380 



Abstract— We consider estimation of 

 mortality rates and growth param- 

 eters from length-frequency data of a 

 fish stock and derive the underlying 

 length distribution of the population 

 and the catch when there is individual 

 variability in the von Bertalanffy 

 growth parameter L„. The model is 

 flexible enough to accommodate 1) any 

 recruitment pattern as a function of 

 both time and length, 2) length-spe- 

 cific selectivity, and 3) varying fish- 

 ing effort over time. The maximum 

 likelihood method gives consistent 

 estimates, provided the underlying 

 distribution for individual variation in 

 growth is correctly specified. Simula- 

 tion results indicate that our method 

 is reasonably robust to violations 

 in the assumptions. The method is 

 applied to tiger prawn data (Penaeus 

 semisulcatus) to obtain estimates of 

 natural and fishing mortality. 



Maximum likelihood estimation of 



mortality and growth with individual variability 



from multiple length-frequency data 



You-Gan Wang 



CSIRO Mathematical and Information Sciences 



65 Brockway Road 



Floreat Park 



Western Australia 6014, Australia 



E-mail address. You-Gan Wangig'csiro.au 



Nick Ellis 



CSIRO Marine Research 



P.O.Box 120 



Cleveland, Queensland 4163, Australia 



Manuscript submitted 5 March 2004 

 to the Scientific Editor's Office. 



Manuscript approved for publication 



9 November 2004 by the Scientific Editor. 



Fish. Bull. 103:380-391 (20051. 



Estimation of growth and mortality 

 is fundamental in fisheries because 

 stock assessment and management 

 rely on these population parameters. 

 Length-frequency-based methods 

 become important when aging tech- 

 niques are either not possible or very 

 expensive. Existing methods such 

 as that of Beverton and Holt (1956) 

 assume that recruitment is continu- 

 ous and constant throughout the year, 

 which leads to a population with an 

 exponentially distributed age struc- 

 ture. Existing modifications to Bever- 

 ton and Holt's method comprise some 

 simple recruitment patterns or distri- 

 butions (Ssentongo and Larkin 1973; 

 Ebert 1980; Hoenig 1987; Wetherall 

 et al. 1987). As pointed out by Vetter 

 (1988), the existing methods for esti- 

 mating mortality in the literature 

 have strong limitations and disadvan- 

 tages. In particular, they require the 

 following assumptions: 



1) each individual follows the same 

 von Bertalanffy growth curve; 



2) the recruitment is either con- 

 tinuous and constant through- 

 out the year (as in Beverton and 

 Holt [1956] and Wetherall et al. 

 [1987]) or is a pulse function (as 

 in Hoenig [1987]); 



3) the total instantaneous mortality 

 rate, z, is constant. 



As pointed out by Sainsbury (1980), 

 it is more realistic to allow individual 

 variability in growth. For example, 

 using tag-recapture data, Wang et al. 

 (1995) found substantial individual 

 variability for the tiger prawn species 

 P. semisulcatus. 



Estimation of mortality relies on 

 the distribution of the lengths, which 

 is determined by the age distribution, 

 mortality rates, and the individual 

 variability in growth rates. If individ- 

 ual variability in growth is ignored, 

 an inappropriate length distribution 

 will be generated, leading to biases 

 in parameter estimates. It is also 

 biologically interesting to quantify 

 the individual variability in growth, 

 which has important implications in 

 fisheries management. Although it is 

 well understood that variability leads 

 to increased uncertainty in estimates, 

 it is less well recognized (among the 

 fisheries community) that variability 

 can also lead to bias. Wang and Ellis 

 (1998) analyzed the effect of ignoring 

 individual variability in a simplified 

 context of constant recruitment and a 

 single length-frequency record. They 

 found that, in the presence of indi- 

 vidual variability, existing methods 

 gave positively biased parameter es- 

 timates. More details about the back- 

 ground can be found in Ebert (1973), 

 Askland (1994), and Wang and Ellis 



