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Measures of success in a class of evolutionary models with fixed population size and structure

Abstract

We investigate a class of evolutionary models, encompassing many established models of well-mixed and spatially structured populations. Models in this class have fixed population size and structure. Evolution proceeds as a Markov chain, with birth and death probabilities dependent on the current population state. Starting from basic assumptions, we show how the asymptotic (long-term) behavior of the evolutionary process can be characterized by probability distributions over the set of possible states. We then define and compare three quantities characterizing evolutionary success: fixation probability, expected frequency, and expected change due to selection. We show that these quantities yield the same conditions for success in the limit of low mutation rate, but may disagree when mutation is present. As part of our analysis, we derive versions of the Price equation and the replicator equation that describe the asymptotic behavior of the entire evolutionary process, rather than the change from a single state. We illustrate our results using the frequency-dependent Moran process and the birth–death process on graphs as examples. Our broader aim is to spearhead a new approach to evolutionary theory, in which general principles of evolution are proven as mathematical theorems from axioms.

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Abbreviations

\(B\) :

Total (population-wide) expected offspring number

\(b_i\) :

Expected offspring number of individual \(i\)

\(d_i\) :

Death probability of individual \(i\)

\(\mathcal{E }\) :

Evolutionary process

\(e_{ij}\) :

Edge weight from \(i\) to \(j\) in the BD process on graphs

\(f_0(x), \, f_1(x)\) :

Reproductive rates of types 0 and 1, respectively, in the frequency-dependent Moran process

\(\mathcal M _\mathcal{E }\) :

Evolutionary Markov chain

\(N\) :

Population size

\(p_{\mathbf{s}\rightarrow \mathbf{s}^{\prime }}\) :

Probability of transition from state \(\mathbf{s}\) to state \(\mathbf{s}^{\prime }\) in \(\mathcal M _\mathcal{E }\)

\(p^{(n)}_{\mathbf{s}\rightarrow \mathbf{s}^{\prime }}\) :

Probability of \(n\)-step transition from state \(\mathbf{s}\) to state \(\mathbf{s}^{\prime }\) in \(\mathcal M _\mathcal{E }\)

\(r\) :

Reproductive rate of type 1 in the BD process on graphs

\(R\) :

Set of replaced positions in a replacement event

\(\mathfrak{R }\) :

Replacement rule

\(s_i\) :

Type of individual \(i\)

\(\mathbf{s}\) :

Vector of types occupying each position; state of \(\mathcal M _\mathcal{E }\)

\(u\) :

Mutation rate

\(w_i\) :

Fitness of individual \(i\)

\(\bar{w}^1, \bar{w}^0, \bar{w}\) :

Average fitness of types 1 and 0, and of the whole population, respectively

\(x_1, x_0\) :

Frequencies of types 1 and 0, respectively

\(\alpha \) :

Offspring-to-parent map in a replacement event

\(\Delta ^\mathrm{sel }x_1\) :

Expected change due to selection in the frequency of type 1

\(\pi _\mathbf{s}\) :

Probability of state \(\mathbf{s}\) in the mutation-selection stationary distribution

\(\pi ^*_\mathbf{s}\) :

Probability of state \(\mathbf{s}\) in the rare-mutation dimorphic distribution

\(\rho _1, \rho _0\) :

Fixation probabilities of types 1 and 0

\(\langle \; \rangle \) :

Expectation over the mutation-selection stationary distribution

\(\langle \; \rangle ^*\) :

Expectation over the rare-mutation dimorphic distribution

References

  1. Allen B, Traulsen A, Tarnita CE, Nowak MA (2012) How mutation affects evolutionary games on graphs. J Theor Biol 299:97–105. doi:10.1016/j.jtbi.2011.03.034

    Google Scholar 

  2. Antal T, Ohtsuki H, Wakeley J, Taylor PD, Nowak MA (2009a) Evolution of cooperation by phenotypic similarity. Proc Natl Acad Sci 106:8597–8600. doi:10.1073/pnas.0902528106

    Article  Google Scholar 

  3. Antal T, Traulsen A, Ohtsuki H, Tarnita CE, Nowak MA (2009) Mutation-selection equilibrium in games with multiple strategies. J Theor Biol 258:614–622. doi:10.1016/j.jtbi.2009.02.010

