## 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.