Models for age structured populations with distributed maturation rates

Journal of

J. Math. Biology (1986) 23: 247-262

@ Springer-Verlag I 1986

Models for age structured populations with distributed maturation rates Richard E. Plant’ and L. T. Wilson2

’ Departments of Mathematics and Entomology, University of California, Davis, CA 95616, USA Department of Entomology, University of California, Davis, CA 95616, USA

Abstract. In the use of age structured population models for agricultural applications such as the modeling of crop-pest interactions it is often essential that the model take into account the distribution in maturation rates present in some or all of the populations. The traditional method for incorporating distributed maturation rates into crop and pest models has been the so-called “distributed delay” method. In this paper we review the application of the distributed delay formalism to the McKendrick equation of an age structured population. We discuss the mathematical properties of the system of ordinary differential equations arising out of the distributed delay formalism. We then discuss an alternative method involving modification of the Leslie matrix.

Key words: Age-structured population models - Distributed maturation rates

- Delay differential equations - Leslie matrix 1. Introduction

In virtually all biological populations, individuals mature at different rates. This variation in maturation rate may be due to genetic factors, environmental factors, or a combination of these. In agroecosystems the maturation rate is strongly influenced by temperature. This influence is usually taken into account by measuring elapsed time in “physiological” units rather than chronological units. The physiological unit provides an approximation to the Arrhenius reaction rate (Brody 1945). If T denotes chronological time elapsed since the initial time T~ and 8 ( 7 ) denotes temperature at time T,then we may define the elapsed physiological time t ( T ) as t(T)=

l:

F ( 8 ( 7 ’ ) )dT’,

(1.1)

where the function F ( 8) represents the influence of temperature on the organism’s growth rate. In practice, the function F ( 8 ) is often taken to be simply F ( 8 ) = ( 8 - 8,)H( 8 - eo), where H is the Heaviside step function and eo is a developmental threshold. For this reason, the units of t are traditionally called “degreedays”. Measuring time in this way has proven a reliable way to incorporate the

248

R. E. Plant and L. T. Wilson

effects of temperature into calculations involving agroecosystems (Wilson and Barnett 1983). Since different individuals in a population may experience different microclimates during their lifetimes, and may respond differently to these microclimates, the variation in maturation rate in systems measured in physiological time units may be quite striking. In the construction of simulation models for interacting populations one generally assumes that all individuals mature at the same rate. In a model that is to be used in an integrated pest management program, however, the consequences of ignoring variation in maturation rate may be serious. For example, if a control action is to be timed around the first emergence of fruit buds then a model that ignored variation in maturation rate of the plant would overestimate the elapsed time before that control action should take place. Stuart and Merkle (1965), Rubinow (1968), and Weiss (1968) each provide early discussions of how the classical McKendrick equation (McKendrick 1926; von Foerster 1959) for an age structured population could be modified to take into account the variation in maturation rates among individuals the same population. Blythe et al. (1984) present a method for dealing with such populations and show how the method may be applied to laboratory populations. Sharpe et al. (1977) discuss the distribution of maturation rates in populations in an agroecosystem. Welch et al. (1978) present one of the first crop models to incorporate distributed maturation rates as a part of its mathematical structure. The method used by Welch et al. to deal with distributed maturation rates has become the accepted standard practice in agricultural crop-pest modeis (e.g. Gutierrez et al. 1985). The method of Welch et al. is based on the so-called “distributed delay” theory of Manetsch (1966). Manetsch’s theory is an expansion upon a well known idea in applied mathematics and numerical analysis, where it is known as the “linear chain trick” (MacDonald 1978). The idea is that solutions of a certain class of integrodiff erential equations are also solutions of an appropriate system of ordinary differential equations. One therefore expresses the effect of maturation in the population in terms of an integrodifferential equation which is then converted into a system of ordinary differential equations. Manetsch (1976) provides an example of how the distributed delay theory may be used in a model of an insect population. He formulates the age distribution as a sequence of stages (e.g. eggs, larvae, pupae, and adults). Individuals enter each stage sequentially and spend a variable amount of time in a given stage. A similar model has also been proposed for cell populations (Takahashi 1968). Manetsch (1980) derives an equation that includes as a special case the relationship between the distributed delay model and the McKendrick equation. Essentially, the distributed delay model may be regarded as a numerical approximation of the McKendrick equation by the method of lines. The consequences of this relationship have not, however, been fully explored. One purpose of this paper is to review this relationship in detail and examine its consequences. If one were to construct a numerical approximation to the McKendrick equation, one would not ordinarily choose the method of lines. The most common approximation involves the Leslie matrix (Leslie 1945). As is the case with the McKendrick equation, however, the Leslie matrix has the property that a cohort

