Hearn and Polachek; Long-term growth rate changes in Thunnus maccoyii 



61 



measured returns in the fitted models below did not indi- 

 cate any systematic pattern. The recaptured fish used in 

 our study ranged in size from 60 to 175 cm, although the 

 number of fish in the larger size ranges was relatively 

 small — less than 2% were larger than 140 cm. (The con- 

 sequences of the small number of fish in the large-size 

 category are discussed below. ) Within both the surface and 

 longline fishery, a range of sizes and age classes is har- 

 vested within a single operation. No indication exists that 

 within the size range encompassed by a cohort at a given 

 age, there existed significant gear or fishery size selectiv- 

 ity. Overall, the above basic assumptions seem reasonable 

 in modeling growth from these SBT tagging data. 



(1991). In this model, fish grow according to one model 

 (or parameter set) up to a certain length and according 

 to another thereafter. In our analyses, we assumed that 

 fish have VBG throughout their lives but grow according 

 to one set of VBG parameters (L_j and k-^) up to length L* 

 and according to a second set (L_.^ and k.-,) at larger sizes, 

 the two-phase VBG model. Thus, the predicted length as a 

 function of time for this model is 



L , l-e-*>"-'°' for t<t* 



L*-(-(L ,-L*) l-e"''-'"-'"' {ort>t* 



Analytical methods 



Models Two basic models were used to analyze growth 

 information from the tag-return data. The first was the 

 simple VBG model: 



where 



/, =L__(l-e-*"-'"'), 



(1) 



= the length that fish grow to asymptotically; 



= the length of a fish at age (or time) t; 



= the exponential rate at which the growth 



rate slows; and 

 = the hypothetical age (or time) when a fish is 



of length zero. 



where t* = the predicted time for a fish to reach L*. 



Note that t* can be solved for in terms of four of the 

 parameters of the model ltQ,k^, L^j, and L*): 



where 



k. 



(o = h + -r^°s 



Mogfl ^ 



^~: 



^«i 



(4) 



and 1,= the length of a fish at the time of tagging, t^ 



When applied to tag-return data, this equation can be 

 used to predict the growth increment as a function of the 

 length at release and the time at liberty: 



As with this simple VBG model. Equation 3 can be solved 

 to predict the growth increment as a function of the 

 release length and the time of liberty 5t = t.2-t{. 



SI = {L_ -l)(l-e 



-kSl) 



(2) 



where SI = the growth increment; 

 St = the time at liberty; and 

 / = the length of release. 



Note, in this study we simplified the growth model by not 

 accounting for seasonal growth. However, data on recap- 

 tured fish with short times at liberty were specifically 

 deleted to ensure that our results were robust after this 

 simplification. 



Preliminary analyses of the tag-return data suggested 

 that a simple and time invariant von Bertalanffy growth 

 model may not provide an adequate description of the 

 growth rate for SBT. These preliminary analyses suggested 



S = 



(5) 



It should be noted that in some of the analyses considered 

 below, the estimate of L^j did not converge (i.e. the esti- 

 mate for L. , was essentially infinite). In such cases, the 

 estimated growth rate is linear, with growth rate /?,, and 

 for the first phase we replaced the von Bertalanffy growth 

 function with a simple linear one: 



« = /?,*, 



1 Growth rates in the 1960s and the 1980s were not 

 equal; 



2 There were systematic deviations from a VBG curve, 

 possibly corresponding to different growth processes or 

 models for adults and juveniles. 



Consequently, in the present study, we considered a more 

 complex model than the simple VBG and conducted sepa- 

 rate, as well as combined, analyses of the tag data from 

 the two periods. The more complex model selected was 

 the two-phase growth model developed by Bayliff et al. 



and 



t* = t^-(L*-l,^)/Ry 



(6) 



Model-fitting procedure A large body of literature exists 

 on statistical approaches for estimating growth from tag- 

 return data (e.g. Fabens, 1965; Sainsbury, 1980; Kirkwood 

 and Somers, 1984; Francis, 1988; James, 1991; Hampton, 

 1991; Wang et al., 1995). The most appropriate approach 

 depends on the error structure assumed for the model. We 

 followed the maximum-likelihood approach and general 

 error structure described by Hampton (1991). The mea- 

 sured growth increment offish "/" is 



