optimal harvesting strategies for a metapopulation

Bulletin of Mathematical Biology Vol. 56, No. 1, pp. 107-127, 1994 Printed in Great Britain

0092-8240/9456.00 + 0.00 Pergamon Press Ltd

© 1993 Society for Mathematical Biology

O P T I M A L H A R V E S T I N G S T R A T E G I E S F O R A M E T A P O P U L A T I O N

GEOFFREY N. TUC~C* and HUGH P. POSSINGHAM~ Applied Mathematics Department, University of Adelaide, G.P.O. Box 498, S.A. 5001, Australia *(E.mail: [email protected]) ~f(E.mail: [email protected])

We consider optimal strategies for harvesting a population that is composed of two local populations. The local populations are connected by the dispersal of juveniles, e.g. larvae, and together form a metapopulation. We model the metapopulation dynamics using coupled difference equations_ Dynamic programming is used to determine policies for exploitation that are economically optimal. The metapopulation harvesting theory is applied to a hypothetical fishery and optimal strategies are compared to harvesting strategies that assume the metapopulation is composed either of single unconnected populations or of one well-mixed population. Local populations that have high per capita larval production should be more conservatively harvested than would be predicted using conventional theory. Recognizing the metapopulation structure of a stock and using the appropriate theory can significantly improve economic gains.

1. Introduction. The exploitation of our remaining renewable natural resources is of vital importance with increasing desires for nature conservation conflicting with harvester's needs for financial survival. This unfortunate but inevitable conflict is analysed in the field of bioeconomics. Bioeconomics combines the fields of biology, mathematics and economics to determine the effects of human utilization on biological resources and provide management guidelines for future exploitation.

The bioeconomic analysis of optimal strategies for the harvesting of a single species was founded by Gordon (1954), Schaefer (1954) and Scott (1955). Clark (1971, 1972, 1973) formalized these papers, producing equations for the economically optimal exploitation of a single population over time. The basic theory has been extended to include various complexities. For example, Clark and Munro (1975) examined the effects of nonlinearity and autonomy in price and costs on optimal policies. Clark (1976) and Goh and Agnew (1978) considered optimal harvesting when there is a delay in recruitment, while Deriso (1980), Schnute (1985) and Murphy and Smith (1990) modelled age- structured populations. Multi-species fisheries were investigated by Hilborn (1976), Mendelssohn (1980), Mesterton-Gibbons (1987, 1988) and Strobele and Wacker (1991).

107

108 G . N . TUCK AND H. P. POSSINGHAM

The structure used for modelling the dynamics of exploited populations has concentrated on either lumping a population's abundance or biomass into a single variable, say x(t), or assigning a single variable, xi(t ), to each age or weight class. The exploited stock is assumed to be a single, well-mixed and spatially homogeneous population. Exceptions are discussed in Hilborn and Walters (1987) and Clark (1990). Hilborn and Walters (1987) simulated stock and fleet dynamics for multi-species or multi-stock fisheries that showed spatial heterogeneity. Using the difference equation model formulated by Deriso (1980) and expanded by Schnute (1985), they were able to investigate the affects of abundance and spatial variability on catch per unit effort. Clark (1990) used a continuous model to simulate an inshore-offshore fishery where the populations are connected by diffusion. For each population an equation is derived that specifies the optimal population level.

In our paper, we assume that the fished population is a single species, composed of two well-mixed, spatially homogeneous sub-populations or local populations. The local populations are connected by the dispersal of juveniles and together form a metapopulation. A metapopulation may have more than two local populations; however, to assist the interpretation of results we concentrate on the two local population case. Our state space is composed of the abundances of these local populations and we assume that the metapopulation is exploited by a single owner or body.

It has become clear that many species occur as metapopulations. In Australia, harvested species like abalone have a well-defined metapopulation structure (Shepherd and Brown, 1993). We cannot expect the environment of a population to be homogeneous, nor to be unstructured spatially. Due to environmental heterogeneity, local populations will experience different conditions and hence population parameters will vary between local populations. The importance of a metapopulation model is that it gives us the ability to model these situations. Levins (1969) and Levins and Culver (1971) formalized the concept of a metapopulation after some earlier work [including MacArthur and Wilson's (1967) work on island biogeography] which had a similar metapopulation framework. Recent works by Roughgarden and Iwasa (1986), Pulliam (1988) and Howe et al. (1991) model metapopulations using deterministic difference equations and look for equilibria and stability conditions for unexploited populations. In this paper we also use deterministic difference equations to model a metapopulation. These equations are then used to determine strategies for exploiting that metapopulation.

Policies for the optimal harvesting of a metapopulation are determined by modifications of the fundamental equation of renewable resources (Conrad and Clark, 1987) or Modified Golden Rule (MGR) equations. An MGR equation is found that specifies the optimal population level (escapement) for each local population. In some circumstances the escapements produced yield

O P T I M A L H A R V E S T I N G S T R A T E G I E S F O R A M E T A P O P U L A T I O N 109

negative harvests. This problem does not arise in conventional single population optimal harvesting theory. We discuss a method for eliminating the possibility of negative harvests.

