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Journal of Mathematical Biology

, Volume 75, Issue 6–7, pp 1735–1774 | Cite as

On the stochastic evolution of finite populations

Article

Abstract

This work is a systematic study of discrete Markov chains that are used to describe the evolution of a two-types population. Motivated by results valid for the well-known Moran (M) and Wright–Fisher (WF) processes, we define a general class of Markov chains models which we term the Kimura class. It comprises the majority of the models used in population genetics, and we show that many well-known results valid for M and WF processes are still valid in this class. In all Kimura processes, a mutant gene will either fixate or become extinct, and we present a necessary and sufficient condition for such processes to have the probability of fixation strictly increasing in the initial frequency of mutants. This condition implies that there are WF processes with decreasing fixation probability—in contradistinction to M processes which always have strictly increasing fixation probability. As a by-product, we show that an increasing fixation probability defines uniquely an M or WF process which realises it, and that any fixation probability with no state having trivial fixation can be realised by at least some WF process. These results are extended to a subclass of processes that are suitable for describing time-inhomogeneous dynamics. We also discuss the traditional identification of frequency dependent fitnesses and pay-offs, extensively used in evolutionary game theory, the role of weak selection when the population is finite, and the relations between jumps in evolutionary processes and frequency dependent fitnesses.

Keywords

Stochastic processes Population genetics Fixation probabilities Perron–Frobenius property Time-inhomogeneous Markov chains Stochastically ordered processes 

Mathematics Subject Classification

92D15 92D25 15B51 60J10 

Notes

Acknowledgements

FACCC was partially supported by FCT/Portugal Strategic Project UID/MAT/00297/2013 (Centro de Matemática e Aplicações, Universidade Nova de Lisboa) and by a “Investigador FCT” grant. FACCC is also indebted to Alexandre Baraviera (Universidade Federal do Rio Grande do Sul, Brazil) and Charles Johnson (College of William and Mary, USA) for useful discussions in preliminary ideas of this work. MOS was partially supported by CNPq under grants # 308113/2012-8, # 486395/2013-8 and # 309079/2015-2. MOS also thanks the hospitality of the Universidade Nova de Lisboa and the partial support under grant UID/MAT/00297/2013. MOS further thanks preliminary discussions of some the ideas in this work with the working group in evolutionary game theory at Universidade Federal Fluminense. Both authors also thank useful comments from Henry Laurie (Cape Town University), Alan Hastings (University of California at Davis), the handling editor, and an anonymous referee which helped to improve the original manuscript.

