Optimal allocation of vessels along a fish migration path

Ecological Modelling, 14 (1982) 229-250 Elsevier Scientific Publishing Company, Amsterdam--Printed in The Netherlands

229

OPTIMAL ALLOCATION OF V E S S E L S ALONG A FISH MIGRATION PATH

B. F R E L E K

Institute of A utomatic Control, Technical University of Warsaw, Warsaw (Poland)

M. GATTO and A. LOCATELLI

Istituto di Elettrotecnica ed Elettronica, Politecnico di Milano, Milano (Italy) (Accepted for publication 10 August 1981)

ABSTRACT Frelek, B., Gatto, M. and Locatelli, A., 1982. Optimal allocation of vessels along a fish migration path. Ecol. Modelling, 14: 229-250. The paper considers a fishery distributed along a migration path ending in a spawning area. With reference to this situation, the optimal allocation of vessels is studied both in the case when full competition among fishermen is allowed and in the case when a sole owner has complete rights to the exploitation of the fish stock. It is shown that, as a result of the full cooperation enforced by the sole owner, vessels tend to concentrate around the most profitable location and the stock exploitation level is less severe than in the competitive case. Moreover, when the number of fishing seasons is sufficiently large, the fishermen's policy converges to a periodic behaviour.

INTRODUCTION

With a few exceptions (see, for instance, Clark (1976) and Hoppensteadt (1976)) no attempts have been made up to now at analysing the management problems arising in connection with the spatial distribution of a fishery. On the other hand, many important fisheries around the world are characterized by a fish stock being distributed over wide areas and exploited by several fleets operating on different fishing grounds. Typical examples are provided by the Pacific salmon populations and the Pacific yellowfin tuna. Therefore, any effort devoted to supplying adequate and new models rationally dealing with such situations should undoubtedly be welcome. Among the many possible problems, a particular, although we believe important, case is the one where the fish stock follows a closed migration 0304-3800/82/0000-0000/$02.75 © 1982 Elsevier Scientific Publishing Company

230

path during its life span ending at a spawning area where fishing is forbidden. Thus harvesting can take place along the migration path only and this obviously implies that upstream vessels create diseconomies to downstream fishermen. Conversely, the existence of a relationship between the spawning stock and the resulting recruitment causes the policy of the downstream vessels to affect the return of the upstream fishermen too. Conflicts are therefore likely to arise unless a sole owner (either a private firm or a government agency) has complete rights to the exploitation and enforces a partially or totally cooperative behaviour. Within this framework, a game-theoretic approach seems to be most appropriate. Although the application of game theory to biological problems is not uncommon (see, e.g. all the papers related to the concept of evolutionary stable strategies like those by Maynard Smith (1974) and by Mirmirani and Oster (1978)), its use, nevertheless, in connection with the optimal management of renewable resources is quite rare, possibly due to the fact that differential games are involved (see, as remarkable exceptions, Clark (1980) and Munro (1977)). With reference to the above-mentioned fish stock migrating along a closed path, this paper will consider two extreme situations. First, the case when each vessel operates per se will be dealt with by making use of the well known concept of Nash strategy; then the authors discuss the opposite case when a sole owner compels the fishermen to a policy aiming at the maximization of the total benefit. No attention, on the contrary, will be devoted to the case when some intermediate form of cooperation among vessels takes place although such situations are well worth being analysed. The main results of the paper can briefly be summarized as follows: (1) The vessels tend to concentrate around the most profitable location when sole ownership is assumed while they tend to spread out in the case of full competition. (2) Although full cooperation obviously entails higher overall economic returns, nevertheless the consequent stock exploitation level turns out to be less severe than in the competitive case. (3) Whenever the planning horizon comprises a sufficiently large number of fishing seasons, the fishermen's policy converges to a periodic behaviour dependent only of the location along the migration path. A FISHERY MODEL

Consider a fish population characterized by the following behaviour: (i) the adults spawn only once in their life in a certain area A (see Fig. 1); (ii)

231

the new generation of fish follows a closed migration path along which the individuals grow up to the moment when they become sexually mature just before entering the spawning area A; (iii) the adults do not leave the spawning area (e.g. because they die, as in the case of salmon). Therefore the stock at any location can be described in terms of a single variable, namely the total number of fish. Introducing a coordinate t, 0 ~< t ~< T, along the migration path (Fig. 1) and an integer i to denote the season, it is possible to define

xi(t ) = number of fish in season i at location t The authors assume that there is a definite relationship between the spawning stock at the end of a season and the recruitment at the beginning of the subsequent season, namely Xi+l(0 ) = F ( x i ( T ) )

(1)

Such a function F(.) is usually referred to as the stock-recruitment relationship and can be given different specific expressions: well-known examples are the relationships introduced by Beverton and Holt (1957) and by Ricker (1975). Moreover, it is assumed that along the migration path the fish experience a mortality which is proportional to the abundance through a coefficient depending on location t only. This mortality is partly due to natural causes and partly to fishing effort which is supposed to be continuously distributed along the migration path. Therefore the following equation can be stated

dx,( t ) / d t = - ( m( t ) + qEi( t ) )xi( t )

(2)

where re(t) is the natural mortality coefficient at location t, Ei(t ) is the effort density in fishing season i at location t and q is the catchability coefficient, supposedly constant. Notice that the coefficient of mortality due to harvesting has been taken as proportional through a constant q to the effort Ei(t ), which is defined as a standardized measure of the density of vessels operating at location t. This

t

Fig. I. The fish migration path and the spawning area A.

