Journal of Mathematical Biology

, Volume 78, Issue 4, pp 1147–1210 | Cite as

A mathematical formalism for natural selection with arbitrary spatial and genetic structure

  • Benjamin AllenEmail author
  • Alex McAvoy


We define a general class of models representing natural selection between two alleles. The population size and spatial structure are arbitrary, but fixed. Genetics can be haploid, diploid, or otherwise; reproduction can be asexual or sexual. Biological events (e.g. births, deaths, mating, dispersal) depend in arbitrary fashion on the current population state. Our formalism is based on the idea of genetic sites. Each genetic site resides at a particular locus and houses a single allele. Each individual contains a number of sites equal to its ploidy (one for haploids, two for diploids, etc.). Selection occurs via replacement events, in which alleles in some sites are replaced by copies of others. Replacement events depend stochastically on the population state, leading to a Markov chain representation of natural selection. Within this formalism, we define reproductive value, fitness, neutral drift, and fixation probability, and prove relationships among them. We identify four criteria for evaluating which allele is selected and show that these become equivalent in the limit of low mutation. We then formalize the method of weak selection. The power of our formalism is illustrated with applications to evolutionary games on graphs and to selection in a haplodiploid population.


Evolution Population genetics Fixation probability Spatial structure Weak selection Markov chain 

Mathematics Subject Classification

91A22 92D15 



BA is supported by National Science Foundation Award #DMS-1715315. AM is supported by the Office of Naval Research, Grant N00014-16-1-2914. We thank Martin A. Nowak for helpful discussions.


  1. Adlam B, Chatterjee K, Nowak M (2015) Amplifiers of selection. Proc R Soc A Math Phys Eng Sci 471(2181):20150,114MathSciNetzbMATHCrossRefGoogle Scholar
  2. Akçay E, Van Cleve J (2016) There is no fitness but fitness, and the lineage is its bearer. Philos Trans R Soc B Biol Sci 371(1687):20150,085CrossRefGoogle Scholar
  3. Allen B, Nowak MA (2014) Games on graphs. EMS Surv Math Sci 1(1):113–151MathSciNetzbMATHCrossRefGoogle Scholar
  4. Allen B, Tarnita CE (2014) Measures of success in a class of evolutionary models with fixed population size and structure. J Math Biol 68(1–2):109–143MathSciNetzbMATHCrossRefGoogle Scholar
  5. Allen B, Traulsen A, Tarnita CE, Nowak MA (2012) How mutation affects evolutionary games on graphs. J Theor Biol 299:97–105. MathSciNetzbMATHCrossRefGoogle Scholar
  6. Allen B, Nowak MA, Dieckmann U (2013) Adaptive dynamics with interaction structure. Am Nat 181(6):E139–E163CrossRefGoogle Scholar
  7. Allen B, Sample C, Dementieva Y, Medeiros RC, Paoletti C, Nowak MA (2015) The molecular clock of neutral evolution can be accelerated or slowed by asymmetric spatial structure. PLoS Comput Biol 11(2):e1004,108. CrossRefGoogle Scholar
  8. Allen B, Lippner G, Chen YT, Fotouhi B, Momeni N, Yau ST, Nowak MA (2017) Evolutionary dynamics on any population structure. Nature 544(7649):227–230CrossRefGoogle Scholar
  9. Antal T, Ohtsuki H, Wakeley J, Taylor PD, Nowak MA (2009a) Evolution of cooperation by phenotypic similarity. Proc Natl Acad Sci 106(21):8597–8600. zbMATHCrossRefGoogle Scholar
  10. Antal T, Traulsen A, Ohtsuki H, Tarnita CE, Nowak MA (2009b) Mutation-selection equilibrium in games with multiple strategies. J Theor Biol 258(4):614–622MathSciNetzbMATHCrossRefGoogle Scholar
