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Fishery Bulletin 97(3), 1999 



obtained qualitatively identical results. Little infor- 

 mation exists on density-dependent relationships for 

 tropical fish; therefore we were forced to use a stan- 

 dard theoretical logistic equation (see Appendix). For 

 simplicity in analysis and in recognition of this knowl- 

 edge gap, we constructed the models with a fixed car- 

 rying capacity of 1000 one-year-old individuals per unit 

 area. We modeled population processes on the basis of 

 density measures in reserve and fishing areas and used 

 j:elative proportions of each to calculate catches and 

 population fecundity. Thus, yields are expressed as kg 

 per year from the whole management area. 



In order to ask the general questions we intended, 

 we used two simple movement assumptions that 

 emphasized the benefits accruing from larval trans- 

 port rather than from adult spillover. Larvae dis- 

 persed widely across reserve boundaries, resulting 

 in an even density of new fish settlement in reserve 

 and nonreserve areas. This assumption does not ne- 

 gate the possibility that larvae drift to the open ocean 

 and become lost — these can be accounted for in lar- 

 val mortality. Rather, the assumption implies that 

 settlement in the reserve and the management area 

 are equally affected by the stock averaged over both 

 areas. We also assumed that adults did not move 

 across reserve boundaries; that is to say fish spent 

 their entire lifetime in the area in which they settled. 

 These assumptions specifically addressed the case 

 where enhanced fecundity within the reserve was 

 exported to fishing areas through larval transport. 

 Thus they complemented previous models that fo- 

 cused on enhancements from adult spillover 

 (Polacheck, 1990; DeMartini, 1993). These models 

 also examined potential increases in reproductive 

 output by means of increased spawning stock bio- 

 mass per recruit. However, they were unable to ex- 

 amine the equilibrium consequences with their par- 

 ticular model construction. 



Our movement assumptions apply to the majority 

 of fishery species on at least some spatial scales. Most 

 aquatic species disperse more widely as larvae than 

 as adults (Boehlert, 1996). Consequently, larvae are 

 more likely to cross boundaries than are adults. As 

 long as individual reserve units stretch beyond the 

 dispersal distance of adults but remain well within 

 the dispersal distance of larvae for a given species, 

 the model assumptions will approximate reality. For 

 large reserve proportions, our assumptions could still 

 be met if the reserve area were partitioned into sev- 

 eral smaller units. Some areas of concern here in- 

 clude ontogenetic and reproductive migrations. To 

 fit the assumptions of this model, reserves must be 

 designed with these movements in mind so that fish 

 are likely to remain in the reserve during the phase 

 of their life in which they are vulnerable to fishing. 



We ran the models over reserve proportions vary- 

 ing from to 99*^ of the management area is-O to 

 0.99), and fishing mortalities varying from 1% to 

 100'^ mortality of fishery-recruited individuals per 

 year (!/=0.01 to 1.00). For each combination of fish- 

 ing mortality and reserve proportion, the model ran 

 until the fish catch — calculated for the whole man- 

 agement area rather than per km''^ of available fish- 

 ing area — had stabilized at the long-term sustain- 

 able yield. The model stored the jaeld, fishing mor- 

 tality, and reserve proportion. It sometimes took 

 hundreds of years to reach stability, and those inter- 

 ested in our model's predictions about the short-term 

 dynamics of reserve creation should refer to Sladek 

 Nowlis and Roberts ( 1997). 



We used these results to determine the optimal 

 reserve proportions and fishing mortalities for indi- 

 vidual fishery species. For each fishing mortality, we 

 found the reserve pi'oportion that maximized sustain- 

 able yields and stored it and the yield. We compared 

 these yields when an optimally-sized reserve was 

 used with the yields without a reserve (s=0) to 

 establish fisheries benefits. We plotted this infor- 

 mation using fishing mortality as an independent 

 variate. 



We also examined the effects of marine fishery re- 

 serves on year-to-year catch variability. Bohnsack 

 ( 1996) suggested that marine fishery reserves could 

 dampen natural fiuctuations in catches, thus mak- 

 ing fisheries more stable and easier to manage. We 

 tested this hypothesis by adding a stochastic compo- 

 nent to lai-val sui-vivorship in our model. These new 

 models drew lai-val survivorship randomly from a 

 normal distribution around the mean larval survi- 

 vorship whose standard deviation we could define. 

 We examined all species over a range of fishing mor- 

 talities and present the results from ;/ = 0.2, 0.4, 0.6, 

 and 0.8 to illustrate the pattern. We also examined 

 some of these conditions at three levels of environ- 

 mental variation, with standard deviations of 5, 10, 

 and 20'^ of the mean larval sui-vivorship, to gain in- 

 sight into whether reserve benefits are influenced 

 by the degree of environmental variability. 



For each possible combination of fishing mortality 

 and reserve proportion, we performed 10 replicate 

 runs of our stochastic models. In each run, we ran 

 the models for 500 years to allow the fisheries to sta- 

 bilize to the maximum extent possible and thus mini- 

 mize the influence of our arbitrarily chosen initial 

 state. The mean and standard deviation of the 

 catches were measured over the next 100 years. We 

 examined the ratio of the standard deviation to the 

 average catch over this period because this measure 

 gave us an estimate of the likelihood of percentage 

 fluctuations in catches rather than absolute changes. 



