388 



Fishery Bulletin 103(2) 



1320 boat-days. The mostly high negative correlations 

 between M and F 89 (equivalently, q) may explain why Z 

 tends to have a smaller standard deviation than either 

 M or F 89 . The results of Figure 3 can be regarded as a 

 sensitivity study on the effect of changing / . The pur- 

 pose of this sensitivity study is not to estimate l but 

 rather to check that the model assumptions have not 

 been violated for the given l . 



The results are fairly similar for the two recruitment 

 models although there are differences: the quasiperi- 

 odic recruitment model gives larger F m estimates and 

 smaller a* estimates. Our method assumes that the 

 recruitment pattern is known without error; therefore 

 the preferred recruitment pattern should be the one 

 with less error. Let us suppose that the true recruit- 

 ment pattern consists of a periodic pattern with random 

 variation both within years and between years. If the 

 within-year variation is sufficiently large in comparison 

 with the between-year variation, then the quasiperiodic 

 pattern should be used. On the other hand, if the be- 

 tween-year variation is large, then the aperiodic pattern 

 is preferred. Based on the objective values (-21og) in 

 Table 2, model 2 with quasiperiodic recruitment pat- 

 tern and fixed k at 4yr _1 appears to be the best model 

 for both males and females. 



Figure 4 shows the 40 length-frequency records for 

 females with the largest total catch. Overlaid is the ex- 

 pected catch (given the total catch) from the model with 

 ik, l v .) fixed at (3, 47.4) for quasiperiodic recruitment 

 (solid line) and for aperiodic recruitment (dashed line). 

 Because the integral for the expected length distribu- 

 tion is singular in the neighbourhood of l , the first few 

 size classes are omitted from the estimation; only data 

 with length above l +2 are used in the estimation. The 

 fit is quite reasonable for most records. It is interest- 

 ing to compare the performance of the two recruitment 

 models. In early 1988, when recruitment occurred later 

 than usual (see Fig. 2), the aperiodic model tracks the 

 data more closely than the quasiperiodic model, espe- 

 cially in March. On the other hand, the quasiperiodic 

 model fits better in October 1990, whereas the aperiodic 

 model predicts higher abundance of small females be- 

 cause of a recuitment "blip" in September, which was 

 perhaps due to sampling variation. 



Discussion 



Methods such as McDonald and Pitcher's (1979), 

 ELEFAN (Pauly et al., 1981), and Sparre's (1987) oper- 

 ate on multiple length-frequency data and attempt to 

 identify cohorts in the frequency pattern. Essentially 

 they estimate the growth parameters by tracing cohorts 

 in time; then they estimate mortality by measuring the 

 evolution in abundance of a cohort. For mortality esti- 

 mation these methods need catch-per-unit-of-effort data. 

 Sparre's method bears some similarity to ours because 

 it attempts to fit the length distribution of a cohort to 

 a normal distribution whose variance is a parameter to 

 be estimated. Our method does not require separation 



of cohorts because samples are assumed to come from a 

 length distribution which may be multimodal. Another 

 advantage of our method is that it is not necessary to 

 have information about sampling effort and thus may 

 greatly reduce the complexity of sampling. However, our 

 approach needs a known recruitment pattern. 



In our application, recruitment was assumed to occur 

 at a fixed length, / , which had to be chosen. We used 

 prior information to constrain l to lie somewhere be- 

 tween 20 mm and 30 mm. We then found the sensitivity 

 of the estimates to changes in l and chose a value that 

 reduced this sensitivity. This choice could be further 

 refined if more accurate constraints were available from 

 other sources. Alternatively, Wang and Somers (1996), 

 who also used / (l to account for continuous recruitment 

 in estimating growth parameters, have provided guide- 

 lines for choosing / . 



Deriso and Parma (1988) and Sullivan et al. (1990) 

 reported methods based on stochastic growth. Sullivan 

 (1992) also applied the Kalman filter approach for es- 

 timating population parameters. Their models differ 

 from ours in the way random variation is incorporated 

 in the growth model. In their models the length incre- 

 ment from one time step to the next follows a distribu- 

 tion whose mean is given by a fixed growth model. As 

 Wang and Thomas (1995) have demonstrated, this is 

 equivalent to assuming that the growth rate changes 

 randomly from time to time. In our model each indi- 

 vidual follows a deterministic growth curve whose L x 

 parameter is chosen from a random distribution. An 

 individual with larger than average growth at one time 

 step will have above-average growth at subsequent time 

 steps. Perhaps further modeling effort could be directed 

 into combining these approaches. 



DeLong et al. (2001) have reported a method for es- 

 timating density-dependent natural mortality and the 

 growth rate from length-frequency data for juvenile 

 winter flounder not subject to fishing mortality. Other 

 growth parameters (L x and the variability of k) were 

 fixed by using information from other sources. Because 

 their data were recorded in the latter half of the year, 

 when recruitment was nearly complete, recruitment 

 was not a complicated issue. In contrast, we had the 

 challenge of a species that recruits all year round. The 

 degree of fit in DeLong et al.'s Figure 5 is comparable 

 to that in our Figure 4. 



Our methods are based on distributional assumptions 

 that must be tested for robustness, because, in practice, 

 the / x distribution of real prawn populations will not 

 equal any of our mathematical distributions. We have 

 found that, even for our ideal model, akin to any other 

 existing model, biases occur for moderate to large co- 

 efficients of variation when violation of distributional 

 assumptions occurs. 



Our model is motivated by the trawl data from the 

 tiger prawn fishery and relies on 1) known recruitment 

 pattern, 2) contrast in commercial fishing effort for 

 estimation of M and F simultaneously, and 3) contrast 

 in sampling times. Requirement 3 is to spread sam- 

 pling effort so that growth and mortality information 



