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  • Letter
  • Open Access

From evolutionarily stable states to eco-evolutionary metastability in evolutionary games with internal mixed equilibrium

Krzysztof Argasiński* and Jacek Miękisz†

Andrew Morozov‡

  • School of Computing and Mathematical Sciences, University of Leicester, Leicester, United Kingdom and Institute of Ecology and Evolution, Russian Academy of Sciences, Moscow, Russia

  • *Contact author: argas1@wp.pl
  • †Contact author: miekisz@mimuw.edu.pl
  • ‡Contact author: am379@leicester.ac.uk
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Phys. Rev. Research 8, L032031 – Published 26 August, 2026

DOI: https://doi.org/10.1103/hsh2-1ldl

Abstract

Evolutionary game theory, using deterministic framework, focuses on deriving conditions that describe the evolutionary stability of Nash equilibria, which are the equilibrium points of the replicator equations. Here, we argue that nonasymptotic metastable states can also play a substantial role in evolutionary games, where the evolving trajectory resides in some state for long time; then, without changing any parameter, it exhibits an abrupt transition to a different state. We consider metastates under the demographic games framework settings. This framework explicitly incorporates the payoff matrix of interactions between individual players into death/birth processes, providing a mechanistic (physical) rationale for the abstract concept of fitness, which is a challenge for the classical game theory. We consider games with an internal equilibrium with mixed strategies. As an insightful case study, we consider the demographic version of the Hawk-Dove game, a paradigmatic model of evolution of aggressive behavior. Apart from eco-evolutionary attractors, the system may have metastable states, such as ghost attractors and crawl-by dynamics. Of particular practical importance is the metastate providing a temporal rescue from the eventual evolutionary suicide. We explain why it is necessary to study transients in evolutionary games and point out consequences of ignoring these phenomena.

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Outline

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Phys. Rev. Research 8, L032031– Published 26 August, 2026
  • Received 24 July 2025
  • Accepted 27 May 2026
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