232

effort density can be regarded as the input (or control) variable in season i. It is natural, therefore, to introduce constraints on such a variable. A realistic assumption is

O<<-Ei(t)<-Emax,

Vi, Vt

(3)

When considering the fishery management problems, basically an economic viewpoint will be assumed; indeed, the interest will be focused on the maximization of the difference between fishermen's revenues and costs. Since the fish grow along the migration path and also prices per unit weight are different in different locations according to market availability, the revenue flow at each location is simply given by p(t)qE,(t)x~(t), p(t) being the price of one fish at location t. As for cost of fishing, it will be assumed that the cost flow is proportional to effort, i.e. given by c(t)E~(t), where c(t) is the cost per unit of effort. Thus the net revenue flow is

Hi(t ) = (qp(t)xi(t) -c(t))Ei(t )

(4)

Hence it follows that the total revenue from the fishery in a given season i is

oi = forlIi(t) dt

(5)

Several situations can arise for the organization of the fishery, depending on the number of different exploiting fleets and on the regulations imposed by the nations having rights to that resource. However, the authors consider only two opposite situations: (1) An infinite number of exploiters, each acting per se, operates on the fishing grounds; this entails the full competition of vessels. (2) A sole owner possesses complete rights to the fishery and wants to maximize the overall benefit; this entails the full cooperation of vessels. Finally, the fact that fishing grounds do not necessarily coincide with all the migration paths can easily be taken into account by setting the price different from zero only in the fishing area. F U L L C O M P E T I T I O N A M O N G VESSELS

This section is devoted to analysing the case when a continuous distribution of competing exploiters is assumed. As from above, the fish population dynamics is described by eqs. 1-2. The problem can then be stated in the following way. Find, in any season, a trajectory xi(.) and a policy Ei(-) such that each exploiter, given the policy of all the others, maximizes his own net revenue flow N

Q(t) = ~ ailXi(t), i=1

N given

233 where a is the seasonal discount factor. Since the policy E~(t) of a single exploiter at location t cannot apparently affect neither x i ( . ) nor xi÷ i("), xi+2('),-.., xN('), the optimization problem which is faced by each exploiter is max

O<~E~(t)<~Emax

oti(qp(t)xi(t)--c(t))Ei(t),

Vi

By first noticing the irrelevance of the term a ~ in the above-stated problem, the following relations between the optimal policy E~(.) and the optimal trajectory x,(.) can be derived p(t)xi(t ) >c(t)/q=

E i ( t ) = Emax

(6a)

p(t)xi(t )
Ei(t ) --0

(6b)

W h e n p ( t ) x , ( t ) = c ( t ) / q over an internal (existence of a singular arc) the expression for E~(t) can be obtained by imposing that eq. 2 is satisfied, namely Ei(t ) : (1/q)(lk(t)/p(t)

-- ~ ( t ) / c ( t ) -- re(t))

(6c)

where ~ will from now on denote the derivative d ~ / d t for any ~(-). Of course constraints (3) must also be fulfilled. As a result, the following feedback law can be derived O,

x i < c / q p or xi = c/tip and p / p - ~ / c - m < 0

1/q(p/p Ei(t ) = M(xi(t),t

) ----

- m),

x i = c / q p and

O<-lg/p--6/c--m<~qEma x Ema x ,

(7)

X i > c / q p or x i = c / q p and

p / p -- ~ / c -- m > qEma x

So, for any given initial condition xl(0 ) in the first considered season, eqs. 2 and 7 together with b o u n d a r y conditions (1) yields a unique optimal trajectory xi(. ). Notice that this trajectory does not depend u p o n the seasonal discount factor as it does not appear in the feedback law (7). Furthermore, it can be shown under suitable assumptions that the sequence of trajectories x~(.) approaches in the long term a periodic behaviour. In fact the following proposition holds. Proposition i

Assume that the stock-recruitment function F ( . ) has the following properties: (i) F ( . ) is increasing and concave, (ii) F ( x ) > t O, Vx>~O and (iii) the equation F ( x ) = x has one non-zero solution. Then lim x i : - ~ ( t ) where

234

~(-) satisfies the equations

dx/dt=

- (m + q M ( x , t ) ) x

x(O)=F(x(T))

(8a) (88)

The proof is only very briefly outlined here. First, it can be proved that %(t) is a monotonic and bounded sequence for all t. Therefore it converges to ~(t). Second, it can be proved that ~(.) satisfies eqs. 8. The above result can conveniently be illustrated by means of the following simple example.