    Google Scholar 

  4. Barbour AD (1976) Quasi-stationary distributions in markov population processes. Adv Appl Prob, pp 296–314

  5. 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 466:1327–1346. doi:10.1098/rspa.2009.0487

    Article  MATH  Google Scholar 

  6. Broom M, Rychtár J (2008) An analysis of the fixation probability of a mutant on special classes of non-directed graphs. Proc R Soc A Math Phys Eng Sci 464:2609–2627. doi:10.1098/rspa.2008.0058

  7. Cannings C (1974) The latent roots of certain markov chains arising in genetics: a new approach, i. haploid models. Adv Appl Prob 6:260–290

    Article  MATH  MathSciNet  Google Scholar 

  8. Cattiaux P, Collet P, Lambert A, Martinez S, Méléard S (2009) Quasi-stationary distributions and diffusion models in population dynamics. Ann Prob 37:1926–1969

    Article  MATH  Google Scholar 

  9. Champagnat N, Ferrière R, Méléard S (2006) Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models. Theor Popul Biol 69:297–321

    Article  MATH  Google Scholar 

  10. Collet P, Martínez S, Méléard S (2011) Quasi-stationary distributions for structured birth and death processes with mutations. Prob Theory Relat Fields 151:191–231. doi:10.1007/s00440-010-0297-4

    Article  MATH  Google Scholar 

  11. Cox J (1989) Coalescing random walks and voter model consensus times on the torus in \(\mathbb{Z}^d\). Ann Prob 17:1333–1366

    Article  MATH  Google Scholar 

  12. Cox JT, Durrett R, Perkins EA (2000) Rescaled voter models converge to super-brownian motion. Ann Prob 28:185–234

    Article  MATH  MathSciNet  Google Scholar 

  13. Cressman R (1992) The stability concept of evolutionary game theory: a dynamic approach. Springer, Berlin

  14. Darroch JN, Seneta E (1965) On quasi-stationary distributions in absorbing discrete-time finite markov chains. J Appl Prob 2:88–100

    Article  MATH  MathSciNet  Google Scholar 

  15. Dieckmann U, Law R (1996) The dynamical theory of coevolution: a derivation from stochastic ecological processes. J Math Biol 34:579–612

    Article  MATH  MathSciNet  Google Scholar 

  16. 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. doi:10.1007/s002850170002

    Article  MATH  MathSciNet  Google Scholar 

  17. 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. Springer, Berlin, Heidelberg Biological and Medical Physics, Biomedical Engineering, pp 5–20

  18. 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. doi:10.1007/s002850050104

    Article  MATH  MathSciNet  Google Scholar 

  19. Durinx M, Metz JAJ, Meszéna G (2008) Adaptive dynamics for physiologically structured population models. J Math Biol 56:673–742

    Article  MATH  MathSciNet  Google Scholar 

  20. Ewens WJ (1979) Mathematical population genetics. Springer, New York

    MATH  Google Scholar 

  21. Falconer DS (1981) Introduction to quantitative genetics. Longman, London

  22. Fehl K, van der Post DJ, Semmann D (2011) Co-evolution of behaviour and social network structure promotes human cooperation. Ecol Lett doi:10.1111/j.1461-0248.2011.01615.x

  23. Fisher RA (1930) The genetical theory of natural selection. Clarendon Press, Oxford

    MATH  Google Scholar 

  24. Fu F, Hauert C, Nowak MA, Wang L (2008) Reputation-based partner choice promotes cooperation in social networks. Phys Rev E 78:026117. doi:10.1103/PhysRevE.78.026117

    Article  Google Scholar 

  25. Gardner A, West SA, Wild G (2011) The genetical theory of kin selection. J Evol Biol 24:1020–1043. doi:10.1111/j.1420-9101.2011.02236.x

    Article  Google Scholar 

  26. Geritz SAH, Kisdi E, Meszéna G, Metz JAJ (1997) Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol Ecol 12:35–57

    Article  Google Scholar 

  27. Grafen A (2000) Developments of the Price equation and natural selection under uncertainty. Proc R Soc London Ser B Biol Sci 267:1223

    Google Scholar 

  28. Gross T, Blasius B (2008) Adaptive coevolutionary networks: a review. J R Soc Interface 5:259–271. doi:10.1098/rsif.2007.1229

