Beverton-Holt model

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

{\ displaystyle n_ {t + 1} = {\ frac {R_ {0} n_ {t}} {1 + n_ {t} / M}}.}

Here 0 is construed as the proliferation rate per generation and K = ( 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 derives from the competition (Brännström & Sumpter 2005), within-year resource limited competition (Geritz & Kisdi 2004) or even the outcome of a source-sink Malthusian patches linked by dispersal-dependent-density (Bravo of Parra et al 2013). The Beverton-Holt model can be generalized to includescramble 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

{\ displaystyle n_ {t} = {\ frac {Kn_ {0}} {n_ {0} + (K-n_ {0}) R_ {0} ^ {- t}}}.}

Because of this structure, the model can be considered as the analogous discrete-time of the continuous-time logistic equation for population growth by Verhulst ; for comparison, the logistic equation is

{\ displaystyle {\ frac {dN} {dt}} = rN \ left (1 – {\ frac {N} {K}} \ right),}

and its solution is

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

References

  • Beverton, RJH; Holt, SJ (1957), On the Dynamics of Exploited Fish Populations , Fishery Investigations Series II Volume XIX, Ministry of Agriculture, Fisheries and Food
  • Brännström, Åke; Sumpter, David JT (2005), “The role of competition and clustering in population dynamics” (PDF) , Proc. R. Soc. B , 272 (1576), pp. 2065-2072, doi : 10.1098 / rspb.2005.3185 , PMC  1559893  , PMID  16191618
  • Bravo from Parra, R .; Marvá, M .; Sánchez, E .; Sanz, L. (2013), “Reduction of discrete dynamical systems with applications to dynamic population models,” Math Model Nat Phenom , 8 (6), pp. 107-129
  • Geritz, Stefan AH; Kisdi, Eva (2004), “The mechanistic underpinning of discrete-time population models with complex dynamics”, J. Theor. Biol. 228 (2), pp. 261-269, doi : 10.1016 / j.jtbi.2004.01.003 , PMID  15094020
  • Ricker, WE (1954), “Stock and Recruitment”, J. Fisheries Res. Board Can. 11 , pp. 559-623

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