Example 1 Let the stock-recruitment function be of the Beverton-Holt type, that is r ( x ) = a x / ( 1 + bx), a = 5.0, b = 3.0 Moreover, let the mortality coefficient and the cost per unit effort be constant and given by m = 0.5 and c = 0.5, respectively. The catchability coefficient q and the maximum effort density Ema x a r e both standardized to 1. Finally, the price function p(-) per one fish is given the form p ( t ) = 5 + 1St and T = 2.0 The optimal solution corresponding to the initial condition x l ( 0 ) = 0.76 is shown in Fig. 2 where, instead of plotting the optimal trajectory x,(-), the 'biovalue' Vi(.) ~ x i ( . ) p ( . ) has been drawn. In the same figure/~(.) is the optimal policy corresponding to the optimal periodic trajectory ~(-). According to eq. 7, Fig. 2 clearly shows that the switching points for E~(-) occur when the biovalue V~(.) crosses the c/q line. Note that for i ~> 2 a singular arc exists in the optimal solution. As a conclusion, in the open-access fully competitive fishery the effort tends to reach an equilibrium distribution. It is worth noting that, if such an equilibrium consists only of pieces of singular arc and pieces where fishing does not take place, then the net revenue flow to each fisherman is zero, i.e. the economic rent is completely dissipated. The similarity between this situation and the well-known bionomic equilibrium first introduced by Gordon (1954) is apparent. An important point for a full understanding of this result is that the cost of effort e(t) should be meant as an opportunity cost, namely the cost of undertaking a particular fishing activity, including the cost of not undertaking the most profitable alternative activity. When the equilibrium is such that effort is maximum along some intervals of the migration path, then the revenue flow is positive along such intervals and the economic rent of the fishery is not dissipated. This may occur when the stock is very productive from an economic viewpoint, i.e. couples a great biological productivity with high selling prices versus fishing costs. In fact, in this case, even when fishing at maximum effort, the biovalue at equilibrium

235

stays above the level given by c(t)/q, at least along some path subintervals (like, e.g. in the previous example). A last remarkable conclusion is that the annual discount factor plays no role in determining the fishermen's policy. This can be summarized by saying that competition among an infinite number of exploiters causes them not to care about the future (myopic behaviour).

5.0 2.0

0.5

Va =V

0.20

t'~

0.5

~

,

1.0

t'z~

1.5

Ca)

___ =t~ f 2.0

t

1.01E1

(b.1)

o'.s

:::t

11o

t'5

2.0 t (b.2)

i ,, 0.5 |E3= E

1.0

t

1.5

0.5

(b.3) t

,ti' 0.5

1.0

1.5

210

t

Fig. 2. The competitive case considered in Example 1. (a) The sequence of optimal biovalues. (bl-3) The corresponding optimal fishing efforts.

FULL COOPERATION AMONG VESSELS

Whenever a private owner possesses complete rights to the fishery or a state agency is created to regulate fishing in a rational way, it is conceivable that a fully cooperative situation is established in order to maximize the total benefit. Nevertheless, such a benefit can be interpreted basically in two different ways, which give rise to two different optimal control problems. First, suppose that not only the economical aspects in the exploitation of the resource are taken into account, but that also some concerns about the preservation of the ecosystem are included in the goals of the planners. In

236

this case the optimal control problem can meaningfully be given the form

maX for( qp( t )x( t ) - c( t ) )E( t ) dt

(9)

subject to

Yc= - ( r e ( t ) + qE(t))x(t) x(O)=F(x(T))

(lOb)

o~g(t)~gmax

(lOc)

(lOa)

The periodicity constraint (10b), by imposing that the number of fish at the beginning of each season is constant, prevents the ecosystem from being too severely exploited. On the other hand, if the economical aspects are prevailing in the planner's viewpoint, the entire optimization horizon should be considered and a suitable discount factor introduced into the performance index. Consistently, the optimization problem can be stated as follows (a denotes the seasonal discount factor) N max

N

Y, a i - I a i = m a x ~ a i- 1r[- ( q p ( t ) x i ( t ) - c ( t ) ) E i ( t ) d t i=1

i=1

(11)

"0

subject to

:t,=-(m+qEi)xi, %+,(0) = F(xi(V)),

i---1,...,N

0 ~ E i ~Emax, x,(0) =given

i = 1,..., N

i = 1.... , N - 1

(12a) (128) (12c) (12d)

Although this second problem is more respectful of economic considerations, one should not, however, conclude that the resulting optimal solution would, anyway, lead to the extinction of the fish stock. Indeed, it will be proved that, in a particular yet meaningful case, the optimal policy basically leads to a periodic behaviour, which coincides with the solution of problem (9), (10) when a approaches unity. In this respect the situation is very much similar to the one described in Proposition 1 for the fully competitive setting.

