198 
Interactions of age-dependent mortality 
and selectivity functions in age-based 
stock assessment models 
Xi He (contact author) 
Stephen Ralston 
Alec D. MacCall 
Abstract — The natural mortality rate 
(M) of fish varies with size and age, 
although it is often assumed to be 
constant in stock assessments. Mis- 
specification of M may bias important 
assessment quantities. We simulated 
fishery data, using an age-based pop- 
ulation model, and then conducted 
stock assessments on the simulated 
data. Results were compared to known 
values. Misspecification of M had a 
negligible effect on the estimation 
of relative stock depletion; however, 
misspecification of M had a large 
effect on the estimation of parame- 
ters describing the stock recruitment 
relationship, age-specific selectivity, 
and catchability. If high M occurs in 
juvenile and old fish, but is misspeci- 
fied in the assessment model, virgin 
biomass and catchability are often 
poorly estimated. In addition, stock 
recruitment relationships are often 
very difficult to estimate, and steep- 
ness values are commonly estimated 
at the upper bound (1.0) and over- 
fishing limits tend to be biased low. 
Natural mortality can be estimated 
in assessment models if M is constant 
across ages or if selectivity is asymp- 
totic. However if M is higher in old 
fish and selectivity is dome-shaped, 
M and the selectivity cannot both 
be adequately estimated because of 
strong interactions between M and 
selectivity. 
Manuscript submitted 18 August 2010. 
Manuscript accepted 17 February 2011. 
Fish. Bull. 109:198-216 (2011). 
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. 
Email address for contact author: xi.he@noaa.gov 
Fisheries Ecology Division 
Southwest Fisheries Science Center 
National Marine Fisheries Service 
National Oceanic and Atmospheric Administration 
110 Shaffer Road 
Santa Cruz, California 95060 
Correctly specifying the instanta- 
neous rate of natural mortality CM) in 
stock assessment models is important 
because misspecification may lead 
to over- or underestimates of criti- 
cal assessment quantities, including 
stock depletion, maximum sustain- 
able yield (MSY), virgin biomass, and 
density dependence (Lapointe et al., 
1989; Thompson, 1994; Mertz and 
Myers, 1997; Punt and Walker, 1998; 
Clark, 1999; Wang et al., 2006). It is 
widely believed that natural mortality 
varies with age or size; young (small) 
fish have higher natural mortality 
rates due to higher predation risks, 
disease, or starvation (Lorenzen, 
1996), whereas older (larger) fish may 
have increased natural mortality with 
senescence or because of cumulative 
reproductive stress (Mangel, 2003; 
Moustahfid et al., 2009). 
In spite of the widely held percep- 
tion that natural mortality varies 
considerably with age, most stock 
assessment models assume that M 
is constant for all ages, mainly be- 
cause there are insufficient data with 
which to estimate natural mortal- 
ity on an age-specific basis. Anoth- 
er reason for assuming constant M 
in stock assessment models is that 
natural mortality is typically highly 
correlated with other key parame- 
ters, including stock recruitment and 
selectivity parameters (Lapointe et 
al., 1992; Thompson, 1994; Schnute 
and Richards, 1995; Fu and Quinn, 
2000), quantities that are often quite 
difficult to estimate with accuracy 
(Maunder et al. 1 ). 
In previous studies a variety of 
approaches have been developed to 
estimate natural mortality, includ- 
ing the use of maximum observed 
age (Hoenig, 1983) and life-history 
parameters (Alverson and Carney, 
1975; Gunderson, 1980; Myers and 
Doyle, 1983; Roff, 1992; Jensen, 
1996; Gunderson, 1997). In other 
studies, life-history data and envi- 
ronmental variables have been com- 
bined to establish empirical relation- 
ships to predict natural mortality 
(Pauly, 1980; Gislason et al., 2010). 
These studies have provided esti- 
mates of natural mortality that can 
be useful for stock assessments but 
these estimates may not be suffi- 
cient for species-specific stock as- 
sessment because of bias (e.g., only 
a subset of possible life histories 
was considered). Other studies have 
shown that unless species-specific 
data were collected before exploi- 
tation of the species, estimates of 
1 Maunder, M. N„ H. H. Lee, and K. R. 
Piner. 2010. A review of natural mor- 
tality, its estimation, and use in fisher- 
ies stock assessment. Unpubl. manuscr, 
35 p. Inter-American Tropical Tuna 
Commission, 8604 La Jolla Shores Drive, 
La Jolla, CA. 
