ON THE RESTRUCTURING OF THE PELLA-TOMLINSON SYSTEM 



R. Ian Fletcher* 



ABSTRACT 



The time-dependent analysis of an earlier work is extended to the equilibrium case of the Pella- 

 Tomlinson system, and the relationships between the equilibrium and nonequilibrium versions of the 

 restructured system are developed. The dual formulations of the conventional analysis are avoided and 

 maximum sustainable yield is separated from the indeterminacy of the system. All arbitrary 

 coefficients are eliminated and the management components incorporated directly into the system 

 equations. The source of the statistical degeneracy in the model is revealed and explicitly formulated, 

 and in the companion article by D. Rivard and L. J. Bledsoe (this issue of the Fishery Bulletin) the 

 restructured model is treated by a new statistical method that subdues the estimation failures 

 associated with past treatments of the Pella-Tomlinson system. 



Because the equilibrium versions of all stock- 

 production models follow from steady-state inte- 

 grations, the strategy of fishery regulation be- 

 comes a strategy of accommodation, so to speak, as 

 determined by a pattern of balanced model states 

 where removals just equal the productivities 

 otherwise surplus to the maintenance needs of the 

 stock. Population status usually enters the process 

 in the simple, robust form of integrated numbers 

 or biomass, and the removals of fishing constitute 

 direct fractions of the whole fishable stock without 

 reference to age or weight distributions. Since the 

 appearance of Schaefer s work (Schaefer 1954) the 

 strategy has been applied to the management of 

 many fisheries. Schaefer devised a rational, 

 linearized method for estimating the parameters 

 of Graham's equilibrium model (Graham 1935) 

 from the actual nonequilibrium yields and effort 

 expenditures of a fishery, a contribution that is 

 often misunderstood. In applying Schaefer's 

 method or like schemes of synthesis, it is not so 

 much that one hopes to observe a pattern of 

 equilibrium levels in a fishery or even expects 

 them to come about, but rather, by knowing the 

 response history of a stock to various exploitation 

 pressures, one might then be guided by the model 

 in bringing a stock, through a sequence of man- 

 agement actions, into a state where some desired 

 level of sustainable yield most likely abides. The 

 philosophy is widely accepted in fisheries man- 

 agement but its application is often censured. 



'Center for Quantitative Science in Forestry, Fisheries and 

 Wildlife, University of Washington, Seattle, WA 98195. 



either on economic or biological grounds (see, for 

 example, Larkin 1977) 



The exploitation model of Pella and Tomlinson 

 ( 1969), as it is customarily thought of, extends the 

 more "basic" model of Graham from a system of 

 second degree in nonlinearity to a flexible or more 

 "general" system of indeterminate degree. The in- 

 creased flexibility comes into the Pella-Tomlinson 

 model through the addition of a single exponential 

 parameter, but the analytical peculiarities that 

 accompany the improvement often lead to 

 paradoxical ends since the equations of the system 

 then permit the simultaneous generation of good 

 data fits and poor parameter estimates (see the 

 commentary of Ricker 1975:323-326 and the 

 treatments of Fox 1971, 1975). This disturbing 

 trait of the statistical model arises from the 

 conflict between the variable (or parametric) cur- 

 vature of the analytical model and the coupling of 

 that curvature, in the conventional formulations, 

 with all the coefficients of the system. As shown in 

 a prior work (Fletcher 1978), those effects may be 

 separated in the time-dependent analysis by re- 

 structuring the system equations so as to accom- 

 modate directly the critical-point coordinates of 

 the system graphs. In this paper we extend the 

 analysis to the equilibrium version of the Pella- 

 Tomlinson system, and we show the relationships 

 between the equilibrium model and the (restruc- 

 tured) time-dependent equations. 



For a stock of mixed age classes, the most 

 difficult problem in applying any equilibrium 

 model will lie, essentially, in the interpretation of 

 time-dependent transitions between idealized 

 states (however momentary, long-enduring, or 



Manuscript accepted March 1978. 



FISHERY BULLETIN: VOL. 76. NO 3. 1978. 



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