The cost imposed on a fishery by a seabird population

J. theor. Biol. (1992) 158, 313-327

The Cost Imposed on a Fishery by a Seabird Population ALASDAIR I. HOUSTON

NERC Unit of Behavioural Ecology, University of Oxford, Department of Zoology, South Parks Road, Oxford OXl 3PS, U.K. (Received on 20 September 1991, Accepted in revisedform on 23 March 1992) Using simple surplus yield models, the effect of a seabird population on the maximum sustainable yield of a fishery is investigated. Two measures of the cost imposed by the birds on the fishery are discussed. The shadow (or marginal) cost is the small change in the fishery's yield that results from a small change in the behaviour of the birds. The total cost is the change in yield to the fishery that would result from removing the seabird population. It is shown that the total cost to the fishery is not necessarily equal to the fish taken by the birds. A general equation for the shadow cost in terms of the bird's functional response is given.

Introduction

Furness (1982) reviews data on the consumption of fish by seabirds, and suggests that there is considerable scope for competition between seabirds and fisheries. The seabirds may reduce the profit of the fishery, and the fishery may reduce the population size of the seabirds. These interactions are discussed in the context of the Shetland sanded fishery by Furness (1990) and Avery & Green (1989, 1990). This paper uses some simple models as a first step towards an exploration of these issues. Two general measures are introduced of the cost to the fishery that results from the fish taken by the seabirds. One measure, denoted by $, is the derivative of the fishery's yield with respect to a parameter that characterizes the seabirds. Following the terminology of economics, $ will be referred to as a shadow, or marginal cost; its exact interpretation depends on the aspect of the seabirds that is under consideration. I will focus on the population size of the seabirds or something that is proportional to population size. In this case, the shadow cost can be thought of as the small change to the fishery's yield that results from a small change in the number of seabirds. The other measure is the total reduction in biomass that results from the presence of the seabirds. The analysis establishes conditions in which the increase in biomass taken by the birds is equal to the biomass lost by the fishery. The effect of the birds and fishery on the characteristic return time of the fish population is also discussed.

The Effect of Seabirds on a Fishery

Given the large amounts of fish consumed by seabirds (Furness, 1982), it is of economic interest to attempt to relate this consumption to the biomass of fish that 313 0022-5193/92/190313 + 15 $08.00/0

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is lost to the fishery. In general, there are several factors that can complicate the relationship between the yield to the fishery and the fish taken by seabirds. For example, the seabirds and the fishery may be taking fish of different sizes or be fishing in different areas. There may also be other marine predators that would take more fish if the seabirds were removed. In the paper it is shown that even in the absence of such effects, the fish taken by the seabirds is not necessarily equal to the fish lost to the fishery. To make this point in a very simple context, surplus yield (or production) models are used (e.g. Pitcher & Hart, 1982). These models are concerned with changes in biomass, without any underlying account of changes in fish size and numbers. This section investigates surplus yield models that include predation on fish by a seabird population. Let the biomass of the fish population be x, with

2=f(x)-YF-Y~,

(1)

wheref(x) = rate of change of the biomass in the absence of the birds and the fishery, Yr = yield to the fishery and YB= yield to the birds. The yield to the fishery will in general depend on both the biomass of fish and the fishing effort EF. It is often assumed that Yr(x, Er) =z(x)Er, where z(x) is a positive increasing function of x. Surplus yield models are based on the maximization of some function of the yield to the fishery and the cost of fishing effort, subject to the equilibrium condition 2 = 0 (see e.g. Clark, 1976; Conrad & Clark, 1987). In the simplest case the yield itself is maximized; the resulting yield is known as the maximum sustainable yield (MSY).

THE MODEL OF SCHAEFER

I now illustrate the two sorts of cost in the context of the model presented by Schaefer (1957). In this model

f(x) = rx(1 - x/K).

(2)

Yr=Erx

(3)

YB EBx

(4)

Assuming that

and =

it follows from eqn (1) and the condition k = 0 that the equilibrium biomass .~(Er) is given by the equation

2(Er)= K(I Er+ Ea).