To facilitate comparisons between the true optimal strategy and strategies developed without recognizing the metapopulation structure of a stock, we define the following local populations according to their per capita larval production. We call a local population that exports a greater per capita number of larvae than it imports a re la t ive e x p o r t e r local population. If costs are considered negligible, we find that relative exporter local populations should be harvested more conservatively than if they were managed as single unconnected populations. A re la t ive impor te r local population imports more larvae per capita than it exports and should be harvested less conservatively. A local population that has the largest (smallest) per capita larval production we call a re la t ive source (s ink) local population. Relative source local populations should be harvested more conservatively than if the metapopulation were managed as one well-mixed single population. Relative sinks should be harvested less conservatively, and should have a smaller optimal escapement than the relative source local population.

To introduce some of the ideas and notations used later in the paper, we summarize Clark's model for the optimal exploitation of a single population (Clark, 1973, 1976, 1990).

Consider an unharvested single population that is spatially unstructured. Assume that this unexploited population is governed by the following recurrence relation,

Rk + 1 = F(Rk) , (1)

where R k is the population's abundance in period k, and F(Rk) is the reproduction curve or stock-recruitment relation. This function determines the abundance in any period from the stock abundance in the previous period. If we assume that harvesting occurs before growth but directly after the population is "surveyed" then equation (1) will become,

Rk + 1 = F ( R k -- Hk)

= F(Sk), (2)

where H k is the harvest taken from the population in period k and R k - H k = S k

is called the escapement (the total stock that escapes capture). We assume that the cost incurred from harvesting a unit of fish when the

stock size is x is c (x ) (a decreasing function of x) and that harvested fish can be sold at a fixed price, p.

Using the escapements, Sk , as the control variable, we maximize the present value of net revenue over T seasons, i.e. maximize

110 G . N . TUCK AND H. P. POSSINGHAM

T

P.v .= y, kn(gk, Sk), (3) k=O

where ~= 1/1 + d is a discounting factor, d is the periodic discount rate or interest rate, and I I (R k, Sk) is the net revenue from a harvest o f H k in period k. The inclusion of a discount rate takes into account the fact that a harvest today is considered of more value than the same catch in the future.

Clark (1976, 1990) solves this problem using the recursive techniques of dynamic programming and his results are presented here for comparison later.

Maximization of the present value expression produces an equation that implicitly defines the first period optimal escapement, So, namely,

1 F'(So)(p--c(F(So))) (4) - - z

p - c ( S o )

Equation (4) is called the fundamental equation of renewable resources. The optimal harveting strategy is an "all or nothing" type of policy. If the initial stock, Ro, is below our optimal escapement, S*, then we do not harvest at all, and if it is above S* then the harvest is H o = R o - S~'. If F(R) is concave and deterministic (Clark, 1971; Reed, 1979), then once the stock is above S ' i t will never fall below S* again. It can be shown that equation (4) also holds for all time horizons T>~ 1 (Clark 1976, 1990) and thus the optimal approach path is the most-rapid approach to S*.

If we assume that the cost of harvesting the stock is negligible, i.e. c(x)= O, then equation (4) becomes,

1 - = F ' ( S * ) , (5) 0~

where S* is the optimal escapement. Assuming that F" (S)<0 for all S then there will be at most one value of S* such that equation (5) holds. For example, suppose the stock-recruitment relation is logistic,

F(S) = 6 S + rS(1 - S/K), (6)

where ~ is the proportion of adults that survive per period, r is a population growth rate and Kis a constant that causes density dependence in the per capita growth rate, then the optimal escapement is,

K K S* - (1 + d - 6 ) , (7)

2 2r

with optimal harvest H* = F ( S * ) - S * ~ > 0. Note that if 1 + d - 6 > r then it is optimal to harvest the whole population, S* = 0. This is a consequence of not

OPTIMAL HARVESTING STRATEGIES FOR A METAPOPULATION 1ll

including costs. When costs are included S* is unlikely to be zero (depending on the cost function chosen) due to the high cost of harvesting a small population.

In the next section, we construct a deterministic model of the metapopulation dynamics using coupled difference equations. An economic framework is developed [following Clark (1976, 1990)] and finite-time dynamic programming is used to determine equations that specify optimal harvesting strategies. We illustrate these results with some specific examples and conclude with some general implications for the harvesting of metapopulations.

2. The Optimal Harvesting of a Metapopulation with Two Local Populations. In this section we extend Clark's model to allow for two local populations which, due to the migration of juveniles between them, form a metapopulation (see Fig. 1). The metapopulation concept introduces spatial heterogeneity into our model. Local populations can have different growth and mortality rates, reflecting geographic variability. We assume that adults do not migrate between local populations. The adults produce juveniles, e.g. larvae, of which a proportion remain within that local population and a proportion migrate to the other local population. The migrating juveniles become members of their new local population and, together with the sedentary juveniles and adults, form the adults of the following generation.

