He et al.: Interactions of age-dependent mortality and selectivity functions in age-based stock assessment models 
211 
the true values (Table 5). For run 16, which had 
the same model configuration as run 12 except 
that four natural mortalities were used in both 
models, the assessment outputs matched very 
poorly with those in the simulation model (Table 
3; Figs. 8 and 9). Spawning outputs of all years, 
including B 0 , estimated by the assessment model 
were much higher than those in the simulation 
model (Table 3; panel 16 in Fig. 8), and selectiv- 
ity was poorly fitted for old fish (panel 16, Fig. 
9). Estimated natural mortalities for old fish 
(M 4 ) showed a bi-modal distribution (Fig. 11), 
and there were strong interactions between es- 
timated M 4 and selectivity for old fish (e.g., fish 
at age 30) (Fig. 12). There was a high proportion 
of cases (394 out of 500, Fig. 12) where M was 
estimated to be very small (mean=0.03). 
Patterns of convergence performance, between 
runs 9 to 12 and between runs 13 to 16, were very 
similar to those runs between runs 1 to 4 because 
these runs had the same setup for selectivities 
(Table 4). 
Discussion 
0.0 0.06 0.12 0.18 0.24 0.30 0.36 0.42 
Natural mortality for 25+ year-old fish 
Figure 11 
Frequency plots of estimated natural mortality for 25+ year- 
old fish (M 4 ) from run 16. True M 4 values ranged from 0.25 
to 0.5, and no prior for steepness (h) was used in the assess- 
ment models. 
1 . 0 - ° 
o 
— i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — 
0.0 0.06 0.12 0.18 0.24 0.30 0.36 0.42 
Natural mortality for 25+ year-old fish 
Figure 12 
Estimated selectivity at age 30 versus natural mortality for 
25+ year-old fish (M 4 ) from run 16. True M 4 values ranged 
from 0.25 to 0.5 and no prior for steepness (/?) was used in the 
assessment models. Outputs were plotted in two separated 
groups based on the estimated M 4 values. The first group had 
estimated M 4 values <0.28 (solid dots) and the second had 
estimated M 4 values >0.28. Means on the graph are mean 
selectivities for age-30 fish. 
Our research has shown that the assumption of 
a constant natural mortality for all ages when 
natural mortality is actually elevated in young and 
old fish can lead to inaccurate estimates of many 
important population and management quantities. 
The manner in which selectivity is modeled is also 
very important in determining which assessment 
parameters are poorly estimated and how these 
interact with one another in the model. 
In general, population depletion was well esti- 
mated, even when mortality and selectivity were 
incorrectly specified in assessment models because 
population depletion is a robust indicator of popu- 
lation status. This is mainly because depletion is 
estimated as the ratio of two quantities (termi- 
nal spawning output divided by virgin spawning 
output), both of which exhibit similar relative bi- 
ases. Estimates of another management variable, 
i.e., the OFL, were consistently biased, although 
95% quantile intervals overlapped zero for some 
runs. These results indicate that OFL may be a 
more precautionary management indicator than 
population depletion. However, more research is 
needed to compare these two indicators because 
OFL depends on F MSY and biomass in the termi- 
nal year and estimates of these two quantities 
were strongly influenced by how natural mortality, 
selectivity and other population parameters were 
modeled in the assessment. 
Our results show that catchability coefficients for 
juvenile and adult surveys can be strongly positively 
biased if natural mortalities are higher in young and 
old fish but are misspecified in the estimation model, 
even when selectivity is correctly specified. If juvenile 
natural mortality is higher than that for adult fish, but 
is assumed to be the same as that for adult fish, catch- 
ability coefficients for juveniles from surveys of prere- 
cruits are poorly estimated. In many stock assessments, 
these coefficients are unknown and are often very small 
numbers because relative abundance is measured in 
