Approximating a closed-form solution for cotton fruiting dynamics

Approximating

a Closed-Form

Solution

for Cotton Fruiting Dynamics GUY L. CURRY, Department

RICHARD

M. FELDMAN,

of Zndusirial Engineering,

AND BRYAN

Texar A&M Uniwrsily,

L. DEUERMEYER Co&e

Station, Texar 77843

ANLI MELVIN E. KEENER Department of Agronomy,

Uniwrsiry

of Missouri at Cohn&a,

Columbia, Missouri 65201

Received 28 March 1980; revised 10 October 1980

ABSTRACT

A nonlinear partial differential equation of the Von Foerster type is used to describe the fruiting dynamics of a cotton crop. An algorithm is developed and convergence proven that yields a functional approximation to the solution of the Von Foerster system of equations. Comparisons are presented which show excellent agreement between the algorithm results and field data for three cotton varieties.

1.

INTRODUCTION

Analytical models of age-dependent population processes utilize the Von Foerster [20] differential equation representation.

frequently Extensions

of the Von Foerster system to include nonlinear birth rates [6,12], resource limitations [ 111, and a mass component [ 18,151 have been accomplished in recent years. Currently, closed-form solutions to the differential-equation representations

are available

for simple structures,

but for most realistic

structures closed-form solutions have yet to be obtained. In general, the differential equation representation is used mainly for concise problem statements. successfully

In some cases, however, Von Foerster systems have been utilized to study limiting behavior including effects of mitigat-

ing influences

on populations

[ 17,6].

For agronomic crop-pest models, transient behavior is of primary concern, and generally for these models closed-form solutions are not possible. Hence it has been necessary to utilize numerical schemes to obtain solutions to these transient problems. The general numerical procedure is to approximate the continuous model with a discretized analog, which is then solved iteratively by a time-step algorithm [23]. This solution procedure is MATHEMATICAL

BIOSCIENCES

54: 91-113

(1981)

91

QElsevier North Holland, Inc., 1981 52 Vanderbilt Ave., New York, NY 10017

0025-5564/&I l/O30091 + 23302.50

GUY L. CURRY ET AL.

92

analogous to the discrete population modeling approaches of Leslie [ 141 and Curry et al. [2]. Successful applications of the Von Foerster approach to cotton crop management have recently appeared. A detailed physiological plant model was developed by Wang et al. [23], and several plant-insect models were studied by Gutierrez et al. [8- lo]. All of these models obtain numerical as opposed to analytical solutions. The purpose of this paper is to present a Von Foerster representation of a cotton crop model which is complex enough to yield excellent agreement with field data while admitting an approximate closed-form solution. The solution scheme is an iterative procedure which exploits segmented contraction mapping properties and utilizes least-squares regression and Gaussian integration to yield a polynomial analytical representation of the age-number structure for the cotton crop fruiting dynamics. The only other closed-form cotton crop model known to the authors is the paper by Wallach [22], which also gives good agreement for total fruiting numbers, but which uses simplistic photosynthate demands and forced-shedding age categories and does not utilize the Von Foerster descriptive structure. The paper is organized as follows. The formulation of the cotton crop fruiting characteristics is given in Sec. 2. The solution structure and computation scheme are provided in Sec. 3. A comparison of the model and field results for three cotton varieties, along with the associated approximate polynomial solutions, their execution times, and their costs, are in Sec. 4. The pertinent theory needed to establish convergence is given in the Appendix. 2.

COTTON CROP MODEL

Several large single plant cotton models have been developed [19,1]. These complex models incorporate a considerable amount of plant process components, but they have little utility as crop simulators due to their single-plant orientation. In 1975 Gutierrez et al. [7] demonstrated that population-ecology concepts could be used to model the cotton crop. In their approach, leaves, fruiting bodies, roots, and stems are treated as continuum on a per-unit-area basis. Following this idea, Jones et al. [13] extended the single-plant model (SIMCOT II) of Baker et al. [I] to a successful crop simulator. By including only the dominant physiological processes for cotton grown in the San Joaquin Valley of California, Wang et al. [23] were able to formulate the cotton crop simulator of Gutierrez et al. [7] in a Von Foerster context. The model developed by Wang et al. treats the crop components of leaves, roots, stems, and fruiting bodies as interacting entities via a photosynthate availability pool.

