Difference equations as models of evolutionary population dynamics

J. M. Cushing

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)103-127
Number of pages25
JournalJournal of biological dynamics
Issue number1
StatePublished - Jan 1 2019


  • 39A28
  • 39A30
  • 39A60
  • 92D15
  • 92D25
  • Darwinian dynamics
  • Population dynamics
  • bifurcation
  • difference equations
  • evolutionary dynamics
  • evolutionary game theory
  • stability

ASJC Scopus subject areas

  • Ecology, Evolution, Behavior and Systematics
  • Ecology


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