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Fishery Bulletin 89(4), 1991 



observed length-frequencies. More recently, digital 

 computers have been used, and the distribution of 

 length-at-age has been assumed to be normal or log- 

 normal (Hasselblad 1966, Kumar and Adams 1977, 

 Macdonald 1969 and 1975, Macdonald and Pitcher 

 1979, Schnute and Fournier 1980, Fournier and Breen 

 1983). Pauly's method (Pauly 1987) of fitting growth 

 curves to observed peaks in a time series of length- 

 frequency data, commonly known as ELEFAN I, has 

 received some recent attention, but it suffers from the 

 assumption that length-at-age does not vary (Hamp- 

 ton and Majkowski 1987). 



Some attempts have been made to study the growth 

 of southern bluefin, using length-frequency data. Ser- 

 venty (1956) plotted the progressions of length-fre- 

 quency means and modes of juvenile age classes caught 

 in Australian coastal waters, but did not attempt to 

 quantify the results as a growth equation. Robins (1963) 

 made the first attempt to quantify growth, obtaining 

 a Walford growth transformation from an analysis of 

 length-frequency modes of juvenile fish. Hearn (1986) 

 identified a seasonal component to growth from anal- 

 yses of similar data. Recently, promising results have 

 been obtained with the application of MULTIFAN, a 

 likelihood-based method for estimating von Bertalanffy 

 growth parameters from length-frequency data, to 

 southern bluefin tuna data (Fournier et al. 1990). 



Length-increment and time-at-liberty data from a 

 tagging experiment provide direct measurements of 

 the growth of individual fish as long as the tag or the 

 tagging procedure does not have a significant effect 

 on growth. Using a von Bertalanffy model and a fit- 

 ting procedure such as that proposed by Fabens (1965), 

 estimates of L^ and K can be obtained, but without 

 additional assumptions no estimate of t„ is available. 



Murphy (1977) analyzed the release and recapture 

 data from 2578 tagged southern bluefin and derived 

 estimates of L m and K that he considered to be more 

 reliable than those previously derived by other workers. 

 Kirkwood (1983) obtained similar estimates from a 

 smaller data set (n 794), excluding fish that had been 

 at liberty for less than 250 days. In addition, he incor- 

 porated age-at-length observations from length-fre- 

 quency modes to the estimation procedure to obtain an 

 estimate of t () . 



None of the studies of southern bluefin growth, and 

 indeed few studies of fish growth in general, take ex- 

 plicit account of variation in growth among individuals 

 by the incorporation of parameters describing such 

 variation into the model. Krause et al. (1967) gave the 

 first thorough treatment of individual variability in 

 growth when deriving conditional probability densities 

 for body weight-at-age of chickens. Sainsbury (1980) 

 recognised the importance of individual variability in 

 fishes showing von Bertalanffy growth, and derived 



equations appropriate for length-at-age and length- 

 increment data if both L^ and K showed individual 

 variation. He also showed that biased estimates of 

 mean growth parameters could result if individual 

 variability in K existed and was ignored. Kirkwood and 

 Somers (1984) developed a simpler model for length- 

 increment data in which only L^ was variable and ap- 

 plied it to two species of tiger prawn. 



A problem with these models (as pointed out by 

 Kirkwood and Somers 1984) is that all the observed 

 "error" is attributed to individual variation in L^, 

 and/or K. It is, of course, reasonable to expect that 

 there will also be error due to some animals not grow- 

 ing exactly according to the von Bertalanffy model, a 

 so-called model error. For standard growth models not 

 incorporating individual variability, all residual error 

 is assumed to be model error. It is also reasonable to 

 expect that, in the case of length-increment data, the 

 initial or release length cannot always be measured ex- 

 actly and therefore will be an additional source of error. 



In this paper, southern bluefin tuna tag-return data 

 are analysed using three existing models, all of which 

 are based on the von Bertalanffy model: the standard 

 model, using the fitting procedure described by Fabens 

 (1965), model (2) of Kirkwood and Somers (1984), and 

 the Sainsbury (1980) model. In addition, models based 

 on the latter two that incorporate model error and 

 release-length-measurement error are derived and ap- 

 plied. The properties and assumptions of each of the 

 models are investigated using computer simulation 

 techniques. 



Methods 



Tagging methods and data 



The primary method used to catch fish for tagging was 

 commercial pole-and-line, using either live or dead bait, 

 although on some occasions trolling was also used. 

 Prior to release, the fork lengths of most fish selected 

 for tagging were measured to the nearest centimeter 

 on a measuring board. While the fish were restrained 

 on the measuring board, one or two numbered tuna 

 tags, each consisting of a molded plastic barbed head 

 with a tubular plastic streamer glued to it (Williams 

 1982), were inserted forward into the musculature at 

 an angle of about 45°, l-2cm below the posterior in- 

 sertion of the second dorsal fin. For double-tagged fish, 

 one tag was inserted on each side of the second dorsal 

 fin. Ideally, the tag barb was anchored behind the 

 second dorsal fin ray supports (pterygiophores). 



The primary data used in this study consist of returns 

 of southern bluefin tagged between 1962 and 1978 that 

 were measured to the nearest centimeter at release, 

 were thought to have reliable dates of release and 