    Google Scholar 

  29. Gyllenberg M, Silvestrov D (2008) Quasi-stationary phenomena in nonlinearly perturbed stochastic systems. Walter de Gruyter, Berlin

    Book  Google Scholar 

  30. Hauert C, Doebeli M (2004) Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428:643–646

    Article  Google Scholar 

  31. Hofbauer J, Sigmund K (1990) Adaptive dynamics and evolutionary stability. Appl Math Lett 3:75–79

    Article  MATH  MathSciNet  Google Scholar 

  32. Hofbauer J, Sigmund K (1998) Evolutionary games& replicator dynamics. Cambridge University Press, Cambridge

    Google Scholar 

  33. Hofbauer J, Sigmund K (2003) Evolutionary game dynamics. Bull Am Math Soc 40:479–520

    Article  MATH  MathSciNet  Google Scholar 

  34. Holley R, Liggett T (1975) Ergodic theorems for weakly interacting infinite systems and the voter model. Annals Probability 3:643–663

    Article  MATH  MathSciNet  Google Scholar 

  35. Imhof LA, Nowak MA (2006) Evolutionary game dynamics in a wright-fisher process. J Math Biol 52: 667–681. doi:10.1007/s00285-005-0369-8

    Article  MATH  MathSciNet  Google Scholar 

  36. Iosifescu M (1980) Finite Markov processes and their applications. Wiley, New York

    MATH  Google Scholar 

  37. Kimura M (1964) Diffusion models in population genetics. J Appl Prob 1:177–232

    Article  MATH  Google Scholar 

  38. Kingman JFC (1982) The coalescent. Stochastic processes and their applications 13:235–248. doi:10.1016/0304-4149(82)90011-4

  39. Koralov L, Sinai Y (2007) Theory of probability and random processes. Springer, Berlin

  40. Lessard S, Ladret V (2007) The probability of fixation of a single mutant in an exchangeable selection model. J Math Biol 54:721–744. doi:10.1007/s00285-007-0069-7

    Article  MATH  MathSciNet  Google Scholar 

  41. Lieberman E, Hauert C, Nowak MA (2005) Evolutionary dynamics on graphs. Nature 433:312–316

    Article  Google Scholar 

  42. Lynch M, Walsh B (1998) Genetics and analysis of quantitative traits. Sinauer, Sunderland

    Google Scholar 

  43. Marshall JA (2011) Group selection and kin selection: formally equivalent approaches. Trends Ecol Evol 26:325–332. doi:10.1016/j.tree.2011.04.008

    Article  Google Scholar 

  44. Maynard Smith J, Price GR (1973) The logic of animal conflict. Nature 246:15–18

    Article  Google Scholar 

  45. Metz JAJ, de Roos AM (1992) The role of physiologically structured population models within a general individual-based modelling perspective. In: L DD, A GL, G HT (eds) Individual-based models and approaches in ecology: populations, communities, and ecosystems. Chapman& Hall, London, pp 88–111

  46. 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 183–231

  47. Mihoc G (1935) On general properties of dependent statistical variables. Bull Math Soc Roumaine Sci 37:37–82

    Google Scholar 

  48. Moran PAP (1958) Random processes in genetics. In: Proceedings of the Cambridge Philosophical Society, vol 54, p 60

  49. Nathanson C, Tarnita C, Nowak M (2009) Calculating evolutionary dynamics in structured populations. PLoS Comp Biol 5:e1000615

    Article  MathSciNet  Google Scholar 

  50. Nowak MA (2006a) Evolutionary dynamics. Harvard University Press, Cambridge

    MATH  Google Scholar 

  51. Nowak MA (2006b) Five rules for the evolution of cooperation. Science 314:1560–1563

    Article  Google Scholar 

  52. Nowak MA, May RM (1992) Evolutionary games and spatial chaos. Nature 359:826–829

    Article  Google Scholar 

  53. Nowak MA, Sasaki A, Taylor C, Fudenberg D (2004) Emergence of cooperation and evolutionary stability in finite populations. Nature 428:646–650

    Article  Google Scholar 

  54. Nowak MA, Tarnita CE, Antal T (2010a) Evolutionary dynamics in structured populations. Philos Trans R Soc B Biol Sci 365:19

