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Fishery Bulletin 106(3) 
6 Project each replicate one year forward by 
applying recruitment and natural mortality 
and taking catch C n . 
7 Repeat steps 2-6 for T years. 
In general, duration T of the projection will 
extend until ACTs based on the next assessment 
can be implemented. The preceding procedure 
gives an ACT for each year in the period, and 
the annual probability of overfishing is kept 
at P*. 
Setting catch levels of gag 
To illustrate the method, we applied PASCL 
to the gag stock off the southeastern United 
States. The stock was most recently assessed 
in 2006 from data through 2004 and a statisti- 
cal catch-age model (Quinn and Deriso, 1999) 
including the Beverton-Holt spawner-recruit 
model (Beverton and Holt, 1957). The stock was 
estimated to be experiencing the effects of over- 
fishing with a biomass at nearly 90% of that at 
maximum sustainable yield (SEDAR, 2006). 
To implement PASCL, we devised a stochastic 
projection model with structure identical to the 
age-based assessment model (SEDAR, 2006), in which 
landings and discards were computed from the Baranov 
(1918) catch equation. The parameter values chosen 
were those used or estimated in the assessment. 
The projection included two sources of uncertainty in 
stock dynamics. One was stochasticity in recruitment, 
assumed to be lognormal about the estimated Bever- 
ton-Holt spawner-recruit model, with parameter values 
from the assessment. The other was uncertainty in the 
estimated final numbers at age (N a 2005 ), which become 
Figure 3 
Probability density (<t>p h ) of the limit reference point (Ej im ), 
defined here as the fishing mortality rate at maximum sus- 
tainable yield. 
a, 2005 
). 
the initial numbers at age in our example (N. 
In some applications, the variance of N a 2005 would be 
estimated during the assessment, but SEDAR (2006) 
provided only point estimates. To include uncertainty, 
we assumed that multiplicative error in the initial num- 
bers at age followed a lognormal distribution with mean 
(in log space) of zero and a standard deviation equal to 
that of recruitment (<7^): 
^ a, 2005 ~ 2005 eX P* V ) ’ 
(3) 
where v~N(ji=0, o=o R ). 
This approach accounts for uncertainty in initial condi- 
tions, while maintaining strong year classes estimated 
in the terminal year of the assessment. 
The first year of the projection was 2005, and new 
regulations on catch levels were implemented in 2008. 
For the projection during the premanagement years 
(2005-07), we applied a fixed level of landings, set to 
the geometric mean of landings from 2002 through 
2004. The duration of the projection was 10 years: three 
premanagement years followed by seven years of man- 
aged catch levels (landings plus discard mortalities). 
Presumably, this duration is generous, spanning a pe- 
riod until the next assessment. 
The stochastic projection model was used to generate 
V=10,000 replicate stocks differing in abundance and 
age structure. This variation, along with imprecise 
management implementation, led to V=10,000 values of 
fishing mortality rate in each year, which were used to 
characterize the fishery’s annual probability density of 
F t . These densities (<j> F ) were quantified nonparametri- 
cally through kernel density estimation with Gaussian 
kernel and bandwidth equal to the kernel’s standard 
deviation (Venables and Ripley, 2002). 
The limit reference point in F was set equal to F MSY 
(Mace, 2001). For this example, the probability den- 
sity of F msy (<j> F) ) was estimated after the assess- 
ment through Bayesian analysis of the Beverton-Holt 
spawner-recruit model, accounting for uncertainty in 
model parameters. A prior distribution was specified 
for steepness (h), the parameter controlling how quickly 
recruitment approaches its unfished level as spawning 
biomass increases. This prior distribution was based on 
meta-analysis of steepness values (Myers et al., 1999) 
from species similar to gag. Species included were those 
considered to be periodic spawners, as defined by Rose 
et al. (2001), and limited to marine or anadromous de- 
mersal fishes, excluding rockfish ( Sebastes spp.) because 
of their uncharacteristically low steepness values. The 
estimated prior distribution was lognormal (SEDAR, 
2004): 
h = exp(x) : x ~ N(/u- -0.33 , a = 0.28). 
(4) 
The resulting posterior distribution of F USY described 
<j> F for use in PASCL (Fig. 3). In this example, (f> Fhm 