Thus, suppose that two interacting local populations are modelled by the following stock-recruitment relation,

Rlk +1 = 61Rlk q- Pl: G: (Rlk) -[- P21G2(R2k) (8)

+ , = + P l , (RI ) + (9)

where Rik + 1 is the number of fish in the ith population at the beginning of the k + l th period. Adult survival in the ith population is represented by 6i and p~j is the proportion of the juveniles produced by population i that recruit to populationj. Assume that a certain proportion of the juveniles of population i, e~, are lost from the system. So p~:-}-pi2-[-ei = 1. The function Gi(Rik ) is the

P:I P

Figure I. A metapopulation with two Iota| populations. The proportion of juveniles migrating from |oca] population i to j in each generation is given by Po-

112 G.N. TUCK AND H. P. POSSINGHAM

recruit production function for population i. For example, we might assume a logistic form for the recruit production (as in the introduction), namely

G~(R~k ) = r ,R,k(1 - R~k/K~) , (10)

where r i is a growth rate and K i is a form of carrying capacity that causes density dependence in the per capita growth rate of local population i.

The local populations are harvested,//~k, and the escapements S~k = Rig -- II~k then grow according to equations (8) and (9) to Rik+x. Thus, including harvesting, equations (8) and (9) become,

Rlk+ 1 = (~ 1Slk -t- pllGa(Slk) + P21G2(S2k) (11)

R2k +1 = (~232k -~- P l 2 G1 (Slk) + P22G2 (Szk)- (12)

Now, using the escapements, Sik , as the control variables, our objective is to maximize the present value of net revenue over T seasons, namely maximize

T 2

P . V . = ~ ek ~ ii~(R~k, Sik) ' (13) k=O i=1

subject to equations (11) and (12) and O<~Sik~Rik. Equation (13) is similar to equation (3) except that the discounted net

revenues from both local populations are added to form our present value expression. As before, c~ is a discounting factor. The net revenue produced in period k from a harvest of Hik from local population i is

I S Rik n,(Rik, X,k)= (p- Ci(X)) dx,

ik (14)

where p is the price of the stock and ci(x ) is the cost of harvesting a unit of stock from local population i when its abundance is x. [See Clark (1990) for a detailed derivation of the single population analogy of equation (14).] The cost of harvesting can be allowed to vary from local population to local population. Local populations may have different harvesting costs associated with them due to factors such as the costs of travelling to the population. However, we shall assume that the price of the harvested stock is independent of its source.

Dynamic programming is used to determine the optimal harvesting strategy. We assign a value function, Jr(Rio, R20), as follows,

T 2

J r ( R i o , R2o)= max ~ c~ k ~ IIi(Rik, Sik ). (15) O<~Sik~Rik k=O i= 1

The value function is the sum of the discounted net revenues from both local

OPTIMAL HARVESTING STRATEGIES FOR A METAPOPULATION 113

populations up until season T maximized by an appropriate choice of the escapements S~k. The value function depends on the initial local population sizes, Rio and R2o.

A recursive equation in terms of the value functions is then obtained from equation (15),

f A \ JT+l(Rxo,R2o) = m a x | )., II,(R,o,g,o)-bCdT(R11,R21)|. (16)

O<~Sio<~Rio i= 1 k /

This expression states that the value function with time horizon T+ 1 is the maximum of the immediate returns in the first period plus the returns from future harvests if the local population's abundances move to R 1 x and R z x . This maximum is achieved by an appropriate choice of the escapements S~o.

Consider first the value function with T= 0, i.e. we wish to maximize our immediate net revenue without any consideration of future generations. Then

2

Jo(Rlo, R2o )= max ~ n~(R~o, S,o) O<~Sio<~Rio i= 1

2

II,(R,o , S~), (17) i = 1

where Sio~ is chosen such that p - c~(S~)= 0 and harvesting a local population from Rio down to Sioo will produce the maximum possible profit from that local population.

If we consider next the time horizon T-- 1 we obtain the following recursive equation,

Jx(Rl°'R2°)-~ o~Sio~RiomaX (~l-Ii(Rio,gio)+~Jo(Rlx,R2x)ti=x

= max Hi(R~o, Sio)+~ I-Ii(R~, S~®) . (18) O~<Sio~Rio i = X i

Equation (18) is maximized by partial differentiation with respect to Sxo and $2o, producing equations (19) and (20),

1 _ 61(p-cx(Rlx))+G'l(S1o ) [pxx(p--cx(R11))+ptz(p--cz(R21))] (19)

P - q ( S l o )

1 _ 62(P--cz(R21))+G~(S2o)[pzl(P--Cx(Rlx))+pzz(p--ca(R21)) ] p_c2($2o ) (20)


The details of the derivation of equations (19) and (20) are in Appendix 1. Note that these equations are generalizations of the optimal harvesting equation for a single population. If we remove migration by setting Pu = 0 for iCj and assign F'(S)=f~+puG~(Si) then equation (4) is recovered. From equations (19) and (20) we find the optimal escapements, S* o and S~" o , for each local population. There are also second order conditions which must hold to ensure a maximum. These conditions are given in Appendix 2.