The problem without discounting Problem (9), (10) is now considered. It can be solved by a straightforward application of the maximum principle (a presentation particularly suited to the management of biological populations can be found in the book by Goh (1980)) provided that the cost c(t) is assumed to be a positive constant.

237

Introduce the Hamiltonian function H(t) = (qp(t)x(t)

- c)E(t) - X(t)(m(t)

+ qE(t))x(t)

(13)

where ?~(t) is the adjoint variable. As H is linear in the control variable E, the optimal effort policy/~(.) will be a 'bang-bang' policy, provided that no singular arcs exist. More precisely, introduce the switching function o(t) = qp(t)x(t)

-- c - - q ~ ( t ) x ( t )

(14)

so that H = o E -- ~ m x

Then /~ (t) = Emax ,

for all t such that o (t) > 0

/? ( t ) = 0,

for all t such that o (t) < 0

It is easy to prove that this problem admits no singular arc if F(0) = 0. In fact, assume by contradiction that o ( t ) - - 0 over an interval; then # would also vanish along the same interval, namely qpx - c - q ~ x -- 0

(15)

qpx + qpYc - q ~ x - q~,~ = 0

(16)

Introducing the adjoint equation ~,= - O H / O x

= -q(p-

~ )E + Am

(17)

it is easy to get from eqs. 10a, 16 and 17 the following result x(~O-pm)

=0

Ruling out the possibility x(t) = 0, which would imply the destruction of the population and is obviously non-optimal under the periodicity constraint, the alternative is that the mortality m ( . ) and the price per fish p ( . ) are such that the differential equation p - p r o = 0 is satisfied over an interval. This possibility can occur only with 'probability zero' and therefore will not be taken into consideration. U p to now the results are in practice the same as those obtained by Clark (1976) with reference to the single cohort Beverton-Holt model. However a main difference, that will now be investigated, is given by the fact that recruitment is not constant, but has to satisfy eq. 10b. By suitably exploiting the features of model (10a) together with constraints (10b) and (10c), the region in the V - t plane (V is biovalue), where any admissible trajectory must lie, can be determined in a fairly easy way. In fact, given x(0), in view of eqs. 10a and 10c, x ( T ) must belong to the interval

s, 2x(o) x(T)

(18)

238

where s, = e x p ( - f 0 r m ( ~ ) d~') S2 = exp(--qEmaxT

)

On the other hand, from eqs. lOb and 18 it follows that

(1/st)x(T) <~F ( x ( r ) ) <- (1/s,s2)x(T)

(19)

Equation (19) allows one to determine the interval Xo(Xr) of initial (final) conditions, which are both non-trivial and admissible (see Fig. 3). The required region can then straightforwardly be constructed as shown in Fig. 3; in fact it is bounded by the two limit trajectories VA(.) and VB(.), which are obtained from the solutions of eqs. 10a and 10b with E(t) = 0 and E(t) = Emax, respectively, for all t.

/=x $152

X

F(X) Xo

L Ca)

I-

xr

-I

vA

p(O)X, .

XT

T

t

Fig. 3. The cooperative case without discount. (a) The intervals X0 and X r of admissible initial and final conditions. (b) The region of admissible trajectories.

239

Apparently, when c/q >>-VA(t) for all t, the solution to the optimal control problem is trivially given by E(t) = 0 for all t and the resulting performance index is zero. Therefore in the following it will be assumed that such a case does not occur. An interesting feature of the optimal solution which is entailed by the periodicity constraint (10b) is the positivity of the adjoint variable for all t ~ [0, T], a property that will be useful later on. First note that ( d / d t ) ( X x ) = - qpEx

(20)

and hence

X(t)x(t)=X(O)x(O)- fotqp(~)E(~)x(~) d~

(21)

Moreover, the periodical nature of the problem implies, as shown by Bittanti et al. (1972), that the adjoint variable must satisfy the boundary conditions

)t(T) = F'(x(r)))t(O)

(22)

where ' denotes the derivative with respect to x. As F(.) is assumed to be increasing, )~(T) and )~(0) are both of the same sign. On the other hand eqs. 21 and 22 easily yield f0rqP ( ~ ) E ( f ) x ( f ) d~"

x(0)= F(x(r))-F'(x(r))x(r) Therefore, the concavity of the stock-recruitment function F(.) implies that X(T) and X(0), and also X(T)x(T) and X(0)x(0), are positive. Since it follows from eq. 21 that X(.)x(.) is decreasing, it can easily be deduced that X(t)x(t) is positive for all t. So, it turns out that the adjoint variable X(t) is positive for any t. As for switching points, i.e. locations where the optimal effort E(.) switches between 0 and Em~x, they satisfy the equation

p(t)x(t) = V ( t ) = c / q + )t(t)x(t)

(23)

With reference to an optimal solution, i t can be deduced that, where the biovalue V(t) is less than c/q+)t(t)x(t), there E(t) is zero; conversely where V(t) is greater than c/q+)t(t)x(t), E(t) is maximum. Since X(.) is positive and X(-)x(.) decreasing, the outcome is as depicted in Fig. 4 (note that )~(.)x(-) is constant where E(t)= 0). Moreover, consider the switching function (14): it is easy to derive that

d = qx(p --pro) Since x(-) > 0, o(. ) is increasing where p - p m is positive, decreasing where

