harvesting discrete nonlinear age and stage structured...

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 57, NO. 1, APRIL 1988

Harvesting Discrete Nonlinear Age and Stage Structured Populations I

W. M. GETZ 2

Communicated by G. Leitmann

Abstract. A natural extension of age structured Leslie matrix models is to replace age classes with stage classes and to assume that, in each time period, the transition from one stage class to the next is incomplete; that is, diagonal terms appear in the transition matrix. This approach is particularly useful in resource systems where size is more easily measured than age. In this linear setting, the properties of the models are known; and these models have been applied to the analysis of population problems. A more applicable setting is to assume that the reproduction, survival, and transition parameters in the model are density dependent. The behavior of such models is determined by the fbrm of this density dependence. Here, we focus on models in which the parameters depend on the value of an aggregated variable, defined to be the weighted sum of the number of individuals in each stage class. In forestry models, for example, this aggregated variable may represent a basal area index; in fisheries models, it may represent a spawning stock biomass, Current age structured nonlinear stock-recruitment fisheries models are a special case of the models considered here. Certain results that apply to age structured models can be extended to this broader class of models. In particular, the questions addressed relate to the minimum number of age classes that need to be harvested to obtain maximum sustainable yield policies and to managing resources under nonequilibrium and stochastic conditions. Application of the model to problems in fisheries, forestry, pest, and wildlife management is also discussed.

Key Words. Resource management, optimal harvesting, maximum sustainable yield.

1 The author would like to thank R. G. Haight for comments and discussions relating to the material presented here. This work was supported by NSF Grant DMS-85-117t7. Professor, Division of Biological Control, University of California, Berkeley, California 94720.

69 0022-3239/88/0400-0069506.00/0 ~ 1988 Plenum Publishing Corporation

70 JOTA: VOL. 57, NO. 1, APRIL 1988

1. Introduction

One of the hardest problems in applying models to managing biological resources is to find the appropriate level of model complexity. It is necessary to find a balance between realism and utility. Sufficient detail must be included in the model to capture the primary dynamical features of the system, especially gross nonlinearities such as saturation effects. Also, the model must be able to at least qualitatively predict the response of the system to any management strategy that may be considered. Too much detail, on the other hand, may reduce the likelihood of finding a satisfactory solution to the management problem by: obscuring the nature of the response of the system to management; making verification of the model output an all absorbing task; making the model too cumbersome to apply in a numerical decision analysis setting (see Ref. 1 for a more detailed discussion).

The construction of models can either be approached in an ad hoc manner or a modeling paradigm can be developed for analyzing a class of problems. Recently, for example, Schnute (Ref. 2) proposed a general theory for the analysis of fisheries catch and effort data that subsumed almost all previous methods as a special case of his more general approach. In doing so, Schnute made transparent the relationship between these various models and the nature of assumptions that had previously been hidden by the particulars of the approach. A desirable property of such paradigms is the facilitation of links between theory and application. This can be done if the same modeling framework, with a little simplification, is amenable to mathematical analysis and yet, with not too much elaboration, provides structurally detailed simulation models. The paradigm should also have application to a broad range of problems; otherwise, the efforts involved in developing a general theory will be largely wasted.

A number of problems arise in biological resource management that involve harvesting or controlling individuals separated into age or life stage classes. Fish are harvested by age (in actuality, size) class; trees are harvested by breast-height diameter class; insects reared for biological control pur- poses are selected by stage class (e.g., production of fruitfly pupae is required for sterile insect release programs); and populations in game parks are controlled by culling individuals of a particular age, size, or sex. Leslie type matrix models (Ref. 3) have been applied to a number of resource management problems, but with limited success because of their linear structure. In fisheries management, a number of studies have modified the Leslie model to include a nonlinear stock recruitment relationship, thereby enhanc- ing their applicability (Refs. 4-6). In forestry management, stand growth simulators that include nonlinear ingrowth functions and nonlinear growth rates between stage classes are used to investigate management questions

JOTA: VOL. 57, NO. 1, APRIL 1988 71

(Ref. 7). The complexity of these stand growth simulators, however, has limited their usefulness as tools for analyzing the dynamics of tree growth and designing and evaluating management policies.

