Hampton: Natural mortality and movement rates of Thunnus maccoyu 



599 



P(x) = P(iv) • [l-e-^tt-*)] 



T23 

 A,' 



P(xi) = P(v) • [l-e- A 3d-^] ?1, 



A3 



simulation model. The HSH model can be similarly 

 tested, with exact recapture times within the deter- 

 mined period of capture simulated by randomly 

 sampling from a truncated exponential distribution 

 (truncated at one year in this case) as shown in Hearn 

 et al. (1987). 



and 



P(xii) = P(v) • [l-e- A 3d-x)] 



P(xiii) = P(v) • e-W- 



M3 

 A3' 



Fj , F 2 , and F 3 may be allowed to vary by specify- 

 ing a constant q or P and dependent f or C (as per Equa- 

 tions 8). An identical process deals with releases into 

 fishery 2. 



To determine which of the possible outcomes (i) to 

 (v) first befalls a tagged fish, a pseudorandom number, 

 a, uniformly distributed on [0, 1] is generated using a 

 computer subroutine (e.g., subroutine URAND given 

 in Forsythe et al. 1977:245). If a<P(i), outcome (i) is 

 chosen; if P(i)<a<[P(i) + P(ii)], outcome (ii) is chosen; 

 if [P(i) + P(ii)]<a<[P(i) + P(ii) + P(iii)], outcome (iii) is 

 chosen; and so on. Additional pseudorandom numbers 

 are generated and further tests relating to outcomes 

 (vi) to (xiii) applied as necessary until the fish is deemed 

 to have been recaptured, died naturally, or survived 

 to the end of the experiment. When the fates of all 

 tagged fish released into both fisheries are determined 

 in this way, the six tag-return vectors, r n , r 12 , r 13 , 

 r 2 i, r 22 , and r 23 are established. The SE model can 

 then be fit to these data and the estimated parameter 

 values compared with the "real" values input to the 



Table 3 



Estimates of the rate of natural mortality (M) and their standard errors (SE) for different report- 

 ing rates (R) obtained from fitting the HSH model to data from experiments 1-4 (see Table 

 1 for descriptions). Separate estimates are given for two tag-shedding models derived for ex- 

 periment 1. All estimates are in units per year. 



Experiment 1 



Experiment 2 Experiment 3 Experiment 4 



R 



Constant 

 shedding rate 



M SE 



Decreasing 

 shedding rate 



M SE 



Constant 

 shedding rate 



M SE 



Decreasing 

 shedding rate 



M SE 



1.0 



0.9 

 0.8 

 0.7 

 0.6 

 0.5 



0.1987 

 0.1896 

 0.1792 

 0.1669 

 0.1519 

 0.1330 



0.0681 

 0.0674 

 0.0665 

 0.0654 

 0.0640 

 0.0620 



0.4038 

 0.3941 

 0.3827 

 0.3693 

 0.3527 

 0.3312 



0.0634 

 0.0625 

 0.0613 

 0.0599 

 0.0580 

 0.0553 



0.2275 

 0.2239 

 0.2194 

 0.2122 

 0.1965 

 0.1148 



0.1740 

 0.1604 

 0.1420 

 0.1158 

 0.0733 

 0.0205 



0.4165 

 0.4091 

 0.4002 

 0.3891 

 0.3748 

 0.3549 



Results 



HSH model 



The jackknife estimates of M and their standard errors 

 obtained by fitting the HSH model to data from ex- 

 periments 1-4 are given in Table 3. If full reporting 

 of tags is assumed, the estimates of M range from just 

 less than 0.2/year to just more than 0.4/year; the esti- 

 mates decrease slightly as reporting rate decreases. 

 It is clear that the tag-shedding model used to weight 

 the returns has a large bearing on the estimate of M 

 obtained. For experiment 2, the best fitting tag- 

 shedding model (constant shedding rate) predicts a very 

 low probability of tag retention after long periods at 

 liberty (Hampton and Kirkwood 1990). Therefore, 

 those returns from the Japanese fishery at liberty for 

 longer than, say, 6 years will receive large weight in 

 the analysis using the HSH method; this is one of the 

 main reasons for the relatively low estimate of M 

 (0.2275/year for a reporting rate of 1.0). In contrast, 

 a decreasing tag-shedding rate model provided the best 

 fit to the double-tagging data from experiments 3 and 

 4. Here, there is little change in the probability of tag 

 retention after about 3 or 4 years at liberty (Hampton 

 and Kirkwood 1990). All returns after this time will, 

 then, receive similar weight from the tag-shedding 

 model; accordingly, relatively high estimates of M are 

 obtained for experiments 3 and 4 (0.4165/year and 



0.4163/year, respectively, 

 for a reporting rate of 1.0). 

 The most direct test of the 

 effect of the different tag- 

 shedding models is the ap- 

 plication of both constant 

 and decreasing shedding- 

 rate models to the analysis 

 of experiment 1 (these tag- 

 shedding models provided 

 equally good fits to experi- 

 ment 1 double-tagging data, 

 and could not be distin- 

 guished on the statistical 

 criterion used by Hampton 

 and Kirkwood 1990). Here, 

 the estimate of M obtained 

 when the constant shed- 

 ding-rate model was used 



Decreasing 

 shedding rate 



M SE 



0.0860 

 0.0815 

 0.0763 

 0.0702 

 0.0628 

 0.0538 



0.4163 

 0.4098 

 0.4023 

 0.3937 

 0.3835 

 0.3710 



0.0646 

 0.0633 

 0.0619 

 0.0601 

 0.0581 

 0.0556 



