Xiao et al.: Instantaneous rate of tag shedding for Galeorhinus galeus and Mustelus antarcttcus 1 73 



Model 



Consider a (single) fish i that is captured, double tagged, and released at time t^^d). The index i can be used to 

 examine the effects of any factor on the instantaneous tag shedding rate. Let A and B indicate the two types of 

 tags and 



P(i,A,B,t(i)) = probability of retaining both tags at time <f J,); 



P(i,A,0,t(i)) = probability of retaining only tag A at time ifij; 



P(i,0,B,t(i)) = probability of retaining only tag B at time t(i); 



P(i, 0,0, t(i)) = probability of retaining neither tag at time <ffj; 



C (i,A,B.t(i>) = probability that it is caught at time t(i) and reported given that it has retained both tags; 



C (i,A,0,t(i)) - probability that it is caught at time t(i) and reported given that it has retained only tag A; 



C (i,0,B,t(i>> - probability that it is caught at time t(i) and reported given that it has retained only tag B; 



C a, 0,0, t(i)J = probability that it is caught at time t(i) and reported given that it has retained neither tag; 



U (i,A,B,t(i)) = probability that it is caught at time t(i) but not reported given that it has retained both tags; 



U (i,A,0,t(i)) = probability that it is caught at time t(i) but not reported given that it has retained only tag A; 



U (i,0,B,t(i)) = probability that it is caught at time t(i) but not reported given that it has retained only tag B; 



U (i,0,0,t(i)) = probability that it is caught at time t(i) but not reported given that it has retained neither 



tag; 



D (i,A,B,t(i)) = probability that it is dead at time t(i) given that it has retained both tags; 



D li,A,0,t(i>> = probability that it is dead at time t(i) given that it has retained only tag A; 



D(i,0,B,t(il) = probability that it is dead at time t(i) given that it has retained only tag B; 



D(i,0,0,t(i)) = probability that it is dead at time t(i) given that it has retained neither tag; 



n(i> = probability that it remains alive after type-I mortality (i.e. mortality due to the immediate 

 effects of tagging and handling); 



pa J) - probability that it retains tag 7 {j=A,B) after type-1 shedding (i.e. tag shedding due to the 

 immediate effects of tagging and handling); 



F(i,t(i)) = instantaneous rate of fishing mortality at time t{i)\ 



M(i,t(i>) = instantaneous rate of natural mortality at time t(i); 



R(i,A,B,t(i)) = probability of reporting given that it is caught at time t(i) and that it has retained both tags 



R(i,A,0,t<i)) = probability of reporting given that it is caught at time t(i) and that it has retained only tag A 



R(i,0,B,t(i)) = probability of reporting given that it is caught at time t(i) and that it has retained only tag B 



R(i,0,0,t(i)) = probability of reporting given that it is caught at time t(i) and that it has retained neither tag 



X{i,A,t(i)) = instantaneous shedding rate of tag A at time t(i>; and 



X(i,B,t(i)) = instantaneous shedding rate of tag B at time <((i. 



We assume that, in the time interval [t(i),t(i)+At], the probability that fish / retaining both tags is caught is 

 F{i,t(i))AtP{i,A,B,t(i}}+0(M), the probability that it is dead is Mli,t(i))AtP(i,A,B,tli)>+0(At>, the probability 

 that it sheds tag A is X(i,A,t(i))AtP(i,A,B,t(i))+0(At), and the probability that it sheds tag B is 

 X(i,B,t(i)>AtP(i,A,B,t(i))+0(AtJ, where O(At)-^0 as zi^^O. It is also assumed that these events are independent 

 with no more than one event occurring in the time interval. Under these assumptions, the probability that 

 fish i retains both tags at time t(i)+At given that it has retained both tags at time t(i) is given by 



Pli,A,B,t(i>+At)^[l-F(i,t(i))At-M(i,t(i))At-Mi,A,t(i))At-Mi,B,t(i))AtJP(i,A,B,t(i))+0(At). 



Taking the limit At^O and letting the dot above a quantity denote the first derivative of that quantity with 

 respect to t(l) yields 



P(i,A,B,t(i))=-[F(i,t(i))+M(i,t(i» +X(i,A,t(i))+X(i,B,t(i))]P(i,A,B,t(i)). 



This and similar arguments yield a tag shedding model of the form 



P{i,A,B,t{i)) = -[F(i,t(i)) + M(i,t(i)) + ?i{i,A,t(i)) + X(i,B,t(i))]P{i,A,B,t{i)) 

 P{i,A.O,t(l)) = -[F(i,t{l)) + M{i,t^i)) + ^i,A,t(i))]P(i.A,0,t(i)) + ^i,B.t(i))P{l,A,B,t^i)) 



