The dynamic effect of a marine reserve

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

The dynamic effect of the reserve is determined by three equations of motion: Fishing ground stock: S : F = rK F S F + X R 1 − S F K F − cES F K F 11 Reserve stock: S : R = rK R S F + X R 1 − S R K R 12 Fishing effort: a \ 0 E : E=apRwE−1=apcS F wK F − 1 13 Eq. 11 comes from Eq. 2 and Eq. 4, Eq. 12 comes from Eq. 3, and Eq. 13 uses the assump- tion that fishing effort adjusts at a rate propor- tional to the rent rate Clark, 1990. With no reserve, the local stability of the two-variable system Eq. 11 and Eq. 13 could be analysed by techniques set out by Clark 1990, who shows that the only two possible types of dynamic be- haviour around the open access equilibrium are a stable focus or a stable node. But with a reserve, it is impossible to analyse the three-equation sys- tem Eqs. 11 – 13 analytically in this way. Instead, we have performed a large number of discrete-time, computer simulations for many dif- ferent parameters including the length of the time step, which was kept short to prevent spurious numerical solutions emerging. Figs. 3 and 4 are representative of the behaviour we found. Fig. 3 shows time paths of stock density on the fishing ground over 40 years, and Fig. 4 the correspond- ing time paths of catch, for the Jamaica case of Table 2, with and without a maximal c. 40 reserve, and with the economic adjustment parameter a = 0.5 or 2. All graphs start from an area with no reserve, and both effort and fish stock being 10 above their open access equi- librium values. Graphs for rent rate are not shown because they parallel those of fishing ground density, since the time derivative of the rent rate can be shown to be pcS : F wK F . Stock density on the reserve is not shown because in all but extreme cases behaviour is similar, with den- sity rising fairly steadily towards 1 i.e. carrying capacity, and reaching over 0.9 within 40 years. Figs. 3 and 4 show that the effect of the reserve on ‘stability’ depends on how stability is mea- sured. In all cases a reserve improves convergence: either creating faster convergence to equilibrium on the fishing ground, or allowing it to happen when the no-reserve system does not converge at all. However, reserves tend to make oscillations more rapid during convergence, with the period length shortening over time. A reserve also makes the first few oscillations larger because the initial level of effort is suddenly confined to the much smaller fish stock on the fishing ground. These faster and deeper initial oscillations mean that some redirection of fishing effort to other loca- tions or occupations during the transition to re- serve equilibrium — a redirection our model ignores — will be very important in securing broad-based support for creating marine reserves. In keeping with the dynamic economic analysis in Sanchirico and Wilen 1996, present dis- counted values of transitional fishing rents are shown in Table 3 for both 5 and 10 discount rates. The effect of the reserve on present value depends on the discount rate, so no generalisation is possible. Finally, mention is due of the possible effect of the reserve on open-access stock extinction. This is a complex issue, and our model excludes sev- eral, potentially relevant features, such as a mini- mum viable population size included by Berck, Fig. 3. Dynamic effect of reserve on stock density on fishing ground. Fig. 4. Dynamic effect of reserve on catch. 1979, the effect of discrete time Bjorndal and Conrad, 1987, a low stock elasticity of produc- tion Bjorndal et al., 1993, and uncertainty over catch levels Lauck et al. 1998. Also, in the multispecies situations of coral reefs, any extinc- tion would probably be of just the economically worthwhile species, rather than of all fish biomass. Nevertheless, in our single species model a reserve obviously does protect against extinction. Even if the fishing ground stock S F becomes momentarily extinct because of a sudden burst of overfishing, it will recover, thanks to the flow of eggs and larvae from the reserve from Eq. 11, S F = 0 [ S: F = rK F K R \ 0.

4. Conclusions