240

v ---

0

~+~x

tl

t2

t3

t4

T

t

Fig. 4. The cooperative case without discount: the qualitative behaviour of an optimal solution.

negative. Hence there exists at most one switching point in between two zeroes (of odd multiplicity) o f / ~ - p r o . Such zeroes can be given a nice interpretation. Consider, in fact, the so-called natural biovalue Vn, i.e. the biovalue p x when the exerted effort is zero. As, in this case, ~ = - m x , it follows that

("n = ( /~/p -- m ) Vn

(24)

Therefore, the zeroes of odd multiplicity of p - p r o coincide with the locations where the natural biovalue has a local minimum or maximum. Thus between a m i n i m u m and a m a x i m u m one switch at most can occur. The results pertaining to the case of fully cooperative vessels without discounting are summarized in the following proposition.

Proposition 2 Consider problem (9), (10) and assume that c(t) is constant, the stockrecruitment function F ( . ) is increasing, concave and such that F ( 0 ) = 0. Then: (1) The optimal solution is of the 'bang-bang' type. (2) The number of switching points is at most equal to the number of local extrema of the natural biovalue V, plus one. (3) At most one switching point can exist between two successive locations where V,(.) attains a local minimum and a local maximum. A somewhat special case, although of some interest, is the one when the price per fish p ( . ) and the mortality m(-) are such that p - p r o is a decreasing function.

241

This is, for instance, the case of a constant m ( . ) and of an increasing and concave p ( . ) . If such a relation holds, then both Vn(') and o(. ) are unimodal and consequently/~(. ) switches at most twice between 0 and Emax. The results presented above are now illustrated by means of the following simple example.

Example 2 Reconsider the fishery specified in Example 1. The optimal periodic cooperative solution is shown in Fig. 5, where, once again, biovalues have been plotted instead of the fish numbers. The figure shows that there are only two switching points t' and t" occurring, respectively, before and after the location t m where the natural biovalue attains its maximum. It is also worth while to note that this cooperative solution is characterized, with respect to the periodic competitive solution l~ of Fig. 2, by a m u c h higher biovalue along the whole migration path. This circumstance will later be proved to occur in general.

10.0 5.0 (a)

2.0 1.0

c/q

0.5

0.2

i

0.5

.... 1. . . . I

1.0

t' tm t" 2.0

1.°t 0.5

1 I

o

t

0.5

I

1.0

I

t'

[

(b) I

t"2.0

t

Fig. 5. The cooperative case without discount considered in Example 2. (a) The natural biovalue Vn and the optimal biovalue 17. (b) The corresponding optimal fishing effort/~.

The problem with discounting The case when the performance index incorporates a discount factor is n o w discussed; therefore the optimization problem is defined by eqs. 11 and

242

12. First, let f~(R~, S~) be the optimal value of the performance index for the 'one season' optimization problem ~,= -(m+qEi)x xi(O):R

i,

(25)

i

xi(r)=s

(26)

i

max for (pqx i - c ) E i dt

(27)

O ~ E i ~ Emax

(28)

Of course, not all the pairs (Ri, S~) can be considered, as eq. 18 must, anyhow, be fulfilled. Thus the following constraint (29)

s1s2R i ~ S i ~ s I R i

must be added. Then the original problem (11), (12) can be restated in terms of an optimal control problem for a discrete-time system, namely N

max ~] a ~ - ~ ( R ~ , S ~ )

(30)

i=1

subject to R,+I=F(S,),

i = 1 .... , N - 1

(31)

R 1 = given and to eq. 29. As for the function f~(-, .) it can easily be proved that it satisfies a partial differential equation. Let us consider the adjoint variable )~, for problem (25)-(28) and note that, as in the problem without discount factor (eqs. 17 and 20), (d/dt)()k,x,) = - qpE, x,

(32)

Furthermore, from eqs. 25 and 26 it follows that E i dt = 1//q l n ( R i s l / S i )

which, in view of eqs. 27 and 32 implies a(R,,s,)=

= X

r(d/dt)(X x3 (0)R, -- X

dt-c

(r)S, -

Eidt

c/q ln(Ris /S,)

(33)

But, as known, X~(0)= Of~/aR~ and X ~ ( T ) = - - ~ / 3 S ~ so that eq. 33 becomes = (3~2/OR,)R, + (Ofl/3S,)S, - c / q l n ( R , s , / S , )

(34)

243 The general solution to eq. 34 can be shown (for instance by means of the method of characteristics) to be of the form ~2( R , , S i ) = R i f ( R i / S ~ ) - c / q ln(Risl/S~)

(35)

where f ( . ) is any continuously differentiable function. Problem (30), (31) can be tackled by applying, for example, the discrete maximum principle (again, see Goh, 1980), thus obtaining the following set of necessary conditions od-'(~f~/OSi) +/~iF' : 0,

