Thompson: Management advice from a simple dynamic pool model 



557 



When pristine growth and recruitment are exactly 

 balanced (K" = 1), Equation (32) reduces to 



F' 



MSY 



(37) 



In the case of Equation (34), Rule I is always an 

 underestimate (i.e., F'^sy is always greater than 1.0). 

 In the case of Equation (35), Rule I is an underestimate 

 whenever q<0.5 and an overestimate whenever q>0.5. 

 In the case of Equation (36), Rule I is always an over- 

 estimate, except in the limiting case where q = 0. In the 

 case of Equation (37), Rule I is an underestimate 

 whenever q< 1/3 and an overestimate whenever q> 1/3. 



Biomass at MSY relative to pristine biomass 



Substituting M + Fmsy for Z in Equation (24) gives 

 B(Fmsy)- Likewise, substituting M for Z in Equation 

 (24) gives B(0). Forming a ratio from these two bio- 

 masses gives 



B(Fmsy) 

 B(0) 



K" + F' 



MSY 



1 



1 

 1-q 



(K" + 1)(F'msy + 1)' 



(38) 



where F'msy is given by Equation (32). 



Equation (38) is illustrated in Figure 2. Note that all 

 of the curves in Figure 2 exhibit the same upper bound 

 (1/e, about 0.37), which is reached in the limit as q ap- 

 proaches 1.0. Thus, Rule II always overestimates Equa- 

 tion (38) by a minimum of about 36%. At values of 

 q>0.5, the biomass ratio is always greater than 0.25, 

 but at lower values of q the ratio can be much smaller. 



Multiplying Equations (32) and (38) gives the ratio 

 MSY/MB(0), which is plotted in Figure 3. This ratio 

 describes a stock's maximum sustainable fishery- 

 induced losses as a proportion of its pristine losses. 

 Alverson and Pereyra (1969) suggested that the MSY/ 

 MB(0) ratio should equal 0.5, a figure obtained by 

 multiplying Rules I and II together. Note that this sug- 

 gestion errs on the high side whenever K" exceeds 1.0, 

 as well as whenever q exceeds ~0.23. 



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). 



B(Fmsy)/B(0) 



0.4 06 



Recruitment Parameter q 



Figure 2 



Ratio of B(Fj,sy) to B(0). For any given K' value, the ratio 

 increases toward a value of 1/e as q approaches 1.0. Higher 

 K" values result in higher values of the ratio, reaching an upper 

 limit as K" approaches infinity. 



MSY/MB(0) 



0.4 0.6 



Recruitment Parameter q 



Figure 3 



Ratio of MSY to the product of M and B(0). For a given K" 

 value, the ratio decreases toward a value of zero as q ap- 

 proaches 1.0. Higher K" values result in lower values of the 

 ratio, reaching a lower limit as K" approaches infinity. 



The parameters to be estimated are q and K". The 

 parameter q can be determined from data on stock 

 biomass and recruitment. Trawl survey estimates of 

 rock sole stock biomass are available for the years 

 1979-88 (Walters and Wilderbuer 1988). In addition, 

 age composition of the stock has been determined for 

 the years 1979-87. In order to obtain an estimate of 

 age composition for 1988, the "iterated age-length 

 key" approach of Kimura and Chikuni (1987) was ap- 

 plied to the 1986 age-length key and the 1988 length- 



