An Analysis of the Evolutionary Dynamic Version of the Dennis Model for Allee Effects

Abstract

This paper studies evolutionary dynamics of a species in the presence of the biological phenomenon known as the strong Allee effect. This phenomenon can be modeled in a multitude of ways but we choose to use the dynamic model developed by Dennis [1989]. We consider an evolutionary game theoretic version of Dennis’ model. This model introduces a dynamical system for population density n and a mean phenotypic trait u. The system depends on several parameters, namely the inherent growth rate r = r (u), the inherent carrying capacity k = k (u), the loss from not mating λ = λ (u), and the population density at which there is a 50% chance of mating which is denoted θ = θ (u). In this paper we obtain sufficient conditions for the stability of the equilibrium points of our evolutionary model, drawing biological punchlines as needed. The results of this paper are summarized in a general theorem. In addition, we provide a specific example to illustrate the application these results as we consider specific functions for the Allee parameters of r, k, λ, and θ.