Booth et al.: Age validation, growth, mortality, and demographic modeling of Tnakis megalopterus 
103 
(IAPE) (Beamish and Fournier, 1981) with the follow- 
ing equation: 
IAPE{%) 
100 - 
N : 
i4.\ x v- x i 
R 
7=1 
X, 
where N 
R 
X « 
X, 
= the number of fish aged; 
= the number of readings; 
= j th vertebral count of the i th fish; and 
= the final agreed age of fish i. 
As with Goosen and Smale (1997), an IAPE calculated 
to be less than 10% was considered acceptable. 
Growth was modeled with the Schnute (1981) growth 
model. This four-parameter model is general and allows 
for the estimation of various nested models. The length 
of a shark at age a, L a , is modeled as 
l q = l£ + (z4-l?) 
l- g - a(a ^> 
l_ e ~ a ( t 2~ t l) 
where t 1 = the youngest fish in the sample; 
t 2 = the oldest fish in the sample; 
L 1 = the estimated length of a fish at age tp 
L 2 = the estimated length of fish at age t 2 , and 
a and j3 are the curvature parameters. 
By setting the parameter (5 to either 1 or — 1, the 
model reduces to either the von Bertalanffy or logistic 
growth model. Both of these nested models have three 
estimated parameters. The von Bertalanffy and logistic 
models are expressed as 
L a =L„(l-e"*< a -^) and L a 
-InL = — In d 2 , 
2 
„ 2 1^/ ~ \ 2 . 
where o - — L ia - L ia I is the maximum likelihood esti- 
,=1 mate of the model variance; 
L ia and L m are the observed and model predicted 
lengths of fish i at age a; and 
n is the number of observed data. 
The most parsimonious model was selected with the 
lowest Akaike information criterion (Akaike, 1973) of 
the form 
A1C = 2(-ln L + p), 
where p = the number of model parameters. 
A likelihood ratio test was used to test the null hy- 
potheses that there is no difference in growth param- 
eters between males and females (Simpfendorfer et al., 
2000; Neer et al., 2005; Natanson et al., 2006). Param- 
eter variability was estimated by using parametric boot- 
strapping with 1000 iterations and the 95% confidence 
intervals were estimated from the bootstrap results by 
using the percentile method (Buckland, 1984). 
Age at maturity 
Age at maturity, t m , was estimated directly from the von 
Bertalanffy growth model as 
1, 
JV 
l Go 
where l$ 0 = the length at maturity obtained from Smale 
and Goosen (1999); and 
, k and t 0 are the von Bertalanffy growth model 
parameters. 
where L x - the theoretical maximum size; 
k = the growth coefficient; and 
t 0 = the theoretical age at zero length. 
The von Bertalanffy ( /3= 1) and logistic (/3=-l) para- 
meters were calculated with the following equations: 
L = 
e at *ll-e at ']J ^ Vl 1 
Natural mortality 
Natural mortality was estimated from the median of 
Pauly’s (1980), Hoenig’s (1983) and Jensen’s (1996) 
empirical models of the form 
M 
Pauly 
exp(-0.0152 - 0.279 In + 0.6543 In /e + 0.463 InT) 
M Hoemg = exp(l.44 - 0.9821n£ max ) 
tr\ — t-i ~\~ tn In 
1 1 a 
I3(e 
L{-L{ 
and K = a. 
The Schnute growth model was fitted to the com- 
bined-sex data and the parameters were estimated by 
nonlinear minimization of a negated normal log-likeli- 
hood function of the form 
M Jensen = 16k ’ 
where L x and k are the Von Bertalanffy growth model 
parameters; 
T - the mean water temperature (estimated 
to be 16°C); and 
t max = the age of the oldest fish sampled (Hoenig, 
1983). 
