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
Article

Abstract

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Axelrod R (1984) The Evolution of Cooperation. Basic Books, New York, USAGoogle Scholar
  2. Axelrod R, Hamilton WD (1981) The evolution of cooperation. Science 211: 1390–1396MathSciNetCrossRefGoogle Scholar
  3. Brännström Å, Dieckmann U (2005) Evolutionary dynamics of altruism and cheating among social amoebas. Proc R Soc Lond Ser B 272: 1609–1616CrossRefGoogle Scholar
  4. Brown SP (1999) Cooperation and conflict in host-manipulating parasites. Proc R Soc Lond Ser B 266(1431): 1899–1904CrossRefGoogle Scholar
  5. Brown SP, Johnstone RA (2001) Cooperation in the dark: signalling and collective action in quorum-sensing bacteria. Proc R Soc Lond Ser B 268: 961–965CrossRefGoogle Scholar
  6. Buss LW (1982) Somatic cell parasitism and the evolution of somatic tissue compatibility. Proc R Soc Lond Ser B 79: 5337–5341Google Scholar
  7. Dao DN, Kessin RH, Ennis HL (2000) Developmental cheating and the evolutionary biology of Dictyostelium and Myxococcus. Microbiology 146: 1505–1512Google Scholar
  8. Dieckmann U, Law R (1996) The dynamical theory of coevolution: a derivation from stochastic ecological processes. J Math Biol 34: 579–612MathSciNetMATHCrossRefGoogle Scholar
  9. Doebeli M, Hauert C (2005) Models of cooperation based on the Prisoner’s Dilemma and the Snowdrift game. Ecol Lett 8: 748–766CrossRefGoogle Scholar
  10. Doebeli M, Hauert C, Killingback T (2004) The evolutionary origin of cooperators and defectors. Science 306: 859–863CrossRefGoogle Scholar
  11. Fortunato A, Queller DC, Strassman JE (2003) A linear dominance hierarchy among clones in chimeras of the social amoeba Dictyostelium discoideum. J Evol Biol 16: 438–445CrossRefGoogle Scholar
  12. Foster KR (2004) Diminishing returns in social evolution: the not-so-tragic commons. J Evol Biol 17(5): 1058–1072MathSciNetCrossRefGoogle Scholar
  13. Geritz SAH (2005) Resident-invader dynamics and the coexistence of similar strategies. J Math Biol 50: 67–82MathSciNetMATHCrossRefGoogle Scholar
  14. Geritz SAH, Kisdi E, Meszéna G, Metz JAJ (1998) Evolutionary singular strategies and the adaptive growth and branching of the evolutionary tree. Evol Ecol 12: 35–57CrossRefGoogle Scholar
  15. Geritz SAH, Gyllenberg M, Jacobs FJA, Parvinen K (2002) Invasion dynamics and attractor inheritance. J Math Biol 44: 548–560MathSciNetMATHCrossRefGoogle Scholar
  16. Gore J, Youk H, van Oudenaarden A (2009) Snowdrift game dynamics and facultative cheating in yeast. Nature 459: 253–256CrossRefGoogle Scholar
  17. Greig D, Travisano M (2004) The Prisoner’s Dilemma and polymorphism in yeast SUC genes. Proc R Soc Lond Ser B 271: S25–S26CrossRefGoogle Scholar
  18. Hamilton WD (1963) The evolution of altruistic behavior. Am Nat 97: 354–356CrossRefGoogle Scholar
  19. Hamilton WD (1964) The genetical theory of social behaviour I, II. J Theor Biol 7: 1–52CrossRefGoogle Scholar
  20. Hamilton WD (1972) Altruism and related phenomena, mainly in social insects. Annu Rev Ecol Syst 3: 193–232CrossRefGoogle Scholar
  21. Hardin G (1968) The tragedy of the commons. Science 162: 1243–1248CrossRefGoogle Scholar
  22. Hauert C, Holmes M, Doebeli M (2002) Volunteering as Red Queen mechanism for cooperation in public goods games. Science 296: 1129–1132CrossRefGoogle Scholar
  23. Hauert C, Holmes M, Doebeli M (2006) Evolutionary games and population dynamics: maintenance of cooperation in public goods games. Proc R Soc Londo Ser B 273: 2565–2570CrossRefGoogle Scholar
  24. Hauert C, Wakano JY, Doebeli M (2008) Ecological public goods games: cooperation and bifurcation. Theor Popul Biol 73: 257–263MATHCrossRefGoogle Scholar