Age structured populations and distributed maturation rates

249

of individuals matures at the same rate. Therefore, if one is faced with a simulation problem in which the inclusion of a distributed maturation rate is important, then the method of lines approach presents the happy occurrence of a situation in which the numerical approximation to the partial differential equation provides a more accurate representation of the behavior of the real system being modeled than does the partial differential equation itself. The ease of formulation of distributed delay models and their relatively accurate representation of the real system are the primary reasons for their popularity in agricultural models. In Sect. 2 of this paper we review the relationship between the method of lines approximation and the McKendrick equation. Much of this material is not really new; portions of it may be found in Manetsch (1976, 1980); Van Sickle (1977), MacDonald (1978), and probably elsewhere as well. The connection between the McKendrick equation and the “distributed delay” model is implicitly recognized by Gutierrez et al. (1984), although they do not explore the consequences of this relation. In Sect. 3 we discuss the eigenstructure of the method of lines approximation and its relation to the numerical solution of the system of differential equations. Section 3 also describes the process of interpolation along the age axis and discusses some of the uses of this process. A major drawback to the application of the method of lines approach to the simulation of populations with distributed maturation rates is that if the distribution has small variance, Le. if members of a cohort mature at rates that are close, but not exactly equal, then the method of lines requires a very large number of coupled ordinary differential equations to simulate the population. In such circumstances an alternative model, first proposed by Slobodkin (1953) and later developed by Werner and Caswell (1977), may be more appropriate. In Sect. 4 we describe this alternative, which is based on a simple and obvious modification of the Leslie matrix, as it relates to our problem. In Sect. 5 we compare these two approaches to the modeling of populations with distributed maturation rates to others that have been proposed. We also give a brief discussion of extensions of the models. 2. Derivation of the equations Consider a single, isolated population that is growing at a rate independent of its density. Let u ( a , t ) be the age density of the population at time t. We emphasize again that for our applications time and age are measured in physiological units. In particular, the value t may be considered to be that recorded by a timetemperature monitoring station located near the population. Thus, an individual born at time to degree days may at time tl be of (physiological) age other than t , - to either because it has experienced a different microclimate than that of the monitoring station, because its response to its environment differs from that of the “average” member of the population, or both. The behavior of the age density may be described by the McKendrick (1926) equation

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R. E. Plant and L. T. Wilson

together with the boundary and initial conditions u(0, t ) = jomP(a,t ) u ( a , t ) da,

(2.2a)

u(a,O)= u"(a).

(2.2b) Here p ( a , t ) and @ ( a ,t ) are the age specific death and birth rates, respectively, and u o ( a ) is the initial age distribution. A complete discussion of the derivation and properties of this equation is given, for example, by Hoppensteadt (1979, and by Nisbet and Gurney (1982). An important property of solutions of the McKendrick equation is that they propagate along characteristic lines. Thus, consider a cohort of individuals born at the same time, which we may take to be t = 0. We may model this cohort at time zero by using a delta function, so that we have (2.3) Since we are considering only the fate of this cohort, to which no individuals are born after time zero, we have p (a, t ) = 0. Suppose for simplicity that p ( a , t ) = p, a constant. It is not hard to show that for t 2 a the solution in this case is u"u) = 6 ( a ) .

u ( a , t ) = 8 ( a - t ) e-CL'.

