Abstract. - a simple dynamic 

 pool model is used to examine the 

 problem of stock-recruitment param- 

 eter uncertainty from a Bayesian 

 perspective. Probabilities associated 

 with different parameter values are 

 used to weight the losses (i.e., oppor- 

 tunity costs to society) associated 

 with any given fishing mortality 

 rate. By choosing appropriate forms 

 for the loss and probability density 

 functions, the model is shown to re- 

 sult in an analytic solution. Because 

 this solution gives the fishing mor- 

 tality rate that maximizes the ex- 

 pected value of the logarithm of sus- 

 tainable yield, it is denoted Fmelsy- 

 The solution is a monotone-decreas- 

 ing function of parameter uncertain- 

 ty, converging on the fishing mortal- 

 ity rate corresponding to maximum 

 sustainable yield as the degree of 

 uncertainty approaches zero. As an 

 empirical illustration, the model is 

 applied to the eastern Bering Sea 

 stock of rock sole Pleuronectes bi- 

 lineatus. 



A Bayesian approach to management 

 advice when stocl<-recruitment 

 parameters are uncertain 



Grant G. Thompson 



Resource Ecology and Fisheries Management Division 



Alaska Fisheries Science Center, National Marine Fisheries Service, NOAA 



7600 Sand Point Way NE, Seattle, Washington 981 15-0070 



Manuscript accepted 4 June 1992. 

 Fishery Bulletin, U.S. 90:561-573 (1992). 



Exploiting a stock at the fishing mor- 

 tality rate (F) associated with max- 

 imum sustainable yield (MSY) is a 

 common fishery management strate- 

 gy. For the most part, three simple 

 propositions are sufficient to justify 

 this strategy: (1) The stock exhibits 

 a sustainable yield determined by the 

 fishing mortality rate, (2) more sus- 

 tainable yield is always preferable to 

 less, and (3) the parameters underly- 

 ing the stock's dynamics are known 

 with certainty. However, parameters 

 governing stock dynamics are typi- 

 cally not known with certainty, and 

 in such cases it is possible to demon- 

 strate that the appropriate F value 

 may be less than the value corre- 

 sponding to MSY (Fmsy)- 



The approach to be used in this 

 demonstration is taken from Baye- 

 sian decision theory (e.g., Raiffa 

 1968, DeGroot 1970). Early applica- 

 tions of Bayesian theory to fisheries 

 problems were presented by Roths- 

 child (1972), Lord (1973, 1976), 

 Walters (1975), and Walters and Hil- 

 bom (1976). Of the many more recent 

 applications, those presented by Lud- 

 wig and Walters (1982), Clark et al. 

 (1985), and Walters and Ludwig 

 (1987) bear most closely on the pres- 

 ent study. 



For simplicity, it will be assumed 

 here that stock dynamics are deter- 

 ministic but governed by parameters 

 which may be imprecisely estimated. 

 This approach is distinct from the 

 more common one of assuming that 

 stock dynamics are the product of a 

 deterministic system (with param- 



eter values given and fixed) modified 

 by a random error term. Important 

 early examples of the latter approach 

 include Ricker (1958), Larkin and 

 Ricker (1964), and Tautz et al. (1969). 

 Ludwig and Walters (1982) and 

 Mangel and Clark (1983) incorporate 

 both approaches in a systematic fash- 

 ion which makes the distinction espe- 

 cially clear. 



The basic model 



Thompson (1992) developed a simple 

 dynamic pool model which can be 

 solved explicitly for Fmsy- Ii terms 

 of biomass per recruit, the model is 

 basically that of Hulme et al. (1947); 

 thus, body weight is taken to be a 

 linear function of age, with intercept 

 ao . The main departure from Hulme 

 et al. is that biomass at recruitment 

 age ar is taken to be proportional to 

 stock biomass raised to a power q 

 (Cushing 1971). With these specifica- 

 tions, sustainable jdeld Y(F) can be 

 written 



Y(F) 



(1) 



l-HK"-fF' 

 (1 + F')2 



1 



l-q 



where M is the instantaneous rate of 

 natural mortality, F' = F/M, p is the 

 proportionality term in the Cushing 

 stock-recruitment relationship, and 

 K" = l/[M(ar - ao)] (which can be in- 

 terpreted in this model as the pristine 

 ratio of growth to recruitment). The 



561 



