FISHERY BULLETIN: VOL. 69, NO. 3 



giving 



Si = J [(C, - C,)/7,]- (11) 



I = 1 



as the appropriate criterion to be minimized. 

 Model 2. Multiplicative Error 



Pi = Pi  en (12) 



so 



or 



C, ^ Ci  621 



(13) 



In Ci = In C, + In e.. (14) 



giving 



2 (In Ci 

 I = 1 



In C,)- (15) 



as the appropriate criterion to be minimized. 

 Model 3. Additive Proportional Error 



P. = Pi + Pi  esi (16) 



so 



giving 



Ci= Ci + C,  €zi 



(17) 



Sa = 2 [(Ci - C,)/C,]- (18) 

 i = 1 



as the appropriate criterion to be minimized. 



Model 1 assumes constant variation at all pop- 

 ulation levels. This is perhaps the least biologi- 

 cally reasonable of the three suggested alterna- 

 tive statistical models since it is easier to conceive 

 that under equilibrium conditions a population 

 will fluctuate more radically near its environ- 

 mentally limited maximum size than at smaller 

 sizes under constant exploitation. Model 1 is 

 usually employed as a statistical model when 

 variation is expected to arise from experimental 

 or measurement error. Assuming adequate sta- 

 tistics of catch and fishing effort exist, it is more 

 likely that variation will arise from environ- 

 mental influences on the parameters of the model. 

 Models 2 and ?> assume that variation in pojui- 



lation size decreases with population size and 

 that variation in catch increases with the size 

 of the catch. Models 2 and 3 approximate the 

 stochastic representation of equation (2) sug- 

 gested by Pella and Tomlinson [their equation 

 (14)] 



dP/dt = 7), l{±)HPt' 



(^) AT,] 



(19) 



where ?ji and tj2 are continuous random variables. 

 Other statistical models obviously could be 

 constructed, such as 



Pi = Pi 



Pi 



(20) 



where c could assume any value — Models 1 and 

 3 are actually special cases with c = or 1 

 respectively. However, this would introduce 

 another parameter to be estimated. The four 

 previously described statistical models will 

 suffice. 



Returning to the point that equation (5) is 

 a numerical approximation of integration, equa- 

 tions (7), (10), (13), (14), and (17) are not 

 strictly true for the Pella-Tomlinson procedure. 

 Accurate representations would include an ad- 

 ditional error term due to linear approximation. 

 However, as provided for, the linear approxi- 

 mation error may be reduced by increasing the 

 value of A' in equation (5). As will be dem- 

 onstrated later, this error is very small in re- 

 lation to the magnitude of the €i even at small 

 values of A^. The choice of N, on the other hand, 

 can be critical to obtaining good estimates of 

 several parameters. 



We now have three alternative statistical 

 models which fulfill the goals of simplicity and 

 biological rationality to various degrees. It re- 

 mains to be determined which of them fulfills 

 the assumptions of least-squares theory for ob- 

 taining the best pai-ameter estimates. 



STOCHASTIC SIMULATION 



An analytical solution for the approi)riate sta- 

 tistical model is not possible since the actual 

 causes of variability and the relationships to 

 their effects on the generalized production model 



572 