Here, we present a general stage structured modeling framework that is simple enough to be amenable to mathematical analysis, provides a skeleton for more elaborate simulation models that are easily integrated into a numerical decision analysis framework, and is general enough to have applications in fishery, forestry, pest, and wildlife management. The modeling paradigm can be characterized as an n-dimensional matrix transition model that includes an input function and includes nonlinearities that depend on m aggregations of the state variables; that is, the functional dependence is on an m-dimensional vector. The formulation has an advan- tage over a general n-dimensional model only if m is smaller than n. First, we outline the general structure of the model. Then, we discuss the problem of finding equilibrium solutions and linearizing the system around equilibrium values. In particular, the former problem is shown to reduce to solving m simultaneous nonlinear algebraic equations. The MSY (maximum sustainable yield) harvesting problem is considered, and the structure of the optimal policy is elucidated. The role of the MSY solution is discussed with regard to harvesting under nonequilibrium conditions. Finally, application of the model to problems in fishery, forestry, pest, and wildlife management is discussed.

2. General Model

Consider a population represented by x = ( x ~ , . . . , xn)', where xi is the number of individuals in the ith stage class. For biological realism, we constrain x to lie in R +", the nonnegative quadrant of R n. Let the parameters s; ~ [0, 1] and p~ c [0, 1] denote the proportion of individuals in stage class i at time t that respectively survive the time interval (t, t + 1] and move into the next stage class at time t + 1. Then, it follows that the proportion of individuals in stage class i at time t that remain in stage class i at time t + 1 is 1 -p~. Let (S)o denote the / j th element of S. Then, define the n x n survival matrix S as

{*i, j = i, i= 1 , . . . , n, ( S ) ° = .0, j # i , i , j= l , . . . , n , (1)

and the n × n transition matrix P as

I (1 - p , ) , j = i, i = 1 , . . . , n,

( P ) o = Pi, j = i - 1 , i = 2 . . . . . n,

{0, j # i , i - 1 , i , j= l , . . . , n . (2)


Also, define an input vector f = (~ . . . . ,fb)', where fl is the number of individuals entering the ith stage class at time t from sources outside the population including births. That is, f l is usually considered as the number of births or, in the fisheries literature, as the number of new recruits, while f~ is attributable to migration processes both into and out of the population. Finally, define a vector z = ( z l , . . . , z , ) ' , where z~(t) is the number of individuals removed from the ith stage class during the time interval ( t, t + 1). Without loss of generality, we will assume that harvesting actually takes place at the end of this time interval, so that the equation for the system dynamics can be written as

x( t+l )=PSx( t ) - z ( t )+f , t = 0 , 1 , 2 , . . . . (3)

In this context, the state variable x( t ) has the interpretation of representing the residual state of the population after harvesting at the end of the transition period (t, t + 1). As long as the elements of P, S, and f are constant, Eq. (3) is linear and time autonomous. In general, however, these elements are nonlinear functions of the state vector x, but often the dependence is on aggregated values of xi summed across i, rather than on the individual values of xi themselves. Thus, define an n × m aggregation matrix A (with elements a~) and an m-dimensional aggregation vector y = (Yl . . . . . y,~)', as a linear transformation of the variables x; that is,

y = Ax. (4)

Assume that the parameters p~, s~, and f depend on y and define

G(y) = P(y)S(y).

Then, it follows that

f (1 -p,(y))s , (y) , j = i, i = 1 , . . . , n, g~(y) = Jp,(y)s,(y), j = i - 1, i = 2 , . . . , n, (5)

[0, j ~ i , i - l , i , j = l , . . . , n .

