Journal of Statistical Physics

, Volume 151, Issue 3–4, pp 637–653 | Cite as

Global Migration Can Lead to Stronger Spatial Selection than Local Migration

Article

Abstract

The outcome of evolutionary processes depends on population structure. It is well known that mobility plays an important role in affecting evolutionary dynamics in group structured populations. But it is largely unknown whether global or local migration leads to stronger spatial selection and would therefore favor to a larger extent the evolution of cooperation. To address this issue, we quantify the impacts of these two migration patterns on the evolutionary competition of two strategies in a finite island model. Global migration means that individuals can migrate from any one island to any other island. Local migration means that individuals can only migrate between islands that are nearest neighbors; we study a simple geometry where islands are arranged on a one-dimensional, regular cycle. We derive general results for weak selection and large population size. Our key parameters are: the number of islands, the migration rate and the mutation rate. Surprisingly, our comparative analysis reveals that global migration can lead to stronger spatial selection than local migration for a wide range of parameter conditions. Our work provides useful insights into understanding how different mobility patterns affect evolutionary processes.

Keywords

Evolutionary dynamics Population structure Mathematical biology Evolutionary game theory 

Notes

Acknowledgements

We are grateful for support from the National Science Foundation/National Institute of Health joint program in mathematical biology (NIH grant no. R01GM078986), the Bill and Melinda Gates Foundation (Grand Challenges grant 37874), the National Institute on Aging (P01-AG031093) and the John Templeton Foundation.

