Skip to main content
SpringerLink
Log in

On the stochastic evolution of finite populations

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  • Altrock PM, Traulsen A (2009) Fixation times in evolutionary games under weak selection. New J Phys 11(1):013012

    Article  Google Scholar 

  • Antal T, Scheuring I (2006) Fixation of strategies for an evolutionary game in finite populations. Bull Math Biol 68(8):1923–1944

    Article  MathSciNet  MATH  Google Scholar 

  • 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 Cooperation

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

  • Ashcroft P, Altrock PM, Galla T (2014) Fixation in finite populations evolving in fluctuating environments. J R Soc Interface 11(100):20140663

    Article  Google Scholar 

  • Atkinson QD, Meade A, Venditti C, Greenhill SJ, Pagel M (2008) Languages evolve in punctuational bursts. Science 319(5863):588

    Article  Google Scholar 

  • Barbosa VC, Donangelo R, Souza SR (2010) Early appraisal of the fixation probability in directed networks. Phys Rev E 82:046114

    Article  Google Scholar 

  • 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)

  • Berman A, Plemmons RJ (1979) Nonnegative matrices in the Mathematical Sciences, Classics in Applied Mathematics, 9. Academic Press, New York, NY

    MATH  Google Scholar 

  • Bru R, Elsner L, Neumann M (1994) Convergence of infinite products of matrices and inner–outer iteration schemes. Electron Trans Numer Anal 2(3):183–193

    MathSciNet  MATH  Google Scholar 

  • Bürger R (2000) The mathematical theory of selection, recombination and mutation. Wiley, Chichester

    MATH  Google Scholar 

  • Cannings C (1974) The latent roots of certain Markov chains arising in genetics: a new approach. I. Haploid models. Adv Appl Probab 6(2):260–290

    Article  MathSciNet  MATH  Google Scholar 

  • 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):264–282

    Article  MathSciNet  MATH  Google Scholar 

  • Carja O, Liberman U, Feldman MW (2014) Evolution in changing environments: modifiers of mutation, recombination, and migration. Proc Nat Acad Sci USA 111(50):17935–17940

    Article  Google Scholar 

  • 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):268–277

    Article  Google Scholar 

  • Chalub FACC, Souza MO (2014) The frequency-dependent Wright–Fisher model: diffusive and non-diffusive approximations. J Math Biol 68(5):1089–1133

    Article  MathSciNet  MATH  Google Scholar 

  • Chalub FACC, Souza MO (2016) Fixation in large populations: a continuous view of a discrete problem. J Math Biol 72(1–2):283–330

    Article  MathSciNet  MATH  Google Scholar 

  • Charlesworth B, Charlesworth D (2010) Elements of evolutionary genetics. Roberts and Company Publishers, Greenhood Village, Colorado

    MATH  Google Scholar 

  • Charlesworth B, Lande R, Slatkin M (1982) A neo-darwinian commentary on macroevolution. Evolution 36(3):474–498

    Article  Google Scholar 

  • Cotterman CW (1940) A calculus for statistico-genetics. PhD thesis, The Ohio State University

  • Crow JF (2001) Shannon’s brief foray into genetics. Genetics 159(3):915–917

    MathSciNet  Google Scholar 

  • Crow JF, Kimura M (1970) An introduction to population genetics theory. Harper International Edition, New York

    MATH  Google Scholar 

  • 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–E5028

    Article  Google Scholar 

  • Daubechies I, Lagarias JC (1992) Sets of matrices all infinite products of which converge. Linear Algebra Appl 161:227–263

    Article  MathSciNet  MATH  Google Scholar 

  • Der R, Epstein C, Plotkin JB (2012) Dynamics of neutral and selected alleles when the offspring distribution is skewed. Genetics 191(4):1331–1344

    Article  Google Scholar 

  • Der R, Epstein CL, Plotkin JB (2011) Generalized population models and the nature of genetic drift. Theor Popul Biol 80(2):80–99

    Article  MATH  Google Scholar 

  • Eldon B, Wakeley J (2006) Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172(4):2621–2633

    Article  Google Scholar 

  • Erwin DH (2000) Macroevolution is more than repeated rounds of microevolution. Evol Dev 2(2):78–84

    Article  Google Scholar 

  • Estep D (2002) Practical analysis in one variable. undergraduate texts in mathematics. Springer, New York

    MATH  Google Scholar 

  • Ethier SN, Kurtz TG (1986) Markov processes. Wiley series in probability and mathematical statistics: probability and mathematical statistics. characterization and convergence. Wiley, New York

    Google Scholar 

  • Ewens WJ (2004) Mathematical population genetics. I: theoretical introduction, 2nd edn. Interdisciplinary mathematics 27. Springer, New York

    Book  MATH  Google Scholar 

  • Felsenstein J (1976) The theoretical population genetics of variable selection and migration. Annu Rev Genet 10(1):253–280