/xi_ , : a ' - l ( ~ / ~ R i ) ,

i = 1 ..... N i : 2,..., U

=0

(36) (37)

(38)

where the/Lis are multipliers while eq. 38 holds because S N is free. Notice that from eqs. 36 and 38 it follows that 3 f ~ / 3 S u = 0 which in turn provides, for any given recruitment, the value of escapement which maximizes the performance index over the last season. Furthermore, by repeatedly using eqs. 34, 36 and 37, it is easy to show that /~, I> 0,

Vi

(39)

3 ~ / ~ S i <<-O,

Vi

(40)

3 ~ / 3 R ; >I O,

Vi

(41)

The inequalities obtained above can be exploited in order to provide a useful characterization of the optimal solution. This can be achieved by introducing the concept of myopic policy, that is a policy which does not care at all about future and therefore assumes the escapement as free in each season. Accordingly, the following equations are satisfied by the resulting recruitments and escapements 3f~/OS i = O, Ri+ , = F ( S i ) ,

(42)

i = 1,..., N

(43)

i = 1 .... , N - - 1

(44)

R I = given

Under the reasonable assumption that f~(., .) is such as to verify the equation

~2~'~/~82 ~ 0,

VR, VS

it is possible to show through eqs. 40-44 that for all is the optimal recruitments, Ris and escapements Sis which satisfy eqs. 31 and 36-38 are bounded from below by the recruitments and escapements resulting from the myopic policy. Moreover, from eqs. 36 and 37 it follows that O~~/~S i -q.-a(O•/ORi+,)r'=

0,

i = 1 ..... N - - 1

(45)

244 Therefore the optimal solution approaches the myopic one as the discount factor a goes to zero. When the planning horizon becomes very large, optimal periodic fishing strategies may turn out to be particularly appealing. If such a strategy exists, then the equation

F' ~f~/~R _ ~f~/~S

1

(46)

a

which can immediately be derived from eq. 45, must be satisfied. In general, it is open to question whether such a periodic optimal policy exists and whether, given any initial recruitment R~, the resulting optimal solution should eventually converge towards a periodic one. In a particular but significant case, however, it can be proved that this circumstance actually occurs. First, assume that the price per fish p(-) and the mortality m(.) are such that p - m p is a decreasing function. Then, as previously remarked, the biovalue V,(.) is unimodal and consequently the optimal effort Ei(.) switches at most twice. Moreover, the switching points can be computed as shown in Clark (1976). Introducing the further assumption that the fishing effort is unbounded, i.e. letting E m a x ---- o0, the coincidence of the two switching points can easily be shown and an explicit form for the function f~(., •) obtained. The optimal policy can now be proved to converge to a periodic one, for any initial recruitment R ~. In fact, when E m a x : o0, the fishing effort, as pointed out by Clark (1976), is apparently concentrated at t-- t* where the natural biovalue V, attains its maximum value. Thus

Ei(t)=,iS(t-t*

)

where 8(. ) is the Dirac delta while ~ is to be suitably chosen so as to satisfy the constraint xi(T ) = S~, given xi(O)=R~. By standard calculations the optimal profit for the ith season is given by f~(R~, Si) =p*(R~ -- S~/s~) - c/q ln(R~Sl/S~)

(47)

where

p* :p(t*)exp(--fot*m("/"

) d*/')

Therefore, the optimal periodic policy (if any) can be obtained by imposing R = F(S) and the fulfilment of eq. 46. Thus the optimal spawning stock must satisfy

p * - - ( c / q ) [ 1 / F ( S )] 1 F'( S ) p*/s, -- ( c / q ) ( 1 / S ) = -d

(48)

245

The feasibility interval for the optimal spawning stock S is given by

cs,/qp* <-S<~ S n

(49)

where S n is the natural spawning stock, i.e., F(Sn) = Sn/s 1. The value for the upper bound is a trivial consequence of the concavity assumption for F(.). On the other hand, it is apparent that any optimal periodic policy should entail a value of the spawning stock not smaller than the one corresponding to a myopic (periodic) policy, which, by requiring that ~f~/~S = 0, yields, in turn, the lower bound in eq. 49. By once again exploiting the fact that the stock-recruitment relationship F(. ) is increasing and concave, it can easily be shown that the left-hand side of eq. 48 is decreasing in the feasibility interval (49), approaches infinity as S goes to csj/qp* and takes on the value s 1 < 1 for S = S n. Therefore eq. 48 admits, for any given o~, a solution which is an increasing function of a. This leads to the fairly obvious conclusion that the smaller the seasonal discount factor a is, the more severe is the exploitation of the fish resource. These results are suitably summarized in Fig. 6. Finally, it can be proved that the optimal harvest policy is the most rapid approach policy to the optimal level of escapement S given by eq. 48. In fact the performance index can be expressed as N