Equations (19) and (20) also hold for all T~> 1 [-proceed by adjusting Clark's (1976, 1990) proof]. This suggests that the optimal first-year escapements are independent of the time horizon considered. However, the escapements will depend on the initial population levels, Rio and R2o. If Rm<S* then population i will have escapement Rm, i.e. there is no harvest from local population i. This assumes that the optimal escapements produce harvests that are positive, or equivalently, that if R~o < S ~ then local population i will experience an increase in abundance when optimally harvesting.

Unlike the analysis of single population exploitation, it is possible that one of the optimal escapements will produce negative harvests even if R m > S*. Thus our globally optimal solution may be infeasible (unless we can produce a negative harvest, by placing stock into the population rather than removing stock). Assuming that a negative harvest is impossible, we should harvest no stock from that local population. Thus we can still maximize J1 as defined by equation (18) (assuming that the concavity conditions in Appendix 2 hold) by setting H~' = 0 and searching along the curve determined by this constraint for the maximum of the surface defined by equation (18). In general it will not be optimal to remain harvesting at S* if local population i has a globally optimal negative harvest. Thus we produce new optimal escapements Si*' and S y . A numerical example of a globally optimal negative harvest is given later in the paper.

3. Discussion of the Two Local Population Results. To facilitate interpreta- tions of the above results, we define two types of local populations according to their per capita larval production. The assumption of negligible costs is then used to produce analytic results that can be readily interpreted. Finally, examples comparing optimal and sub-optimal strategies for the costs and no costs case are examined.

3.1. Local population classifications. Before proceeding with the no costs theory, we make two biological classifications of local populations according to their per capita larval production, i.e. the number of larvae produced per individual in a local population.

Firstly, consider a local population that exports a greater per capita number of larvae to the other local population than it imports. We call such a local


population a relative exporter local population. Mathematically it is a local population i with

riPij > rjpji .

Similarly, a local population that has greater importation than exportation we call a relative importer local population and it is mathematically defined by reversing the above inequality. Note that if local population i is a relative exporter then local population j is a relative importer.

Consider a local population whose per capita larval production is greater than the other local population's per capita larval production. We call this local population a relative source local population and mathematically a relative source local population i has

r io - ai) > r j(1 -- e j).

A relative sink is the local population that has the smaller per capita larval production and it has r i O - e ~ ) < r j ( 1 - e j ) . If local population i is a relative source then population j is automatically a relative sink.

3.2. No costs case. Assume that the cost of harvesting the resource is negligible, then equations (19) and (20) respectively become,

1 - = 61 + G'~(S~) (P11 + P 1 2 ) (21) 0~

1 - = 62 + G 2 ( S ~ ) (P21 + Pa2)- (22) c(

Assume that G['(Si)< 0 so that these equations determine no more than one solution for S*. For example, assume logistic growth for G~(S~k ) of the form seen in equation (10). In this case the optimal escapements are,

S~" - K1 K 1 (1 - ~ - d - ~ l ) (23) 2 2rl (Pll -4-P12)

K~ (1 + d - ~ 2 ) S* - -2 ~ (24)

2r2 (P21 -'[-P22) '

and the conditions for a maximum are satisfied. A comparison of the above equations with the escapement derived for the

optimal economic harvesting of a single population shows that if we set r =- ri(Pu + Pij) then the equations are of the same general form.

We would now like to know when the metapopulat ion escapements are


larger (or smaller) than the escapement used if we are harvesting the local populations as single unconnected populations or as one well-mixed population. For the following analysis we assume that the optimal escapements defined by equations (23) and (24) produce positive harvests.

Assume that the two local populations have K 1 = K 2 and that the local populations are harvested as two unconnected single populations. Observa- tions of a local population would suggest that its growth rate, r~s, would be measured as r~s=riPu+rjpji if we did not know that the population was connected by migration to another local population. The escapement from metapopulation harvesting theory will be larger than that from unconnected single population harvesting theory, S* > Si*, if rip~j > rjpj~. If this inequality holds for local population i then the reverse is automatically true for population j, i.e. if S* > Si* then S* < S~*. We conclude that relative exporter populations will be over-harvested if unconnected, single population harvesting theory is the preferred guide to managing the stock, while relative importer populations will be under-harvested. The sum of the optimal escapements from both local populations will be greater than the sum of the escapements when the populations are believed to be unconnected single populations, S* + S* > S~s + S~s, if rgp~j > r~pj~ and r~p u < r~p~j (see Appendix 3). Thus if the metapopulation is a relative exporter-importer system with the relative exporter local population retaining fewer of its larvae per capita than the importer local population, we should leave more of the total stock than if the metapopulation were managed as two unconnected populations.

