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Evolutionary Dynamics

Conference paper
Part of the NATO Science for Peace and Security Series B: Physics and Biophysics book series (NAPSB)

Abstract

Evolutionary dynamics in finite populations reflects a balance between Darwinian selection and random drift. For a long time population structures were assumed to leave this balance unaffected provided that the mutants and residents have fixed fitness values. This result indeed holds for a certain (large) class of population structures or graphs. However, other structures can tilt the balance to the extend that either selection is eliminated and drift rules or drift is eliminated and only selection matters.

In nature, however, fitness is generally affected by interactions with other members of the population. This is of particular importance for the evolution of cooperation. The essence of this evolutionary conundrum is captured by social dilemmas: cooperators provide a benefit to the group at some cost to themselves, whereas defectors attempt to exploit the group by reaping the benefits without bearing the costs of cooperation. The most prominent game theoretical models to study this problem are the prisoner’s dilemma and the snowdrift game. In the prisoner’s dilemma, cooperators are doomed if interactions occur randomly. In structured populations, individuals interact only with their neighbors and cooperators may thrive by aggregating in clusters and thereby reducing exploitation by defectors. In finite populations, a surprisingly simple rule determines whether evolution favors cooperation: b > c k that is, if the benefits b exceed k-times the costs c of cooperation, where k is the (average) number of neighbors. The spatial prisoner’s dilemma has lead to the general belief that spatial structure is beneficial for cooperation. Interestingly, however, this no longer holds when relaxing the social dilemma and considering the snowdrift game. Due to the less stringent conditions, cooperators persist in populations with random interactions but spatial structure tends to be deleterious and may even eliminate cooperation altogether.

In many biological situations it seems more appropriate to assume a continuous range of cooperative investment levels instead of restricting the analysis to two a priori fixed strategic types. In the continuous prisoner’s dilemma cooperative investments gradually decrease and defection dominates just as in the traditional prisoner’s dilemma. In contrast, the continuous snowdrift game exhibits rich dynamics but most importantly provides an intriguing natural explanation for phenotypic diversification and the evolutionary origin of cooperators and defectors. Thus, selection may not always favor equal contributions but rather promote states where two distinct types co-exist — those that fully cooperate and those that exploit. In the context of human societies and cultural evolution this could be termed the Tragedy of the Commune because differences in contributions to a communal enterprise have significant potential for escalating conflicts on the basis of accepted notions of fairness.

Keywords

Evolutionary game theory evolutionary graph theory social dilemmas prisoner’s dilemma snowdrift game structured populations continuous games evolutionary branching 

