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Fishery Bulletin 106(3) 
overfishing in a probabilistic sense by keep- 
ing the expectation of overfishing below a 
preset level (e.g., 0.1), presumably satisfying 
the new requirement of the MSRA. The ap- 
proach is intended for setting annual catch 
levels while accommodating uncertainties in 
future stock dynamics, assessment results, 
and in the implementation of management 
measures. 
Materials and methods 
Probability-based approach to setting 
catch levels (PASCL) 
The proposed method acts as a harvest-con- 
trol rule. It is a probability-based approach 
to setting catch levels (PASCL), incorporat- 
ing uncertainties in future stock dynamics, 
assessment results, and management imple- 
mentation. Given these uncertainties, PASCL 
sets annual target levels of catch consistent 
with the level of risk considered acceptable by 
managers. The method is based on the ratio- 
extended approach to setting target reference 
points (REPAST) of Prager et al. (2003), but 
is considerably revised 1) to establish reference points 
in catch, rather than in fishing mortality rate, and 2) to 
add a stock-projection component, which is needed to set 
catch for more than one year after a stock assessment. 
The new method is a general framework that can incor- 
porate details of almost any stock that is assessed. It is 
illustrated with gag ( Mycteroperca microlepis), a grouper 
found off the southeastern United States. 
Uncertainty in stock dynamics is represented by a 
stochastic projection model. The projection allows the 
setting of annual catch levels for more than one year 
and, if necessary, can account for a lag between the 
final year of assessment data and the first year of man- 
agement implementation. The projection model need not 
carry the assumption of equilibrium dynamics and can 
include any source of process or estimation uncertainty 
deemed appropriate, as with projections commonly used 
in fishery management. Sources often considered are 
recruitment dynamics and initial numbers of fish at 
age. Modeling nonequilibrium population dynamics, 
as here, is critical for developing harvest strategies 
(Hauser et al., 2006). 
Stock assessment results generally include estimates 
of uncertainty. A key stock assessment result used 
in PASCL is the estimate of F hm , the limit reference 
point of fishing mortality rate (F) and its associated 
uncertainty, described by a probability density function 
(PDF), which can be either parametric or nonparamet- 
ric. If a PDF of F lim is unavailable, PASCL can use a 
point estimate, but ignoring that source of uncertainty 
can make overfishing more likely (Prager et al., 2003). 
Another basic assessment result, the estimate of stock 
abundance at age (with the corresponding estimate of 
uncertainty), is used to initialize stock replicates in 
stochastic projection with PASCL. 
Uncertainty in implementation stems from managers 
having only partial control of the catch (Rosenberg and 
Brault, 1993; Caddy and McGarvey, 1996; Prager et al., 
2003). A target catch may not be met precisely if catch 
is monitored with delay, catch is managed indirectly 
through input controls, regulations are poorly enforced, 
or fishing behavior is unpredictable. 
In PASCL, as in REPAST (Prager et al., 2003), the 
level of risk acceptable to managers (P*) is quantified 
and explicit. In our study, risk is defined as the prob- 
ability of overfishing in any year t, i.e., as Pr(F ( >F lim ). 
A small value of P* corresponds to risk-averse manage- 
ment. Always, P*< 0.5 should hold, because at P*=0.5, 
overfishing is expected in half of all years. When P* is 
defined as a constant probability, the risk of overfishing 
in at least one of T years grows with the time horizon 
as 1-(1-P*) T (Fig. 1). 
In a simple formulation, the limit fishing mortal- 
ity rate F lim could be represented by a point estimate. 
Then, the probability (here equated to risk) of overfish- 
ing in year t would be a function of Fj im and the PDF 
of F t (<p Ft ): 
Pr( F ( > Fij m ) = [ <t> Fi (F)dF = l-nF hm ), (1) 
where F lim ) = the cumulative distribution of F t evalu- 
ated at F lim . 
A catch level can then be set to position the distri- 
bution of F t so that the desired risk is achieved; i.e., 