COTTON FRUITING DYNAMICS

93

Cotton production is affected by insect pests, with the major pests in Texas (boll weevil, fleahopper, budworm, and bollworm) feeding mainly on fruiting bodies and not the other plant structures. The emphasis of the model presented in this paper is therefore oriented towards predicting the time evolution of the age-number density of fruiting bodies. In this context a simplification is possible. Fruiting bodies are generally considered to be of first priority for the photosynthates available for growth. Since the other plant structures are relatively constant during the plant reproductive period [7,8], maintenance photosynthate demand during this period can be approximated by a constant. Thus, photosynthate demand during this period is controlled by fruiting bodies. Because photosynthate availability is a function of leaf area, the availability can also be considered constant during the period of fruit-generated stress. Therefore, the dynamics of leaves, stems, and roots need not be described when modeling the fruiting structures. This simplification to the Von Foerster-type system is possible only when biologically justified as above and numerically validated, as will be shown in the final section. COTTON

FRUITING

The rate at which plants initiate flower buds increases and then peaks shortly after the first buds appear, and declines thereafter. The initiation rate is a function of the difference between the availability of photosynthate (assumed constant) and demand for the photosynthate (a function of the age density of current fruit). Fruiting bodies are abscised from the plants in an apparently random fashion in the absence of photosynthate limitations. The appearance of fruit causes photosynthate demand to increase rapidly, which in turn causes a rapid decrease in bud initiation. Bud initiation ceases at approximately the time peak fruit counts occur, which coincides with the first occurrence of a balance between photosynthate availability and fruiting-body growth demand. Subsequent to this instant in time, forced shedding occurs, so that demand and availability equilibrate. Only fruiting bodies within a restricted age category, namely the medium-size floral buds up to young fruit, are abscised. For simplicity in the model, fruiting structures are abscised until demand and availability are balanced, starting at the medium-size floral buds. In addition, plants are assumed to be determinant; that is, once bud initiation ceases, it cannot start again. The plant photosynthate capacity is not only a function of its leaf area, but also a function of several site and cultural factors including soil type, nitrogen and water availability, planting density, and variety. The ability to predict capacity (and final yield) for a given site is currently beyond the capabilities of even the most complex cotton crop models. However, grower experience can be used to estimate yields of known varieties in the absence

GUY L. CURRY

94

ET AL.

of insect pests or environmental disasters. The model to follow employs a site calibration parameter which specifies the peak number of floral buds per plant during the growing season. This plant characteristic is typically measured in production systems by scouting personnel (Texas Cotton Scouting Program). MODEL The number density of the fruiting structures as a function of age and time is denoted by, N( t, a) for 0 < a < t. That is, the number of fruit between ages a, and a, at time t is obtained by integrating N( t, .) between these ages:

s

(1)

P2N(t,a)da. aI

The mortality of fruiting bodies is apparently random with mean death (dropping) rate ~(a) for a fruit of age a. Thus, in the absence of forced shedding, the Von Foerster equation for the number density is

(2) where both time and age are measured on a physiological time scale using a degree-day or a rate summation rule [2] which implies da/dt = 1. The birth (bud initiation) rate at time t is denoted by B(t). Integrating Eq. (2) along characteristics and using the fact that the plant has no fruit at time t = 0 yields N(t,o)=B(t-a)exp(

for

-iap(x)dx)

O
(3)

The birth rate at time t is a function of the plant photosynthate demand D(t), which is given by D(t)=I’p(a)N(t,a)du 0

for

t>O,

(4)

where p(s) is the age-dependent fruiting-structure photosynthate demand function. One such photosynthate demand function is displayed in Fig. 1. The linear segments of p( .) describe the experimental fruit demand rates reported in SIMCOT II [I] and are indicated in the figure. The linear rates for bud demand generally agree with the authors’ experimental bud growth data, as do the relative magnitudes of the bud and fruit peak demands.

95

COTTON FRUITING DYNAMICS

0.6

-

0.6

-

ii

2

x

.

c’

I

-

f

-

z g0.4 _ g 0.2

bud

/’

-

of’; 0

’

2

/-

fruit

~

----------

’

:

.

’

a2

;

I

’ a4

0.6

FRACTIONAL

0.8

L

1.0

AGE

FIG. 1. Age-dependent relative photosynthate demand function p(s) compared with demand used in SIMCOT n (0) [1).