(5)

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The fishing effort ~ the condition

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that maximizes the sustainable yield can now be found from d

- - (.~(Er)Er) = O, dEr

from which it follows that r-E~ E* = - 2

(6)

Substituting ~F into eqn (5) we find that the biomass x* under the MSY effort is (7)

x* = K(1 - EB/r) /2.

It follows from eqns (3), (6) and (7) that the MSY, I~F, is given by the equation = K r / 4 - KEn(1 - E s / 2 r ) / 2 .

(8)

From eqn (8), the shadow or marginal cost, ;t, is 7~= d Y*FF/dEB= - K ( 1 - E n / r ) /2.

(9)

By comparing eqns (7) and (9) it can be seen that d l~v/dEn = - x * .

(10)

The standard analysis of this model (e.g. Clark, 1976; Conrad & Clark, 1987) does not include seabirds and finds E*v= r / 2 and l~v = Kr/4. Equations (6) and (8) reduce to these results when E n = 0 . From eqn (7) it can be seen that the biomass is a decreasing function of En, falling from K / 2 when En = 0 to 0 when E s = !'. From eqn (8) we can define a total cost which is the lost biomass fl*(En) that results from the presence of birds given that the fishery adopts the MSY policy = ( r - E n ) / 2 [eqn (6)]. It follows from this definition that fl*( Es) = (I - E n / 2 r ) K E n / 2 .

(11)

At the MSY, the birds are taking x* EB = ( 1 -- E n / r ) K E B / 2 ,

(12)

which is less than fl*(EB). The difference is f l * ( E s ) - x * E n = K E 2 / 4 r . The reason for the difference between fl*(En) and x*E~ can be seen from Fig. 1. In this figure, the yield curve is the total yield Y to the birds and the fishery. If we define the total effort E to be EB + EF, then Y = EK(1 - E / r ) ,

where K(1 - E / r ) is the equilibrium biomass, ~. For any value of En and Er, ~ is the slope of the line from the origin to the yield-effort curve. In the absence of the birds, yield to the fishery is maximized when Er = r/2. N o w assume that the birds adopt fishing effort En < r/2. If the fishery adopts an effort E r such that En + E e = r/2, the total yield is maximized, but the fishery's yield is small [Fig. l(a)]. The fishery's best

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Fie. 1. An illustration of why the biomass lost by the fishery is greater than the biomass taken by the birds in the Schaefer model. The figure shows the total yield Y= Y.+ Y~ as a function of total effort E = E . + EF for a given value of Es. (a) In the absence of the birds, the fishery could obtain a maximum yield of Kr/4 by adopting an effort of r/2. If the birds use an effort EB and the fishery adopts an effort E r such that E B + E r = r / 2 then the fishery's yield is substantially reduced. (b) The fishery's best effort E~F= ( r - E B ) / 2 , which means that E* = E~-+ EB = r/2 + Ea/2. The solid straight lines show the resulting yield to the fishery and yield to the birds. 8 " is the biomass lost by the fishery. The broken lines show the effect of the fishery using an effort of r/2. The total effort is now r/2+EB and the lost biomass is/31.

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effort results in a total yield that is less than the maximum possible yield, as is shown in Fig. l(b). It can be seen from the figure that when the birds are removed, the biomass gained by the fishery is greater than the biomass that the birds were taking. We can also define a lost biomass fl~(En) that results if the fishery always uses Er = r/2. This is the effort that maximizes the sustainable yield in the absence of seabirds. [It is illustrated by the broken lines in Fig. l(b).] Equation (5) gives the equilibrium value of x as a function of Er. When Ep= r/2, the lost biomass as a result of seabird effort En is

[3,( En) = KEn/2,

(13)

which is greater than fl*(En). In fact, fl,(En)- fl*(En)= KEg/4r, which is the same as p*(EB)-x*EB. It follows from eqn (12) that the yield to the seabirds when the fishery adopts the optimal effort is a non-monotonic function of En. When En is less than r/2, Yn increases with En, whereas when En is greater than r/2, Yn decreases with En. A qualitatively similar result holds when Er = r/2; Yn increases with En when En is less than r/4 and decreases with En when En is greater than 1"/4. It cannot be assumed that it is always advantageous to adjust fishing effort in response to the effort of the birds. In the model of Fox (1970),

f(x) = rx(-ln (x/K)).

