594 



Fishery Bulletin 89(4), 1991 



Table 2 



Tag-shedding models and parameter estimates (from Hamp- 

 ton and Kirkwood 1990) used in the estimation of natural mor- 

 tality rate and related parameters of southern bluefin tuna. 



Parameter 

 estimates 



Experiment Tag-shedding model 



Q(t) = p e" 

 b 



Q(t) = 



b + At 



Q(t) = Pe-" 



b 



Q(t) = 



b + At 



b + At 



J 



0.29 



1.00 0.26 



0.17 



0.78 0.26 



1.04 0.36 



0.93 



0.98 



e Mtj 



N R - I 



If there is a long-term mortality associated with bear- 

 ing tags, this will be incorporated into M and be in- 

 distinguishable from it. An initial mortality due to tag- 

 ging simply reduces the effective number of releases; 

 if known, it can be included in the model in an iden- 

 tical fashion to R. 



As discussed in Hearn et al. (1987), an estimate of 

 the standard error of M is not available by conventional 

 means because the estimate is conditional on the dis- 

 tribution of the data t,. There is also no guarantee 

 that the estimator is unbiased. A statistical tool that 

 is commonly used in applied statistics both to reduce 

 the bias of an estimator and to provide an approximate 

 standard error is the jackknife (Cox and Hinkley 1974). 

 This technique is not described in detail here; suffice 

 to say that it is based on N separate estimates of M 

 that are obtained by removing, in turn, each tagged 

 fish (whether recaptured or not) from the data. The 

 mean of these estimates is the jackknife estimate of 

 M, and their standard error is the jackknife estimate 

 of the standard error of M. 



The SE model: An extension 

 of the Sibert model 



This model, unlike the HSH model, is fit to data 

 grouped into time intervals, rather than individual, 

 exact recapture times. In common with traditional 



single-release tag-attrition models (as described in 

 detail, for example, in Seber 1973 and Wetherall 1982) 

 that have been used extensively in the analysis of mark- 

 recapture data, the SE model is based on classical 

 population dynamics theory, as embodied in the Bara- 

 nov eatch equation (Baranov 1918). 



The SE model extends the work of Sibert (1984), who 

 developed a method of analyzing a tagging experiment 

 in which tagged fish are released (not necessarily 

 simultaneously) into two geographically separate fish- 

 eries that interact through the movement of fish. Sibert 

 accomplished this basically by adding a spatial dimen- 

 sion and incorporating movement rates into the Bara- 

 nov catch equation. Tag returns and catch or effort 

 from the two fisheries, by time interval, form the obser- 

 vations to which the model is fit, yielding estimates of 

 movement rates between the two fisheries, and esti- 

 mates of M, catchability coefficient (when effort is 

 used), and average population size (when catch is used) 

 for each fishery. Recently, Hilborn (1990) developed a 

 model similar to Sibert's in terms of estimation of 

 movement rates; however, natural mortality rate is not 

 specified in the population dynamics and therefore can- 

 not be estimated using the Hilborn model. 



To facilitate analyses of southern bluefin tagging ex- 

 periments, Sibert's two-fishery model is extended in 

 this paper to incorporate a third recapture fishery (the 

 Japanese longline fishery). It is now assumed that there 

 is movement of tagged fish from fisheries 1 and 2 (the 

 release fisheries) into fishery 3. Movement between 

 fisheries 1 and 2 can take place in both directions, but 

 movement into fishery 3 is assumed to be permanent, 

 i.e., there is no possibility of movement back to either 

 fishery 1 or 2 once a fish has moved to fishery 3. This 

 restriction of the model is adopted to avoid the neces- 

 sity of estimating two additional movement parameters 

 about which little information is available in the ab- 

 sence of releases into fishery 3 (large-scale tagging in 

 the longline fishery is not feasible for southern bluefin 

 tuna). However, this assumption is consistent with the 

 known migratory behavior and the age composition of 

 catches of southern bluefin from the Australian and 

 Japanese fisheries (Hampton 1989). 



In addition to the normal assumptions regarding tag 

 loss, equal vulnerability of tagged and untagged fish 

 must be assumed since catch and/or effort statistics are 

 required for the analysis. It is also assumed, for sim- 

 plicity, that M is constant over time within recapture 

 fisheries. 



Model derivation Following release, the tagged fish 

 fall into six categories: the numbers released into 

 fishery 1 that are at large in fishery 1 (N n ), fishery 2 

 (N 12 ), and fishery 3 (N 13 ), and the numbers released 

 into fishery 2 that are at large in fishery 1 (N 2 i), 



