Thompson Variations on a simple dynamic pool model 



727 



Equation 38 collapses to Equation 5. However, at any 

 positive value of K\ ",„,, Equation 5 will tend to overesti- 

 mate the true value of B(F') to some extent. Conversely, 

 Equation 6 will tend to underestimate the true value 

 of F't/gy. Figure 5 shows loci of -10% bias in Equation 

 6's estimate of F' MSY . Points above and to the left of the 

 curves result in a bias of less than 10% (absolute value). 

 For example, a stock with Af=0.2, o„=-l, and a m „=24 

 would have a K", wx value of 0.2. For such a stock, Equa- 

 tion 6's estimate of F' MSY would be within 10% of the 

 correct value for any value of K ">0.5 so long as q was 

 less than about 0.53. 



Conclusion 



Four modifications to the base model presented 

 by Thompson (1992) have been considered 

 (Beverton-Holt recruitment, generalized von 

 Bertalanffy growth, divergent ages of recruitment 

 and maturity, and finite maximum age). The first 

 three modifications all increase the degree of the 

 polynomial solution for F MSY (Table 2), while the 

 fourth modification renders a polynomial solu- 

 tion impossible. 



In order to make the Cushing stock-recruit- 

 ment form of the model comparable to the 

 Beverton-Holt form, an acceptable approximation 

 can often be made by equating the pristine stock- 

 recruitment points and placing the other (non- 

 zero) intersection of the stock-recruitment curves 

 at a fairly low level (e.g., at 10% of the pristine 

 biomass level or at 50% of the pristine recruit- 

 ment level). 



In order to make the linear growth form of the 

 model comparable to the generalized von 

 Bertalanffy growth form, an acceptable approxi- 

 mation can often be made by equating the weights 

 at recruitment and the pristine biomass-per- 

 recruit ratios. 



When the ages of recruitment to the fishery 

 and to the mature stock diverge sufficiently or 

 when the maximum age in the stock is sufficiently 

 low, the base model can produce a significantly 

 biased estimate of F MSY . Except for the case in 

 which the age of recruitment to the fishery pre- 

 cedes the age of recruitment to the mature stock, 

 though, it is helpful to note that the base model 

 always errs on the conservative side. 



In conclusion, it appears that simple models 

 (at least the base model considered here) may 

 often perform adequately even when the true dy- 

 namics of the system follow more complicated 

 formulae. This tends to confirm the results of 

 studies by Silliman (1971), Roff (1983), and 

 Ludwig and Walters ( 1985), who also found that simple 

 models could perform at least as well as more complex 

 versions in a variety of situations. 



Acknowledgments 



I would like to thank James Balsiger, Nicholas Bax, 

 Roderick Hobbs, Daniel Kimura, and Richard Methot 

 of the Alaska Fisheries Science Center for reviewing 

 all or portions of this paper in various stages of devel- 

 opment. Comments provided by Ian Fletcher of the 

 Great Salt Bay Experimental Station were especially 



