Lopez Quintero et af.: Incorporating uncertainty into a length-based estimator of natural-mortality 
363 
ture about the species under study and incorporating 
this information in the copula method. In addition, the 
method proposed is not limited to the use of Charnov 
et al. (2013) estimator as the underpinning model to 
relate M and growth parameters. A method for esti- 
mating M addressing uncertainty and considering en- 
vironmental factors such temperature (e.g., Hewitt and 
Hoenig, 2005) can also be used. 
As pointed out in the introduction, M is a key pa- 
rameter in modeling any animal population but, it is 
crucial for harvested fish populations. Natural mortal- 
ity affects these populations concurrently and continu- 
ously with fishing mortality to yield the total mortality 
rate, which determines the decay in the abundance of 
a population over time and therefore the size of the 
stock. Estimation of M within stock assessment mod- 
els is difficult, and resultant estimates are usually im- 
precise (Vetter, 1988; Gavaris and Ianelli, 2002). Stan- 
dard practice is therefore to use a constant value for 
M across sizes or ages — a value that is derived from 
indirect methods when fitting a population model. Most 
of the current stock assessment models are age or size 
based, and therefore the incorporation of an age or size 
constant value for M is misleading and may introduce 
a critical source of bias in abundance estimates. Recent 
stock assessment models recognize the importance of 
incorporating size-dependent mortality, and thus incor- 
porating uncertainty in size-based models has become 
highly recommended (e.g., Clark, 1999; Fu and Quinn, 
2000; Siegfried and Sanso 1 ; Gislason et al., 2010; Lee 
and Chang 3 ). We agree with Quiroz et al. (2010) in 
the sense that uncertainty, as reported here for size- 
based M estimates, can then be integrated into stock 
assessment models and used for management analysis 
through methods such as: 1) sensitivity analysis, where 
an assessment is conducted repeatedly for several val- 
ues of size-based M (see McAllister et al., 1994; Pat- 
terson, 1999) drawn from the empirical distributions 
using copula methods; 2) Bayesian framework, by set- 
ting the empirical distributions as prior distributions 
of size-based M; and 3) state-space models, where un- 
certainty in M is incorporated as one of the random 
components regulating the stochasticity in the popula- 
tion dynamics (i.e. the model process error; see Millar 
and Meyer, 2000). 
Acknowledgments 
The authors are grateful to the Instituto de Fomento 
Pesquero for providing access to the data used in this 
work. J. Contreras-Reyes was supported partially be- 
ginning in 2016 by Comision Nacional de Investigation 
3 Lee, H.-H., and Y. J. Chang. 2013. Age-structured natural 
mortality for Pacific blue marlin based on meta-analysis and 
an ad hoc model. Working document ISC/13/BILLWG-1/07. 
Submitted to the International Scientific Committee for Tuna 
and Tuna-like Species in the North Pacific Ocean. [Avail- 
able from website.] 
Cientffica y Tecnologica (CONICYT) doctoral scholar- 
ship number 21160618. R. Wiff was funded beginning 
in 2014 by the Centro de Ecologfa Aplicada y Sustent- 
abilidad (CAPES) project CONICYT FB 0002. We are 
sincerely grateful to the 3 anonymous reviewers for 
their comments and suggestions that greatly improved 
an early version of this manuscript. 
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