(2.4) Equation (2.4) implies that every member of the cohort matures at the same rate. To approximate the system (2.1,2.2) we use the numerical method of lines (e.g. Madsen and Sincovec 1973). In this method, which is most commonly used for parabolic equations rather than hyperbolic ones such as (2.1), one obtains a system of ordinary differential equations by discretizing in the age variable but not the time variable. Our derivation is a special case of one given by Manetsch (1980). Let {a,} be a sequence of values of a, with ~ , < a , + and ~ ao=O. Let k, = a, -a,-,, n = 1,2,. . . . Let u , ( t ) be the approximation to u(a,, t ) obtained by the following procedure. We use the discrete approximation

Substituting (2.5) into (2.1) and replacing the approximate equality with equality yields

where p, = p((a,, t ) . We approximate the boundary condition (2.2a) by replacing the integral with a series. This yields (2.7a) where Pn(t ) = P(a,,

t).

Finally, the initial condition (2.2b) is simply replaced by u,(O) = u O ( a , ) ,

n = 1,2,. . . .

(2.7b)

Equations (2.6,2.7) constitute the method of lines, or linear chain approximation to (2.1,2.2).

Age structured populations and distributed maturation rates

25 1

Let us compare the behavior of a cohort of individuals born at time zero in the model (2.6,2.7) with that already obtained using the McKendrick equation. Assume that kl = k2 = = k, that p l= p2= * * = p, that u,(O)= l/k, and that u,(O) = 0, n = 2,3, . . . .Then one may show, for example using Laplace transforms, that

u,,(t)=G(t;n,k)e-@‘

(2.8)

where

Since k = a , / n , the function G(t ; n, a , / n ) defines a gamma density in the variable t with mean a, and variance ( ~ , ) ~ / This n . is a well known result, given, for example, by Manetsch (1976) and MacDonald (1978). As n approaches infinity, the solution (2.8) of Eqs. (2.6) approaches the solution (2.4) of Eq. (2.1). Manetsch (1980) points out an alternative representation for the solution (2.8). This is 1 u,(t)=-P(n-l, k

t / k ) e-p‘,

(2.10)

where

A” A!

P(m, A ) =- e*#

(2.1 1 )

The function P(myA ) has the form of a Poisson distribution in m. Therefore, for any fixed t, the average value of n is 1 + t / k . This is another indication that a cohort of individuals in Eqs. (2.6,2.7) propagates through maturity at the same average rate as that of a cohort in Eq. (2.1,2.2). Furthermore, since X P ( n 1, t / k ) = 1 it follows that the total number of individuals in the population at time t is L’ku,( t ) = e-@*.This is the same value as that obtained from the solution (2.4) of Eq. (2.1). The variance of the distribution P ( n - 1, t / k ) representing the cohort solution is o = t / k . This is the variance in n ; the variance in t may be obtained by multiplying by k2. Of more biological significance perhaps is the difference in physiological age between the most and least mature members of a cohort at some time t. This is roughly six times the standard deviation, or Van Sickle (1977) shows that the average time for an individual to reach age a,, conditional on surviving to that age, is kn/(kp + 1). For p = 0 this reduces to kn, and as p increases, the average time decreases. This may be interpreted as showing that premature death takes a greater toll on individuals who spend a greater time in the juvenile state. We conclude this section by noting a few properties of Eqs. (2.6,2.7). First, the equations obey the same conservation law as the McKendrick equations. If we let x( t ) denote the total population at time t, then (2.12)

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R. E. Plant and L. T. Wilson

Differentiating Eq. (2.12) and substituting from (2.6) for du,/dt, one finds that (2.13) In other words, the rate of change of the population equals the birth rate minus the death rate. A second property is that we may rewrite the equations in terms of numbers of individuals in an age class. Let q n ( t ) = k,,u,(t). Then q , ( t ) is the number of individuals between ages and a, at time t. Since the mesh width k, is constant in t, q, obeys the same differential equation as u,, so that Eq. (2.6) may be written in this variable, with obvious modifications to Eq. (2.7). This is the form used by Gutierrez and Baumgaertner (1984) and Gutierrez et al. (1984). Finally, if we keep the second order term in the approximation (2.5) then, assuming all the k, are identical and equal to k leads to a u au - - p u + - - k a2u -+-a t aa 2 aa2’

(2.14)

Thus the linear chain model represents to a second order approximation a convection diffusion equation for a population in which individuals are “diffusing” along the age axis as well as maturing. Such a model was originally proposed by Stuart and Merkle (1965) and has been discussed by Auslander et al. (1974). A major theoretical advantage of the linear chain model over the diffusion model (2.14) for most applications is that individuals in the linear chain model cannot age in the negative direction.