    Article  Google Scholar 

  • Fisher RA (1922) On the dominance ratio. Proc R Soc Edinburgh 42:321–341

    Article  Google Scholar 

  • Fisher RA (1930) The distribution of gene ratios for rare mutations. Proc R Soc Edinburgh 50:214–219

    MATH  Google Scholar 

  • Fontdevila A (2011) The dynamic genome: a Darwinian approach. Oxford University Press, Oxford

    Book  Google Scholar 

  • Frazzetta TH (2012) Flatfishes, turtles, and bolyerine snakes: evolution by small steps or large, or both? Evol Biol 39(1):30–60

    Article  Google Scholar 

  • Fudenberg D, Imhof LA (2012) Phenotype switching and mutations in random environments. Bull Math Biol 74(2):399–421

    Article  MathSciNet  MATH  Google Scholar 

  • 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):352–363

    Article  MATH  Google Scholar 

  • Gillespie JH (1972) The effects of stochastic environments on allele frequencies in natural populations. Theor Popul Biol 3(3):241–248

    Article  MATH  Google Scholar 

  • Gillespie JH (1973) Natural selection with varying selection coefficients—a haploid model. Genet Res 21(2):115–120

    Article  Google Scholar 

  • Gillespie JH (1991) The causes of molecular evolution. Oxford University Press, Oxford

    Google Scholar 

  • Gokhale CS, Traulsen A (2010) Evolutionary games in the multiverse. Proc Nat Acad Sci USA 107(12):5500–5504

    Article  MathSciNet  Google Scholar 

  • Grinstead C, Snell J (1997) Introduction to probability. American Mathematical Society, Providence

    MATH  Google Scholar 

  • Gzyl H, Palacios JL (2003) On the approximation properties of Bernstein polynomials via probabilistic tools. Boletín de la Asociación Matemática Venezolana 10(1):5–13

    MathSciNet  MATH  Google Scholar 

  • Haldane JBS, Jayakar SD (1963) Polymorphism due to selection of varying direction. J Genet 58:237

    Article  Google Scholar 

  • Hamilton WD (1970) Selfish and spiteful behaviour in an evolutionary model. Nature 228(5277):1218

    Article  Google Scholar 

  • 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 1999

  • Hartle DL, Clark AG (2007) Principles of population genetics. Sinauer, Massachussets

    Google Scholar 

  • Hennion H (1997) Limit theorems for products of positive random matrices. Ann Probab 25(4):1545–1587

    Article  MathSciNet  MATH  Google Scholar 

  • Hilbe C (2011) Local replicator dynamics: a simple link between deterministic and stochastic models of evolutionary game theory. Bull Math Biol 73(9):2068–2087

    Article  MathSciNet  MATH  Google Scholar 

  • Imhof LA, Nowak MA (2006) Evolutionary game dynamics in a Wright–Fisher process. J Math Biol 52(5):667–681

    Article  MathSciNet  MATH  Google Scholar 

  • Johnson CR, Tarazaga P (2004) On matrices with Perron–Frobenius properties and some negative entries. Positivity 8(4):327–338

    Article  MathSciNet  MATH  Google Scholar 

  • Karlin S, Lieberman U (1974) Random temporal variation in selection intensities: case of large population size. Theor Popul Biol 6(3):355–382

    Article  MathSciNet  MATH  Google Scholar 

  • Karlin S, Levikson B (1974) Temporal fluctuations in selection intensities: case of small population size. Theor Popul Biol 6(3):383–412

    Article  MathSciNet  MATH  Google Scholar 

  • Karlin S, Taylor TM (1975) A first course in stochastic processes, 2nd edn. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London

    MATH  Google Scholar 

  • Karlin S, Taylor HM (1981) A second course in stochastic processes, 2nd edn. Academic Press, New York-London

    MATH  Google Scholar 

  • Keilson J, Kester A (1977) Monotone matrices and monotone Markov processes. Stoch Proc Appl 5(3):231–241

    Article  MathSciNet  MATH  Google Scholar 

  • Kimura M (1954) Process leading to quasi-fixation of genes in natural populations due to random fluctuation of selection intensities. Genetics 39(3):1943–2631

    Google Scholar 

  • Kimura M (1962) On the probability of fixation of mutant genes in a population. Genetics 47:713–719

    Google Scholar 

  • Kimura M (1983) The neutral theory of molecular evolution. University Press, Cambridge

    Book  Google Scholar 

  • Kimura M, Ohta T (1969) Average number of generations until extinction of an individual mutant gene in a finite population. Genetics 63(3):701

    Google Scholar 

  • Klenke A, Mattner L (2010) Stochastic ordering of classical discrete distributions. Adv Appl Probab 42(2):392–410

    Article  MathSciNet  MATH  Google Scholar 

  • Lewin M (1971) On nonnegative matrices. Pac J Math 36(3):753–759

    Article  MATH  Google Scholar 

  • 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 386:166–176

    Article  MathSciNet  MATH  Google Scholar 

  • 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 282:20150061

    Article  Google Scholar 

  • Maynard Smith J (1998) Evolutionary genetics. Oxford University Press, Oxford

    Google Scholar 

  • McCandlish DM, Epstein CL, Plotkin JB (2015) Formal properties of the probability of fixation: identities, inequalities and approximations. Theor Popul Biol 99:98–113

