The evolutionarydynamics of a population model with a strong Allee effect

Jim M. Cushing; Jim M. Cushing

doi:10.3934/mbe.2015.12.643

Mathematical Biosciences and Engineering

2015, Volume 12, Issue 4: 643-660. doi: 10.3934/mbe.2015.12.643

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The evolutionarydynamics of a population model with a strong Allee effect

Jim M. Cushing ¹

1.
Department of Mathematics, Interdisciplinary Program in Applied Mathematics, 617 N Santa Rita, Tucson, Arizona, 85721

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).

Keywords:

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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Abstract

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