556 



Fishery Bulletin 90(3). 1992 



Dividing Equation (28) through by b(F, ar) gives the 

 ratio between the two components of stock production, 

 i.e., growth and recruitment: 



1 



G(F) ^ 



b(F, a,) Z(ar-ao)' 



(29) 



In the case of a pristine stock, Equation (29) reduces 



to 



G(0) 



b(0, ar) M(ar-ao) 



= K". 



(30) 



In other words, K" is simply the pristine ratio of 

 growth to recruitment. At values of K">1 pristine pro- 

 duction is dominated by growth, while at K" = 1 the two 

 components of pristine production are equal, and at 

 values of K"<1 pristine production is dominated by 

 recruitment. 



Fishing mortality at maximum sustainable yield 



Differentiating Equation (25) with respect to F and setting the resulting expression equal to zero gives the follow- 

 ing equation for F^sy • 



q+1 



. ar - ao , 



\+M + ] 



q+1 - (6q-2)M 



-I- + M- 



^T — ao/ \ ar-ao / 



MSY 



M. 



2q 



(31) 



Using F' to denote the ratio F/M, Equation (31) can be simplified via Equation (26) to 



- (q-t-1) K" -h 1 -^ ^(q-Hl)^ K"2 -t- (6q-2) K" -h 1 



F'mSY = T 1- 



2q 



(32) 



Figure 1 illustrates the behavior of F'msy as a func- 

 tion of q for four values of K" (0, 1, 3, and °°). Note that 

 F'msy can deviate substantially from the value of 1.0 

 suggested by Rule I. The locus of parameter values for 

 which Rule I holds under Equation (32) is 



q = 



1 



K"-(-2 



(33) 



implying that q must be less than 0.5 in order for Rule 

 I to hold. 



When q=l. Equation (32) falls to zero. As q ap- 

 proaches zero, Equation (32) approaches an upper limit 

 F'max defined by 



K" + l 



K"-l' 



(34) 



The limits of Equation (32) as K" approaches zero and 

 infinity are, respectively. 



and 



1-q 



hm Fmsy = 



K"-(i q 



1-q 



hm F MSY = • 



K^<» 1-i-q 



(35) 



(36) 



