A modified Schaefer – Gordon model of the equilibrium of a marine fishery reserve

area not in the reserve, and thus to dissipate all rent profit in equilibrium. Secondly, our model is based on an ad hoc growth function which implicitly reflects two age classes with completely different movements, as in Brown and Roughgarden 1997: eggs and larvae, which are dispersed evenly throughout the reserve and fishing ground, and adults, which do not migrate at all. These are appropriate assumptions for coral reef fish, and also most shellfish such as abalone, scallops and clams, and some demersal fish. They might also be empirically acceptable approximations for the whole systems of marine reserves now being suggested McGlade et al., 1997; Halfpenny and Roberts, 1998 to protect overfished migratory species in temperate areas such as the North Sea. This is why we call our model one of a ‘marine’ reserve, not just a ‘reef’ reserve; though clearly there are many existing or proposed marine reserves, particularly in temper- ate waters, to which it would not apply. Finally, for simplicity, the model is spatially homoge- neous, and so ignores biological variation within a management area. It is the combination of open access fishing economics, implicitly age-dependent dispersal, and analytical simplicity which distinguishes our model from other models of marine reserves. We derive a simple analytic formula, requiring only two data points for any management area, for the proportional size of a reserve which maximises sustainable catch. In Section 2, we develop the model, compare its assumptions to those of other models, derive its equilibrium theoretical proper- ties, and consider their empirical plausibility in three Caribbean locations; data are not yet avail- able to test them completely. In Section 3, we simulate the dynamic effects of reserve creation and Section 4 concludes.

2. A modified Schaefer – Gordon model of the equilibrium of a marine fishery reserve

Here we describe our model in Sections 2.1 and 2.3, compare it to other models in Section 2.4, briefly consider a couple of important ‘off-model’ effects in Section 2.5, and then consider some empirical applications in Section 2.6. We start with the standard Schaefer – Gordon assumptions of neglecting both the age structure of fish popu- lations, and the complex ecological interdepen- dencies within coral reefs, such as predator – prey relationships and inter- or intra-specific competi- tion among organisms. Instead we assume a sin- gle, undifferentiated fish biomass and then make two key modifications, as follows. 2 . 1 . Dependence of carrying capacity on area, and of catch on density A standard assumption is that the carrying capacity K for the biomass of a certain fish species is determined once the species and a broad geo- graphical area are defined: for example, the North Sea Herring, or the Pacific Whiting in Conrad 1995. Another is that the catch often, if rather inappropriately, called harvest R in tonnesyear, say per unit of fishing effort E is proportional to the total fish stock S in tonnes, irrespective of the sea area over which the stock is spread: R = qES, where q is the catchability parameter Clark, 1990. Neither assumption is appropriate for coral reefs, where the dominant fish species are largely non-migratory as adults and tied to small areas of reef Polunin and Roberts, 1996. So instead we assume that the carrying capacity K for biomass of any species of a given sea surface area of a reef is proportional to its area A, and that catch R from the reef per unit effort E expended is pro- portional to the areal stock density SA. The catch function then becomes R = cESK, 1 where c is a catchability parameter with different size and units from q, and SK is the stock density in relation to carrying capacity of the area. 2 . 