Equation (3) can now be written as

x( t + 1) = G(y(t))x(t) - z ( t ) + f(y(t)). (6)

3. Equilibrium Analysis

Consider the solution to Eq. (6) under equilibrium conditions, that is, subject to the constraints

x ( t + 1) =x ( t ) , t = 0 , 1 , 2 , . . . . (7)


When z(t) has the value ~ for all t = 0, 1, 2 , . . . , then it follows from Eqs. (6) and (7) that the corresponding values ~ and ~ satisfy the equation

= ( I - G(~))-l(f(~) -~) (8)

or, from Eq. (4),

= A ( I - G(~))-l(f(~) -~) . (9)

In making the solution to these two equations as explicit as possible, we will invoke the following lemma.

Lemma 3.1. If a matrix G with elements gg, i , j = 1 , . . . , n, satisfies g~ ~ 0, for i = 1 , . . . , n, and go = 0, for all i and for j ~ i, i - 1, then the L/th element of the inverse matrix G-* satisfies

[0, if i <j ,

(G-,)/j.~_/(_I)-j i ~ i " (gr+l._rl, i f i>-J • (10) ~. g i+l i r=j \ grr /

This result is easily obtained by using forward substitution to solve the system of linear equations o~ = GI3. Note that we do not need to define g,+l, in the above expression because, when it occurs in the product, it will cancel with the term outside the product. Also note that, when i =j , the right-hand side of Eq. (10) reduces to 1/g,~. Direct application of this 1emma to G as defined in Eq. (5) yields the following result:

I 0,1 i f i < j , ( ( I - G ) - I ) q = rr prs,. (11)

i f i>_j. I PiSi l~j=. 1 - (1 - - p r ) s r '

If we substitute A~ for ~ in Eq. (8), then we obtain a set of n nonlinear simultaneous algebraic equations which can be solved numerically to obtain the equilibrium vector ~. Since we have assumed that m, the dimension of y, is much smaller than n, it is much better to solve Eq. (9) for ~ first and then substitute ~ in Eq. (8) to obtain ~ than it is to solve Eq. (8) directly.

From Eqs. (8) and (11), it can be shown for i= 1 , . . . , n that

1 ' ' p , ( ~ ) s , ( ~ ) " =p, ~ ( f : (Y)-~) 1-I (12) x, (~ ' )s i ( i ) j 1 ,=j 1-(1-p,(~))sr(~)"

Note that the above are a system of equilibrium balance equations in the number of individuals in each stage class. The product term for the ith stage class determines the proportion of individuals that make it through from stage classes j to i, while ~(~) and ~ are the number of individuals added to and subtracted from stage class j, j = 1 , . . . , i. Finally, it follows


from Eq. (4) for k = 1 , . . . , m that

Yk = a,k 5" (fj(~)-- ~) . (13) i~l ~kPi(y)si(y) j~=l r=j 1 -- (1 --pr(~r))Sr(y)/I

The system of equations (13) is implicit in ~. For particular choices of ~, solutions ~ may not be unique or in R +m. In general terms, solutions of Eq. (13) correspond to fixed points of the mapping [see the right-hand side of Eq. (9)]

y = Fz(y) --- A ( I - G(y)) - l ( f (y) - z ) . (14)

Define /3i =pi(~), and similarly define si and f~. If the equilibrium solution is to be biologically meaningful, then it follows that the elements of the control and residual state vector must satisfy

ffg->O, i= 1 , . . . , n, (15a)

~i>--O, i = l , . . . , n . (15b)

From Eq. (12), this implies for i= 1 , . , . , n that

p,.s,. > p,.s,. . . . . . -- • ~ -->0. (16)

j=l r=j 1 - - t i - -p , . ) S r j=l r=g 1--(1--~r)Sr

Note that '

[I ~ >0 , (17) 1 - (1--E)Sr--

for all i and j -< i, since pr and s~ ~ [0, 1] for all r. To examine the stability of 2, we need to expand Eq. (6) around the

equilibrium solution (~, ~). Define

Ax(t) = x ( t ) - ~ , Az(t) = Z(t)--~.. (18)

Since

Oyk/OX i = aki , i = 1 . . . . , n, k = 1 , . . . , m,

then it follows that

Ax( t+ 1) = (G+/ - t )Ax( t ) - Az(t) + / (Ax( t ) , (19)

where G, H, /£ are the matrices defined by

d = (20)

~=, ~ ao, (21a)

+ ~ ) . (21b) o g,(y)

~1 a° \ ~ Oyr

Thus, the stability properties are determined by the matrix G + H +/ ( .