3. Mathematical properties of the linear chain approximation

In this section we consider some of the mathematical properties of the linear chain approximation reviewed in the last section. In particular, we discuss the eigenstructure of the system, and the interpolation of values of u ( a , t ) . The eigenvalue equation of the system (2.6,2.7) may be written 03

k;’

C

PnknZ,

- Z,

k,’[z,_l-

z,]

[n=1

I

- p.1~1-V Z ~=O,

-p , ~ , VZ, = 0,

n = 2,3, . . . ,

-

(3.1)

where the vector z = [z1z2 -IT is an eigenvector corresponding to the eigenvalue v. Let v1 be the eigenvalue with largest real part. If vl is real then the components of the normalized eigenvector corresponding to this eigenvalue are the proportions of the population at the stable age distribution. The second of Eqs. (3.1) may be rearranged and iterated to yield n

z,=

n (l+km[pm+v])-lz*.

m=2

Substituting this into the first of Eqs. (3.1) yields the eigenvalue equation

Age structured populations and distributed maturation rates

253

The analogy of this equation to the eigenvalue equation of the McKendrick equation (e.g. Hoppensteadt 1975) is evident. To continue our analysis, we consider the finite model obtained by truncating the age distribution vector at n = N. Letting u ( t ) = [ u l (t ) u N ( t ) I T , we may write the linear system (2.6,2.7) in the form d u / d t = Au, where

Wh

re

K,

= l/k,,.

Le

The matrix A may be written as A = B + diag( K m a x + pmax), where PI+SI

B=

...

P2

K2

62

0

K3

0 63

0

...

.

PN

* ' '

0 0

KN

6N-

* * '

(3.6)

and 6, = p m a x - p , + ~ m a xThe -~n matrix . B is positive, and therefore may be analyzed by classical Perron-Frobenius arguments (e.g. Gantmacher 1959; Pollard 1973). Indeed, the argument used to prove Theorem 4.3.4 on p. 41 of Pollard (1973) may be used without modification to show that the matrix B of Eq. (3.6) is positive regular if and only if the greatest common divisor of the numbers of the columns containing positive Pn is one. It then follows from the PerronFrobenius theorem (e.g. Pollard 1973, p. 39) that the matrix B has a positive eigenvalue of algebraic multiplicity one, corresponding to left and right eigenvectors with positive elements, that is greater in absolute value than any other eigenvector. If A is an eigenvalue of B, then v = A - K,,,pmaX is an eigenvalue of A. Therefore, if B is positive regular then the matrix A will have at most one positive eigenvalue vl which is also the eigenvalue with maximal real part, and this eigenvalue will correspond the unique eigenvector all of whose components are positive. Another point is worth noting. In many populations, and especially those involved in typical agricultural problems, the intrinsic death rate p,, is only mildly = K N = K , = p m a x - p m l n and P n = dependent on age. If we let K , = K~ = (pm~x-pn)/(pmax-pmi") then we may write K

0

B( E ) =

.

o

(3.7)

...

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R. E. Plant and L. T. Wilson

