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Evolutionary game dynamics in a Wright-Fisher process.

Lorens A Imhof1, Martin A Nowak

  • 1Statistische Abteilung, Universität Bonn, Germany. limhof@uni-bonn.de

Journal of Mathematical Biology
|February 8, 2006
PubMed
Summary
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In evolutionary game dynamics, the 1/3 law states that selection favors replacing strategy B with A when the unstable equilibrium frequency is below 1/3, particularly in finite populations.

Area of Science:

  • Evolutionary game theory
  • Population genetics
  • Mathematical biology

Background:

  • Evolutionary game dynamics in finite populations are modeled using the stochastic Wright-Fisher process.
  • This process involves frequency-dependent selection and discrete generations with offspring production proportional to payoff.

Purpose of the Study:

  • To quantify frequency-dependent selection by comparing absorption probabilities to random drift.
  • To derive conditions for selection favoring one strategy over another using total positivity.

Main Methods:

  • Modeling evolutionary game dynamics with a frequency-dependent, stochastic Wright-Fisher process.
  • Analyzing a symmetric two-strategy game (A and B) in a constant-sized population.
  • Utilizing the concept of total positivity to derive selection conditions.

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Main Results:

  • The Markov process has two absorbing states: all A or all B.
  • Conditions for selection favoring A or B were derived.
  • The 1/3 law for weak selection was established: selection favors A over B if the unstable equilibrium frequency of A is less than 1/3.

Conclusions:

  • The 1/3 law provides a condition for strategy replacement in evolutionary games under weak selection.
  • Total positivity is a useful mathematical tool for analyzing selection in finite populations.
  • Understanding these dynamics is crucial for predicting evolutionary trajectories.