554 



Fishery Bulletin 90(3). 1992 



numbers and biomass, respectively) is given by 



N(0) = 



n(0, a^) 



(9) 



and 



B(0) 



W(0) n(0, ar 

 M 



(10) 



Inconsistency of Rules I and II 



Tlie argument of Francis (1974) can be generalized to 

 address more fully the compatibility of Rules I and II. 

 The method to be used is as follows: First, it will be 

 shown that if Rules I and II were to hold simultaneously 

 with the properties of simple dynamic pool models, 

 these rules would imply a particular result. It will then 

 be shown that this result is incompatible with a major 

 subset of the family of simple dynamic pool models, 

 thus proving that Rules I and II are also incompatible 

 with this subset. 

 Rule II and Equation (10) imply 



B(Fmsy) = 



W(0) n(0, a,) 

 2M 



(11) 



b(FMSY. ar) 

 b(0, a,) 



r(B(FMSY)) 

 r(B(0)) 



(15) 



Now let the discussion be restricted to models in 

 which the first derivative of the stock-recruitment rela- 

 tionship is uniformly nonnegative. In such cases. Equa- 

 tion (15) indicates that the left-hand side of Equation 

 (14) is less than or equal to 1 if equilibrium stock 

 biomass decreases as a function of F [i.e., if B(Fmsy) 

 <B(0), then r(B(FMSY))<i'(B(0))]. To examine the 

 conditions under which this occurs, let Equation (8) be 

 rewritten 



B(F) 



W(F) r(B(F)) 



w(ar) Z 

 Equation (16) can be differentiated as follows: 



(16) 



dB(F) 

 dF 



r(B(F)) 



/dW(F)', 

 Zi-^] - W(F) 



w(ar) Z-W(F) 



dr(F(F)) 

 dB(F) 



(17) 



Equation (8) implies that B(Fmsy) must also con- 

 form to 



B(Fmsy) = 



W(FMSY)n(FMSY,a,) 



Fmsy+M 



(12) 



Solving Equations (11) and (12) for F^sy gives 



The numerator in Equation (17) is negative whenever 

 dW(F)/dF<0, which is easily shown to be true when- 

 ever w(a) is monotone increasing, a characteristic 

 typical of all commonly used growth functions (Schnute 

 1981). 



Thus, it follows that dB(F)/dF will likewise be nega- 

 tive whenever the denominator in Equation (17) is 

 positive; that is, whenever 



f2W(FMsv) n(F„.., a,) _ > 



W(0) n(0, a,) 



w(a,)Z ^ dr(F(F)) 



W(F) 



dB(F) 



(18) 



Next, Rule I and Equation (13) imply 



By Equation (16), the left-hand side of (18) can be 

 rewritten as the ratio of r(B(F)) to B(F), giving 



n(FMSY. ar) 



W(0) 



(14) 



n(0, a,) W(Fmsy) 



The left-hand side of Equation (14) can be rewritten 



n(FMSY, ar) w(ar) n(FMSY. ar) 



n(0, ar) 



w(ar) n(0, ar) 



r(B(F)) ^ dr(B(F)) 

 B(F) dB(F) 



(19) 



Given that the discussion has been restricted to 

 models with stock-recruitment relationships that are 

 nondecreasing (nonnegative first derivative), a suffi- 

 cient condition for Equation (19) to hold is for dr(B 

 (F))/dB(F) to be nonincreasing (nonpositive second 



