4 and 6 wk. This means that maximum efficiency of nitrogen recovery is
62% for this harvest interval. Actual efficiency of plant N recovery, E, is defined by









Results are shown in Figure 21, where the line and curve are drawn from Eqs. (87) and (88),
respectively. This confirms the linear-exponential relationship between Q and At predicted by the
expanded growth model. It can be shown from calculus that peak harvest interval (Atp) for
maximum Q can be estimated from

1 a 1 3.40
Atp =-- -_ 8.54wk (89)
y 8/ 0.075 0.709

which is confirmed in Figure 21. Response is relatively insensitive in the range 7 < At < 10 wk.

CONCLUSIONS

The extended logistic model of seasonal response of biomass yield, plant N uptake, and plant
N concentration to applied N has been shown to describe field data of switchgrass grown at
Ames, IA rather well. The model predicts a linear relationship between plant N concentration
and plant N uptake, which has been confirmed from the data.
The expanded growth model has been shown to describe accumulation of biomass with
calendar time by the process of photosynthesis. A linear relationship between biomass
accumulation and the growth quantifier defined in the model has been confirmed. Parameter
values appear to be the same as those for a different grass grown at Tifton, GA. The model
assumes a linear relationship between biomass/plant N ratio and biomass accumulation with
calendar time, which has been confirmed from the data. This phase relation implies that biomass
accumulation by photosynthesis is the rate limiting step in plant growth, and that plant nitrogen
accumulates in virtual equilibrium with biomass. This is consistent with results for other crops as
well.
Growth curves have been generated for various harvest intervals from the expanded growth
model (Figures 8, 11, 13, and 16). Distribution of dimensionless probability function (Z) vs.
calendar time (t) generated straight lines with p = 26 wk and -5o- = 8 wk as assumed in the
expanded growth model (Figures 9, 12, 14, 17, and 19). Cumulative growth quantifier (Q) vs.
calendar time (t) followed the simple probability function (Figures 10, 15, 18, and 20) as
predicted by the theory. Dependence of seasonal growth quantifier on harvest interval followed a
linear-exponential function, as also predicted by the theory. These results are consistent with
results obtained for the warm-season perennial coastal bermudagrass (Overman and Scholtz,
2002).
Maximum biomass production occurs for a harvest interval of approximately 12 wk (Figure
3). Other characteristics of the growth model have been discussed by Overman (2006b).

REFERENCES

Abramowitz, M. and I.A. Stegun. 1965. Handbook of Mathematical Functions. Dover
Publications. New York, NY. 1046 p.
Madakadze, I.C., K.A. Stewart, P.R. Peterson, B.E. Coulman, and D.L. Smith. 1999. Cutting
frequency and nitrogen fertilization effects on yield and nitrogen concentration of
switchgrass in a short season area. Crop Science 39:552-557.
Overman, A.R. 1998. An expanded growth model for grasses. Communications in Soil Science
and Plant Analysis 29:67-85.









Overman, A. R. 2006a. A Memoir on Crop Yield and Nutrient Uptake. University of Florida.
Gainesville, FL. 116 p. (46 Tables and 62 Figures).
http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00072010&v=00001
Overman, A. R. 2006b. A Memoir on Crop Growth: Accumulation ofBiomass and Mineral
Elements. University of Florida. Gainesville, FL. 386 p. (84 Tables and 184 Figures).
http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00072283&v=00001
Overman, A.R. and K.H. Brock. 2003. Confirmation of the expanded growth model for a warm-
season perennial grass. Communications in Soil Science and Plant Analysis 34:1105-1117.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Taylor & Francis. Philadelphia, PA. 328 p.
Overman, A.R., S.R. Wilkinson, and D.M. Wilson. 1994. An extended model of forage grass
response to applied nitrogen. Agronomy J. 86:617-620.









Results are shown in Figure 21, where the line and curve are drawn from Eqs. (87) and (88),
respectively. This confirms the linear-exponential relationship between Q and At predicted by the
expanded growth model. It can be shown from calculus that peak harvest interval (Atp) for
maximum Q can be estimated from

1 a 1 3.40
Atp =-- -_ 8.54wk (89)
y 8/ 0.075 0.709

which is confirmed in Figure 21. Response is relatively insensitive in the range 7 < At < 10 wk.