(14)

The fishing effort that maximizes sustainable yield in the Fox model is given by the equation

E~=r,

(15)

and the MSY is

= rK exp ( - E ~ / r - 1).

(16)

In comparing the two models, we see that in the Schaefer model ~ decreases as EB increases, whereas in the Fox model E*v is independent of EB.

An Illustrative Example The Peruvian fishery for the anchoveta Engraulis ringens was at one time the largest fishery in the world (Idyll, 1973). As well as supporting the fishery, the anchoveta was also the food of a vast seabird community, comprised primarily of the Peruvian cormorant (or guanay) Phalacrocorax bougainvillii, the Peruvian booby (or piquero) Sula oariegata and the Peruvian brown pelican (or alcatras) Pelecanus occidentalis thagus (Nelson, 1978; Duffy, 1983a, b). These are known collectively as the Peruvian guano birds. Schaefer (1970) analysed the Peruvian system as if the fishery and the birds constituted a combined fishery. Schaefer assumed that Yr = qEFx and Yn = EBx; for the purposes of a simple illustration I will take q = 1. Schaefer concluded that the maximum sustainable yield for the combined fishery was about

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10 million metric tons. Table I gives for r = 5 and K = 8 million metric tons (a) the yield Y~Fto the fishery, the loss fl*(EB), the yield Ys to the birds and the equilibrium biomass x* under the fishery's best policy, and (b) the corresponding quantities when E r = r / 2 . It can be seen that the difference between fl*(EB) and YB increases as EB increases. The cost of not adjusting the effort to EB also increases as E~ increases, as can be seen by comparing fl*(Es) and fll(Es). As has been shown above, these quantities are equal and are proportional to Es:. Furthermore, the policy o f always adopting fishing effort EF = r/2 eliminates the stock at Es = r/2, whereas the optimal policy still has a positive yield at this point. To get some idea of the effect of the fishery parameters, note that when r = 2 and K = 20 million metric tons, the yield when E B = 0 is again 10 million metric tons. When EB=0.5, fl*=4.375, Y~=3.75 and fl~ = 5 ; when Es = 1, f l * = 7 . 5 , Y s = 5 and flj = I 0 . From 1965 to 1968, Schaefer estimated that the guano birds were taking about 0.7 million metric tons. For r = 5, K = 8, E s = O .18 gives Ys =0.694 when the optimal fishing effort is used. The corresponding value o f Es for r = 2 , K = 2 0 is 0.072. In these cases, the loss that results from using a fishing effort E~= r/2 is about 0-01 million metric tons. Between 1961 and 1965 the guano bird population was higher than it was later in the 1960s, and at its peak Schaefer estimated that nearly 3 million metric tons a year would be

TABLE 1 The yieM YF to the fishepy in the Schaefer model, together with the cost and the biomass YB taken by the birds and equilibrium value o f x. r = 5, K = 8 million metric tons. The table entries are in millions o f metric tons (a) Fishery uses MSY effort E~ given by eqn (6). The loss [3*(Ea) is given by eqn ( 11) Ea 0 0.5 1-0 1.5 2-0 2.5 3.0 3-5

Yv 10.0 8.1 6.4 4-9 3-6 2.5 1-6 0.9

fl*(Es) 0 1.9 3.6 5.1 6.4 7.5 8-4 9.1

Yn 0 1.8 3.2 4.2 4.8 5.0 4.8 4.2

x* 4.0 3.6 3-2 2.8 2.4 2.0 1-6 1.2

(b) Fishery uses constant effort EF= r/2. The loss 131(Ea)is given by eqn (13)

