FISHERY BULLETIN: VOL. 76. NO. 3 



has any value on the interval 0<n<l, then 

 steady-state CPUE will exist for all magnitudes of 

 effort (save/" = ^). Trajectories of Equation ( 14) are 

 similar to those of Equation ( 12), and any graph of 

 (12) will represent the corresponding graph of (14) 

 (CPUE as a function of fishing intensity) with the 

 substitution of U for 5^=, Pt ^or F, and the ratio 

 ym/U^ for ym/By^. 



DISCUSSION 



For all its adaptability, the Pella-Tomlinson 

 system has serious, inherent limitations and it 

 cannot be viewed as a perfectly generalized model 

 of exploitation-productivity relationships. It is, 

 instead, a nonlinear, first-order, wholly empirical 

 system of open degree that admits of a convenient 

 flexibility in a minimum number of terms. Prop- 

 erly regarded, a particularization of the system 

 will accommodate an arbitrary prototype to the 

 extent that the system graphs might be geometri- 

 cally accommodating to the data. 



Experience with the model has shown that un- 

 realistic estimates of coefficients are likely to 

 occur when the data lie in confined or badly scat- 

 tered patterns over ranges of effort and yield. The 

 tendency to unrealistic estimates arises from the 

 conflict between graph curvature, as controlled by 

 exponent n, and the coupling of n with the 

 coefficients of the system. Heretofore, the exact 

 relationships between exponent n and the man- 

 agement components have been obscured by the 

 conventional casting of the system. But with the 

 independent parameters and the restructured 

 equations, much of the parametric uncertainty as- 



sociated with previous statistical treatments can 

 be circumvented. As we have seen, maximum sus- 

 tainable yield m bears no essential relationship to 

 exponent n , and m may be wholly separated from n 

 in all the system equations. And despite the fact 

 that parameters m, p, By: share no interdepen- 

 dence (any one may be varied without change in 

 the value of the others), the parametric ratiop/fioc 

 determines n in the relationship (4). But n in turn 

 prescribes the curvature ( hence the fit) of every 

 graph of the system. As indicated by Figure 3, 

 exponent n exhibits a dismaying sensitivity to 

 perturbations in ratio p/By. The variational re- 

 sponse in n, for a perturbation of 10^ inp/Boc, is of 

 the order of n near n = 1, and the instability 

 increases asp/fi.^ -► 1. Therefore, when an esti- 

 mation procedure depends solely on a general 

 curve-fitting statistic, poor parameter estimates 

 are likely to follow, owing to stochastic displace- 

 ment of datum points at biomass levels remote 

 from locationsp andBx- In the article that follows, 

 Rivard and Bledsoe ( 1978) address such problems 

 directly and their work illustrates certain advan- 

 tages of the restructuring treated here. 



LITERATURE CITED 



Fletcher, R. I. 



1975. A general solution for the complete Richards func- 

 tion. Math. Biosci. 27:349-360. 



1978. Time-dependent solutions and efficient parameters 

 for stock-production models. Fish. Bull., U.S. 76:477- 

 488. 

 FOX, W. W., JR. 



1971. Random variability and parameter estimation for 

 the generalized production model. Fish. Bull., U.S. 

 69:569-580. 



Figure 3. — Graph of the relationship between parameter ratiop/B^; and exponent n of the Pella-Tomlinson system, which 



indicates the slow convergence of the ratio for increasing n. 



520 