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References

  1. 1..
    Albert, R. and Barabási, A.-L. (2002) Statistical mechanics of complex networks, Rev. Mod. Phys. 74, 47–97.CrossRefADSGoogle Scholar
  2. 2.
    Barabasí, A.-L. and Albert, R. (1999) Emergence of scaling in random networks, Science 286, 509–512.CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bollobás, B. (1995) Random Graphs, New York, Academic.Google Scholar
  4. 4.
    Clutton-Brock, T. H., M. J. O’Riain, P. N. M. B., Gaynor, D., Kansky, R., Griffin, A. S., and Manser, M. (1999) Selfish sentinels in cooperative mammals, Science 284, 1640–1644.CrossRefADSGoogle Scholar
  5. 5.
    Clutton-Brock, T. H. and Parker, G. A. (1995) Punishment in animal societies, Nature 373, 209–216.CrossRefADSGoogle Scholar
  6. 6.
    Connor, R. C. (1996) Partner preferences in by-product mutualisms and the case of predator inspection in fish, Anim. Behav. 51, 451–454.CrossRefGoogle Scholar
  7. 7.
    Dawes, R. M. (1980) Social dilemmas, Ann. Rev. Psychol. 31, 169–193.CrossRefGoogle Scholar
  8. 8.
    Dieckmann, U. and Law, R. (1996) The dynamical theory of coevolution: a derivation from stochastic ecological processes, J. Math. Biol. 34, 579–612.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Doebeli, M. and Hauert, C. (2005) Models of cooperation based on the Prisoner’s Dilemma and the Snowdrift game, Ecol. Lett. 8, 748–766.CrossRefGoogle Scholar
  10. 10.
    Doebeli, M., Hauert, C., and Killingback, T. (2004) The evolutionary origin of cooperators and defectors, Science 306, 859–862.CrossRefADSGoogle Scholar
  11. 11.
    Dugatkin, L. A. (1996) Tit for tat, by-product mutualism and predator inspection: a reply to Connor, Anim. Behav. 51, 455–457.CrossRefGoogle Scholar
  12. 12.
    Flood, M. (1958) Some experimental games, Management Science 5, 5–26.zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Geritz, S. A. H., Kisdi, E., Meszéna, G., and Metz, J. A. J. (1998) Evolutionary singular strategies and the adaptive growth and branching of the evolutionary tree, Evol. Ecol. 12, 35–57.CrossRefGoogle Scholar
  14. 14.
    Greig, D. and Travisano, M. (2004) The Prisoner’s Dilemma and polymorphism in yeast SUC genes, Biol. Lett. 271, S25–S26.Google Scholar
  15. 15.
    Hamilton, W. D. (1964) The genetical evolution of social behaviour I, J. Theor. Biol. 7, 1–16.CrossRefGoogle Scholar
  16. 16.
    Hamilton, W. D. (1971) The geometry of the selfish herd, J. Theor. Biol. 31, 295–311.CrossRefGoogle Scholar
  17. 17.
    Hardin, G. (1968) The tragedy of the commons, Science 162, 1243–1248.CrossRefADSGoogle Scholar
  18. 18.
    Hauert, C. (2006) Spatial effects in social dilemmas, J. Theor. Biol. 240, 627–636.CrossRefMathSciNetGoogle Scholar
  19. 19.
    Hauert, C. (2007) VirtualLabs: Interactive Tutorials on Evolutionary Game Theory, http://www.univie.ac.at/virtuallabs.
  20. 20.
    Hauert, C., De Monte, S., Hofbauer, J., and Sigmund, K. (2002) Volunteering as red queen mechanism for cooperation in public goods games, Science 296, 1129–1132.CrossRefADSGoogle Scholar
  21. 21.
    Hauert, C. and Doebeli, M. (2004) Spatial structure often inhibits the evolution of cooperation in the Snowdrift game, Nature 428, 643–646.CrossRefADSGoogle Scholar
  22. 22.
    Hauert, C., Holmes, M., and Doebeli, M. (2006a) Evolutionary games and population dynamics: maintenance of cooperation in public goods games, Proc. R. Soc. Lond. B 273, 2565–2570.CrossRefGoogle Scholar
  23. 23.
    Hauert, C., Holmes, M., and Doebeli, M. (2006b) Evolutionary games and population dynamics: maintenance of cooperation in public goods games, Proc. R. Soc. Lond. B 273, 3131–3132.CrossRefGoogle Scholar
  24. 24.
    Hauert, C., Michor, F., Nowak, M., and Doebeli, M. (2006c) Synergy and discounting of cooperation in social dilemmas, J. Theor. Biol. 239, 195–202.CrossRefMathSciNetGoogle Scholar
