FISHERY BULLETIN: VOL. 73. NO. 1 



3) Yield and fishing effort 



(Kq q ^X"! 1 



(5) 



(6) 



The critical points, useful as management impli- 

 cations and previously derived by Pella and Tom- 

 linson (1969), are: 



/■.p. = K{^ - l)/9 



Popt = [KI{mH)] 



m - I 



(7) 

 (8) 



and 



m - I m - I 



H[K/(mH)] - [K"'/imH)] , (9) 



where f^^^ is the amount of fishing effort required 

 to produce Ymax, the maximum sustainable aver- 

 age yield (MSAY),^ and Popt is the equilibrium 

 population size obtained atf^^^. Figure 1 demon- 

 strates the flexibility of the generalized stock pro- 

 duction model with three values for m (0.5, 2.0, 

 4.0); each curve has the same value for Pmax and 

 Y 



-* max • 



In utilizing the production model for analysis of 

 the status of a particular population, the usual 

 basic assumptions are that 1) the model is being 

 applied to a closed single unit population, 2) the 

 concept of equilibrium conditions^ applies to the 

 population under analysis, and 3) the age-groups 

 being fished have remained, and v^ll continue to 

 remain, the same. If one is able to obtain data 

 which represent equilibrium conditions at three 

 or more population levels, then no additional as- 

 sumptions are needed to fit the production model. 

 In most fishery data sets, however, no real period 

 of equilibrium conditions will exist. Using data 

 from the transitional states of a population re- 

 quires the additional assumptions that both 1) 

 time lags in processes associated with population 

 change and 2) deviations from the stable age 



*i^max is usually referred to as the maximum sustainable jaeld 

 (MSY). The term MSY, however, does not convey that in reality 

 the yield will fluctuate due to changes in the population even if 

 the fishing effort and catchabillty coefficient remain constant. 

 Hence, the "equilibrium yield" curve represents a curve of yield 

 that is sustainable at some average level. 



*The definition of equilibrium adopted here, essentially that of 

 Beverton and Holt (1957), is: 0ven a constant rate of fishing, 

 including zero, a population will achieve a state where, on the 

 average, it will not change in size or characteristics. 



structure at any population level have negligible 

 effects on the production rate, Ptg (P)< (Schaefer 

 and Beverton 1963). 



Schaefer (1954, 1957) pioneered the use of 

 transitional state data for fitting a production 

 model (the logistic form) to catch and fishing effort 

 data. Schaefer's (1957) method for estimating the 

 parameters consisted of approximating differen- 

 tial equation (1) with two finite difference equa- 

 tions and then iteratively solving them. Pella and 

 Tomlinson (1969) greatly improved upon 

 Schaefer's method by demonstrating that a catch 

 history of a fishery could be predicted from the 

 fishing effort history, initial estimates of the pro- 

 duction model parameters, and the integrated 

 form of Equation (1). Then final parameter esti- 

 mates could be obtained by a pattern search rou- 

 tine which finds those parameters which minimize 

 the residual sum of squared differences between 



UJ 



POPULATION SIZE 



FISHING EFFORT 



FISHING EFFORT 



Figure 1.— Equilibrium relationships of the generalized stock 

 production model for three values of m. (A) Equilibrium yield 

 and population size; (B) population size and fishing effort; 

 (C) equilibrium yield and fishing effort. 



24 



