104 
Fishery Bulletin 109(1 ) 
Demographic modeling 
The demographics of female T. megalopterus was mod- 
eled with an age-structured matrix model (Caswell, 
2001) of the form n t+1 =Axn t , where n t is a vector of 
numbers at age at time t and A is the Leslie projection 
matrix. In matrix formulation the model is expressed as 
N u+1 
<Pi 
02 
03 
0f m „ 
N u 
•^ 27+1 
Si 
0 
0 
0 
0 
N 2 , t 
= 
0 
S 2 
0 
0 
0 
X 
0 
0 
0 
0 
0 
0 
0 
0 
y 
N t,„ax.t 
\ w / 
where S a and tj> a = age-dependent survivals and fecundi- 
ties, respectively; and 
t max = maximum age considered in the 
analysis. 
The annual population growth rate (A), stable age 
distribution (w), and age-specific reproductive value (v) 
vectors were obtained by solving the equations Aw=Aw 
and v*A=Aw, where * is the complex conjugate trans- 
pose function. In the solutions, A is the common domi- 
nant eigenvalue and w and v are the corresponding 
right and left eigenvectors. The reproductive value vec- 
tor was normalized in relation to the age-1 value. 
The conditional intrinsic rate of increase (Gedamke 
et al., 2007) was calculated as r - InA, the average age 
of mothers of newborn individuals in a population with 
a stable age distribution as T-( w,v)=w'v, and the 
average number of female offspring per female during 
her lifespan as _R 0 =exp(rT) . 
The sensitivity of A to changes in the demographic 
parameters provides an indication of which parameter 
has the largest impact on the population growth rate. 
Sensitivity can be measured in either relative (as “sen- 
sitivity”) or absolute terms (as “elasticity”). Both forms 
of sensitivity were calculated from the individual values 
of the Leslie matrix, a t •, the population growth rate, 
and the left-right eigenvectors as 
3 A v i w j _ vw 
,J da t ■ (w,v) (w,v) 
Elasticities, or 
din A 
din a, ■ 
were calculated as 
such that 
IX‘u = i- 
i j 
Elasticities were summarized by age 
E ,='L e l ,j’ 
i 
fertilities 
tmax 
E 1 = X e 0 ,j ’ 
juvenile survival 
tmax «50 
^ = 11',, 
i= 2 7=1 
and adult survival 
E 2=l'f 
‘=2 7 =a 50 
(Mollet and Cailliet, 2002). 
Model implementation 
Age-dependent survival was estimated as a function of 
both age-independent instantaneous natural mortality 
(M), age-dependent selectivity (£ a ), and fully recruited 
fishing mortality (F), such that S 0 =exp(- M-^F). A maxi- 
mum age of t max - 26 was used in the analysis. All param- 
eters used in the analysis are summarized in Table 1. 
Age-dependent fertility was estimated as the number 
of embryos surviving to age-1 per female in a calendar 
year, E a , and was weighted by the proportion of females 
that were mature at age a, y/ a , such that 0 Q = S 0 y/ a E a . 
Because maturity, selectivity, and number of embryos 
are length- rather than age-dependent, all age-depen- 
dent values were calculated from the corresponding 
lengths predicted from the von Bertalanffy growth 
model for combined sexes. 
Both maturity-at-length, and selection-at-length, 
were modeled as logistic ogives as 
^=(l + exp(-(/-/ 5 V o)/<5 v ')) 
and , x 
^ = (l + exp(-(/-Z| 0 )/^)j , 
respectively, where l^ 0 and /g 0 are the lengths at which 
shark were 50% mature or at which 50% of sharks were 
selected by the fishery. The inverse rates of maturation 
and selection are denoted as d^'and S respectively. 
Because these rates were not available from the litera- 
ture, they were assumed to be 2% of their corresponding 
50% values. These rates were considered reasonable 
given other maturity and selectivity studies (Booth, 
unpubl. data). Length at 50% selectivity was estimated 
as 1326 mm TL, which is the mean length of sharks 
(n=252) measured from recreational anglers (Smale, 
unpubl. tag and release data). The number of female 
embryos per adult female at length l in a calendar year, 
given a gestational period of 20 months and a sex ratio 
of 1:1, was calculated as 
