Beverton-Holt model

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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,

$n_{t+1} = \frac{R_0 n_t}{1+ n_t/M}.$

Here $R 0$ is interpreted as the proliferation rate per generation and $K = (R 0 - 1) M$ is the carrying capacity of the environment. The Beverton–Holt model was introduced in the context of fisheries by Beverton & Holt (1957). Subsequent work has derived the model under other assumptions such as contest competition (Brännström & Sumpter 2005) or within-year resource limited competition (Geritz & 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).

Despite being nonlinear, the model can be solved explicitly, since it is in fact an inhomogeneous linear equation in $1 / n$ . The solution is

$n_t = \frac{K n_0}{n_0 + (K - n_0) R_0^{-t}}.$

Because of this structure, the model can be considered as the discrete-time analogue of the continuous-time logistic equation for population growth introduced by Verhulst; for comparison, the logistic equation is $d N / d t = r N (1 - N / K)$ , and its solution is

$N(t) = \frac{K N(0)}{N(0) + (K - N(0)) e^{-rt}}.$

[edit] References

Beverton, R. J. H. & Holt, S. J. (1957), On the Dynamics of Exploited Fish Populations, Fishery Investigations Series II Volume XIX, Ministry of Agriculture, Fisheries and Food