References

  1. 1.
    Lebowitz, J.L., Penrose, O.: Cluster and percolation inequalities for lattice systems with interactions. J. Stat. Phys. 16, 321–337 (1977) MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    De Masi, A., Ferrari, P.A., Lebowitz, J.L.: Reaction-diffusion equations for interacting particle systems. J. Stat. Phys. 44, 589–644 (1986) ADSMATHCrossRefGoogle Scholar
  3. 3.
    Maynard-Smith, J., Szathmáry, E.: The Major Transitions in Evolution. Oxford University Press, New York (1998) Google Scholar
  4. 4.
    Nowak, M.A.: Evolutionary Dynamics. Harvard University Press, Cambridge (2006) MATHGoogle Scholar
  5. 5.
    Nowak, M.A.: Five rules for the evolution of cooperation. Science 314, 1560–1563 (2006) ADSCrossRefGoogle Scholar
  6. 6.
    Doebeli, M., Hauert, C.: Models of cooperation based on the Prisoner’s Dilemma and the Snowdrift game. Ecol. Lett. 8, 748–766 (2005) CrossRefGoogle Scholar
  7. 7.
    Nowak, M.A.: Evolving cooperation. J. Theor. Biol. 299, 1–18 (2012) CrossRefGoogle Scholar
  8. 8.
    Castellano, C., Fortunato, S., Loreto, V.: Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591–646 (2009) ADSCrossRefGoogle Scholar
  9. 9.
    Axelrod, R., Hamilton, W.D.: The evolution of cooperation. Science 211, 1390–1396 (1981) MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Szabó, G., Töke, C.: Evolutionary prisoner’s dilemma game on a square lattice. Phys. Rev. E 58, 69 (1998) ADSCrossRefGoogle Scholar
  11. 11.
    Abramson, G., Kuperman, M.: Social games in a social network. Phys. Rev. E 63, 030901 (2001) ADSCrossRefGoogle Scholar
  12. 12.
    Szabó, G., Hauert, C.: Phase transitions and volunteering in spatial public goods games. Phys. Rev. Lett. 89, 118101 (2002) ADSCrossRefGoogle Scholar
  13. 13.
    Traulsen, A., Röhl, T., Schuster, H.G.: Stochastic gain in population dynamics. Phys. Rev. Lett. 93, 028701 (2004) ADSCrossRefGoogle Scholar
  14. 14.
    Santos, F.C., Pacheco, J.M.: Scale-free networks provide a unifying framework for the emergence of cooperation. Phys. Rev. Lett. 95, 098104 (2005) ADSCrossRefGoogle Scholar
  15. 15.
    Hauert, C., Szabó, G.: Game theory and physics. Am. J. Phys. 73, 405–414 (2005) ADSMATHCrossRefGoogle Scholar
  16. 16.
    Gomez-Gardenes, J., Campillo, M., Floria, L.M., Moreno, Y.: Dynamical organization of cooperation in complex topologies. Phys. Rev. Lett. 98, 108103 (2007) ADSCrossRefGoogle Scholar
  17. 17.
    Ohtsuki, H., Nowak, M.A., Pacheco, J.M.: Breaking the symmetry between interaction and replacement in evolutionary dynamics on graphs. Phys. Rev. Lett. 98, 108106 (2007) ADSCrossRefGoogle Scholar
  18. 18.
    Roca, C.P., Cuesta, J.A., Sánchez, A.: Evolutionary game theory: temporal and spatial effects beyond replicator dynamics. Phys. Life Rev. 6, 208–249 (2009) ADSCrossRefGoogle Scholar
  19. 19.
    Galla, T.: Intrinsic noise in game dynamical learning. Phys. Rev. Lett. 103, 198702 (2009) ADSCrossRefGoogle Scholar
  20. 20.
    Arenas, A., Camacho, J., Cuesta, J., Requejo, R.: The joker effect: cooperation driven by destructive agents. J. Theor. Biol. 279, 113–119 (2011) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Black, A.J., Traulsen, A., Galla, T.: Mixing times in evolutionary game dynamics. Phys. Rev. Lett. 109, 028101 (2012) ADSCrossRefGoogle Scholar
  22. 22.
    Szabó, G., Fáth, G.: Evolutionary games on graphs. Phys. Rep. 446, 97–216 (2007) MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    Maynard-Smith, J., Price, G.R.: The logic of animal conflict. Nature 246, 15–18 (1973) CrossRefGoogle Scholar
  24. 24.
    Maynard-Smith, J.: Evolution and the Theory of Games. Cambridge University Press, Cambridge, UK (1982) CrossRefGoogle Scholar
  25. 25.
    Weibull, J.W.: Evolutionary Game Theory. MIT Press, Cambridge (1995) MATHGoogle Scholar
  26. 26.
    Hofbauer, J., Sigmund, K.: Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge, UK (1998) MATHCrossRefGoogle Scholar
