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 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).

References

Beverton RJH and Holt SJ (1957). On the Dynamics of Exploited Fish Populations.
Brännström A and Sumpter DJ (2005) The role of competition and clustering in population dynamics. Proc Biol Sci. Oct 7 272(1576):2065-72 [1]
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.