  11. Benaïm M, Schreiber SJ (2018) Persistence and extinction for stochastic ecological difference equations with feedbacks. arXiv preprint arXiv:1808.07888
  12. Blume LE (1993) The statistical mechanics of strategic interaction. Games Econ Behav 5(3):387–424MathSciNetzbMATHCrossRefGoogle Scholar
  13. Broom M, Hadjichrysanthou C, Rychtář J (2010) Evolutionary games on graphs and the speed of the evolutionary process. Proc R Soc A Math Phys Eng Sci 466(2117):1327–1346MathSciNetzbMATHCrossRefGoogle Scholar
  14. Bürger R (2000) The mathematical theory of selection, recombination, and mutation. Wiley, LondonzbMATHGoogle Scholar
  15. Bürger R (2005) A multilocus analysis of intraspecific competition and stabilizing selection on a quantitative trait. J Math Biol 50(4):355–396MathSciNetzbMATHCrossRefGoogle Scholar
  16. Cattiaux P, Collet P, Lambert A, Martinez S, Méléard S, San Martín J (2009) Quasi-stationary distributions and diffusion models in population dynamics. Ann Probab 37(5):1926–1969MathSciNetzbMATHCrossRefGoogle Scholar
  17. Cavaliere M, Sedwards S, Tarnita CE, Nowak MA, Csikász-Nagy A (2012) Prosperity is associated with instability in dynamical networks. J Theor Biol 299:126–138MathSciNetzbMATHCrossRefGoogle Scholar
  18. Champagnat N, Ferrière R, Méléard S (2006) Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models. Theor Popul Biol 69(3):297–321zbMATHCrossRefGoogle Scholar
  19. Chen YT (2013) Sharp benefit-to-cost rules for the evolution of cooperation on regular graphs. Ann Appl Probab 23(2):637–664MathSciNetzbMATHCrossRefGoogle Scholar
  20. Chen YT (2018) Wright-Fisher diffusions in stochastic spatial evolutionary games with death–birth updating. Ann Appl Probab 28:3418–3490MathSciNetzbMATHCrossRefGoogle Scholar
  21. Chen YT, McAvoy A, Nowak MA (2016) Fixation probabilities for any configuration of two strategies on regular graphs. Sci Rep 6(39):181Google Scholar
  22. Chotibut T, Nelson DR (2017) Population genetics with fluctuating population sizes. J Stat Phys 167(3–4):777–791MathSciNetzbMATHCrossRefGoogle Scholar
  23. Cohen D (1966) Optimizing reproduction in a randomly varying environment. J Theor Biol 12(1):119–129CrossRefGoogle Scholar
  24. Constable GW, Rogers T, McKane AJ, Tarnita CE (2016) Demographic noise can reverse the direction of deterministic selection. Proc Natl Acad Sci 113(32):E4745–E4754CrossRefGoogle Scholar
  25. Cox JT (1989) Coalescing random walks and voter model consensus times on the torus in \({\mathbb{Z}}^d\). Ann Probab 17(4):1333–1366MathSciNetzbMATHCrossRefGoogle Scholar
  26. Cox JT, Durrett R, Perkins EA (2013) Voter model perturbations and reaction diffusion equations. Asterisque 349Google Scholar
  27. Crow JF, Kimura M (1970) An introduction to population genetics theory. Harper and Row, New YorkzbMATHGoogle Scholar
  28. Cvijović I, Good BH, Jerison ER, Desai MM (2015) Fate of a mutation in a fluctuating environment. Proc Natl Acad Sci 112(36):E5021–E5028CrossRefGoogle Scholar
  29. Débarre F (2017) Fidelity of parent-offspring transmission and the evolution of social behavior in structured populations. J Theor Biol 420:26–35MathSciNetzbMATHCrossRefGoogle Scholar
  30. Débarre F, Hauert C, Doebeli M (2014) Social evolution in structured populations. Nat Commun 5:4409CrossRefGoogle Scholar
  31. Dercole F, Rinaldi S (2008) Analysis of evolutionary processes: the adaptive dynamics approach and its applications. Princeton University Press, PrincetonzbMATHGoogle Scholar
  32. Dieckmann U, Doebeli M (1999) On the origin of species by sympatric speciation. Nature 400(6742):354–357CrossRefGoogle Scholar