0 0.5 1.0 1.5 2.0 2.5

I0 8.0 6.0 4.0 2.0 0

0 2.0 4.0 6.0 8.0 I0.0

0 1.6 2.4 2-4 1.6 0

4.0 3.2 2.4 1-6 0.8 0

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consumed. It can be seen from Table 1 that the loss from using Er = r/2 is about 0-4 million metric tons in this region. Table 1 also shows the non-monotonic relationship between En and the yield to the birds discussed above. When the fishery uses the optimal effort, the table shows that a yield to the birds of 4.2 million metric tons can arise because En = 1-5 and x = 2-8 or because EB=3-5 and x = 1-2. In the former case, the loss to the fishery is 5-1 million metric tons, whereas in the latter it is 9.1 million metric tons. Similarly, when Er = r/2, it can be seen that Es = 0.5 and Ea = 2.0 both result in a yield to the seabirds of 1.6 million metric tons. These examples make it clear that the biomass taken by the birds does not uniquely determine the loss to the fishery.

A General Analysis of the Shadow Cost In the analysis of the models of Schaefer and Fox, it was assumed that YB= EBx. This section gives a general analysis of the way in which the form of YB determines the small change in the yield to the fishery that results from a small change in the seabird population. This quantity is denoted by A. It is an example of what is known in economics as a shadow cost. F r o m eqn (1) it follows that at equilibrium

Yv=f(x)-Ys. The yield to the birds depends on x and on the seabird population size b, so that we have

Yr(x, b ) = f ( x ) - Ys(x, b).

(17)

The biomass x*(b) that maximizes the sustainable yield is given by the condition

oYr

-Ox (x, b)=

~

oYn

(x) - ~ x

(x, b)= 0

(18)

when x = x*(b). The shadow cost, ~., is given by

d YF(x*(b),b)

d---b

OYF X dx* =~-x (,b)-~(b)

+OYv x --~(,b)

(19)

where the partial derivatives are evaluated at x=x*(b). It follows from eqn (18) that the first term on the right-hand side of eqn (19) is zero, and from eqn (17)

OYF x 0~(

OYn (x,b) ,b)=-

0---b-

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so that A,= d YF(x*(b), b) - -O Ys x b) , . db ---~ ( ' ~,=x tb)

(20)

Equation (10) is a special case of this result when Es=b.

The Form of the Yield to the Seabirds

The yield to the seabirds is likely to depend on various aspects of their biology, including the availability of fish stocks other than that being exploited by the fishery. If the fishery stock is the only food source available to the birds, then a given population of birds will have to take a given amount of fish if it is to survive. In this case we have YB=be

(21)

where b is the population size and c is the fish captured by each bird. It follows from eqn (20) that the shadow cost is - c . The total loss in biomass can be found by integrating the shadow cost between 0 and b and is the biomass eaten by the birds. It might be argued that eqn (21) will not hold because the birds will switch to another prey species when x is low. There are, however, several cases in which one bird species changes its diet in such circumstances and another does not. For example, in the face of a shortage of sandeels around Shetland, the arctic tern (Sterna paradisaea) and the puffin (Fratercula artica) seem to be more dependent on sandeels than the common tern (Sterna hirundo) and the gannet (Sula bassana) (Martin, 1989; Uttley et al., 1989). As fish biomass decreases, the birds may work harder (i.e. fish for more hours per day) in order to maintain the necessary intake. In line with this, Hamer et al. (1991) report that the great skua (Catharacta skua) on Shetland has reduced the time spent on territory in response to a decrease in the abundance of sandeels. Thus, if the bird population is fixed then eqn (21) is justified. It can be argued, however, that the bird population is not really fixed but depends on the biomass of the fish population. If the equilibrium bird population is proportional to the fish biomass and each bird takes a given amount of fish, then Ys is proportional to x, i.e. eqn (4) holds. Looking at this question more generally, the prey captured by a population of predators as a function of the abundance x of prey can be represented in terms of the combined effect of the numerical response and the functional response of the predators (e.g. Taylor, 1984). The numerical response gives the number of predators in an area as a function of x, whereas the functional response gives the number of prey consumed by a predator as a function of x. Given that there is no spatial structure in the models considered in this paper, it is reasonable to assume a fixed population of seabirds and to concentrate on the functional response of the seabirds. A type II functional response is a decelerating function of abundance, whereas a type III functional response is accelerating when x is small and decelerating when x is