2 . The ecological linkage between reser6e and fishing ground A reserve denoted R of carrying capacity K R is created within a total management area of capac- ity K, which is normalised to 1, so K R also mea- sures the proportional reserve size. The fishing ground denoted F that occupies the rest of the management area has carrying capacity K F = 1 − K R . The stocks on the management area, reserve and fishing ground are, respectively, S, S R , and S F = S − S R , as depicted in Fig. 1. The manage- ment area is spatially homogeneous, so the reserve can be anywhere in it and need not be contiguous; we say nothing about edge effects or the optimal shape of the reserve. The scale K of the manage- ment area also plays no role in the model, but in practice there will be both a minimum worthwhile scale, below which relative enforcement costs be- come excessive; and a maximum worthwhile scale, probably exceeding tens of kilometres for coral reefs, above which the dispersal effects mentioned below would be weaker Roberts, 1997b. The model then has to describe the ecological linkages between reserve and fishing ground in some way. There are three main linkages: 1. The density of adult fish in the reserve will rise after fishing is banned there. This causes greater egg production per unit of reef area, and net export of eggs and larvae via pelagic dispersal to the surrounding fishing ground Bohnsack, 1992; Sluka et al., 1997; Nowlis and Roberts, 1999. The dispersal of eggs and larvae is actually very complex and remains poorly understood Boehlert, 1996; Roberts, 1997b, but we follow Nowlis and Roberts 1997 and assume that eggs and larvae are dispersed by sea currents evenly throughout the management area. It is then the average fish density over the area, which determines egg and larval density in both reserve and fishing ground. However, we ignore the effect of the reserve on fish size. Since larger female fish are much more fecund than smaller ones, egg output per unit biomass will in fact be higher in the reserve Sluka et al., 1997. In this respect our model will thus understate the reserve’s effect, and we briefly discuss this ‘off-model’ in Section 2.5. 2. There will be some net flow of adult fishes from the reserve to the fishing ground, caused by the greater population density in the re- serve Rakitin and Kramer, 1996; Russ and Alcala, 1996. However, most studies of coral reef fishes have found that movements of adults and juveniles are limited, even among species for which higher mobility was origi- nally suspected Holland et al., 1996; Roberts, 1996. Tagging studies suggest high site fidelity and very limited movements typically tens of metres with only a few percent of individuals moving out of reserves Holland et al., 1993; Corless et al., 1999. Reef fishes are especially reluctant to cross areas of sand or open water, and so reserves with their boundaries coincid- ing with reef boundaries will retain almost all of the fish within them Kramer and Chap- man, 1999; Chapman and Kramer, 1999. An extremely small reserve in St. Lucia less than 3 ha had more than double the biomass of Fig. 1. Definition of management area, reserve and fishing ground. reef fish inside it in adjacent fished areas after only 3 years of protection, suggesting a high level of retention of fish Roberts and Hawkins, 1997. Our implicit assumption is therefore that adult fish migration is zero be- tween reserve and fishing ground. 3. The reserve will cause changes in species com- position, both within the marine reserve and on the fishing ground, but we ignore these in our model. Fishing pressure can sequentially eliminate species from multispecies stocks on reefs, according to their vulnerability to cap- ture and intrinsic rate of population increase. At low fishing intensities and hence high stock densities, the biomass is usually dominated by carnivorous species, but these are generally easier to catch because they are bolder and more inquisitive, so they tend to disappear first when fishing intensities increase Roberts, 1997a. This is economically important since they are larger and more valuable as food, commanding a higher market price per unit weight Polunin and Roberts, 1993. In Sec- tion 2.5, we also discuss this effect ‘off-model’. We implicitly combine assumption i of uni- form dispersal of eggs and larvae with assumption ii of completely sedentary adults by modifying the logistic growth function Clark, 1990 as follows: Assumption: The absolute rate of natural in- crease of fish biomass in the fishing ground is G F X F = rK F X F + X R 1 − X F K F 2 where r is the intrinsic growth rate of the biomass. The rationale for this is that the stock relevant to juvenile growth in the fishing ground at rate r is not S F , but K F S F + S R , since K F is the propor- tion of the eggs and larvae from the total manage- ment area stock S F + S R which will be in the fishing ground, given uniform dispersal. But the carrying capacity that ultimately limits growth is that of the fishing ground alone, hence the 1 − S F K F factor in Eq. 2. Unlike other models of marine