References

  1. Altrock PM, Traulsen A (2009) Fixation times in evolutionary games under weak selection. New J Phys 11(1):013012CrossRefGoogle Scholar
  2. Antal T, Scheuring I (2006) Fixation of strategies for an evolutionary game in finite populations. Bull Math Biol 68(8):1923–1944MathSciNetCrossRefMATHGoogle Scholar
  3. Archetti M, Scheuring I (2012) Review: game theory of public goods in one-shot social dilemmas without assortment. J Theor Biol 299:9–20 Evolution of CooperationMathSciNetCrossRefMATHGoogle Scholar
  4. Arrow KJ (1989) A “dynamic” proof of the Frobenius–Perron theorem for Metzler matrices. Probability, statistics, and mathematics, Pap. in Honor of Samuel Karlin, 17–26 (1989)Google Scholar
  5. Ashcroft P, Altrock PM, Galla T (2014) Fixation in finite populations evolving in fluctuating environments. J R Soc Interface 11(100):20140663CrossRefGoogle Scholar
  6. Atkinson QD, Meade A, Venditti C, Greenhill SJ, Pagel M (2008) Languages evolve in punctuational bursts. Science 319(5863):588CrossRefGoogle Scholar
  7. Barbosa VC, Donangelo R, Souza SR (2010) Early appraisal of the fixation probability in directed networks. Phys Rev E 82:046114CrossRefGoogle Scholar
  8. Berg C (1990) Positive definite and related functions on semigroups. In: The analytical and topological theory of semigroups, Conf., Oberwolfach/Ger. 1989, De Gruyter Expo. Math. 1, 253–278 (1990)Google Scholar
  9. Berman A, Plemmons RJ (1979) Nonnegative matrices in the Mathematical Sciences, Classics in Applied Mathematics, 9. Academic Press, New York, NYMATHGoogle Scholar
  10. Bru R, Elsner L, Neumann M (1994) Convergence of infinite products of matrices and inner–outer iteration schemes. Electron Trans Numer Anal 2(3):183–193MathSciNetMATHGoogle Scholar
  11. Bürger R (2000) The mathematical theory of selection, recombination and mutation. Wiley, ChichesterMATHGoogle Scholar
  12. Cannings C (1974) The latent roots of certain Markov chains arising in genetics: a new approach. I. Haploid models. Adv Appl Probab 6(2):260–290MathSciNetCrossRefMATHGoogle Scholar
  13. Cannings C (1975) The latent roots of certain Markov chains arising in genetics: a new approach, II. Further haploid models. Adv Appl Probab 7(2):264–282MathSciNetCrossRefMATHGoogle Scholar
  14. Carja O, Liberman U, Feldman MW (2014) Evolution in changing environments: modifiers of mutation, recombination, and migration. Proc Nat Acad Sci USA 111(50):17935–17940CrossRefGoogle Scholar
  15. Chalub FACC, Souza MO (2009) From discrete to continuous evolution models: a unifying approach to drift-diffusion and replicator dynamics. Theor Popul Biol 76(4):268–277CrossRefGoogle Scholar
  16. Chalub FACC, Souza MO (2014) The frequency-dependent Wright–Fisher model: diffusive and non-diffusive approximations. J Math Biol 68(5):1089–1133MathSciNetCrossRefMATHGoogle Scholar
  17. Chalub FACC, Souza MO (2016) Fixation in large populations: a continuous view of a discrete problem. J Math Biol 72(1–2):283–330MathSciNetCrossRefMATHGoogle Scholar
  18. Charlesworth B, Charlesworth D (2010) Elements of evolutionary genetics. Roberts and Company Publishers, Greenhood Village, ColoradoMATHGoogle Scholar
  19. Charlesworth B, Lande R, Slatkin M (1982) A neo-darwinian commentary on macroevolution. Evolution 36(3):474–498CrossRefGoogle Scholar
  20. Cotterman CW (1940) A calculus for statistico-genetics. PhD thesis, The Ohio State UniversityGoogle Scholar
  21. Crow JF (2001) Shannon’s brief foray into genetics. Genetics 159(3):915–917MathSciNetGoogle Scholar
  22. Crow JF, Kimura M (1970) An introduction to population genetics theory. Harper International Edition, New YorkMATHGoogle Scholar
  23. Cvijović I, Good BH, Jerison ER, Desai MM (2015) Fate of a mutation in a fluctuating environment. Proc Nat Acad Sci USA 112(36):E5021–E5028CrossRefGoogle Scholar
  24. Daubechies I, Lagarias JC (1992) Sets of matrices all infinite products of which converge. Linear Algebra Appl 161:227–263MathSciNetCrossRefMATHGoogle Scholar
