Stability in a stock-recruitment model of an exploited fishery

Stability

in a

Stock-Recruitment B. S. GOH Mathematics NedIan& Received

Model

Department,

of an Exploited

The Universig

Western Australia,

Fishery

of Western Australia,

6009, Australia

26 March 1976; revised 24 August 1976

ABSTRACT Simple and effective tests for the global stability of a stock-recruitment model are described. The results are applied to models of exploited fisheries. It is shown that a constant quota policy for harvesting a fish population always creates an unstable equilibrium. This implies that such a policy could easily lead to the destruction of a fishery. In the Bicker model if the “maximum sustainable yield” constant-quota harvesting policy is used, a single equilibrium is created and it is unstable. On the other hand, in the same model if the “maximum sustainable yield” constant effort harvesting policy is used, the exploited model is always globally stable. This result applies irrespective of the behavior of the unexploited model. It is interesting that this global stability of the exploited population when the optimal constant-effort harvesting policy is used, applies even when the natural dynamics has chaotic solutions. An important application of this result is that regular density independent mortalities due to environmental factors can stabilize a population which otherwise exhibits chaotic behavior. Global stability provides an ideal property for a constant effort harvesting policy. It means it can be safely implemented without monitoring the population density.

1.

INTRODUCTION

Recently May [l] showed that the solutions of the Ricker [2] stock-recruitment model for a fish population possess an interesting range of behavior. If r is the intrinsic growth rate, May showed that the Ricker stock-recruitment model has (i) a globally stable equilibrium when 2 > r > 0, (ii) stable cycles when 2.692> r >2 and (iii) chaotic solutions when r > 2.692. These results are of great interest for those involved with the management of fisheries. Many commercially important fishes have a high fecundity. For example, the fecundity of a female yellowfin tuna is three million eggs [3]. Also, a tenfold or a hundredfold variation in the recruitment to a fishery is not uncommon. In the past such fluctuations in the MATHEMATICAL

BIOSCIENCES

0 Elsevier North-Holland,

Inc., 1977

33, 359-372 (1977)

359

360

B. S. GOH

recruitment have been attributed solely to environmental variations. It is plausible that the large variations in the recruitment of a fish population in the time-recruitment space is a manifestation of chaotic solutions. In support of this explanation, we note that deviations from the corresponding stock-recruitment models of fish populations in the stock-recruitment space are nowhere as spectacular [4]. A characteristic feature of a model with chaotic solutions is that the relationship between stock and recruitment in the stock-recruitment space can be very simple and regular, but the solutions of the model in the time-recruitment space are chaotic. We can distinguish two types of constant (open loop) harvesting policies for a fishery. Firstly, a constant quota policy prescribes a constant number of fish which can be harvested per unit time or generation. Secondly, we can prescribe a constant amount of fishing effort which is applied to catch the fish. The first type of policy requires that the technology to harvest the fish should be very efficient and the capacity to harvest the fish should be in excess. An example of this type of policy is the quota policy of the International Whaling Commission [5-Q For many fisheries the realistic way to prescribe a constant harvesting policy is to fix the applied fishing effort. This is a measure of the number of boats and the number of hours they are in service. In a stock-recruitment model of a fishery we shall approximate a constant effort policy by prescribing that a constant fraction of a year class is harvested impulsively at recruitment. This is when a year class becomes vulnerable to fishing. This approximation is a standard approach in fishery analysis even when the harvesting actually takes place over a significant fraction of the life span of a year class. It has not been sufficiently emphasized in the past that there is a crucial qualitative difference between the two types of constant harvesting policies. A constant quota policy cannot be globally stable. This means errors in measurements of the density of the population or a year class can lead rapidly to the destruction of the fishery. When the density of a year class is low, a constant quota policy could lead to the complete harvest of a year class. If the constant quota policy has been deduced from a “maximum sustainable yield” analysis, then we shall show that the Ricker model has no stable equilibrium. If the constant quota is less than this critical constant quota, the Ricker model has two equilibria. The higher level equilibrium may be stable while the lower level equilibrium is unstable. The logistic model of a fishery has exactly the same qualitative behavior when a constant quota policy is used [9, lo]. On the other hand, a constant fraction harvesting policy creates a globally stable equilibrium for a large range of values of the control parameter. In particular, if the constant fraction harvesting policy is specified in order to create the “maximum sustainable yield” equilibrium, the

STABILITY IN A STOCK-RECRUITMENT

MODEL

361

exploited fishery is globally stable. It is interesting to note that this result applies even when the natural dynamics has chaotic solutions. The logistic model of a fishery shows similar behavior when a constant effort harvesting policy is used [ 111. 2.