CONCLUSIONS

The extended logistic model of seasonal response of biomass yield, plant N uptake, and plant
N concentration to applied N has been shown to describe field data of switchgrass grown at
Ames, IA rather well. The model predicts a linear relationship between plant N concentration
and plant N uptake, which has been confirmed from the data.
The expanded growth model has been shown to describe accumulation of biomass with
calendar time by the process of photosynthesis. A linear relationship between biomass
accumulation and the growth quantifier defined in the model has been confirmed. Parameter
values appear to be the same as those for a different grass grown at Tifton, GA. The model
assumes a linear relationship between biomass/plant N ratio and biomass accumulation with
calendar time, which has been confirmed from the data. This phase relation implies that biomass
accumulation by photosynthesis is the rate limiting step in plant growth, and that plant nitrogen
accumulates in virtual equilibrium with biomass. This is consistent with results for other crops as
well.
Growth curves have been generated for various harvest intervals from the expanded growth
model (Figures 8, 11, 13, and 16). Distribution of dimensionless probability function (Z) vs.
calendar time (t) generated straight lines with p = 26 wk and -5o- = 8 wk as assumed in the
expanded growth model (Figures 9, 12, 14, 17, and 19). Cumulative growth quantifier (Q) vs.
calendar time (t) followed the simple probability function (Figures 10, 15, 18, and 20) as
predicted by the theory. Dependence of seasonal growth quantifier on harvest interval followed a
linear-exponential function, as also predicted by the theory. These results are consistent with
results obtained for the warm-season perennial coastal bermudagrass (Overman and Scholtz,
2002).
Maximum biomass production occurs for a harvest interval of approximately 12 wk (Figure
3). Other characteristics of the growth model have been discussed by Overman (2006b).

REFERENCES

Abramowitz, M. and I.A. Stegun. 1965. Handbook of Mathematical Functions. Dover
Publications. New York, NY. 1046 p.
Madakadze, I.C., K.A. Stewart, P.R. Peterson, B.E. Coulman, and D.L. Smith. 1999. Cutting
frequency and nitrogen fertilization effects on yield and nitrogen concentration of
switchgrass in a short season area. Crop Science 39:552-557.
Overman, A.R. 1998. An expanded growth model for grasses. Communications in Soil Science
and Plant Analysis 29:67-85.









Table 1. Dependence of seasonal biomass yield (Y), plant nitrogen uptake (Nu), and plant
nitrogen concentration (Nc) on applied nitrogen (N) and harvest interval (At) for switchgrass
grown at Montreal, Quebec, Canada (1996). Data are averages of three cultivars.1
N Y N, Nc Y Nu Nc Y Nu Nc
kg ha-' Mg ha' kg ha-1 gkg- Mgha'1 kg ha1 g kg-1 Mgha' kg ha' gkg-1
At= 4 wk At= 6 wk At= 15 wk
0 4.87 69.1 14.2 6.01 69.0 11.5 8.30 42.2 5.08
75 6.84 108.6 15.9 7.80 106.6 13.7 9.66 52.1 5.39
150 8.75 155.1 17.7 10.01 157.1 15.7 11.25 66.9 5.95
'Data adapted from Madakadze et al. (1999).


Table 2. Dependence of standardized yield (Y*) and standardized plant nitrogen uptake (Nu*) on
applied nitrogen (N) and harvest interval (At) for switchgrass at Montreal, Quebec, Canada.
N Y* Nu*
kg ha-' Mg ha' kg ha'
At= 4 wk At=6wk At= 15wk At=4wk At=6wk At=15wk
0 10.94 13.50 18.64 253 253 155
75 10.93 12.46 15.43 247 243 119
150 11.26 12.88 14.47 250 253 108
avg(stddev) 11.04(0.19) 12.95 (0.52) 16.18 (2.18) 250(3.0) 250(5.8) 127 (24.6)

Table 3. Dependence of standardized yield parameter (Ay*) and plant N parameter (An*) on
harvest interval (At) for switchgrass at Montreal, Quebec, Canada.
At Ay Ay* An An* Ncm
wk Mg ha1 Mg ha- kg ha1 kg ha'1 g kg'
4 11.04 14.90 250 337 22.6
6 12.95 20.31 250 392 19.3
15 16.18 49.84 127 391 7.85









Table 4. Coupling of biomass yield (Y) and plant nitrogen accumulation (N,) with calendar time
(t) and applied nitrogen (N) for switchgrass grown at Montreal, Quebec, Canada (1996).1
At t AY Y ANu Nu AY Y ANu Nu AY Y ANu Nu
wk wk Mg ha-1 kg ha-1 Mg ha'1 kg ha- Mg ha'1 kg ha'
N= 0 N= 75 kg ha-' N= 150 kg ha1
4 23.1 0 0 0 0 0 0
2.54 37.8 3.21 48.3 3.87 66.7
27.1 2.54 37.8 3.21 48.3 3.87 66.7
1.36 21.2 2.22 42.1 3.08 62.3
31.1 3.90 59.0 5.43 90.4 6.95 129.0
0.97 10.1 1.41 18.2 1.80 26.1
35.4 4.87 69.1 6.84 108.6 8.75 155.1