Finally, note that Eq. (6), subject to the equilibrium condition (7), can be used to express z in terms of x and y; that is,

z = ( G ( y ) - I ) x - f ( y ) . ( 22 )

4. Maximum Sustainable Yield

Let w ~ R" be a vector of coefficients such that w~ is the net return from harvesting an individual in the ith stage class. Fixed costs and marginal costs that represent deviations from constant returns to scale are not considered in this section, so that the total value of the harvest is

V(z(t)) = w'z(t). (23)

Under equilibrium conditions, we have seen that particular choices of z constrain x, and hence y [through Eq. (4)], to satisfy Eq. (22). Thus, we can state the equilibrium harvesting problem, recalling inequalities (15) and (16), as follows.

Maximum Sustainable Yield Problem

(MSY) max [ V(z) = w'z], y ~ R + m z ~ R +n

subject to Eq. (14) and to

( [ I p~(y)s~(y) ~ -< ~ ( [ I pr(y)Sr(y) ~f~(y) j=~ ~=j 1--(1--pr(y))sr(y)/ZJ J=' r=j 1 - - ( ~ s r ( y ) ] ~

(24)

holding for i = 1 , . . . , n. Although this problem does not specify the form of the nonlinearities

in the m-dimensional vector y, it is possible to prove the following result.

Theorem 4.1. Suppose that at most k elements of the input vector f(y) are not identically equal to zero. If the MSY problem has an optimal solution, then it has an optimal solution z ° in which at most m + k of its elements are nonzero.

Proof. Conceptually, the MSY problem can be separated into a linear programming problem, parametrized by y, and the nonlinear problem of finding the best solution in the parametrized solution set. The linear programming problem, for fixed y c R +m, is to find a solution zy that maximizes V(z) over all z c R ÷" subject to Eq. (14) and Ineq. (24). The nonlinear


problem is to find a solution yO that maximizes the value V(z~) over all y~ R +". Hence, if we can show that, for any y~ R+% the existence of a solution of the linear programming problem implies the existence of a solution that has at most m + k nonzero elements, the theorem is proven.

Suppose that lneq. (24) holds for given y~ R +m and some i such that f (y)---0. It follows from Ineq. (17) and the nonnegativity of the elements of z that

i-1 i-1 pr(y)Sr(y ) E ~ jE

j=l r=~ 1 - ( 1 - p ~ ( y ) ) s r ( y )

1--(1--p~(y))Sr(y) ~ ~=j p~(y)Sr(y) --< Zj

pr(y)s,(y) ~=~ =" 1--(1--pr(y))s~(y)

< l(1--pr(y))sr(y) ~ f~ l~i pr(y)Sr(y) -- pr(y)sr(y) j=l r=j 1--(1 --pr(y))Sr(y)'

since it is assumed that (24) holds for i;

1--(1--pr(y))sr(y) i--1 r~ pr(y)sr(y) -< E f j

e r ( y ) s , ( y ) j=l r=:

since f / = 0;

i--1 i--I p r ( y ) s r ( y )

j=~ 1 - (1 -Pr(Y)I&(Y)'

cancelling terms. That is, Ineq. (24) holds for i - 1 . Thus, Ineq. (24) holding for i = n

implies that this inequality holds for i = n - 1, n - 2 . . . . . n - l, where n - 1 is the last element in f (y ) that is not identically zero. Similarly, Ineq. (24) holding for i = n - l - 1 implies that the inequalities below it hold until the second last nonidentically zero element of f(y) is encountered. In this way, all but k inequalities in (24) are redundant. Thus, the linear programming problem can be reduced to a formulation that has m equality constraints given by Eq. (14) and k of the inequality constraints in (24) that correspond to the nonidentically zero elements of f(y). By the fundamental theorem of linear programming, if a solution exists, then a basic solution exists (that is, a solution having at most m + k nonzero elements), thereby proving the theorem. []

This generalizes the MSY bimodal harvesting result obtained by Getz (Ref. 4) and Reed (Ref. 6) in the context of harvesting fish populations.


