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 \[ N_{y=1,a} = \begin{cases} R^{\textrm{eq}} \exp(\delta_{y=1} - 0.5 \tau^2) & a = 0\\ R^{\textrm{eq}} \times l(Z^{\textrm{eq}})_a \times \exp(\delta^{\textrm{init}}_a-0.5\tau^2) & a = 1, \ldots, A\\ \end{cases} \] where the \(R^{\textrm{eq}}\) is the equilibrium recruitment \(l(Z^{\textrm{eq}})_a\) is the survival:
\[ l(Z^{\textrm{eq}})_a = \begin{cases} 1 & a = 0\\ \exp(-\sum_{i=0}^{a-1}Z^{\textrm{eq}}_i) & a = 1, \ldots, A-1\\ \dfrac{\exp(-\sum_{i=0}^{a-1}Z^{\textrm{eq}}_i)}{1 - \exp(-Z^{\textrm{eq}}_A)} & a = A, \end{cases} \] where \(Z^{\textrm{eq}}_a = M_{y=1,a} + \sum_f v_{y=1,a,f} F^{\textrm{eq}}_f\) is the equilibrium total mortality rate.
Unfished conditions are modeled by setting \(Z^{\textrm{eq}}_a = M_{y=1,a}\).
The model estimates \(F^{\textrm{eq}}_f\) and \(R^{\textrm{eq}}\) from the equilibrium catch in weight (\(Y^{\textrm{eq}}_f\), user-provided) such that
\[ Y^{\textrm{eq}}_f = \sum_a\left[\frac{v_{y=1,a,f} F^{\textrm{eq}}_f}{Z^{\textrm{eq}}_a}R^{\textrm{eq}}l_a(1 - \exp(Z^{\textrm{eq}}_a))w_a\right] \] 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 \(\phi^{\textrm{eq}} = \sum_a l(Z^{\textrm{eq}})_a \times f_a \times m_a\) is the spawners-per-recruit calculated from equilibrium survival, spawning output \(f_a\) (typically from weight as a proxy) and maturity \(m_a\) at age.
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 be estimated, with \(\tau\) as the corresponding standard deviation.
By default, the model initializes with an unfished population, i.e., no fishing mortality and no deviations from the equilibrium age structure.