6 23.1 0 0 0 0 0 0
3.86 42.0 4.46 60.3 5.98 89.0
29.1 3.86 42.0 4.46 60.3 5.98 89.0
2.15 26.9 3.34 46.3 4.03 68.1
35.3 6.01 68.9 7.80 106.6 10.01 157.1

15 23.1 0 0 0 0 0 0
8.30 42.2 9.66 52.1 11.25 66.9
38.0 8.30 42.2 9.66 52.1 11.25 66.9
'Data adapted from Madakadze et al. (1999).


Table 5. Estimates of the growth quantifier (Q) with calendar time (t) for the expanded growth
model.
t x erfx exp(-x2) AQi Q
wk
23.1 0.2375 0.264 0.9452 0
1.259
27.1 0.7375 0.703 0.5805 1.259
1.078
31.1 1.2375 0.9198 0.2162 2.337
0.598
35.4 1.7750 0.9880 0.0428 2.935

23.1 0.2375 0.264 0.9452 0
1.988
29.1 0.9875 0.837 0.3771 1.988
1.140
35.4 1.7750 0.9880 0.0428 3.128

23.1 0.2375 0.264 0.9452 0
3.317
38.0 2.100 0.9970 0.0122 3.317










Table 6. Dependence of biomass accumulation (Y) on the growth quantifier (Q), applied nitrogen
(N) and harvest interval (At) for switchgrass grown at Montreal, Quebec, Canada.
At t Q Y
wk wk Mg ha1
N, kg ha-1
0 75 150


23.1
27.1
31.1
35.4

23.1
29.1
35.4

23.1
380


0
1.26
2.34
2.94

0
1.99
3.13

0
332


0
2.54
3.90
4.87

0
3.86
6.01

0
8 O


1Yield data adapted from Madakadze et al. (1999).


0
3.21
5.43
6.84

0
4.46
7.80

0
9.66


0
3.87
6.95
8.75

0
5.98
10.01

0
11.25


Table 7. Estimates for standardized yield factor (A *) for each yield factor (A) at applied nitrogen
(N) and harvest interval (At) for switchgrass grown at Montreal, Quebec, Canada.
N A A*
kg ha1 Mg ha-' Mg ha-'
At, wk At, wk
4 6 15 4 6 15
0 1.70 1.93 2.50 3.82 4.33 5.62
75 2.35 2.42 2.91 3.75 3.87 4.65
150 2.98 3.14 3.39 3.83 4.04 4.36
avg (std dev) ----- ----- ----- 3.80 (0.04) 4.08 (0.23) 4.88 (0.66)


11.25


33 830









Table 8. Simulation of the growth quantifier (AQi) with calendar time (t) for the expanded
growth model for harvest interval At = 4 wk.
t x erfx exp(-x2) AQi
wk


3.1
4.1
5.1
6.1
7.1

7.1
8.1
9.1
10.1
11.1

11.1
12.1
13.1
14.1
15.1

15.1
16.1
17.1
18.1
19.1

19.1
20.1
21.1
22.1
23.1

23.1
24.1
25.1
26.1
27.1

27.1
28.1
29.1
30.1
31.1


-2.2625
-2.1375
-2.0125
-1.8875
-1.7625

-1.7625
-1.6375
-1.5125
-1.3875
-1.2625

-1.2625
-1.1375
-1.0125
-0.8875
-0.7625

-0.7625
-0.6375
-0.5125
-0.3875
-0.2625

-0.2625
-0.1375
-0.0125
0.1125
0.2375

0.2375
0.3625
0.4875
0.6125
0.7375

0.7375
0.8625
0.9875
1.1125
1.2375


-0.9985
-0.9973
-0.9954
-0.9924
-0.9873

-0.9873
-0.9794
-0.9676
-0.9503
-0.9258

-0.9258
-0.8923
-0.8478
-0.7905
-0.7191

-0.7191
-0.6327
-0.5313
-0.4163
-0.2895

-0.2895
-0.1542
-0.0140
0.1264
0.2625

0.2625
0.3918
0.5095
0.6136
0.7030

0.7030
0.7774
0.8374
0.8843
0.9199


0.00598
0.0104
0.0174
0.0284
0.0448

0.0448
0.0685
0.1015
0.1458
0.2031

0.2031
0.2742
0.3587
0.4549
0.5591

0.5591
0.6660
0.7690
0.8606
0.9334

0.9334
0.9813
0.9998
0.9874
0.9452

0.9452
0.8769
0.7885
0.6872
0.5805

0.5805
0.4752
0.3771
0.2901
0.2162


0.0000
0.0002
0.0004
0.0008
0.0019

0.000
0.001
0.004
0.009
0.019

0.000
0.010
0.029
0.061
0.111

0.000
0.046
0.125
0.243
0.405

0.000
0.130
0.328
0.591
0.907

0.000
0.224
0.526
0.880
1.258

0.000
0.235
0.515
0.804
1.078









Table 8 (continued).