N

o~i-'f~(R~,Si) = ~ a ~ - ' [ P * ( R , - S ~ / s , ) - c / q l n ( R ~ s , / S i ) ] i=l

i=l N

I

= p * R , - c / q l n R ~ + ~ a'-tW(S,,a) i=1

+ aN-l[--p*(S N/s, ) + c/q ln(S N/s 1)] where

W( S~,a) = -p*S~/s, + c/q ln( S~/s,) + a[ p*r( s~) - c/q ln F( S~)] Hence, it is apparent that the maximum of the performance index is achieved when S N = cs 1 / q p *

and the S is, i = 1,..., N - 1, equal the value which maximizes W(-). As W(.) can be shown to be concave within the feasibility interval (49), such a value can obviously be obtained by setting W'(S) = 0, which, however, is nothing but eq. 48. Therefore, the optimal policy consists of: (i) driving the stock to S (the solution of eq. 48). at the end of the first fishing season; (ii) applying the constant policy which maintains the resource at such a level during the

246

..... ..... / " .-"'I //.111

~'~

Natural biovatue Discount (x ~-0 Discount ~ = 0 .......... Competitive sotution

"\

I -"--... -...... \ .,. ,--......

/ .//'// / . . . . . .

','7 I

O

I

t~

T

t

Fig. 6. The case w h e n Emax is u n b o u n d e d : typical shapes of competitive and cooperative solutions.

remaining seasons; (iii) exploiting the stock down to cst/qp* in the last season. In view of these results, the main feature of the optimal policy is its constancy during the N - 2 intermediate seasons. Moreover, by noticing that W(S, 1) = f2(F(S), S), it is apparent that when a = 1 the constant part of the optimal policy coincides with the solution of problem (9), (10), with E m a x = O0.

COMPARISON BETWEEN THE COMPETITIVE AND COOPERATIVE SITUATIONS

The aim of this section is to compare the competitive and cooperative policies which have previously been analysed. In particular, it will be shown that cooperative policies, although providing a greater economic return, entail a less severe exploitation of the fish stock since the resource is characterized by a higher biovalue along the whole migration path. In the previous section two different optimal control problems have been considered: accordingly, the comparison between competitive and cooperative policies will first be performed in the case where no discount factor is present and a periodicity constraint has to be satisfied. Let the cost per unit effort c be constant and I~(.) be the biovalue corresponding to the periodic solution of the competitive problem, namely 1,1(t) = p(t)~(t), where ~(.) is the solution of eqs. 8. Consider the optimal solution ~(.) of problem (9), (10) and the related biovalue IT'(t) =p(t)~c(t); then it is easy to prove that

l~(t)<-l~(t),

t E [ 0 , T]

(50)

In fact, recall that in the cooperative case the fishing effort is given by the

247

following feedback law

E(t) = £ ( V , t ) = {0,

Emax,

V < c / q + Xx V> c/q + ~x

(Sl)

As for the competitive policy, define

£(V, t) = M ( V / p , t)

(52)

where 37/(., .) is given by eq. 7. From eqs. 7, 51 and 52 and the positivity of the adjoint variable ~(-), it follows

vv, vt

(53)

In order to prove that I;'(t)~< Iv(t), consider the sequence V/(.)= xi(.)p(.), VI(O) = 17"(0), where xi(.) is a sequence of competitive solutions. In view of eq. 53 it turns out that

Vl(t)<-iv(t),

t ~[0, T]

In particular, V~(T)<~ IV(T); hence, recalling that the stock recruitment function F(.) is increasing and IV(.) satisfies a periodicity constraint

v2(0)

Iv(0) = v,(0)

Therefore, by iteration, V,(t)< Iv(t), t e l 0 , T] which, in view of Proposition 1, finally implies eq. 50. Even when the discount factor is considered, it is possible to show that the cooperative policy is to be preferred to a competitive one both from the economic and species preservation point of view. To this end, first recall that, under a mild assumption (~2~~/~$2)~0), a non-myopic policy ensures higher escapements and recruitments and therefore higher biovalues along the migration path than a myopic policy. The latter, in turn, entails a less severe exploitation of the resource than the competitive policy. In fact, a myopic behaviour is characterized by setting the fishing effort to zero where the biovalue is less than c/q. Indeed, in the opposite case a negative return rate would follow at that location while contemporarily the returns of the downstream fishermen would be lowered. On the other hand, the competitive policy (see eq. 7) requires that the effort is maximum where the biovalue is greater then c/q. Therefore, given the same biovalue, the competitive policy sets the fishing effort to a level which is anyway bigger than the corresponding myopic one. It follows that, as a further result, given the same initial recruitment R~, the biovalue resulting from the implementation of a non-myopic cooperative policy is in any season and in any location higher than the one consequent to the adoption of a competitive policy. This statement can obviotisly be extended to the comparison between optimal

