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Evolutionary and dynamic stability in continuous population games.

Ilan Eshel1, Emilia Sansone

  • 1School of Mathematics, Tel Aviv University, Tel Aviv, Israel. illan@post.tau.ac.il

Journal of Mathematical Biology
|May 17, 2003
PubMed
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Asymptotic stability in evolutionary dynamics depends on the chosen topology. A new Maximum Shift Topology and Continuous Replicator Stability (CRSS) condition offer a more relevant stability measure for natural selection processes.

Area of Science:

  • Evolutionary Game Theory
  • Mathematical Biology
  • Population Genetics

Background:

  • Asymptotic stability of replicator dynamics is crucial for understanding evolutionary processes.
  • Existing stability conditions, like Strong Uninvadability, are sensitive to the chosen topology on strategy spaces.
  • The variational distance topology has limitations in capturing stability for continuous traits.

Purpose of the Study:

  • To investigate the impact of different topologies on asymptotic stability in replicator dynamics.
  • To introduce and analyze the Maximum Shift Topology for continuous quantitative traits.
  • To establish a new, relevant stability condition for evolutionary processes under this topology.

Main Methods:

  • Analysis of replicator dynamics over a continuum of pure strategies.

Related Experiment Videos

  • Introduction and application of the Maximum Shift Topology.
  • Derivation of the Continuous Replicator Stability (CRSS) condition.

    • Main Results:

    • Strong Uninvadability is insufficient for asymptotic stability under variational distance topology for small deviations.
    • The Maximum Shift Topology provides a more relevant framework for continuous traits.
    • Continuous Replicator Stability (CRSS) is a necessary and sufficient condition for asymptotic stability under the Maximum Shift Topology.

    Conclusions:

    • The choice of topology significantly influences the assessment of asymptotic stability in evolutionary dynamics.
    • CRSS, under the Maximum Shift Topology, offers a more robust and biologically relevant measure of stability for continuous traits.
    • This framework advances the understanding of evolutionary selection processes in quantitative traits.