355 
National Marine 
Fisheries Service 
NOAA 
Fishery Bulletin 
fh- established in 1881 
Spencer F. Baird 
First U.S. Commissioner 
of Fisheries and founder 
of Fishery Bulletin 
Incorporating uncertainty into a length-based 
estimator of natural mortality in fish populations 
Freddy O. Lopez Quintero 1 
Javier E. Contreras-Reyes (contact author ) 2 ' 3 
Rodrigo Wiff 4 
Email address for contact author: javier.contreras@ifop.cl 
Abstract- — -Natural mortality (M) is 
one of the most important life histo- 
ry attributes of functioning fish pop- 
ulations. The most common methods 
to estimate M in fish populations 
provide point estimates which are 
usually constant across sizes and 
ages. In this article, we propose a 
framework for incorporating uncer- 
tainty into the length-based estima- 
tor of mortality that is based on von 
Bertalanffy growth function (VBGF) 
parameters determined with Bayes- 
ian analysis and asymmetric error 
distributions. Two methods to incor- 
porate uncertainty in M estimates 
are evaluated. First, we use Mar- 
kov chains of the estimated VBGF 
parameters directly when comput- 
ing M and second, we simulate the 
posterior distribution of VBGF pa- 
rameters with the copula method. 
These 2 approaches were applied 
and compared by using the exten- 
sive database available on age and 
growth for southern blue whiting 
(Micromesistius australis) harvested 
in the southeast Pacific. The copula 
approach provides advantages over 
Markov chains and requires far 
less computational time, while con- 
serving the underlying dependence 
structure in the posterior distribu- 
tion of the VBGF parameters. The 
incorporation of uncertainty into 
length-based estimates of mortality 
provides a promising way for model- 
ing fish population dynamics. 
Manuscript submitted 30 August 2016. 
Manuscript accepted 11 April 2017. 
Fish. Bull. 115:355-364 (2017). 
Online publication date: 18 May 2017. 
doi: 10.7755/FB.115.3.6 
The views and opinions expressed or 
implied in this article are those of the 
author (or authors) and do not necessarily 
reflect the position of the National 
Marine Fisheries Service, NOAA. 
1 Departamento de Matematica 
Instituto Venezolano de Investigaciones Cientfficas 
Carretera Panamericana, Km 11, Altos de Pipe 
Estado Miranda 1020-A, Venezuela 
2 Division de Investigacion Pesquera 
Instituto de Fomento Pesquera 
Avenida Almirante Blanco Encalada 839 
Valparaiso 2361827, Chile 
Natural mortality ( M ) rate is one of 
the most important parameters shap- 
ing the population dynamics of fish 
populations (Siegfried and Sanso 1 ; 
Brodziak et al., 2011). It is defined as 
the death rate of fish due to causes 
other than fishing, such as predation, 
senescence, cannibalism, starvation, 
and other natural factors. Despite 
the key importance of M in fish and 
fisheries modeling, this parameter is 
extraordinarily difficult to estimate 
accurately. Methods for determin- 
ing M in fish populations generally 
entail one of 2 approaches: 1) direct 
methods which estimate M from ob- 
servations on survival, with methods 
derived from tagging or telemetry 
experiments, 2) indirect methods 
that estimate mortality from other, 
more easily obtained parameters, of- 
ten from life history traits, such as 
age and growth, and maturity. Direct 
methods provide the most precise es- 
timates of M, but those approaches 
1 Siegfried, K. I., and B. Sanso. 2009. A 
review for estimating natural mortality 
in fish populations. Southeast Data, As- 
sessment, and Review SEDAR 19-RD29, 
31 p. [Available from website.] 
3 Instituto de Estadfstica 
Universidad de Valparaiso 
Avenida Gran Bretana 1111 
Valparaiso 2360102, Chile 
4 Center of Applied Ecology and Sustainability 
Pontificia Universidad Catolica de Chile 
Avenida Libertador Bernardo O'Higgins 328 
Santiago 8331150, Chile 
are data intensive and usually cost 
prohibitive and therefore preclude 
their application to a large number 
of fish stocks. Indirect methods are 
therefore commonly applied because 
they are easy, fast, and cheap to 
implement for most harvested fish 
populations. 
Indirect estimators are based on 
correlations of M in well-studied 
stocks with other life history attri- 
butes, such as individual growth, 
longevity and maturity. The underly- 
ing assumption in all indirect meth- 
ods is that the relationship between 
M and other life history parameters 
is the same for thoroughly studied 
stocks and lesser-studied ones where 
this method is usually applied. For 
most indirect methods, an estimate 
of M is computed, which is invariant 
across age and size classes, although 
this parameter is dependent on age 
and size (Gislason et al., 2010). The 
relationship between M and age and 
size is usually defined as a negative 
exponential function in which early 
life stages include much greater mor- 
tality than later ones, especially af- 
ter reaching sexual maturity. Never- 
theless, M estimators, whether they 