31.1 1.2375 0.9199 0.2162 0.000
32.1 1.3625 0.9460 0.1562 0.150
33.1 1.4875 0.9646 0.1094 0.306
34.1 1.6125 0.9774 0.07426 0.451
35.4 1.7750 0.9879 0.04283 0.602

35.4 1.7750 0.9879 0.04283 0.000
36.4 1.9000 0.9928 0.02705 0.050
37.4 2.0250 0.9957 0.01656 0.107
38.4 2.1500 0.9975 0.00983 0.147
39.4 2.2750 0.9987 0.00565 0.167

39.4 2.2750 0.9987 0.00565 0.000
40.4 2.4000 0.99931 0.00315 0.011
41.4 2.5250 0.99968 0.00170 0.015
42.4 2.6500 0.99981 0.00089 0.029
43.4 2.7750 0.99990 0.00045 0.034

43.4 2.7750 0.999910 0.000453 0.0000
44.4 2.9000 0.999959 0.000223 0.0005
45.4 3.0250 0.999983 0.000106 0.0011
46.4 3.1500 0.999993 0.000049 0.0020
47.4 3.2750 0.999999 0.0000220 0.0022









Table 9. Distribution of Z vs. t for harvest interval At = 4 wk.
t AQi Q F Z Q
wk
3.1 0.000 0.00000 --- 0.0001
0.002
7.1 0.002 0.00044 -2.380 0.0021
0.019
11.1 0.021 0.00458 -1.842 0.019
0.111
15.1 0.132 0.0288 -1.346 0.124
0.405
19.1 0.537 0.117 -0.842 0.511
0.907
23.1 1.444 0.315 -0.341 1.396
1.258
27.1 2.702 0.589 0.159 2.653
1.078
31.1 3.780 0.824 0.658 3.742
0.602
35.4 4.382 0.9555 1.203 4.364
0.167
39.4 4.549 0.99193 1.702 4.545
0.034
43.4 4.583 0.99935 2.270 4.581

oo 4.586 1 --- 4.586









Table 10. Simulation of the growth quantifier (AQi) with calendar time (t) for the expanded
growth model for harvest interval of At = 6 wk.
t x erfx exp(-x2) AQi
wk


5.1
6.1
7.1
8.1
9.1
10.1
11.1

11.1
12.1
13.1
14.1
15.1
16.1
17.1

17.1
18.1
19.1
20.1
21.1
22.1
23.1

23.1
24.1
25.1
26.1
27.1
28.1
29.1

29.1
30.1
31.1
32.1
33.1
34.1
35.4


-2.0125
-1.8875
-1.7625
-1.6375
-1.5125
-1.3875
-1.2625

-1.2625
-1.1375
-1.0125
-0.8875
-0.7625
-0.6375
-0.5125

-0.5125
-0.3875
-0.2625
-0.1375
-0.0125
0.1125
0.2375

0.2375
0.3625
0.4875
0.6125
0.7375
0.8625
0.9875

0.9875
1.1125
1.2375
1.3625
1.4875
1.6125
1.7750


-0.9954
-0.9924
-0.9873
-0.9794
-0.9676
-0.9503
-0.9258

-0.9258
-0.8923
-0.8478
-0.7905
-0.7191
-0.6327
-0.5313

-0.5313
-0.4163
-0.2895
-0.1542
-0.0140
0.1264
0.2625

0.2625
0.3918
0.5095
0.6136
0.7030
0.7774
0.8374

0.8374
0.8843
0.9199
0.9460
0.9646
0.9774
0.9879


0.0174
0.0284
0.0448
0.0685
0.1015
0.1458
0.2031

0.2031
0.2742
0.3587
0.4549
0.5591
0.6660
0.7690

0.7690
0.8606
0.9334
0.9813
0.9998
0.9874
0.9452

0.9452
0.8769
0.7885
0.6872
0.5805
0.4752
0.3771

0.3771
0.2901
0.2162
0.1562
0.1094
0.07426
0.04283


0.0000
0.0002
0.0011
0.003
0.006
0.012
0.022

0.000
0.010
0.029
0.061
0.111
0.184
0.283

0.000
0.082
0.215
0.403
0.644
0.934
1.260

0.000
0.224
0.639
0.993
1.371
1.747
2.100

0.000
0.199
0.422
0.640
0.832
0.991
1.146









Table 10 (continued).


