25 Fleet Schedules
Fleet schedules are arrays generated by PopulateFleet() from the fleet subcomponent objects of a Fleet object. Each fleet has three schedule types: selectivity, retention, and discard mortality. All arrays carry Sim × Age × Year × Area dimensions; as in Chapter 24, the simulation index is omitted from equations and all calculations are performed independently per fleet.
Size-based schedules are converted to the age dimension via the ALK or AWK (see Section 24.3).
Model equations below use \(x\) as a generic variable representing age, length, or weight depending on the model chosen. The resulting fully-dimensioned arrays are denoted with subscripts (e.g. \(v_{a,t}\), \(r_{a,t}\)).
25.1 Selectivity
Selectivity \(v_{a,t}\) is the probability that an individual of age \(a\) in year \(t\) is encountered by the gear, relative to the most vulnerable age class (\(\max_a v_{a,t} = 1\)). It is stored in Fleet@Selectivity@MeanAtAge. Available models via SelectivityModels():
Logistic (SelectivityAtAge, SelectivityAtLength, SelectivityAtWeight): \[v(x) = \frac{1}{1 + \exp\!\left(-\dfrac{\ln 19}{x_{95} - x_{50}}(x - x_{50})\right)}\]
where \(x_{50}\) and \(x_{95}\) are the ages (or lengths, or weights) at 50% and 95% selectivity.
Knife-edge (SelectivityKnifeEdgeAge, SelectivityKnifeEdgeLength): \[v(x) = \begin{cases} 0 & x < x_t \\ 1 & x \ge x_t \end{cases}\]
where \(x_t\) is the threshold age or length.
Double-normal (DoubleNormal, DoubleNormalWeight): \[v(\ell) = 2^{-\left(\dfrac{\ell - \ell_{fs}}{s}\right)^{\!2}}, \quad s = \begin{cases} s_l & \ell \le \ell_{fs} \\ s_r & \ell > \ell_{fs} \end{cases}\]
The ascending and descending standard deviations are derived from three user-supplied parameters:
\[s_l = \frac{\ell_{fs} - \ell_5}{\sqrt{-\log_2 0.05}}, \qquad s_r = \frac{\max(\ell) - \ell_{fs}}{\sqrt{-\log_2 V_{\max}}}\]
where \(\ell_5\) is the length at 5% selectivity, \(\ell_{fs}\) is the length at full selectivity (the peak), and \(V_{\max} \in [0,1]\) is the selectivity at the maximum length class. Setting \(V_{\max} = 1\) gives \(s_r = \infty\), producing a flat-topped (asymptotic) curve; \(V_{\max} < 1\) produces a dome.
25.2 Retention
Retention \(r_{a,t}\) is the probability that a selected individual is retained (kept), conditional on being encountered. It is stored in Fleet@Retention@MeanAtAge. If no retention object is specified, all selected fish are assumed fully retained (\(r_{a,t} = 1\)).
Available models via RetentionModels() mirror the selectivity families:
Logistic (RetentionAtAge, RetentionAtLength, RetentionAtWeight): \[r(x) = \frac{\texttt{MaxRet}}{1 + \exp\!\left(-\dfrac{\ln 19}{x_{95} - x_{50}}(x - x_{50})\right)}\]
MaxRet \(\in (0, 1]\) sets the asymptotic retention (default 1). Values less than 1 allow for partial retention at large sizes.
Knife-edge (RetentionKnifeEdgeAge, RetentionKnifeEdgeLength): \[r(x) = \begin{cases} 0 & x < x_t \\ 1 & x \ge x_t \end{cases}\]
Double-normal (DoubleNormalRetention, DoubleNormalRetentionWeight): same form as the selectivity double-normal with parameters LR5, LFR, and Rmaxlen (or WR5, WFR, Rmaxweight).
25.3 Discard Mortality
Discard mortality \(m_{a,t}\) is the proportion of discarded fish that die. If no discard mortality object is specified, all discards are assumed to survive (\(m_{a,t} = 0\)).
Discard mortality can be supplied as a direct MeanAtAge array, a MeanAtLength array (converted to the age dimension via the ALK by PopulateDiscardMortality()), or a constant scalar applied uniformly across ages and years. The populated object stores both MeanAtAge and MeanAtLength; the age-based array is used in the dynamics.
25.4 Combining Schedules
The three schedules combine to give the total mortality rate on encountered fish and the composition of the catch. For an individual of age \(a\):
- Landed with probability \(v_{a,t} \cdot r_{a,t}\)
- Discarded and dies with probability \(v_{a,t} \cdot (1 - r_{a,t}) \cdot m_{a,t}\)
- Discarded and survives with probability \(v_{a,t} \cdot (1 - r_{a,t}) \cdot (1 - m_{a,t})\)
The total fishing mortality apportioned to age \(a\) is therefore proportional to \(v_{a,t}\bigl[r_{a,t} + (1 - r_{a,t})\,m_{a,t}\bigr]\). The full dynamics are developed in Chapter 27 and Chapter 30.