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Fishery Bulletin 101(2) 



i?,, = the last complete band; and 

 i?,i J = the next-to-last complete band. 



Mean MIR was plotted against month to determine trends 

 in band formation. A single factor analysis of variance was 

 used to test for differences in MIR among months. 



Chi-square tests of likelihood ratios (Kimura, 1980; Cer- 

 rato, 1990) implemented by using SAS code were used to 

 determine whether there were differences between sexes. 

 Theoretical longevity was estimated as the age at which 

 gS'X of L^ is reached (5(ln2)/A'; Fabens, 1965; Cailliet et 

 al., 1992). 



Estimation of size at maturity 



Median total length at maturity for male and female 

 sharks was determined by fitting a logistic model, Y=\l 

 d-i-e" '"*''-''^'), where y=the binomial maturity data (imma- 

 ture=0, mature=l; Mollet et al., 2000) and X=total length 

 (mm). Median total length at maturity was expressed as 

 MTL= -alb. The model was fitted using least squares non- 

 linear regression (S-Plus 2000, 2000). 



Estimation of natural mortality, 

 productivity, and elasticity 



The instantaneous rate of natural mortality (M) was 

 estimated by five indirect life-history methods described 

 extensively elsewhere (see Cortes, 2002, and references 

 therein). Four of the five methods (Pauly, 1980; Chen and 

 Watanabe, 1989; and two methods by Jensen, 1996) use 

 parameters estimated through the von Bertalanffy growth 

 model. The fifth method (Peterson and Wroblewski, 1984) 

 estimates M based on body mass. All required parameter 

 estimates (age at maturity, maximum age, L^, K, tg) were 

 taken from the aging section of the present study. Mean 

 annual water temperature (21.8°C), which is needed in the 

 Pauly method, was taken from Brusher and Ogren (1976). 

 Body mass of finetooth sharks at age was estimated by con- 

 verting age into length through the growth model derived 

 in the present study, and length into weight through the 

 power relationship given in Castro (1993). 



Population growth rates and productivity were estimated 

 by two methods that complement each other. Productivity 

 (i.e. rebound |)olential as defined in Smith et al., 1998) was 

 calculated by a modified demogi-aphic technique that incor- 

 porates concepts of density dependence (Smith et al., 1998). 

 In this method, rebound potentials or productivities (r,) 

 are calculated at the population level producing maximum 

 sustainable yield iMSY), which is assumed to occur at Z = 

 1.5 M or, alternatively, at Z = 2 M (Z=total instantaneous 

 mortality rate). 



Two methods that assume density independence also 

 were used. Life tables allowed calculation of mean genera- 

 tion times [A), and Leslie matrix population models were 

 used to estimate population growth rates (A=e'') and to 

 calculate elasticities (proportional sensitivities; Caswell. 

 2001 ). Elasticities for fertility, juvenile survival, and adult 

 survival were obtained by summation of matrix element 

 elasticities across relevant age classes (e.g. fertility elastic- 



ity is the sum of all first-row elasticities) and sum to 1. The 

 fertility term in matrix methodology includes survival to 

 age-1 (see Cortes, 2002, for details). 



Probability density functions (pdfs) were developed to 

 describe age at maturity, maximum age, M, survivorship 

 at age (S^=e"'^'), and fecundity at age im^_) for females, fol- 

 lowing in part the method and rationale in Cortes (2002). 

 Two scenarios, based on the results of the two equations 

 used to describe growth, were considered. 



Scenario 1 Age at maturity was represented by a trian- 

 gular distribution with 4.3 yr as the likeliest value and 

 ±1 yr (3.3, 5.3 yr) as lower and upper bounds. Maximum 

 age was represented by a linearly decreasing distribution 

 scaled to a total relative probability of 1. The likeliest value 

 corresponded to the age of the oldest animal aged in the 

 age and growth study (8 yr) and the lower bound was the 

 theoretical estimate of longevity (14.4 yr). 



Natural mortality (M) for adults for the Smith et al. 

 (1998) method was represented by a uniform distribu- 

 tion ranging from 0.162 (minimum) to 0.499 (maximum). 

 Conversely, annual survivorship at age in the density-inde- 

 pendent model was represented by a uniform distribution 

 that ranged from the minimum estimate (0.607=?" ^^^) to a 

 maximum, generally corresponding to the estimate derived 

 through the weight-based method (which ranged from 

 0.722/yr at age to 0.850/yr at age 15). Fecundity at age 

 was assumed to follow a normal distribution with a mean 

 of 4.036 and SD=0.793, with the lower and upper bounds of 

 2 and 6 reflecting the range of litter sizes reported for this 

 species (Castro, 1993). We assumed a 1:1 male-to-female 

 ratio, that lOO'^f of females were reproductively active after 

 reaching maturity, and a reproductive cycle of 2 yr. The 

 percentage of mature females at age was estimated from 

 the logistic model. 



Scenario 2 The age-at-maturity and fecundity-at-age 

 distributions were identical to those in scenario 1. The 

 only difference in the pdf describing maximum age in this 

 scenario compared to that used in scenario 1 was that the 

 theoretical estimate of longevity (9.9 yr) was lower Natural 

 mortality for adults for the Smith et al. ( 1998) method was 

 also represented by a uniform distribution ranging from 

 a minimum of 0.174 to a maximum of 0.528. Survivorship 

 at age in the density-independent model was represented 

 by a uniform distribution that ranged from the minimum 

 estimate (0.590=c "■''-■'*) to a maximum, generally corre- 

 sponding to the estimate derived through the weight-based 

 method (which ranged from 0.689/yr at age to 0.840/yr 

 at age 10). 



The simulation and projection process involved ran- 

 domly selecting a set of life-history traits from the pdfs de- 

 scribing each individual trait and calculating productivity 

 ti-y) in the modified demographic technique and population 

 growth rates (A), generation times (,\), and fertility, juve- 

 nile survival, and adult survival elasticities in the life table 

 and matrix population model approach. This process was 

 repeated 10. ()()() times, yielding frequency distributions, 

 means, medians, and confidence intei-\'als (calculated as the 

 2.5th and 97.5th percentiles) for parameter estimates. All 