The bud initiation rate B(i) is a function availability and is given for t > 0 by A-D(t)

B(t)=P

[ 1.4/l-D(f)

of excess photosynthate

1’ +

where /3 is a variety-dependent parameter which represents the maximum bud initiation rate, and the notation [x] + denotes the maximum of x and 0. Equation (5) is an empirically determined function based on the experimental data of the authors. [It can be written as a more general function of the demand, e.g. B(t)= G(D(t)), and this is the approach used in the Appendix, where the convergence theory is established.] When demand D(t) exceeds the photosynthate supply A, then not only is bud initiation stopped [B(t) = 0] but all fruit within a “shedding window” is dropped. The shedding window is the dynamically changing range of ages that must be abscised by the plant in order to balance supply and demand, The smallest possible age that can be shed is predetermined as a*, and the window is given by the open interval (a*, s(t)). The upper limit, s(t), is

96

GUY L. CURRY

ET AL.

defined for f > 0 by s(l)=inf(s>o*:JCu

u’p(n)N(t,a)da+l’~(a)N(r,a)do~A s

. I

(6)

The time at which forced shedding first occurs is denoted by f* and is defined by t*=inf{t:

s(t)>a*}.

(7)

After the shedding window has “opened” there will be a time at which it will no longer be needed. Even after the window has closed there will be a region containing no fruit because of previously shed fruit. That is, once the density along a characteristic is zero, it will remain so thereafter. Therefore a “zero-fruit” region must be defined which includes a current shedding window if it exists and the effect from previous windows. For each t > t* let (a*, s*(t)) denote the interval of zero fruit, where s*(t) is defined, for f > t*, by s*(t)=sup{a:a=u+s(t-u)forO
(8)

Combining equations (8) and (3) it is seen that the number density for

so -

190

160

170

180 JULIAN

190

200

210

220

230

DATE

FIG.2. The birth-ratefunction superimposed on the characteristic lines along which fruiting structures maturetogetherwith the regionof no fruit due to forced shedding.

COlTON

FRUITING

DYNAMICS

cotton fruiting structures

97

is given, for 0
Figure 2 displays the nonzero support region for iV(., .). The birth-rate function B(-) is superimposed on the time axis to indicate the functional values and range. In the algorithm used to obtain B(e), the demand must be obtained as a function of B(a) at points in time before the birth rate becomes zero. Substituting equations (9) into (4), o(t) is defined, for t < t*, by o(t)=l>(a)B(t-a)exp(

3.

SOLUTION

-/oa~(x)dx)

da.

(W

PROCEDURE

The exact solution of the nonlinear Von Foerster system given by equations (8) and (9) is not known. However, a closed-form approximate solution is possible through an iterative procedure which utilizes curve fits. A complete analysis of the convergence of this procedure is provided in the Appendix. The algorithm is initialized by defining an initial birth-rate function B,,(.) and a grid G, over the time axis. At the kth iteration, a birth-rate function Bk( .) and a grid Gk over the time axis will be established. Utilizing Bk_ 1(-) in Eq. (IO), the demand function Dk( .) will be calculated at the grid points of Gk through numerical (Gaussian) integration. Once Dk(.) is obtained, Bk( -) is determined by a least-squares curve fit to the points obtained from Eq. (5) over the nonzero support of the grid Gk. The grid G, is identical to G,, except for the addition of time points close to the time at which Bk( 0) becomes zero, namely t *. This successive approximation procedure is utilized only over the interval [0, t*], since determinate plants are assumed. For notational purposes let 8,(e) denote the function obtained only at the grid points via Eq. (5), and let B,(.) denote the continuous function obtained via the curve fit to 8,(e). The computational algorithm can now be stated formally. ALGORITHM

Step 0: Specify an initial grid G, on the time axis and an initial birth-rate function B,-,(-). Set the iteration counter k-0 and the convergence tolerance E.

98

GUY L. CURRY

grid

0 150

160

point

170

solutions

160

ET AL.

l

190

DATE

FIG. 3.

Successive iterates for the birth-rate function B.(.)

compared with the final

grid-point solutions.

Step 1: Let k = k+ 1, and compute gk( .) at each grid point of Gk by Gaussian integration of Eq. (10) and evaluation of (5). Step 2: Obtain the refined grid Gk by truncating Go at the first point for which 8, is nonpositive. Additional points are included in Gk in the last segment to give a better approximation for the zero point, 8, being evaluated over this grid refinement. Step 3: Estimate the continuous birth-rate function Bk( .) by curvefitting 8,(e) on the grid Gk with Bk(f) -0 for f greater than the rightmost point in G,. Step 4: Compute the maximum error 5 over Gk between iterations from

If 5
COTTON FRUITING

150

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DYNAMICS

160

170 JULIAN

180

190

DATE

FIG. 4. The accuracy of the final approximating polynomiaLscompared with the final grid-point solution.

polynomial for estimating Bk(-),the maximum error 5 on the grid Gk generally becomes less than 0.001 within six iterations, starting with a constant function for B,,(s). Figure 3 displays iterative estimates Bk(-)and the converged values of the birth-rate function for five iterations; further iterates are indistinguishable from B5(*). The cotton modeling problem under consideration does not require a high-degree polynomial for the curve-fitting procedure. Above a fourthdegree polynomial, the solution differences are negligible. Figure 4 displays the limiting solutions using polynomials of first through fourth degrees. The accuracy of the curve-fitting procedure, of course, affects the minimal error that can be obtained with respect to convergence on B(-).However, convergence to the level permitted by the curve-fitting accuracy occurs for each polynomial degree. Specific applications to three cotton varieties are discussed in the following section. 4.

