WETHERALL: ANALYSIS OF DOUBLE-TAGGING EXPERIMENTS 



which are unaffected by Type I losses, and in fact 

 will yield an estimate of 77 as well as the usual 

 estimates of F and X. Since the conditions re- 

 quired for this scheme will not often be encoun- 

 tered, it will usually be necessary to conduct a 

 multiple-release experiment, with at least one 

 preseason release, in order to obtain separate 

 estimates of F, X, and 77. Models appropriate to 

 this situation have also been extensively devel- 

 oped by Paulik (1963). 



However, even in the multiple-release models 

 the Type II tagging mortality and Type II shed- 

 ding will generate an underestimate of the true 

 exploitation rate. Thus while the problems im- 

 posed by Type I losses may be circumvented by 

 more elaborate experimental designs, the unde- 

 sired effects of Type II losses remain. Two reme- 

 dies are possible: 1) The single-tagging may be 

 supplemented with a double-tagging experiment 

 and other special studies to estimate Type II 

 components and determine correction factors, or 

 2) double-tagging may be used exclusively to esti- 

 mate mortality rates unaffected by Type I and 

 Type II shedding. Both strategies are treated be- 

 low. We note here that even when tag shedding 

 and mortality are not the chief concerns of a tag- 

 ging experiment, double-tagging is often em- 

 ployed simply to increase expected recovery 

 rates (e.g., Hynd 1969; Bayliff 1973). 



MODELS OF DOUBLE-TAGGING 



We restrict our attention to the case where 

 members of the population are marked with two 

 tags differing in position of attachment and pos- 

 sibly type (call these Type A and Type B). We 

 assume the burden of carrying both tags is equal 

 to the stress of carrying either one alone. Fur- 

 ther, we assume that the probabilities of loss are 

 the same for each tag of a specified type and inde- 

 pendent of the status of the other tag. Suppose a 

 cohort of fish is double-tagged at time 0. For any 

 fish still alive at time t, the probability that the 

 Type A tag has been shed can be stated as 



n A (0 --■- 1 - pa0a(O 

 where #a(0 = exp 



can be dropped, i.e., the common probability of 

 shedding by time t is 



>(— JLa(u) duj. 



An analogous expression exists with respect to 

 tag B. Where shedding rates are assumed to be 

 the same for tag Types A and B the subscripts 



il(t) = 1 ■- P g{t). 



(2) 



If we set L(u) = L(constant), Equation (2) em- 

 bodies the assumptions of Bayliff and Mobrand 

 (1972)— Type II shedding is a simple Poisson 

 process with an identical constant rate for each 

 tag, so that each tag has the same probability of 

 shedding by time t. Moreover, in due course all 

 surviving fish will have shed both tags as long as 

 L>0, i.e., ft(oo) - 1. 



The validity of this particular set of assump- 

 tions has recently been challenged by results of 

 tag-shedding studies with northwest Atlantic 

 bluefin tuna, Thunnus thynnus, (Baglin et al. 

 1980) and with southern bluefin tuna, T. mac- 

 coyii, (Kirkwood 1981). In the former case, it was 

 found that the Type II shedding rate increased 

 with time. In Kirkwood's analysis it was appar- 

 ent that the Type II rate decreased markedly 

 over time. Therefore, it clearly would be advan- 

 tageous to construct a model permitting time- 

 dependent Type II shedding rates. Kirkwood 

 approached this problem by attacking the com- 

 mon assumption of uniform shedding probabili- 

 ties among all fish in the cohort. Instead, he con- 

 sidered the Type II shedding rate for each tag 

 applied to be constant over time, but further as- 

 sumed that the rate for each tag was a random 

 variable with specified probability density. In 

 this light, the deterministic model at Equation 

 (2) is replaced by the expectation J(t) = E[fl(t)] 

 — 1 — p E[g{t)]. The average time-varying shed- 

 ding rate at time t may now be defined as 



*(t) 



E{g(t)  L] 

 E{g(t)} 



where the expectations are taken with respect to 

 the probability density of L. Following standard 

 principles of reliability theory, this may be re- 

 duced to 



nn = - 



d\nE{g(t)} 



dt 



Under Kirkwood's assumptions V(t) will de- 

 crease with time as long as there is variation in 

 shedding rate among tags, i.e., there will be a 

 continuous culling of tags with relatively high 

 shedding probabilities. This concept is clearly an 



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