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large (Holling, 1966; Taylor, 1984). I will assume that the functional response is of the form ax" a + X m"

This gives a type II functional response when m = 1 and a type III functional response when m > 1. If the population size of the seabirds is b then rB--

bax '~ a + x"

Figures 2 and 3 illustrate the consequences of maximizing sustainable yield in this system w h e n f ( x ) = r x ( 1 - x / K ) . In Fig. 2, m = 1 (i.e. there is a type II functional response) whereas in Fig. 3 rn = 2 (i.e. there is a type III functional response). Each figure is based on two values of a (solid line: a = 3 , broken line: a = 5). From eqn (18), x* as a function of b can be found. This is shown in Figs 2(a) and 3(a). It can be seen that in Fig. 2(a) the biomass crashes to zero once a critical value of b is reached. In Fig. 3(a) x* drops smoothly when a = 5 but jumps from a high value to a low value when a = 3. In this case there can be more than one local maximum sustainable yield. It is assumed that the fishery always chooses the global maximum sustainable yield. The jump occurs when the fishery moves from one local maximum to the other. The resulting MSY, I~s, and the yield to the birds, ~ , as functions of b are shown in Figs 2(b) and 3(b). In Fig. 2(b) Y*r falls smoothly to zero but Ys rises until shortly before the stock collapses. In Fig. 3(b) the jump in the stock results in a jump in Y~s. In the other case Y* rises and then falls in a smooth fashion. These yields are plotted against each other in Figs 2(c) and 3(c). In each figure it can be seen that a given yield to the fishery may not uniquely determine the MSY of the fishery. Figures 2(d) and 3(d) show the shadow cost as a function of b and Figs 2(e) and 3(e) show the total cost as a function of b. In Fig. 2(e) the cost reaches its maximum possible value when the stock has been eliminated.

Characteristic Return T i m e s

As well as looking at the optimal policy for the fishery and the cost imposed by the birds, we can find the system's characteristic return time TR (May et al., 1974). This is an indication of how quickly a perturbation from the equilibrium dies away. It is shown in Appendix A that for the Schaefer model with Yr = Erx, TR is of the order of 1 / ( r - E r - E s ) . It follows that in the absence of the fishery for small perturbations TR ~--1/(r-- En) whereas in the presence of the fishery with E*r given by eqn (6) TR~-2/(r-En),

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FIG. 2. Maximization o f sustainable yield Y F = f ( x ) - Yn w h e n f ( x ) = rx( I - x / K ) and YB= b a x / ( a + x) with a = 1, r = 5, K = 8, a = 3 (solid lines) and a = 5 (broken lines). (a) The biomass x* as a function o f b. (b) T h e yield to the fishery and the yield to the birds as a function of b. (c) T h e yield to the fishery as a function of the yield to the birds. (d) The shadow cost as a function o f b. (e) The cost as a function o f b.

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FIG. 3. Maximization o f sustainable yield Yr=f(x)- Ys when f(x)=rx(1-x/K) and YB=bax2/ (a+x z) with a = 1, r=5, K = 8 , a=3 (solid lines) and a = 5 (broken lines). (a) The biomass x* as a function o f b. (b) The yield to the fishery and the yield to the birds as a function o f b. (c) The yield to the fishery as a function o f the yield to the birds. (d) The shadow cost as a function of b. (e) The cost as a function of b.

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i.e. the return time is doubled, so that if the fishery adopts the MSY effort ~ r then a small perturbation takes twice as long to die away. Appendix A also considers T~ for the Schaefer model when YB= B. If the fishery adopts the effort that maximizes the sustainable yield, then TR --~1/(,'/2 -- 2B/K).

