RANDOM VARIABILITY AND PARAMETER ESTIMATION 

 FOR THE GENERALIZED PRODUCTION MODEL' 



William W. Fox, Jr.^ 



ABSTRACT 



Three alternative statistical models are proposed for estimating the parameters of the generalized pro- 

 duction model by the method of least squares. A stochastic representation of the generalized produc- 

 tion model is consti'ucted and simulation (or the Monte Carlo Method) is employed to infer the effects 

 of random variability on the variation in catch. The use of residuals examination for selecting the 

 appropriate statistical model for least-squares estimation of the generalized production model param- 

 eters is demonstrated for the yellowfin tuna fishery in the eastern tropical Pacific Ocean. In both the 

 simulation and actual fishery, statistical Model 3 — assuming catch residual variance is proportional to 

 the catch squared — best fulfills the assumptions of least-squares theory and should, therefore, provide 

 the best least-square parameter estimates. 



Mathematical models are powerful tools which 

 are being used increasingly in resource man- 

 agement. A knowledge of mathematics allows 

 a resource manager to construct from gathered 

 data a representation of the real system and, 

 coupled with statistical theory, allows estima- 

 tion of the parameters of his model. Then, as 

 is impossible in the real system, a manager may 

 experiment on his model and derive outcomes 

 which aid decisions about management of the 

 real system. Results of model experimentation 

 usually depend greatly on the formulation of the 

 model and to some degi-ee on the accuracy of 

 the parameter estimates. Often precise statisti- 

 cal parameter estimation lags behind mathemati- 

 cal formulation, primarily because many math- 

 ematical models are robust, i.e., decisions are 

 independent of parameter accuracy. This is one 

 reason for the development of detei-ministic 

 rather than stochastic models. However, it 

 seems that it is always desirable to obtain the 

 best possible parameter estimates from the data 

 at hand. 



' Quantitative Science Paper No. 16. A series pre- 

 pared under the general sponsorship of the Quantitative 

 Ecology and Natural Resource Management Program 

 supported by Ford Foundation Grant Number 68-183. 



- Center for Quantitative Science in Forestry, Fish- 

 eries and Wildlife, University of Washington, Seattle, 

 Wash. 98105. 



A simple case of Bernoulli's equation has been 

 suggested as a model for the growth of an or- 

 ganism by Richards (1959), Chapman (1961), 

 and Taylor (1962) 



dx/dt 



Ha-,'" — Kxt 



(1) 



where x, represents either weight or length at 

 time t, and H, K, and m are parameters which 

 may be given some physiological significance. 

 Recently equation (1) has been advanced inde- 

 pendently by Chapman (1967) and Pella and 

 Tomlinson (1969) as a simple model for assess- 

 ing the relation between exploitation and yield 

 (or catch) from a living resource 



dP/dt = HPr — KP, — qfP, for m < 1 



(2) 

 dP/dt = — HP,'" + KP, — qfP, for m > 1 



where P, is the population size (biomass or num- 

 bers), / is the amount of fishing effort, q is the 

 coefficient of catchability, and H, K, and m are 

 parameters. It is assumed that / is constant 

 over the time period that equation (2) is used. 

 Therefore, qf = F, the instantaneous fishing 

 mortality coefficient, and qfP, = C, the catch. 

 Equation (2), referred to herein as the gener- 

 alized production model after Pella and Tomlin- 

 son (1969), includes the logistic model used by 



Manuscript accepted February 1971. 



FISHERY BULLETIN: VOL. 69, NO. 3, 1971. 



569 



