FISHERY BULLETIN: VOL. 80. NO. 4 



attractive alternative to the Bayliff-Mobrand 

 model. 



Problems in which the instantaneous loss rate 

 is treated as a random variable arise in a variety 

 of contexts ranging from bioassay to analysis of 

 labor turnover in corporations. Because of its 

 unimodality and mathematical tractability, the 

 distribution often selected to describe this varia- 

 tion is the gamma distribution with mean A and 

 variance k 2 /b (e.g., see Bartholomew 1973:186 

 or McNolty et al. 1980). As Kirkwood (1981) 

 showed, for the tag-shedding problem, this 

 choice leads to 



J(t) = 1 - p 



b + kt, 



(3) 



so that 



nt) 



bk 



b + kt 



The Bayliff-Mobrand model is now seen as a 

 special deterministic case; when b — °°, J(t) 

 — 1 - p exp(— kt) and ^(0 — k. Kirkwood con- 

 sidered a further elaboration of Equation (3) by 

 assuming only a fraction of the tags, <5, will have a 

 nonzero probability of shedding; the remainder 

 are regarded as permanently attached. In this 

 event the expected probability of shedding by 

 time t is 



J(t) = 8 



1-P 



b + kt, 



(4) 



While this approach significantly advances 

 the realism and flexibility of tag-shedding the- 

 ory, it fails to account for the apparent increase 

 in average shedding rate as observed in the 

 Atlantic bluefin tuna. Thus, although permitting 

 variation in shedding rate among tags, it still 

 considers the rate for each tag to be constant over 

 time. 



This condition is not apt to hold. As Kirkwood 

 (1981) himself pointed out, plastic dart tags may 

 become so firmly imbedded and overgrown by 

 tissue as time passes that the probability of shed- 

 ding approaches zero. This is most apt to occur in 

 species which grow slowly, such as the southern 

 bluefin tuna. On the other hand, it is well known 

 that various metallic tags may corrode with time 

 and their shedding probabilities increase. Plas- 

 tic tags also deteriorate. 



Accordingly, consider the Type II shedding 

 rate to be a function of time, L(t). A relatively 



simple model for this situation is L{t) = a + fit {y ~ l \ 

 permitting a wide variety of forms for the instan- 

 taneous shedding process. In general, all three 

 parameters of this model could be specified as 

 random variables. Thus with a > 0, /3 > 0, and 

 — oo < y < oo the probability of shedding might 

 increase over time for some fish in a cohort, de- 

 crease for others, and be constant for the remain- 

 der. However, to simplify the analysis assume 

 here that y is fixed and identical for all members 

 of the cohort. Now if a and fi are independently 

 distributed as gamma random variables we have 



J(t) 

 and 



= 8 



1 - Pi 



bk_ 

 b + kt 



b + kt 



yc 



+ 



ye it 

 yc + gf 



yc + tjt' 



y-V 



(5) 



where the new symbols are £ , the expected value 

 of )8, and c, the reciprocal of the squared coeffi- 

 cient of variation of p. Hence, if between-tag var- 

 iability in a and /3 approaches zero, 



J(t) - 5<1 - p exp 



kt + £t y 



y 



and V(t) - A + Zt ly ~ u . In the basic model at Equa- 

 tion (4), for tags still in place at time t the condi- 

 tional probability of shedding in the interval (t, 

 t + dt) is independent of t. In the extended model 

 at Equation (5), this conditional probability may 

 also increase or decrease with t depending on 



7- 



While elaboration of the tag-shedding equa- 

 tions in this manner is straightforward, it is 

 doubtful whether a very clear discrimination be- 

 tween such parameter-laden models is possible 

 given the usual recapture statistics. Distinctions 

 between the extended models are reduced by the 

 integration of the shedding processes over sev- 

 eral recapture periods and are further obscured 

 by sampling variation. 



However, on the basis of these conceptual 

 models of the tag-shedding process we can now 

 write the well-known equations describing the 

 expected number of tags of a specified type still 

 attached at time t. For N,t(0) fish initially double- 

 tagged with Types A and B tags, let 



S(t) = w  exp 



(-\z(u)di) 



690 



