598 



Fishery Bulletin 89(4), 1991 



period i (p 12 {i> = ' ), p 13(i) is the probability of recovery in fishery 3 during period i (p 13(i , = - — ), and 



N, 



01 



N 



01 



Pl'l = 2! Pll(i) + Pl2(i) + Pl3(i)- 

 i=l 



An equivalent function, f 2 ({r 21 (j), >"22(i)> r 23(i)}) can De written for releases into fishery 2. Estimates of param- 

 eters may then be found by minimizing 



L = -log e [fi({r 11(1) , r 12( i), r 13(i) }) • f 2 ({r 2 i(i), r 22 (i), r 23(i) })] 



with an estimate of the variance-covariance matrix 

 found using the inverse-Hessian method (Bard 1974). 

 Simulation trials using the model described below 

 showed that unbiased results are obtained using the 

 maximum-likelihood technique, whereas unbiased re- 

 sults could not be guaranteed with the least-squares 

 approach. In the applications of the SE method pre- 

 sented here, the maximum-likelihood estimation tech- 

 nique is used. 



Simulation model 



In order to examine the behavior of the HSH and SE 

 estimates, a simulation model was developed. The 

 simulation model determines the fate of each tagged 

 fish released into the two fisheries in a probabilistic 

 fashion. For the moment, consider only releases into 

 fishery 1. During the first time period after release, 

 a tagged fish will either: 



(i) be recaptured in fishery 1; 

 (ii) die from natural causes in fishery 1; 

 (iii) survive in fishery 1 to the end of the first period 



and then be subject to all possibilities in the next 



time period; 

 (iv) migrate to fishery 2 at time x<l; 

 (v) migrate to fishery 3 at time x < 1 ; 

 (vi) given (iv), be recaptured in fishery 2 at time 



y(x<y<i); 

 (vii) given (iv), die from natural causes in fishery 2 



at time y (x<y<l); 

 (viii) given (iv), survive in fishery 2 until the end of 



time period 1 and then be subject to possibilities 



(vi) through (x) in the next time period; 

 (ix) given (iv), migrate back to fishery 1 at time 



y (x<y < 1) and then be subject to all possibilities 



for the remainder of time period 1; 

 (x) given (iv), migrate to fishery 3 at time y (x<y 



<1) and then be subject to possibilities (xi) 



through (xiii) in fishery 3 for the remainder of 



time period 1; 



(xi) given (v), be recaptured in fishery 3 at time 



y_(x<y<l); 

 (xii) given (v), die from natural causes in fishery 3 at 



time y (x<y<l); 

 (xiii) given (v), survive in fishery 3 until the end of 



time period 1 and then be subject to possibilities 



(xi) through (xiii) in the next time period. 



The probabilities of each of these events occurring 

 are (omitting, for convenience, the time subscript): 



P(i) = [1-e-Ai] *A 

 A] 



P(ii) = [l-e- A i] 



P(iii) = e" A i, 



Mi 



A/ 



P(iv) = [l-e- A i] — , 

 Ai 



P(v) = [l-e- A i] =^, 



Ai 



P(vi) = P(iv) - [l-e- A 2<i-*)] -£, 



A 2 



P(vii) = P(iv)  [l-e-^d-*)] 



P(viii) = P(iv) • e - A 2d- x ), 



P(ix) = P(iv) • [l-e- A 2"- x >] 



M, 



T 21 