  33. Dieckmann U, Law R (1996) The dynamical theory of coevolution: a derivation from stochastic ecological processes. J Math Biol 34(5):579–612MathSciNetzbMATHCrossRefGoogle Scholar
  34. Diekmann O, Gyllenberg M, Metz JAJ, Thieme HR (1998) On the formulation and analysis of general deterministic structured population models I. Linear theory. J Math Biol 36:349–388. MathSciNetzbMATHCrossRefGoogle Scholar
  35. Diekmann O, Gyllenberg M, Huang H, Kirkilionis M, Metz JAJ, Thieme HR (2001) On the formulation and analysis of general deterministic structured population models II. Nonlinear theory. J Math Biol 43:157–189. MathSciNetzbMATHCrossRefGoogle Scholar
  36. Diekmann O, Gyllenberg M, Metz J (2007) Physiologically structured population models: towards a general mathematical theory. In: Takeuchi Y, Iwasa Y, Sato K (eds) Mathematics for ecology and environmental sciences, biological and medical physics, biomedical engineering. Springer, Berlin, pp 5–20Google Scholar
  37. Doebeli M, Ispolatov Y, Simon B (2017) Towards a mechanistic foundation of evolutionary theory. eLife 6:e23,804CrossRefGoogle Scholar
  38. Durinx M, Metz JAJ, Meszéna G (2008) Adaptive dynamics for physiologically structured population models. J Math Biol 56(5):673–742MathSciNetzbMATHCrossRefGoogle Scholar
  39. Durrett R (2014) Spatial evolutionary games with small selection coefficients. Electron J Probab 19(121):1–64. MathSciNetzbMATHCrossRefGoogle Scholar
  40. Eshel I, Feldman MW, Bergman A (1998) Long-term evolution, short-term evolution, and population genetic theory. J Theor Biol 191(4):391–396CrossRefGoogle Scholar
  41. Ewens WJ (2004) Mathematical population genetics 1: theoretical introduction, vol 27, 2nd edn. Springer, New YorkzbMATHCrossRefGoogle Scholar
  42. Faure M, Schreiber SJ (2014) Quasi-stationary distributions for randomly perturbed dynamical systems. Ann Appl Probab 24(2):553–598. MathSciNetzbMATHCrossRefGoogle Scholar
  43. Fisher R (1930) The genetical theory of natural selection. Clarendon Press, OxfordzbMATHCrossRefGoogle Scholar
  44. Fotouhi B, Momeni N, Allen B, Nowak MA (2018) Conjoining uncooperative societies facilitates evolution of cooperation. Nat Hum Behav 2:492–499CrossRefGoogle Scholar
  45. Fudenberg D, Imhof LA (2006) Imitation processes with small mutations. J Econ Theory 131(1):251–262. MathSciNetzbMATHCrossRefGoogle Scholar
  46. Geritz SAH, Kisdi E, Meszéna G, Metz JAJ (1997) Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol Ecol 12(1):35–57CrossRefGoogle Scholar
  47. Gould SJ, Lloyd EA (1999) Individuality and adaptation across levels of selection: How shall we name and generalize the unit of Darwinism? Proc Natl Acad Sci 96(21):11,904–11,909CrossRefGoogle Scholar
  48. Gyllenberg M, Parvinen K (2001) Necessary and sufficient conditions for evolutionary suicide. Bull Math Biol 63:981–993. zbMATHCrossRefGoogle Scholar
  49. Gyllenberg M, Silvestrov D (2008) Quasi-stationary phenomena in nonlinearly perturbed stochastic systems. Walter de Gruyter, BerlinzbMATHCrossRefGoogle Scholar
  50. Haccou P, Iwasa Y (1996) Establishment probability in fluctuating environments: a branching process model. Theor Popul Biol 50(3):254–280zbMATHCrossRefGoogle Scholar
  51. Haccou P, Jagers P, Vatutin VA (2005) Branching processes: variation, growth, and extinction of populations. Cambridge University Press, Cambridge. zbMATHCrossRefGoogle Scholar
  52. Haldane J (1924) A mathematical theory of natural and artificial selection. Part I. Trans Camb Philos Soc 23:19–41Google Scholar
  53. Hammerstein P (1996) Darwinian adaptation, population genetics and the streetcar theory of evolution. J Math Biol 34(5–6):511–532zbMATHCrossRefGoogle Scholar
