AbStr3Ct.— A simple dynamic 

 pool model (the "base model"), de- 

 fined by a linear weight-at-age rela- 

 tionship and a Gushing (convex 

 power) stock-recruitment relation- 

 ship, results in an explicit solution 

 for the fishing mortality rate corre- 

 sponding to maximum sustainable 

 yield F MSY . This solution's sensitivity 

 can be examined by comparing it to 

 solutions derived under alternative 

 model specifications. Four such 

 modifications are considered here: 1) 

 replacing the Gushing stock-recruit- 

 ment equation with an equation of 

 the Beverton-Holt form; 2) general- 

 izing from linear growth to a flexible 

 form of von Bertalanffy growth; 3) 

 allowing the ages of recruitment to 

 the fishery a and the mature stock 

 a m to diverge; and 4) allowing for a 

 finite maximum age in the stock. Ex- 

 act polynomial solutions for F MSY are 

 derived for each specification (except 

 the fourth), and the potential bias 

 introduced by use of the base model 

 is examined for each. In all cases, 

 the solution to the base model is 

 within 10% of the solution to the al- 

 ternative model under a range of pa- 

 rameter values. 



Variations on a simple dynamic 

 pool model 



Grant G. Thompson 



Resource Ecology and Fisheries Management Division 



Alaska Fisheries Science Center, National Marine Fisheries Service, NOAA 



7600 Sand Point Way NE., Seattle. WA 98 1 I 5-0070 



Manuscript accepted 24 June 1993. 

 Fishery Bulletin 91:718-731 ( 1993). 



Some fishery models can be solved 

 analytically (i.e., by mathematical 

 manipulation of the equations), while 

 others can be solved only numerically 

 (i.e., by the brute force of computer 

 simulation). One advantage of ana- 

 lytic models is that the generality of 

 their solutions is more straightfor- 

 wardly addressed. For example, it is 

 easy to show that the stock size as- 

 sociated with maximum sustainable 

 yield (MSY) in a Schaefer (1954) 

 model is always one-half the pristine 

 stock size; this property follows di- 

 rectly from the assumption of logis- 

 tic growth upon which the model is 

 based. Such generality is more diffi- 

 cult to demonstrate for simulation 

 models, however. For instance, if a 

 particular simulation showed that 

 MSY was obtained at a stock size 

 equal to one-half the pristine level, 

 there would be no way to tell whether 

 this result was general, except by re- 

 peated trial and error with different 

 values for the input parameters. 



Another example of an analytic so- 

 lution is the one obtained by Thomp- 

 son (1992) for the fishing mortality 

 rate at MSY (F Msy ) in his simple dy- 

 namic pool model. Because this solu- 

 tion is an analytic one, it is com- 

 pletely general in the sense that it 

 will hold whenever the underlying as- 

 sumptions of the model hold, regard- 

 less of parameter values. Of course, 

 the underlying assumptions may not 

 hold in a particular instance, which 

 raises the question: How sensitive is 

 the solution to those assumptions? 

 The purpose of this paper is thus to 

 examine the sensitivity of Thomp- 



son's (1992) solution relative to the 

 underlying assumptions of that 

 model. This will be accomplished by 

 developing four reasonable modifica- 

 tions to the base model proposed by 

 Thompson and by examining the 

 range of errors that might likely be 

 encountered if the base model were 

 employed in situations where one of 

 the modifications would have been 

 more appropriate. 



Review of the base model 



Thompson ( 1992, see also Jensen, 

 1973) defined a simple dynamic pool 

 model as one that reflects the follow- 

 ing assumptions: 1) cohort dynamics 

 are of continuous-time form; 2) vital 

 rates are constant with respect to 

 time and age; 3) fish mature and re- 

 cruit to the fishery continuously and 

 at the same invariant ("knife-edge") 

 age; 4) mean body weight at age is 

 determined by age alone; 5) the stock 

 (or population) consists of the pool of 

 recruited individuals; 6) maximum 

 age is infinite; 7) the stock is in an 

 equilibrium state determined by the 

 fishing mortality rate; and 8) recruit- 

 ment is determined by stock biomass 

 alone. Within the framework pro- 

 vided by these assumptions, particu- 

 lar models are distinguished by the 

 forms assigned to the weight-at-age 

 and stock-recruitment functions. 



As an example, Thompson (1992) 

 developed a particular simple dy- 

 namic pool model than can be solved 

 explicitly for F MSY . In terms of biom- 

 ass per recruit, the model is basically 



718 