STABILITY

OF UNEXPLOITED

MODEL

Let N, be the number of fish in a year class when it is recruited to the fishery. This is when the year class becomes vulnerable to fishing. A year class is the cohort of fish born in the same year. A stock-recruitment model relates the density of a year class at recruitment to the density of one or more year classes in the previous generation. This type of model was first developed by Ricker [2]. For simplicity, and to make the mathematical analysis tractable, I shall assume that the density of a recruitment is only dependent on the density of a single year class in the previous generation. Moreover the density of the parent stock is also measured at the same age as the recruitment. Let N, be the density of a year class in the tth generation. We have N r+,=N,f(N,).

A popular

(1)

model of this class is N t+1 = N,expr[l-

N,/K],

(2)

where r and K are positive parameters. If the fishery contains non-overlapping generations, the parameters r and K are called the intrinsic growth rate and the carrying capacity, respectively. When a fish population contains many year classes at the same time, these parameters do not have these simple interpretations. The model (2) has a unique feasible equilibrium at N,= K. Assume that the general model displayed in (1) has also a unique feasible equilibrium at N = K. A good candidate to serve as a Lyapunov function for the model (1) is V=$(NF-This function AC=

K')- K*ln(N,/K).

has a unique global minimum V[N,+,l-

(3)

at N, = K. We have

V[N,]=iN:[f*(N,)-l]-K*lnf(N,).

(4)

For large values of N,/K, N,f(N,)and j(N,) should be small because of density dependent mortalities due to intraspecific competition. It follows that the dominant term in (4) is then - NF/2, and hence AV, would be

362

B. S. GOH

negative. In a viable population which does not possess the “Allee effect” (see, e.g., p. 208 in Odum [ 121) when N,/ K is small, f(N,) should be greater than one. Then the dominant term in AV, is K*lnf(N,). Thus in a viable single species population without the Allee effect, AV, should be negative for small values of N,/ K. This discussion suggests that the V[N,] displayed in (3) is a very good candidate to serve as a Lyapunov function for a single species population with non-overlapping generations. Even if A V, fails to be negative in a neighborhood of the equilibrium Iv,= K, we could expect that the function V[N,] would enable us to deduce that solutions to the model (1) are ultimately bounded. This means solutions to the model (1) which initiate inside a region B (co) will remain in it, while solutions which initiate at a feasible point but outside B (03) will ultimately enter the region B (co). For the model (2) we have AV,=kNF[exp2r(l-N,/K)-l]-K*[r(l-N,/K)]. The model (2) is globally stable in the region { N,IN1 > 0} if A V, is negative for all N, >0 and N,# K. This condition is satisfied if A V, has a unique global maximum at N, = K. It is difficult to prove thiwondition in a purely analytical way. We establish that AV, has a unique global maximum at N, = K in two steps. Firstly we show A V, has a local maximum at N, = K. Next, we show numerically that outside a small neighborhood of N, = K, A V, is’ negative. This is achieved by evaluating A V, and A V,‘[N,]. The function A V,‘[N,] gives us some assurance that A V, does not change rapidly. For the model (2) we have AV;[N,]=N =+ AV,” [N,]=exp2r

AV;[K]=O.

(7)

-I-4$N,expZr

+2N:(Xrexp2r(l-:I, =a AV,“[K]=2r(-2+r). Hence A V, has a local maximum

(9)

at N, = K if 2>r>O.

(10)

For the parameter values I= 1.9oand K= 1.O, the functions V[ N,] and A V, are plotted in Fig. 1. I have also evaluated A V, and A V:[ N,] at a large

STABILITY IN A STOCK-RECRUITMENT

MODEL

363

2

1

0

-1

-2

FIG. 1. Lyapunov function V(N,) and A V, for the model N,, , = N, exp 1.9 (1 -N,).

number of points where N, >3. At all these points AV, and AV; are negative. Hence model (3) is globally stable when r = 1.9 as A V, is negative for all N, > 0 and N, # K. I have also examined A V, for other values of r which satisfy (10). The conclusion is the same. It appears that the above analysis is the first mathematically satisfactory proof that the model (2) is globally stable when 2> r >O. Previously, the