reserves cited in the Introduction, ours has no explicit term for migration between reserve and fishing ground because we assume that the biomass of eggs and larvae that disperse out of the reserve is negligible, and adult fish are seden- tary. The rationale for formula Eq. 2 is thus rather rough, but not greatly more so than that of the logistic function itself, which does not specify whether S should be population number May, 1981 or population biomass Clark, 1990, and does not explicitly take account of growth in body size. Ultimately Eq. 2 is a formula that should be judged on its empirical usefulness. By the same arguments, stock growth in the reserve is assumed to be: G R S R = rK R S F + S R 1 − S R K R 3 With no catch from the reserve, the equilibrium reserve stock is S R = K R . From Eq. 1, the catch from the fishing ground is R = cES F K F 4 The fishing ground stock will be in biological equilibrium when growth G F equals catch R. From Eqs. 2 and 4, with K F = 1 − K R and S R = K R , this will be when r1 − K R S F + K R [1 − S F 1 − K R ] = c ES F 1 − K R . 5 2 . 3 . Finding the maximal size of the reser6e under bionomic equilibrium We then set the objective of finding the bio- nomically maximal size of the reserve, by which we mean the proportional size which maximises the open access, equilibrium catch from the fishing ground. This is a crucial and debatable assumption. Why would one care about whether none, half or all of a management area is turned into a reserve, if rent remains zero in all cases? Here we adopt the political economy approach of Sanchirico and Wilen: in the context of a develop- ing economy, sustainable increases in catch and effort and hence in nutrition and livelihoods are seen as politically desirable for their own sake. We also assume that both the price p of fish and unit cost, w, of effort are constant. If instead we assume that price depends on catch, or that unit cost depends on effort, the algebra below becomes intractable. Using the standard assumption Gor- don, 1954; Clark, 1990 that the fishery makes zero rent profit in an open access, and denoting open access equilibrium variables by O , rent = pR O − wE O = 0, so R O = wE O p 6 Maximising catch is thus the same as maximising effort here. Together, Eq. 4 and Eq. 6 give R O = c E O S F O 1 − K R = wE O p hence S F O 1 − K R = j, where j: = wpc is assumed to be B 1. 7 We call j the relative cost of catching fish the cost of effort divided by fish price times catchabil- ity. We assume j B 1 because j ] 1 means that fish are not worth catching at all under open access. In the initial open access equilibrium with no reserve, K R = 0, S F O becomes S O , which is also the stock density since K = 1, and Eq. 7 becomes S O = j = wpc. 8 This initial stock density is in fact the only bio- nomic information needed to find the maximal size of the reserve in this model. Also, note that with or without the reserve, a ceteris paribus reduction in fishermen’s wages or improvement in fishing technology either of which will lower w will cause resource degradation a lower stock density in the fishing ground. Eq. 5 is true in any equilibrium, so we can add O superscripts to S F and E there and substi- tute for S F O from Eq. 7 to give r1 − K R [j1 − K R + K R ]1 − j = cE O j or E O = [rp1 − jw] [j + 1 − 2jK R − 1 − jK R 2 ]. 9 This predicts the equilibrium effort and hence catch, from Eq. 6 given any reserve proportion K R . It is a negative quadratic in K R , so its local maximum will also be the global one. The sign of the partial derivative, E O K R = [rp1 − jw] [1 − 2j − 21 − jK R ], as both j and K R vary, shows that if 12 5 j B 1, a reserve will reduce equilibrium effort and catch, and so is not worth- while in terms of our objectives. But if j B 12, a reserve will increase effort and catch until K R = K R O : = 12 − j1 − j = 12 − S O 1 − S O , 10 where we denote values in maximal catch-max- imising open access equilibrium by O . This max- imal reserve proportion K R O rises from 0 to 12 as j falls from 12 to 0. For K R \ K R O , effort and catch fall, becoming the same as in the no-reserve equilibrium when K R = 2K R O . Our main result is thus: If a management area’s initial stock density S O in relation to carrying capacity is in an open access fishing equilibrium hence S O B 1, then making a proportion K R of the area into a no-take marine reserve will in- crease equilibrium catch on the remaining fishing ground if and only if S O