Discussion

Although the models presented in this paper are very simple, they serve to illustrate certain effects which do not seem to have been commented on in the discussion of the interaction between a fishery and a population of seabirds. Perhaps the most basic point is that the fish eaten by the birds is not necessarily equal to the fish lost by the fishery. For the models presented in this paper, the loss to the fishery is greater than or equal to the fish taken by the seabirds, but in more complicated models the loss to the fishery could be less than the fish eaten by the seabirds. One obvious variation on the model presented here which would produce this effect would involve the seabirds capturing one species of fish which is in competition with the species of fish taken by the fishery. Another possibility is that the fish population has densitydependent growth or reproduction, so that the action of the birds in reducing the biomass improves the growth or reproduction of the remaining fish. Now assume that we are concerned with the profit achieved by the fishery. The fishery can maximize its yield by thinning out the fish early in the season and then harvesting from the improved population at a later time. If, however, the thinning procedure is very expensive, then the profit to the fishery may be greater when the birds do the thinning than when the fishery does. I have based the analysis on the maximization of sustainable yield. This has been criticized as a criterion for managing a fishery (e.g. Larkin, 1977), but that is not its r61e in this paper. It is merely providing a simple economic criterion in terms of which the effect of a population of seabirds can be evaluated. Many of the basic effects described in the paper are found in other contexts (e.g. maximization of some function of yield and effort). The advantage of working with the maximization of yield is that the loss to the fishery is directly comparable to the fish taken by the seabirds. The analysis of the Schaefer model illustrates that the loss to the fishery can depend on the fishery's behaviour. If the fishery always adopts behaviour that would maximize the sustainable yield in the absence of the birds then its loss is greater than if it changes its fishing policy as a function of the bird's activity. It cannot be concluded, however, that it will always be advantageous for the fishery to adjust its behaviour in this way. In the Fox model the optimal behaviour for the fishery does not depend on what the birds are doing. I have worked with a marginal or shadow cost that gives the change in the yield to the fishery as a result of a small change in the population size of the seabirds. This measure can be thought of as an economic analogue of the competition coefficient that gives the change in the population size of one species as a result of a

