FOX: FITTING THE GENERALIZED STOCK PRODUCTION MODEL 



The difference method involves writing Equa- 

 tion (1) as a finite difference equation for the pro- 

 duction model in terms of catch per unit effort and 

 the estimates for a , |3 , and m as 



^At/, 



9 M 





a 



J u. -f.u. 



(17) 



for each year i;M is taken as one unit, Equation 

 (17) is divided through by (/,, summed over the 

 n - 2 yr that A [7, can be estimated, and then 

 solved for q^ , 



where 



(18) 



(19) 



This method has provided reasonable estimates 

 with the logistic (m = 2) and Gompertz {m—>l) 

 forms of the production model for several fisheries 

 (Fox 1970). 



Pella and Tomlinson (1969) observed that Equa- 

 tion (19) can be a poor estimator of the change in 

 stock size during year i under certain circum- 

 stances. The integral method avoids this problem 

 by writing Equation ( 17) as a differential equation 



dU 

 : = q dt, (20) 



ui-^ -r + 4[/"'-^) 



where f* , the effective effort having been exerted 

 between years i and i +1, is estimated by 



r = (/; +/",.i)/2. (21) 



The integral of Equation (20) after rearranging 

 some terms is 



q, = \n[\{zUy"'+ 4 )/uu! :T + ^mzm -z) (22) 



where 

 2 = -d//? - f*. 



The fact that Equation (22), as an estimator of g, 

 gives negative values when the stock changes in 

 one direction, depending on whether m is greater 

 or less than 1, is remedied by taking the absolute 

 value of q. Also, since q is constrained against 

 being less than zero, the geometric mean will 

 probably be a better estimator than the arithmetic 



mean (this will be demonstrated to be so in at least 

 one case), such that 



q, = e 



n - 1 



2 In I o. I Kn - 1) 

 ( = 1 ' 



(23) 



becomes the integral estimator. 



Variability Measures 



Some measure of the variability of the parame- 

 ter estimates can be made using the "delta" 

 method (Deming 1943). If S is the weighted re- 

 sidual sum of squares for the final parameter es- 

 timates, a variability index matrix, V, is com- 

 puted by 



V = {X'WX)'^SI{n - 3) 



(24) 



where W is sltx n hy n diagonal matrix of the 

 statistical weights, X is an n by 3-parameter ma- 

 trix of first partial derivatives of Equation (11) 

 with respect to each parameter (given in the Ap- 

 pendix). The diagonal elements of V are variabil- 

 ity indices of the parameter estimates and the 

 off-diagonal elements of V are covariability indi- 

 ces. Since Equation (11) is nonlinear, the indepen- 

 dent variable is not without error, the errors in the 

 dependent variable are correlated, and the statis- 

 tical weights are random variables, it is virtually 

 impossible to make probability statements about 

 the accuracy of the parameter estimates (Draper 

 and Smith 1966). However, V gives some index of 

 the variability inherent in the data which is useful 

 largely for comparative purposes between differ- 

 ent fisheries and data sets. For convenience, an 

 error index may be formulated as 



E, = [100 ^V {X)]lx 



(25) 



where X is the estimated parameter and V (X) is 

 its corresponding variability index. Variability 

 and error indices of Y max/opt. and t/opt also may be 

 computed by the "delta" method (see Appendix) 

 and the elements of V (Equation 24). 



Program PRODFIT 



A computer program PRODFIT, in FORTRAN 

 IV language, was written to perform the calcula- 

 tions described above. A brief description of the 

 program's options and mode of operation is given 

 below. 



DATA INPUT OPTION. Option l.—A catch 



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