35.4
36.4
37.4
38.4
39.4
40.4
41.4

41.4
42.4
43.4
44.4
45.4
46.4
47.4


1.7750
1.9000
2.0250
2.1500
2.2750
2.4000
2.5250

2.5250
2.6500
2.7750
2.9000
3.0250
3.1500
3.2750


0.9879
0.9928
0.9957
0.9975
0.9987
0.99931
0.99968

0.99968
0.99981
0.999910
0.999959
0.999983
0.999993
0.999999


0.04283
0.02705
0.01656
0.00983
0.00565
0.00315
0.00170


0.00170
0.00089
0.000453
0.000223
0.000106
0.000049
00000220


0.000
0.050
0.107
0.147
0.167
0.186
0.196

0.000
0.016
0.017
0.019
0.020
0.021
0021


Table 11. Distribution of Z vs. t for harvest interval At = 6 wk.
t AQi Q F Z 0
wk
5.1 0.000 0.000 --- 0.001
0.022
11.1 0.022 0.0044 -1.842 0.021
0.283
17.1 0.305 0.0606 -1.097 0.289
1.260
23.1 1.565 0.3111 -0.349 1.528
2.100
29.1 3.665 0.7286 0.430 3.565
1.146
35.4 4.811 0.9565 1.210 4.784
0.196
41.4 5.007 0.9958 1.865 5.012
0.021
47.4 5.028 1 ------ 5.028









Table 12. Simulation of the growth quantifier (AQi) with calendar time (t) for the expanded
growth model for harvest interval of At = 8 wk.
t x erfx exp(-x2) AQ1
wk


7.1
8.1
9.1
10.1
11.1
12.1
13.1
14.1
15.1

15.1
16.1
17.1
18.1
19.1
20.1
21.1
22.1
23.1

23.1
24.1
25.1
26.1
27.1
28.1
29.1
30.1
31.1

31.1
32.1
33.1
34.1
35.4
36.4
37.4
38.4
39.4


-1.7625
-1.6375
-1.5125
-1.3875
-1.2625
-1.1375
-1.0125
-0.8875
-0.7625

-0.7625
-0.6375
-0.5125
-0.3875
-0.2625
-0.1375
-0.0125
0.1125
0.2375

0.2375
0.3625
0.4875
0.6125
0.7375
0.8625
0.9875
1.1125
1.2375

1.2375
1.3625
1.4875
1.6125
1.7750
1.9000
2.0250
2.1500
2.2750


-0.9873
-0.9794
-0.9676
-0.9503
-0.9258
-0.8923
-0.8478
-0.7905
-0.7191

-0.7191
-0.6327
-0.5313
-0.4163
-0.2895
-0.1542
-0.0140
0.1264
0.2625

0.2625
0.3918
0.5095
0.6136
0.7030
0.7774
0.8374
0.8843
0.9199

0.9199
0.9460
0.9646
0.9774
0.9879
0.9928
0.9957
0.9975
0.9987


0.0448
0.0685
0.1015
0.1458
0.2031
0.2742
0.3587
0.4549
0.5591

0.5591
0.6660
0.7690
0.8606
0.9334
0.9813
0.9998
0.9874
0.9452

0.9452
0.8769
0.7885
0.6872
0.5805
0.4752
0.3771
0.2901
0.2162

0.2162
0.1562
0.1094
0.07426
0.04283
0.02705
0.01656
0.00983
0.00565


0.000
0.001
0.004
0.009
0.019
0.034
0.058
0.093
0.142

0.000
0.046
0.125
0.243
0.405
0.612
0.861
1.146
1.456

0.000
0.224
0.526
0.880
1.258
1.635
1.988
2.302
2.571

0.000
0.150
0.306
0.451
0.602
0.686
0.750
0.793
0.818









Table 12 (continued).


