FOX: RANDOM VARIABILITY AND PARAMETER ESTIMATION 



are unknown. However, a commonly used ap- 

 proach, simulation (or the Monte Carlo method) , 

 may be employed to infer probable eflfects of 

 variability and lead to selection of the "best" 

 statistical model. This simulation study con- 

 sisted of constructing a stochastic (or proba- 

 bilistic) analogue of the generalized production 

 model and then simulating the catches at var- 

 ious levels of constant fishing effort. Inferences 

 will be drawn about the propriety of all four 

 statistical models from residual variation pro- 

 duced in the catches. Also, the sensitivity of 

 catch residual variation to parameter variation 

 will be demonstrated. 



The generalized production model can be writ- 

 ten in a form that is more easily discussed bio- 

 logically 



dP/dt = P,K[{P„ 

 — qfPt 



— F,'"-') /•?«"'-'] 



(21) 



The signs ( + or — ) are set for convenience 

 assuming m > 1. The usual biological in- 

 terpretation of the constants is as follows: 

 K is "the intrinsic rate of natural increase", 

 P = (K/H)''""~^^ is the asymptotic environ- 

 mentally limited maximum population size or 

 "carrying capacity", and m is the determinant 

 of the pi'oportion of P^ at which the maximum 

 rate of production occurs. The stochastic an- 

 alogue of equation (21) is 



dP/dt 



PtK{(TT^- 



- yfP. 



— P,--') /tt^-'] 



(22) 



|. are stochastic variables with 

 m, q\ re- 



where [k, tt, (jl, y ^ 



expected values (or means) \K, P 



spectively, and distributions and variances to 



be specified. The parameters of equation (21) 



were considered to be stochastic variables since 



they are actually average conditions determined 



by many environmental inter-relationships. 



The distributions and variances of the sto- 

 chastic variables are unknown as are their 

 expected values to be estimated from the fishery 

 data. Some broad inferences about the distri- 

 butions can be made, however, from biological 

 and mathematical implications of the production 

 model. The "intrinsic rate of natural increase", 



K, was assumed to be approximately normally 

 distributed [■~N (K.a-i)], because K is the re- 

 sultant rate of a linear combination of rates-— 

 birth rate — death rate (P in numbers) , or birth 

 rate + growth rate — death rate (P in biomass) 

 — so may be either positive or negative at any 

 given time. Negative values for tt and y are 

 biologically and physically meaningless so they 

 were assumed to be approximately log-normally 

 distributed [~logN(P^, 0--2) and ~logN(f/,cr^3) 

 respectively]. The integrated forms of equa- 

 tions (2), (21), or (22) do not exist for ni = 1; 

 therefore /x was assumed to be given by [1 -f 

 (m — 1 ) ^] where ^ was assumed to be approx- 

 imately log-normally distributed with a mean 

 of one [~logAMl, o-'4)]. This resulted in fi 

 having a mean of m with a range of minus in- 

 finity to one, or one to plus infinity, depending 

 on whether m is less or greater than one. 



Integrating equation (22) from Po to Pt yields 



— Po'-" 



„U— ,/) (1-1, 



(23) 



this is the stochastic analogue of equation (4). 

 Expected values and arbitrary variances 

 (tr-i, ah, 0-^3, o-'i) were chosen to allow: 



The expected values were rounded approximate 

 values obtained in the example following this 

 section. In the same manner as the previously 

 described Pella-Tomlinson technique, equations 

 (5) and (23) were used to simulate a 48-year 

 catch history at each of 13 levels of fishing effort. 

 The continuous stochastic variable case was ap- 

 proximated by setting A'' = 10 in equation (5). 

 At each iteration, the stochastic variables 

 i^ K, n, IX, y \ were drawn at random from their 

 respective probability distributions, produced 



573 