RESULTS

AND CONCLUSIONS

The iterative solution procedure described in Sec. 3 yields good results for the fruiting-age distribution of a cotton crop. The algorithm results in

GUY L. CURRY

100

ET AL.

two functions: one approximates the birth-rate function B, and the second approximates the function defining the upper limit of the abscised fruitingpoint regions, s*. The accuracy of the final approximating function is dictated by the tolerance of the intermediate curve-fitting procedures and the grid spacing utilized along the time axis for the contraction-mapping procedure, as indicated by Theorem 2 of the Appendix. From a cropmodeling point of view, a refinement of the time-axis grid much beyond the field sampling frequency does not appear warranted. Although almost all cotton varieties will initiate a secondary growth phase once the initial fruit load has matured, production of secondary regrowth is discouraged in contemporary Texas agricultural practice. The reasons for this production practice are that Hefiothis spp. insecticide resistance and the increasing costs of insecticides along with the inherent boll weevil infestations over most of Texas have forced growers into a short-season system in Texas [21]. Thus, the secondary regrowth phase of the Stoneville variety (Fig. 6) was not modeled, and all cotton was assumed determinate. Two additional aspects of crop modeling not addressed here concern the impact of insect damage on fruiting structures and the plant photosynthate supply. If the photosynthate supply A is considered as a function of time, A(t) merely replaces A in Eqs. (5)-(6). Dynamic feedback between the

TABLE

1

Model Parameter Values and Experimental Means and Standard Deviations ( p , o ) Symbol

Parameter Init. to flowera

Flower to mature fruita Peak bud9 Peak fNit*

TAMCOT SP37

(27.12.34)

40 (42.5,3.56)

(43.3Yl.87)

21 17

Natural mortality (o-0.83) dev. (0.83- 1 .O) dev.

25 12.5

&a)

Bud initiation

P

Fruit/bud

a

demand

Stoneville 2 13

26 (25.7, 2.52)

O.Olf/day 0

o.o2/day 0

1.8/&y

2/daY

1

1.26

*Values are given in chronological days; one day averaged 14.4 day-degrees during the experimental data collection period.

COTTON FRUITING

fruiting structures and is not treated fruiting structures Eq. (3) is replaced

101

DYNAMICS

and supply, however, is a much more difficult problem in this paper. For insect damage where the effect on the can be described as a time-dependent function p(a, t), by the more general form given by a

N(t, a)=B(r-a)exp

(1 -

j~(x, t-a+x)

dx

0

for

I

O
For situations where the insect population is closely tied to the plant population, the problem is more difficult, requiring a two-dimensional contractive mapping procedure. The appropriate algorithms and associated convergence theory for this situation have not been established. The model developed in Sections 2 and 3 was tested against field data on the seasonal fruiting characteristics for two cotton varieties, TAMCOT SP37 and Stonevilfe 213. The relevant parameters for the two varieties are given in Table 1. The difference in fruit weights of the two varieties was compensated for by scaling the fruit photosynthate demand of Fig. 1 by the factor (Ylisted in Table 1. The model and experimental results are presented in Fig. 5 and 6. These figures contain the data and their 95% confidence

?.m,

jj

160

160

200

JULIAN

_-

-

- 220

--

-

240

DATE

Fro. 5. Model comparisons with 1974 experimental data mean and 95% confidence limits for TAMCOT SP37 grown in the Brazes River Valley near College Station, Texas.

