Thompson. Variations on a simple dynamic pool model 



719 



the same as that of Hulme et al. (1947), where body 

 weight is assumed to be a linear function of age (e.g., 

 Richards, 1969): 



w(a) 



\a r -aoJ 



(1) 



where a represents age, a r is the age of recruitment, a u 

 is the age intercept, w(a) represents individual weight 

 at age, and w r is the weight at recruitment. 



For a given recruitment level, stock biomass in the 

 model is given by 



*". me^f) 



(2) 



where M is the instantaneous rate of natural mortal- 

 ity, F'=FIM, B(F') is the equilibrium stock biomass 

 obtained under a relative fishing mortality rate off, 

 b(F',a r ) is the equilibrium biomass at a=a r obtained 

 under a relative fishing mortality rate of F\ and 



K" 



M(a-a n ) 



(3) 



which can be interpreted in this model as the pristine 

 ratio of growth to recruitment (Thompson, 1992). 



Thompson (1992) extended the model described in 

 Equation 2 by incorporating a stock-recruitment rela- 

 tionship of the convex power form suggested by Gushing 

 (1971): 



biF'.a r )=pBlF'F, 



(4) 



where p and q are constants and 0<g<l. In the limit- 

 ing case of g=0, recruitment is constant, while in the 

 other limiting case of q=l, recruitment is proportional 

 to biomass. 



Substituting Equation 4 into Equation 2 and rear- 

 ranging terms gives the following equation for equilib- 

 rium stock biomass: 



*"-[(fr)C-i£rf)I 



(5) 



Multiplying both sides of Equation 5 by MF' then 

 gives the equation for sustainable yield, which is maxi- 

 mized at 



F'msy 



-iq+DK"+l+^l(.q+l) 2 K" 2 +(6q-2)K"+l 



2q 



-1, (6) 



where F' MSY = F MSY /M. 



In the special case where q=0. Equation 6 reduces to 

 the solution forF,' , ( = F, IM): 



F' 



K"+l 

 K'-l 



(7) 



A common rule of thumb is that F' MSY should equal 1 

 (Alverson and Pereyra, 1969; Thompson, 1992). The 

 locus of parameter values for which this rule holds 

 precisely is given by 



1 



q= . (8) 



K" + 2 



Another behavior of interest is the ratio of B(F' MSY ) 

 to B(0). Here, this ratio has a lower limit of (K"-\)l 

 OK" ) (at g=0) and an upper limit of 1/e (at q=l ). 



The base model presented in Equations 1-6 can be 

 modified in a number of ways. Although such modifi- 

 cations may make it more difficult to obtain an ex- 

 plicit solution for F MSY , they may also provide some 

 guidance as to the generality of the base model's be- 

 havior. Four modifications will be considered here: 

 1) replacing the Gushing stock-recruitment equation 

 with an equation of the form suggested by Beverton 

 and Holt ( 1957); 2) generalizing from linear growth to 

 a flexible form of von Bertalanffy (1938) growth; 3) 

 allowing the ages of recruitment to the fishery a f and 

 the mature stock a m to diverge; and 4) allowing for a 

 finite maximum age in the stock. For each modifica- 

 tion (except the fourth), polynomial solutions for F' MSY , 

 F' max , and the locus at which F' MSY = 1 will be derived, 

 and the upper limit to the ratio B(Fj /s ,-)/B(0) will be 

 presented. The potential bias introduced by use of the 

 base model will also be examined graphically for each 

 modification. 



Beverton-Holt recruitment 



The choice of stock-recruitment relationship can have 

 an appreciable impact on the resulting estimate of F MSY 

 (Kimura, 1988). For comparative purposes, the stock- 

 recruitment relationship of Beverton and Holt (1957) 

 can be substituted for the Cushing form used in the 

 base model. It will prove convenient to parametrize 

 the Beverton-Holt equation as follows: 



b(F\a r ) = 



QB(F') 



(9) 



PB(F')+ 1 



where Q represents the slope of the curve at the origin 

 and P represents the ratio between the slope at the 

 origin and the height of the asymptote. 



Substituting Equation 9 into Equation 2 and solv- 

 ing for fi(F') gives 



B(F')= l~ 



(i)( 



Q(l + K" + F') 

 M(l+F') 2 



')■ 



10) 