    Article  Google Scholar 

  55. Nowak MA, Tarnita CE, Wilson EO (2010b) The evolution of eusociality. Nature 466:1057–1062

    Article  Google Scholar 

  56. Ohtsuki H, Hauert C, Lieberman E, Nowak MA (2006) A simple rule for the evolution of cooperation on graphs and social networks. Nature 441:502–505

    Article  Google Scholar 

  57. Pacheco JM, Traulsen A, Nowak MA (2006) Active linking in evolutionary games. J Theor Biol 243: 437–443. doi:10.1016/j.jtbi.2006.06.027

    Google Scholar 

  58. Pacheco JM, Traulsen A, Nowak MA (2006) Coevolution of strategy and structure in complex networks with dynamical linking. Phys Rev Lett 97:258103

    Article  Google Scholar 

  59. Perc M, Szolnoki A (2010) Coevolutionary games-a mini review. BioSystems 99:109–125

    Article  Google Scholar 

  60. Price GR (1970) Selection and covariance. Nature 227:520–521

    Article  Google Scholar 

  61. Price GR (1972) Extension of covariance selection mathematics. Ann Human Genet 35:485–490

    Article  Google Scholar 

  62. Queller D (1992) A general model for kin selection. Evolution 376–380

  63. Queller DC (2011) Expanded social fitness and Hamilton’s rule for kin, kith, and kind. Proc Natl Acad Sci 108:10792–10799. doi:10.1073/pnas.1100298108

    Article  Google Scholar 

  64. Rand DG, Arbesman S, Christakis NA (2011) Dynamic social networks promote cooperation in experiments with humans. Proc Natl Acad Sci 108:19193–19198. doi:10.1073/pnas.1108243108

    Article  Google Scholar 

  65. Rice S (2008) A stochastic version of the price equation reveals the interplay of deterministic and stochastic processes in evolution. BMC Evol Biol 8:262. doi:10.1186/1471-2148-8-262

    Article  Google Scholar 

  66. Rice SH, Papadopoulos A (2009) Evolution with stochastic fitness and stochastic migration. PLoS ONE 4:e7130. doi:10.1371/journal.pone.0007130

    Article  Google Scholar 

  67. Roca CP, Cuesta JA, Sánchez A (2009) Effect of spatial structure on the evolution of cooperation. Phys Rev E 80:046106. doi:10.1103/PhysRevE.80.046106

    Article  Google Scholar 

  68. Rousset F, Ronce O (2004) Inclusive fitness for traits affecting metapopulation demography. Theor Popul Biol 65:127–141. doi:10.1016/j.tpb.2003.09.003

    Article  MATH  Google Scholar 

  69. Santos FC, Pacheco JM (2005) Scale-free networks provide a unifying framework for the emergence of cooperation. Phys Rev Lett 95:98104

    Article  Google Scholar 

  70. Santos FC, Santos MD, Pacheco JM (2008) Social diversity promotes the emergence of cooperation in public goods games. Nature 454:213–216

    Article  Google Scholar 

  71. Shakarian P, Roos P, Johnson A (2012) A review of evolutionary graph theory with applications to game theory. Biosystems 107:66–80. doi:10.1016/j.biosystems.2011.09.006

    Article  Google Scholar 

  72. Simon B (2008) A stochastic model of evolutionary dynamics with deterministic large-population asymptotics. J Theor Biol 254:719–730

    Google Scholar 

  73. Sonin I (1999) The state reduction and related algorithms and their applications to the study of markov chains, graph theory, and the optimal stopping problem. Adv Math 145:159–188. doi:10.1006/aima.1998.1813

  74. Sood V, Antal T, Redner S (2008) Voter models on heterogeneous networks. Phys Rev E 77:41121

    Article  MathSciNet  Google Scholar 

  75. Sood V, Redner S (2005) Voter model on heterogeneous graphs. Phys Rev Lett 94:178701

    Article  Google Scholar 

  76. Szabó G, Fáth G (2007) Evolutionary games on graphs. Phys Rep 446:97–216

    Article  MathSciNet  Google Scholar 

  77. Szolnoki A, Perc M, Szabó G (2008) Diversity of reproduction rate supports cooperation in the prisoner’s dilemma game on complex networks. Eur Phys J B Condens Matter Complex Syst 61:505–509