  27. 27.
    Frank, S.A.: Foundations of Social Evolution. Princeton University Press, Princeton (1998) Google Scholar
  28. 28.
    Cressman, R.: Evolutionary Dynamics and Extensive Form Games. MIT Press, Cambridge (2003) MATHGoogle Scholar
  29. 29.
    Skyrms, B.: The Stag Hunt and the Evolution of Social Structure. Cambridge University Press, Cambridge, UK (2003) CrossRefGoogle Scholar
  30. 30.
    Kimura, M.: Evolutionary rate at the molecular level. Nature 217, 624–626 (1968) ADSCrossRefGoogle Scholar
  31. 31.
    Kimura, M.: The Neutral Theory of Molecular Evolution. Cambridge University Press, Cambridge, UK (1983) CrossRefGoogle Scholar
  32. 32.
    Lieberman, E., Hauert, C., Nowak, M.A.: Evolutionary dynamics on graphs. Nature 433, 312–316 (2005) ADSCrossRefGoogle Scholar
  33. 33.
    Antal, T., Redner, S., Sood, V.: Evolutionary dynamics on degree-heterogeneous graphs. Phys. Rev. Lett. 96, 188014 (2006) ADSCrossRefGoogle Scholar
  34. 34.
    Nowak, M.A., May, R.M.: Super infection and the evolution of parasite virulence. Proc. - Royal Soc., Biol. Sci. 255, 81–89 (1994) CrossRefGoogle Scholar
  35. 35.
    Pfeiffer, T., Schuster, S., Bonhoeffer, S.: Cooperation and competition in the evolution of ATP-producing pathways. Science 292, 504–507 (2001) ADSCrossRefGoogle Scholar
  36. 36.
    Turner, P.E., Chao, L.: Prisoner’s dilemma in an RNA virus. Nature 398, 441–443 (1999) ADSCrossRefGoogle Scholar
  37. 37.
    Taylor, P.D., Jonker, L.B.: Evolutionary stable strategies and game dynamics. Math. Biosci. 40, 145–156 (1978) MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Nowak, M.A., Sasaki, A., Taylor, C., Fudenberg, D.: Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 646–650 (2004) ADSCrossRefGoogle Scholar
  39. 39.
    Traulsen, A., Claussen, J.C., Hauert, C.: Coevolutionary dynamics: from finite to infinite populations. Phys. Rev. Lett. 95, 238701 (2005) ADSCrossRefGoogle Scholar
  40. 40.
    Fudenberg, D., Nowak, M.A., Taylor, C., Imhof, L.A.: Evolutionary game dynamics in finite populations with strong selection and weak mutation. Theor. Popul. Biol. 70, 352–363 (2006) MATHCrossRefGoogle Scholar
  41. 41.
    Altrock, P.M., Traulsen, A.: Deterministic evolutionary game dynamics in finite populations. Phys. Rev. E 80, 011909 (2009) MathSciNetADSCrossRefGoogle Scholar
  42. 42.
    Taylor, C., Fudenberg, D., Sasaki, A., Nowak, M.A.: Evolutionary game dynamics in finite populations. Bull. Math. Biol. 66, 1621–1644 (2004) MathSciNetCrossRefGoogle Scholar
  43. 43.
    Imhof, L.A., Nowak, M.A.: Evolutionary game dynamics in a Wright-Fisher process. J. Math. Biol. 52, 667–681 (2006) MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Traulsen, A., Pacheco, J.M., Nowak, M.A.: Pairwise comparison and selection temperature in evolutionary game dynamics. J. Theor. Biol. 246, 522–529 (2007) MathSciNetCrossRefGoogle Scholar
  45. 45.
    Nowak, M.A., May, R.M.: Evolutionary games and spatial chaos. Nature 359, 826–829 (1992) ADSCrossRefGoogle Scholar
  46. 46.
    Nakamaru, M., Matsuda, H., Iwasa, Y.: The evolution of cooperation in a lattice structured population. J. Theor. Biol. 184, 65–81 (1997) CrossRefGoogle Scholar
  47. 47.
    van Baalen, M., Rand, D.A.: The unit of selection in viscous populations and the evolution of altruism. J. Theor. Biol. 193, 631–648 (1998) CrossRefGoogle Scholar
  48. 48.
    Hauert, C., Doebeli, M.: Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428, 643–646 (2004) ADSCrossRefGoogle Scholar
  49. 49.
    Ohtsuki, H., Hauert, C., Lieberman, E., Nowak, M.A.: A simple rule for the evolution of cooperation on graphs and social networks. Nature 441, 502–505 (2006) ADSCrossRefGoogle Scholar
  50. 50.
    Ohtsuki, H., Nowak, M.A.: Evolutionary games on cycles. Proc. R. Soc. B 273, 2249–2256 (2006) CrossRefGoogle Scholar
  51. 51.
