Xiao et al.: Instantaneous rate of tag shedding for Galeorhinus galeus and Mustelus antarcticus 



181 



male) combined -log-likelihood of 45.0582 ( = 18.1029+ 

 26.9553). The increase in value of the -log-likelihood 

 function for an extra parameter is, again, statistically 

 significant (x2iooi69=2x*47. 9105-45.0582)^5. 7046), 

 again suggesting significant differences in tag shed- 

 ding rates between sexes for gray Petersen discs. No- 

 tice, in these cases, that tag shedding rates for males 

 nearly doubled those for females. For the second tag- 

 ging experiment, no differences in tag shedding rates 

 were found among sexes for either species of shark 

 (Table 4). 



The shedding rate of Petersen discs for the school 

 shark was very high. When combined with a 50-mm- 

 long and 23-mm-wide internal tag (J-tag), white 

 Petersen disc (W-tag) had a shedding rate of 

 Af(,B,^fiiJ=0.2829(±0.1104)/yr or 100x(l-e-0 2829) = 

 24.64%/yr for males, and A('/,fi,;f/J)=0.5816{+0.2946)/ 

 yr or 44.10%/yr for females (rows 1-3, Table 3). When 

 combined with a 50-mm-long and 22-mm-wide inter- 

 nal tag (L-tag), gray Petersen disc (G-tag) had a shed- 

 ding rate of Af/,S,mi)= 1. 1439 (±0.2534 )/yr or 68. 14%/ 

 yr for males and Af/,B,<ffJJ=0.5202 (±6.1016)/yr or 

 40.56%/yr for females (rows 7-10, Table 3). When 

 combined with a 35-mm-long and 10-mm-wide inter- 

 nal tag (S-tag), gray Petersen disc (G-tag) had a shed- 

 ding rate of Af;,fi,m))=4.5992 (±1.0705 )/yr or 98.99%/ 

 yr for males and X(i,B,tfi))=2.3509 (±0.4955 )/yr or 

 90.477f/yr for females (rows 15-18, Table 3). Other 

 combinations of tag type and tagging position for the 

 first tagging experiment did not yield reliable (in 

 accuracy and precision) estimates of tag shedding 

 rate because of insufficient data. 



For the second tagging experiment, tag shedding 

 rates varied considerably for both species of sharks 

 (rows 1-10 and 22-30, Table 4). However, dart tags 

 had a higher shedding rate than either Roto or Jumbo 

 tags. For example, for male gummy shark tagged in 

 the fin, dart tags had an instantaneous shedding rate 

 of 0.5642 (±0.3260)/yr and Jumbo tags 0.2133 

 (±0.2125)/vr (row 2, Table 4). For either gummy or 

 school shark, the shedding rate of dart tags placed 

 in the fin was about half that of dart tags placed in 

 the muscle (rows 1-10 and 22-30, Table 4). 



Discussion 



We developed a simple tag shedding model (Equa- 

 tions 1-4) to account for the effects of time at liberty, 

 sex, size, tag position, and other factors and used a 

 special case to estimate the instantaneous shedding 

 rates of Petersen discs, Roto tags, and dart tags in 

 two species of sharks. It can be used to estimate the 

 shedding rates of two tags, singly or in combination, 

 and has two interesting features. In Equation 1, both 



F(i,t(i)) and M(i,t(i)) are independent of the 16 state 

 variables. This independence ensures thatP(iA,B,t(i)), 

 P(iA,0,t(i)), P(i,0^,t(i» and P(i,0,0,t(i)) are all express- 

 ible as a product (Equation 2), which in turn ensures 

 that terms involving ffz.ffzJi and M(i,t(i>) in the like- 

 lihood function (Equation 3 or 4) are cancelled out. 

 Thus, as in Xiao (1996a), our tag shedding model 

 applies, even when F(i,t(i)) and M(i,t(i)) are arbitrary 

 functions of time t(i). On the other hand, if fishing 

 and natural mortalities depend on the state variables 

 of tags A and B, then terms in P(i,A,B,t(i)), 

 P(iA,0,t(i)),P(i,0,B,t(i» and P(i,0,0,t(i))invo\ying{ouT 

 fishing mortalities F(i,A,B,t(i)), F(i,A,0,t(i)), 

 7^f;,0,B,^fiiJ and Ff(,0,0,«/Ji and four natural mortali- 

 ties M(i,A,B,t(i)), M(i,A.O,t(i>), M{i,0,B,t(i)) and 

 M(i,0,0,t(i)) cannot be factored out. Then, for esti- 

 mation of parameters by maximizing Equation 3, 

 particular functional forms of all the eight mortali- 

 ties must be hypothesized. This tag shedding model 

 is more general but more data-demanding. The other 

 interesting feature of our tag shedding model is that 

 Equation 3 is independent of probabilities of report- 

 ing R(i,A,B,t(i)), R(i,A,0,t(i)), R(i,0,B,t(i)) and 

 R(i,0,0,t(i)) if these probabilities are identical, arbi- 

 trary functions of time t(i) because of the way they 

 enter Equation 3. 



Statistically significant differences in shedding 

 rates of Petersen discs between male and female 

 school sharks were detected when many fish were 

 recaptured. We do not know why such differences 

 existed but we postulate that male sharks have a 

 higher tag shedding rate because they are more ac- 

 tive and would tend to rub off the tags and that fe- 

 male sharks have a lower tag shedding rate because 

 they are larger and have thicker fins. An external 

 fin tag, such as a Petersen disc, is shed only after its 

 pin or locking mechanism has cut through the fin. 

 The larger the tagged fish, the thicker is its fin and 

 hence the farther the distance its pin or locking 

 mechanism has to cut through to the posterior edge 

 of the fin. Consequently, larger animals have lower 

 shedding rates. Thus, sex is confounded in its effects 

 with size. That is probably why the length at release 

 of school sharks did not affect the shedding rates of 

 Petersen discs within a wide size range examined, 

 although the loss of anchor tags (Floy tags) was size- 

 dependent for striped bass Morone saxatilis 

 ( Waldman et al., 1990 ) but size-independent for lake 

 trout Salvelinus namaycush (Fabrizio et al., 1996). 

 We could not detect differences between sexes with 

 fewer recaptures, however, because the use of Equa- 

 tion 1 or 2 to resolve sexual differences in tag shed- 

 ding rate requires many recaptures (see below). 



Shedding rates of Petersen discs, Roto tags, and 

 dart tags did not change with time at liberty. Some 



