Abstract. - Von Bertalanffy 

 growth models appropriate for fit- 

 ting to length-increment data by 

 maximum likelihood are described. 

 Models incorporating variation in 

 growth among individuals, release- 

 length-measurement error, and model 

 error are developed and fit to south- 

 ern bluefin tuna Thunnus maccoyii 

 tag-return data. On the basis of like- 

 lihood ratio tests, a model in which 

 individual variation in growth is rep- 

 resented by variation in L^ and 

 which explicitly incorporates model 

 error is selected as the most ap- 

 propriate model for these data. The 

 parameter estimates obtained were 

 Ml. = 186.9cm, o L J = 218.8cm 2 , K 

 = 0.1401/year, and o e 2 = 15.25cm 2 . 

 Analyses of simulated data suggest 

 that biased estimates of growth 

 parameters can result if model error 

 is not explicitly included in von Ber- 

 talanffy models incorporating indi- 

 vidual variation in growth. 



Estimation of Southern 

 Bluefin Tuna Thunnus maccoyii 

 Growth Parameters from Tagging 

 Data, using von Bertalanffy Models 

 Incorporating Individual Variation 



John Hampton 



CSIRO Division of Fisheries. Marine Laboratories 



GPO Box 1538. Hobart, Tasmania 7001, Australia 



Present address: South Pacific Commission, BP D5. Noumea Cedex, New Caledonia 



Manuscript accepted 15 April 1991. 

 Fishery Bulletin, U.S. 89:577-590 (1991). 



Knowledge of the growth of a fish 

 and, in particular, a mathematical 

 description of the increase in length 

 or weight with time, is important for 

 understanding its population and 

 fishery dynamics. Also, fish growth 

 has been used directly or indirectly 

 to calculate catch age composition 

 (Hayashi 1974, Baglin 1977, Kume 



1978, Skillman and Shingu 1980, 

 Majkowski and Hampton 1983), mor- 

 talities (Beverton and Holt 1957, 

 Pauly 1987) and yield-per-recruit 

 (Beverton and Holt 1957, Ricker 

 1975) and to make inter- and intra- 

 population comparisons (Brousseau 



1979, Goldspink 1979, Kohlhorst et 

 al.1980, Francis 1981). 



Techniques for studying fish growth 

 in length fall into three categories: (i) 

 Direct measurements of age (such as 

 those obtained by counting periodic 

 protein and calcium depositions in 

 scales, otoliths, vertebrae, fin rays, 

 or some other hard tissue) and length; 

 (ii) analysis of time series of length- 

 frequency data (sometimes called 

 modal progression analysis); and (iii) 

 analysis of length-increment and time- 

 at-liberty data from a mark-recapture 

 experiment. In each case, a growth 

 model is usually fit to the data. Vari- 

 ous models have been proposed for 

 fitting to these types of data (e.g., 

 Brody 1927 and 1945, Ford 1933, 

 Walford 1946, Richards 1959, Knight 

 1969), but by far the most used model 



in fisheries research is that of von 

 Bertalanffy (1938). This model, orig- 

 inally formulated on physiological 

 considerations, has three parameters 

 that have the biological interpreta- 

 tions of average maximum length 

 (L^), the average rate at which L m 

 is approached (K), and the theoretical 

 average time at which length would 

 be zero if growth had always oc- 

 curred according to the model (t ). 



The study of growth by direct mea- 

 surements of age (reviewed by Bage- 

 nal 1974, Brothers 1979, Beamish 

 1981, Prince and Pulos 1983) and 

 length is not possible for many spe- 

 cies. In particular, good estimates of 

 the ages of older fish are frequently 

 hard to come by. This was the case 

 for Yukinawa's (1970) study of south- 

 ern bluefin tuna Thunnus maccoyii 

 age and growth from presumed an- 

 nual rings on scales. He examined the 

 scales of 2240 fish within the length 

 range of 38- 184 cm, but was able to 

 age only about 15% of fish larger 

 than 120 cm, and none of those larger 

 than 153cm. Similarly, Thorogood 

 (1986) was unable to age significant 

 numbers of large southern bluefin 

 from examinations of their otoliths. 



A major aspect of length-frequency 

 analysis is the identification of age 

 classes in the data. To do this, Hard- 

 ing (1949) and Cassie (1954) used 

 probability paper, and Tanaka (1956) 

 fit parabolas to the logarithms of 



577 



