FOX: FITTING THE GENERALIZED STOCK PRODUCTION MODEL 



the observed and predicted catches. While these 

 two estimation methods are very different in their 

 degree of sophistication, they are fundamentally 

 the same in that both methods utilize the predic- 

 tion of population transitional state changes by 

 the production model. For convenience, this ap- 

 proach will be subsequently referred to as the 

 transition prediction approach. 



Gulland (1961) established a second approach to 

 fitting production models with transitional state 

 data. Gulland's approach estimates the level of 

 fishing effort which, if equilibrium obtained, 

 would produce, on the average, the observed level 

 of catch per unit effort in each year of the fishery. 

 Then the set of paired catches per unit effort and 

 estimated equilibrium fishing effort units are 

 fitted to one of the equilibrium relationships given 

 by, or derived from, Equation (4), (5), or (6). This 

 approach will be referred to subsequently as the 

 equilibrium approximation approach. 



Clearly, the transition prediction and equilib- 

 rium approximation approaches are basically dif- 

 ferent. The transition prediction approach is obvi- 

 ously intimately based upon the transition state 

 population assumptions. On the other hand, the 

 degree to which the equilibrium approximation 

 approach is dependent on these assumptions is 

 unclear. This paper presents a least-squares 

 method and a computer program PRODFIT, 

 which uses the equilibrium approximation ap- 

 proach to estimate the parameters (and indices of 

 their variability) of the generalized stock produc- 

 tion model. A weighting procedure for providing 

 improved estimates of equilibrium fishing effort 

 and an estimator of the catchability coefficient are 

 developed. The utility and performance of com- 

 puter program PRODFIT is illustrated by fitting 

 deterministic and stochastic data from a simu- 

 lated pandalid shrimp population. Some cursory 

 comparisons between the equilibrium approxima- 

 tion and transition prediction approaches are 

 made by repeating the pandalid shrimp simulated 

 data fits with GENPROD, the computer program 

 written by Pella and Tomlinson (1969). 



FITTING METHOD 



The equilibrium approximation approach was 

 first outlined in Gulland (1961), but is more fully 

 explained in Gulland (1969:120). Gulland's 

 method involves relating the annual catch per 

 unit effort in year i, Ui, to the fishing effort aver- 



aged over some number of years, T. Gulland 

 (1961) first defined T as the mean life expectancy 

 of an individual in the fishable population, orZ ~^, 

 where Z is the instantaneous total mortality 

 coefficient and the value of Z "^ is rounded off to 

 the nearest integer. Subsequently, Gulland (1969) 

 defined T as the average fishable duration of a 

 year class (again to the nearest whole year) — he 

 provided the following example: if recruitment is 

 at 4 yr and if most of the catch in year / consists of 4 

 to 9 yr-old fish, then the average fishable duration 

 is about 3 yr so U, would be related to an average 

 off,, fi - 1 and/*! - 2. The general formulation for 

 the averaged fishing effort in year i is 



l = h 2^ 



(7) 



J = I 



r + 1 



A discussion of the rationale for, and performance 

 of, Gulland's averaging method is given by Gul- 

 land (1969:120). 



Weighted Average 

 Fishing Effort Method 



In this paper a different tack is taken which 

 results in approximating equilibrium fishing ef- 

 fort with a weighted average. The catch per unit 

 effort of the incoming year class J in year i, U,j, is 

 related to the amount of effort in year i; that of the 

 previous year class, C/, j _ i , is related to the fishing 

 effort in years i and / - 1; that of the year class 

 which entered 2 yr previously, [/, j - 2 . is related to 

 the fishing effort in years i,i — 1, and i - 2; and so 

 forth. The catch per unit effort of the total fishable 

 population, assuming equal catchability, is 



+ f/, 



k + 1 



for k year classes. For the simplest case where the 

 incoming year class is recruited at the beginning 

 of each year's fishing season, therefore, 



[/, - {A: • f, + (^ - 1) • /; _ 1 + 



(8) 



Equation (8) suggests a weighted average of 

 fishing effort over the total number of years that a 

 year class contributes significantly to the fishery, 

 or 



+ 



f=[k  /•, +(^ -1)./;. 

 [k + {k - I) +    + i] 



+ f. 



k + 



(9) 



25 