  54. Holley RA, Liggett TM (1975) Ergodic theorems for weakly interacting infinite systems and the voter model. Ann Probab 3(4):643–663MathSciNetzbMATHCrossRefGoogle Scholar
  55. Hull DL (1980) Individuality and selection. Annu Rev Ecol Syst 11(1):311–332CrossRefGoogle Scholar
  56. Ibsen-Jensen R, Chatterjee K, Nowak MA (2015) Computational complexity of ecological and evolutionary spatial dynamics. Proc Natl Acad Sci 112(51):15,636–15,641Google Scholar
  57. Jeong HC, Oh SY, Allen B, Nowak MA (2014) Optional games on cycles and complete graphs. J Theor Biol 356:98–112MathSciNetCrossRefGoogle Scholar
  58. Kemeny JG, Snell JL (1960) Finite Markov chains, vol 356. van Nostrand, PrincetonzbMATHGoogle Scholar
  59. Kimura M (1964) Diffusion models in population genetics. J Appl Probab 1(2):177–232MathSciNetzbMATHCrossRefGoogle Scholar
  60. Kimura M et al (1968) Evolutionary rate at the molecular level. Nature 217(5129):624–626CrossRefGoogle Scholar
  61. Kingman JFC (1982) The coalescent. Stoch Processes Appl 13(3):235–248MathSciNetzbMATHCrossRefGoogle Scholar
  62. Korolev KS (2013) The fate of cooperation during range expansions. PLoS Comput Biol 9(3):e1002,994MathSciNetCrossRefGoogle Scholar
  63. Kussell E, Leibler S (2005) Phenotypic diversity, population growth, and information in fluctuating environments. Science 309(5743):2075–2078CrossRefGoogle Scholar
  64. Lambert A (2006) Probability of fixation under weak selection: a branching process unifying approach. Theor Popul Biol 69(4):419–441zbMATHCrossRefGoogle Scholar
  65. Lehmann L, Rousset F (2009) Perturbation expansions of multilocus fixation probabilities for frequency-dependent selection with applications to the Hill–Robertson effect and to the joint evolution of helping and punishment. Theor Popul Biol 76(1):35–51zbMATHCrossRefGoogle Scholar
  66. Lehmann L, Mullon C, Akcay E, Cleve J (2016) Invasion fitness, inclusive fitness, and reproductive numbers in heterogeneous populations. Evolution 70(8):1689–1702CrossRefGoogle Scholar
  67. Lessard S, Ladret V (2007) The probability of fixation of a single mutant in an exchangeable selection model. J Math Biol 54(5):721–744MathSciNetzbMATHCrossRefGoogle Scholar
  68. Lessard S, Soares CD (2018) Frequency-dependent growth in class-structured populations: continuous dynamics in the limit of weak selection. J Math Biol 77(1):229–259. MathSciNetzbMATHCrossRefGoogle Scholar
  69. Leturque H, Rousset F (2002) Dispersal, kin competition, and the ideal free distribution in a spatially heterogeneous population. Theor Popul Biol 62(2):169–180zbMATHCrossRefGoogle Scholar
  70. Lewontin RC (1970) The units of selection. Annu Rev Ecol Syst 1(1):1–18CrossRefGoogle Scholar
  71. Lieberman E, Hauert C, Nowak M (2005) Evolutionary dynamics on graphs. Nature 433(7023):312–316CrossRefGoogle Scholar
  72. Lindholm AK, Dyer KA, Firman RC, Fishman L, Forstmeier W, Holman L, Johannesson H, Knief U, Kokko H, Larracuente AM et al (2016) The ecology and evolutionary dynamics of meiotic drive. Trends Ecol Evol 31(4):315–326CrossRefGoogle Scholar
  73. Maciejewski W (2014) Reproductive value in graph-structured populations. J Theor Biol 340:285–293MathSciNetCrossRefGoogle Scholar
  74. Malécot G (1948) Les Mathématiques de l’Hérédité. Masson et Cie, ParisGoogle Scholar
  75. McAvoy A, Hauert C (2016) Structure coefficients and strategy selection in multiplayer games. J Math Biol 72(1–2):203–238MathSciNetzbMATHCrossRefGoogle Scholar
  76. McAvoy A, Adlam B, Allen B, Nowak MA (2018a) Stationary frequencies and mixing times for neutral drift processes with spatial structure. Proc Ro Soc A Math Phys Eng Sci.