The metapopulation could also be managed as though it were a single, well- mixed population. In making a comparison of the optimal escapements for the local populations we estimate the escapements for the local populations from the single population theory by dividing the single optimal escapement by two (we are assuming K 1 = K 2 ) . For simplicity, assume that 61 = 6 2 and that the growth rate measured for the single merged population is

r l ( P l l q-P12) -1- r2(P22 + P21) FL ~ 2

The escapement from local population i will be larger than the estimated optimal escapement from single population exploitation, S * > S * / 2 , if r i ( p , + p i j ) > r L , i.e. ri(1--e~)>rj(1--ej). We conclude that, to harvest the metapopulation as a single well-mixed population will over-harvest relative source populations and under-harvest relative sink populations. The sum of the optimal escapements from both local populations will never be greater than the escapement derived from the well-mixed single population exploitation, i.e. S~' + S~ ~ S* (see Appendix 4). Thus a metapopulation that is managed as a single well-mixed population may be harvested too conservatively.


Finally, assume that we have recognized the metapopulation structure of the stock. We would like to know how a particular local population's escapement will differ from other local populations' escapements.

For example, assuming K 1 = K 2 and 31 = 3 2 and that ri(p, + Pij) > rj(pjj + Pji), (or r~(1--e3>rj(1--e)), then S * > S * . This means that relative source populations should be harvested more conservatively than relative sink populations. If a local population has no juvenile production at all or there is total juvenile wastage from that local population, i.e. 1 - ez = 0, then that local population should be completely harvested, S* = 0. For example, if migration were uni-directional such that Pi~ > 0 and Pji = 0 then we should fully harvest local population j. Uni-directional migration may be observed in a river hatchery/put and take fishery where the source is the hatchery and the sink local population is the fishing ground. Oceanic currents may also produce juvenile migration that is uni-directional.

The escapement for local population i will also be larger than that of local population j, S* > S*, if K~ > Kj or 6 i > 6j, if all other population parameters remain equal. [-These conditions are analogous to increasing a single population's size or adult survival, as can be seen from equation (7).] Thus local populations that produce a large per capita number of larvae, have a large density dependence parameter Ki, or have low adult mortality, should be harvested more conservatively than those local populations that do not. However, if one local population has, say, a larger K~ but is a relative sink then there will be a trade-off in escapement sizes.

3.3. No costs case: examples of metapopulations with two local populations. In the following examples we assume that the initial population size is the stable equilibrium of the unharvested population, Rio =/~i and equilibrium harvesting behaviour is then considered. The local populations' parameters are r l=r2=lO00, Kl=K2=400 ,000 and 6 1 = t ~ 2 = 0 . 0 0 1 . The migratory parameters, Pij, are given in the examples and the discount rate is 10%.

Consider a metapopulation whose local populations are indistinguishable except that P12 >P21, i.e. local population 1 is a relative source and exporter population. We can express the migratory parameters using a migration matrix where the (i, j)th entry is the proport ion of juveniles migrating from local population i to local populat ionj . In this example let the migration matrix be:

= (0.001 0.003) P' \0.001 O.OOaJ"

In Fig. 2 we plot the contours of the objective function given by equation (18), along with the harvests from each local population that are produced

118 G . N . T U C K AND H. P. POSSINGHAM

from using particular escapements. The optimal escapements derived from equations (23) and (24) are S~ = 145,050 and S~ = 90,100 (see Table 1) which is the maximum observed from Fig. 2. We note that the unharvested population has a stable equilibrium at /~=148 ,028 and /~=334 ,710 and that the equilibrium harvests per period produced using the optimal escapements are HI" = 17,351 and H* = 257,149. Thus we protect the relative source population and heavily harvest the relative sink population.

The single escapement derived if the metapopulation is believed to be a single, well-mixed population is S~ =253,467, which, as shown above, is greater than the sum of the two escapements from metapopulation harvesting,

zero harvest contours m a x i m u m

, , . , . , , . . <~ . ; , ' , . , . . , / - " . , . .: . - / - . . . " / ' ' /," " ,<' , _.3-- -_ , ~ "-

,' .;:'..,,,.')<',,, , , . , > ,. , , . . . . I > . , . , I ..;-.<. / / / s . t / / / / / J .K , - , : . .' . .'

. " , I , 1' ," , "". ' " ' " " .. ." ." . - b U - , , i , ~ ' . i i / / • , , ' t ,,"

' , . , ,., , '.-.-:~" , " , . ' < . , , - , - . , . - - !=~- .. ',1

, . . . \ :~. ',., ~ ~ , . , >~ . . . . . ~ ..... , /

/ i ~ ' ~ ' - ~ ~ - - ' ~ i ' J

" i ~ ' ' '""""|"'~" " " ' "q ' i I - ' i

50000 i00000 150000 200000

200000

150000

i00000 $2

50000

s 1

Figure 2. The objective function and harvest contours as func t ions of escapements S 1 a n d S 2 for P1 with negligible costs. The objective function, defined by equation (18), is represented by the ( - - - ) c o n t o u r s , ha rves t s H 1 by ( . . . . . . . . ) and H 2 by