  25. Der R, Epstein C, Plotkin JB (2012) Dynamics of neutral and selected alleles when the offspring distribution is skewed. Genetics 191(4):1331–1344CrossRefGoogle Scholar
  26. Der R, Epstein CL, Plotkin JB (2011) Generalized population models and the nature of genetic drift. Theor Popul Biol 80(2):80–99CrossRefMATHGoogle Scholar
  27. Eldon B, Wakeley J (2006) Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172(4):2621–2633CrossRefGoogle Scholar
  28. Erwin DH (2000) Macroevolution is more than repeated rounds of microevolution. Evol Dev 2(2):78–84CrossRefGoogle Scholar
  29. Estep D (2002) Practical analysis in one variable. undergraduate texts in mathematics. Springer, New YorkMATHGoogle Scholar
  30. Ethier SN, Kurtz TG (1986) Markov processes. Wiley series in probability and mathematical statistics: probability and mathematical statistics. characterization and convergence. Wiley, New YorkGoogle Scholar
  31. Ewens WJ (2004) Mathematical population genetics. I: theoretical introduction, 2nd edn. Interdisciplinary mathematics 27. Springer, New YorkCrossRefMATHGoogle Scholar
  32. Felsenstein J (1976) The theoretical population genetics of variable selection and migration. Annu Rev Genet 10(1):253–280CrossRefGoogle Scholar
  33. Fisher RA (1922) On the dominance ratio. Proc R Soc Edinburgh 42:321–341CrossRefGoogle Scholar
  34. Fisher RA (1930) The distribution of gene ratios for rare mutations. Proc R Soc Edinburgh 50:214–219MATHGoogle Scholar
  35. Fontdevila A (2011) The dynamic genome: a Darwinian approach. Oxford University Press, OxfordCrossRefGoogle Scholar
  36. Frazzetta TH (2012) Flatfishes, turtles, and bolyerine snakes: evolution by small steps or large, or both? Evol Biol 39(1):30–60CrossRefGoogle Scholar
  37. Fudenberg D, Imhof LA (2012) Phenotype switching and mutations in random environments. Bull Math Biol 74(2):399–421MathSciNetCrossRefMATHGoogle Scholar
  38. Fudenberg D, Nowak MA, Taylor C, Imhof LA (2006) Evolutionary game dynamics in finite populations with strong selection and weak mutation. Theor Popul Biol 70(3):352–363CrossRefMATHGoogle Scholar
  39. Gillespie JH (1972) The effects of stochastic environments on allele frequencies in natural populations. Theor Popul Biol 3(3):241–248CrossRefMATHGoogle Scholar
  40. Gillespie JH (1973) Natural selection with varying selection coefficients—a haploid model. Genet Res 21(2):115–120CrossRefGoogle Scholar
  41. Gillespie JH (1991) The causes of molecular evolution. Oxford University Press, OxfordGoogle Scholar
  42. Gokhale CS, Traulsen A (2010) Evolutionary games in the multiverse. Proc Nat Acad Sci USA 107(12):5500–5504MathSciNetCrossRefGoogle Scholar
  43. Grinstead C, Snell J (1997) Introduction to probability. American Mathematical Society, ProvidenceMATHGoogle Scholar
  44. Gzyl H, Palacios JL (2003) On the approximation properties of Bernstein polynomials via probabilistic tools. Boletín de la Asociación Matemática Venezolana 10(1):5–13MathSciNetMATHGoogle Scholar
  45. Haldane JBS, Jayakar SD (1963) Polymorphism due to selection of varying direction. J Genet 58:237CrossRefGoogle Scholar
  46. Hamilton WD (1970) Selfish and spiteful behaviour in an evolutionary model. Nature 228(5277):1218CrossRefGoogle Scholar
  47. Harmer GP, Abbott D, Taylor PG, Parrondo JMR (2000) Parrondo’s paradoxical games and the discrete Brownian ratchet. In: Abbott, D and Kish, LB (ed) Unsolved problems of noise and fluctuations, volume 511 of AIP Conference Proceedings, pp 189–200. 2nd international conference on unsolved problems of noise and fluctuations (UPoN 99), Adelaide, Australia, 12–15 Jul 1999Google Scholar
  48. Hartle DL, Clark AG (2007) Principles of population genetics. Sinauer, MassachussetsGoogle Scholar
  49. Hennion H (1997) Limit theorems for products of positive random matrices. Ann Probab 25(4):1545–1587MathSciNetCrossRefMATHGoogle Scholar
