Beverton–Holt model
The Beverton–Holt model is a classic discrete-time population model which gives the expected number n t+1 of individuals in generation t + 1 as a function of the number of individuals in the previous generation,
Here R0 is interpreted as the proliferation rate per generation and K = M is the carrying capacity of the environment. The Beverton–Holt model was introduced in the context of fisheries by Beverton & Holt. Subsequent work has derived the model under other assumptions such as contest competition, within-year resource limited competition or even as the outcome of a source-sink Malthusian patches linked by density-dependent dispersal. The Beverton–Holt model can be generalized to include scramble competition. It is also possible to include a parameter reflecting the spatial clustering of individuals.
Despite being nonlinear, the model can be solved explicitly, since it is in fact an inhomogeneous linear equation in 1/n.
The solution is
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
and its solution is