364

B. S. GOH

function V= (N, - K)’ was used as a Lyapunov function. It is not a suitable Lyapunov function for all positive values of N,, because it does not tend to infinity as N,-+O+ In comparison, the function V displayed in (3) tends to infinity as N,+cc or N,+O+ . For large values of N,/K the dominant term in (3) is NF/2. Hence it is a measure of the energy embodied in the standing biomass. If AI’, is negative at such densities, that means energy is dissipated by the population. For small values of N,/K the dominant term in (3) is K’ln(N,/ K). This is again proportional to the energy embodied in the standing biomass. The negative sign in front of this term implies that if AV, is negative at such densities, energy is absorbed by the population from external sources. In a viable single species population it is obvious that on balance the population must absorb energy when its density is low and must dissipate energy when its density is high. This principle for constructing Lyapunov functions for biological populations has proved very effective for constructing suitable Lyapunov functions for populations described by continuous time models 1131. Suppose (1) has a unique feasible equilibrium at N*. A Lyapunov function for (1) which is constructed according to the above principle is V= Nt-2N*+(N*)2/N,. This function N,+O + . Along

has a global solutions

minimum

at N *. It tends

(11) to infinity

as N,-+cc

or

of (l),

(12) For N* > N, >O, A V, is negative if f(N,) > 1 and f(N,) < (N*)2/ N:, [i.e., N,f( N,) < ( N*)2/ N,]. For N, > N*, A V, is negative if f( N,) < 1 and N,f( N,) > (N*)2/ N,. These conditions imply that any stock-recruitment relationship that lies inside the hatched region in Fig. 2 leads to a globally stable population. A complex stock-recruitment relationship which satisfies these conditions is displayed in Figure 2. Unfortunately the model (2) does not lie inside the hatched region in Fig. 2 for any value of r. The function N,expr(l -N,/ K) tends to zero faster than the function (N*)2/N, as N, tends to infinity. However, it is pertinent to ask whether in real populations N, expr( 1- N,/ K) gives the correct description of a stock-recruitment relationship when N,/ K is large. The model (2) has chaotic solutions when r >2.692. From this it is tempting to conclude that a species with a high fecundity and whose population is described by (1) would probably have chaotic solutions.

STABILITY

IN A STOCK-RECRUITMENT

MODEL

365

N*

FIG. 2. Any single species population whose hatched region and passes through the equilibrium

stock-recruitment point. is globally

relation stable.

is inside the

However, Fig. 2 shows clearly this is not the case. The recruitment could be very large when the stock in the previous generation is very small. It follows that a species with a very high fecundity could have a globally stable population if it sustained the right density dependent mortalities. 3.

STABILITY

IN AN EXPLOITED

FISHERY

For simplicity it is assumed that harvesting takes place impulsively at recruitment. This is a realistic assumption for some salmon fisheries in which harvesting is completed in a few weeks. Two types of variables may be used to describe an exploited population: the densities of recruitment just before and just after harvesting. In view of the fact that the possible destruction of a population depends directly on the post-harvest density, we

366

B. S. GOH

shall use it as the state variable. Another advantage of using this variable is that it is easier to study the effects of harvesting policy. At time t+ 1 let z++, be the number of fish harvested. At time t let S, be the number of fish left after harvesting. It follows from the model (1) that S ,+I=s,f(s,j-~,+,.

In an equilibrium

(13)

state S,,

The so called “maximum 9, satisfies the condition

sustainable

yield” (MSY) equilibrium,

denoted

~[s,Jws,]=o *

f(&)+W’(&)=

by

(15) 1.

(16)

If we set the price of fish equal to one, the cost of harvesting zero and the discount factor equal to zero, the optimal economic tion policies derived by Clark [14] imply that the control level

i?=@(S)-s

equal to exploita-

(17)

gives the dynamic optimal control policy for maximum biomass yield except during an initial transition period. Thus by coincidence the steady state optimal policy for maximum biomass yield is also the dynamic and long term optimal harvesting policy except during an initial transition period. For the model (2) Eq. (16) gives (l-rS/K)expr(l-S/K)=l.

(18)

It appears this equation cannot be solved analytically. (18) imply that the optimal harvesting level is ii=

pf(Q,

Equations

(17) and

(19)

if f(s>=expr(l -S/K). If this optimal harvesting level is prescribed as a constant quota policy, the stock-recruitment relationship for the exploited fish population is S,+,=w-(St)-

pf(Q.

(20)

STABILITY

FIG. 3.

IN A STOCK-RECRUITMENT

Instability

of the “maximum

367

MODEL

sustainable

yield” constant

quota

policy.

