A Darwinian Ricker Equation

Jim M. Cushing

Research output: Chapter in Book/Report/Conference proceedingConference contribution

5 Scopus citations


The classic Ricker equation xt + 1= bxtexp (- cxt) has positive equilibria for b> 1 that destabilize when b> e2 after which its asymptotic dynamics are oscillatory and complex. We study an evolutionary version of the Ricker equation in which coefficients depend on a phenotypic trait subject to Darwinian evolution. We are interested in the question of whether evolution will select against or will promote complex dynamics. Toward this end, we study the existence and stability of its positive equilibria and focus on equilibrium destabilization as an indicator of the onset of complex dynamics. We find that the answer relies crucially on the speed of evolution and on how the intra-specific competition coefficient c depends on the evolving trait. In the case of a hierarchical dependence, equilibrium destabilization generally occurs after e2 when the speed of evolution is sufficiently slow (in which case we say evolution selects against complex dynamics). When evolution proceeds at a faster pace, destabilization can occur before e2 (in which case we say evolution promotes complex dynamics) provided the competition coefficient is highly sensitive to changes in the trait v. We also show that destabilization does not always result in a period doubling bifurcation, as in the non-evolutionary Ricker equation, but under certain circumstances can result in a Neimark-Sacker bifurcation.

Original languageEnglish (US)
Title of host publicationProgress on Difference Equations and Discrete Dynamical Systems - 25th ICDEA, 2019
EditorsSteve Baigent, Martin Bohner, Saber Elaydi
Number of pages13
ISBN (Print)9783030601065
StatePublished - 2020
Event25th International Conference on Difference Equations and Applications, ICDEA 2019 - London, United Kingdom
Duration: Jun 24 2019Jun 28 2019

Publication series

NameSpringer Proceedings in Mathematics and Statistics
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017


Conference25th International Conference on Difference Equations and Applications, ICDEA 2019
Country/TerritoryUnited Kingdom


  • Chaos
  • Darwinian Ricker equation
  • Evolutionary game theory
  • Ricker equation

ASJC Scopus subject areas

  • General Mathematics


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