WETHERALL: ANALYSIS OF DOUBLE-TAGGING EXPERIMENTS 



be the probability of survival to time t, where 

 1 — 7r = probability of immediate mortality and 

 Z(n) = H(u) — L(u) is the time dependent instan- 

 taneous death rate. Then the expected number of 

 fish bearing a single tag of Type A at time t is 

 N*(t) = N d (0) S(t) J B (0(1 - Mt)). An analogous 

 expression may be written for Ns(t), and if the 

 two tags are considered identical, the subscripts 

 A and B may be dropped to yield 



NM) = mt) + N B (t) 



= 2NA0) S(t) J(t) (1 - J(t)), 



the expected number of fish still bearing a single 

 tag at time t. 



The expected number of double-tagged fish at 

 time t when A and B types are differentiated is 

 N d (t) = N d (0) S(t)(l - Mt))(l - Mt)) or when 

 no distinction is made between A and B types, 

 N d (t) = N d (0) S(t)(l - J(t)f. In any event, the 

 total number of fish expected at time t with one 

 or two tags remaining is N.(t) = N s (t) + N d (t). 



The processes described above are not directly 

 observable, so inferences about the shedding 

 rates must be made on the basis of catch statis- 

 tics. When recapture effort is exerted during a 

 time interval we assume it is applied continu- 

 ously. During a period of length A, beginning at 

 time t, the expected number of recaptures of 

 tagged fish in category j is therefore 



n+&i 



E{r jr ) = / F(u) Nj(u) du 



(6) 



The standard procedures for estimating shed- 

 ding rate parameters, and many of those to be 

 described shortly, rely on a sequence of ratios of 

 the estimated or observed number of recaptures 

 from the various categories during successive 

 fishing periods. It is clear from the equations 

 above that such ratios will be functions of r, and 

 the shedding parameters only, and independent 

 of Fj, N (l (0) and any parameters of the survival 

 function S(r ( ). 



Further, the ratios will be unaffected by non- 

 recovery or nonreturn as long as these processes 

 operate at constant levels with respect to re- 

 captures during a given time interval and at 

 the same rates for each tagged fish category. 

 Throughout this paper we assume this is so. 

 However, this latter condition is one which could 

 be violated easily, particularly if catches are not 

 inspected carefully for tag recaptures. Where 

 fish are handled individually there may be no 

 difference in recovery rates between single- and 

 double-tagged fish. Otherwise, recovery rates 

 may be greater in double-tagged individuals. 

 Once tagged fish are recovered, there may be 

 further problems with respect to return rates. 

 Laurs et al. (1976) in a study of shedding rates in 

 North Pacific albacore, T. alalunga, and Myhre 

 (1966) in experiments with Pacific halibut, Hip- 

 poglossus stenolepis, allowed for the possibility 

 that a certain proportion of double-tagged recov- 

 eries would be misreported as having only a 

 single tag. [For example, a fisherman might 

 pocket one of the tags as a souvenir, or one tag 

 might be simply lost after recapture.] 



where j = A, B, s, d,  . 



To complete the integral at Equation (6) it has 

 been customary to make two key assumptions at 

 this juncture (Chapman et al. 1965). First, we as- 

 sume the fishing mortality rate, F(u), is a step 

 function constant within each recapture inter- 

 val, i.e., F(u) = F, for t,< u < t, + A, Second, we 

 assume the average value of N : (u) during the in- 

 terval is approximately equal to Nj{t, + A,/2). 

 [This approximation is generally quite good for 

 A, of 1 yr or less. If Nj(u) is linear over the interval 

 the relation is exact regardless of A,.] Under 

 these conditions the set of recapture equations 

 becomes 



E(r fi ) = F % A, Nj( Ti ) 

 where r, = t,  + A, /2 . 



(7) 



ESTIMATION OF 



SHEDDING RATES AND 



PARAMETERS 



In the analysis of tag-shedding data a broad 

 range of objectives may be pursued, and these 

 give rise to a variety of estimation problems and 

 approaches. Fundamentally, of course, the 

 analyst wishes to correct systematic bias in esti- 

 mates of basic population parameters caused by 

 tag loss. There are several ways to do this. Where 

 concurrent single-tagging and double-tagging 

 experiments are conducted, information on shed- 

 ding rates from the double-tagging may be used 

 to compute adjustment factors, which in turn are 

 applied to recoveries from the primary single- 

 tagging study. Thus in single-tag estimation pro- 

 cedures based on Equation (1), for example, r, 

 would simply be replaced by 



691 



