Bertignac et al.; Estimates of exploitation rates for Thunnus ala/unga from tagging data 



427 



gregate in order to reduce the potential bias in the 

 other parameter estimates of interest. We defined a 

 fleet-specific availability coefficient, 0,f, where the 

 index t,j refers to the number of time periods at lib- 

 erty of release group i in time period^. Equation (3) 

 then becomes 



^■Jf = l^'./'is.T, 



E.. 



(4) 



(p, f is therefore a proxy variable related to time at 

 liberty that we used to correct, in an approximate 

 way, for the effects of changing availability of the 

 tagged population over time. However, 0, ^ need not be 

 estimated for all t,j. Rather, ranges of ^^ may be speci- 

 fied for which the 0, f are considered to be constant at 

 either (completely unavailable) or 1 (fully available). 

 A scheme for constraining 0, f was devised from 

 preliminary inspection of the data, and some knowl- 

 edge of albacore dispersion (Laurs and Lynn, 1977), 

 growth ( Laurs and Wetherall, 1981 ), and gear selec- 

 tivity (Bartoo and Holts, 1993) in the North Pacific 

 (Table 3). The scheme provided three time periods 

 for spatial mixing of tagged albacore with respect to 

 the surface fishery fleets and the "other" fleet (which 

 has characteristics similar to the surface fishery 

 fleets), and eight time periods to allow recruitment 

 of tagged albacore to the longline fishery. During 

 these periods, 0, ^ was estimated from the data. We 

 assumed that tagged albacore remain fully available 

 to the longline fishery during the constrained period 

 for (<), / =1 for ^,,>8). For the surface fishery and 

 "other" fleets, this may not be a good assumption, 

 because these fishing methods may not have fully 

 selected albacore of larger size (Bartoo and Holts, 

 1993). Therefore, we assumed that albacore were 

 fully available during periods 4-11 after release {4>tj 

 = 1 for 3<^,,<12) but were completely unavailable to 

 these fleets after 15 time periods (0, / =0 for ^,,>15). 



To allow for a gradual decline in availability during 

 the intermediate period (ll<^y<16), (p^^ was esti- 

 mated from the data. 



Parameter estimation 



We used a multinomial likelihood function to fit the 

 various models to the tagging data. A derivation of 

 the likelihood function as applied to tagging data is 

 given in Kleiber and Hampton ( 1994). We minimized 

 the negative log of this function to obtain the pa- 

 rameter estimates, i.e. by minimizing (omitting terms 

 dependent only on the data) 



-I 



(i?,-I/,.)ln 



R, 



X/,*ln 





(5) 



where the k subscript indicates an individual recap- 

 ture stratum (combining recapture period, fleet, and 

 time at liberty dimensions). Minimization was car- 

 ried out with a quasi-Newton routine (Otter Re- 

 search, 1991 ). The variance-covariance matrix of the 

 estimated parameters was estimated from the in- 

 verse of the Hessian matrix (Bard, 1974). The vari- 

 ance of quantities that are functions of the estimated 

 parameters (such as exploitation rates) were deter- 

 mined by the delta method (Seber, 1973). 



Results 



Model fits 



Eight different model formulations, based on differ- 

 ent combinations of constant or variable availabil- 

 ity, constant or seasonal catchability, and presence 

 or absence of multiyear effects on catchability, were 

 fitted to the North Pacific albacore tagging data. The 

 total numbers of estimated parameters and the maxi- 

 mum log-likelihood function values for each fit are 

 shown in Table 4. It is clear that the addition of vari- 

 able availability, seasonal catchability, and multiyear 

 effects on catchability all result in highly significant 

 improvements in fit. The model incorporating all 

 three effects, model 8, is suggested as the most ap- 

 propriate of the tested models on the basis of likeli- 

 hood-ratio tests (Kendall and Stuart, 1979). Model 6, 

 which did not incorporate multiyear catchability ef- 

 fects, was the next preferred model. 



Examples of plots of observed and predicted tag 

 returns, aggregated over tag release groups, are 

 shown in Figure 6 for model 2 and in Figure 7 for 

 model 8. As expected, there are large discrepancies 