    Grafen, A.: An inclusive fitness analysis of altruism on a cyclical network. J. Evol. Biol. 20, 2278–2283 (2007) CrossRefGoogle Scholar
  52. 52.
    Taylor, P.D., Day, T., Wild, G.: Evolution of cooperation in a finite homogeneous graph. Nature 447, 469–472 (2007) ADSCrossRefGoogle Scholar
  53. 53.
    Antal, T., Ohtsuki, H., Wakeley, J., Taylor, P.D., Nowak, M.A.: Evolution of cooperation by phenotypic similarity. Proc. Natl. Acad. Sci. USA 106, 8597–8600 (2009) ADSCrossRefGoogle Scholar
  54. 54.
    Tarnita, C.E., Antal, T., Ohtsuki, H., Nowak, M.A.: Evolutionary dynamics in set structured populations. Proc. Natl. Acad. Sci. USA 106, 8601–8604 (2009) ADSCrossRefGoogle Scholar
  55. 55.
    Nathanson, C.G., Tarnita, C.E., Nowak, M.A.: Calculating evolutionary dynamics in structured populations. PLoS Comput. Biol. 5, e1000615 (2009) MathSciNetCrossRefGoogle Scholar
  56. 56.
    Tarnita, C.E., Ohtsuki, H., Antal, T., Fu, F., Nowak, M.A.: Strategy selection in structured populations. J. Theor. Biol. 259, 570–581 (2009) MathSciNetCrossRefGoogle Scholar
  57. 57.
    Fu, F., Nowak, M.A., Hauert, C.: Invasion and expansion of cooperators in lattice populations: prisoner’s dilemma vs. Snowdrift games. J. Theor. Biol. 266, 358–366 (2010) MathSciNetCrossRefGoogle Scholar
  58. 58.
    Tarnita, C.E., Wage, N., Nowak, M.A.: Multiple strategies in structured populations. Proc. Natl. Acad. Sci. USA 108, 2334–2337 (2011) ADSCrossRefGoogle Scholar
  59. 59.
    Allen, B., Nowak, M.A.: Evolutionary shift dynamics on a cycle. J. Theor. Biol. 311, 28–39 (2012) MathSciNetCrossRefGoogle Scholar
  60. 60.
    Allen, B., Traulsen, A., Tarnita, C.E., Nowak, M.A.: How mutation affects evolutionary games on graphs. J. Theor. Biol. 299, 97–105 (2012) MathSciNetCrossRefGoogle Scholar
  61. 61.
    van Veelen, M., García, J., Rand, D.G., Nowak, M.A.: Direct reciprocity in structured populations. Proc. Natl. Acad. Sci. USA 109, 9929–9934 (2012) ADSCrossRefGoogle Scholar
  62. 62.
    van Veelen, M., Nowak, M.A.: Multi-player games on the cycle. J. Theor. Biol. 292, 116–128 (2012) CrossRefGoogle Scholar
  63. 63.
    Fu, F., Tarnita, C.E., Christakis, N.A., Wang, L., Rand, D.G., Nowak, M.A.: Evolution of in-group favoritism. Sci. Rep. 2, 460 (2012) Google Scholar
  64. 64.
    Nowak, M.A., Tarnita, C.E., Antal, T.: Evolutionary dynamics in structured populations. Philos. Trans. R. Soc. B 365, 19–30 (2010) CrossRefGoogle Scholar
  65. 65.
    Macy, M.W., Flache, A.: Learning dynamics in social dilemmas. Proc. Natl. Acad. Sci. USA 99, 7229–7236 (2002) ADSCrossRefGoogle Scholar
  66. 66.
    Masuda, N., Ohtsuki, H.: A theoretical analysis of temporal difference learning in the iterated Prisoner’s Dilemma game. Bull. Math. Biol. 71, 1818–1850 (2009) MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    Santos, F.C., Pacheco, J.M., Lenaerts, T.: Evolutionary dynamics of social dilemmas in structured heterogeneous populations. Proc. Natl. Acad. Sci. USA 103, 3490–3494 (2006) ADSCrossRefGoogle Scholar
  68. 68.
    Santos, F.C., Santos, M.D., Pacheco, J.M.: Social diversity promotes the emergence of cooperation in public goods games. Nature 454, 213–216 (2008) ADSCrossRefGoogle Scholar
  69. 69.
    Szolnoki, A., Perc, M., Danku, Z.: Towards effective payoffs in the prisoner’s dilemma game on scale-free networks. Physica A 387, 2075–2082 (2008) ADSCrossRefGoogle Scholar
  70. 70.
    Gracia-Lázaro, C., Ferrer, A., Ruiz, G., Tarancón, A., Cuesta, J.A., Sánchez, A., Moreno, Y.: Heterogeneous networks do not promote cooperation when humans play a Prisoner’s Dilemma. Proc. Natl. Acad. Sci. USA. doi:10.1073/pnas.1206681109 (2012) Google Scholar
  71. 71.
    Skyrms, B., Pemantle, R.: A dynamic model of social network formation. Proc. Natl. Acad. Sci. USA 97, 9340–9346 (2000) ADSMATHCrossRefGoogle Scholar
  72. 72.
    Ebel, H., Bornholdt, S.: Coevolutionary games on networks. Phys. Rev. E 66, 056118 (2002) ADSCrossRefGoogle Scholar
  73. 73.