    Article  MATH  Google Scholar 

  78. Tarnita CE, Antal T, Ohtsuki H, Nowak MA (2009) Evolutionary dynamics in set structured populations. Proc Natl Acad Sci 106:8601

    Article  Google Scholar 

  79. Tarnita CE, Ohtsuki H, Antal T, Fu F, Nowak MA (2009) Strategy selection in structured populations. J Theor Biol 259:570–581. doi:10.1016/j.jtbi.2009.03.035

    Google Scholar 

  80. Tarnita CE, Wage N, Nowak MA (2011) Multiple strategies in structured populations. Proc Natl Acad Sci 108:2334–2337. doi:10.1073/pnas.1016008108

    Article  Google Scholar 

  81. Taylor C, Fudenberg D, Sasaki A, Nowak MA (2004) Evolutionary game dynamics in finite populations. Bull Math Biol 66:1621–1644. doi:10.1016/j.bulm.2004.03.004

    Article  MathSciNet  Google Scholar 

  82. Taylor P, Lillicrap T, Cownden D (2011) Inclusive fitness analysis on mathematical groups. Evolution 65:849–859. doi:10.1111/j.1558-5646.2010.01162.x

    Article  Google Scholar 

  83. Taylor PD, Day T, Wild G (2007a) Evolution of cooperation in a finite homogeneous graph. Nature 447: 469–472

    Google Scholar 

  84. Taylor PD, Day T, Wild G (2007b) From inclusive fitness to fixation probability in homogeneous structured populations. J Theor Biol 249:101–110

    Article  MathSciNet  Google Scholar 

  85. Taylor PD, Jonker LB (1978) Evolutionary stable strategies and game dynamics. Math Biosci 40:145–156

    Article  MATH  MathSciNet  Google Scholar 

  86. Traulsen A, Pacheco JM, Nowak MA (2007) Pairwise comparison and selection temperature in evolutionary game dynamics. J Theor Biol 246:522–529. doi:10.1016/j.jtbi.2007.01.002

    Article  MathSciNet  Google Scholar 

  87. van Baalen M, Rand DA (1998) The unit of selection in viscous populations and the evolution of altruism. J Theor Biol 193:631–648

    Article  Google Scholar 

  88. van Veelen M (2005) On the use of the Price equation. J Theor Biol 237:412–426

    Article  Google Scholar 

  89. van Veelen M, García J, Sabelis MW, Egas M (2012) Group selection and inclusive fitness are not equivalent; the price equation vs. models and statistics. J Theor Biol 299:64–80. doi:10.1016/j.jtbi.2011.07.025

    Article  Google Scholar 

  90. Wakeley J (2009) Coalescent Theory: an introduction. Roberts& Co, Greenwood Village

    Google Scholar 

  91. Weibull JW (1997) Evolutionary game theory. MIT Press, Cambridge

    Google Scholar 

  92. Woess W (2009) Denumerable Markov chains: generating functions, boundary theory, random walks on trees. European Mathematical Society, Zürich

    Book  Google Scholar 

  93. Wu B, Zhou D, Fu F, Luo Q, Wang L, Traulsen A, Sporns O (2010) Evolution of cooperation on stochastic dynamical networks. PLoS One 5:1560–1563

    Google Scholar 

  94. Zhou D, Qian H (2011) Fixation, transient landscape, and diffusion dilemma in stochastic evolutionary game dynamics. Phys Rev E 84:031907. doi:10.1103/PhysRevE.84.031907

    Article  Google Scholar 

  95. Zhou D, Wu B, Ge H (2010) Evolutionary stability and quasi-stationary strategy in stochastic evolutionary game dynamics. J Theor Biol 264:874–881

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors thank Martin A. Nowak, our editor Mats Gyllenberg, and an anonymous referee for helpful conversations and suggestions. B.A. is supported by the Foundational Questions in Evolutionary Biology initiative of the John Templeton Foundation.

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Correspondence to Benjamin Allen.

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Allen, B., Tarnita, C.E. Measures of success in a class of evolutionary models with fixed population size and structure. J. Math. Biol. 68, 109–143 (2014). https://doi.org/10.1007/s00285-012-0622-x

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Keywords

  • Evolution
  • Stochastic
  • Axioms
  • Fixation probability
  • Evolutionary success
  • Price equation

Mathematics Subject Classification (2000)

  • 92D15
  • Problems related to evolution