Abstract
ABSTRACTWe 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.
Highlights
Difference equations have a long history of use as discrete time models of population dynamics
We describe a methodology that a modeller can use to account for evolutionary changes to the coefficients in any difference equation population model of interest
In case (b) the population is not threatened with extinction from the extinction equilibrium (0, 0), and so we focus on case (a) when the extinction equilibrium loses stability as b0 increases through the critical value b∗0
Summary
Difference equations have a long history of use as discrete time models of population dynamics. This primary and basic bifurcation results in the creation of positive (survival) equilibria whose stability, at least near the bifurcation point, is dependent on the direction of bifurcation. The occurrence of secondary bifurcations is highly dependent on the specific properties of the nonlinearities in β(x)x and σ (x)x
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