  25. 25.
    Hauert, C. and Szabó, G. (2005) Game theory and physics, Am. J. Phys. 73, 405–414.CrossRefADSGoogle Scholar
  26. 26.
    Hauert, C., Traulsen, A., Brandt, H., Nowak, M. A., and Sigmund, K. (2007) Via freedom to coercion: the emergence of costly punishment, Science 316, 1905–1907.CrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Hofbauer, J. and Sigmund, K. (1998) Evolutionary Games and Population Dynamics, Cambridge, Cambridge University Press.zbMATHGoogle Scholar
  28. 28.
    Huang, A. S. and Baltimore, D. (1977) Comprehensive Virology, Vol. 10, Chapt. Defective Interfering Animal Viruses, pp. 73–116, New York, Plenum.Google Scholar
  29. 29.
    Karlin, S. and Taylor, H. M. (1975) First Course in Stochastic Processes, 2nd edition London, Academic.zbMATHGoogle Scholar
  30. 30.
    Killingback, T. and Doebeli, M. (2002) The continuous prisoner’s dilemma and the evolution of cooperation through reciprocal altruism with variable investment, Am. Nat. 160, 421–438.CrossRefGoogle Scholar
  31. 31.
    Koonin, E. V., Wolf, Y. I., and Karev, G. P. (2006) Power Laws, Scale-Free Networks and Genome Biology, Molecular Biology Intelligence Unit, Springer, New York, NY.Google Scholar
  32. 32.
    Lieberman, E., Hauert, C., and Nowak, M. A. (2005) Evolutionary dynamics on graphs, Nature 455, 312–316.CrossRefADSGoogle Scholar
  33. 33.
    Magurran, A. E. and Higham, A. (1988) Information transfer across fish shoals under predator threat, Ethology 78, 153–158.CrossRefGoogle Scholar
  34. 34.
    Maruyama, T. (1970) Effective number of alleles in a subdivided population, Theor. Pop. Biol. 1, 273–306.CrossRefMathSciNetGoogle Scholar
  35. 35.
    Matsuda, H., Ogita, N., Sasaki, A., and Sato, K. (1992) Statistical mechanics of populations, Prog. Theor. Phys. 88, 1035–1049.CrossRefADSGoogle Scholar
  36. 36.
    Maynard Smith, J. (1982) Evolution and the Theory of Games, Cambridge, Cambridge University Press.zbMATHGoogle Scholar
  37. 37.
    Maynard Smith, J. and Price, G. (1973) The logic of animal conflict, Nature 246, 15–18.CrossRefGoogle Scholar
  38. 38.
    Maynard Smith, J. and Szathmáry, E. (1995) The Major Transitions in Evolution, Oxford, W. H. Freeman.Google Scholar
  39. 39.
    Metz, J. A. J., Geritz, S. A. H., Meszena, G., Jacobs, F. J. A., and van Heerwaarden, J. S. (1996) Adaptive dynamics: a geometrical study of the consequences of nearly faithful replication, In S. J. van Strien and S. M. Verduyn Lunel (eds.), Stochastic and Spatial Structures of Dynamical Systems, Amsterdam, North Holland, pp. 183–231.Google Scholar
  40. 40.
    Milinski, M. (1987) Tit for tat in sticklebacks and the evolution of cooperation, Nature 325, 433–435.CrossRefADSGoogle Scholar
  41. 41.
    Milinski, M. (1996) By-product mutualism, tit-for-tat and cooperative predator inspection: a reply to Connor, Anim. Behav. 51, 458–461.CrossRefGoogle Scholar
  42. 42.
    Milinski, M., Semmann, D., Krambeck, H.-J., and Marotzke, M. (2006) Stabilizing the Earths climate is not a losing game: supporting evidence from public goods experiments, Proc. Natl. Acad. Sci. USA 103, 3994–3998.CrossRefADSGoogle Scholar
  43. 43.
    Moran, P. A. P. (1962) The Statistical Processes of Evolutionary Theory, Oxford, UK, Clarendon.zbMATHGoogle Scholar
  44. 44.
    Nash, J. (1951) Non-cooperative games, Annals of Mathematics 54, 286–299.CrossRefMathSciNetGoogle Scholar
  45. 45.
    Neu, H. C. (1992) The Crisis in Antibiotic Resistance, Science 257, 1064–1073.CrossRefADSGoogle Scholar
  46. 46.
    Nowak, M. A. (2006) Evolutionary Dynamics, Cambridge MA, Harvard University Press.zbMATHGoogle Scholar
  47. 47.
    Nowak, M. A. and May, R. M. (1992) Evolutionary Games and Spatial Chaos, Nature 359, 826–829.CrossRefADSGoogle Scholar
  48. 48.