  77. McAvoy A, Fraiman N, Hauert C, Wakeley J, Nowak MA (2018b) Public goods games in populations with fluctuating size. Theor Popul Biol 121:72–84. zbMATHCrossRefGoogle Scholar
  78. Metz JAJ, de Roos AM (1992) The role of physiologically structured population models within a general individual-based modelling perspective. In: DeAngelis DL, Gross LA, Hallam TG (eds) Individual-based models and approaches in ecology: populations, communities, and ecosystems. Chapman & Hall, London, pp 88–111CrossRefGoogle Scholar
  79. Metz JA, Geritz SA (2016) Frequency dependence 3.0: an attempt at codifying the evolutionary ecology perspective. J Math Biol 72(4):1011–1037MathSciNetzbMATHCrossRefGoogle Scholar
  80. Metz J, Nisbet R, Geritz S (1992) How should we define ‘fitness’ for general ecological scenarios? Trends Ecol Evol 7(6):198–202. CrossRefGoogle Scholar
  81. Metz JAJ, Geritz SAH, Meszéna G, Jacobs FA, van Heerwaarden JS (1996) Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In: van Strien SJ, Lunel SMV (eds) Stochastic and spatial structures of dynamical systems. KNAW Verhandelingen, Afd., Amsterdam, pp 183–231zbMATHGoogle Scholar
  82. Nowak MA, May RM (1992) Evolutionary games and spatial chaos. Nature 359(6398):826–829CrossRefGoogle Scholar
  83. Nowak MA, Sasaki A, Taylor C, Fudenberg D (2004) Emergence of cooperation and evolutionary stability in finite populations. Nature 428(6983):646–650CrossRefGoogle Scholar
  84. Nowak MA, Tarnita CE, Antal T (2010a) Evolutionary dynamics in structured populations. Philos Trans R Soc B Biol Sci 365(1537):19CrossRefGoogle Scholar
  85. Nowak MA, Tarnita CE, Wilson EO (2010b) The evolution of eusociality. Nature 466(7310):1057–1062CrossRefGoogle Scholar
  86. Ohtsuki H, Hauert C, Lieberman E, Nowak MA (2006) A simple rule for the evolution of cooperation on graphs and social networks. Nature 441:502–505CrossRefGoogle Scholar
  87. Okasha S (2006) Evolution and the levels of selection. Oxford University Press, OxfordCrossRefGoogle Scholar
  88. Pacheco JM, Traulsen A, Nowak MA (2006a) Active linking in evolutionary games. J Theor Biol 243(3):437–443. MathSciNetCrossRefGoogle Scholar
  89. Pacheco JM, Traulsen A, Nowak MA (2006b) Coevolution of strategy and structure in complex networks with dynamical linking. Phys Rev Lett 97(25):258,103CrossRefGoogle Scholar
  90. Parsons TL, Quince C (2007a) Fixation in haploid populations exhibiting density dependence I: the non-neutral case. Theor Popul Biol 72(1):121–135zbMATHCrossRefGoogle Scholar
  91. Parsons TL, Quince C (2007b) Fixation in haploid populations exhibiting density dependence II: the quasi-neutral case. Theor Popul Biol 72(4):468–479zbMATHCrossRefGoogle Scholar
  92. Parsons TL, Quince C, Plotkin JB (2010) Some consequences of demographic stochasticity in population genetics. Genetics 185(4):1345–1354CrossRefGoogle Scholar
  93. Parvinen K, Seppänen A (2016) On fitness in metapopulations that are both size-and stage-structured. J Math Biol 73(4):903–917MathSciNetzbMATHCrossRefGoogle Scholar
  94. Pavlogiannis A, Tkadlec J, Chatterjee K, Nowak MA (2018) Construction of arbitrarily strong amplifiers of natural selection using evolutionary graph theory. Commun Biol 1(1):71CrossRefGoogle Scholar
