Beverton-Holt model

The Beverton-Holt model is a classic discrete-time population model which gives the expected number (or density) of individuals $$n_{t+1}$$ in generation $$t+1$$ as a function of the number of individuals in the previous generation,  $$n_{t+1} = \frac{R_0 n_t}{1+ \frac{n_t}{k}}. $$  Here $$R_0$$ is interpreted as the proliferation rate per generation and $$(R_0-1) k $$ is the carrying capacity of the environment. The Beverton-Holt model was introduced in the context of the fisheries by Beverton & Holt (1957). It is the discrete-time analog of the continuous time logistic equation for population growth created by Pierre Verhulst. Subsequent work has derived the model under other assumptions such as contest competition (Brännström & Sumpter 2005) or within-year resource limited competition (Geritz and Kisdi 2004). The Beverton-Holt model can be generalized to include scramble competition (see the Ricker model, the Hassell model and the Maynard-Smith Slatkin model). It is also possible to include a parameter reflecting the spatial clustering of individuals (see Brännström & Sumpter 2005).