Ricker model

The Ricker model is a classic discrete population model which gives the expected number (or density) of individuals $a_{t+1}$ in generation $t+1$ as a function of the number of individuals in the previous generation,

$a_{t+1}=a_{t}e^{r(1-{\frac {a_{t}}{k}})}$

Here $r$ is interpreted as an intrinsic growth rate and $k$ as the carrying capacity of the environment. The Ricker model was introduced in the context of the fisheries by Ricker (1954). Subsequent work has derived the model under other assumptions such as scramble competition (e.g. Brännström & Sumpter 2005) or within-year resource limited competition (Geritz and Kisdi 2004). The Ricker model is a limiting case of the Hassell model (Brännström & Sumpter 2005) which takes the form

$a_{t+1}=a_{t}{\frac {e^{r}}{(1+a_{t})^{b}}}.$

where $b$ should be taken equal to $b=(k+1)/k$ to obtain the same carrying capacity.

References

Brännström A and Sumpter DJ (2005) The role of competition and clustering in population dynamics. Proc Biol Sci Oct 7 272(1576):2065-72
Geritz SA and Kisdi E (2004). On the mechanistic underpinning of discrete-time population models with complex dynamics. J Theor Biol. 2004 May 21;228(2):261-9.
Ricker, WE (1954). Stock and recruitment. Journal of the Fisheries Research Board of Canada.