Lopez Quintero et al.: Incorporating uncertainty into a length-based estimator of natural-mortality 
361 
K 
Histograms of the asymptotic length (L„) and growth rate coefficient ( K) of the von Berta- 
lanffy growth function (A-B) from simulations based on posterior distributions and (C-D) 
from the copula approach. To make the histograms easy to compare, they were built to have 
a total area of 1 (“density” on the y-axis). 
tion between K and L„. Note the copula method keeps 
the correlation structure reported in Siegfried and 
Sanso (2006) and Lopez Quintero et al. (2017). These 
simulated values of and Lji\ from both, Markov 
chain or from copula, will be used in the expression 
M-at-length estimation. 
Figure 5 shows the simulated M derived from 
Equations 10 and 11, by using both the Markov chains 
and copula approaches. Uncertainty at each age was 
incorporated by simulating the posterior distribu- 
tions of each VBGF parameter. The uncertainty for the 
method error was incorporated through the random 
variable i] (as explained in the Materials and meth- 
ods section). The empirical distributions related to the 
copula method showed a similar shape to that with 
the Markov chains method because we have assumed 
marginals of the Gaussian family. The empirical dis- 
tributions behavior can change depending on the class 
of copula used and the available information for mar- 
ginals (dos Santos Silva and Lopes, 2008). Moreover, it 
has been assumed that M follows a slow exponential 
decay with length (Gedamke and Hoenig, 2006). On 
the other hand, recent empirical and theoretical work 
has shown that M represents a decreasing exponential 
function of length in fish populations (Gislason et al., 
2010; Charnov et al., 2013). Specifically, for both meth- 
ods, the percent change in the median values from the 
first to the sixth year is around 57%. It is important 
to note that the most of the methods used to estimate 
M consider this parameter as constant across ages and 
lengths within species (e.g. Pauly, 1980; Hewitt and 
Hoenig, 2005). These indirect methods are based on the 
assumption that M remains relatively constant in fish 
after they reach sexual maturity. 
Discussion 
We presented a mortality estimator that incorporates 
2 sources of variability or uncertainty. One of them 
is associated with the VBGF parameters (trait error) 
and the other is related to the mortality model (model 
error) which is related to uncertainty in the original 
method described in Charnov et al. (2013). In relation 
