Carrillo-Colin et al.: Bayesian estimation of the age and growth of Rhinoptera steindachneri 17 
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Figure 7 
The posterior probability distribution of (A) theoretical 
maximum length (DW..), (B) length or disc width (DW) 
at birth (DW,), and (C) the completion growth parameter 
(g) predicted with the Gompertz growth model for golden 
cownose rays (Rhinoptera steindachneri) in the southern 
Gulf of California in Mexico. The model was fit to length- 
at-age data pooled for both sexes of golden cownose rays 
sampled during 2008-2014. 
test of other growth models was not done because the ver- 
sions of both models tested (VBGF and GM) conveniently 
include length at birth, an appropriate inclusion for vivip- 
arous species (Cailliet et al., 2006). 
The multi-model approach to estimation of growth 
parameters allows selection of the model that best fits 
the data (Katsanevakis, 2006). Although growth models 
with multiple parameters may be useful for estimating 
growth of elasmobranchs (Richards, 1959; Chapman, 
1961; Schnute, 1981), the multi-model approach suffers 
from testing models with more parameters, some of which 
might have no biological meaning as is the case with the 
theoretical age when length is zero in the VBGF (Cailliet 
and Goldman, 2004). In addition, researchers may even- 
tually be conditioned to ignore factors that potentially 
bias the selection of data collected from a fishery when 
fitting models to that data. 
Use of fishery-dependent data can potentially lead to a 
distorted growth curve through sampling bias due to gear 
selectivity and migration (Ricker, 1979; Moulton et al., 
1992). Such biases can potentially straighten the growth 
curve and change the magnitude of parameters because 
of their autocorrelation (Gallucci and Quinn, 1979). Such 
an effect, known as the phenomenon of apparent change 
in growth rate, is common in studies of age and growth 
of chondrichthyans (Lee, 1912; Walker et al., 1998). The 
occurrence of the phenomenon of apparent change in 
growth rate in our study may have been caused by the 
susceptibility of fish of certain sizes to be caught by the 
fishing gears used during our sampling campaign (Moul- 
ton et al., 1992). 
Distorted growth curves are common for shark species 
in fisheries that use highly size-selective fishing gear and 
high fishing intensity (Walker, 1998). In the samples of 
golden cownose ray taken from the artisanal fishery, small 
individuals (<55 cm DW) predominated. Individuals older 
than 4 years could escape from fishing gears because of 
their large size (>59 cm DW). Their shape means they 
avoid the proper length selectivity of gill nets and, there- 
fore, are easily captured or entangled. In contrast, length 
selectivity of fishing gear is more likely to work for thinner 
guitarfish species with a more pointed shape (Marquez- 
Farias, 2005). However, the inclusion of golden cownose 
rays caught incidentally in the shrimp trawl fishery in 
the GOC allowed an increase in size ranges because it has 
been reported that trawl nets do not allow their escape 
(Garcés-Garcia et al., 2020). Another explanation for size 
classes being misrepresented in our sample could be size- 
dependent migration, and it should be investigated in the 
future. We believe bias was minimal in our sample because 
we included individuals in all classes within the known 
size range for the golden cownose ray and combined the 
DW for both sexes. 
The Bayesian approach applied in our study was par- 
ticularly useful for exploring the effect of uncertainty in 
modeling life history characteristics (Cortés et al., 2015; 
Dono et al., 2015) and is likely to be useful for examining 
life history traits of other data-deficient batoid species. 
For instance, Rolim et al. (2020) implemented a Bayesian 
approach to assess estimates of life history characteris- 
tics derived from a growth model. This approach included 
more realism in the process of fitting the model to data and 
accounting for uncertainty in the other growth parameters. 
The convergence of the growth parameters DW,, g, and 
DW, was satisfactorily achieved, considering the well- 
defined posterior distribution profile of each parameter. 
The informative prior distribution for DW, favored model 
fitting because it helped select realistic birth size values 
and because it represents the y-axis intercept (Rolim 
et al., 2020). The size at birth of elasmobranchs is often 
well-defined and known; therefore, it is easy to judge 
whether the DW, from the fitted model is a reasonable 
value (Cailliet et al., 2006). However, using prior values 
and uncertainty seems to be a better alternative than 
using fixed values for the length at birth and solving only 
for g and DW, (Smart et al., 2016). 