39.4
40.4
41.4
42.4
43.4
44.4
45.4
46.4
47.4


2.2750
2.4000
2.5250
2.6500
2.7750
2.9000
3.0250
3.1500
3.2750


0.9987
0.99931
0.99968
0.99981
0.99991
0.999959
0.999983
0.999993
0.999999


0.00565
0.00315
0.00170
0.000892
0.000453
0.000223
0.000106
0.000049
0.0000220


0.000
0.011
0.015
0.029
0.032
0.034
0.036
0.037
0037


Table 13. Distribution of Z vs. t for harvest interval At = 8 wk.
t AQi Q F Z Q
wk
7.1 0.000 0.000 --- 0.0025
0.142
15.1 0.142 0.0283 -1.358 0.136
1.456
23.1 1.598 0.3181 -0.335 1.530
2.571
31.1 4.169 0.8298 0.674 4.092
0.818
39.4 4.987 0.9926 1.723 4.979
0.037
47.4 5.024 1 ------ 5.024









Table 14. Simulation of the growth quantifier (AQi) with calendar time (t) for the expanded
growth model for harvest interval of At = 12 wk.
t x erfx exp(-x2) Ai
wk


11.1
12.1
13.1
14.1
15.1
16.1
17.1
18.1
19.1
20.1
21.1
22.1
23.1

23.1
24.1
25.1
26.1
27.1
28.1
29.1
30.1
31.1
32.1
33.1
34.1
35.4

35.4
36.4
37.4
38.4
39.4
40.4
41.4
42.4
43.4
44.4
45.4
46.4
474


-1.2625
-1.1375
-1.0125
-0.8875
-0.7625
-0.6375
-0.5125
-0.3875
-0.2625
-0.1375
-0.0125
0.1125
0.2375

0.2375
0.3625
0.4875
0.6125
0.7375
0.8625
0.9875
1.1125
1.2375
1.3625
1.4875
1.6125
1.7750

1.7750
1.9000
2.0250
2.1500
2.2750
2.4000
2.5250
2.6500
2.7750
2.9000
3.0250
3.1500
3.2750


-0.9258
-0.8923
-0.8478
-0.7905
-0.7191
-0.6327
-0.5313
-0.4163
-0.2895
-0.1542
-0.0140
0.1264
0.2625

0.2625
0.3918
0.5095
0.6136
0.7030
0.7774
0.8374
0.8843
0.9199
0.9460
0.9646
0.9774
0.9879

0.9879
0.9928
0.9957
0.9975
0.9987
0.99931
0.99968
0.99981
0.99991
0.999959
0.999983
0.999993
0.999999


0.2031
0.2742
0.3587
0.4549
0.5591
0.6660
0.7690
0.8606
0.9334
0.9813
0.9998
0.9874
0.9452

0.9452
0.8769
0.7885
0.6872
0.5805
0.4752
0.3771
0.2901
0.2162
0.1562
0.1094
0.07426
0.04283

0.04283
0.02705
0.01656
0.00983
0.00565
0.00315
0.00170
0.00089
0.000453
0.000223
0.000106
0.000049
0.0000220


0.000
0.010
0.029
0.061
0.111
0.184
0.283
0.411
0.570
0.758
0.972
1.205
1.450

0.000
0.224
0.526
0.880
1.258
1.635
1.988
2.302
2.571
2.789
2.960
3.089
3.204

0.000
0.050
0.107
0.147
0.167
0.186
0.196
0.206
0.210
0.212
0.213
0.214
0.214


. ~ ~ -










Table 15. Distribution ofZvs. t for harvest interval At = 12 wk.
t AQi Q F Z
wk
11.1 0.000 0.0000 -------- 0.020
1.450
23.1 1.450 0.2979 -0.374 1.480
3.204
35.4 4.654 0.9560 1.207 4.633
0.214
47.4 4.868 1 ------ 4.868









Table 16. Simulation of the growth quantifier (AQi) with calendar time (t) for the expanded
growth model for harvest interval of At = 16 wk.
t x erfx exp(-x2) AQ,
wk

15.1 -0.7625 -0.7191 0.5591 0.000
16.1 -0.6375 -0.6327 0.6660 0.046
17.1 -0.5125 -0.5313 0.7690 0.125
18.1 -0.3875 -0.4163 0.8606 0.243
19.1 -0.2625 -0.2895 0.9334 0.405
20.1 -0.1375 -0.1542 0.9813 0.612
21.1 -0.0125 -0.0140 0.9998 0.861
22.1 0.1125 0.1264 0.9874 1.146
23.1 0.2375 0.2625 0.9452 1.456
24.1 0.3625 0.3918 0.8769 1.782
25.1 0.4875 0.5095 0.7885 2.109
26.1 0.6125 0.6136 0.6872 2.424
27.1 0.7375 0.7030 0.5805 2.717
28.1 0.8625 0.7774 0.4752 2.979
29.1 0.9875 0.8374 0.3771 3.206
30.1 1.1125 0.8843 0.2901 3.394
31.1 1.2375 0.9199 0.2162 3.547