  25. Hofbauer J, Sigmund K (1998) Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge, UKMATHGoogle Scholar
  26. Hutchinson GE (1961) The paradox of the plankton. Am Nat 95: 137–145CrossRefGoogle Scholar
  27. Kagel JH, Roth AE (1995) The Handbook of Experimental Economics. Princeton University Press, Princeton, NJ, USAGoogle Scholar
  28. Killingback T, Doebeli M, Knowlton N (1999) Variable investment, the Continuous Prisoner’s Dilemma, and the origin of cooperation. Proc R Soc Lond Ser B 266: 1723–1728CrossRefGoogle Scholar
  29. Kun A, Boza G, Scheuring I (2006) Asynchronous snowdrift game with synergistic effect as a model of cooperation. Behav Ecol 17: 633–641CrossRefGoogle Scholar
  30. Mar G, Denis PS (1994) Chaos in cooperation – continuous-valued Prisoner’s Dilemmas in infinite-valued logic. Int J Bifurcat Chaos 4: 943–958MATHCrossRefGoogle Scholar
  31. Maynard Smith J (1982) Evolution and the Theory of Games. Cambridge University Press, Cambridge, UKMATHGoogle Scholar
  32. Maynard Smith J, Szathmáry E (1995) The Major Transitions in Evolution. W. H. Freeman & Co., Oxford, UKGoogle Scholar
  33. Metz JAJ, Nisbet RM, Geritz SAH (1992) How should we define “fitness” for general ecological scenarios?. Trends Ecol Evol 7: 198–202CrossRefGoogle Scholar
  34. Metz JAJ, Geritz SAH, Meszéna G, Jacobs FJA, van Heerwaarden JS (1996) Adaptive dynamics: a geometrical study of the consequences of nearly faithful reproduction. In: Strien SJ, Lunel SMV (eds) Stochastic and spatial structures of dynamical systems. North Holland, Amsterdam, pp 183–231Google Scholar
  35. Nowak MA (2006) Five rules for the evolution of cooperation. Science 314: 1560–1563CrossRefGoogle Scholar
  36. Parvinen K (2010) Adaptive dynamics of cooperation may prevent the coexistence of defectors and cooperators and even cause extinction. Proc R Soc Lond Ser B (in press)Google Scholar
  37. Poulin R, Vickery WL (1995) Cleaning symbiosis as an evolutionary game: to cheat or not to cheat?. J Theor Biol 175: 63–70CrossRefGoogle Scholar
  38. Rainey PB, Rainey K (2003) Evolution of cooperation and conflict in experimental bacterial populations. Nature 425: 72–74CrossRefGoogle Scholar
  39. Raper KB (1984) The Dictyostelids. Princeton University Press, Princeton, NJ, USAGoogle Scholar
  40. Rapoport A (1966) The game of chicken. Am Behav Sci 10: 10–28CrossRefGoogle Scholar
  41. Ross-Gillespie A, Gardner A, Buckling A, West SA, Griffin AS (2009) Density dependence and cooperation: theory and a test with bacteria. Evolution 63: 2315–2325CrossRefGoogle Scholar
  42. Strassmann JE, Zhu Y, Queller DC (2000) Altruism and social cheating in the social amoeba Dictyostelium discoideum. Nature 408: 965–967CrossRefGoogle Scholar
  43. Sugden SR (1986) The Economics of Rights, Cooperation and Welfare. Blackwell Publishing, Oxford, UKGoogle Scholar
  44. Sumpter DJT, Brännström Å (2008) Synergy in social communication. In: Hughes D (eds) Social communication. Oxford University Press, Oxford, pp 191–209CrossRefGoogle Scholar
  45. Trivers RL (1971) The evolution of reciprocal altruism. Q Rev Biol 46: 35–57CrossRefGoogle Scholar
  46. Turner PE, Chao L (2003) Escape from Prisoner’s Dilemma in RNA phage phi 6. Am Nat 161: 497–505CrossRefGoogle Scholar
  47. Wilson DS (1980) The Natural Selection of Populations and Communities. Benjamin-Cummings, Menlo Park, CA, USAGoogle Scholar
  48. Wilson DS, Dugatkin LA (1997) Group selection and assortative interactions. Am Nat 149: 336–351CrossRefGoogle Scholar

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

Personalised recommendations