06model06_ricker.qmd

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## Model 6. Ricker Curve with Lognormal Error {.unnumbered}

The Ricker curve [@ricker1954d] with lognormal error is a parametric model of recruitment generation where survival to recruitment age is density dependent and subject to stochastic variation. *The Ricker curve with lognormal error model depends on spawning biomass and is time invariant.*

The Ricker curve with lognormal error generates recruitment as

$$
\begin{split}
&\hat{r}(t) = \alpha \cdot b_S(t-1) \cdot e^{-\beta \cdot b_S(t-1)}\ \cdot e^w \\
& where\ w \sim N(0,\sigma_w^2), \hat{R}(t) =  c_R \cdot \hat{r}(t),\ and\ B_S(t) = c_B \cdot b_S (t)
\end{split}
\tag{27}\label{eq:27}
$$

The stock-recruitment parameters $\alpha$, $\beta$, and $\sigma_w^2$ and the conversion coefficients for recruitment $C_R$ and spawning stock biomass $c_B$ are specified by the user. Here it is assumed that the parameter estimates for the Beverton-Holt curve have been estimated in relative units of recruitment $r(t)$ and spawning biomass $b_s(t)$ which are converted to absolute values using the conversion coefficients. It is important to note that the expected value of the lognormal error term is not unity but is $exp\left(\dfrac{1}{2}\sigma^2_w\right)$. To generate a recruitment model that has a lognormal error term that is equal to 1, premultiply the parameter $\alpha$ by $exp\left(-\dfrac{1}{2}\sigma^2_w\right)$; this mean correction applies when the lognormal error used to fit the Ricker curve has a log-scale error term $w$ with zero mean.