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small change in the population size of another species (e.g. May, 1974). Both these concepts arc based on the system satisfying an equilibrium condition. Equation (20) shows how the shadow cost depends on the yield Ya to the seabirds. This result is illustrated in some simple cases in which YB is the functional response of an individual bird multiplied by the number of birds, b. In reality, things may not be this simple--for example, G6tmark el al. (1986) found that the fishing success of black-hcaded gulls (Larus ridibundus) increased with the number of gulls hunting. This would require a more complicated form of YB, but the general principle embodied in eqn (20) will still hold. When Ya=bc [eqn (21)], the fish lost by the fishery is the same as the fish taken by the birds. Analysis of the Schaefer model when Ya = Esx or Ya = bx"/(a + x") shows that when the fishery maximizes its sustainable yield, then Ya first increases and then decreases as a function of increasing Ea or b. This non-monotonic relationship arises becausc when Es or b is small, the effect of increase in EB or b more than compensates for the decrease in equilibrium biomass. As the Ea or b is increased further, equilibrium biomass drops so rapidly that YB declines. (For some parameter values for the non-linear form of YB, the equilibrium biomass jumps from a high to a low value at a critical value of b.) What this means is that the amount of fish taken by the seabirds may not uniquely determine the increase in fish taken by the fishery that would result from eliminating the seabird population. In developing the ideas in this paper, I benefited from suggestions made by Wayne Thompson, who also commented on several earlier versions and produced the figures. Helpful comments were also received from John McNamara and an anonymous referee. My thanks to them all. I was supported by the Natural Environment Research Council. REFERENCES AVERY, M. & GREEN, R. (19895. New Sci. 122, 28-29. AVERY, M & GREEN, R. (1990). New Sci. 128, 50-51. CLARK, C. W. (19765. J. math. Biol. 3, 381-391. CONRAD,J. M. & CLARK,C. W. (1987). Natural Resource Economics. Cambridge: Cambridge University Press. DUFF-Y, D. C. (1983a). Biol. Cons. 26, 227-238. DUFFY, D. C. (1983b). Auk 100, 800-811. FOX, W. W. JR (19705. Trans. Am. Fish. Soc. 99, 80-88. FURNESS, R. W. (19825. Ado. mar. Biol. 20, 225-307. FURNESS, R. W. (1990). Ibis 132, 205-217. GOTMARK, F., WINKLER,D. W. & ANDERSSON,M. (1986). Nature, Lond. 319, 589-591. HAMER, R. C., FURNESS, R. W. t~ CALDOW, R. W. G. (1991). J. Zool. 223, 175-188. HOLL1NO, C. S. (1966). Mere. Entomol. Soe. Can. 48, 1-86. IDYLL, C. P. 0973). Sci. Am. 228, 22-29. LARKIN, P. A. (1977). Trans. Ant. Fish. Soc. 196, l - I I . M AR'nTq, A. R. (1989). Bird Study 36, 170- ] 80. MAY, R. M., CONWAY, G. R., HASSELL, M. P. & SOt.rrHWOOD, T. R. E. (1974). J. Anita. Ecol. 43, 747-770. NELSON, J. B. (1978). The Sulidai: Gannets and Boobies. Oxford: Oxford University Press. PITCHER, T. J. & HART, P. J. B. (1982). Fisheries Ecology. London: Croom Helm. SCHAEFER, M. B. (19575. J. Fish. Res. Board Can. 14, 669-681. SCHAEFER,M. B. (1970). Trans. Am. Fish. Soc. 9, 461-467. TAYLOR, R. J. (1984). Predation. London: Chapman and Hall. UTTLEY, J., MONAGHAN, P. & WHITE, S. 09895. Ornis Scan. 20, 273-277.

326

A . I . HOUSTON APPENDIX A The Characteristic Return Time TR

Let 2 be the equilibrium value of x. Then we can look at magnitude of the perturbation x - 2 . For small perturbations, we have approximately d/dt(x-

2) = - ( 1 / T R ) ( x - - 2),

(A. 1)

for a suitable constant TR. Integrating this equation shows that x - 2 decays as exp (--t/TR), so that TR is the time for x - 2 to fall to e -1 of its initial value. TR is called the characteristic return time (May et al., 1974). When x = f ( x ) for some function f, d/dt(x-

2) = f ( x ) - f ( 2 ) ,,"~f'(2)(x - 2).

(A.2)

It follows from eqns (A.l) and (A.2) that (A.3)

TR = - 1I f ' ( 2 ) .

In the Schaefer model, f ( x ) = rx(1 - x / K )

- Y r - YB.

(A.4)

If YF = EI:x and YB= EBx, then from eqns (A.3) and (A.4), (A.5)

TR = I / ( r - E F - E B ) .

If Yr = EFX and YB= B then the analysis is not quite so clean. There are two possible equilibrium values 2,, and 22, with 0 < 2 t < 2 2 < K . The stable value is 22, which is given by the equation ( r - Er) + ( ( E r - r) 2 - 4 r B / K ) I/2 22 -

(A.6)

2r/K

In this case, f'(x) = r- 2rx/K-

EF,

(A.7)

so that f ' ( ~ ) = --( (EF-- r) 2 -- 4 r B / K ) '/2.

Thus from eqn (A.3) TR = ((EF-- r) 2 -- 4 r B / K ) -~/2.

(A.8)

EFFECTS

OF SEABIRDS

ON A FISHERY

327

A much simpler equation for T~ results if the MSY effort Er* = r/2 - 2 B / K is adopted. The resulting equilibrium is x* = K / 2 and so from eqn (A.7) f ' ( x * ) = - E * = - r / 2 + 2B/K,

and hence TR = 1/(r/2 - 2 B / K ) .

(A.9)