In Fig. 3, the model (13) is plotted with u,, , = U. Clearly if this constant quota policy is used the population is rapidly driven to extinction when the initial state is less than c Figure 3 displays such a sequence of events. If u,, , is set equal to a constant which is less than ii, the model (13) with f(S,) = expr( 1 - S,/K) has two equilibria. The higher equilibrium may be stable relative to certain perturbations. If the unexploited population has chaotic solutions, it is likely that both equilibria are unstable and the

B. S. GOH

368 population equilibrium

is rapidly driven to extinction. In all cases the lower is unstable, and if the population is less than the lower

level level

equilibrium, the population will be rapidly driven to destruction. At time t + 1 let a constant fraction of the recruitment be harvested impulsively at recruitment. Denote this fraction by E,,,.At time t let S, be the population after harvesting. We have SI+ 1= An equilibrium

is created

(I- Et+, )W(St ).

at a feasible

point

S* if E,,, is chosen

so that

(22)

Cl-E,+,)f(S*)=l for all values

of t. Denote

such a value of E,,,by E*. Equation

(22) implies

E*= 1- I/f(S*). This is positive

and meaningful

For the model

if f(S*)

(21) with f(S,)=expr(l

(23)

> 1. - S,/K),

Eq. (23) gives

E*=l-exp[-r(l-S*/K)].

(24)

Clearly it is possible to create an equilibrium at any point in the region { S,l K > S,> 0) by choosing the corresponding value of E,,, given in (24). A good choice for S* is 9, which satisfies (18). For the model (21) with f(S,) = expr( 1 - S,/ K), the condition (16) implies

(25)

The expression on the left hand side of (26) is none other than the value of which creates an equilibrium at %. Denote this value of E,,,by E. By

E=r,?/K. With E, as the control variable, the constant effort displayed in (27) is the dynamic and long term optimal maximum biomass yield except during an initial period. assumption the environment is deterministic.

(27) harvesting policy control policy for This is under the

STABILITY IN A STOCK-RECRUITMENT

Equation

369

(24) implies I-E=exp[

When E,,, becomes

MODEL

-T(I-g/K)].

(28)

is set equal to 2, the model (21) with f(S,)=expr(l-

.S,+,=S,exp[

The conditions

S,/K)

$-:I.

(27) and (29) imply

sI+1 =S,exp[E(l-rS,/KE)]. Comparing

(30)

(30) with the model (2) we deduce that it is globally

stable if

2>E>O.

(31)

This is always satisfied. By definition E,,, is a fraction which is harvested impulsively at time t + 1. Hence

of the recruitment

l>E>O.

(32)

The new equilibrium is at KE/r. This result on the stability of the optimal constant harvesting policy is valid for all positive values of r. It is interesting that it is valid even when the unexploited population possesses chaotic solutions. For an illustration of this result, consider S ,+l=(1-E,+l)exp4(1-S,).

The unexploited case

population

has chaotic

population

because

r =4.0.

i?= 0.9525.

%=0.2381, The exploited

solutions

(33)

is described

(34)

by

S 1+1 = S,exp[0.9525(1-4.19953)].

It is globally stable, It is interesting recruitment is harvested.

In this

to observe that a high percentage

(35) of the

B. S. GOH

370

This result on the global stability of the optimal constant effort harvesting policy has only been established for the Ricker stock-recruitment relationship where f( S,) = expr( 1 - St/K). However, it is very plausible that this result applies to a large variety of stock-recruitment relations. For other stock-recruitment relations an effective way to examine the stability of an exploited population is to employ the results in the previous section, which are embodied in Fig. 2. We note that the model (21) can be rewritten in the form S 1+1

=

S,F(SnE,+, 1.

(36)

This is of the same form as the model (1). Suppose an effort policy E,+,(S,) is prescribed. This could be a complicated but fixed function of S,. It is clearly advantageous to prescribe E,,, as a function of S,, which is usually known with some precision at time t + 1. Assume F[S,, E,+,(S,)]= 1 at a unique feasible point S*. Plot the graphs of S,, , = St and S,, , = (S*)2/S, as in Fig. 2. If S,F[S,, E,+,(S,)] is contained inside the hatched region as in Fig. 2, the exploited population is globally stable. This algorithm will undoubtedly be a very useful and robust rule for the management of fisheries which can be described by a difference equation model as displayed in (36). 4.

