06model10_beverton-holt-autocorrected.qmd

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## Model 10. Beverton-Holt Curve with Autocorrelated Lognormal Error {.unnumbered}

The Beverton-Holt curve with autocorrelated lognormal errors is a parametric model of recruitment generation where survival to recruitment age is density dependent and subject to serially-correlated stochastic variation. *The Beverton-Holt curve with autocorrelated lognormal error model depends on spawning biomass and is time-dependent.*

The Beverton-Holt curve with autocorrelated lognormal error generates recruitment as

$$
\begin{split}
\hat{r}(t) &= \frac{\alpha \cdot b_s (t-1)}{\beta + b_s (t-1)} \cdot e^{\varepsilon_t} \\
& where\ \varepsilon_t = \phi\varepsilon_{t-1} + w_t \ \ with\ w_t \sim  N(0,\sigma^2_w), \\ 
& \hat{R}(t) = c_r \cdot \hat{r}(t), and\ B_s(t) = c_B \cdot b_S(t)
\end{split}
\tag{30}\label{eq:30}
$$

The stock-recruitment parameters $\alpha$, $\beta$, $\phi$, $\varepsilon_0$, and $\sigma^2_w$ and the conversion coefficients
for recruitment $c_R$ and spawning stock biomass $c_B$ are specified by the user. The parameter $\varepsilon_0$ is the log-scale residual for the stock-recruitment fit in the time prior to the projection. If this value is not known, the default is to set $\varepsilon_0 = 0$.