The matrix B ( 0 ) is a Leslie matrix, and the eigenvalues of Leslie matrices have been well studied (e.g. Pollard 1973). From a well known theorem on the continuity of eigenvalues (e.g. Wilkinson 1965, p. 65), it follows that as E tends to zero, the eigenvalues of B(E ) approach those of B(0). In applications, the relative magnitude of the eigenvalues of A may be of considerable importance. The numerical solution of the density dependent versions of Eqs. (2.6,2.7) likely to arise in applications involves the integration of a large system of highly structured ordinary differential equations. It is well known (e.g. Madsen and Sincovec 1973) that the system of differential equations derived by applying the method of lines to the diffusion equation is sti& that is, the moduli of the eigenvalues with largest and smallest moduli differ by several orders of magnitude. Specifically, for large N the smallest magnitude eigenvalue has magnitude approximately rr’, and the largest magnitude eigenvalue has magnitude approximately 4N2 (Madsen and Sincovec 1973). The numerical solution of stiff systems of ordinary differential equations is quite difficult unless special solution methods are used (e.g. Gear 1971). Accordingly, it is of interest to know whether a transport equation such as (2.1) exhibits the property of stiffness under the method of lines approximation. The eigenvalues of the matrix A are difficult to compute in general. In the special case in which Pn = 0 for all n, however, it is trivial to show that they are given by v,, = - ( K , , + p u n )Therefore . we are led to expect that if the death rates are widely scattered, the eigenvalues will be, too. As was mentioned earlier, we are primarily concerned with applications in which the death rates are roughly independent of age. Therefore, we must look to the other potential sources of variation in the eigenvalues: the birth rates and the size of the matrix. To attempt to study the effect of the birth rates and matrix size we constructed a model matrix A and numerically determined the eigenvalues of this matrix for a range of parameter values. The elements of the matrix are based roughly on life table data of Carey and Bradley (1982) for the spider mite Tetrunychus urticue Koch on cotton. The birthrates Pn are set as follows. Let N = 3 N 1 , then Pn = 0, n = l , ..., N1;P,,=6Pf, n = N , + l , ..., 2 N l ; Pn=4/& n = 2 N , + 1 , ..., N. The factor Pf is a test parameter; for Pf = 1 the birthrates provide a rough approximation to Carey and Bradley’s data. The deathrates are set at p = 0.05. The eigenvalues of this matrix were computed numerically using the EISPACK subroutines (Smith et al. 1976) for various values of N and Pr. Table 1 shows the results of the computations. These results indicate that while increasing N and/or the deviation in the birthrates with age does tend to increase the spread Table 1

9 48 99 48 48

1 .o 1 .o 1 .o 5.0 10.0

0.93 3.35 6.65 3.49 3.56

0.22 0.21 0.26 0.42 0.50

Age structured populations and distributed maturation rates

255

in the magnitudes of the eigenvalues, in neither case is this increase substantial. Accordingly, we may expect that the linear chain model may be satisfactorily integrated using numerical integration schemes for nonstiff system of differential equations. Let us now consider the interpolation of values of u , ( t ) to obtain a representation of values of u ( a , t ) for an age u that is not a mesh value a,. Recall that the solution u , ( t ) of the basic system of Eqs. (2.6'2.7) is given by u,( t ) = G( t ; n, k ) e-wf

(3.8)

where (3.9)

Suppose we wish to interpolate to a value u(u, t ) where a = (1 - O)u, + Ban+,for some fixed value of n. The most natural interpolation would be one that interpolates the distribution (3.9), giving u,+l_e(t)=G(t;[l-e]nfe[n+l),

k ) e-@'.

(3.10)

The form of Eq. (3.10) suggests that we use a weighted geometric mean of the form u( a, t ) = u,( f)(l-o)u,+l(t y .

(3.11)

We shall now show that for sufficiently large n this does indeed give a good approximation to Eq. (3.10). Substituting from (3.8) and (3.9) into (3.11) yields

where v = n + 1 - 8. This will have the proper form for large n provided Iim r ( n ) o - ' T ( n + l ) e = T ( n + l - e ) .

,-roo

(3.13)

Stirling's formula yields

Let us define L , ( n ) and L,(n) by (3.15)

Then we must show that lim L,( n ) = L,( n).

n+m

(3.16)

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R. E. Plant and L. T. Wilson

Taking logarithms of L , and L2 and subtracting yields, after a little algebra,

In L1-In L ~ (1 = - e ) ( n + ; )I n ( n / ( n + = (1 - O ) ( n -;)[

e + n+e

e ) ) + e(n+:) In((n + l ) / ( n + e))

o(n-2) (3.17)