GUY

102

j

L. CURRY ET AL.

,f,

160

IS0

220

200

240

_-

__ _- --

ooo

o

n o”

_-

--

_-

cum. drop _-

160

I

180

I

200 JULIAN

I

220

I

7

240

DATE

FIG. 6. Model comparisons with 1974 experimental data mean and 95% confidence limitsfor Stoneville 213 grownin the Brazes River Valley near College Station, Texas.

intervals. As a secondary check of the modeling approach, published data on an Acala variety grown in the San Joaquin Valley in California [7] are compared in Fig. 7. The coefficients of the polynomial solutions for the three varieties are given in Table 2. To utilize the model for predicting the number-age distribution of fruit, two types of parameters are necessary: variety-dependent parameters and site-dependent parameters. Once the variety-dependent values have been obtained, apparently only two constants are needed to predict responses at different sites. These two parameters, peak number of buds and peak number of fruit, allow the model to be readily adapted to a wide range of site and agronomic conditions. In general, the iterative solution scheme converges in six iterations or less starting from a constant birth rate. The solution is fairly insensitive to the degree of the curve fit (a least-squares polynomial procedure was used) and to the time-axis grid spacing, which was established at four days to coincide with the experimental sampling period. The iterative solution method is guaranteed to converge to the solution of the Von Foerster system, within the curve fit and finite iteration tolerances, only on the specified grid points. The proposed iterative approximation-solution scheme has potential for solving more complex Von Foerster systems while yielding closed-form

‘=

0

. 500

0

l

,,

/

‘.

0

, 1000

1500

PHYSIOLOGIAL

TIME

P

,

2000

2500

‘\

Fro. 7. Model comparisons with 1973 experimental data for Acala grown in the San Joaquin Valley near Corcoran, California (Data from Gutierrez et al.[7D.

TABLE 2 Fourth-Deesee Polynomial Coefficients for Solution Approximations Stoneville’

Acalab

1.2106 8.9805x 10 -3 -2.2053x 10 -’ 1.1164x10-4 -1.9307x10-6

1.4134 2.9106x 10 -* -6.1429x 1O-3 3.1108~10-~ -5.1831 x 1O-6

0.99212 2.0779x10-2 -5.0762x10-’ 2.8663 x 10 -4 -5.4481x10-6

3.73750x -3.4386~ 1.1855~ - 1.8153~ 0.10417

7.2837 x lo3 -6.7828 x IO* 2.351 x 10 -0.35891 2.0396x 1O-3

2.0394x lo4 -2.2433x103 9.214x 10 - 1.6742 0.011364

37-45

32-38

TAhICOTa B(f) Coeff. Const. t 12 13 14 s*(t) cbeff. Const. t t* 13 t4 Nonlinear s*(t) domain: Linear s*(t) for

41-46 t>46: 27+(t-46)

105 lo4 IO3 10

t>45: 26+(t-45)

‘Time t is in chronological days [l day= 14.4 day-degrees Q]. bTime I is in chronological days [ ls day- 13.9 day-degrees (C)I.

t>38: 28+(r-38)

104

GUY

L. CURRY ET AL.

approximate analytical solutions. The model is programmed in APL. and executed on an AMDAHL 47O/V6 machine with a Tektronix 4015 graphics terminal. The execution time averages 7 cpu seconds with a total cost, including graphic output, of $2.80 per seasonal run. APPENDIX PROBLEM The successive-approximation algorithm described in Sec. 3 iteratively seeks the birthrate function B which solves a functional equation of the form

FB=B.

(11)

In order to define IF, it is necessary to formalize the definitions given earlier in Eqs. (4) and (5). It is assumed throughout that all functions are contained in Z.,(Z), the class of all bounded measurable functions, where Z=[O, r-1 and T_ is sufficiently large so that f* EZ. Define the functional B by XEZ

‘Wx)=G(u(x)),

(12)

uEL,(Z),

where G is a real-valued function with the following properties: (1) G:R++R+, (2) G is Lipschitz continuous with constant (modulus) K, 0
00.

These properties are the only restrictions on G necessary to establish the convergence of the algorithm. It is not difficult to show that the specific form for G used in this paper [Eq. (5)] satisfies these conditions with a Lipschitz constant K=2S@/A. Further, let the functionals 89 and IF be defined by XEZ,

Du(x)=loa(u)e_~.o(x-o)do,

UEL,(Z),

(13)

where p( .) is a bounded nonnegative real-valued function, and F=BD.

(14)

The interpretation of (14) is that given u EL,(Z),

Fu)(x)=PPu))(xh

XEZ.