B 12 and 0 B K R B 21 2 − S O 1 − S O . Catch will be maximised at the midpoint of this range, where reserve size K R = 1 2 − S O 1 − S O . However, as shown below, the dynamic, pre- equilibrium impact of the reserve will be to reduce catch initially, creating an obvious transitional problem for fishermen and hence fishery man- agers to overcome. Henceforth, we assume the initial stock density S O = j B 12 unless otherwise stated. This is realistic for reef fisheries that are moderately to heavily exploited Polunin and Roberts, 1996. Table 1 gives the equilibrium values of stock, effort and catch without the reserve K R = 0 and with a maximal reserve K R = K R O ; open access stock comes from Eq. 7, effort from Eq. 9, and catch from Eq. 6. Also given are corresponding values for statically zero-discounting optimal management by controlling effort to maximise sustainable rent instead of creating a reserve an approach which we ignored because of its likely information and enforcement costs in coral reef situations. Appropriate inequalities between the values are also noted: the maximal reserve enables a greater catch than the static optimum 14 r \ 141 − j 2 r, but using proportionally more effort 1 4 rpw \ 121 − jjrpw and with a lower overall stock density 12 B 1 + j2. The in- crease in effort is inevitable because of the con- Table 1 Comparison of theoretical equilibrium results Static optimum: No reserve, max- Variable Maximal reserve plus open access to No reserve, open access imum sustainable rent denoted O fishing ground denoted O 1+j2 \ Overall stock density 1 2 \ j S F + S R Effort E B 1 2 1−jjrpw 1 4 rpw \ 1−jjrpw B 1 4 r 1 4 1−j 2 r \ Catch R 1−jjr 1 4 1−j 2 rp Rent pR−wE \ = straint of banning fishing on part of the manage- ment area. And there is some empirical evidence for the increase in effort compared to the no-re- serve, open-access equilibrium, for Goodridge et al. in press observed an increase in effort follow- ing the creation of reserves in St Lucia, though this was before final equilibrium had been reached. Also, since Eq. 7 gives S F O K F O = j B 12, the stock density in the fishing ground is less than half that in the reserve where S R K R = 1. So there will indeed be net flow of eggs and larvae from the reserve, although this is only implicit in our model. 2 . 4 . Comparison with existing models Having set out our model, we can now compare it more precisely to the existing literature. It is the combination of open access fishing economics, age-dependent dispersal, and analytical simplicity that distinguishes our model. Polacheck 1990 examined the effect of permanently closing part of a fishery using a model with several age classes derived from Beverton and Holt 1957, but nei- ther of these models contained any economic ele- ments; neither does Lauck et al. 1998, which models the role of marine reserves in coping with irreducible scientific uncertainty in marine ecosys- tems combined with only probabilistic control of catch levels. Holland and Brazee 1996 included many age classes and maximised the present value of catches, but their maximisation did not allow any economic response of fishing effort to changes in rents. Brown and Roughgarden’s 1997 model was specifically designed for temperate species such as scallops and clams, but could also be applied to coral reefs since it assumed mobile larvae and stationary adults, as we do. Their model has explicit terms for these two age classes explicitly, so it is technically more sophisticated than ours; but its assumption of dynamically optimal fishing management is less applicable to developing country situations, for reasons noted earlier, and results in no recommendation for optimal reserve size. Other existing models ignore any age effects on migration rates, and are thus less relevant to the coral reefs, but they have many interesting fea- tures which we have ignored. Thus Hannesson 1998 includes discrete as well as continuous time, which allows season length to be studied; Sanchirico and Wilen 1996, 2000 consider source-sink as well as density-dependent systems of migration; and Conrad 1999 extends Homans and Wilen’s 1997 model of regulated open access to include a reserve or sanctuary. Conrad’s model is aimed at the marine sanctu- aries of the US, and is less applicable to the tropical reef situation because it assumes that dispersal of fish is independent of age, and that catches are targeted and regulated by controlling season length, rather than just banned. In con- trast to our assumption of spatial homogeneity, Brown and Roughgarden and Sanchirico and Wilen also address the question of ‘patchiness’, where some parts of the management area have different biological features, which perhaps influ- ence where reserves should be sited. Which is the more appropriate assumption is an empirical matter. 