Letting n tend to infinity establishes Eq. (3.16), which in turn establishes our result. For moderate values of n the approximation of L 2 ( n ) with L , ( n ) is a good one. For example, for 0 = ;, if n = 5 then LI and L2 differ by about ten percent, while i f n = 10 then they differ by about one percent. Finally, it must be noted that unless u, and u,+[ differ significantly, the geometric interpolation of Eq. (4.2) gives almost exactly the same result as ordinary algebraic interpolation. For example, if w, and u,+[ differ by ten percent, then the two interpolation methods give values that differ by less than one percent. In short, we may expect either form of interpolation to give a numerically satisfactory result in most cases. One application of interpolation is obvious and is used by Gutierrez and Baumgaertner (1984) and Gutierrez et al. (1984). Insect populations are often subdivided into their various life stages, and indeed many insects differ in their pest status in different life stages. Similarly, in many agricultural crops significant events, such as flowing, occur at a certain physiological age. In both of these cases, the appropriate age may lie between two age mesh points, and in such cases interpolation is obviously appropriate. Another use for interpolation is in the dynamic variation of the variance of the distribution G ( t ;n, k ) of Eq. (3.8). If k is fixed then this variance is equal to ka,, and therefore is directly proportional to k for fixed a,. Suppose that one wishes to model a system in which the variance depends on a vector q of parameters. The components of q may include such things as the nutritional value of the plant (which may in turn be affected, for example, by irrigation), the abundance of predators, and so forth. This dynamic variation in variance must be accomplished by a dynamic variation in k. This variation may in turn be accomplished in a stepwise manner by holding t fixed, interpolating to a new mesh, and then letting the system evolve at the new mesh points. 4. The modified Leslie matrix model Although the linear chain model discussed in Secs. 2-4 is highly useful in many cases, it has one serious drawback that limits its usefulness in a certain class of systems. As can be seen by examining Eq. (2.10), the variance of the distribution representing the cohort solution is fixed once the mesh spacing k is determined. Indeed, this variance is proportional to the mesh width, k. If the variance is small relative to maximum life span, then the linear chain model will require a very large number of nodal points. This makes numerical solution of the problem time consuming and in extreme cases may exceed the capacity of the computer. In this section we discuss an alternative model that permits the simulation of

Age structured populations and distributed maturation rates

257

populations with small variance in the distribution representing the cohort solution. The model results from a simple and obvious modification of the Leslie matrix (Leslie 1945). The idea of modifying the Leslie matrix to represent populations whose members age at different rates was first proposed by Slobodkin (1953). Later, such modifications were studied in detail by Werner and Caswell (1977) in connection with the modeling of populations of annual plants. It is easy to see that one may add positive elements to the Leslie matrix on and to the left of the diagonal to achieve the effect of distributed maturation. We study the simplest such modification. In keeping with the usual Leslie matrix formulation, we consider the primary dependent variable to be q ( t ) = [ q l ( t ) ,. . . ,qn(t), . . .]', where q n ( t )is the number of individuals between ages ( n - l ) h and nh at time t. We assume that the vector q ( t ) satisfies the equation q ( t + h ) = Aq(t).

(4.1)

Consider the infinite matrix

where O < E < p n / 2 for all n. In this model a fraction E of each age class passes through two age classes in one time step and a second fraction does-not leave its current age class. The remainder move up one age class. Note that this model shares with the linear chain model the property that no individuals can lose age. If the distribution of the maturation rate changes with age, then the population may be represented by a sequence of systems of the form (4.1), one for each life stage. We now derive the solution, analogous to (2.8) of Eq. (4.2) for the case of a cohort with no death. The birth and survival rates are set at P,, = 0 and p,, = 1, and the initial conditions are ql(0) = 1, qn(0)= 0, n > 1. As in Sect. 2, we write the solution in terms of probability distribution. Consider the one dimensional random walk of a particle along the set of positive integers. Let x,,, denote the position of the particle at time m,and suppose that x l = 1. Assume that the particle's motion is governed by the following equations: Pr{x,+l = x,}

= E,

+ I} = 1 - 2 ~ , Pr{x,,,+, = x,,, + 2) = Pr{x,,,+l = x,

E,

Pr{x,+l = x j , j # 0 , 1 , 2 } = 0. Let Q(n; m,E ) = Pr{x, = n } . It is evident that the solution to Eq. (4.2) under the

I

258

R. E. Plant and L. T. Wilson

conditions prescribed above is q , ( m h ) = Q ( n ; m,E ) . (4.3) The probability distribution governing the motion of the particle is the trinomial (e.g. Fisz 1963), and it is straightforward to show that

Q(n; m , ~ ) =

C

i+Zj=n-l

m! ( 1 -2E)iEm-i. l ! j ! ( m- i-j)!