  95. Pelletier F, Clutton-Brock T, Pemberton J, Tuljapurkar S, Coulson T (2007) The evolutionary demography of ecological change: linking trait variation and population growth. Science 315(5818):1571–1574CrossRefGoogle Scholar
  96. Peña J, Wu B, Arranz J, Traulsen A (2016) Evolutionary games of multiplayer cooperation on graphs. PLoS Comput Biol 12(8):e1005,059CrossRefGoogle Scholar
  97. Perc M, Szolnoki A (2010) Coevolutionary games—a mini review. BioSystems 99(2):109–125CrossRefGoogle Scholar
  98. Philippi T, Seger J (1989) Hedging one’s evolutionary bets, revisited. Trends Ecol Evol 4(2):41–44CrossRefGoogle Scholar
  99. Price GR (1970) Selection and covariance. Nature 227:520–521CrossRefGoogle Scholar
  100. Rand DA, Wilson HB, McGlade JM (1994) Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics. Philos Trans R Soc B Biol Sci 343(1305):261–283. CrossRefGoogle Scholar
  101. Roth G, Schreiber SJ (2013) Persistence in fluctuating environments for interacting structured populations. J Math Biol 69(5):1267–1317. MathSciNetzbMATHCrossRefGoogle Scholar
  102. Roth G, Schreiber SJ (2014) Pushed beyond the brink: Allee effects, environmental stochasticity, and extinction. J Biol Dyn 8(1):187–205. MathSciNetCrossRefGoogle Scholar
  103. Rousset F, Billiard S (2000) A theoretical basis for measures of kin selection in subdivided populations: finite populations and localized dispersal. J Evol Biol 13(5):814–825CrossRefGoogle Scholar
  104. Sample C, Allen B (2017) The limits of weak selection and large population size in evolutionary game theory. J Math Biol 75(5):1285–1317MathSciNetzbMATHCrossRefGoogle Scholar
  105. Sandler L, Novitski E (1957) Meiotic drive as an evolutionary force. Am Nat 91(857):105–110CrossRefGoogle Scholar
  106. Santos FC, Pacheco JM (2005) Scale-free networks provide a unifying framework for the emergence of cooperation. Phys Rev Lett 95(9):98,104CrossRefGoogle Scholar
  107. Schoener TW (2011) The newest synthesis: understanding the interplay of evolutionary and ecological dynamics. Science 331(6016):426–429CrossRefGoogle Scholar
  108. Schreiber SJ, Benaïm M, Atchadé KAS (2010) Persistence in fluctuating environments. J Math Biol 62(5):655–683. MathSciNetzbMATHCrossRefGoogle Scholar
  109. Silvestrov D, Silvestrov S (2017) Nonlinearly perturbed semi-Markov processes. Springer, ChamzbMATHCrossRefGoogle Scholar
  110. Simon B, Fletcher JA, Doebeli M (2013) Towards a general theory of group selection. Evolution 67(6):1561–1572CrossRefGoogle Scholar
  111. Starrfelt J, Kokko H (2012) Bet-hedging—a triple trade-off between means, variances and correlations. Biol Rev 87(3):742–755CrossRefGoogle Scholar
  112. Szabó G, Fáth G (2007) Evolutionary games on graphs. Phys Rep 446(4–6):97–216MathSciNetCrossRefGoogle Scholar
  113. Tarnita CE, Taylor PD (2014) Measures of relative fitness of social behaviors in finite structured population models. Am Nat 184(4):477–488CrossRefGoogle Scholar
  114. Tarnita CE, Antal T, Ohtsuki H, Nowak MA (2009a) Evolutionary dynamics in set structured populations. Proc Natl Acad Sci 106(21):8601–8604CrossRefGoogle Scholar
