Natanson and Deacy: Changes in vertebral band-pair deposition rates with ontogeny in Carcharhinus plumbeus 
55 
Table 2 
Log-likelihood ratio values and estimates of von Bertalanffy growth function param¬ 
eters for 5 versions of the GROTAG model (Francis, 1988) fitted to tagging data for 149 
sandbar sharks tagged between 1965 and 2013 in the western North Atlantic Ocean. 
The mean annual growth rates were calculated at 60 cm FL and 160 cm FL, and 
standard errors of the mean (SEs) are given. For each version of the model, a param¬ 
eter was added. For a significant (P< 0.05) improvement in fit, the introduction of one 
extra parameter must increase the log-likelihood function (k) by at least 1.92 (Fran¬ 
cis, 1988). An asterisk (*) indicates fixed parameters. There are no data for models 1 
and 5 because of failed fits. AIC=Akaike information criterion; s=standard deviation 
of measurement error; m=mean measurement error; p =a contamination probability. 
Parameter 
Symbol (unit) 
Model 
1 
2 
3 
4 
5 
Log likelihood 
1 
-550.3 
-541.1 
-539.1 
AIC 
1108.6 
1092.2 
1094.1 
Mean growth rates 
G 60 (cm/y) 
14.17 
11.00 
10.65 
SE 
0.86 
Gi6o (cm/y) 
1.67 
1.67 
1.67 
SE 
0.48 
Growth variability 
V 
0* 
0.098 
0.166 
0.149 
0* 
Measurement error 
s (cm) 
9.260 
8.167 
7.647 
m (cm) 
0* 
0* 
5.164 
5.288 
0* 
Outliers 
P 
0* 
0* 
0* 
0.022 
larger individual, the OTC mark was at the third band 
pair past the birth band (Fig. 1). The smaller of these 
2 sharks (CM1031) was at liberty for 15.7 years, recap¬ 
tured at a size of 156 cm FL, and classified as mature. 
The larger shark (CM1027), however, was recaptured 
at a size of 147 cm FL after 10 years at liberty and, 
on the basis of size, likely immature (estimated size at 
50% maturity was 154.9 and 151.6 cm FL for females 
and males, respectively; Baremore and Hale, 2012). 
These 2 sharks had total band-pair counts of 16 and 
13, which are just over the number of band pairs at 
which Andrews et al. (2011) detected a shift from an¬ 
nual to non-annual band-pair deposition (10-12 years). 
Size at both tagging and recapture appears to influ¬ 
ence the amount of age underestimation in our OTC 
specimens. The percentage of time underestimated by 
band-pair counts (23.5-58.0%) increased in relation 
to size at recapture and secondarily decreased with 
size at tagging, findings that are consistent with the 
conclusion that larger sharks (approaching maturity) 
have slower growth and deposit fewer band pairs than 
smaller sharks (young of the year and juveniles). For 
example, 2 sharks (CM1027 and CM1029) at liberty for 
approximately the same time but tagged at different 
sizes and, therefore, at different ages, had different 
outcomes regarding degree of age underestimation. In 
examined vertebral sections of the smaller of these 2 
sharks (CM 1027), 3 band pairs were visible before the 
OTC mark and deposition of band pairs before and af¬ 
ter the OTC mark was annual. This individual’s total 
age at recapture was 13.0 years, close to the age up 
to which band-pair deposition has been validated as 
annual for the sandbar shark (12 years; Andrews et 
al., 2011). In contrast, the other shark (CM1029) was 
tagged at a larger size and at an estimated age of 10.0 
years, based on band-pair count before the OTC mark. 
The estimated total age for this shark was 19.5 years 
at recapture (10 band pairs + 9.5 years at liberty), past 
the validated ages for annual deposition (10-12 years). 
Band-pair deposition in this individual had slowed; 
therefore, its age was underestimated. 
Casey et al. (1985) aged the sandbar shark using 
vertebral band-pair counts and tag-recapture data. 
Their data indicated slower growth rates from us¬ 
ing tag-recapture data than from counting vertebral 
band pairs, leading them to suggest that ages based 
on vertebral band-pair counts underestimate true age. 
These findings were reinforced by the results of Casey 
and Natanson (1992) and Andrews et al. (2011), who 
used tag-recapture data and bomb radiocarbon dating, 
respectively. At the time of these earlier studies, the 
use of vertebral data was considered more robust than 
other methods, and age estimations based on vertebral 
band-pair counts were generally accepted over results 
from other methods of age determination. In particular, 
the older ages at maturity and longevity estimated by 
using models with tag-recapture data were not consid¬ 
ered reliable partly as a result of the low (often 
estimated by using this method because of a lack of 
large sharks reliably measured at tagging and recap¬ 
ture), which is highly linked to the k value (von Ber¬ 
talanffy, 1938). 
