Difference equations as models of evolutionary population dynamics

Abstract

We describe the evolutionary game theoretic methodology for extending a difference equation population dynamic model in a way so as to account for the Darwinian evolution of model coefficients. We give a general theorem that describes the familiar transcritical bifurcation that occurs in non-evolutionary models when theextinction equilibrium destabilizes. This bifurcation results in survival (positive) equilibria whose stability depends on the direction of bifurcation. We give several applications based on evolutionary versions of some classic equations, such as the discrete logistic (Beverton–Holt) and Ricker equations. In addition to illustrating our theorems, these examples also illustrate other biological phenomena, such as strong Allee effects, time-dependent adaptive landscapes, and evolutionary stable strategies.

Keywords

• population dynamics
• evolutionary dynamics
• bifurcation
• stability
• darwinian dynamics
• evolutionary game theory
• difference equations