Beverton–Holt model

Short description: Discrete-time population model

The Beverton–Holt model is a classic discrete-time population model which gives the expected number n _t+1 (or density) of individuals in generation t + 1 as a function of the number of individuals in the previous generation,

[math]\displaystyle{ n_{t+1} = \frac{R_0 n_t}{1+ n_t/M}. }[/math]

Here R₀ is interpreted as the proliferation rate per generation and K = (R₀ − 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), within-year resource limited competition (Geritz & Kisdi 2004) or even as the outcome of a source-sink Malthusian patches linked by density-dependent dispersal (Bravo de la Parra et al. 2013). 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

[math]\displaystyle{ n_t = \frac{K n_0}{n_0 + (K - n_0) R_0^{-t}}. }[/math]

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

[math]\displaystyle{ \frac{dN}{dt} = rN \left( 1 - \frac{N}{K} \right), }[/math]

and its solution is

[math]\displaystyle{ N(t) = \frac{K N(0)}{N(0) + (K - N(0)) e^{-rt}}. }[/math]

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 
• Brännström, Åke; Sumpter, David J. T. (2005), "The role of competition and clustering in population dynamics", Proc. R. Soc. B 272 (1576): 2065–2072, doi:10.1098/rspb.2005.3185, PMID 16191618, PMC 1559893, http://www.math.uu.se/~david/web/BrannstromSumpter05a.pdf 
• Bravo de la Parra, R.; Marvá, M.; Sánchez, E.; Sanz, L. (2013), "Reduction of discrete dynamical systems with applications to dynamics population models", Math Model Nat Phenom 8 (6): 107–129, http://oa.upm.es/49338/1/INVE_MEM_2013_269104.pdf 
• Geritz, Stefan A. H.; Kisdi, Éva (2004), "On the mechanistic underpinning of discrete-time population models with complex dynamics", J. Theor. Biol. 228 (2): 261–269, doi:10.1016/j.jtbi.2004.01.003, PMID 15094020, Bibcode: 2004JThBi.228..261G 
• Ricker, W. E. (1954), "Stock and recruitment", J. Fisheries Res. Board Can. 11: 559–623