    Nowak, M. A., Michor, F., and Iwasa, Y. (2003) The linear process of somatic evolution, Proc. Natl. Acad. Sci. USA 100, 14966–14969.CrossRefADSGoogle Scholar
  49. 49.
    Nowak, M. A., Sasaki, A., Taylor, C., and Fudenberg, D. (2004) Emergence of cooperation and evolutionary stability in finite populations, Nature 428, 646–650.CrossRefADSGoogle Scholar
  50. 50.
    Nowak, M. A. and Sigmund, K. (1998) Evolution of indirect reciprocity by image scoring, Nature 393, 573–577.CrossRefADSGoogle Scholar
  51. 51.
    Ohtsuki, H., Hauert, C., Lieberman, E., and Nowak, M. A. (2006) A simple rule for the evolution of cooperation on graphs and social networks, Nature 441, 502–505.CrossRefADSGoogle Scholar
  52. 52.
    Pitcher, T. (1992) Who dares, wins: the function and evolution of predator inspection behavior in shoaling fish, Neth. J. Zool. 42, 371–391.CrossRefGoogle Scholar
  53. 53.
    Poundstone, W. (1992) Prisoner’s Dilemma, New York, Doubleday.Google Scholar
  54. 54.
    Rainey, P. B. and Rainey, K. (2003) Evolution of cooperation and conflict in experimental bacterial populations, Nature 425, 72–74.CrossRefADSGoogle Scholar
  55. 55.
    Saunders, C. D. and Hausfater, G. (1988) The functional significance of baboon grooming behavior, Annals N. Y. Acad. Sci. 525, 430–432.CrossRefADSGoogle Scholar
  56. 56.
    Slatkin, M. (1981) Fixation probabilities and fixation times in a subdivided population, Evolution 35, 477–488.CrossRefGoogle Scholar
  57. 57.
    Stammbach, E. and Kummer, H. (1982) Individual contributions to a dyadic interaction: an analysis of baboon grooming, Anim. Behav. 30, 964–971.CrossRefGoogle Scholar
  58. 58.
    Sugden, R. (1986) The Economics of Rights, Co-operation and Welfare, Oxford and New York, Blackwell.Google Scholar
  59. 59.
    Szabó, G. and Hauert, C. (2002a) Evolutionary prisoner’s dilemma with optional participation, Phys. Rev. E 66, 062903.CrossRefADSMathSciNetGoogle Scholar
  60. 60.
    Szabó, G. and Hauert, C. (2002b) Phase transitions and volunteering in spatial public goods games, Phys. Rev. Let. 89, 118101.CrossRefADSGoogle Scholar
  61. 61.
    Taylor, C., Iwasa, Y., and Nowak, M. A. (2006) A symmetry of fixation times in evolutionary dynamics, J. Theor. Biol. 243, 245–251.CrossRefMathSciNetGoogle Scholar
  62. 62.
    Taylor, P. D., Day, T., and Wild, G. (2007) Evolution of cooperation in a finite homogeneous graph, Nature. 447, 469–472.CrossRefADSGoogle Scholar
  63. 63.
    Traulsen, A., Claussen, J. C., and Hauert, C. (2005) Coevolutionary dynamics: from finite to infinite populations, Phys. Rev. Lett. 95, 238701.CrossRefADSGoogle Scholar
  64. 64.
    Trivers, R. L. (1971) The evolution of reciprocal altruism, Q. Rev. Biol. 46, 35–57.CrossRefGoogle Scholar
  65. 65.
    Turner, P. E. and Chao, L. (1999) Prisoner’s dilemma in an RNA virus, Nature 398, 441–443.CrossRefADSGoogle Scholar
  66. 66.
    Turner, P. E. and Chao, L. (2003) Escape from prisoner’s dilemma in RNA phage Φ6, Am. Nat. 161, 497–505.CrossRefGoogle Scholar
  67. 67.
    van Baalen, M. and Rand, D. A. (1998) The unit of selection in viscous populations and the evolution of altruism, J. Theor. Biol. 193, 631–648.CrossRefGoogle Scholar
  68. 68.
    Velicer, G. J., Kroos, L., and Lenski, R. E. (2000) Developmental cheating in the social bacterium Myxococcus xanthus, Nature 404, 598–601.CrossRefADSGoogle Scholar
  69. 69.
    Vogelstein, B. and Kinzler, K. W. (1998) The Genetic Basis of Human Cancer, Toronto, McGraw-Hill.Google Scholar
  70. 70.
    Watts, D. J. and Strogatz, S. H. (1998) Collective dynamics of ‘small world’ networks, Nature 393, 440–442.CrossRefADSGoogle Scholar

Copyright information

© Springer Science + Business Media B.V 2008

Authors and Affiliations

  1. 1.Program for Evolutionary DynamicsHarvard UniversityCambridgeUSA

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