( - . . . . . )_ The c o n t o u r increment is 25,000, starting f rom zero_

Tab le 1. M a n a g e m e n t pol icy comparisons with migration matrix Px and negligible costs, Table escapements and harvests are rounded to the nearest

thousand x 103

Metapopulation U n c o n n e c t e d single

populations Well-mixed single

population

S] 145 90 - - S~ 90 145 - - S~ota I 235 235 S ~ = 2 5 3

H ~ 17 72 - - H ~ 257 157 - - H~ota 1 274 229 H ~ = 2 6 6

O P T I M A L HARVESTING STRATEGIES FOR A M E T A P O P U L A T I O N 119

S~ + S~ = 235,150. The estimated harvest is H~ = 266,267 which is less than the total harvest from metapopulation exploitation, H? + H I = 274,500. Thus if we manage the population believing it to be a single, well-mixed population we will be under-exploiting the resource. If the metapopulation is managed assuming it to be two unconnected single populations then the escapements and harvests derived for each local population are S~'s=90,100 and S*s = 145,050 with harvests H~s = 72,246 and H~s = 156,961. The total harvest, H~'s+H]s=229,207 is less than the total harvest from metapopulation harvesting. Note that the escapements are the direct opposite of those derived for metapopulation harvesting. This is due to the difference in the growth rate terms. In the metapopulat ion theory local population i has r=ri(p,+pij) whereas ris = rip" + rjpji in the unconnected single population case. Thus, if this metapopulation is exploited assuming that the local populations are unconnected single populations then we will be under-harvesting the relative sink-importer local population while over-harvesting the relative s o u r c ~ exporter local population.

As an example that produces negative harvests on applying the escapements from equations (23) and (24), consider a local population that has P22 = P 12 and Pal and P2a equal and extremely small, i.e. few recruits remain within local population 1 and few migrate there.

Consider the following migration matrix:

_;o.oool o.oo2 2- \0 .0001 0.002 ]"

Equations (23) and (24) produce equal optimal escapements for both local populations, S a - S 2 = 95,333. The local populations have unharvested stable equilibria /~1=10,982 and /~2=219,651. If we attempt to use the above escapements, local population 1 does not approach S~ but actually decreases, indicating a negative harvest. From Fig. 3 we see that negative harvests are indeed produced from local population 1. To find the feasible optimal escapements we set H f ' = 0 and find the new maximum of equation (18). In this example the new optimal escapements are S*' = 8333 and S~" = 100,174, with harvests H~"= 0 and H * ' = 66,419 (see Table 2). Thus we protect the relative exporter local population and harvest the relative importer local population.

Due to the negative harvests produced, in this case it is difficult to make a comparison with the escapements and harvests produced from single well- mixed population management. However, we can make a comparison if the metapopulation is assumed to be two unconnected single populations. The escapement for local population 1 is negative and thus is set to zero, while S~s = 145,050. The harvests produced are H* s = 9245 and H~' s = 39,998 with a total harvest considerably less than that achieved using metapopulation


h a r v e s t i n g . T h u s , loca l p o p u l a t i o n 1 is o v e r - h a r v e s t e d whi le loca l p o p u l a t i o n 2 is u n d e r - h a r v e s t e d . T h e h a r v e s t for loca l p o p u l a t i o n 1 is n o t ze ro even t h o u g h its e s c a p e m e n t is ze ro due to the s e a s o n a l m i g r a t i o n of j uven i l e s f r o m loca l p o p u l a t i o n 2.

N o t e t h a t if p ~ = P 2 1 = 0 in the a b o v e e x a m p l e t h e n loca l p o p u l a t i o n 1 is d o o m e d to e x t i n c t i o n ( S ] = 0) a n d we h a r v e s t l oca l p o p u l a t i o n 2 as a s ingle p o p u l a t i o n .

positive harvest maximum

zero h a r v } s t c o n t o / g loba l / /max imum

, i,( . / . ,, )k , • t ' i r t ~x

I t / I t I ~ Lt

, i x x ~ , . x . ~ ' , , x /

50000 I00000 150000 200000

200000

150000

i00000 $2

50000

S1

Figure 3. The objective function and harvest contours as functions of escapements S 1 and S z for P2 with negligible costs. The objective function, defined by equation (18), is represented by the ( . . . . ) contours, harvests H 1 by ( . . . . . . . . ) and H 2 by (- . . . . . . ). The global maximum produces a negative harvest from local population 1. The positive harvest maximum is found along the H a = 0 contour. The contour

increment is 25,000, starting from zero.

Table 2. Management policy comparisons with migration matrix P2 and negligible costs. Table escapements and

harvests are rounded to the nearest thousand x 10 3

Metapopulation Unconnected single

populations

S~' 8 0 S~ 100 145 S,~ota I 108 145

H* 0 9 H* 66 40 H~ota I 66 49

O P T I M A L HARVESTING STRATEGIES FOR A M E T A P O P U L A T I O N 121

3.4. Costs case: examples of metapopulations with two local populations. We now consider the previous two examples when costs are no longer considered to be negligible. The cost function used is,

ci(xi)- ai q i x i '

where a i = 5000 and qi = 1.3 × 10- 5 for i = 1, 2. The price of a unit of fished stock is p -= 7000.