These fisheries models, as discussed below, are a special case of Eq. (6) with m = l a n d k = l .

Since resources do not serendipitously start out at MSY conditions and, further, stochastic events continually displace resources from equilibrium conditions, MSY solutions are not in of themselves the most appropriate policies for managing resources. Rather, they provide a reference point for the design of dynamic harvesting policies, preferably feedback or adaptive policies, that can respond to unforeseen changes in the state of the resource.

The concept of using feedback or adaptive policies is fundamental to managing stochastic resources. Over the past 10-15 years, Walters, Hilborn, and Ludwig have been active in developing a theory of adaptive resource management (for a comprehensive review, see Ref. 8). They distinguish between passive and active adaptive prescriptions for management strategies, where active prescriptions include evaluating the amount of information that is obtained from implementing a particular management strategy with respect to improving estimates of parameters in the resource model. In theory, this approach has great merit. In practice, actively adaptive policies may lead to strategies that increase yield variability between years (current yield may be sacrificed to increase estimated future yields), and this may be unacceptable when considering some of the broader socioeconomic issues related to the problem.

Another problem that arises when trying to relate equilibrium strategies to dynamic situations concerns choosing a value T for the length of the management planning horizon. The tradeoff between short-term and long- term gains (solutions for small versus large T) is an essential part of the management problem. Short-term gains are more attractive to individuals, even in regulated resources, because of the uncertainty associated with the future; one feels more confident having money in the bank than being guaranteed a share in an uncertain resource. Hence, resources tend to be economically overexploited in the long run (Ref. 9). The interests of the society at large are less myopic and, hence, regulatory agencies have been created to ensure the long-term profitability of the resource. This raises the questions, addressed in the next section, of how to select a suitable value for T and how to deal with boundary conditions that arise when planning over a finite-time horizon.

5. Dynamic Harvesting

Consider the question of optimally harvesting a resouce, modeled by Eq. (6), that time t = 0 has some arbitrary initial state x (0 )= Xo. The first


problem that arises is the question of choosing a planning horizon. This problem is better understood if we examine the relationship between infinite- time horizon formulations and formulations in which the time horizon is fixed at a value T. One problem with finite-time horizon formulations is that optimal solutions will generally leave the resource in an overexploited state at time T, unless some endpoint condition, such as x ( T ) = x r , is specified. A key issue is selecting suitable values for T and xT.

The value of future revenues obtained from a resource, or any invest- ment for that matter, is usually discounted by economists to reflect the real growth rate that can be expected on current investments of capital (taking inflation into account). The first problem that we will consider here is a biomass maximization problem or, equivalently, a value maximization problem over a finite-time horizon [0, T] in which the discount rate is assumed to be zero.

Recalling Eq. (23), define the value of the harvest of [0, T] to be

T--1 iT(x0)= E w'z(t). (25)

t=0

Note that this value depends not only on our choice of z ( 0 ) , . . . , z ( T - 1 ) , but also on the value Xo as explicitly indicated on the right-hand side of expression (25). To avoid the problem of allowing the resource to be overexploited at time T, constrain

x ( r ) = x °. (26)

Now, consider the problem of selecting z ( 0 ) , . . . , z ( T - 1 ) to maximize Jr(xo), subject to Eq. (6), for t = 1 , . . . , T, and the final time constraint relationship (26). The first question that arises is a controllability question; that is, can the system be driven from xo to x (T) in T time steps? For linear systems, a controllability theory exists that is not directly extendible to nonlinear systems. We can, however, approach the problem from another direction by defining a family of sets X j, in terms of Eq. (6) and the constraints

~(t)>_O, i = l , . . . , n , t = O , 1,2 . . . . . (27a)

~i(t)>-O, i = l , . . . , n , t = O , 1 , 2 , . . . , (27b)

as follows:

X j" = {x c R +" that can be driven to x ° in j time steps}. (28)

Clearly, X ° = x ° _ X 1 ___ X 2 c • •. ; and, if Xo ~ X j, but xo~ X J-~, then j is the minimum number of time steps in which Xo can be driven to x °, assuming that x ° is the unique solution to the MSY problem defined in the previous section.