2 . 5 . Does the reser6e gi6e any economic benefit ? Here we consider two ‘off-model’ economic benefits of marine reserves that strengthen the case for them, and point to further research needed. Firstly, consider the reserve’s economic benefit in terms of fishing. The derivation of the maximal reserve size Eq. 10 assumed both a constant price of fish in common with almost all existing models and a constant unit cost of effort, so indeed there is apparently no net fishing benefit from the increase in catch achieved by the reserve. The higher fish catch gives no benefit to con- sumers because the price of fish does not fall; and the higher effort expended is of no benefit to producers for example, to fishermen, since the unit payment does not rise. All the reserve achieves in the model is a larger catch from the management area, but with all rent still being dissipated by open access. However, a maximal reserve would actually give an economic gain in practice because of four features ignored by our model. Consider Fig. 2, where as before the no-reserve equilibrium is de- noted O , and the reserve equilibrium is denoted O . On the demand side, our model ignores the fact that at a national or regional if not local level, a lower price would be needed to induce people to buy more fish, giving a downward slope of the demand curve for fish. But it also ignores the increase in average fish size and improvement in species composition that the reserve will produce, as described in Section 2.2i and iii above. Together, these two features would mean that creating a reserve would shift a falling demand schedule out from, say, D O p to some schedule further to the right. The two features ignored on the supply side are the increase in payment per unit needed to induce more fishing effort at a national or regional level, and the higher catchability of the larger fish re- sulting from the reserve, as described in Section 2.2iii. Together, these would mean that the re- serve shifts a rising supply schedule outwards from, say, S O p to a schedule further right. The net effect of these outward shifts could be that the fish price rises, stays the same, or falls. Our simplifying assumption shown on Fig. 2, that the schedules shift to D O p and S O p which intersect at an unchanged price p, is therefore a special but perfectly acceptable case. The assump- tion of a constant unit cost w is similarly restric- tive but acceptable, though it does amount to subsuming an increase in catchability c as a de- crease in w instead. Fig. 2 shows that the reserve increases both consumer and producer surplus, giving a more conventional economic case for the reserve, though our maximal reserve size will gen- erally not be the economic optimum. Fig. 2. Economic benefit of a maximal reserve. Table 2 Bioeconomic variables and maximal reserve sizes Bonaire Parameter Belize St. Lucia Jamaica heavily fished moderately fished intensively fished no fishing 3.241 a Stock in 150 m 2 ‘count’ kg b 1.883 8.85 1.447 1 0.3662 0.2127 Hence initial stock density S = j = wpc = relative 0.1636 cost of catching fish – 0.2111 Hence 0.5−j1−j, maximal catch-maximising 0.3649 0.4022 reserve proportion, from Eq. 10 14j1−j, predicted factor by which reserve will – 1.077 1.493 1.827 increase catch and effort, from Table 1 a Initial daily catch kgperson day c 7.6 – 5 2.86 300 300 – 300 a Length of season days fishingperson year – Hence catch per unit effort RE tperson year 2.280 1.500 0.858 6.226 7.052 Hence c, catchability = RES tperson year 5.245 – 2530 4100 – 4250 a p = price USt, 1998 values d – 5768 Hence w = unit cost of effort = pRE, assuming 6150 3647 open access equilibrium person year e 1.11 a Intrinsic rate of growth r per year f 1.15 – 1.20 a Empirically measured values; all other values are derived from them. b Fish stocks in each area were estimated from direct, underwater, visual census data collected as described in Roberts 1995b. Figures are based on a large number of replicated counts between 70 and 330 made at 5 and 15 m intervals at each site. c Figures are based on trap catch data from