Thompson: Bayesian approach to management advice from a simple dynamic pool model 



571 



for comparison with the results of Ludwig and Walters. 

 As Equations (26-28) indicate, a variety of parameter 

 combinations allow for B(Fmsy) to be less than half of 

 B(Fmelsy)- Since the results presented by Ludwig 

 and Walters (1982) were derived from a numbers-based 

 model, Equation (27) is particularly relevant. Under 

 this equation, a v' value greater than 1/3 is sufficient 

 to guarantee that the stock size at MSY will be less 

 than half the stock size at MELSY, regardless of the 

 value of m. At values of m>0.5, a v' value of 0.227 is 

 sufficient. 



Clark et al. (1985) found that the relationship be- 

 tween the myopic Bayes and certainty-equivalent solu- 

 tions depended on the model used. In the special case 

 where recruitment is independent of stock size, for 

 example, they found that the myopic Bayes solution 

 always exceeded the certainty equivalent solution. For 

 the same model, the authors also found that the myopic 

 Bayes solution always increased with the level of uncer- 

 tainty. These results are precisely the opposite of those 

 obtained in the present study, where F^elsy is always 

 less than Fmsy arid decreases monotonically with v'. 

 In their "full cohort model" with a stock-recruitment 

 relationship, however, Clark et al. (1985) obtained 

 results similar to those of the present study. In one 

 example, the myopic Bayes solution prescribed a 

 30-50% reduction in F relative to the certainty- 

 equivalent solution. Using yet another model, Walters 

 and Ludwig (1987) also found that the myopic Bayes 

 solution was a monotone-decreasing function of 

 uncertainty. 



Conclusion 



This paper describes an approach for treating the prob- 

 lem of parameter uncertainty in a systematic fashion. 

 Although fisheries are often managed as though stock 

 parameters are known with certainty, it would be 

 preferable to develop a management approach more 

 consistent with the fact that such certainty is the ex- 

 ception rather than the rule. Such an approach was 

 developed here in the context of Bayesian decision 

 theory. When applied to the particular model pre- 

 sented, this approach indicates that the optimal fishing 

 mortality rate Fmelsy (Eq. 20) is always less than 

 Fmsy (Eq. 2) except in the limiting case where q is 

 known with certainty (Fig. 4). 



This result provides formal support for the intuitive 

 conclusion (e.g., Kimura 1988) that fishing mortality 

 should be strongly constrained when the stock-recruit- 

 ment relationship is uncertain. Similarly, Equation (25) 

 indicates that if recruitment is highly dependent on 

 stock size (specifically, if m exceeds 0.5), Fmelsy will 

 always be less than the natural mortality rate. 



The rock sole example illustrates the basic con- 

 servatism of the Fmelsy approach. In this example, 

 Fmelsy was less than Fmsy by about 40%. Given that 

 neither the Fmsy value (0.121) nor the fit from the 

 stock-recruitment regression (Fig. 6) was atypical of 

 groundfish stocks, the ratio between Fmelsy and Fmsy 

 in this example provides a practical illustration of the 

 extent to which an explicit accounting for uncertainty 

 can influence management strategy. The magnitude of 

 the effort reduction prescribed in this example is 

 similar to results described by Ludwig and Walters 

 (1982) and Clark et al. (1985). The confirmatory nature 

 of these studies may suggest that the conventional 

 wisdom regarding optimal exploitation rates should be 

 reexamined. At the very least, the Fmelsy approach 

 provides a low-end estimate of the maximum accept- 

 able harvest rate and a warning against taking Fmsy 

 estimates too seriously. 



A great deal of the conservatism resulting from the 

 Fmelsy approach as developed here stems from the 

 assumption that all values of q are logically possible, 

 despite the fact that a q value of 1 results in extinction 

 under any level of fishing. One alternative might be to 

 examine q in the context of life-history theory, to deter- 

 mine if it is possible to justify some other upper limit 

 on the logically permissible range. A related alternative 

 would be to use a nonuniform prior in estimating P(q). 

 The assumption of a uniform prior may be overly 

 pessimistic, since fishery biologists often have an in- 

 tuitive feel for stock-recruitment parameters, even in 

 the absence of data for a particular stock. Such infor- 

 mation could be used to define an alternate prior pdf. 

 Another possibility would be to establish an empirical 

 prior based on the results of other stock-recruitment 

 studies, but this would likely require a fairly elaborate 

 weighting scheme so that stock-recruitment param- 

 eters from the most dissimilar stocks or environments 

 would have the least influence on the form of the 

 resulting pdf. 



An additional factor which may add to the conser- 

 vatism of the Fmelsy strategy as developed here is 

 the use of the myopic Bayes solution rather than an 

 actively adaptive solution. An actively adaptive solu- 

 tion would attempt to anticipate and make use of 

 changes in available information resulting from alter- 

 native management actions (e.g., Walters and Hilborn 

 1976, Smith and Walters 1981, Ludwig and Walters 

 1982, Ludwig and Hilborn 1983, Clark et al. 1985, 

 Walters 1986, Milliman et al. 1987, Walters and Lud- 

 wig 1987, Parma 1990, Parma and Deriso 1990). How- 

 ever, myopic Bayes (or similar) solutions often perform 

 nearly as well as their actively adaptive counterparts 

 (Mendelssohn 1980, Walters and Ludwig 1987, Parma 

 1990, Parma and Deriso 1990), and if the myopic Bayes 

 solution is reestimated each year, the result is a 



