Hampton: Natural mortality and movement rates of Thunnus maccoyu 



597 



and 



R»F 



r 22(t) = 



2 r 2 



1 + ab 



N 2 2(t)+bN 2 i(t) 



(l_e-«At) _ b 



N2i(t)-aN 22 (t) 



(l_e-vAt) 



?23(t) = R3F3 



L 23 



Ao - A, 



N 2 2(t) + N 2 3(t) 



2-^3 



(1-e-AsAt) t 23 (l-e-A 2 At) 



N 2 2 ( t) - 



A2-A3 



(7) 



Equations (6) and (7) describe the dynamics of tagged 

 fish as a function of the parameters Mj , M 2 , M 3 , F l , 

 F 2 , F 3 , T 12 , T 13 , T 2] , and T 23 . Fishery-specific report- 

 ing rates, Rj, R 2 , and R 3 , have been introduced and 

 are assumed known. The tag-shedding correction fac- 

 tor, w t , (Eq. 2) can be introduced in a similar fashion 

 to R (i.e., as a multiplier of F). Note that w t now 

 refers to the probability of retaining at least one tag 

 at the midpoint of period t + At and may differ accord- 

 ing to fishery of release. 



In practice, it is likely that the Fs will vary with time. 

 If catch or effort data are available by time period (in- 

 dexed by i), Fi, F 2 , and F 3 can be reparameterized as 



Fn = qifn ~ 



F 2i = q 2 4i ~ 



'3i 



Qsf; 



Cii 

 Pi' 



^2i 

 P 2 ' 



c 3i 



3i 



(8) 



where q 1} q 2 , and q 3 are catchability coefficients for 

 the three fisheries, f li? f 2i , and f 3i are the fishing 

 efforts in the three fisheries in time period i, P l , P 2 , 

 and P 3 are mean population sizes available to the 

 three fisheries over the course of the tagging experi- 

 ment, and Cjj, C 2 i, and C 3i are catches in the three 

 fisheries during period i (expressed in the same units 



as P). Variable F can now be accommodated without 

 the addition of extra parameters to the model. This, 

 as noted above, involves the assumption that the 

 tagged and untagged fish are equally vulnerable from 

 the moment of release. The choice of which param- 

 eterization to use will ultimately depend on the data 

 available and how one views the relationship between 

 catch, effort, catchability, and population size. In 

 fisheries for surface schooling tunas, effective effort 

 is extremely difficult to quantify, and no such estimates 

 are available for the southern bluefin fisheries. For this 

 reason, I have preferred the parameterization using 

 catch in this paper, although for ease of presentation 

 of simulation results, this has been done by assuming 

 the catch to be an index of effective effort and esti- 

 mates of q, rather than P, obtained. (By so doing, P 

 is in fact the reciprocal of q.) Note that Equations (8) 

 are approximations in the case of catch data, and re- 

 quire that the populations be close to equilibrium for 

 unbiased estimates to be obtained. 



Sibert (1984) used a least-squares technique to ob- 

 tain estimates of the various parameters for the two- 

 fishery model, and employed a square-root transforma- 

 tion as a weighting scheme for observations within the 

 four tag-recovery categories. Several methods of 

 weighting the four individual sums of squares were also 

 tested. Difficulties can arise in choosing the most ap- 

 propriate weighting scheme, both for observations 

 within and between return categories. These problems 

 can be avoided by using a maximum-likelihood tech- 

 nique based on multinomial probabilities (Seber 1973). 

 Here, a likelihood function can be constructed for each 

 fishery of tag release, e.g., for releases into fishery 1, 



n 01 ! (l-Pr^Noi-i n{p;y;;;p^;)p; 3 3 ;;;} 



fl ({ r ll(i). r 12(i). r 13(i)}) = 



i = l 



I r ll(i)-' r 12(i)! i"l3(i 



, (Noj-n,)! 



where n, is the total number of returns from fishery 1 releases, up to and including period k, p 11(i) is the probabil- 



ity of recovery in fishery 1 during period i (p 



ll(i) = 



■li(i) 



N 



)> Pi2(i) is the probability of recovery in fishery 2 during 



01 



