Zheng et al Catch-length analysis for crab populations 



577 



crab pot fisheries. The model incorporates stochas- 

 tic growth, in which individual crabs molt with an 

 annual molting probability and gradual recruitment 

 over length. We applied our catch-length analysis to 

 RKC populations in Bristol Bay and off Kodiak Is- 

 land, Alaska. Bristol Bay RKC have supported one 

 of the most valuable fisheries in the United States. 

 The Kodiak RKC fishery was also very valuable be- 

 fore it collapsed in the early 1980's, and the popula- 

 tion has failed to recover since then. We selected these 

 stocks because both populations have been inten- 

 sively studied during the last three decades and be- 

 cause data are available to estimate biological pa- 

 rameters such as growth and mortality. Moreover, 

 the availability of annual trawl or pot survey data 

 allowed us to compare population abundances esti- 

 mated from the catch-length analysis and from sur- 

 veys to determine the reliability of the catch-length 

 analytical methods that we developed. 



Methods 



surveys conducted by the National Marine Fisheries 

 Service (NMFS) (Stevens et al. 1 ). Standardized catch 

 per unit of effort (CPUE) from pot surveys and from 

 tag and recovery data for Kodiak legal male crabs 

 was provided by Peterson et al. ( 1986). We estimated 

 mean catchability of the Kodiak pot survey using the 

 Petersen mark-recapture equation and annual com- 

 mercial catch, tag, and recovery data from 1973 to 

 1979. This allowed us to estimate annual legal male 

 abundances by dividing relative abundance (stan- 

 dardized survey CPUE) by the catchability. These 

 abundance estimates derived from survey data were 

 not used in catch-length analyses but were instead 

 compared to final results of catch-length analyses. 



Population model 



The population model is similar to the length-based 

 model for Bristol Bay RKC developed by Zheng et al. 

 ( 1995 ). Mean growth increment per molt for length class 

 i, G , is assumed to be a linear function of mean cara- 

 pace length i of the length class just before molting: 



Data requirement 



Data required for the model include annual length- 

 frequency of commercial catches by shell condition, 

 total annual catch and effort, the mean and variance 

 in growth increment per molt, and natural mortal- 

 ity. Catch-length frequency and total fishing effort are 

 available for most crab populations in Alaska. In this 

 study we defined fishing effort as total annual pot lifts, 

 i.e., the product of the total number of pots fished and 

 the number of times each pot was deployed and re- 

 trieved in the fishery. The mean and variation in growth 

 increment per molt can be estimated from tagging data, 

 which are available for some crab populations. 



In our study, we estimated the mean and varia- 

 tion in growth increments per molt for Bristol Bay 

 and Kodiak RKC from studies by Balsiger (1974), 

 Powell ( 1967 ), and Weber and Miyahara ( 1962 ). The 

 model was fitted to RKC fishery data for Bristol Bay 

 from 1974 to 1993 and for Kodiak from 1964 to 1982. 

 Owing to low abundance, the Bristol Bay fishery was 

 closed in 1994, and the Kodiak fishery has been closed 

 since 1983. The RKC fisheries in Alaska harvest only 

 male crabs, and the minimum legal carapace length 

 (CD is 135 mm for Bristol Bay RKC and 147 mm for 

 Kodiak. Accordingly, we modeled only legal male 

 crabs. The length frequency of catches was summa- 

 rized by 5-mm intervals, and the largest length group 

 included crabs >165 mm CL for Bristol Bay and >182 

 mm CL for Kodiak. 



For comparisons, area-swept estimates of Bristol 

 Bay legal male abundance were obtained from trawl 



G, =a + bi, 



(1) 



where a and b are the intercept and slope. Param- 

 eters a and b were estimated from Weber and 

 Miyahara ( 1962 ) to be 13. 140 and 0.018 for Bristol Bay, 

 whereas we estimated them from Powell's ( 1967 ) data 

 as 41.047 and -0.159 for waters off Kodiak Island. 



For flexibility, we chose a gamma distribution to 

 describe the variation in growth increment per molt: 



g(x\a l ,p) = x"--V" ll /(l3 a T(a i )), 



(2) 



where .r is growth increment per molt, a ; and /3 are 

 parameters, and a t = GJ ji. The expected proportion 

 of crabs molting from length class V to length class i 

 is equal to the integral of the gamma function over the 

 length interval [i., L) of the receiving length class i: 



'2-1 



P Vi = \g{x\a i ,P)dx, 



(3) 



where i is the mean length of length class i'. For the 

 last length class L, P LL - 1. The variation in growth 

 increment per molt, p, for both populations was set to 

 0.75, which produced a transitional matrix similar to 

 that estimated by Balsiger ( 1974 ) for Bristol Bay RKC. 



1 Stevens, B. G., R. A. Macintosh, J. A. Haaga, and J. H. Bower- 

 man. 1993. Report to industry on the 1993 eastern Bering 

 Sea crab survey. Alaska Fish. Sci. Center., Natl. Mar. Fish. 

 Serv., NOAA, Seattle, WA, Proc. Rep. 93-14, 53 p. 



