Goldman and Musick: Growth and maturity of Lamna ditropis 



281 



soft Inc., Seattle, WA) to estimate parameters. The von 

 Bertalanffy growth function is 



L, =L„[l-exp(-^(^ -/(,))], 



where L, = length at age 7'; 



L^ = asymptotic or maximum length; 

 k = the growth coefficient; and 

 t^ = age or time when length theoretically equals 

 zero. 



Growth parameters were estimated for the sexes sepa- 

 rately and combined. A likelihood ratio test was used to 

 determine whether differences between female and male 

 growth parameters were significant or if a single set of 

 growth parameters described the data better (Kimura, 

 1980; Quinn and Deriso, 1999; Haddon, 2001) (SAS, 

 vers. 8.0, SAS, Gary, NO. 



Back-calculation is a method for describing the growth 

 history of each individual sampled, and numerous varia- 

 tions in methods exist (see Francis, 1990, for a thorough 

 review and Goldman, 2004, for a description and ap- 

 plication to elasmobranchs). Back-calculations estimate 

 lengths-at-previous-ages for each individual and should 

 be used if sample sizes are small and if samples have not 

 been obtained from each month (Goldman, 2004; Cailliet 

 and Goldman, 2004). Cailliet and Goldman (2004) stated 

 "the proportional relationship between animal length 

 or disk width and the radius of the vertebral centrum 

 among different length animals within a population is 

 used as a basis for empirical relationships regarding 

 population and individual growth, as is the distance 

 from the focus to each annulus within a given centrum." 

 Hence, choosing the appropriate method (based on the re- 

 lationship between animal length or disk width and the 

 radius of the vertebral centrum) is critical if back-calcu- 

 lated data are to be used for obtaining accurate life his- 

 tory parameter estimates from growth function models. 

 Because smaller size classes were not well represented 

 in our female sample and because our male sample size 

 was small, lengths at previous ages were back-calculated 

 from centrum measurements for both sexes and fitted 

 with the von Bertalanffy growth function. 



The relationship between OR and PCL for ENP salm- 

 on sharks was used to determine the most appropriate 

 method for back-calculating previous length-at-age. To 

 our knowledge, however, no studies on elasmobranch 

 fishes have examined multiple back-calculation methods 

 for statistical or biological accuracy in relation to ver- 

 tebral sample data. To that end, four different propor- 

 tion methods were used and compared with our sample 

 length-at-age data. We first applied the standard Dahl- 

 Lea direct proportions method (Carlander, 1969): 



L,{LJCR,.}xCR^, 



where L, = length at ring 'f ; 



L, = length at capture; 



CR^ = centrum radius at capture; and 



CR^ = centrum radius at ring '/'. 



(1) 



Next, we applied two modified versions of the Dahl-Lea 

 method that use parameter estimates from the specific 

 linear and quadratic fits that described the PCL-CR rela- 

 tionship. The linear-modified Dahl-Lea method (Francis, 

 1990) is 



L, = L. X 



[{a + hCR,)/{a + bCR^.)], 



(2) 



where 'a' and 'h' are the linear fitted parameter esti- 

 mates. The quadratic-modified Dahl-Lea method (Fran- 

 cis, 1990) is 



L, = L,, X [ia + bCR, + cCRf)/{a + bCR,. + cCR^ )], 



(3) 



where 'a' 'b.' and V are the quadratic-fitted parameter 

 estimates. 



Kicker (1992) applauded Francis's (1990) back-calcu- 

 lation review paper, but like Campana (1990) suggested 

 that the point of origin of proportional back-calcula- 

 tions should be related to a biologically derived inter- 

 cept (i.e., length at birth). We, therefore, also applied 

 Campana's (1990) "size-at-birth-modified" Fraser-Lee 

 equation: 



L, = L +[(Ci?, -Ci?,,)x(L,,-L;j„.„, )/(C/e,.-Ci?B„,,, )], (4) 



where L^,,.,,, = length at birth; and 



^^Biiih - centrum radius at birth. (Based on Tan 

 aka, 1980, 62.5 cm PCL was used for 



Von Bertalanffy growth parameter estimates were 

 obtained from all individual and mean back-calculated 

 length-at-age, and from a combination of back-calculated 

 lengths-at-age and our sample data. A likelihood ratio 

 test was used for all scenarios to determine whether 

 differences between female and male growth parameters 

 were significant or if a single set of growth parameters 

 described the data better (Kimura, 1980; Quinn and 

 Deriso, 1999; Haddon, 2001) (SAS vers. 8.0, SAS, Gary, 

 NC). Back-calculated length-at-age results from all four 

 methods were examined to see which best reflected our 

 vertebral sample data. 



The reproductive tracts of 64 female and 14 male 

 salmon sharks were examined visually to assess their 

 reproductive status. Females ranged in size from 71 to 

 209 cm PCL, and males ranged from 63 to 187 cm PCL. 

 Clasper lengths were obtained from 12 of the males 

 (from 91 to 187 cm PCL). Gross analysis of reproduc- 

 tive tracts and maturity determinations were made ac- 

 cording to the methods of Pratt (1988), Gilmore (1993), 

 Pratt and Tanaka (1994), Hamlett (1999) and Hamlett 

 and Koob (1999). 



Median precaudal length-at-maturity (MPCL) was 

 determined by first coding female (n = 64) and male 

 (n=14) maturity data into binary form, with 0=imma- 

 ture and l=mature. The binary data were fitted with a 

 logistic regression model ("GLM" in S-Plus, Professional 

 Release 1, Mathsoft Inc., Seattle, WA). The median 

 precaudal length-at-maturity was then estimated as 



