FISHERY BULLETIN: VOL. 80, NO. 4 



lation, losses may be caused by fishing mortality 

 or recapture, natural mortality, tagging mortal- 

 ity (mortality induced by the application or 

 presence of the tag), permanent emigration, or 

 by tag shedding. In addition, recaptured tags are 

 considered "lost" if not detected in the catch and 

 recovered, or, when recovered, if not returned or 

 reported. 



bined effects of fishing mortality, F(x), natural 

 mortality, M(x), instantaneous tag shedding, 

 L(x), and remaining losses, G(x). The usual as- 

 sumptions are that the Type II losses operate in 

 the manner of independent Poisson processes 

 with constant rates and that the recovery and 

 reporting rates also are constant. Under these 

 conditions the model takes the familiar form 



E{r) = 77AUO)| 



F + X 



) (l - expHF + X)A,)) (< 



exp(-(F + X)t)) (1) 



') 



Beverton and Holt(1957) recognized twokinds 

 of losses, which they designated Type I and Type 

 II (these are called Type A and Type B by Ricker 

 1975). Type I losses are those which, in effect, re- 

 duce the number of tags initially put out. They 

 result from the pulse of tagging mortality and 

 tag shedding occurring immediately after re- 

 lease (or in a relatively brief period following 

 release) and from the nonrecovery and nonreport- 

 ing of tag recaptures. Type II losses are those 

 happening steadily and gradually over an ex- 

 tended period following release of the tagged 

 fish. 



These relations may be stated more succinctly 

 in a simple mathematical model. Let E(r,) denote 

 the expected number of returns of tags recap- 

 tured in the ith time interval following the re- 

 lease of iY,(0) single-tagged fish. Then 



where 17 

 X 



TT P I 



M + L + G. 



A variety of estimation schemes based on this 

 equation have been developed, notably by Paulik 

 (1963). These have been reviewed along with 

 other mark-and-recapture approaches by Cor- 

 mack (1968) and Seber (1973). The importanceof 

 assumptions on Type I and Type II losses in these 

 procedures depends on which parameters are of 

 central concern in the experiment. In fisheries 

 applications the parameter most often focused on 

 is the fishing mortality rate, F. Paulik's single- 

 release regression model with constant A, for 

 estimating F and the exploitation rate, n=(F/(F 

 + M)) [1 - exp(-(F+ M)A)] stems directly from 

 Equation (1). In this situation, if Type I losses are 

 present the model will estimate -qF rather than 



E(r,) = n p N S (Q) / F{u) J(u) exp (- / H(x) dx)du 



u \ / 



where 



t, 



A, 



1 - TV 



1 ~ P 



time at the beginning of in- 

 terval i 



length of time interval i 



probability that a tag is lost 

 due to immediate tagging 

 mortality 



probability that a tag is shed 

 immediately following re- 

 lease 



instantaneous fishing mor- 

 tality rate at time u 



probability that a tag recap- 

 tured at time u is not re- 

 covered and reported (a 

 Type I loss) 

 H(x) = total instantaneous rate of 

 Type II tag loss at time x. 



Here H(x) represents the unspecified com- 



F(u) 

 1 - { (u) 



F. Subsequent estimates of X will be too large. 

 Further, estimates of the exploitation rate will 

 be negatively biased, and if these are used along 

 with N s (0) to estimate total population sizes, such 

 estimates will be inflated. Of course, this is the 

 general effect of Type I losses on Petersen esti- 

 mates. 



If Type II tagging mortality or Type II tag 

 shedding occur, the estimate of F'from this sin- 

 gle-release model will not be affected, but the 

 estimate of n will be less than the true exploita- 

 tion rate of the unmarked population. 



Sometimes all recaptures are made during 

 subintervals of equal length imbedded and ir- 

 regularly spaced within the total recapture peri- 

 od (e.g., in a salmon fishery with a complex pat- 

 tern of open and closed periods). In this case, 

 Paulik shows that the single-release model based 

 on Equation (1) will give estimates of Fand X 



688 



