## 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.