Wang and Ellis: Maximum likelihood estimate of mortality and growth from multiple length-frequency data 



385 



A Female recruitment 



cc 0-1 



300- 

 250- 

 200- 

 150- 

 100- 

 50- 



r 



1986 



1987 



1988 



1 989 



1990 



1991 



D Male recruitment 



1986 



1987 



1989 



1990 



1991 



C Commercial fishing effort 



_aJL 



\ 



jK 



1986 



1987 



1989 



1990 



1991 



Date 



Figure 2 



(A) Quasiperiodic Isolid line) and aperiodic (dashed line) recruitment patterns for female tiger 

 prawns (Penaeus esculetus) in the study area; (B) quasiperiodic and aperiodic recruitment pat- 

 terns for female tiger prawns in the study area; (C) the weekly fishing effort pattern in the 

 study area. 



therefore uncertain and so the lag parameter adopts the 

 role of estimating this uncertainty. 



We do have sampling effort information, so that it 

 would be reasonable to consider incorporating into the 

 likelihood the Poisson term for the total catch as men- 

 tioned in section 3. Information on total catch per oc- 

 casion would improve estimates of mortality. However, 

 preliminary analysis found that there was a mismatch 

 of the expected total catch with the observed total 

 catch. Therefore, it appears to be unrealistic to assume 

 that the catch is proportional to the sampling effort. 

 In the subsequent data analysis we use the form of the 

 log-likelihood in Equation 11, which uses the shape of 

 the observed distribution and takes the total catch as 

 given. 



We have estimated all the parameters k, l at , a,, M, 

 q, and the lag simultaneously (model 1). To achieve 

 a better understanding of the data, we also estimate 

 parameters for a range of fixed values of M (model 3). 



This is common practice in the fisheries literature (e.g. 

 Sullivan, 1992). Estimates of q for corresponding values 

 of M can be useful in some contexts where the outcome 

 of an analysis is insensitive to the joint pairs (M, q) 

 (Somers and Wang, 1996). Taking the rough values 

 of Somers and Wang (1996) and Wang and Die (1996) 

 as a guide, we choose the values M=l, 2, and 3yr~ 1 . 

 The utility of considering a range of values of M ap- 

 plies equally to considering a range of values for {k, 

 IJ. Somers and Kirkwood (1991), Wang et al. (1995) 

 and Wang (1998) have all reported estimates of ik, l^) 

 for this species, and we would like to incorporate this 

 information. However, it is well known that estimates 

 of the growth parameters are strongly correlated. We 

 therefore considered a range of feasible pairs (k. I , I, 

 and estimated the remaining parameters under model 

 2. The fixed values we used were, for males, (2, 39.3), 

 (3, 37.7), and (4, 36.1), and for females, (2, 53.1), (3, 

 47.4), and (4, 41.7). These values were obtained by a 