We now state and prove a theorem that generalizes a result obtained in the more limited context of fishery cohort models (Ref. 10).

Theorem 5.1. I f the set X ~ is convex and x(t) is constrained to lie in X ~ for t = 0 , . . . , T - 1 , then the solution that maximizes expression (25), subject to the dynamic equation (6), the endpoint condition (26), and the constraints (27) drives Xo to x ° in the first time step and keeps it there for all t = 1 , . . . , T; that is, i f the optimal trajectory is denoted by x*(t), then

x * ( t ) = x °, t - l , . . . , T. (29)

Proof. Suppose that x ( j - 2 ) = x l ~ X 1, x ( j - 1 ) is constrained to belong to X 1, and x( j ) = x °. Consider the two time step problem

max W(x(j-1))=w'[z(j-2)+z(j-1)], (30) z(j--2),z(j--1)

subject to Eq. (6) holding for t = j - 2 , j - 1 [note that W is a function of x ( j - 1), since x( j - 1) is the only state in this problem that can vary]. Since X l is convex, all x ( j - 1) c X 1 can be represented by

x( j - 1) = (1 -- e)X°+ EX, (31)

for some x~ (Xl -{x°} ) and some e z [0, 1]. From Eq. (6), it follows that z ( j - 2 ) and z ( j - 1 ) can be solved to obtain

z( j - 2 ) = G(y ~)x ~ + f ( y ' ) - (1 - e)x ° - ex, (32a)

z ( j - 1) = G(y~)((1 - E)x°+ Ex) +f(y~) - x °, (32b)

where

y l = A x 1 and y ~ = A [ ( 1 - e ) x ° + ¢ x ] .

Thus, from Eqs. (31) and (32), it follows that W ( x ( j - 1 ) ) in expression (30) can be considered to be a function to(E), with to(0)= W(x °) and to(l) = W(x), for all x~ ( x l - { x ° } ) . Hence, if it can be shown that w(0)-> w(1), then it follows that the optimal solution corresponds to E = 0; that is, from the identity (31), the optimal value for x ( j - 1) is x * ( j - 1)= x °.

From Eqs. (30) and (32), it follows that to(0) - t o ( l ) can be reduced to

to (0) - to (1) = w'([(G(y °) - I )x ° +f(yO)] _ [(G(y) - I )x + f (y ) ) ]).

But the right-band side of this equation is nonnegative, because x ° maximizes w'z subject to Eq. (22) (see the MSY problem discussed in Section 4). Hence, x * ( j - 1 ) = x °, and the theorem follows by the principle of optimality and repeated iteration of this result f o r j = T, T - 1 . . . . . 2 (that is, using dynamic programming). []


Under the assumptions of Theorem 5.1, the MSY solution is an integral part of the solution of the dynamic problem. It is not clear, however, how much the assumptions can be weakened and a comparable theorem still hold. For example, if x o c X j but xo~X j-~, then under what conditions would it be optimal to reach x ° in j time steps. Constraining x(t), in Theorem 5.1, to lie in X 1 for all t may be more restrictive than necessary. The above theorem also has limited application, because of the linear form of the value function JT given by Eq. (25), but serves to indicate that extremal equilibrium solutions may often play a central role in solutions to dynamic problems, as discussed in the context of continuous two-dimensional systems in Ref. 11.