105

CO’ITON FRUITING DYNAMICS

If u is an initial “guess” of the birth-rate function, then Fu is the result of a single iteration of the successive-approximation algorithm. Since closed-form representations of IFUare not possible, the algorithm described in Sec. 3 fits a polynomial to Fu rather than working with Fu itself. Details of this step and the associated functionals will be described later in this Appendix. Before this additional notation is developed, the existence of a solution to (11) is first established. Let y be defined by y= supp(x)e-“. XEI The assumptions on p( .) assure that y < co [in fact y is approximately 2.0 using the function p (0) described in Sec. 21. Since the interval Z is finite, it is possible to select a finite cover {Ii; i=l , . . . , M} of Z satisfying Ii =[q_,,

T.),

To =O,

si=yK(T,-q_,)
(‘5)

(‘6)

Let s=max{&; 1,2,..., M}. Next, the following “norms” are defined. Let uEL,(Z), and i,2 ,..., M. Then

Il4lr, = sup Ie(t)l, WO, Til

(‘7)

and

llu1l1;= sup lo(t I EI,

EXISTENCE

(‘8)

THEORY

Typically, the approach used to establish the existence of a solution to a functional equation such as (11) is to first demonstrate that the functional is contractive (as, for example, [5]). Once this has been done, existence can be established by appealing to one of several versions of the Banach fixed-point theorem 141. Unfortunately, F is not contractive over the entire range of I, so an alternative approach is needed. The approach taken below shows that if UEL,(Z), then {F”u; n=l, . . . }, where F”u=F(F”-‘u) with F”u=u, forms a Cauchy sequence. The existence of a limit B of (F”u} is a consequence of the completeness of L,(Z).

GUY L. CURRY ET AL.

106

Lemma 1 below shows that IF is “piecewise” contractive; that is, over each of the intervals Ii, F satisfies a Lipschitzian relationship. LEMMA

1

Let u,

vEL,(Z).

Then

Let i, 1 < i c M, be fixed and t E[O, q]. First,

Proof.

I(~v)(t)-(~u)(t)l=I(~~V)(f)-(~~u)(t)l ~K(pv-Dull~,.

(19)

Now, J(lDv)(t)-(Du)(t)l

G~‘p(t-a)e-~(‘-“+(a)-u(a)[~

Hence,

IlDv-Dull,

QY

i

vj-L)Il-+,.

j-l

The final result follows by using the definition of Sj and then taking the supremum of (19) over [0, Ti 1. n Next define the coefficient function f by

j-o, a”-‘(k)(k+l)...(k+j-1)/j!, for k-

j=1,2,...,M-1,

1,2,3, . . . .

LEMMA Let n

2

and k be arbitrary positive integers. Then, 5 6f(r, r-=Q

k-

I)=f(u,

k).

COTTON FRUITING

107

DYNAMICS

Proof. The proof is by induction. Only the general step is provided here. Assume the lemma is true for values 1,2,. . , , n - 1 and for any k. Then, by the inductive hypothesis, 2

6f(r, k-

l)=f(n,

k)+6f(n+

1, k-

1)

r-0

LEMMA

3

Let vEL,(I)

and k> 1. Then

IF k+'V-FkVII~,

<

2 f(i-j,k)Gj((Fv-vl(,,,

i- 1,2 ,..., M.

j-1

Proof. The proof is by induction. Only a sketch of the general inductive step is provided here. Assume the inequality is true for values 1,2,. . . , k - 1 M. Then, using Lemma 1, Lemma 2, and the fact that and all i=1,2,..., 6 > Sj for allj,

=

LEMMA

1:

j- I

$llFv-v)lI,_f(i-j,

k).

n

4

Let v EL,(I),

let n > m 2 1 be arbitrary integers, and let i, i= I, 2,. . . , M,

be fixed. Then,

Proof.

Applying

the triangle

inequality

to the left side of the above

108

GUY

L. CURRY ET AL.

inequality yields

k-m

The final result is obtained by applying Lemma 3 to each of the terms in the summation. n LEMMA Let

(a)

5

r, r=O,l,..., 5

M - 1, be fixed.

f(r, n)=(l

Then

-S)++‘),

PI-0 (b)

lim f(r,n)=O, n--r- m

Proof.

and

By assumption, 0 < 6 < 1. Thus, . g S4=(1-q-‘. q-0

Upon differentiating both sides k times, (k!)-’

5

. (q-k+l)6q-k,(l-8)-(k+‘).

q(q-l)..

q-k The function f is obtained by identity. This fact establishes quence of the convergence of of the terms in the summation THEOREM

making the appropriate substitution into this part (a) of the Lemma. Part (b) is a consethe series in (a). Part (c) is true because each is positive. H

I

Let UEL,(Z). Then {lF”u; n=l, . . . } forms a Cauchy sequence in L,(Z) and has a limit B. Furthermore, B is the unique function which satisfies B=IFB.

Proof.

For any n, m with n > m > 1, Lemma 4 with i=m asserts that I(F”+‘u-F”‘ujl,

< 5

i

j-1

<

5 j-

f(M-j,

k)Gjl(Fu-u(l,j

k-m

GjllFu-oil,, 1

% k-m

f(M-/‘,k)

1 .

