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  $$a_{t+1} = k_1 \frac{a_t}{ (1+k_2 a_t)^{c}}. $$  When $$c=1$$ the Hassell model is simply the Beverton-Holt model.