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| The '''Beverton–Holt model''' is a classic discrete-time [[population modeling|population model]] which gives the [[expected value|expected]] number ''n''<sub> ''t''+1</sub> (or density) of individuals in generation ''t'' + 1 as a function of the number of individuals in the previous generation,
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| : <math>n_{t+1} = \frac{R_0 n_t}{1+ n_t/M}. </math>
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| Here ''R''<sub>0</sub> is interpreted as the proliferation rate per generation and ''K'' = (''R''<sub>0</sub> − 1) ''M'' is the [[carrying capacity]] of the environment. The Beverton–Holt model was introduced in the context of [[fisheries management|fisheries]] by [[Ray Beverton|Beverton]] & [[Sidney Holt|Holt]] (1957). Subsequent work has derived the model under other assumptions such as [[competition (biology)|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 [[John Maynard Smith|Maynard Smith]]–Slatkin model). It is also possible to include a parameter reflecting the spatial clustering of individuals (see Brännström & Sumpter 2005).
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| Despite being nonlinear, the model can be solved explicitly, since it is in fact an inhomogeneous linear equation in 1/''n''.
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| The solution is
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| : <math>
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| n_t = \frac{K n_0}{n_0 + (K - n_0) R_0^{-t}}.
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| </math> | |
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| 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 [[Pierre Verhulst|Verhulst]]; for comparison, the logistic equation is
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| : <math>\frac{dN}{dt} = rN \left( 1 - \frac{N}{K} \right),</math> | |
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| and its solution is
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| : <math> | |
| N(t) = \frac{K N(0)}{N(0) + (K - N(0)) e^{-rt}}.
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| </math>
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| ==References==
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| * {{Citation
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| | last = Beverton
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| | first = R. J. H.
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| | author-link =
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| | last2 = Holt
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| | first2 = S. J.
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| | year = 1957
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| | title = On the Dynamics of Exploited Fish Populations
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| | edition =
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| | volume =
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| | series = Fishery Investigations Series II Volume XIX
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| | publication-place =
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| | place =
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| | publisher = Ministry of Agriculture, Fisheries and Food
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| | pages =
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| | page =
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| | id =
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| | isbn =
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| | doi =
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| | oclc =
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| | url =
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| | accessdate =
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| }}
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| * {{Citation
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| | last = Brännström
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| | first = Åke
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| | last2 = Sumpter
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| | first2 = David J. T.
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| | author-link =
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| | year = 2005
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| | title = The role of competition and clustering in population dynamics
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| | periodical = Proc. R. Soc. B
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| | series =
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| | publication-place =
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| | place =
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| | publisher =
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| | volume = 272
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| | issue = 1576
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| | pages = 2065–2072
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| | url = http://www.math.uu.se/~david/web/BrannstromSumpter05a.pdf
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| | doi = 10.1098/rspb.2005.3185
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| | oclc =
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| | accessdate =
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| | pmid = 16191618
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| | pmc = 1559893
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| }}
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| | |
| * {{Citation
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| | last = Geritz
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| | first = Stefan A. H.
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| | author-link =
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| | last2 = Kisdi
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| | first2 = Éva
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| | year = 2004
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| | title = On the mechanistic underpinning of discrete-time population models with complex dynamics
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| | periodical = J. Theor. Biol.
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| | series =
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| | publication-place =
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| | place =
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| | publisher =
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| | volume = 228
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| | issue = 2
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| | pages = 261–269
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| | url =
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| | doi = 10.1016/j.jtbi.2004.01.003
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| | oclc =
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| | accessdate =
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| | pmid = 15094020
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| }}
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| *{{Citation
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| | last = Ricker
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| | first = W. E.
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| | author-link = Bill Ricker
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| | year = 1954
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| | title = Stock and recruitment
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| | periodical = J. Fisheries Res. Board Can.
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| | series =
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| | publication-place =
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| | place =
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| | publisher =
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| | volume = 11
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| | issue =
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| | pages = 559–623
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| | url =
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| | issn =
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| | doi =
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| | oclc =
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| | accessdate =
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| }}
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| {{DEFAULTSORT:Beverton-Holt model}}
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| [[Category:Demography]]
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| [[Category:Biostatistics]]
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| [[Category:Fisheries science]]
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| [[Category:Stochastic processes]]
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