.

(4.4)

The expected value and variance of the distribution Q may be computed by noting that the position x,,, of the particle may be expressed as one plus the sum of m independent, identically distributed random variables ti, where the probability distribution of the tiis Pr{t = 0) = E , Pr{ 5 = I } = 1 - 2 ~and , Pr{t = 2) = E . The mean of this distribution is 1, and the variance is 2 ~ Therefore . the expected value of n under the distribution Q( n ; m, E ) is 1 + m, and the variance of n is 2 m ~ Substituting . m = t / h into these values, we see that at a fixed time t the solutions q n ( t )may be represented by a probability distribution in n whose mean is 1+ t / h , and whose variance is 2 ~ t / hIt. follows from the central limit theorem (e.g. Fisz 1963) that for fixed t, for small h the distribution of the u, approximates that of a normal, as does the poisson distribution of Eq. (2.11). The presence of the parameter E in the variance allows for the control of this variance for a fixed stepsize h. Since E must take on values less than p n / 2 , it is evident that the modified Leslie matrix model of this section is most useful for problems in which the maturation rate is tightly distributed about the mean: this provides a nice complement to the linear chain model, which is most useful for populations whose maturation rate is widely distributed. The modified Leslie model may be in principal be used for problems in which the variance of the probability distribution representing the solution is high. As a practical matter, however, such applications would require a large value for the stepsize h, which would render the solution relatively inaccurate. Also, unlike the linear chain model, the cohort solution qn(t ) in t of the modified Leslie model cannot be represented by a probability distribution since the rows of the matrix A of Eq. ( 4 . 1 ) do not sum to one. Finally, the arguments of Sect. 3 regarding the eigenvalues and eigenvectors of the matrix B of Eq. (3.6) may be repeated for the matrix A of Eq. (4.1). To determine how the matrix Eq. (4.2) relates to the linear chain method we consider Eq. (2.14), the McKendrick equation with an added diffusion term, which represents a second order approximation to the linear chain model. To construct a discrete time approximation to (2.14) we write u(a+h, t+h)-u(a, t + h ) U(U, t+h)-u(a, t ) h h k u ( a + h , t ) - 2 ~ ( t~) +, u ( a - h , t ) . = -/.Lu(a, t ) + T h2

+

(4.5)

Rearranging this equation yields u(a

I

k + h, t + h ) = -u( 2h

k a - h, t ) + (1 - k / h - p h ) u ( a , t ) + - u ( 2h

q

+ h, t ) .

(4.6)

I I

Age structured populations and distributed maturation rates

259

The boundary condition (2.7a) is discretized in the obvious way to yield the equation

U ( t +h ) = A U ( t ) where u,( t ) = u(nh, t), and U(t ) = [ u , ( t ) , u2(I), . . .IT. modified Leslie matrix of Eq. (4.2), with &

=-

k

2h’

p,, = 1 -p,h.

(4.7) The matrix A is the

(4.8)

Equation (4.7) is analogous to Eq. (4.2) except that it is written in the vector U ( t )= q(t)/h. Therefore the linear chain system of Eqs. (2.6) and (2.7) with k, = k and the modified Lesli matrix equation of Eq. (4.2) with stepsize h are both approximated to second order by the McKendrick equation with diffusion, Eq. (2.14), and may be related through Eqs. (4.8). This relationship is consistent with the mean and variance terms of the Poisson and trinomial distributions describing the behavior of a cohort with no death in the respective models. 5. Discussion

