Selectivity and mortality

Fleet selectivity

For fleets, selectivity is defined by blocks (indexed by \(b\)) which can then be assigned to any fleet \(f\) and year \(y\) to allow for time-varying selectivity. By default, each fleet is assigned to its own block for all years (no time-varying selectivity).

For flat-topped selectivity in block \(b\), two parameters are used and expressed in terms of length units: the length of 5% selectivity (\(L^5_b\)) and the length of full selectivity \(L^{\textrm{FS}}_b\).

For dome selectivity, a third parameter, the selectivity at \(L_{\infty}\), \(V^{L_{\infty}}_b\) is also used. Length-based selectivity is converted to age-based selectivity in the age-structured model as:

\[ v_{y,a,b} = \begin{cases} 2^{-[(L_{y,a} - L^{\textrm{FS}}_b)/(\sigma^{\textrm{asc}}_b)]^2} & \textrm{if } L_{y,a} < L^{\textrm{FS}}_b\\ 1 & \textrm{if logistic and } L_{y,a} \ge L^{\textrm{FS}}_b,\\ 2^{-[(L_{y,a} - L^{\textrm{FS}}_b)/(\sigma^{\textrm{des}}_b)]^2} & \textrm{if dome and } L_{y,a} \ge L^{\textrm{FS}}_b \end{cases} \]

where \(L_{y,a}\) is the mean length-at-age and \(\sigma^{\textrm{asc}}_b = (L^5_b - L^{\textrm{FS}}_b)/\sqrt{-\log_2(0.05)}\) and \(\sigma^{\textrm{des}}_b = (L_{\infty} - L^{\textrm{FS}}_b)/\sqrt{-\log_2(V^{L_{\infty}}_b)}\) control the shape of the ascending and descending limbs, respectively, of the selectivity function.

In this parameterization, length-based selectivity is constant over time. The corresponding age-based selectivity within each block is constant so long as growth is not time-varying.

Selectivity can also be parameterized where \(v_{y,a,b}\) are free independent parameters. In this case, selectivity does not vary among years.

Total mortality \(Z\) in year \(y\) and for age \(a\) is the sum of fishing mortality \(F\) from all fleets and natural mortality \(M\),

\[ Z_{y,a} = M_{y,a} + \sum_f v_{y,a,f} F_{y,f},\]

where \(v_{y,a,f}\) is the fleet selectivity after assigning blocks to fleets.

Index selectivity

Index selectivity is constant over time and is denoted as \(v_{a,s}\), using either logistic, dome, or free parameterizations.