Shertzer et al.: Probability-based approach to setting annual catch levels 
227 
Figure 2 
Flowchart of method to compute annual catch level. P* is the acceptable risk of overfishing, F lim is 
the limit reference point in fishing mortality rate, N is the number of replicate projections, ACT is the 
target annual catch level, CV is the coefficient of variation of management implementation, F t is the 
fishing mortality rate in year t of a single projection replicate, and P=Pr{F t >F hm ) is the probability 
of overfishing in year t associated with a trial ACT. 
Pr (F t >F hm )=P*. That catch level becomes the annual 
catch level in the sense of the MSRA. 
The full formulation used here is slightly more com- 
plex (and realistic) in that F lim is described by its PDF, 
4> Fi . Then, the probability of overfishing is computed 
as 
Pr(F t > F hm ) = 
J J <t> Ft (0)de 
( F)dF , 
( 2 ) 
The goal of PASCL is to set annual catch levels such 
that Pr {F t >F lim )=P* in each year of a multiyear se- 
quence. Extensions from the formulations described by 
Equations 1 and 2 are twofold: D use of output controls 
(catches) for management, and 2) a management time 
frame of more than one year. In what follows, we as- 
sume that PASCL is used to compute annual catch 
targets (ACTs). 
The approach is implemented through a projection 
model (Fig. 2) with the following steps: 
where 9 = a dummy variable. 
Equation 2 is the weighted sum of probabilities com- 
puted by Equation 1 for all possible values of F lim . 
Again, the distribution of F t can be positioned so that 
Pr(F t >F lim )=P*. 
An assumption of Equation 2 is that Fj im and F t are 
independent. If correlation is observed or suspected, the 
probability of overfishing could be computed from the 
bivariate PDF <f> P „ , 
r UimFf’ 
Pr( F t > Fjjju ) = J J F( dOdF. ( 3 ) 
o F 
Although Equation 3 is more general, estimation of 
<t>F h F from data may seldom be possible. Fortunately, 
in many applications, Equation 2 will be a suitable 
approximation (see Discussion section). 
1 Initialize N replicates of the stock, each different in 
abundance and age structure, to reflect uncertainty 
in the estimated current state of the stock. 
2 Given implementation uncertainty in controlling 
catch, each ACT will be the central tendency of a 
probability distribution <p c . Choose a trial value 
of p, and draw N values \C 1 ... C N 1 from the cor- 
responding distribution. Catch C ; is the catch 
taken from stock replicate N { , C 2 from N 2 , and 
so forth. 
3 To combine uncertainties in the state of the stock 
and implementation, compute for each replicate the 
fishing mortality rate that yields C n . This produces 
N values of F t to define its empirical probability 
density (< p F ). 
4 Given <p Ft and </> F| , compute P=Pr(F t >F ]im ) from 
Equation 2. 
5 Using a numerical optimization method, adjust p 
until P=P*. The adjusted p is that year’s ACT. 
