The evolutionarydynamics of a population model with a strong Allee effect

  • Received: 01 May 2014 Accepted: 29 June 2018 Published: 01 April 2015
  • MSC : Primary: 92D25, 92D15; Secondary: 37N25.

  • An evolutionary game theoretic model for a population subject to predationand a strong Allee threshold of extinction is analyzed using, among othermethods, Poincaré-Bendixson theory. The model is a nonlinear, planeautonomous system whose state variables are population density and the meanof a phenotypic trait, which is subject to Darwinian evolution, thatdetermines the population's inherent (low density) growth rate (fitness). Atrade-off is assumed in that an increase in the inherent growth rate resultsin a proportional increase in the predator's attack rate. The main resultsare that orbits equilibrate (there are no cycles or cycle chains ofsaddles), that the extinction set (or Allee basin) shrinks when evolutionoccurs, and that the meant trait component of survival equilibria occur atmaxima of the inherent growth rate (as a function of the trait).

    Citation: Jim M. Cushing. The evolutionarydynamics of a population model with a strong Allee effect[J]. Mathematical Biosciences and Engineering, 2015, 12(4): 643-660. doi: 10.3934/mbe.2015.12.643

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  • An evolutionary game theoretic model for a population subject to predationand a strong Allee threshold of extinction is analyzed using, among othermethods, Poincaré-Bendixson theory. The model is a nonlinear, planeautonomous system whose state variables are population density and the meanof a phenotypic trait, which is subject to Darwinian evolution, thatdetermines the population's inherent (low density) growth rate (fitness). Atrade-off is assumed in that an increase in the inherent growth rate resultsin a proportional increase in the predator's attack rate. The main resultsare that orbits equilibrate (there are no cycles or cycle chains ofsaddles), that the extinction set (or Allee basin) shrinks when evolutionoccurs, and that the meant trait component of survival equilibria occur atmaxima of the inherent growth rate (as a function of the trait).
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    © 2015 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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