568 



Fishery Bulletin 90(3). 1992 



where s% is the estimated variance of q given by 



(33) 



For the present application, the solution given by 

 Press (1989) needs to be modified in only one respect. 

 His suggested form for g] (q) implies a uniform distri- 

 bution over the entire real line, whereas here P(q) has 

 been specified a priori to be zero for all values less than 

 or greater than 1. Given Equations (30) and (31), this 

 implies that the suggested uniform shape for gi(c|) 

 should be truncated outside the range to 1. This in 

 turn implies that hi(q | x,y) should also be truncated 

 outside the range to 1 (and rescaled appropriately). 



Strictly speaking, then, P(q) follows a truncated t 

 distribution in this approach, rather than the hypothe- 

 sized beta. However, a beta distribution can be made 

 to approximate the truncated t by solving for m and 

 v as follows: 



/: 



q hi(q I x,y) dq 



m = 



; 



(34) 



hi(q I x,y) dq 



and 



;: 



(q-m)2 hi(q | x,y) dq 



V = 



r hi(q I x,y) dq 







(35) 



Applying the model to rock sole 



As an illustration of the approach suggested above, the 

 model can be applied to the eastern Bering Sea stock 

 of rock sole Pleuronectes bilineatus. This stock is ex- 

 ploited by a multispecies flatfish fishery, and is also the 

 target of an important roe fishery (Walters and Wilder- 

 buer 1988). 



The parameters to be estimated are K", m, and v'. 

 Thompson (1992) estimated K" for this stock at a value 

 of 3.279, and described a set of stock and recruitment 

 data (n = 7) which can be used to estimate m and v'. Fit- 

 ting Equation (29) to these data gives q = 0.235 and 

 s"q = 0.114 (Fig. 6). Substituting these parameters into 

 Equations (34) and (35) gives m = 0.369 and v = 0.057, 

 with v' = 0.243. The relationship between the truncated 

 t distribution defined by these values and the beta ap- 

 proximation is shown in Figure 7 (i?- = 0.97). 



With parameter values K" = 3.279, m = 0.369, and 

 v' = 0.243, Equation (20) gives F'melsy = 0.365. Multi- 

 plying through by M (set at 0.2 by Walters and Wilder- 

 buer 1988) gives F'melsy =0.073. Substituting m for 

 q in Equation (2) yields F'msy =0.607, or Fmsy =0.121. 

 This value of Fmsy differs somewhat from the value 

 of 0.176 given by Thompson (1992), which was based 

 on the least-squares estimate of q (q) instead of the 



Recruitment biomass (thousands of t) 



1 2 0,3 0-4 0.5 6 7 8 9 10 



stock biomass (millions of t) 



Figure 6 



Stock-recruitment data and curve for eastern Bering Sea rock 

 sole Pleuronectes bilineatus. Age-3 biomass (lagged 3 yr) is 

 plotted against stock biomass for the years 1979-88. The curve 

 is the least-squares fit. 



Probability density 



t distribution 



0.1 2 3 4 5 6 7 8 0.9 1.0 



Stock-recruitment exponent q 



Figure 7 



Comparison of truncated t and beta pdfs for the stock- 

 recruitment exponent q in the eastern Bering Sea rock sole 

 PleuroTwctes bilineatus example. 



