16 Fishery Bulletin 119(1) 
Table 1 
Comparison of the Gompertz and von Bertalanffy growth models fit to length-at-age data for 
golden cownose rays (Rhinoptera steindachneri) sampled between 2008 and 2014 in the south- 
ern Gulf of California in Mexico. Posterior values and 95% credible intervals (lower quartile: 
2.5%; upper quartile: 97.5%) are provided from the fit of the growth models with a Markov 
chain Monte Carlo algorithm to data pooled for both sexes of golden cownose rays. The model 
parameters are disc width (DW) at birth (DW,, in centimeters), theoretical maximum length 
(DW.,, in centimeters), the completion growth parameter (g, year !), and the growth coefficient 
(k, year |). The standard deviations (SDs) and median values are given in addition to mean 
values. The Watanabe—Akaike information criterion (WAIC) was used in model selection. 
Model Parameter Mean SD 
Gompertz 
von Bertalanffy 
Disc width (cm) 
T T T 
5 6 YY & 
Age (years) 
Figure 6 
Estimated growth curve for golden cownose rays (Rhinoptera stein- 
dachneri) in the southern Gulf of California in Mexico, from use of the 
Gompertz growth model for sexes combined. The black circles repre- 
sent observed lengths at age for rays sampled during 2008-2014, and 
the gray shaded area indicates 95% credible intervals of estimates. 
cownose ray is 21 years (Fisher et al., 2013). Vertebrae are 
considered a valid structure for aging the golden cownose 
ray, as indicated by the proportional growth of the verte- 
brae to body size observed in our study. Also, values of 
IAPE and CV for band counts among readers were within 
the standard threshold for age determination in elasmo- 
branchs (Campana, 2001). 
The assumption of an annual pair of bands in the ver- 
tebrae is supported by the results of MI and edge percent- 
age analyses and has been validated for the golden cownose 
2.5% Median 97.5% WAIC 
1459.82 
39.29 40.00 40.71 
94.19 100.67 110.43 
0.12 0.15 0.18 
4.12 4.49 4.92 
1462.26 
39.20 39.93 40.65 
104.06 117.41 140.85 
0.05 0.08 0.10 
4.15 4.51 4.95 
ray by using biomarkers (Osuna-Soto, 2016). In 
addition, the same pattern of band formation has 
been observed for other species of the same genus 
(Neer and Thompson, 2005; Fisher et al., 2018). 
An annual pattern of band pairs has also been 
described for other batoids in the tropical and 
subtropical Pacific Ocean (Davis et al., 2007; Hale 
and Lowe, 2008; Mejia-Falla et al., 2014). Although 
there are more precise methods for validation of 
periodicity of band formation for aquatic organisms 
(e.g., biomarkers and mark and recapture), MI and 
edge percentage analyses provide the advantage 
of producing cost-effective and comparable results 
(Campana, 2001). 
We found that the GM best described the 
growth of the golden cownose ray in the GOC for 
sexes combined, as indicated by the WAIC. Such a 
model includes an inflection point that represents 
a change in growth rate (Joung et al., 2011); this 
change in growth rate is assumed to be associ- 
ated with a difference in ontogenetic feeding 
patterns between juveniles and adults. The GM 
may be a better option when the volume of an 
organism greatly expands with age, as has been observed 
for myliobatiform rays (Cailliet and Goldman, 2004; Neer 
and Thompson, 2005) and for elasmobranchs that have a 
change in growth rate after age 0 (Smart et al., 2016). 
Results of several studies indicate that the conventional 
VBGF does not always satisfactorily describe the growth 
of chondrichthyans (Cailliet et al., 2006), and the authors 
of some studies have emphasized the need to explore dif- 
ferent nonlinear models to describe growth (Bishop et al., 
2006; Natanson et al., 2007). In our study, an exhaustive 
