Initial population distribution

The population numbers at age \(N_{y=1,a}\) in the first year of the model \(y=1\) for ages \(a=0,1,\ldots,A\) is \[ N_{y=1,a} = \begin{cases} R^{\textrm{eq}} \exp(\delta_{y=1} - 0.5 \tau^2) & a = 0\\ R^{\textrm{eq}} \exp(-\sum_{i=1}^{a}Z^{\textrm{eq}}_i)\exp(\delta^{\textrm{init}}_a-0.5\tau^2) & a = 1, \ldots, A-1\\ \dfrac{R^{\textrm{eq}} \exp(-\sum_{i=1}^{a}Z^{\textrm{eq}}_i)\exp(\delta^{\textrm{init}}_a-0.5\tau^2)}{1 - \exp(-Z^{\textrm{eq}}_A)} & a = A, \end{cases} \] where the \(R^{\textrm{eq}}\) is the equilibrium recruitment and \(Z^{\textrm{eq}}_a = M_{1,a} + \sum_f v_{1,a,f} F^{\textrm{eq}}_f\) is the equilibrium total mortality rate. Unfished conditions are modeled by setting \(F^{\textrm{eq}}_f = 0\). The model estimates \(F^{\textrm{eq}}_f\) from the provided equilibrium catch in weight \(Y^{\textrm{eq}}_f\) using the yield curve. Once \(Z^{\textrm{eq}}_a\) is obtained, then the equilibrium recruitment is calculated as:

\[ R^{\textrm{eq}} = \begin{cases} \dfrac{\alpha^{\textrm{B}}\phi^{\textrm{eq}} - 1}{\beta^{\textrm{B}}\phi^{\textrm{eq}}} & \textrm{if Beverton-Holt stock-recruit relationship}\\ \dfrac{\log(\alpha^{\textrm{R}}\phi^{\textrm{eq}})}{\beta^{\textrm{R}}\phi^{\textrm{eq}}} & \textrm{if Ricker stock-recruit relationship} \end{cases}, \] where \(\phi^{\textrm{eq}}\) is the spawners-per-recruit when total mortality at age is \(Z^{\textrm{eq}}_a\).

From steepness \(h\), \(\alpha^{\textrm{B}} = \frac{4h}{(1-h)\phi_0}\), \(\beta^{\textrm{B}} = \frac{5h-1}{(1-h)B^S_0}\), \(\alpha^{\textrm{R}} = \frac{(5h)^{1.25}}{\phi_0}\), \(\beta^{\textrm{R}} = \frac{\log(5h)}{B^S_0}\), where \(\phi_0\) and \(B^S_0\) are unfished spawners-per-recruit and unfished spawning biomass, respectively.

The initial recruitment deviations \(\delta^{\textrm{init}}_a\) can also be estimated, with \(\tau\) as the corresponding standard deviation. By default they are not estimated.