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Continuous Probabilistic Analysis to Evolutionary Game Dynamics in Finite Populations

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Abstract

Evolutionary game dynamics of two strategies in finite population is studied by continuous probabilistic approach. Besides frequency dependent selection, mutation was also included in this study. The equilibrium probability density functions of abundance, expected time to extinction or fixation were derived and their numerical solutions are calculated as illustrations. Meanwhile, individual-based computer simulations are also done. A comparison reveals the consistency between theoretical analysis and simulations.

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References

  • Antal, T., Scheuring, I., 2006. Fixation of strategies for an evolutionary game in finite populations. Bull. Math. Biol. 68, 1923–1944.

    Article  MathSciNet  Google Scholar 

  • Axelrod, R., 1984. The Evolution of Cooperation. Basic Books, New York.

    Google Scholar 

  • Brandt, H., Hauert, C., Sigmund, K., 2006. Punishing and abstaining for public goods. Proc. Natl. Acad. Sci. USA 103, 495–497.

    Article  Google Scholar 

  • Doebeli, M., Hauert, C., 2004. Models of cooperation based on the Prisoner’s Dilemma and the Snowdrift game. Ecol. Lett. 8, 748–766.

    Article  Google Scholar 

  • Gardiner, C.W., 1983. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, Berlin.

    MATH  Google Scholar 

  • Hauert, C., De Monte, S., Hofbauer, J., Sigmund, K., 2002. Volunteering as red queen mechanism for cooperation in public goods games. Science 296, 1129–1132.

    Article  Google Scholar 

  • Hofbauer, J., Sigmund, K., 1998. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Hofbauer, J., Schuster, P., Sigmund, K., 1979. A note on evolutionarily stable strategies and game dynamics. J. Theor. Biol. 81, 609–612.

    Article  MathSciNet  Google Scholar 

  • Imhof, L.A., Nowak, M.A., 2006. Evolutionary game dynamics in a Wright–Fisher process. J. Math. Biol. 52, 667–681.

    Article  MATH  MathSciNet  Google Scholar 

  • Imhof, L. A, Fudenberg, D., Nowak, M.A., 2005. Evolutionary cycles of cooperation and defection. Proc. Natl. Acad. Sci. USA 102, 10797–10800.

    Article  Google Scholar 

  • Lieberman, E., Hauert, C., Nowak, M.A., 2005. Evolutionary dynamics on graphs. Nature 433, 312–316.

    Article  Google Scholar 

  • Maynard Smith, J., 1982. Evolution and the Theory of Games. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Nowak, M.A., 2006. Five rules for the evolution of cooperation. Science 314, 1560–1563.

    Article  Google Scholar 

  • Nowak, M.A., May, R.M., 1992. Evolutionary games and spatial chaos. Nature 359, 826–829.

    Article  Google Scholar 

  • Nowak, M.A., Sigmund, K., 2004. Evolutionary dynamics of biological games. Science 303, 793–799.

    Article  Google Scholar 

  • Nowak, M.A., Sasaki, A., Taylor, C., Fudenberg, D., 2004. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 646–650.

    Article  Google Scholar 

  • Taylor, P.D., Jonker, L.B., 1978. Evolutionarily stable strategies and game dynamics. Math. Biosci. 40, 145–156.

    Article  MATH  MathSciNet  Google Scholar 

  • Taylor, C., Fudenberg, D., Sasaki, A., Nowak, M.A., 2004. Evolutionary game dynamics in finite populations. Bull. Math. Biol. 66, 1621–1644.

    Article  MathSciNet  Google Scholar 

  • Traulsen, A., Claussen, J.C., Hauert, C., 2005a. Coevolutionary dynamics: from finite to infinite populations. Phys. Rev. Lett. 95, 238701.

    Article  Google Scholar 

  • Traulsen, A., Sengupta, A.M., Nowak, M.A., 2005b. Stochastic evolutionary dynamics on two levels. J. Theor. Biol. 235, 393–401.

    Article  MathSciNet  Google Scholar 

  • Traulsen, A., Pacheco, J.M., Imhof, L.A., 2006a. Stochasticity and evolutionary stability. Phys. Rev. E 74, 021905.

    Article  MathSciNet  Google Scholar 

  • Traulsen, A., Nowak, M.A., Pacheco, J.M., 2006b. Stochastic dynamics of invasion and fixation. Phys. Rev. E 74, 011909.

    Article  Google Scholar 

  • Traulsen, A., Nowak, M.A., Pacheco, J.M., 2007a. Stochastic payoff evaluation increases the temperature of selection. J. Theor. Biol. 244, 349–356.

    Article  MathSciNet  Google Scholar 

  • Traulsen, A., Pacheco, J.M., Nowak, M.A., 2007b. Pairwise comparison and selection temperature in evolutionary game dynamics. J. Theor. Biol. 246, 522–529.

    Article  MathSciNet  Google Scholar 

  • Weibull, J.W., 1995. Evolutionary Game Theory. MIT Press, Cambridge.

    MATH  Google Scholar 

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Correspondence to Meng Gao.

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Gao, M. Continuous Probabilistic Analysis to Evolutionary Game Dynamics in Finite Populations. Bull. Math. Biol. 71, 1148–1159 (2009). https://doi.org/10.1007/s11538-009-9397-7

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  • DOI: https://doi.org/10.1007/s11538-009-9397-7

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