COTTON FRUITING

109

DYNAMICS

Each of the terms in brackets can be made arbitrarily small as shown in Lemma 5. Hence, for any E> 0, there exist n and m sufficiently large, with n > m, such that j=1,2

It then follows that for n and m sufficiently

,...,

M.

large,

and thus {F”u} is a Cauchy sequence. Since L,(Z) converges to (say) B EL,( I). Now, to show that B= FB, let n > 1. Then,

is complete,

{F”o}

IIB-FBII,d)IB--*u(l,+IJIF”u-FBI(,. But, since each of the terms on the right-hand side goes to zero as n gets large, it must be true that B = IFB. It remains to prove that B is unique. Suppose it is not. Then there is a function uEL.,(Z) such that { lF”u} converges to (say) Z? and i?#B. The proof above shows that 8= F8. Successive application of IF yields F”B=B

and

F”+‘B=8

for any n > 1. Now, using Lemma 3,

Also.

II+Bllr, Let e>O be arbitrary. that

GIIF8-BII,=Q<,.

Then, from Lemma 5 there exists ni sufficiently

f(M-jyn,)s&, J

j=1,2

,..., M.

large

110

GUY L. CURRY ET AL

Putting all these facts together shows that

which establishes that B = B, since E is arbitrary.

n

Theorem 1 asserts that given any arbitrary selection of an initial birth-rate function B,, exact successive approximations (P” B,) will eventually converge to the true birth-rate function B, which is the unique solution to lFB = B. The convergence is guaranteed for any grid satisfying the assumptions (15) and (16) and any T,,. It is also important to emphasize that no restrictive assumptions concerning mappings defined in (12)-( 14) were made to obtain convergence. The parameters and form of these mappings were determined a priori from biological considerations. Unfortunately, while Theorem 1 assures convergence, it is impossible to use this exact successive-approximation procedure, since closed-formed representations for the functions IF”u are not available. To circumvent this practical difficulty, a polynomial fit was used in Sec. 3 to obtain a closedform expression (the polynomial) to approximate F”u at each iteration. Theorem 2 below demonstrates that this polynomial-based successiveapproximation algorithm does, in a sense, converge. The theorem provides an upper bound on the distance between the solution obtained from the algorithm and the theoretical limit B. Before stating Theorem 2, some preliminaries and notation are provided. APPROXIMA

TION THEOR Y

Let V(q, G) be the set of polynomials of degree 9 that can be generated by a particular finite grid set G. Let the functional P map L,(Z) into V(q,G). That is, given uEZ&(Z), let PuEV(q, G) be the polynomial approximation (obtained via a least-squares procedure) to v using a polynomial of degree 17and the grid G. Furthermore, let A(n, G) be the maximal approximation error:

In general, A(v, G) is decreasing in 7. (A more general statement of this problem can be found in [3].) When no confusion can arise, the notation A= A(q, G) and V= V(q, G) will be employed to simplify expressions. Now, define the functional F : I,,( Z)+V(q, G) by F-W.

(20)

That is, given v EL,(Z), the function lFu is computed for values in G. Using this information, the polynomial approximation to this function P(Fv) is

COTTON

FRUITING

111

DYNAMICS

determined. This two-step procedure is combined into the single composite mapping, F, which forms the basis for the polynomial successiveapproximation algorithm. It is now possible to make a formal statement of the procedure used in Sec. 3. ALGORITHM Step 0: Select 11, G, B,, EL,(I), E > 0, and set n= 1. (There is a lower bound on E; indeed, E cannot be chosen smaller than the constant appearing in Theorem 2 below.) Step 1: Evaluate F”+i& = F(g”B,,). Step 2: If (1!?“+iB,, - t?“B,, [I, Q E, stop and set B = gn+‘BO, If not, set n to n+ I and return to step 1.

The formal proof of the algorithm is contained in the following lemma and theorem, LEMMA 6 Let 9

and let B solve B = IFB. Then for any k > 1 and

and G be fixed,

UEL,(I). (a) I)~ku-BIII,

921.i 2 /(i-l,n)+,$l n=l

(b) jl~ku-BII,

~2A.(1--8)-~+

f(i-~,k)~jllu-BIIIj;

2 f(i-j,k)$lJu-BJJ,,. j- 1

Proof The proof is by induction on k. Only the general step is provided here. Therefore, assuming that the lemma is true for indices 1,2,. . . , k - 1, it is proven true for index k. Proof of (a):

<2A+6

i )~~*-‘u-_(Fk-‘B~~,, 1-l i

<2A

1+ 2 1-I

+ i I-1

k-l

x

&=(l-l,n)

+ i

n==l

I-1

i

&f(,--j,k-1)

j-1

f: ~f(~-j,k-1)6,Ilu-Bll,,, j-l

which follows from Lemma 3, Lemma 4, and the inductive hypothesis applied to (a). The final result follows by interchanging the order of summation, applying Lemma 2, and using the fact that f(r, I)= 1 for all r.

