Thompson: Management advice from a simple dynamic pool model 



555 



derivative). Therefore, for all simple dynamic pool 

 models in which r(B(F)) is nondecreasing and dr(B 

 (F))/dB(F) is nonincreasing, the left-hand side of Equa- 

 tion (14) is less than or equal to 1.0. 



Turning to the right-hand side of Equation (14), note 

 that this expression is necessarily greater than 1.0 

 whenever dW(F)/dF<0, a condition which has already 

 been noted to hold whenever w(a) is monotone 

 increasing. 



Summarizing the argument, then, it has been shown 

 that Rules I and II cannot hold simultaneously for any 

 simple dynamic pool model in which the first derivative 

 of the stock-recruitment relationship is uniformly non- 

 negative, the second derivative of the stock-recruit- 

 ment relationship is uniformly nonpositive, and the first 

 derivative of the weight-at-age relationship is uniformly 

 positive. 



Example of a simple dynamic 

 pool model 



Growth, bJomass, recruitment, and yield 



As an alternative to Rules I and II, it is possible to 

 examine the behavior of Fmsy/M and B(Fmsy)''B(0) 

 explicitly for a particular model. The model to be ex- 

 amined here incorporates a linear weight-at-age func- 

 tion (Schnute 1981). Let 



w(a) = w(ar 



a — a0 



a,- — a* I 



(20) 



where ao represents the age intercept. 

 Biomass at age is then 



b(F,a) = 



b(F, a^)(a-ao)e-z(a-ar) 



a,, -ao 



(21) 



The stock-recruitment relationship of Gushing (1971) 

 will be used to complete the model, giving recruitment 

 as a power function of stock size: 



b(F, a,) = pB(F)q, 



(23) 



where p and q are constants, and 0<q<l. In the 

 limiting case of q = 0, recruitment is constant, while in 

 the other limiting case of q= 1, recruitment is propor- 

 tional to biomass. 



The Gushing stock-recruitment relationship has the 

 advantage of rendering Equation (22) explicitly solv- 

 able. Substituting Equation (23) into Equation (22) and 

 rearranging terms gives the following equation for 

 equilibrium stock biomass: 



B(F) = 



1 + 



1 



Z(a,. -ao)/J 



1 



(24) 



Multiplying both sides of Equation (24) by F then 

 gives the equation for yield Y(F) shown below: 



Y(F) 



[P 



1 + 



1 



Z(ar-ao)/- 



l-q 



(25) 



A partitioning of stocl< production 



From this point on, it will prove helpful to make use 

 of a new parameter K", defined as follows: 



K" = 



1 



M(ar-ao) 



(26) 



The parameter K" has a special biological interpreta- 

 tion in the context of the present model. To develop 

 this interpretation, first multiply Equation (22) through 

 by Z, yielding: 



For a given value of b(F, a^), biomass at age can be 

 integrated from a = ar to a = °° to obtain the correspon- 

 ding equilibrium stock size. Equation (21) can be in- 

 tegrated by parts, giving the following expression for 

 equilibrium stock biomass (Hulme et al. 1947): 



B(F) 



(a-ao) e-^'^'-^'r) da 



fb(F, a,)\ 



1 -1- 



1 



Z (a^ - ao ) 



(22) 



Z B(F) = b(F, a,) 1 -h 



Z (a^ - ao ) I 



(27) 



Assuming no immigration or emigration, stock losses 

 due to mortality must equal stock gains due to recruit- 

 ment and growth at equilibrium (Russell 1981). Since 

 the left-hand side of Equation (27) represents losses due 

 to mortality, the right-hand side must equal the sum 

 of equilibrium recruitment and growth. Therefore, 

 Equation (27) can be rearranged to define equilibrium 

 stock growth G(F) as follows: 



G(F) = Z B(F) 



b(F, a^) 



b(F,ar) = — -• (28) 



Z(ar -ao) 



