PARAMETER ESTIMATION FOR THE PELLA-TOMLINSON STOCK 

 PRODUCTION MODEL UNDER NONEQUILIBRIUM CONDITIONS 



D. RivARD AND L. J. Bledsoe' 



ABSTRACT 



To estimate the parameters of the Pella-Tomlinson model, as restructured by Fletcher in this issue, we 

 suggest a derivative-free version of the Levenberg-Marquardt algorithm, along with an algorithm that 

 locates starting values for the iterative procedure. The iterative method of Levenberg-Marquardt was 

 applied to two versions of the restructured model: five parameters were estimated in the first version 

 and three in the second, the latter preventing degeneracy of the model to exponential form. We discuss 

 in particular the causes of the degeneracies associated with previous applications of the model. Such 

 faults lie, inherently, with the mathematical indeterminacy of the system equations themselves, so 

 that all nonlinear estimation methods will tend to be inefficient in the absence of external constraints. 

 The effectiveness of the Levenberg-Marquardt method was evaluated by Monte-Carlo simulation. As 

 examples, we analyzed catch-effort data from the yellowfin tuna fishery of the eastern Pacific and 

 catch-effort data from the Pacific halibut fishery (Area 2 of the International Pacific Halibut Commis- 

 sion). 



Parameter estimation has been the greatest 

 source of difficulty in applying the generalized 

 stock-production model to management schemes, 

 and the problem has attracted considerable atten- 

 tion. Pella and Tomlinson ( 1969) fitted the model 

 to the catch-effort history of a fishery under 

 nonequilibrium conditions by means of a search 

 algorithm, and although good graphical fits are 

 generally obtained by that procedure, unreason- 

 able parameter estimates are frequently gener- 

 ated owing to the lack of internal constraints on 

 parameter values (see Ricker 1975, example 13.6). 

 Fox ( 1971) constructed a stochastic representation 

 of the generalized production model and employed 

 simulation to infer the effects of random variabil- 

 ity in catch data. Fox suggested that variation in 

 catch increases with the size of the catch (additive 

 proportional error) and he gave a new formulation 

 of the minimization criterion for the Pella- 

 Tomlinson procedure. Walter (1975) suggested a 

 graphical method for calculating the coefficients of 

 the Graham-Schaefer model. Walter's procedure 

 requires the plotting of catch per effort against 

 effort data and then correcting for disequilibrium 

 of the fishery. Fox (1975) also described a proce- 

 dure for fitting the Pella-Tomlinson model that 



'Center for Quantitative Science in Forestry, Fisheries and 

 Wildlife, University of Washington, Seattle, WA 98195. 



Manuscript accepted March 1978. 



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



requires equilibrium approximations. And finally, 

 Schnute (1977) derived linear and nonlinear 

 methods for finding estimates of the coefficients of 

 the Schaefer model; his method also includes a 

 way of measuring the uncertainty of the esti- 

 mates. 



Fletcher ( 1978b) presented a reparametrization 

 of the generalized production model and explains 

 how the tendency to ill-determined parameter es- 

 timates arises from a conflict between variable 

 graph curvature and its coupling with the 

 coefficients of the system. In this paper, we take 

 advantage of that restructuring and examine the 

 use of a derivative-free version of the 

 Levenberg-Marquardt numerical optimization 

 algorithm, together with a Runge-Kutta differen- 

 tial equation solver, to estimate parameters in 

 Fletcher's differential form of the Pella-Tomlinson 

 model. Estimates of the variability in the 

 coefficients are also provided, and the complete 

 estimation procedure is analyzed by a Monte- 

 Carlo simulation. The estimation problem is 

 finally reformulated to prevent ill determination 

 of the parameters and degeneracy of the model to 

 exponential form. 



MODEL AND NOTATION 



As indicated by Fletcher (1975, 1978b), the 

 generalized production model can be generated by 

 the single differential equation 



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