578 



Fishery Bulletin 94(3). 1996 



The molting probability for a given length class 

 and time / is modeled by a logistic function: 



in 



= 1- 



1 



l + <p,e~ 



Hi 



where and co t are parameters, and i is the mean 

 length of length class i. Different logistic functions 

 were used to describe the molting probabilities dur- 

 ing different periods. The molting probabilities dur- 

 ing different years were grouped to produce a good 

 fit. Three groups were selected for the Bristol Bay 

 RKC: group 1 includes 1974-76, 1980, 1987-88, and 

 1992-93; group 2 covers 1981-86; and group 3 in- 

 cludes 1977-79 and 1989-91. Two groups were made 

 for the Kodiak RKC: group 1 includes 1964-65, 1972, 

 and 1980-82 and group 2 includes 1966-71 and 

 1973-79. 



Recruitment into the exploitable population is 

 separated into two components: 1) total recruits for 

 year / entering the exploitable population. R r and 2 ) 

 the proportion of recruits belonging to each length 

 class, U . This is expressed by 



R..=R, U. 



(5) 



where U i is described by a gamma distribution for 

 flexibility, such as in Equations 2 and 3, with pa- 

 rameters a r and fi r . 



To reduce the number of parameters, we assumed 

 that new-shell and old-shell crabs have the same 

 natural mortality and probability of molting the fol- 

 lowing year (Zheng et al., 1995). The annual abun- 

 dance of new-shell crabs is the combined result of 

 growth, mortality, and recruitment: 



N 



\iN il +0, J )e- M - 



+ R,~i,.i< 



3vi 



III 



1(1) 



where TV t and ( are the respective abundances of 

 new-shell and old-shell crabs in length class i on 1 

 July in year t; CN t t and CO t t are the commercial 

 catches of new-shell and old-shell crabs in length 

 class i, year t, and fishing period p; M is instanta- 

 neous natural mortality; and T is the period from 

 1 July to the mid-point of fishing period p in year t. 

 All recruits are new-shell crabs. Old-shell crabs are 

 crabs that did not molt the previous year: 



M , M = 



W i+u+ O i+u )e- M 

 I 



<CAU.p + 



CO„ u „if 



lT,j,-l)M 



(!-'«,.,,: 



(7) 



The new-shell and old-shell crabs in the last length 

 class were lumped together. 



There was only one fishing period each year for 

 the Bristol Bay RKC fishery and for the Kodiak RKC 

 fishery prior to 1974. A second fishing period that 

 targeted large crabs occurred for the Kodiak fishery 

 each year from 1974 to 1982. The catch in the sec- 

 ond fishing period was about 10% of that in the first 

 period. We did not use the fishing efforts in the sec- 

 ond period because the efforts targeted only a small 

 proportion of the population and were relatively 

 small. 



For the first fishing period, the catch by length was 

 estimated as 



CN iittl = TC til s itl N itl /TB ttl , 

 CO UA =TC tA s iA O^ITB tA , 



(8) 



where TC t ; is total annual observed catch for the 

 first fishing period, and TB f , is estimated exploit- 

 able abundance on 1 July in year t , i.e. 



TB, 



s.AN.,+0.,) 



(9) 



and s j is the selectivity coefficient in length class i 

 for the first fishing period. Parameters s 7 ; and s 2 « 

 were estimated for each population, and selectivity 

 coefficients for length class 3 or larger were set to 

 one. 



The annual fishing effort for the first fishing pe- 

 riod was 



f t =TC, A KqMB tA 



(10) 



where q is catchability coefficient for the fishery and 

 MB t j is mean exploitable abundance during the first 

 fishing period in year t and approximated as 



MB lA =TB lA 



e '''• '" - 0.57T, 



(11) 



During the second fishing period for the Kodiak 

 RKC fishery, there were minimum size limits of 168 

 mm CL from 1974 to 1977, and 157.5 mm CL from 

 1978 to 1982. Two parameter values, 6 sl and .,, were 

 estimated for the two minimum size limits: 



= (e 



1)1 s L , 



(12) 



