Beverton–Holt 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,

: $n\_\{t+1\}\; =\; frac\{R\_0\; n\_t\}\{1+\; n\_t/M\}.$

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

: $frac\{dN\}\{dt\}\; =\; rN\; left(\; 1\; -\; frac\{N\}\{K\}\; ight),$

and its solution is

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

References

* Citation
last = Beverton
first = R. J. H.
author-link =
last2 = Holt
first2 = S. J.
year = 1957
title = On the Dynamics of Exploited Fish Populations
edition =
volume =
series = Fishery Investigations Series II Volume XIX
publication-place =
place =
publisher = Ministry of Agriculture, Fisheries and Food
pages =
page =
id =
isbn =
doi =
oclc =
url =
accessdate =

* Citation
last = Brännström
first = Åke
last2 = Sumpter
first2 = David J. T.
author-link =
year = 2005
title = The role of competition and clustering in population dynamics
periodical = Proc. R. Soc. B
series =
publication-place =
place =
publisher =
volume = 272
issue = 1576
pages = 2065–2072
url = http://www.math.uu.se/~david/web/BrannstromSumpter05a.pdf
issn =
doi = 10.1098/rspb.2005.3185
oclc =
accessdate =

* Citation
last = Geritz
first = Stefan A. H.
author-link =
last2 = Kisdi
first2 = Éva
year = 2004
title = On the mechanistic underpinning of discrete-time population models with complex dynamics
periodical = J. Theor. Biol.
series =
publication-place =
place =
publisher =
volume = 228
issue = 2
pages = 261–269
url =
issn =
doi = 10.1016/j.jtbi.2004.01.003
oclc =
accessdate =

*Citation
last = Ricker
first = W. E.
author-link = Bill Ricker
year = 1954
title = Stock and recruitment
periodical = J. Fisheries Res. Board Can.
series =
publication-place =
place =
publisher =
volume = 11
issue =
pages = 559–623
url =
issn =
doi =
oclc =
accessdate =