# 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}}. $ <P> 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).

### 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*.