Thompson Management advice from a simple dynamic pool model 



559 



Robustness of the rules of thumb 



Neither Rule I nor Rule II is particularly robust when 

 applied to simple dynamic pool models in general or 

 to the model developed here in particular. Rule I can 

 drastically over- or underestimate the true relationship 

 between Fmsy and M. When q exceeds 0.5, Rule I con- 

 sistently overestimates the ratio between Fmsy and M, 

 whereas when q is less than 0.5, the ratio can range 

 both well above and well below the value suggested by 

 Rule I. 



Although these results do not provide much theoret- 

 ical support for Rule I, it is still possible that Rule I 

 holds as an empirical generalization (it turned out to 

 be fairly close in the case of eastern Bering Sea rock 

 sole, for example). If Rule I does hold as an empirical 

 generalization, Equation (33) indicates that this implies 

 an inverse relationship between the relative importance 

 of growth in pristine production (K") and the degree 

 of density-dependence in the stock-recruitment rela- 

 tionship (q). Further work is necessary to see if such 

 an inverse relationship is supported on the basis of life 

 history or other theory. 



Rule II consistently overestimates the ratio between 

 B(Fmsy) and B(0) in the model presented here (Eq. 

 38, Fig. 2). In the case of eastern Bering Sea rock sole. 

 Rule II was off by 51%. The problem with Rule II stems 

 from the "diminishing returns" nature of the relation- 

 ship between F and B(F), wherein successive increases 

 in F result in less and less of an impact on biomass. 

 Rule II, on the other hand, was inspired by the Schaefer 

 (1954) model, in which the relationship between F and 

 B(F) is linear (i.e., it exhibits constant returns to scale). 



Interestingly, the upper asymptote displayed in Fig- 

 ure 2 corresponds exactly to the asymptote observed 

 in a pair of surplus production models proposed by Pella 

 and Tomlinson (1969, reparametrized by Fletcher 1978) 

 and Fowler (1981), models that are conceptually very 

 different from the one presented here. Mathematical- 

 ly, the isomorphism stems from the fact that all three 

 models involve functions that raise a parameter x to 

 an exponent of the form 1/(1 -x). The fact that this 

 result can be obtained from both surplus production 

 and dynamic pool models indicates that it may be 

 worthy of further investigation. 



Since the rule of thumb setting MSY/MB(0) equal 

 to 0.5 was derived by multiplying Rules I and II, it is 

 affected by the upward bias inherent in Rule II. This 

 is reflected in the eastern Bering Sea rock sole ex- 

 ample, where the estimated value for the MSY/MB(0) 

 ratio was only 0.216. It appears that the "MSY/MB(0) 

 rule" can be a good approximation only when Rule I 

 results in a major underestimate, which in the context 

 of the model developed here requires two things: (1) 

 Recruitment must be relatively independent of stock 



size, and (2) pristine production must be relatively 

 dependent on recruitment (Fig. 3). Another conse- 

 quence of this relationship is that Rule I can never hold 

 when MSY/MB(0) = 0.5, and vice-versa. This conclusion 

 stands in stark contrast to the traditional view which 

 holds that the MSY/MB(0) rule derives from Rule I. 

 Instead, it seems more likely that the two are mutual- 

 ly exclusive, at least in the context of simple dynamic 

 pool models. 



Acknowledgments 



I would like to thank James Balsiger, Nicholas Bax, 

 Roderick Hobbs, Daniel Kimura, Richard Methot, and 

 Thomas Wilderbuer of the Alaska Fisheries Science 

 Center for reviewing all or portions of this paper in 

 various stages of development. Comments provided by 

 Ian Fletcher of the Great Salt Bay Experimental Sta- 

 tion were especially helpful. Three anonymous review- 

 ers also supplied constructive suggestions. 



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