FITTING THE GENERALIZED STOCK PRODUCTION MODEL BY 

 LEAST-SQUARES AND EQUILIBRIUM APPROXIMATION ^ 



William W. Fox, Jr.^ 

 ABSTRACT 



A least-squares method for fitting the generalized stock production to fishery catch and fishing effort 

 data which utilizes the equilibrium approximation approach is described. A weighting procedure for 

 providing improved estimates of equilibrium fishing effort and an estimator of the catchability 

 coefficient are developed. A computer program PRODFIT for performing the calculations is presented. 

 The utility and performance of PRODFIT is illustrated with data from a simulated pandalid shrimp 

 population. 



The production model approach to fish stock as- 

 sessment is simply an adaptation of the Lotka- 

 Volterra population equations into the situation 

 of a population exploited by man. The earliest 

 such adaptation was by Graham (1935) in assess- 

 ing the potential production from North Sea fish 

 stocks. The major development of this approach in 

 fisheries management, though, is due to Schaefer 

 (1954, 1957) who initiated it as a management tool 

 for the yellowfin tuna fishery of the eastern tropi- 

 cal Pacific Ocean. While there has been an at- 

 tempt at a detailed extention of the production 

 model approach to multispecies fisheries (Lord 

 1971), the usual application has been on a single 

 species stock. 



Mathematical formulation of the production 

 model begins with the general differential equa- 

 tion 



dPIdt = P,g (P,) - PMO 



(1) 



where P, is the population size at time ^ Ptg (Pf ) is 

 the population production function encompassing 

 the effects of reproduction and natural mortality 

 (and growth in weight if biomass is the population 

 unit), and h (/",) is the fishing mortality coefficient 

 exerted by/", units of fishing effort. Fishing effort 

 is assumed to be standardized from nominal 

 fishing effort such that qf^ = F^ , where F, is the 

 instantaneous coefficient of fishing mortality and 

 <7 is a constant (the catchability coefficient), giving 

 QftPf - dCldt, the rate of catch. At equilibrium, 

 that is dPIdt = 0, the catch rate equals the produc- 



'Adapted, in part, from a Ph.D. dissertation, College of 

 Fisheries, University of Washington, Seattle, WA 98195. 



^Southwest Fisheries Center, National Marine Fisheries Ser- 

 vice, NOAA, P.O. Box 271, La Jolla, CA 92037. 



tion rate such that an equilibrium yield, Y , is 

 obtained 



Y = qfP = PgiP). 



(2) 



The most general assumptions about the form of 

 PfgiPf) are that it should 1) approach zero as P, 

 approaches some environmental capacity, P^ax' 

 and 2) increase to some maximum at a population 

 size smaller than the environmentally limited 

 size. Practically, the function should be simple, 

 since in any case the approach is a gross 

 simplification of population dynamics. The most 

 fiexible, simple function advanced for Ptg (P,) is a 

 simple case of Bernoulli's equation (Chapman 

 1967; Pella and Tomlinson 1969) 



PtgiPt) =HPr-KP, 



(3) 



where H, K, and m are constant parameters.^ 

 Equation (3) includes the logistic function when 

 m = 2 (Schaefer 1954, 1957) and the Gompertz 

 function [K'P^ - H'P.lnPJ as m^l (Fox 1970). 

 Equation (3), hereafter referred to as the 

 generalized stock production model after Pella 

 and Tomlinson (1969), approaches zero at Pmax 

 = {K/H) !'•"' 1' and has a maximum Popt = 



[m^'^'-'^n -P^ax- 



Three equilibrium relationships can be derived 

 by the substitution of Equation (3) in Equation (2) 

 to obtain 



1) Yield and population size 



Y =HP"' - KP, 



2) Population size and fishing effort 



(4) 



^When formulated as in Equation (3), H and K are positive for 

 m < 1, but are negative for m > 1. 



Manuscript accepted May 1974. 



FISHERY BULLETIN: VOL. 73, NO. 1, 1975. 



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