Transformation Ricker equation The classical Ricker equation for modelling density-dependent population growth is:
$N_{t+1} = N_t * e^{r * \left(1-\frac{N_t}{k}\right)}$
where $N_t$ is the initial number of individuals (starting population) and $N_{t+1}$ is the number of individuals in generaltion $t+1$ (after one time step). $k$ is the so called carrying capacity defining the upper asymtotic level of maximum growth (see http://en.wikipedia.org/wiki/Ricker_model)
However the equation is sometimes also given as:
$R = \alpha * S * e^{-\beta * S}$
where $S$ is the initial number of individuals (Spawners, similar to $N_t$) and $R$ is the number of recruits after one time step (probably equal to $N_{t+1}$ or $N_{t+1}$-$N_{t}$; this has not been clear to me from literature).
So my question is how are the variables $r$ and $k$ of the first equation related to $\alpha$ and $\beta$ of the second equation? And is possible to calculate $r$ and $k$ once $\alpha$ and $\beta$ are known?

Replace sequences with functions:
$$ N_{t} \longrightarrow  N(t)$$
Consider $R$ as rate of population growth:
$$R(t)=N'(t)$$
If
$$N(t+1)=N(t) exp(r(1-\frac{N(t)}{k}))$$
Then for small $h \approx 0$ you can linearize the equation:
$$N(t+h)=N(t)+hN(t) exp(r(1-\frac{N(t)}{k}))$$
Then simply:
$$R(t)=N'(t)=\frac{N(t+h)-N(t)}{h}=N(t) exp(r(1-\frac{N(t)}{k}))$$
and
$$R(t)=exp(r/k)N(t) exp(-N(t)/k)$$
Which is in the form you want.
$$N(t) \rightarrow S$$
$$exp(\frac{r}{k}) \rightarrow \alpha$$
$$-\frac{1}{k} \rightarrow \beta$$
