390 



Fishery Bulletin 103(2) 



are in the data. We fitted a variety of different models. 

 The objective function -21og( likelihood) values in Table 

 2 should be used only as guidelines and should not 

 drive the analysis or be used for model selection. Tiger 

 prawns are subject to very high total mortality and 

 hence are short-lived species. Our method is also ap- 

 plicable to longer-lived species. However, for application 

 to other fisheries, some modification of the model may 

 be necessary to incorporate relevant information in the 

 model. Simulation studies may have to be carried out 

 to see how reliable the modified version is for param- 

 eter estimation because many factors, such as growth 

 rate and commercial effort patterns, will determine if 

 parameter estimates can be found or how reliable they 

 are if they can be found. 



We aim to obtain growth and mortality parameter 

 estimates simultaneously. However, this may be too 

 ambitious, especially for short-lived species unless 

 other information can be incorporated to assist esti- 

 mation. For instance, Ebert (1973) found estimation 

 of even two parameters (natural and fishing mortal- 

 ity) unreliable and had to assume one of them. This 

 is perhaps why natural mortality is assumed to be 

 known in traditional cohort analysis. Also Askland's 

 method (1994), one of the most recent cohort-analysis 

 methods, requires a known M. Nevertheless, in prac- 

 tice, ik, l.,) may be estimated from different types of 

 data. The results based on model 2 (assuming {k, l^) 

 are known) indicate that both M and F can then be 

 estimated more reliably when there is substantial 

 contrast in the effort pattern. Another assumption 

 is that catchability does not change over time. This 

 may not be necessarily true when new technology is 

 introduced into the fishery (Bishop et al., 2000). The 

 assumption that growth parameters are known greatly 

 reduces the complexity of estimating the remaining 

 unknown parameters and improves the performance 

 of the proposed methods. 



We have chosen to allow only l m to be random because, 

 unlike tag-recapture data, the length-frequency data do 

 not have multiple measures from each individual. Each 

 individual is measured only once. Therefore, it might be 

 problematic to allow random K and correlation between 

 K and L^. Such an attempt using length-frequency 

 data may lead to misleading conclusions because the 

 conclusion will be model-driven instead of data-driven. 

 Parameter estimates obtained by fixing M as a constant 

 are deemed more reliable. 



We provided a framework for length-frequency da- 

 ta analysis that incorporates continuous recruitment, 

 selectivity, and time-dependent fishing mortality. We 

 have also provided guidelines for how to compute the 

 likelihood function, which depends on rather delicate 

 integrals. Such a model would be very useful for many 

 fisheries because such unified models are not available 

 in the literature. Our work provides a sensible case 

 study. Application of our method may require incorpora- 

 tion of specific information in a fishery. We believe our 

 model, which generalizes the traditional model and is 

 somewhat complicated, has provided us with some use- 



ful results for future stock assessment and evaluation 

 of management strategies. 



Acknowledgments 



This research project was partly supported by the Fisher- 

 ies Research and Development Corporation of Australia. 

 We gratefully acknowledge the helpful suggestions and 

 comments of David Die, Andre Punt, Neil Loneragan, 

 and two anonymous referees. 



Literature cited 



Askland, M. 



1994. A general cohort analysis method. Biometrics 

 50:917-932. 

 Beverton, R. J. H., and S. J. Holt. 



1956. A review of methods for estimating mortality rates 

 offish in exploited fish populations, with special refer- 

 ence to sources of bias in catch sampling. Rapp. P.-V. 

 Reun., Cons. Int. Explor. Mer 140: 67-83. 

 Bishop, J., D. Die, and Y.-G. Wang. 



2000. A generalized estimating equations approach for 

 analysis of fishing power in the Australian Northern 

 Prawn Fishery. Aust. N. Z. J. Stat. 42:159-177. 



DeLong, A. K., J. S. Collie, C. J. Meise, and J. C. Powell. 



2001. Estimating growth and mortality of juvenile 

 winter flounder, Pseudopleuronectes americanus, 

 with a length-based model. Can. J. Fish. Aquat. Sci. 

 58:2233-2246. 



Deriso, R. B., and A. M. Parma. 



1988. Dynamics of age and size for a stochastic population 

 model. Can. J. Fish. Aquat. Sci. 45:1054-1068. 

 Ebert, T. A. 



1973. Estimating growth and mortality rates from size 

 data. Oecologia 11:281-298. 



1980. Estimating mortality from growth parameters and a 

 size distribution when recruitment is periodic. Limnol. 

 Oceanogr. 26:764-769. 



Hoenig, J. M. 



1987. Estimation of growth and mortality parameters 

 for use in length-structure stock production models. In 

 Length-based methods in fisheries research (D. Pauly 

 and G. R. Morgan, eds.), p. 121-128. ICLARM Conf. 

 Proc. 13. 



McDonald, P. D„ and T. J. Pitcher. 



1979. Age groups from size-frequency data: a versatile and 

 efficient method of analyzing distribution mixtures. J. 

 Fish. Res. Board Can. 36:987-1001. 



Pauly, D., J. Ingles, and T. Neal. 



1981. Application to shrimp stocks of objective methods 

 for the estimation of growth, mortality and recruit- 

 ment-related parameters from length-frequency data 

 (ELEFAN I and III. In Penaeid shrimps: their biology 

 and management (J. A. Gulland and B. J. Rothschild, 

 eds.), p. 220-234. Fishing News Books Ltd., Farnham, 

 England. 



Powell, D. G. 



1979. Estimation of mortality and growth parameters 

 from the length frequency of a catch. Rapp. P.-V. Reun., 

 Cons. Int. Explor. Mer 175:167-169. 