31.1 1.2375 0.9199 0.2162 0.000
32.1 1.3625 0.9460 0.1562 0.150
33.1 1.4875 0.9646 0.1094 0.306
34.1 1.6125 0.9774 0.07426 0.451
35.4 1.7750 0.9879 0.04283 0.602
36.4 1.9000 0.9928 0.02705 0.686
37.4 2.0250 0.9957 0.01656 0.750
38.4 2.1500 0.9975 0.00983 0.793
39.4 2.2750 0.9987 0.00565 0.818
40.4 2.4000 0.99931 0.00315 0.835
41.4 2.5250 0.99968 0.00170 0.844
42.4 2.6500 0.99981 0.00089 0.851
43.4 2.7750 0.99991 0.000453 0.855
44.4 2.9000 0.999959 0.000223 0.856
45.4 3.0250 0.999983 0.000106 0.857
46.4 3.1500 0.999993 0.000049 0.858
47.4 3.2750 0.999999 0.0000220 0.858









Table 17. Simulation of the growth quantifier (AQi) with calendar time (t) for the expanded
growth model for harvest interval of At = 2 wk.
t x erfx exp(-x2) AQi Q F Z Q
wk

1.1 -2.5125 -0.99951 0.00181 --- 0.0000 --- --- ---
3.1 -2.2625 -0.9985 0.00598 0.0001 0.0001 --- --- ---
5.1 -2.0125 -0.9954 0.0174 0.0004 0.0005 0.00012 -2.600 0.0004
7.1 -1.7625 -0.9873 0.0448 0.0011 0.0016 0.00040 -2.480 0.0016
9.1 -1.5125 -0.9676 0.1015 0.0040 0.0056 0.00140 -2.120 0.0058
11.1 -1.2625 -0.9258 0.2031 0.0116 0.0172 0.00430 -1.858 0.0170
13.1 -1.0125 -0.8478 0.3587 0.0289 0.0461 0.0115 -1.618 0.0452
15.1 -0.7625 -0.7191 0.5591 0.111 0.157 0.0392 -1.245 0.108
17.1 -0.5125 -0.5313 0.7690 0.125 0.282 0.0705 -1.040 0.232
19.1 -0.2625 -0.2895 0.9334 0.215 0.497 0.124 -0.817 0.446
21.1 -0.0125 -0.0140 0.9998 0.328 0.825 0.206 -0.580 0.774
23.1 0.2375 0.2625 0.9452 0.441 1.266 0.316 -0.339 1.218
25.1 0.4875 0.5095 0.7885 0.526 1.792 0.448 -0.093 1.726
27.1 0.7375 0.7030 0.5805 0.554 2.346 0.586 0.154 2.310
29.1 0.9875 0.8374 0.3771 0.515 2.861 0.715 0.401 2.832
31.1 1.2375 0.9199 0.2162 0.422 3.283 0.821 0.650 3.266
33.1 1.4875 0.9646 0.1094 0.306 3.589 0.897 0.895 3.580
35.4 1.7750 0.9879 0.04283 0.225 3.814 0.9535 1.188 3.807
37.4 2.0250 0.9957 0.01656 0.107 3.921 0.9802 1.453 3.912
39.4 2.2750 0.9987 0.00565 0.039 3.960 0.9900 1.645 3.964
41.4 2.5250 0.99968 0.00170 0.015 3.975 0.99375 1.778 3.987
43.4 2.7750 0.99991 0.000453 0.017 3.992 0.99800 2.040 3.996
45.4 3.0250 0.999984 0.000106 0.007 3.999 0.99975 2.440 3.999
47.4 3.2750 0.999999 0.0000220 0.001 4.000 1 ------- 4.000



Table 18. Dependence of the seasonal growth quantifier (Q) on harvest interval (At) for
switchgrass grown at Montreal, Quebec, Canada.
At Q Q*
wk

2 4.000 4.647 4.155
4 4.586 6.190 4.627
6 5.028 7.885 4.887
8 5.024 9.154 4.984
12 4.868 11.973 4.845
16 4.405 14.625 4.444








List of Figures


Figure 1. Response of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) to
applied nitrogen (N) and harvest interval (At) for switchgrass grown at Montreal, Quebec,
Canada. Data adapted from Madakadze et al. (1999). Curves drawn from Eqs. (23) through (25)
with Ay(4 wk) = 11.04,Ay(6 wk) = 12.95, and Ay(15 wk) = 16.18 Mg ha-'; An(4 wk) = 250, An(6
wk) = 250, and An(15 wk) = 127 kg ha-; and Ncm(4 wk) = 22.6, Ncn(6 wk) = 19.3, and Ncm(15
wk) = 7.85 g kg-1.