This paper discusses two alternative methods for the simulation of age structured populations whose members mature at different rates. In both methods the population is represented at discrete points along the age axis. The dynamic population variable at point n may be taken to represent either the value of the age density (denoted u,( t ) = u(a,, t ) ) ,or the number of individuals between ages a,-l and a, (denoted q,(f)). In the model problem consisting of a simulation of the evolution of a cohort born at time zero and aging with no death, both models have a solution that may be represented by a probability distribution in n for any given t. The models are complementary in the sense that each is best suited for a particular range of values of the variance of the representative probability distribution. Although the “linear chain” model discussed in Sects. 2 and 3 was derived as a numerical approximation to the McKendrick equation, it is clear that it could also be developed from scratch with no reference to partial differential equation. In this context it represents a compartmental model of the type commonly used in many applications (e.g. Jacquez 1972). The linear chain and modified Leslie matrix models are related through the convection-diffusion equation which approximates each of them to second order. The discussion in this paper has focused entirely on the linear model in which birth and death rates are independent of population density. The extension to more interesting and relevant problems in which density dependence is included is completely straightforward. Since the models are to be solved numerically anyway, one simply includes whatever one wants in the birth and death rate functions in the numerical solution. Indeed, one can develop a theory of density dependent distributed maturation models analogous to that of Gurtin and MacCamy (1974) or Smouse and Weiss (1975). The theory as it applies to the distributed maturation rate models discussed in this paper does not appear to provide any significant insights specific to distributed maturation rate models.

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One can also consider the interaction of two or more populations, each of which consists of individuals with distributed maturation rates. As has often been pointed out (e.g. Hastings 19831, the interaction of age structured populations is a complicated affair. Hastings’ study touches upon the question of the effect of distributed maturation rates on a predator prey interaction. Although the results are complicated, one may say as a rule of thumb that the distribution of maturation rates often appears to have a stabilizing effect on this interaction. A major intended application of distributed maturation rate models is in the simulation of agroecosystems. In this context it is common to include in the complete system submodels representing components of the crop, such as leaves, roots, fruit, etc., and submodels representing the major, usually arthropod, pests. The submodels for the pests may be drawn directly from the equations developed in Sects. 2 or 4 of this paper. The submodels for the crop, however, would have a slightly different form due to the fact that plant organs do not directly “give birth” to other plant organs. In the case of the modified Leslie matrix model of Sect. 4, an appropriate representation for an age structured population of plant organs would have the form q ( t + h ) = Aq( t ) -tb,

(5.1)

where the matrix A has the form of Eq. (4.2) but with all the Pn terms identically equal to zero, and the vector b has a birthrate function in the first element and zeroes in all other elements. Other models for populations with distributed maturation rates have been proposed. One of the first models developed in this context is the McKendrick equation with diffusion, Eq. (2.14) (e.g. Stuart and Merkle 1965; Auslander et al. 1974). This model, however, has the property that given any maturation rate, either positive or negative, some fraction of the population is maturing at a rate that exceeds the given rate in sign and magnitude. In other words, some individuals are losing maturity at an arbitrarily large rate. Both of the models described in this paper have the property that no individuals mature in the negative direction. Other models for populations with distributed maturation rates have been developed by Sharpe et al. (1977) and by Blythe et al. (1984). The model of Blythe et al. is particularly interesting in that it uses a distribution consisting of a shifted gamma density. Although the description of Blythe et al. focuses on the adult population of a laboratory colony of insects, their model could be extended in a straightforward way to cover the agricultural applications described here, in which several life stages are typically important to the model. This extension would, however, rob the model of Blythe et al. of some of its attractive simplicity. In summary, the models discussed in this paper, together with the model of Blythe et al., represent any array of tools that can be used to simulate models with distributed maturation rates. Each model represents the variance in maturation rate of a cohort of individuals in terms of a different probability distribution (Poisson, shifted gamma, or trinomial). At the level of resolution of data available for most ecosystems, however, it is unlikely that this difference will have much significant effect. The choice of which model to use for a particular application depends on the nature of the application, as well as the preference of the modeler.

Age structured populations and distributed maturation rates

26 1

Acknowledgements. We are grateful to J. R. Carey, A. P. Gutierrez, and A. M. Hastings for many helpful discussions. This research was supported by the University of California Statewide Integrated Pest Management Program.

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