558 



Fishery Bulletin 90(3). 1992 



frequency distribution. Assuming that rock sole recruit 

 at age 3 (Walters and Wilderbuer 1988), these data 

 provide seven years of information on the stock- 

 recruitment relationship. Fitting Eq. (23) to these 

 seven points (assuming lognormal error, Fig. 4) gives 

 q = 0.235. 



The composite parameter K" can be estimated from 

 its constituent parameters SLy, ao, and M (Eq. 26). 

 Walters and Wilderbuer (1988) set a^ = 3 and M = 0.2. 

 The parameter ay can be derived by regressing a line 

 through the mean weights-at-age, as shown in Figure 

 5 (R'~ =0.904). This gives an ao value of 1.475 years, 

 implying a K" value of 3.279. 



With these parameter values, Equation (32) gives 

 F'msy = 0.880, or Fmsy =0-176. This estimate of Fmsy 

 compares favorably with the value of 0.155 that 

 Walters and Wilderbuer (1988) derived from a surplus 

 production model. It is relatively close to (within 12% 

 of) the value indicated by Rule I. 



However, Rule II does not fare so well in this ex- 

 ample. Equation (38) estimates the ratio between 

 B(Fmsy) and B(0) at a value of 0.245, 51% below the 

 value predicted by Rule II. 



Discussion 



The topic of this paper, management advice from a 

 simple dynamic pool model, has been considered from 

 the perspective of how two commonly used rules of 

 thumb compare with simple dynamic pool models in 

 general, and how they compare with one such model 

 in particular. 



Choice of functional forms 



Within the family of simple dynamic pool models, a 

 particular model is defined by its stock-recruitment 

 and growth functions. As Paulik (1973) and Ricker 

 (1979) state, the choice of functional form for these 

 two processes is largely a matter of convenience. The 

 linear growth and Gushing stock-recruitment functions 

 have been chosen for the proposed model, in part 

 because of the tractability they confer. For example, 

 their use permits explicit specification of Fmsy (im- 

 possible in other known examples of simple dynamic 

 pool models, except in the special case where Fmsy = 

 F,„ax). Another advantage is economy of parametriza- 

 tion: only two parameters (K" and q) are required. The 

 main disadvantage is the possibility that the simplicity 

 of these functional forms might ignore critical 

 behaviors. 



The linear growth assumption is probably the more 

 controversial of the two choices. The primary criticism 

 of the linear growth equation is that it implies a con- 



Recruitment Biomass (thousands of t) 



00 0.1 0.2 0.3 04 O.S 06 07 0.8 0.9 1.0 



stock Biomass (millions of t) 



Figure 4 



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. 



stant growth rate, whereas other commonly used func- 

 tions exhibit decreasing growth rates at upper ages 

 (Beverton and Holt 1957), usually manifested in the 

 form of an upper asymptote. In practice, however, the 

 absence of an asymptote may be inconsequential or 

 even preferable (Knight 1968, Ricker 1979) for two 

 reasons: (1) In exploited populations, individuals may 

 only rarely survive to reach the portion of the growth 

 curve where a marked decrease in growth rate would 

 be most discernible; and (2) in functional forms that in- 

 corporate an asymptote, this parameter is often poor- 

 ly estimated, being highly correlated with at least one 

 other parameter in the equation. 



