194 
Fishery Bulletin 116(2) 
N 
t+l,a 
Rt 
iV t , a V Mt+St - Ft) 
iV u _i e - (Mt+S - lFt ) +iV t>A _ 1 e' (Mt+SuFt } 
ifa=0 
ifl<a<A-l, (1) 
if a=A 
number of years for the simulated stock to rebuild to 
the target biomass when a period of reduced data could 
affect the performance of the estimation method to cor¬ 
rectly estimate the stock size and status. The catch of 
fish of age a during year t in numbers was given by 
where N tA 
Rt 
St,a 
A 
Ft 
M t 
the number of fish of age at the start of the 
year t ; 
the number of age-0 fish at the start of year 
t; 
the selectivity during year t for fish of age 
a; 
the plus group (i.e., the oldest age group 
modeled, set equal to age 70); 
the instantaneous fishing mortality rate 
during year l; and 
the instantaneous rate of natural mortality 
during year t. 
Natural mortality for year is defined as 
M t = Me-°- 5a ™ +£ ™, (2) 
where M = the mean value of natural mortality; 
a M = the standard error of the annual deviations 
in natural mortality; and 
t;'’ 1 - the autocorrelated lognormal deviation in 
natural mortality for year t: 
ef = P£ t -i + V 1 - P\ 4 >t ~ MO;0mX 
where p = the level of autocorrelation associated with 
natural mortality; and 
<j) t = the deviation in natural mortality for year t. 
The time-invariant natural mortality case 
assumed 0 M =O and hence ef=0. 
The number of age-0 fish is related to spawning bio¬ 
mass according to the Beverton-Holt stock recruitment 
relationship (Beverton and Holt, 1957): 
R t = 
4hR 0 SB t ^-0.5a|+ef 
SB 0 (l-h) + SB t (5h-l) e 
ef ~ iV(0;a|), (4) 
where R 0 = the number of age-0 fish when the popula¬ 
tion is in an unfished state; 
SB 0 = the unfished spawning biomass; 
SB t - the spawning biomass at the start of the 
spawning season in year t ; 
a R = the standard deviation of recruitment in log 
space; and 
h - steepness. 
A nonequilibrium starting condition was created by 
applying the numbers-at-age (combined with the natu¬ 
ral mortality calculations for the number of years equal 
to the maximum age before the start of fishing) with 
variation in recruitment from the Beverton-Holt stock 
recruitment relationship. Historical catches for years 
1-50 were generated so that the populations were at 
0.15$i?o in year 50, a state that would allow correct 
detection by the estimation method that the simu¬ 
lated stocks were in an overfished state. Additionally, 
the simulated populations would require an extended 
_ *-k,a^t AT _ - 
*> a ~~ S/f Q T? 
A. 
(5) 
+ S t>& F t 
The observation model was used to generate a fish¬ 
ery CPUE index for each year t: 
I t =QR t e-°- 5a * +el ~ N(0;Of), 
( 6 ) 
where Q = the catehability coefficient; 
Of = the standard deviation of catehability in log 
space; and 
B t - the vulnerable biomass available to the fish¬ 
ery in the middle of year t: 
5 t = Zl 1 M; a S! t>a Ar t;a e- 0 - 5(Mt+St ' aFt) ! 
where w a = the weight of a fish of age a. 
(7) 
The length- and age-composition data for the fishery 
were assumed to be multinomially distributed (for de¬ 
tails, see the “Data scenarios” section). Age-determina¬ 
tion error was assumed to be normally distributed with 
ages subject to a 5% standard deviation by age. 
The fishery selectivity was modeled by using the 
double normal parameterization (for details, see Meth- 
ot and Wetzel, 2013), which is a flexible form that al¬ 
lows selectivity to range in shape from asymptotic to 
dome-shaped. Fishery selectivity during the histori¬ 
cal period (years 1-50) was assumed to be asymptotic 
(Fig. 1, A and C). Fishery selectivity shifted to a dome¬ 
shaped (in contrast with the historical asymptotic) 
form (Fig. 1, B and D) within the operating model dur¬ 
ing the period that the simulated stock was estimated 
to be below the target biomass (0.40 SB 0 ). Once the 
population was estimated to have recovered to above 
the target biomass, fishery selectivity reverted to the 
asymptotic form. The shift in selectivity was designed 
as a way to mimic a change in the behavior of fisher¬ 
men that results from an overfished designation (e.g., 
1) the creation of rockfish conservation areas that pro¬ 
tect portions of the stock, or 2) areas of known specific 
habitat that are avoided by fishermen, or 3) areas as¬ 
sociated with high abundance of the overfished stock). 
The change in shape of the selectivity curve depended 
on the estimated stock status rather than on the true 
status from the operating model (i.e., changes in be¬ 
havior of fishermen modeled by a change in selectivity 
were assumed to be driven by management restrictions 
based on the perception of the simulated stock by the 
estimation method rather than on the true unobserv¬ 
able state of the simulated stock). 
The estimation method 
Stock synthesis, an integrated statistical catch-at-age 
model (Methot and Wetzel, 2013), was the estimation 
