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Beyond replicator dynamics: From frequency to density dependent models of evolutionary games.

Vlastimil Křivan1, Theodore E Galanthay2, Ross Cressman3

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This study links evolutionary game theory and population dynamics, showing stable coexistence of Hawk and Dove phenotypes is possible. The findings challenge classic models by incorporating population size and interaction times.

Keywords:
Contest competitionEvolutionary game theoryExploitative competitionHawk–Dove gamePair formationPopulation dynamics

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Area of Science:

  • Evolutionary Game Theory
  • Population Dynamics
  • Theoretical Ecology

Background:

  • Classic evolutionary game theory models (e.g., Hawk-Dove) often assume pair-wise contests and random mating (Hardy-Weinberg equilibrium).
  • Population dynamics models typically focus on single individuals and use mass action for interactions, differing from game theory assumptions.
  • A gap exists in integrating these two approaches to understand evolutionary stable strategies in more realistic population structures.

Purpose of the Study:

  • To link game theoretic and population dynamics models for evolutionary analysis.
  • To investigate conditions for stable coexistence of different phenotypes (e.g., Hawk and Dove) by combining distribution and population dynamics.
  • To assess the impact of population size, interaction times, and density-dependent fitness on evolutionary equilibria.

Main Methods:

  • Developed a combined model integrating distribution dynamics (phenotype frequencies) with population dynamics (birth/death rates).
  • Analyzed the interior Nash equilibrium of the Hawk-Dove game under varying population sizes and interaction durations.
  • Separated individual fitness into birth and death components for singles and pairs, and incorporated density-dependent competition.

Main Results:

  • Stable coexistence of Hawk and Dove populations can be achieved by combining distribution and population dynamics, even without assuming negative density-dependent fitness.
  • The Nash equilibrium is sensitive to population size and the duration of interactions between phenotypes.
  • In models with density-dependent fitness (resource competition), Dove populations often go extinct, regardless of Hawk fighting costs.

Conclusions:

  • Integrating population dynamics with evolutionary game theory provides a more nuanced understanding of phenotype coexistence.
  • Classic evolutionary game theory assumptions regarding interaction times and payoffs may not apply to real populations.
  • The interplay between distribution and population dynamics, particularly the relative time scales, is crucial for stable evolutionary outcomes.