Tribuzio et al.: Age and growth of Squalus acanthias in the Gulf of Alaska 
123 
Table 2 
Growth models fitted to spiny dogfish ( Squalus acanthias) length-at-age (L t ) data. Parameters are: asymptotic length (L a ), the 
growth coefficient (k), length at birth (L 0 ), age at size zero (f 0 ), a phase change parameter (A t ) for the two-phase model, age at 
transition (t h ), magnitude of the maximum difference between model 1 and the two phase model (h), time increment from previ- 
ous t value (5), and the inflection point of the logistic curve (a). 
Model number 
Model name 
Model equation 
Reference 
1 
vB 1 
L , 
= Z, oo (l-e" t( '"' o) ) 
von Bertalanffy (1938) 
2 
vB 2 
L 
t 
ll 
i 
1 
TO 
a- 
Fabens (1965) 
3a-3c 
Two-phase vB with L 0 
L 
= A- s +b.-Aj*( i-v 4 -'*""-'’). 
This study 
1 + /ope« h -<) 
4 
Gompertz 
L 
t 
= Le 1 1 
Ricker (1975) 
L 
5a-5c 
Two-parameter Gompertz 
L t 
= V ] >G = 
In — 
L 
Mollet et al. (2002) 
L 
6 
Logistic 
L 
t 
, -k(t-a) 
l + e [ 1 
Ricker (1979) 
follow a logistic pattern and remain in the second phase. 
Another problem we encountered fitting the two-phase 
model was that the typical differential form of the vB 
equation can result in a decrease in length at the tran- 
sition between phases. To prevent this unlikely result 
the difference equation form of the vB equation (Gulland 
1969) was used in this analysis. 
Model parameters for equations describing the num- 
ber of worn bands or growth were fitted by nonlinear 
least-squares regression or ordinary least-squares re- 
gression, and confidence intervals were estimated by 
a bootstrap procedure with 5000 replicates by using R 
statistical software (R, vers. 2.10.0, www.r-project.org). 
Confidence intervals (95%) for parameter estimates 
were based on the lower and upper 2.5 th percentile of 
the bootstrap replications. Parameters were considered 
significantly different if the 95% confidence intervals 
did not overlap. To evaluate best model fit for the male 
and female datasets, Akaike information criteria (AIC) 
and model summary statistics were calculated (Burn- 
ham and Anderson, 2004). 
Fraser-Lee back-calculation methods (Francis, 1990; 
Campana, 1990; Goldman et al., 2006). The Fraser-Lee 
method produced results that on an individual level 
could be quite unreasonable (large negative ages), but 
on average were more biologically reasonable than either 
of the Dahl-Lea methods. Further, growth model results 
with either of the Dahl-Lea methods were unreasonable 
(L x of >150 cm TL ext ), therefore, we used the Fraser-Lee 
method for our data. Thus, the following equation was 
used to estimate back-calculated length-at-age data: 
TL, = TL, + 
[EBD,-EBD c )(TL c -TL bh 
birth , 
EBD c - EBD birth 
(3) 
where TL i = the back calculated length; 
TL C = the length at capture; 
TL birth = the length at birth; 
EBD i = the enamel base diameter at band /; 
EBD c - the enamel base diameter at capture; and 
EBD hirth = the enamel base diameter at birth. 
Back-calculation methods 
Owing to a paucity of specimens with EBD< 3.5 mm, 
back-calculation methods were used to fill in the size 
range missing from samples. The spine diameter at each 
band along the spine (hereafter called “band diameters”) 
was measured from a random subsample of 153 unworn 
spines for use in the estimation of worn bands (Eqs. 1-4); 
spiny dogfish with unworn spines tend to be smaller and 
younger than those with worn spines. We examined the 
Dahl-Lea, linear Dahl-Lea, and size at birth modified 
Results 
Sample collection 
A total of 1713 spiny dogfish were sampled over the four 
years of the study (585 males, 1128 females, Table 1) of 
which 537 male and 1062 female spines were usable. 
Lengths ranged from 56 to 99 cm TL ext for males, and 
56 to 123 cm TL ext for females. The x 2 test revealed no 
significant differences between the mean length at age 
