Journal of Mathematical Biology

, Volume 63, Issue 2, pp 263–281 | Cite as

Consequences of fluctuating group size for the evolution of cooperation

  • Åke Brännström
  • Thilo Gross
  • Bernd Blasius
  • Ulf Dieckmann


Studies of cooperation have traditionally focused on discrete games such as the well-known prisoner’s dilemma, in which players choose between two pure strategies: cooperation and defection. Increasingly, however, cooperation is being studied in continuous games that feature a continuum of strategies determining the level of cooperative investment. For the continuous snowdrift game, it has been shown that a gradually evolving monomorphic population may undergo evolutionary branching, resulting in the emergence of a defector strategy that coexists with a cooperator strategy. This phenomenon has been dubbed the ‘tragedy of the commune’. Here we study the effects of fluctuating group size on the tragedy of the commune and derive analytical conditions for evolutionary branching. Our results show that the effects of fluctuating group size on evolutionary dynamics critically depend on the structure of payoff functions. For games with additively separable benefits and costs, fluctuations in group size make evolutionary branching less likely, and sufficiently large fluctuations in group size can always turn an evolutionary branching point into a locally evolutionarily stable strategy. For games with multiplicatively separable benefits and costs, fluctuations in group size can either prevent or induce the tragedy of the commune. For games with general interactions between benefits and costs, we derive a general classification scheme based on second derivatives of the payoff function, to elucidate when fluctuations in group size help or hinder cooperation.

Mathematics Subject Classification (2000)

00A69 92B05 92D15 


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Åke Brännström
    • 1
    • 4
  • Thilo Gross
    • 2
  • Bernd Blasius
    • 3
  • Ulf Dieckmann
    • 4
  1. 1.Department of Mathematics and Mathematical StatisticsUmeå UniversityUmeåSweden
  2. 2.Max-Planck-Institute for Physics of Complex SystemsDresdenGermany
  3. 3.Institute for Chemistry and Biology of Marine EnvironmentOldenburg UniversityOldenburgGermany
  4. 4.Evolution and Ecology ProgramInternational Institute for Applied Systems Analysis (IIASA)LaxenburgAustria

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