Catch, Landings, and Discards
The fishery catch depends on the fraction of the population that is vulnerable to harvest or the exploitable stock size. Catch at age by fleet (fleets are indexed by \(v\)) is determined by the Baranov catch equation (e.g., Quinn II and Deriso 1999). The catch of age-\(a\) fish in year \(t\) by fleet \(v\) is \(C_{V,a}(t)\) .
\[ C_{V,a}(t)= \frac{F_{v,a}(t)}{M_a(t)+F_{v,a}(t)}{\Bigl[ 1-e^{-M_a(t)-F_{v,a}(t)}\Bigr]} \cdot N_a(t) \tag{1}\]
To account for age-specific discarding of fish, let \(P_{v,D,a}(t)\) be the proportion of age-\(a\) fish that are discarded by fleet \(v\) in year \(t\), and let \(W_{v,L,a}(t)\) and \(W_{v,D,a}\) be the average weight at age-\(a\) in year \(t\) for landed and discarded fish, respectively. Then, if discarding is included in the projections, the total landed weight of fish caught by fleet \(v\) in year \(t\), denoted by \(L_v(t)\), is
\[ L_V(t)=\sum_{a=1}^{A} \cdot {\Bigl[ 1 - P_{v,D,a}(t) \Bigr]} \cdot W_{v,L,a(t)} \tag{2}\]
Similarly, the total weight of discarded fish in year \(t\), denoted by \(D_y(t)\), is
\[ D_v(t) = \sum_{a=1}^{A}C_{v,a}(t) \cdot P_{v,D,a}(t) \cdot W_{v,D,a}(t) \tag{3}\]