A more applicable formulation of the value function could take the form

T--1

JT(Xo) = E ~'R(x(t), z ( t ) )+L(x (T) , T), (33) t = O

where R (x(t), z(t)) is the net revenue obtained during the ( t + 1)th harvesting period, ~ is a discount factor, and L(x(T), T) represents a value of the resource as a function of its state at time T. Note that, when the discount rate is zero, the discount factor 8 is 1.

The form of the revenue function R(x, z) depends on the nature of the problem, but it is not always clear from the problem what form L(x, T) should take. In particular, how should this function depend on T; and, more generally, how will the solution to the problem of maximizing expression (33) subject to Eq. (6) behave as T-~oQ These two questions have been addressed elsewhere (Ref. 12). In the context of systems modeled by Eq. (6), an appropriate form for L(x, T) is

6T L(x, T) -- R(x, z(x)),

1-/~

where R(x, z) is the same revenue function as before, but z is no longer independent; it has a form, denoted here by z(x), determined by the equilibrium relationship in Eq. (22). In this case, under appropriate stability assumptions, it is shown (Ref. 12) that the optimal solutions converge in a natural way to a discounted extremal equilibrium solution determined by considering the current value Hamiltonian associated with Pontryagin's maximum principle.

So far, our discussion has focused around deterministic dynamic models. There are a number of obvious ways in which Eq. (6) can be extended to a stochastic setting. For example, the elements of the matrices S(y) and P(y) can be made stochastic; but, for many systems, it may be reasonable to regard the primary source of stochasticity as being associated


with the input vector f(y). In general, stochastic optimization problems are much harder to analyze than their deterministic counterparts. Because Eq. (6) is easily linearized, as presented in Eqs. (18)-(21) (for small m, which is one of the motivations for the formulation, relatively few partial deriva- tives enter into the linearized equation), a number of techniques, such as Kalman filtering, used to obtain approximate optimal controls, may be much more easily implemented than in the case of general nonlinear models.

6. Conclusions

It is clear from the material presented here that the system of equations (6), although allowing general nonlinearities to be included in the elements of f and G, has enough structure to yield some insights into resource optimization problems as traditionally formulated. However, as discussed in the introduction, it is also important that the model be applicable to several classes of problems. Since Eq. (6) includes life table models (Leslie matrix formulation, see Ref. 3) as a special case, the approach presented in the previous sections is sufficiently broad to encompass management problems in all areas of population biology where age or size is a central factor. These include problems in fisheries (Refs. 4, 5, 10), forestry (Refs. 7, 13), pest (Refs. 14, 15), and wildlife management (Refs. 16, 17).

In forests, for example, trees are often classified into diameter classes, and average costs and profits are ascribed to harvesting a tree in a particular diameter class. Thus, in Eq. (6), the variable xi would represent the number of individuals in stage class i. It may be adequate to set m = 1 and let y (now, a scalar variable; hence, the omission of boldface notation) represent a basal area index; that is, Y=F.7=I wix~ measures the proportion of the stand covered by trees (w~ the average cross-sectional area of a tree in the ith diameter class). One would expect the elements si(y) and Pi(y) of the survival and transition matrices, defined by identities (1) and (2), to be decreasing functions of y. Only the first element of the input vector f(y) would be nonzero and would represent the ingrowth function. As such, one would expect f l (y) to initially increase as seed production increases (with increasing y), but then decrease as crowding begins to affect the establish- ment of saplings. Recent results (Ref. 13) indicate that Eq. (6) simulates stand growth as accurately as far more detailed simulation models, but is much less cumbersome to embed in numerical algorithms for the purpose of management analyses.

In fisheries, age rather than size classes are typically used, even though age is indirectly obtained through correlations with size measurements. By definition, all individuals move up one age class in one time period, so that


the transition elements Pi = 1, i = 1 , . . . , n - 1. Only p, is not necessarily 1, because the nth class represents all individuals of age n or greater [see Eq. (2)]. In most studies, migration is not considered and the only nonzero element of f(y) is the stock-recruitment relationship fl(Y). Invariably, the vector y is a scalar index y representing the fecundity, egg potential, or biomass of the spawning stock. Usually, the survival parameters si are assumed to be independent of y, so that f~(y) is the only nonlinearity that appears in Eq. (6). Also, the control vector z usually has the form z = vQx, where v is a scalar effort variable and Q is a diagonal matrix of catchability coefficients ( Q ) , = qi, i = 1 , . . . , n. Since the control variable v is a scalar v, Theorem 4.1 is no longer applicable. An MSY calculation can still be made, as though full control were possible, and then compared with the MSY value obtained under restricted control, thereby allowing the cost of the reduction in controllability to be gauged (see Ref. 4).

