Dynamics equations

After setting the equilibrium population age distribution in the first year of the model, the population abundance \(N_{y,a}\) in subsequent years is \[ N_{y,a} = \begin{cases} R_y \exp(\delta_y - 0.5 \tau^2) & a = 0\\ N_{y-1,a-1} \exp(-Z_{y-1,a-1}) & a = 1, \ldots, A - 1,\\ N_{y-1,a-1} \exp(-Z_{y-1,a-1}) + N_{y-1,a} \exp(-Z_{y-1,a}) & a = A \end{cases} \] where \(R_y\) is the recruitment, \(\delta_y\) are recruitment deviates, and \(A\) is the maximum-age as the plus-group.

Recruitment is modelled as \[ R_y = \begin{cases} \dfrac{\alpha^{\textrm{B}} B^S_{y-1}}{1 + \beta^{\textrm{B}}B^S_{y-1}} & \textrm{if Beverton-Holt stock-recruit relationship}\\ \alpha^{\textrm{R}} B^S_{y-1} \exp(-\beta^{\textrm{R}} B^S_{y-1})& \textrm{if Ricker stock-recruit relationship} \end{cases}. \]

The spawning biomass is \(B^S_y\) is \[B^S_y = \sum_a w_{y,a} m_{y,a} N_{y,a},\] where \(m_{y,a}\) and \(w_{y,a}\) are the maturity at age and weight at age, respectively.