He et at: Interactions of age-dependent mortality and selectivity functions in age-based stock assessment models 
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natural mortality are impractical, if not impossible, 
to derive from fishery or survey data, because of the 
interaction between fishing and natural mortality 
(Vetter, 1988; Quinn and Deriso, 1999). Clark (1999) 
examined the effects of incorrectly specifying M for a 
simple age-structured stock assessment and concluded 
that errors in M mainly affect estimates of fishing 
mortality and abundance, but not estimates of age- 
specific selectivity. 
In most regions of the world, where statistical 
stock assessments of single species provide the ba- 
sis for management advice (Worm et al., 2009), it is 
commonplace to assume a constant natural mortal- 
ity rate for all exploitable ages or sizes (or for both). 
Moreover, natural mortality is also typically assumed 
to be constant over time and identical among regions 
(Punt, 2003; Yin and Sampson, 2004; PFMC, 2008). 
Uncertainty in the use of constant natural mortal- 
ity in these assessment models is usually evaluated 
by an approach that is similar to likelihood profiles, 
where M values are changed and other parameters 
are fixed. However, this approach is highly dependent 
on the specific model structure and parameter set- 
tings being evaluated. For example, if stock recruit- 
ment relationships or selectivity functions are fixed 
in an assessment model, likelihood profile methods 
on natural mortality can provide only the validity of 
the model fitted to fixed values of natural mortality 
and not the validity of the model for its interactions 
with other model parameters. 
In this study, we compare stock assessment re- 
sults among simulated populations with different 
natural mortality schedules. The simulation data 
were generated with an age-based population model 
characterized by exploitation from a single fishery 
with a constant selectivity pattern over an extended 
period of time, representing somewhat ideal condi- 
tions. Simulations were crafted to reflect conditions 
in the U.S. west coast groundfish fishery — the source 
of most available fishery data for the last 40 years, a 
period when fishing intensity was high in the early 
years and has been low in recent years. In the simu- 
lation operating model, two different natural mortal- 
ity patterns were used: 1) constant natural mortality 
for all ages; and 2) elevated values of M in both 
juvenile and old fish. The data, along with sampling 
errors, were input into the assessment models. In the 
assessment model, natural mortality was assumed to 
be known and constant for all ages, estimated to be 
constant for all ages, or was estimated to follow an 
age-specific pattern. Estimated quantities from the 
assessments were then compared with the simulation 
models that generated the data. Important assess- 
ment results, e.g., stock depletion and stock-recruit 
relationships, were then compared to evaluate the 
effect of misspecification of natural mortality and se- 
lectivity on stock assessment estimates. In addition, 
results from the assessment models were compared 
with and without an informative parameter prior for 
spawner-recruit steepness parameters. 
Methods 
Simulation model 
The simulation or operating model in this study was 
an age-structured population model with a max age 
of 30 years. The last age was an age plus group. The 
Beverton-Holt stock-recruitment function was used 
to model stock recruitment. In particular, the “steep- 
ness” parameterization of Mace and Doonan 2 with 
/; = 0.6 was used (see also Dorn, 2002). Recruitment 
variability was lognormal with a R set equal to 0.5 
and lognormal survey variability set to equal 0.25. 
Specifics of the simulated population dynamics are 
presented in the Appendix. Base values for biologi- 
cal, fishery, and modeling parameters are presented 
in Table 1. Because of variability in recruitment and 
catchability, the model was run for 260 years, with 
the first 200 years as a “burn-in” period with no fish- 
ing to minimize the effect of initial conditions in the 
model. Only the last 40 years of data were provided 
for the assessment model. 
Biological parameters, including growth, fecundity, 
and the length-weight relationship were patterned 
after widow rockfish ( Sebastes entomelas) off the U.S. 
west coast (He et al., 2009). Although widow rockfish 
shows differences between the sexes in biological pa- 
rameters, the same values were used for both sexes 
to simplify the model. 
We modeled two different functional types of age- 
dependent natural mortality ( M ) in the simulations 
including 1) constant natural mortality for all ages 
(0.15/yr); and 2) high M in both juvenile and old fish 
(Table 2; Fig. 1). The annual sample size for age com- 
positions was 500 for all simulations — a size that 
ensured that informative age composition data were 
available to the assessment models. 
We used only one fishery in the operating model, 
and catches began in the last 40 years of the simula- 
tions. Fishing mortalities (F) were modeled as propor- 
tions of F msy , which varied over time. During the first 
20 years, fishing at F MSY occurred, and in the last 20 
years fishing mortality was 10% of F MSY . Two types 
of fishery selectivity patterns were simulated, i.e., 
simple asymptotic logistic and double normal curves 
(Fig. 2). The later was moderately dome-shaped and 
is implemented in the stock synthesis model (Methot, 
2009a), a widely used stock assessment model. In all 
simulations the ascending limbs of the two selectiv- 
ity curves were constrained to be similar, i.e., 50% of 
individuals were selected at age 8. The above specific 
values and patterns in M, F, and selectivity were 
based on typical life history patterns of fish, and 
fishing patterns, off the U.S. west coast (e.g., those 
of widow rockfish). 
2 Mace, P. M., and I. J. Doonan. 1988. A generalized bioeco- 
nomic simulation model for fish population dynamics. New 
Zealand Fishery Assessment Res. Doc., 88/4, 21 p. MAF 
Fisheries, Greta Point, Wellington, New Zealand. 