    Zimmermann, M.G., Eguíluz, V.M., San Miguel, M.: Coevolution of dynamical states and interactions in dynamic networks. Phys. Rev. E 69, 065102 (2004) ADSCrossRefGoogle Scholar
  74. 74.
    Santos, F.C., Pacheco, J.M., Lenaerts, T.: Cooperation prevails when individuals adjust their social ties. PLoS Comput. Biol. 2, e140 (2006) ADSCrossRefGoogle Scholar
  75. 75.
    Hanaki, N., Peterhansl, A., Dodds, P.S., Watts, D.J.: Cooperation in evolving social networks. Manag. Sci. 53, 1036–1050 (2007) MATHCrossRefGoogle Scholar
  76. 76.
    Fu, F., Hauert, C., Nowak, M.A., Wang, L.: Reputation-based partner choice promotes cooperation in social networks. Phys. Rev. E 78, 026117 (2008) ADSCrossRefGoogle Scholar
  77. 77.
    Du, F.Q., Fu, F.: Partner selection shapes the strategic and topological evolution of cooperation. Dyn. Games Appl. 1, 354–369 (2011) MathSciNetMATHCrossRefGoogle Scholar
  78. 78.
    Perc, M., Szolnoki, A.: Coevolutionary games—A mini review. Biosystems 99, 109–125 (2010) CrossRefGoogle Scholar
  79. 79.
    Traulsen, A., Nowak, M.A.: Evolution of cooperation by multilevel selection. Proc. Natl. Acad. Sci. USA 103, 10952–10955 (2006) ADSCrossRefGoogle Scholar
  80. 80.
    Wild, G., Traulsen, A.: The different limits of weak selection and the evolutionary dynamics of finite populations. J. Theor. Biol. 247, 382–390 (2007) MathSciNetCrossRefGoogle Scholar
  81. 81.
    Antal, T., Traulsen, A., Ohtsuki, H., Tarnita, C.E., Nowak, M.A.: Mutation-selection equilibrium in games with multiple strategies. J. Theor. Biol. 258, 614–622 (2009) MathSciNetCrossRefGoogle Scholar
  82. 82.
    Tarnita, C.E., Antal, T., Nowak, M.A.: Mutation-selection equilibrium in games with mixed strategies. J. Theor. Biol. 261, 50–57 (2009) MathSciNetCrossRefGoogle Scholar
  83. 83.
    Sabeti, P.C., et al.: Detecting recent positive selection in the human genome from haplotype structure. Nature 419, 832–837 (2002) ADSCrossRefGoogle Scholar
  84. 84.
    Nowak, M.A., Sigmund, K.: The evolution of stochastic strategies in the prisoner’s dilemma. Acta Appl. Math. 20, 247–265 (1990) MathSciNetMATHCrossRefGoogle Scholar
  85. 85.
    Metz, J.A.J., Geritz, S.A.H., Meszena, G., Jacobs, F.J.A., van Heerwarden, J.S.: Adaptive dynamics,a geometrical study of the consequences of nearly faithful reproduction. In: van Strien, S.J., Verduyn Lunel, S.M. (eds.) Stochastic and Spatial Structures of Dynamical Systems. K. Ned. Akad. Van Wet. B, vol. 45, pp. 183–231. North-Holland, Amsterdam (1996) Google Scholar
  86. 86.
    Dieckmann, U., Law, R.: The dynamical theory of coevolution: a derivation from stochastic ecological processes. J. Math. Biol. 34, 579–612 (1996) MathSciNetMATHCrossRefGoogle Scholar
  87. 87.
    Dieckmann, U., Law, R., Metz, J.A.J.: The Geometry of Ecological Interactions: Simplifying Spatial Complexity. Cambridge University Press, Cambridge, UK (2000) CrossRefGoogle Scholar
  88. 88.
    Reichenbach, T., Mobilia, M., Frey, E.: Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games. Nature 448, 1046–1049 (2007) ADSCrossRefGoogle Scholar
  89. 89.
    Helbing, D., Yu, W.: The outbreak of cooperation among success-driven individuals under noisy conditions. Proc. Natl. Acad. Sci. USA 106, 3680–3685 (2009) ADSCrossRefGoogle Scholar
  90. 90.
    Wu, T., Fu, F., Zhang, Y.L., Wang, L.: Expectation-driven migration promotes cooperation by group interactions. Phys. Rev. E 85, 066104 (2012) ADSCrossRefGoogle Scholar
  91. 91.
    Kimura, M., Weiss, G.H.: The stepping stone model of population structure and the decrease of genetic correlation with distance. Genetics 49, 561–576 (1964) Google Scholar
  92. 92.
    Levin, S.A.: Dispersion and population interactions. Am. Nat. 108, 207–228 (1974) CrossRefGoogle Scholar
  93. 93.
    Boyd, R., Richerson, P.J.: The evolution of reciprocity in sizable groups. J. Theor. Biol. 132, 337–356 (1988) MathSciNetCrossRefGoogle Scholar
  94. 94.
