An Analysis of the Evolutionary Dynamic Version of the Dennis Model for Allee Effects
Publisher
The University of Arizona.Rights
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.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.Type
textElectronic Thesis
Degree Name
B.S.Degree Level
bachelorsDegree Program
Honors CollegeMathematics