  50. Hilbe C (2011) Local replicator dynamics: a simple link between deterministic and stochastic models of evolutionary game theory. Bull Math Biol 73(9):2068–2087MathSciNetCrossRefMATHGoogle Scholar
  51. Imhof LA, Nowak MA (2006) Evolutionary game dynamics in a Wright–Fisher process. J Math Biol 52(5):667–681MathSciNetCrossRefMATHGoogle Scholar
  52. Johnson CR, Tarazaga P (2004) On matrices with Perron–Frobenius properties and some negative entries. Positivity 8(4):327–338MathSciNetCrossRefMATHGoogle Scholar
  53. Karlin S, Lieberman U (1974) Random temporal variation in selection intensities: case of large population size. Theor Popul Biol 6(3):355–382MathSciNetCrossRefMATHGoogle Scholar
  54. Karlin S, Levikson B (1974) Temporal fluctuations in selection intensities: case of small population size. Theor Popul Biol 6(3):383–412MathSciNetCrossRefMATHGoogle Scholar
  55. Karlin S, Taylor TM (1975) A first course in stochastic processes, 2nd edn. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-LondonMATHGoogle Scholar
  56. Karlin S, Taylor HM (1981) A second course in stochastic processes, 2nd edn. Academic Press, New York-LondonMATHGoogle Scholar
  57. Keilson J, Kester A (1977) Monotone matrices and monotone Markov processes. Stoch Proc Appl 5(3):231–241MathSciNetCrossRefMATHGoogle Scholar
  58. Kimura M (1954) Process leading to quasi-fixation of genes in natural populations due to random fluctuation of selection intensities. Genetics 39(3):1943–2631Google Scholar
  59. Kimura M (1962) On the probability of fixation of mutant genes in a population. Genetics 47:713–719Google Scholar
  60. Kimura M (1983) The neutral theory of molecular evolution. University Press, CambridgeCrossRefGoogle Scholar
  61. Kimura M, Ohta T (1969) Average number of generations until extinction of an individual mutant gene in a finite population. Genetics 63(3):701Google Scholar
  62. Klenke A, Mattner L (2010) Stochastic ordering of classical discrete distributions. Adv Appl Probab 42(2):392–410MathSciNetCrossRefMATHGoogle Scholar
  63. Lewin M (1971) On nonnegative matrices. Pac J Math 36(3):753–759CrossRefMATHGoogle Scholar
  64. Lorenzi T, Chisholm RH, Desvillettes L, Hughes BD (2015) Dissecting the dynamics of epigenetic changes in phenotype-structured populations exposed to fluctuating environments. J Theor Biol 386:166–176MathSciNetCrossRefMATHGoogle Scholar
  65. Mathew S, Perreault C (2015) Behavioural variation in 172 small-scale societies indicates that social learning is the main mode of human adaptation. P Roy Soc B-Biol Sci 282:20150061CrossRefGoogle Scholar
  66. Maynard Smith J (1998) Evolutionary genetics. Oxford University Press, OxfordGoogle Scholar
  67. McCandlish DM, Epstein CL, Plotkin JB (2015) Formal properties of the probability of fixation: identities, inequalities and approximations. Theor Popul Biol 99:98–113CrossRefMATHGoogle Scholar
  68. Melbinger A, Vergassola M (2015) The impact of environmental fluctuations on evolutionary fitness functions. Sci Rep 5:15211CrossRefGoogle Scholar
  69. Moran PAP (1962) The statistical process of evolutionary theory. Clarendon Press, OxfordMATHGoogle Scholar
  70. Nåsell I (2011) Extinction and quasi-stationarity in the stochastic logistic sis model. Springer, Berlin, HeidelbergCrossRefGoogle Scholar
  71. Nassar RF, Cook RD (1974) Ultimate probability of fixation and time to fixation or loss of a gene under a variable fitness model. Theor Appl Genet 44(6):247–254CrossRefGoogle Scholar
  72. Noutsos D (2006) On Perron–Frobenius property of matrices having some negative entries. Linear Algebra Appl 412(2–3):132–153MathSciNetCrossRefMATHGoogle Scholar
  73. Nowak MA (2006) Evolutionary dynamics: exploring the equations of life. The Belknap Press of Harvard University Press, Cambridge, MAMATHGoogle Scholar
  74. Nowak MA, Sasaki A, Taylor C, Fudenberg D (2004) Emergence of cooperation and evolutionary stability in finite populations. Nature 428(6983):646–650CrossRefGoogle Scholar
