Reference Point Thresholds

Feasible Simulations

A feasible simulation is defined as one where the all landings quotas by fleet can be harvested in each year of the projection time horizon. An infeasible simulation is one where the exploitable biomass is insufficient to harvest at least one landings quota in one or more years of the time horizon. All simulations are feasible for projections where population harvest is based solely on fishing mortality values. For projections that specify landings quotas by fleet in one or more years, the feasibility of harvesting the landings quota is evaluated using an upper bound on \(F\) that defines infeasible quotas relative to the exploitable biomass (Appendix). For purposes of summarizing projection results, the total number of simulations is denoted as \(K_{TOTAL}\) and the total number of feasible simulations is denoted as \(K_{FEASIBLE}\).

Biomass Thresholds

The user can specify biomass thresholds for spawning biomass \(\bigl(B_{S,THRESHOLD}\bigr)\), mean biomass \(\bigl(\overline{B}_{THRESHOLD}\bigr)\), and total stock biomass \(\bigl(B_{T,THRESHOLD}\bigr)\) for Sustainable Fisheries Act (SFA) policy evaluation. These biomass thresholds can be specified using the input keyword REFPOINT (Tables 2 and 3). If the REFPOINT keyword is used in an AGEPRO model, then projected biomass values are compared to the input thresholds through time. Probabilities that biomasses meet or exceed threshold values are computed for each year. In addition, the probability that biomass thresholds were exceeded in at least one year within a single simulated population trajectory is computed. If the user specifies fishing mortality-based harvesting with no landings quotas, then the SFA-threshold probabilities are computed over the entire set of simulations. Let \(K_B(t)\) be the number of times that projected biomass \(B(t)\) meets or exceeds a threshold biomass \(B_{THRESHOLD}\) in year \(t\). The counter \(K_B(t)\) is evaluated for each year and biomass series (spawning, mean, or total stock). Given that \(K_{TOTAL}\) is the total number of feasible simulation runs, the estimate of the annual probability that \(B_{THRESHOLD}\) would be met or exceeded in year \(t\) is

\[ \Pr(B(t)) \ge B_{THRESHOLD} = \frac{K_B(t)}{K_{TOTAL}} \tag{50}\label{eq:50} \]

Note that this also provides an estimate of the probability of the complementary event that biomass does not exceed the threshold via

\[ \Pr(B(t) < B_{THRESHOLD}) = 1 - \Pr(B(t) \ge B_{THRESHOLD} ) = 1 - \frac{K_B(t)}{K_{TOTAL}} \tag{51}\label{eq:51} \]

Next, if \(K_{THRESHOLD}\) denotes the number of simulations where biomass exceeded its threshold at least once, then the probability that \(K_{THRESHOLD}\) would be met or exceeded at least

\[ \Pr(\exists{t} \in [1,2,\dots,Y]\ such\ that\ B(t) \ge B_{THRESHOLD}) = \frac{K_{THRESHOLD}}{K_{TOTAL}} \tag{52}\label{eq:52} \]

If the user specifies landings quota-based harvesting in one or more years, then the SFA-threshold probabilities can be computed over the set of feasible simulations. In this case, the year-specific conditional probability that \(B_{THRESHOLD}\) would be met or exceeded for feasible simulations is

\[ \Pr(B(t)) \ge B_{THRESHOLD} = \frac{K_B(t)}{K_{FEASIBLE}} \tag{53}\label{eq:53} \]

Note that the counter \(K_B(t)\) can only be incremented in a feasible simulation. In contrast, the joint probability that \(B_{THRESHOLD}\) would be met or exceeded for the entire set of simulations is given by (\(\ref{eq:52}\)) and the probability that \(B_{THRESHOLD}\) would be met or exceeded at least once during the projection time horizon is given by (\(\ref{eq:53}\)).

Fishing Mortality Thresholds

The user can specify the fishing mortality rate threshold for annual total fishing mortality \((F_{THRESHOLD})\) calculated across all fleets using the keyword REFPOINT. In this case, projected total \(F\) values are compared to the \(F_{THRESHOLD}\) through time. Probabilities that fishing mortalities exceed threshold values are computed for each year in the same manner as for biomass thresholds (see Biomass Thresholds). In particular, if \(K_F(t)\) is the number of times that fishing mortality \(F(t)\) exceeds the threshold fishing mortality \(F_{THRESHOLD}\) in year \(t\), then the annual probability that the fishing mortality threshold is exceeded is

\[ \Pr(F(t) > F_{THRESHOLD}) = \frac{K_F(t)}{K_{TOTAL}} \tag{54}\label{eq:54} \] and the complementary probability that the fishing mortality threshold is not exceeded is

\[ \Pr(F(t) \le F_{THRESHOLD}) = 1 - \frac{K_F(t)}{K_{TOTAL}} \tag{55}\label{eq:55} \]