DISCUSSION

In an exploited population there is a crucial qualitative difference between a constant quota harvesting policy and a constant effort harvesting policy. This difference is important because measurements of the density of a population always contain some errors and the population is subjected to environmental perturbations. If a constant quota policy is specified from a “maximum sustainable yield” analysis, then it will create an unstable equilibrium in a Ricker stock-recruitment model or in a logistic model of a population [9]. If a constant quota policy which is less than this critical constant quota policy is used, usually two equilibria are created. The lower level equilibrium is unstable. If the population density is below the lower level equilibrium, the population will rapidly be driven to destruction. The higher level equilibrium may be stable relative to some finite perturbations from equilibrium, it it may be unstable. In particular, if the unexploited population displays chaotic behavior, the exploited population may not have any stable feasible equilibrium. It was previously mentioned that the harvesting policies of the International Whaling Commission provide a good example of constant quota policies. Such policies were used prior to 1974. It should be stressed that not all the conclusions in the previous sections apply directly to the harvesting

STABILITY

IN A STOCK-RECRUITMENT

371

MODEL

of whales. In whaling there is no realistic choice between a constant quota policy and a constant effort policy. This is because international whaling exploits many species at the same time. An important reason for the quota policy is that it converts the management problem into a problem of managing a collection of single species fisheries. A way to circumvent the instability of a quota policy is to employ an optimal feedback control policy, as developed in [9]. Currently the harvesting policy of the International Whaling Commission is prescribed and implemented in a feedback form. It is updated each year [5]. In a Ricker stock-recruitment model, if the optimal (maximum sustainable yield) constant effort harvest policy is used, the exploited population is globally stable irrespective of the value of intrinsic growth rate r. Thus the results apply even when the unexploited population displays chaotic behavior. In such a case the exploited population has a better chance of survival than the unexploited population because the globally stable equilibrium is larger than the minimum values of the population densities when the unexploited population undergoes chaotic behavior. An important biological implication of this result is that regular density independent mortalities caused by environmental factors can provide the crucial mechanism which stabilizes a population which otherwise possesses chaotic solutions because of intense density dependent mortalities. This provides an interesting argument to the classical debate on the relative importance of density dependent and density independent mortalities in the regulation of a population. The global stability property of a constant effort harvesting policy applies to a very general class of continuous time models of a single species population [ 111. Let e be a constant harvesting effort applied to a population which is described by g= N+(N) in the absence of fishing. Thus we have N= N[+(N)-e].

(37)

Assume that the model (37) has a unique means of the Lyapunov function V= N-

N*-

we can deduce that (37) is globally G(N)-e>O

feasible

N*ln(N/N*)

equilibrium

at N*. By

(38)

stable if for

N*>N>O

(39)

N>N*.

(40)

and G(N)-e
for

372

B. S. GOH

Clearly these are very general and flexible conditions on the function G(N). It is possible to generalize this result by letting e be a function of N. The simplest way to verify the conditions (39) and (40) and the assumption that the equilibrium N* is unique is by graphing the function +(N)e against N. Global stability is an ideal property for an exploited population. It implies that it is not necessary at all to monitor the population density if a constant effort harvesting policy which leads to global stability is used. Hence it can provide a very robust management policy. REFERENCES

2 3 4

6 7 8 9 10 11 12 13 14

R. M. May, Biological populations with non-overlapping generations: stable points, stable cycles, and chaos, Science 186, 645-647 (1974). W. E. Ricker, Stock and recruitment, J. Fish. Re. Board Can. 11, 5599623 (1954). D. H. Cushing, The dependence of recruitment on parent stock in different groups of fishes, J. Cons. Perm. int. Explor. Mer. 33, 34&362 (1971). D. H. Cushing and J. G. K. Harris, Stock and recruitment and the problem of density dependence, Rapp. P-V. Reun. Cons. int. Expl. Mer 164, 142-155 (1973). Twenty-sixth Report of the International Commission on Whaling, London, 1976. R. Gambell, The unendangered whale, Nature 250, 454455 (1974). S. J. Holt, Whales: conserving a resource, Nature 251, 366367 (1974). N. Myers, The whaling controversy, Am. Sci. 63, 448455 (1975). B. S. Goh, Optimal control of a fish resource, Malay. Sci. 5, 65-70 (1969/70). F. Brauer and D. A. Sanchez, Constant rate population harvesting: equilibrium and stability, Theor. Popul.. Bioi. 8, 12-30 (1975). B. S. Goh, Optimization, Stabi& and Non-vulnerability in the Management of Ecosystems, Elsevier, to be published. E. P. Odum, Fundamentals of Ecology, 3rd ed.,Saunders, Philadelphia, 1971. B. S. Goh, Global stability in many species systems, Am. Nat., to be published. C. W. Clark, Mathematical bioeconomics, in Mathematical Problems in Biology (P. van den Driessche, Ed.), Springer, New York, 1974, pp. 29451