For the first example, where the migration matrix is P1, equations (19) and (20) yield optimal escapements, S~' = 156,169 and S~' = 121,953 (see Table 3). The optimal harvests are H~ = 23,956 and H* = 248,532. Thus we still harvest the relative source populat ion conservatively.

The escapement derived from well-mixed single populat ion harvesting is S* = 269,996, which is no longer greater than the sum of the escapements from metapopulat ion harvesting, S* + S* =278,122. The harvest produced using S~ is H* = 266,895, which is less than the total harvest from metapopulat ion harvesting, H* + H * = 272,488. Thus, we not only leave more stock but we also harvest more than if the populat ion was managed as a well-mixed single population. If the local populations are believed to be unconnected then the optimal escapements for each local population are S~'~=114,121 and S~'~= 158,141 with harvests H*~=63,175 and H~'~= 182,323. Once again, the sum of the metapopulat ion escapements is greater than the sum of the escapements if the local populations were managed believing them to be unconnected S*~ + S~" s = 272,262. The total harvest, H*~ + H~'~ = 245,498, is also less than the harvest from metapopulat ion management. Thus, in comparison to alternative management schemes, metapopulat ion harvesting not only leaves more of the stock behind but it also increases the combined harvest from the local populations.

Using the migration matrix P2 the optimal escapements from equations (19)

Table 3. Management policy comparisons with migration matrix P1 and costs included. Table escapements and harvests are rounded to the nearest thousand

× 103


populations Well-mixed single

population

S~ 156 114 - - S~ 122 158 S~ot~ 1 278 272 S~=270

H~' 24 63 H~" 249 182 - - H~ota I 273 245 H~ = 267

122 G.N. TUCK AND H. P, POSSINGHAM

and (20) are S~=118,129 and S~'=118,397 (see Table 4). As before, this produces a negative harvest in local population 1. Setting H~"=0 and searching for the maximum of equation (18), we produce new optimal escapements S~"= 9509 and S~"= 124,400 with harvest H * ' = 65,713.

If the local populations were managed believing them to be unconnected single populations then the escapements produced would be S~'s=7164 and S*~= 158,141 with harvests H~'s=3109 and H~'~=47,328. The total harvest, H~'~ +H*~= 50,437 is less than the harvest achieved using metapopulation harvesting theory.

4. Conclusion. It is clear that populations possess a distinct spatial structure. In this paper we introduce the concept of a metapopulation into the optimal harvesting literature. We no longer assume that the fished stock is a single homogeneous population, but assume it is composed of many interacting subpopulations or local populations. These local populations have their own peculiar growth and death characteristics. Together the local populations form a dynamic heterogeneous unit that is connected by migrating juveniles and referred to as a metapopulation.

Modelling metapopulation dynamics with discrete, coupled difference equations, we optimize the present value of net revenues derived from each local population. The maximization uses dynamic programming techniques following Clark (1976). We show that the equations for optimal harvesting are generalizations of the fundamental equation of renewable resources derived for the exploitation of a single population.

To facilitate our understanding of the system, results are derived assuming negligible costs, a logistic recruit production function and adult mortality, and density dependence parameters that are equivalent for both local populations. It is true that costs will rarely, if ever, be negligible, but the general conclusions yield simple insights into the costs case.

Table 4. Management policy comparisons with migration matrix P2 and costs included. Table escapements and harvests

are rounded to the nearest thousand x 10 3


populations

S~ 10 7 S* 124 158 S*olal 134 165

H* 0 3 H* 66 47 H~ota I 66 50


The assumption of logistic growth for the juvenile production curve is merely for mathematical convenience. Many alternative forms could be used (Ricker, 1954; Beverton and Holt, 1957; Schnute, 1985). The assumption that K 1 = K 2 and 61 = c52 is also for simplicity. The main extension to existing literature is the addition of juvenile migration. Therefore, we focused on the effect of the per capita larval production and the migration terms on optimal harvesting strategies.

It is important to realize the limitations of our model. We have assumed that there is a single owner or body that can control the fishery, that the fishery is not subject to variability, that there is no age-structure or delays in recruitment and that no malleability in vessel capital exists. All of these factors would improve the realism of future models, but possibly to the detriment of our understanding of the implications of metapopulation structure on harvesting strategies.

It is difficult to measure the parameters in the model, determine the correct recruit production function and finally produce optimal escapements with which a fishery manager can be satisfied. However, if one can recognize the metapopulation structure of the stock, then some basic rules of thumb are worth emphasizing. Local populations that export a greater per capita number of larvae than they import, defined as relative exporter local populations, should be harvested more conservatively than if they were managed as unconnected single populations. A local population with the largest per capita larval production is called a relative source local population and should be more conservatively harvested than if it were managed as a well-mixed single population. Relative source local populations should also have the largest escapement in the metapopulation.

