and fishing effort history, ^Ci , /", | , of i = 1 . . .n 

 years length and a vector of significant year 

 class numbers [k, | are read in. There may be 

 embedded zeros, if they are true zeros and do not 

 simply reflect a lack of information. The only 

 real problem with unreal zeros, however, occurs 

 in the estimation of g^. The catch per unit effort 

 vector is computed internally and the averaged 

 fishing effort vector is computed by Equation (9) 

 with SUBROUTINE AVEFF. 

 Option 2 . — If one wishes to compute the aver- 

 aged fishing effort vector by another method or 

 if data are obtained which represent equilib- 

 rium conditions, then this option is selected and 

 the vectors of catch per unit effort and averaged 

 (or equilibrium) fishing effort |t/, ,//} are read 

 in directly. No estimate of q- can be made, how- 

 ever. 



STARTING VALUES OPTION. Option 1 .— 

 Initial estimates of the parameters are com- 

 puted in SUBROUTINE INEST and the user 

 provides the starting estimate for m, either 0, 1, 

 or 2. Option 2. — Occasionally the data are so 

 variable that INEST does not provide compati- 

 ble starting values for the parameters. In this 

 case, or in any case, the user may opt to enter 

 directly all the initial parameter estimates. 



MODEL OPTION. The user may allow PROD- 

 FIT to estimate m to any desired precision. Fre- 

 quently, however, the data are so variable that 

 no significant reduction in the residual sum of 

 squares is obtained by varying m . The user then 

 has the option to fix m at 2, the logistic model 

 (Schaefer 1957); at 1, the Gompertz model (Fox 

 1970); or at 0, the asymptotic yield model. 



WEIGHTING OPTION. The user may select the 

 statistical weights as Equation (16) or may 

 choose to not weight the observations, i.e., 

 Wi = 1 for all i. 



CATCHABILITY COEFFICIENT. The catch- 

 ability coefficient, q, is estimated by Equation 

 (22), but both the geometric and arithmetic av- 

 erages are computed. 



Program PRODFIT uses an adaptation of the 

 same pattern search optimization routine, MIN, 

 as contained in GENPROD (Pella and Tomlinson 

 1969) to locate the least-squares parameter esti- 

 mates. A more sophisticated Taylor series ap- 



FISHERY BULLETIN: VOL. 73, NO. 1 



proach (Draper and Smith 1966) was attempted 

 initially, but severe distortion of the sum-of- 

 squares space prevented reasonable convergence. 

 In order to facilitate termination of the searching 

 procedure, the sum-of-squares space is searched 

 with m and a transformation of the parameters a 

 and |3to 



Ur 



l/(m - 1) 



a 



(g - a.m){alm) 

 m0 



l/(m 



(26) 



(27) 



where Umax is the unexploited population size in 

 terms of catch per unit effort. Neither Umax nor 

 Y max change greatly with moderate changes in m . 



The output of PRODFIT provides a listing of the 

 input data, the transformed data, initial parame- 

 ter estimates, the iterative solution steps, the final 

 estimates of a, |3, and m and their variability indi- 

 ces, the management implications of the final 

 model Umax, U opt, f opt, and Ymax and their variabil- 

 ity indices, the observed and predicted values and 

 error terms, and estimates of the catchability 

 coefficient, q. 



A listing of program PRODFIT and a user's 

 guide are available on request from the author. 



COMPARATIVE EXAMPLES OF THE 



EQUILIBRIUM APPROXIMATION 



METHODS 



Two methods of averaging fishing effort which 

 attempt to approximate equilibrium conditions 

 have been presented, the unweighted method 

 (Equation 7) and the new weighted method (Equa- 

 tion 9). In order to compare these two methods, 

 catch histories for a simulated pandalid shrimp 

 fishery (Fox 1972) were generated using a 

 generalized exploited population simulation 

 model GXPOPS (Fox 1973). It should be noted, 

 however, that the comparisons are, for the most 

 part, simply illustrative. It is virtually impossible 

 to demonstrate conclusively which is the better 

 method because there is an infinite choice of life 

 histories, parameter values, fishing effort his- 

 tories, and stochastic variation representations. 



Equilibrium values for the unexploited popula- 

 tion biomass in terms of catch per unit effort 

 (^max), the maximum equilibrium yield {Ymax), 

 and optimum fishing effort (/"opt), were determined 

 empirically by running the simulation model 



28 



