FOX: FITTING THE GENERALIZED STOCK PRODUCTION MODEL 



Table 8. — Empirical and estimated parameters for the five replicated stochastic catch 

 histories using the equilibrium approximation and transition prediction approaches. 



'Estimated, Table 1. 



^Assumed value 



^Program PRODFIT; /( = 4, weighted estimates option. 



"Standard error of the mean. 



^Program GENPROD; KK = 3, DEL = 3. weighted estimates. The program was modified slightly from 

 the version of Pella and Tomllnson (1969) by replacing /opt with Vmax as one of the determining 

 parameters to allow fittmg the case where rfi = (i.e. ?opt = =c at m =0). Identical solutions were 

 obtained for the remaining three cases with either version. 



random and with constant expectation and vari- 

 ance), the observed catch being within 20% of the 

 expected catch with probability 0.95, and the 

 fishing effort being known without error. The 

 maximum error for the equilibrium approxima- 

 tion approach was +54% (replicate 2) and for the 

 transition prediction approach was +67% (repli- 

 cate 5). The problem with these maximum errors 

 (as well as an additional replicate of the transi- 

 tion prediction approach) was estimating m as 

 0.0, where Ymax occurs at infinite fishing effort. It 

 is not unreasonable, however, to obtain m = 0.0 

 since the data series is so short and the best value 

 for m is about 0.60. Considering these results and 

 the true relationship between yield and effort 

 (Figure 2) it would be prudent to adopt an alter- 

 native m estimation strategy for short data series. 



Alternative strategies which could be adopted 

 for short data series are 1) to consistently assume 

 one of the special cases of the generalized stock 

 production model, either the logistic form (m = 2) 

 or the Gompertz form (m -^ 1), or 2) fit both special 

 cases and select the one with the least sum of 

 squared errors. Table 9 presents the parameters 

 estimated by the two approaches through fixing 

 the value for m at 1 (actually 1.001) and 2. For 

 comparative purposes, the results of these alter- 

 native strategies are summarized in Table 10. Fix- 

 ing m at 1 or 2 resulted in average estimates of 

 ^max nearer the empirical value with less vari- 

 ability than obtained by allowing m to be freely 

 estimated for both the equilibrium approximation 



and transition prediction approaches. The empiri- 

 cal value of m is 0.6; hence assuming m ^ 1 

 produced estimates nearer the empirical value of 

 ^max than assuming m — 2. For any given data set, 

 however, one could not determine a priori which 

 value of m to assume. The strategy of fitting both 

 m ^^ 1 and m = 2 and then selecting that which 

 provided the least-squares parameter estimates 

 worked very well in comparison with freely es- 

 timating m under three criteria: 1) more accurate 

 average estimate, 2) smaller average percentage 

 error, and 3) smaller maximum overestimate. 

 Comparing the equilibrium approximation and 

 transition prediction approaches with the same 

 three criteria reveals that the equilibrium approx- 

 imation approach was superior [ 1) 0.5% vs. 5.2%, 

 2) 3.6% vs. 8.5%, and 3) 3.6% vs. 18.4%)]. 



DISCUSSION 



The simple, illustrative calculations on the 

 simulated pandalid shrimp population, of course, 

 did not determine which of the approaches was 

 better for general use in fitting the generalized 

 stock production model. However, some additional 

 guidance can be gained through examining some 

 of their relative weaknesses with regard to the 

 number of data points and the number of 

 parameters they require. 



The moving average of fishing effort in the 

 equilibrium approximation approach results in 

 the exclusion of points at the beginning of the data 



33 