    Taylor, P.D.: Altruism in viscous populations: an inclusive fitness model. Evol. Ecol. 6, 352–356 (1992) CrossRefGoogle Scholar
  95. 95.
    Durrett, R., Levin, S.A.: The importance of being discrete (and spatial). Theor. Popul. Biol. 46, 363–394 (1994) MATHCrossRefGoogle Scholar
  96. 96.
    Rousset, F.: Genetic Structure and Selection in Subdivided Populations. Princeton University Press, Princeton (2004) Google Scholar
  97. 97.
    Killingback, T., Bieri, J., Flatt, T.: Evolution in group-structured populations can resolve the tragedy of the commons. Proc. R. Soc. B 273, 1477–1481 (2006) CrossRefGoogle Scholar
  98. 98.
    Boyd, R., Richerson, P.J.: Voting with your feet: payoff biased migration and the evolution of group beneficial behavior. J. Theor. Biol. 257, 331–339 (2009) MathSciNetCrossRefGoogle Scholar
  99. 99.
    He, Q., Mobilia, M., Täber, U.C.: Spatial rock-paper-scissors models with inhomogeneous reaction rates. Phys. Rev. E 82, 051909 (2010) ADSCrossRefGoogle Scholar
  100. 100.
    Nowak, M.A., Tarnita, C.E., Wilson, E.O.: The evolution of eusociality. Nature 466, 1057–1062 (2010) ADSCrossRefGoogle Scholar
  101. 101.
    Hauert, C., Imhof, L.: Evolutionary games in deme structured, finite populations. J. Theor. Biol. 299, 106–112 (2011) MathSciNetCrossRefGoogle Scholar
  102. 102.
    Maruyama, T.: Effective number of alleles in a subdivided population. Theor. Popul. Biol. 1, 273–306 (1970) MathSciNetCrossRefGoogle Scholar
  103. 103.
    Maruyama, T.: Stepping stone models of finite length. Adv. Appl. Probab. 2, 229–258 (1970) MathSciNetMATHCrossRefGoogle Scholar
  104. 104.
    Ohtsuki, H.: Evolutionary games in Wright’s island model: kin selection meets evolutionary game theory. Evolution 64, 3344–3353 (2010) CrossRefGoogle Scholar
  105. 105.
    Abbot, P., et al.: Inclusive fitness theory and eusociality. Nature 471, E1–E4 (2011) ADSCrossRefGoogle Scholar
  106. 106.
    Nowak, M.A., Tarnita, C.E., Wilson, E.O.: Nowak, et al. reply. Nature 471, E9–E10 (2011) ADSCrossRefGoogle Scholar
  107. 107.
    Nowak, M.A., Tarnita, C.E., Wilson, E.O.: A brief statement about inclusive fitness and eusociality. http://www.ped.fas.harvard.edu/IF_Statement.pdf/ (2011)
  108. 108.
    Gammaitoni, L., Hanggi, P., Jung, P., Marchesoni, F.: Stochastic resonance. Rev. Mod. Phys. 70, 223–287 (1998) ADSCrossRefGoogle Scholar
  109. 109.
    Ren, J., Wang, W.X., Qi, F.: Randomness enhances cooperation: a resonance-type phenomenon in evolutionary games. Phys. Rev. E 75, 045101 (2007) ADSCrossRefGoogle Scholar
  110. 110.
    Szabó, G., Vukov, J., Szolnoki, A.: Phase diagrams for an evolutionary prisoner’s dilemma game on two-dimensional lattices. Phys. Rev. E 72, 047107 (2005) ADSCrossRefGoogle Scholar
  111. 111.
    Vukov, J., Szabó, G., Szolnoki, A.: Cooperation in the noisy case: Prisoner’s dilemma game on two types of regular random graphs. Phys. Rev. E 74, 067103 (2006) ADSCrossRefGoogle Scholar
  112. 112.
    Fu, F., Wu, T., Wang, L.: Partner switching stabilizes cooperation in coevolutionary Prisoner’s dilemma. Phys. Rev. E 79, 036101 (2009) MathSciNetADSCrossRefGoogle Scholar
  113. 113.
    Weiss, R.A., McMichael, J.A.: Social and environmental risk factors in the emergence of infectious diseases. Nat. Med. 10, S70–S76 (2004) CrossRefGoogle Scholar
  114. 114.
    Brockmann, D., Hufnagel, L., Geisel, T.: The scaling laws of human travel. Nature 439, 462–465 (2005) ADSCrossRefGoogle Scholar
  115. 115.
    Wakeley, J.: Coalescent Theory: An Introduction. Roberts & Company Publishers, Greenwood Village (2008) Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Program for Evolutionary DynamicsHarvard UniversityCambridgeUSA
  2. 2.Department of Health Care PolicyHarvard Medical SchoolBostonUSA
  3. 3.Department of MathematicsHarvard UniversityCambridgeUSA
  4. 4.Department of Organismic and Evolutionary BiologyHarvard UniversityCambridgeUSA

Personalised recommendations