Zheng et at Catch-length analysis for crab populations 



579 



where i is the mean length of length class i, and i' 

 and L are the mean lengths of the first and last length 

 classes. This selectivity function was chosen, based 

 on the differences of the catch-length frequencies 

 between the first and second fishing periods. 



The catch by length for the second fishing period 

 was estimated as 



CN, l2 = TC l2 s, l2 (N I ,e- TllM -CN ljA )/TB t2 , 

 CO, l2 = TC l2 s, l2 (O IJ e- T " M -CO, IA )/TB, 2 , (13> 



where TC t ., is total annual observed catch for the 

 second fishing period and TB t ., is estimated exploit- 

 able abundance just after the first fishing period in 

 year t, i.e. 



ra„2=£ 



((N,,+0,,)e 



T, ,M \ 



-CN, 



CO 



i.t.l I 



(14) 



Parameter estimation 



Model parameters were estimated under the assump- 

 tion of log normally distributed measurement or ob- 

 servation errors for length compositions of catches 

 and annual fishing efforts. In Alaska, catch data are 

 legally required for sales transactions between fish- 

 ermen and processors, and it is generally believed 

 that total annual crab catches in Alaska are fairly 

 accurately reported. Thus, no measurement error 

 was imposed on total annual catch for the catch- 

 length analysis. A nonlinear least squares approach 

 was used to minimize the residual sum of squares 

 (RSS) of length compositions of catches and annual 

 effort: 



RSS = Y, 



(ln(CAT, , p + c> - ln(CW, , p + C))" 



(ln(CO,, p +c)-ln(CO,, p +c))" 

 +A 2 ^ (ln(/;+l)-ln(/, +1))" 



(15) 



crabs to avoid taking the logarithm of zero and to 

 reduce the impact of extremely small catches on pa- 

 rameter estimation. Generally, c should be relatively 

 small compared to total catch, but estimation may 

 fail to converge for a very small c. 



The subroutine DBCLSF of IMSL FORTRAN 

 (IMSL, 1991) was used to perform nonlinear least- 

 squares parameter estimation through a modified 

 Levenberg-Marquardt algorithm and a finite-differ- 

 ence Jacobian. All parameters were bounded to be 

 nonnegative. 



The following model parameters were estimated 

 for each population: recruits for each year, except the 

 first year; total abundance in the first year; param- 

 eters a r and /3 r ; molting probability parameters 0, and 



a> t ; selectivity parameters s ; v s 9 ; . 



6 sl and 0,„; and 



catchability coefficient q. Starting in the second year, 

 the abundances by length, sex, and shell condition 

 were computed recursively from 1) the abundances 

 by length and shell condition in the first year, 2) an- 

 nual recruitment, 3) catch, and 4) model parameters. 

 To reduce parameters further, we used the observed 

 frequencies of length and shell classes from first-year 

 catch data to approximate the true catch frequen- 

 cies. Thus, we had to estimate only total abundance 

 of male crabs for the first year. 



Initial values of parameters were approximated by 

 using catch data. A 40% harvest rate was used to 

 convert total catch in the first year into total abun- 

 dance and the sum of the new-shell catch in the first 

 four length classes each year into recruitment (Zheng 

 et al., 1995). Initial values were interactively up- 

 dated; the estimated parameters for the first run 

 were used as the initial parameters for the second 

 run, and so on, until no further reduction of total 

 RSS could be made. 



Because we did not know the ratio between the 

 variance of catch by length and the variance of fish- 

 ing effort, we could not use a maximum likelihood 

 approach. Instead, we used seven error weighting 

 factors (A): 0, 1, 2, 3, 5, 7, and 10 to conduct alterna- 

 tive catch-length analyses and compared the results 

 for different weighting factors. 



We did not estimate natural mortality but used 

 several natural mortalities for comparisons. Kruse 

 and Collie 2 and Collie 3 used 0.3 as instantaneous 

 natural mortality for the Bristol Bay and Kodiak le- 



where CN t and CO i t are observed catches for new- 

 shell and old-shell crabs in length class i, year t, and 

 fishing period p\ f, is observed fishing effort in year t 

 for the first fishing period each year; X is an error 

 weighting factor for fishing effort relative to catch 

 composition; andc is a constant set equal toO.OlxlO 6 



2 Kruse, G. H., and J. S. Collie. 1991. Preliminary application 

 of a population size estimation model to the Bristol Bay stock 

 of red king crabs. Alaska Dep. Fish Game, Comm. Fish. Div., 

 Reg. Information Rep. 5J91-09, 25 p. 



3 Collie, J. S. 1991. Estimating the abundance of king crab 

 populations from commercial catch and research survey 

 data. Rep. to Alaska Dep. Fish Game. Univ. Alaska, Fairbanks, 

 Rep. 91-03, 27 p. 