In conclusion, it is clear that nonlinearities are an essential component of biological population processes. By incorporating these nonlinearities into a mathematical setting that retains some analytical tractability, new insights can be gained into a number of classical resource management problems.

References

1. GETZ, W. M., Interfacing Biology and Systems Analysis in Pest Management, Systems Analysis in Fruitfly Management, Edited by M. Mangel, Springer- Vertag, Heidelberg, West Germany, 1986.

2. SCrtNUTE, J., A General Theory for Analysis of Catch and Effort Data, Canadian Journal of Fisheries and Aquatic Sciences, Vol. 42, pp. 414-429, 1985.

3. LESLIE, P. H., On the Use of Matrices in Certain Population Mathematics, Biometrika, Vol. 35, pp. 183-212, 1945.

4. GETZ, W. M., Optimal Harvesting of Structured Populations, Mathematical Biosciences, Vol. 48, pp. 279-292, 1980.

5. LEVlN, S. A., and GOODYEAR, C. P., Analysis of an Age-Structured Fishery Model, Journal of Mathematical Biology, Vol. 9, pp. 245-274, 1980.

6. REED, W. J., Optimum Age-Specific Harvesting in a Nonlinear Population Model, Biometrics, Vol. 36, pp. 579-593, 1980.

7. HAIGHT, R. G., BRODIE, J. D., and DAHMS, W. G., A Dynamic Programming Algorithm for Optimization of Lodgepole Pine Management, Forest Science, Vol. 31,321-330, 1985.

8. WALTERS, C. J., Adaptive Management of Renewable Resources, Macmillan, New York, New York, 1986.

9. GORDON, H. S., The Economic Theory of a Common Property Resource: The Fishery, Journal of Political Economics, Vol. 62, 124-142, 1954.


10. GETZ, W. M., Optimal and Feedback Strategies for Managing Multicohort Popula- tions, Journal of Optimization Theory and Applications, Vol. 46, pp. 505-514, 1985.

11. HAURIE, A., Stability and Optimal Exploitation over an Infinite-Time Horizon of Interacting Populations, Optimal Control Applications and Methods, Vol. 3, pp. 241-256, 1982.

12. OETZ, W. M., Modeling for Biological Resource Management, Modeling and Management of Resources under Uncertainty, Edited by T. L. Vincent, Y. Cohen, W. J. Grantham, G. P. Kirkwood, and J. M. Skowronski, Springer-Verlag, Heidelberg, West Germany, 1987.

13. HAIGHT, R. G., and GETZ, W. M., m Comparison of Stage-Structured and Single-Tree Models for Stand Management, Natural Resource Modelling (to appear).

14. CAREY, J. R., and VARGAS, R. I., Demographic Analysis oflnsect Mass Rearing: A Case Study of Three Tephritids, Journal of Economic Entomology, Vol. 78, pp. 523-527, t983.

15. PLANT, R. E., The Sterile Insect Technique: A Theoretical Perspective, Systems Analysis in Fruitfly Management, Edited by M. Mangel, Springer-Verlag, Heidel- berg, West Germany, 1986.

16. FLIPSE, E., and VELING, E. J. M., An Application of the Leslie Matrix Model to the Population Dynamics of the Hooded Seal, Cystophora Crista Erxleben, Ecological Modelling, Vot. 24, pp. 43-59, 1984.

17. STARFIELD, A. M., and BLELOCH, A. L., Building Models for Wildlife Manage- ment, Macmillan, New York, New York, 1986.

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