Figure 2. Phase plots of biomass yield (Y) and plant N concentration (Nc) vs. plant N uptake (Nu)
for three harvest intervals (At) for switchgrass grown at Montreal, Quebec, Canada. Data adapted
from Madakadze et al. (1999). Curves drawn from Eqs. (27), (30), and (33); lines drawn from
Eqs. (28), (31), and (34).

Figure 3. Dependence of parameters for yield (Ay), plant N uptake (An), and plant N
concentration (Ncm) on harvest interval (At) for switchgrass grown at Montreal, Quebec, Canada.
Curves drawn from Eqs. (41) through (43).

Figure 4. Correlation of cumulative plant N uptake (Nu) with cumulative biomass (Y) for applied
nitrogen (N) and harvest interval (At) for switchgrass grown at Montreal, Quebec, Canada. Data
adapted from Madakadze et al. (1999). Lines drawn from Eqs. (46) through (54).

Figure 5. Dependence of plant N concentration (Nc) on applied nitrogen (N) and harvest interval
(At) for switchgrass grown at Montreal, Quebec, Canada. Data are from Table 4. Curves drawn
from Eq. (25) with Ncm(4 wk) = 22.6, Ncm(6 wk) = 19.3, and Ncm(15 wk) = 7.85 g kg-1.

Figure 6. Correlation of cumulative biomass (Y) with cumulative growth quantifier (Q) for
applied nitrogen (N) and harvest interval (At) for switchgrass grown at Montreal, Quebec,
Canada. Data adapted from Madakadze et al. (1999). Lines drawn from Eqs. (57) through (65).

Figure 7. Dependence of yield factor (A) on applied nitrogen (N) and harvest interval (At) for
switchgrass grown at Montreal, Quebec, Canada. Data are from Table 7. Curves drawn from Eq.
(66) with Ao(4 wk) = 3.80, A.0(6 wk) = 4.08, Aoo(15 wk) = 4.88 Mg ha"1.

Figure 8. Dependence of the growth quantifier (AQ) on calendar time (t) for harvest interval (At)
of 4 wk. Curves drawn from Table 8.

Figure 9. Distribution of dimensionless probability function (Z) vs. calendar time (t) for harvest
interval (At) of 4 wk. Values from Table 9; line drawn from Eq. (76).

Figure 10. Distribution of seasonal growth quantifier (Q) vs. calendar time (t) for harvest interval
(At) = 4 wk. Values from Table 10; curve drawn from Eq. (77).

Figure 11. Dependence of the growth quantifier (AQ) on calendar time (t) for harvest interval
(At) of 6 wk. Curves drawn from Table 11.








Figure 12. Distribution of dimensionless probability function (Z) vs. calendar time (t) for harvest
interval (At) of 6 wk. Values from Table 11; line drawn from Eq. (78).

Figure 13. Dependence of the growth quantifier (AQ) on calendar time (t) for harvest interval
(At) of 8 wk. Curves drawn from Table 12.

Figure 14. Distribution of dimensionless probability function (Z) vs. calendar time (t) for harvest
interval (At) of 8 wk. Values from Table 13; line drawn from Eq. (80).

Figure 15. Distribution of seasonal growth quantifier (Q) vs. calendar time (t) for harvest interval
(At) = 8 wk. Values from Table 13; curve drawn from Eq. (81).

Figure 16. Dependence of the growth quantifier (AQ) on calendar time (t) for harvest interval
(At) of 12 wk. Curves drawn from Table 14.

Figure 17. Distribution of dimensionless probability function (Z) vs. calendar time (t) for harvest
interval (At) of 12 wk. Values from Table 15; line drawn from Eq. (82).

Figure 18. Distribution of seasonal growth quantifier (Q) vs. calendar time (t) for harvest interval
(At) = 12 wk. Values from Table 15; curve drawn from Eq. (83).

Figure 19. Distribution of dimensionless probability function (Z) vs. calendar time (t) for harvest
interval (At) of 2 wk. Values from Table 17; line drawn from Eq. (84).

Figure 20. Distribution of seasonal growth quantifier (Q) vs. calendar time (t) for harvest interval
(At) = 2 wk. Values from Table 18; curve drawn from Eq. (85).

Figure 21. Dependence of seasonal growth quantifier (Q) and transformed growth quantifier
(Q*) on harvest interval (At). Values from Table 19. Line and curve drawn from Eqs. (87) and
(88), respectively.



















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