GUY L. CURRY

112

ET AL.

Proof of (bj: This follows directly from (a) by applying part (b) of Lemma 5 to the first term in (b). n THEOREM Let

q

2

and G be given, and let B satisfy IFB = B. Then, for any v E L,( I). lim [lp”v-B((, n+oo

Proof.

<2A(q,G)(1-S)-“.

From Lemma 6, I]h-BII,

<2A.(l-6)

-M+

5 j=

f(~-j,~)~jll~--BIII, 1

for every n > 1. But the second term goes to zero for every j, from Lemma 5, since Jjv-BIJ,,
COTTON 7

FRUITING

DYNAMICS

113

A. P. Gutierrez, L. A. Falcon, W. Loew, P. A. Leipzig, and R. van den Bosch, An analysis of cotton production in California: A model for acala cotton and the effects of defoliators on its yield, Environ. Entomol. 4: 125- 136 (1975).

8

9

10

A. P. Gutierrez, T. F. Leigh, Y. Wang, and R. D. Cave, An analysis of cotton production in California: Qgus hespencr (Heteroptera: Miridae) injury-an evaluation, Can. Entomot, 109:1375-1386 (1977). A. P. Gutierrez, G. D. Butler, Jr., Y. Wang, and D. Westphal, The interaction of pink bollworm (Lepidoptera: Gelichiidae), cotton, and weather: A detailed model, Can. Entomol. 109:1456- 1468 (1977). A. P. Gutierrez, Y. Wang and R. Daxl, The interaction of cotton and boll weevil: A study in co-adaptation, Can. Entomol. 111(3):357-366 (1979).

11

A. Haimovici, On the growth of a population dependent resources and pollution, Math. Eiosci. 43:213-237 (1979).

12

F. Hoppensteadt, Mathematical theories of populations: demographics, genetics and epidemics, Regional Conference Series in Applied Mathematics, SLAM, Philadelphia, 1975. J. W. Jones, L. G. Brown, J. D. Hesketh, COTCROP: A computer model for cotton

13

14 15 16

17

on ages and involving

growth and yield, in Predicting Photoynthesis for Eco-Models (J. D. Hesketh, Ed.), 1980, Chapter 10. P. H. Leslie, On the use of matrices in certain population mathematics, Biometrika, 35:213-245 (1945). G. Oster, and Y. Takahashi, ModeIs for age-specific interactions in a periodic environment, Ecol. Mono. 44:483-501 (1974). F. Rodolphe, H. El Shishing, and J. C. Gnillon, Modelisation de deux populations d aleurobes ravageurs de cultures, in AFCET: Moaklisation et Maitrise a& Systekes (C. R. Gong., Ed.), 1:527-535 (1977). M. Rote&erg, Equilibrium and stability in populations age-specific, J. Theoret. Biol. 541207-224 (1975).

whose interactions

are

18

J. W. Sir&o, and W. Streiffer, A new model for age-size structure of a population, Ecology 48:910-918 (1967).

19

H. N. Stapleton, D. R. Buxton, F. L. Watson, D. J. Nolting, and D. N. Baker, Cotton: A computer simulation of cotton growth, Tech. Bull. 206, Agri. Exp. Stat., Univ. of Arizona, Tucson, 1973, 124 pp.

20

H. Von Foerster, Some remarks on changing pop_ulations, in 17heKinetics of Celhdar Prolifeation (F. Stohlman, Jr., Ed.), Grune and Stratton, New York, 1959, pp. 382-407. J. K. Walker, R. E. Frisbie, and G. A. Niles, A changing perspective: Heliothis in short-season cotton in Texas, Bull. EntomoI. Sot. Amer. 24(3):385-391 (1978).

21 22 23

D. Wallach, An empirical mathematical model of a cotton crop subject to damage, unpublished manuscript, Department of Botany, Hebrew Univ. of Jerusalem, 1979. Y. Wang, A. P. Gutierrez, G. Oster, and R. Daal, A general model for plant growth and development: Coupling plant-herbivore interactions, Can. Enromol. 109: 13591374 (1977).