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Fishery Bulletin 91(4), 1993 



which is the analogue to Equation 5 for the Beverton- 

 Holt recruitment case. 



Polynomial solution 



Multiplying Equation 10 through by MF', differentiat- 

 ing with respect to F\ and setting the resulting ex- 

 pression equal to zero gives the following polynomial 

 solution for F' MSY : 



F MS /+3F', ISY 2 +[3+(K"-l)Q'\F', s ^l-{K"+l)Q'=0, (11) 



where Q' = Q/M. 



It is possible to solve Equation 11 explicitly for F' MSY . 

 For the case where K">1, 



[«fa-«fr J H] 



For the case where K"=l, 



(12) 



Q' = 



(F' USY +lf 



Fi S y+l-(F' MSY -l)K" 



115) 



Another difference between this model and the base 

 model is that here F' MSY reaches zero at Q'=1/(K"+1), 

 whereas F' MSY in the base model does not reach zero 

 until q=l. Still another difference is that here the up- 

 per limit to the ratio B{F' VSY )/B(0) is 0.5, contrasted 

 with 1/e in the base model. In both models, this limit 

 is reached as F' MSY approaches zero. 



Finally, the behavior of this modification differs from 

 that of the base model in that extinction is possible 

 here, owing to the Beverton-Holt curve's finite slope at 

 the origin. Extinction occurs here at 



FL 



Q' + ^(4K" + Q')Q' 



-1. 



(16) 



The relative fishing mortality rate described by Equa- 

 tion 16 need not be unrealistically high. For example, 

 it will be less than F' max whenever the following rela- 

 tionship holds: 



Q'< 



4A'" 



K" 



(17) 



FW=(2Q') W -1. 



(13) 



For the case where K"<1 

 2 



r MSY 



^pf^cos ||) COS' [(gj~) 

 \<l-ff">Q'J J _1 • 



(14) 



The parameter Q' functions inversely to q in the 

 sense that Equations 11-14 reduce to Equation 7 as 

 Q' approaches infinity (the F'„ mx case), whereas Equa- 

 tion 6 does so as q approaches zero. As Q' increases, 

 F' MSY increases monotonically, whereas F' MSY decreases 

 with increasing q in the base model. 



As with the base model, Equations 11-14 contain 

 F'msy = 1 as a special case. Here this is obtained when 

 Q' = 4 or when K" approaches infinity. This contrasts 

 somewhat with the base model, where keeping F' MSY at 

 a constant value of 1.0 required an inverse relation- 

 ship between q and K". However, it should be pointed 

 out that F' MSY = 1 is a very special case in the Beverton- 

 Holt form of the model, since this turns out to be the 

 only constant value of F' MSY that does not imply some 

 sort of relationship between Q' and K". In fact, a di- 

 rect relationship between Q' and K" is required for all 

 constant values ofF' MSY > 1, as described below: 



Bias resulting from the assumption of 

 Cushing recruitment 



Assuming that the stock-recruitment relationship follows 

 the Cushing form when it actually follows a Beverton- 

 Holt form can lead to a biased estimate of F MSY . To com- 

 pare stock-recruitment curves, Kimura (1988) observed 

 that a two-parameter function can be defined by any 

 two points on the curve. In his example, Kimura used 

 hypothetical stock-recruitment "observations" at the pris- 

 tine biomass level and at one-half the pristine biomass 

 level. Kimura conjectured that recruitment might be 

 reduced to about 90% of the pristine level when biom- 

 ass has been reduced by 50% relative to its own pris- 

 tine level, a suggestion which has been endorsed by 

 others (e.g., Clark, 1991). Given the other parameters 

 used in his example, Kimura found that the F MSY value 

 under a Beverton-Holt stock-recruitment relationship 

 was much less than the value under a Cushing rela- 

 tionship fit to the same two stock-recruitment points. 



However, there is no reason to believe a priori that 

 a Cushing relationship is necessarily less conservative 

 than a Beverton-Holt relationship in terms of its asso- 

 ciated F MSY value. Note that Equation 6 can be solved 

 explicitly for q as a function of K" and F' MSY as follows: 



K"+1-(K'-1)F' MSY 

 (K" + 1+F' MSY W MSY +1) 



(18) 



