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Fishery Bulletin 115(3) !j 
Figure 2 
Image of a transverse section of the centrum of a vertebra of a nar- 
rownose smooth-hound ( Mustelus schmitti ) caught in Anegada Bay, 
Argentina, in 2008. The white dots indicate growth bands (narrow 
and hypermineralized). The first dot from the centrum is the birth 
mark. 
We modeled growth by using a multimodel approach, a 
method recommended over the use of a single-model ap- 
proach (Cailliet et al., 2006; Katsanevakis and Marave- 
lias, 2008; Ogle, 2015). To fit the age data, 7 candidate 
growth models were chosen a priori: 3 variants of the 
von Bertalanffy growth function (VBGF) (von Berta- 
lanffy, 1938), plus Francis and Mooij parameterizations 
of the VBGF model (Table 1; Francis, 1988; Mooij et 
al., 1999); the logistic function (Ricker, 1976); and the 
Gompertz function (Gompertz, 1825) with the parame- 
terization of Ricker (1976). The 3 variants of the VBGF 
model used the original, traditional, and fixed length- 
at-birth parameter (L 0 ), respectively. The advantage of 
the model with the Francis parameterization lies in the 
noncorrelation between the growth coefficient ( K) and 
the asymptotic length (ZO parameters (Francis, 1988). 
The model with the Mooij parameterization estimates 
the (Gj n j t ) parameter, which has a clear biological inter- 
pretation as the initial growth rate in length per year; 
in contrast, the VBGF growth parameter K cannot be 
interpreted biologically (Mooij et al., 1999). 
For the fixed-L 0 variant model and the model with 
the Mooij parameterization, L 0 was calculated by fol- 
lowing Conrath (2005) and using a value that repre- 
sents a compromise between the largest embryo and 
the smallest free-living individual captured. For the 
original VBGF and logistic models, L 0 was estimated 
by the nonlinear least-squares routine. The use of L 0 
is biologically meaningful for chondrichtyan 
fishes (Cailliet et al., 2006). The parameter 
can be compared directly between mod- 
els (the Francis parameterization of the 
VBGF allows for calculation of L«, and K\ 
Ogle, 2015 ). In contrast, the parameters of 
growth completion of each model, K (VBGF), 
G (Gompertz), g (logistic), and G init (Mooij), 
cannot be compared because they are mea- 
sures of different processes. 
For each model, the parameter estimates 
that were best fitted to the data were calcu- 
lated by using the nls function in the statis- 
tical software R, vers. 3.2.4 (R Core Team, 
2016) and specific functions from the FSA 
package, vers. 0.8.1, in R (Ogle, 2015). 
Back-calculation methods are used to de- 
scribe the growth history (lengths at previ- 
ous ages) of each individual fish, and cur- 
rently there are many approaches in use: 
Francis (1988) provides a thorough revi- 
sion of back-calculation methods, and Gold- 
man et al. (2012) and Cailliet et al. (2006) 
provide a review of these methods applied 
to chondrichthyan fishes. We employed a 
Fraser— Lee approach to back calculate the 
length-at-age data, with which we fitted our 
models to estimate the growth parameters, [ 
and compared length at ages calculated 
from our observational data. 
To assess the fit of each model, we used a 
bias-corrected Akaike information criterion 
(AICc) and the Bayesian information criterion (BIC) 
(Burnham and Anderson, 2002; Katsanevakis, 2006; 
Zhu et al., 2009; Lopez Cazorla et al., 2014). The model 
with the lowest AICc and BIC values was chosen as the 
most appropriate to describe the growth of narrownose 
smooth-hound. 
Differences, by sex, in estimates of growth param- 
eters from the selected model were assessed with i 
likelihood ratio tests (Rimma, 1980) and extra sum- 
of-squares tests (Ritz and Streibig, 2008). To identify 
which growth parameter differ between sexes, we fitted 
iterative models: the most complex model represents ! 
the case where all 3 growth parameters differ among 
sexes. The simplest model represents the case where 
none of the parameters differ between females and 
males. Between these 2 extremes, we built 3 models 
where 2 parameters differ and 3 models where only 1 i 
parameter differs between sexes (Ogle, 2015). Protocols : 
for selection of AICc and BIC were used to select the 
most appropriate model, as described previously. 
To test the overall growth performance, the growth 1 
performance index (<J>) was calculated (Pauly, 1984) by 
employing the parameters from the selected model: 
4> = lcgio^ + 21og 10 ZO (1) 
This index is useful because it reduces the correlation be- 
tween L„ and K, and that reduction is desirable for com- 
parisons of growth among studies of different species. 