248

periodic solutions (of course provided that such a solution exists in the cooperative case) by simply considering as initial recruitment the one corresponding to the periodic cooperative solution. This last result is again summarized in Fig. 6 with reference to the case when Ema x is unbounded. Finally, again with reference to the comparison between optimal periodic solutions, suppose that the natural biovalue Vn(. ) is unimodal. Then both in the competitive and in the cooperative case the fishing effort E is not zero only along one interval whose length is denoted by A and A, respectively. Moreover, let S and S be the spawning stocks corresponding to the competitive and cooperative periodic solutions. It is now straightforward to obtain S=s,F(g)

exp(-- f; ;+~E dt)

S= slF (S ) e x p ( - qEmaxT~)

(54) (55)

Since F(.) is concave and increasing, F(0) -- 0 and S ~
F(S)/S

(56)

As

li+ aE dt <. Emax/~ , eqs. 54-56 imply that A ~
This paper has considered both the open access and the regulated fisheries when the fish resource is distributed along a known migration path. It has been shown that in both cases an equilibrium is attained in the long run, the cooperative situation being characterized by a greater profitability and a less severe exploitation of the stock. However, it is open to question whether such a cooperative policy could actually be enforced by a state agency. In view of this fact it might therefore be of interest to discuss the alternative (not considered here) of driving the open access fishery towards the adoption of a so-called Pareto strategy but, unfortunately, the problem of finding the set of such strategies does not seem to be a simple solution in general. Another result refers to the spatial allocation of vessels: it turns out that competition causes the fishermen to be active on a wider area. It is curious to

249 notice that if vessels are viewed as predators of the same resource, then this conclusion somehow contradicts the well-known principle of competitive exclusion (see, e.g. Volterra (1927) and, more recently, Armstrong and McGehee (1980)). In fact, it is not competition but cooperation which excludes the predators (vessels in our case) operating in non-profitable locations. M a n y an assumption which has been introduced into the fishery model considered here might be modified slightly in order to generate new problems which deserve further investigation. For instance, what happens if vessels operate from harbours which are not located too near the migration path, so that transportation costs should be taken into account? Or, what policy would be implemented when vessels aggregate to constitute several different fleets, assuming that vessels belonging to the same fleet cooperate while competition arises among fleets? Another interesting situation to be analysed would be the one when complete rights to the fishery are owned by a nation so that one or more fleets are bestowed special privileges. In this case the appropriate approach should possibly be the detection of strategies which take into account the leading role of some fleets (see, e.g. the well-known concept in game theory of Stackelberg strategies, Cruz (1975)). Finally, it would be worth while to pay attention also to the case when the adults, after reproduction, leave the spawning area thus creating a generation overlap along the migration path. ACKNOWLEDGEMENTS The authors are grateful to Professor S. Rinaldi for having suggested the problem and for useful discussions. The work has been partially supported by Centro di Teoria dei Sistemi (C.N.R.) and Ministero della Pubblica Istruzione. REFERENCES Armstrong, R.A. and McGehee, R., 1980. Competitive exclusion. Am. Natur., 115: 151-170. Beverton, R.J.H. and Holt, S.J., 1957. On the Dynamics of Exploited Fish Populations. U.K. Ministry of Agriculture, Fisheries and Food, 533 pp. Bittanti, S., Fronza, G., Guardabassi, G. and Maffezzoni, C., 1972. A maximum principle for periodic optimization. Ric. Automat., 3: 170-179. Clark, C.W., 1976. Mathematical Bioeconomics.Wiley, New York, NY, 352 pp. Clark, C.W., 1980. Restricted access to common-property fishery resources: a game-theoretic analysis. In: P.T. Liu (Editor), Dynamic Optimization and Mathematical Economics. Plenum, New York, NY, pp. 117-132. Cruz, J.B., Jr., 1975. Survey of Nash and Stackelbergequilibrium strategies in dynamic games. Ann. Econ. Social Meas., 4: 339-344.

250 Goh, B.S., 1980. Management and Analysis of Biological Populations. Elsevier, Amsterdam, 288 pp. Gordon, H.S., 1954. The economic theory of a common property resource: the fishery. J. Polit. Econ., 62: 124-142. Hoppensteadt, F.C., 1976. Optimal exploitation of a spatially distributed fishery. Unpublished manuscript. Maynard Smith, J., 1974. The theory of games and the evolution of animal conflicts. J. Theor. Biol., 47: 209-221. Mirmirani, M. and Oster, G., 1978. Competition, kin selection and evolutionary stable strategies. Theor. Pop. Biol., 13: 304-339. Munro, G.R., 1977. Canada and Extended Fisheries Jurisdiction in the Northeast Pacific: Some Issues in Optimal Resources Management. Department of Economics, University of British Columbia, Vancouver, BC. Ricker, W.E., 1975. Computation and Interpretation of Biological Statistics of Fish Populations. Department of the Environment, Fisheries and Marine Service, Ottawa, Ont., 382 PP. Volterra, V., 1927. Variazioni e fluttuazioni del numero di individui in specie animali conviventi. R. Comit. Talass. Italiano, Memoria 131, Venezia.