The authors would like to thank Professor E. O. Tuck for his helpful thoughts on an earlier draft. Special thanks are also due to Jemery Day for his assistance and to the anonymous referees for their suggestions.

A P P E N D I X 1

To derive equations (19) and (20) we partially differentiate equation (18) with respect to the variables Sto and $2o. Define V(Rlk, R2k ) as follows,

2

V(Rlk, R2k) = ~ Ili(Rik, S, oo) i = 1

= ( p - c , ( x ) ) dx i = 1 i ~


where

Rlk=61Slk l +PllGt(S1k-1)+P21G2(S2k-1)

R2k=62S2k-l+P22G2(Szk 1)+P12Gl(Slk 1)"

Thus, noting that,

~l-Ii(Rio, Sio) OS, o

( p - c~( Sio) )

and

OV(Rlt , R21) - (p-cl(R11))(61 +PtlG'l(S~o))+ (P-c2(R20)P12G'1(S~o)

~Slo

and similarly for aV(Rl t , R 2 1 ) / ~ 8 2 o , w e obtain

63J(Rlo, R2o) --

3Slo (p -c l (S lo ) )q -~[ (p- -Cl (R l t ) ) (61 q-p,lG'l(Slo))

+ (p-cz(R21))p~zGi(S,o)] = 0

OJ(Rlo, R20)

~S,o -- -- (p -- c2(S20)) + aE(p -- c2(R 21)) (62 + PzzG'2(S20))

+(p-c , (Rl l ) )P21G2(S2o)] =0.

After some rearranging we arrive at equations (19) and (20).

A P P E N D I X 2

To ensure that we have found a maximum, rather than a minimum or a saddle point, we must have,

Jsloslo=Cl(Slo)+a[G'~(Slo){P(Pll +P12)-(cl(Rlt)P11+c2(R21)P12)}

+ c'l(Rl 1)(61 +PtlG'I(Sao)) 2 -c~(R21)(P12G'l(Slo)) z] < 0

and

Js~os~o = c~($20) + aEG~(S20) {P(P22 + P21)- (c2(R21)P22 + ca(R11)P21)}

+ c'2(R2x ) (6 z +Pz2G'2(S20)) 2 - c ' l ( R 1 ,)(P2,G'2(S20)) 2] < 0

while also JsloSloJs2o&o - (Jsxos2o) 2 > 0 where,

JsloS2o = -~[c'l(RH)G2(Szo)P21(61 + P,IG'I(Slo))

+ c'2(R20G[(S~o)p~2(62 +P2zG'2(S~o))]-


APPENDIX 3

In finding conditions 6t = ~2 = 6 and assign M~ = r~pq. Consider

(S* + &*) -- (S*~ + S~) =

K (1 + d - 6 ) [- 1 + L M l l -Jr M21

under which * * * * S 1 -~ S 2 > Sis ~- S2s we assume that K 1 = K 2 = K and

1 1 1 ] / M22+M~2 M~+Maz Mzz+Mzl

If we multiply through to produce a positive common denominator for the square bracket terms, we need only consider the numerator,

(M22 +M12 ) (M~, +M12 ) (M22 +M2~)+ (M~, + M2a ) (Mll +M,E ) (M22 +M2~ )

--(MI1 +M2a) (M22 + M~2) (M22 + M2~)-- (Mll +M2~) (M22 + M,z) (Mll + MaE)

= [ (Ml l +M~2) (3422 + M z l ) {Mll +M12 +M22 + Mza}]

-[(MH + M2t)(M22 + Mt2){Mll + M,2 + Mee + M21}]

= {Ml~ + M,z + M22 + M21} [MllM2, + M22M,2-- M,1M~2-- M22M2,]

{Mlt + M12 + M22 + M21 } 2 a = [rlr2PllP2~+rlr2P22ptz--rlpllp12--r2P22P21]

={MlI + M~E + M22 + M21} [(rEP2~-r~P~2)(rxPtl-r2P22)].

Thus for (S* + S*) - (S~'~ + S*~) > 0 we require riPij > rjp~ and rip~ < rjpjj for i = 1, 2 and j = l, 2 with j :~ i.

APPENDIX 4

To prove that S* + S* ~< S~ where S* is determined assuming that the populat ion is a well-mixed single population and that there are no costs in harvesting, we first define the expressions S, + S 2 and S* as follows,

S~+S~=K K( l+d-6 ) [ 1 1 1 2 rl(1---el)+r2(1-g2~

2K(1+d-6) S~=K

rl(1-q)+r2(1--e2)

Let Ag = r j(1 - e j). Thus,

S~+S*-S*=K(I+d-6 ) )II~_A 2 +

= K(l + d-6) [-~ 8AxAz - (Ax4(~ A2)2A z - (A , + A2)2A1. ]

2 .

/2~n~ + A2~Aln2/


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R e c e i v e d 14 J a n u a r y 1993

R e v i s e d 2 A p r i l 1993

optimal harvesting strategies for a metapopulation

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