  115. Tarnita CE, Ohtsuki H, Antal T, Fu F, Nowak MA (2009b) Strategy selection in structured populations. J Theor Biol 259(3):570–581. MathSciNetzbMATHCrossRefGoogle Scholar
  116. Tarnita CE, Wage N, Nowak MA (2011) Multiple strategies in structured populations. Proc Natl Acad Sci 108(6):2334–2337. CrossRefGoogle Scholar
  117. Tavaré S (1984) Line-of-descent and genealogical processes, and their applications in population genetics models. Theor Popul Biol 26(2):119–164MathSciNetzbMATHCrossRefGoogle Scholar
  118. Taylor PD (1990) Allele-frequency change in a class-structured population. Am Nat 135(1):95–106MathSciNetCrossRefGoogle Scholar
  119. Taylor PD, Frank SA (1996) How to make a kin selection model. J Theor Biol 180(1):27–37CrossRefGoogle Scholar
  120. Taylor P, Day T, Wild G (2007a) From inclusive fitness to fixation probability in homogeneous structured populations. J Theor Biol 249(1):101–110MathSciNetCrossRefGoogle Scholar
  121. Taylor PD, Day T, Wild G (2007b) Evolution of cooperation in a finite homogeneous graph. Nature 447(7143):469–472CrossRefGoogle Scholar
  122. Traulsen A, Hauert C, De Silva H, Nowak MA, Sigmund K (2009) Exploration dynamics in evolutionary games. Proc Natl Acad Sci 106(3):709–712zbMATHCrossRefGoogle Scholar
  123. Uecker H, Hermisson J (2011) On the fixation process of a beneficial mutation in a variable environment. Genetics 188(4):915–930CrossRefGoogle Scholar
  124. Van Cleve J (2015) Social evolution and genetic interactions in the short and long term. Theor Popul Biol 103:2–26zbMATHCrossRefGoogle Scholar
  125. van Veelen M (2005) On the use of the price equation. J Theor Biol 237(4):412–426MathSciNetCrossRefGoogle Scholar
  126. Wakano JY, Nowak MA, Hauert C (2009) Spatial dynamics of ecological public goods. Proc Natl Acad Sci 106(19):7910–7914CrossRefGoogle Scholar
  127. Wakano JY, Ohtsuki H, Kobayashi Y (2013) A mathematical description of the inclusive fitness theory. Theor Popul Biol 84:46–55zbMATHCrossRefGoogle Scholar
  128. Wakeley J (2009) Coalescent theory: an introduction. Roberts & Company Publishers, Greenwood VillagezbMATHGoogle Scholar
  129. Wardil L, Hauert C (2014) Origin and structure of dynamic cooperative networks. Sci Rep 4:5725CrossRefGoogle Scholar
  130. Waxman D (2011) A unified treatment of the probability of fixation when population size and the strength of selection change over time. Genetics 188(4):907–913CrossRefGoogle Scholar
  131. Williams GC (1966) Adaptation and natural selection: a critique of some current evolutionary thought. Princeton University Press, PrincetonGoogle Scholar
  132. Wu B, Zhou D, Fu F, Luo Q, Wang L, Traulsen A (2010) Evolution of cooperation on stochastic dynamical networks. PLoS ONE 5(6):e11,187. CrossRefGoogle Scholar
  133. Wu B, Gokhale CS, Wang L, Traulsen A (2012) How small are small mutation rates? J Math Biol 64(5):803–827MathSciNetzbMATHCrossRefGoogle Scholar
  134. Wu B, Traulsen A, Gokhale CS (2013) Dynamic properties of evolutionary multi-player games in finite populations. Games 4(2):182–199MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsEmmanuel CollegeBostonUSA
  2. 2.Program for Evolutionary DynamicsHarvard UniversityCambridgeUSA

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