  75. Orr HA (2009) Fitness and its role in evolutionary genetics. Nat Rev Genet 10(8):531–539CrossRefGoogle Scholar
  76. Osipovitch DC, Barratt C, Schwartz PM (2009) Systems chemistry and Parrondo’s paradox: computational models of thermal cycling. New J Chem 33(10):2022–2027CrossRefGoogle Scholar
  77. Pagel M, Venditti C, Meade A (2006) Large punctuational contribution of speciation to evolutionary divergence at the molecular level. Science 314(5796):119–121CrossRefGoogle Scholar
  78. Parrondo JMR, Harmer GP, Abbott D (2000) New paradoxical games based on Brownian ratchets. Phys Rev Lett 85(24):5226–5229CrossRefGoogle Scholar
  79. Peacock-López E (2011) Seasonality as a parrondian game. Phys Lett A 375(35):3124–3129MathSciNetCrossRefMATHGoogle Scholar
  80. Phillips GM (2003) Interpolation and approximation by polynomials. CMS Books in Mathematics. Springer, New YorkCrossRefMATHGoogle Scholar
  81. Proulx SR, Adler FR (2010) The standard of neutrality: still flapping in the breeze? J Evol Biol 23(7):1339–1350CrossRefGoogle Scholar
  82. Reed FA (2007) Two-locus epistasis with sexually antagonistic selection: a genetic parrondo’s paradox. Genetics 176(3):1923–1929CrossRefGoogle Scholar
  83. Ressel P (1987) Integral representations on convex semigroups. Math Scand 61:93–111MathSciNetCrossRefMATHGoogle Scholar
  84. Schuster P (2011) The Mathematics of Darwin’s theory of evolution: 1859 and 150 years later. In: Chalub FACC, Rodrigues JF (ed) Mathematics Of Darwin’S legacy, mathematics and biosciences in interaction, pp 27–66. Conference on mathematics of Darwin’s legacy, Univ Lisbon, Lisbon, PORTUGAL, 23–24 Nov 2009Google Scholar
  85. Shannon CE (1940) An algebra for theoretical genetics. PhD thesis, Massachussets Institute of Technology, Cambridge, MA. Ph.D. thesis in MathematicsGoogle Scholar
  86. Tan S, Lü L, Yu X, Hill D (2012) Monotonicity of fixation probability of evolutionary dynamics on complex networks. In: IECON 2012-38th annual conference on IEEE industrial electronics society, pp 2337–2341. IEEEGoogle Scholar
  87. Tarazaga P, Raydan M, Hurman A (2001) Perron–Frobenius theorem for matrices with some negative entries. Linear Algebra Appl 328(1–3):57–68MathSciNetCrossRefMATHGoogle Scholar
  88. Taylor HM, Karlin S (1998) An introduction to stochastic modeling, 3rd edn. Academic Press Inc., San DiegoMATHGoogle Scholar
  89. Traulsen A, Pacheco JM, Imhof LA (2006) Stochasticity and evolutionary stability. Phys Rev E 74:021905MathSciNetCrossRefGoogle Scholar
  90. Traulsen A, Pacheco JM, Nowak MA (2007) Pairwise comparison and selection temperature in evolutionary game dynamics. J Theor Biol 246(3):522–529MathSciNetCrossRefGoogle Scholar
  91. Uecker H, Hermisson J (2011) On the fixation process of a beneficial mutation in a variable environment. Genetics 188(4):915–930CrossRefGoogle Scholar
  92. Waxman D, Welch J (2005) Fisher’s microscope and Haldane’s ellipse. Am Nat 166(4):447–457CrossRefGoogle Scholar
  93. Williams PD, Hastings A (2013) Stochastic dispersal and population persistence in marine organisms. Am Nat 182(2):271–282 PMID: 23852360CrossRefGoogle Scholar
  94. Wright S (1931) Evolution in mendelian populations. Genetics 16(2):97–159Google Scholar
  95. Wright S (1937) The distribution of gene frequencies in populations. Proc Nat Acad Sci USA 23:307–320CrossRefMATHGoogle Scholar
  96. Wright S (1938) The distribution of gene frequencies under irreversible mutations. Proc Nat Acad Sci USA 24:253–259CrossRefMATHGoogle Scholar
  97. Yakushkina T, Saakian DB, Bratus A, Hu C-K (2015) Evolutionary games with randomly changing payoff matrices. J Phys Soc Jpn 84(6):064802CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Departamento de Matemática and Centro de Matemática e Aplicações, Faculdade de Ciências e TecnologiaUniversidade Nova de LisboaCaparicaPortugal
  2. 2.Departamento de Matemática AplicadaUniversidade Federal FluminenseNiteróiBrasil

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