Abstract
An evolutionary game theoretic model for a population subject to predation and a strong Allee threshold of extinction is analyzed using, among other methods, Poincaré-Bendixson theory. The model is a nonlinear, plane autonomous system whose state variables are population density and the mean of a phenotypic trait, which is subject to Darwinian evolution, that determines the population's inherent (low density) growth rate (fitness). A trade-off is assumed in that an increase in the inherent growth rate results in a proportional increase in the predator's attack rate. The main results are that orbits equilibrate (there are no cycles or cycle chains of saddles), that the extinction set (or Allee basin) shrinks when evolution occurs, and that the meant trait component of survival equilibria occur at maxima of the inherent growth rate (as a function of the trait).
Highlights
There is a growing literature on the modeling and analysis of Allee effects in population dynamics and related fields
One basic notion is that of a strong Allee effect, which is a dynamic scenario in which both extinction and survival attractors simultaneously exist
The ordinary differential equation (4) arises in various contexts concerning the dynamics of a biological population, for example in an interaction with a predator
Summary
There is a growing literature on the modeling and analysis of Allee effects in population dynamics and related fields (e.g., the management of renewable resources and endangered species, ecosystem dynamics, the and the spread of epidemics). One basic notion is that of a strong Allee effect, which is a dynamic scenario in which both extinction and survival attractors (usually equilibria) simultaneously exist. This bi-stability is usually considered to be the defining hallmark of an Allee effect and is a property that is often of central interest in studies of Allee effects. The basin of attraction of the extinction state (the Allee basin) is a region in which a population is threatened with extinction This is the primary motivation for studying models that include mechanisms that result in a strong Allee effect
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