Sary 1995 for Jamaica, Goodridge et al. in press for St. Lucia, and one of the author’s CR field notes from Belize. d Values are converted to 1998 from Polunin and Roberts 1993 for Belize, from Sary 1995 for Jamaica, and from personal observation for St. Lucia. e Includes labour and costs of traps, fuel, boat depreciation etc. f Chosen to be most appropriate to the species dominating catches at each site, and based on limited available data given by Nowlis and Roberts 1997. The second economic benefit of the reserve which deserves a mention is that the original and often still prime purpose of conservationist pro- posals for no-take reserves is to improve the ecological ‘value’, and thereby also the tourism potential, of the marine reserves themselves. This is positively related to the biomass see Man et al., 1995; Nowlis and Roberts, 1997, 1999, for mod- elling studies; Alcala, 1989; Polunin and Roberts, 1993; Jennings and Polunin, 1995; Roberts, 1995b; McClanahan and Kaunda-Arara, 1996, for empirical results; and Cesar et al., 1997 for an economic analysis. Such value, which could per- haps be partially estimated by a survey of willing- ness to pay by reef-diving tourists, would obviously lead to numerically calculated, optimal reserve sizes being larger than on grounds of fishing benefit alone. 2 . 6 . Typical empirical 6alues To see what our model might mean in practice, we give in Table 2 some typical values from three Caribbean fisheries with different levels of fishing intensity. We report the measured stock density S O without a reserve, which given no contrary evidence we assume to be in open access equi- librium, and then use Eq. 10 to calculate the maximal reserve proportion K R O , finding values in a 20 – 40 range, and the predicted increase in catch, finding values in 10 – 80 range. We also report the measured catch per unit effort RE, and derive the catchability parameter c; and we report the price of fish p, and derive the cost of effort w consistent with open access equilibrium we have no direct data for w. Finally, for infor- mation for example, in calculating R or E in Table 1 we report intrinsic growth rates. Foot- notes give the sources of the data. The initial stock densities shown are clearly in the region where the model predicts a fishery benefit from a reserve. However, such reserves as exist have either not been established for the several years that it takes to reach equilibrium, or have been managed with communal access restric- tions rather than by open access Alcala, 1989; Goodridge et al., in press. So our main predic- tions, which are that the stock density on the fishing ground is unchanged by the reserve, but catch and effort rise by a factor 14j1 − j, remain untested empirically. However, there is rough empirical support for our model from three sources. Firstly, the three derived figures for catchability are within an ac- ceptable range of variation for what are consid- ered to be fundamentally similar reef ecosystems. Secondly, the ratio of the inferred unit costs for Belize and Jamaica 57683647 = 1.582 is fairly close to the ratio of their per capita GNPs 2450 1440 = 1.701, from WRI, 1996, even though GNP per capita obviously may not reflect the living standards of fishermen. Thirdly, the initial stock density in Conrad’s case study was 0.60 = j, so that our model concurs with his qualita- tive prediction that a reserve will reduce catches in that case. But his reserve proportion was 43.3 = K R , so Eq. 9 would predict a catch equal to 73.1 of what it was before the reserve, whereas his result was a fall to 57.9. Also, our results differ markedly from Hannesson’s Hannesson, 1998 calculation of a maximal reserve size of about 75 for effectively a relative cost of catching j = 0.15, similar to our figure for Ja- maica where we find a maximal reserve size of about 40. If our result is empirically valid, an average increase of 30 in catch from overexploited coral reef fisheries could well be feasible. A conservative estimate is that a third of all reefs worldwide are threatened by overexploitation Bryant et al., 1999. Applying this to the 18 billion annual catch value calculated at the outset, and noting that proportionally less of this will be coming from over-exploited reefs, we estimate conserva- tively that reef fishery reserves could increase the annual world catch by the order of one billion US dollars per year at current prices.

3. The dynamic effect of a marine reserve