    Article  MATH  Google Scholar 

  • Melbinger A, Vergassola M (2015) The impact of environmental fluctuations on evolutionary fitness functions. Sci Rep 5:15211

    Article  Google Scholar 

  • Moran PAP (1962) The statistical process of evolutionary theory. Clarendon Press, Oxford

    MATH  Google Scholar 

  • Nåsell I (2011) Extinction and quasi-stationarity in the stochastic logistic sis model. Springer, Berlin, Heidelberg

    Book  Google Scholar 

  • 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):247–254

    Article  Google Scholar 

  • Noutsos D (2006) On Perron–Frobenius property of matrices having some negative entries. Linear Algebra Appl 412(2–3):132–153

    Article  MathSciNet  MATH  Google Scholar 

  • Nowak MA (2006) Evolutionary dynamics: exploring the equations of life. The Belknap Press of Harvard University Press, Cambridge, MA

    MATH  Google Scholar 

  • Nowak MA, Sasaki A, Taylor C, Fudenberg D (2004) Emergence of cooperation and evolutionary stability in finite populations. Nature 428(6983):646–650

    Article  Google Scholar 

  • Orr HA (2009) Fitness and its role in evolutionary genetics. Nat Rev Genet 10(8):531–539

    Article  Google Scholar 

  • Osipovitch DC, Barratt C, Schwartz PM (2009) Systems chemistry and Parrondo’s paradox: computational models of thermal cycling. New J Chem 33(10):2022–2027

    Article  Google Scholar 

  • Pagel M, Venditti C, Meade A (2006) Large punctuational contribution of speciation to evolutionary divergence at the molecular level. Science 314(5796):119–121

    Article  Google Scholar 

  • Parrondo JMR, Harmer GP, Abbott D (2000) New paradoxical games based on Brownian ratchets. Phys Rev Lett 85(24):5226–5229

    Article  Google Scholar 

  • Peacock-López E (2011) Seasonality as a parrondian game. Phys Lett A 375(35):3124–3129

    Article  MathSciNet  MATH  Google Scholar 

  • Phillips GM (2003) Interpolation and approximation by polynomials. CMS Books in Mathematics. Springer, New York

    Book  MATH  Google Scholar 

  • Proulx SR, Adler FR (2010) The standard of neutrality: still flapping in the breeze? J Evol Biol 23(7):1339–1350

    Article  Google Scholar 

  • Reed FA (2007) Two-locus epistasis with sexually antagonistic selection: a genetic parrondo’s paradox. Genetics 176(3):1923–1929

    Article  Google Scholar 

  • Ressel P (1987) Integral representations on convex semigroups. Math Scand 61:93–111

    Article  MathSciNet  MATH  Google Scholar 

  • 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 2009

  • Shannon CE (1940) An algebra for theoretical genetics. PhD thesis, Massachussets Institute of Technology, Cambridge, MA. Ph.D. thesis in Mathematics

  • 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. IEEE

  • Tarazaga P, Raydan M, Hurman A (2001) Perron–Frobenius theorem for matrices with some negative entries. Linear Algebra Appl 328(1–3):57–68

    Article  MathSciNet  MATH  Google Scholar 

  • Taylor HM, Karlin S (1998) An introduction to stochastic modeling, 3rd edn. Academic Press Inc., San Diego

    MATH  Google Scholar 

  • Traulsen A, Pacheco JM, Imhof LA (2006) Stochasticity and evolutionary stability. Phys Rev E 74:021905

    Article  MathSciNet  Google Scholar 

  • Traulsen A, Pacheco JM, Nowak MA (2007) Pairwise comparison and selection temperature in evolutionary game dynamics. J Theor Biol 246(3):522–529

    Article  MathSciNet  Google Scholar 

  • Uecker H, Hermisson J (2011) On the fixation process of a beneficial mutation in a variable environment. Genetics 188(4):915–930

    Article  Google Scholar 

  • Waxman D, Welch J (2005) Fisher’s microscope and Haldane’s ellipse. Am Nat 166(4):447–457

    Article  Google Scholar 

  • Williams PD, Hastings A (2013) Stochastic dispersal and population persistence in marine organisms. Am Nat 182(2):271–282 PMID: 23852360

    Article  Google Scholar 

  • Wright S (1931) Evolution in mendelian populations. Genetics 16(2):97–159

    Google Scholar 

  • Wright S (1937) The distribution of gene frequencies in populations. Proc Nat Acad Sci USA 23:307–320

    Article  MATH  Google Scholar 

  • Wright S (1938) The distribution of gene frequencies under irreversible mutations. Proc Nat Acad Sci USA 24:253–259

    Article  MATH  Google Scholar 

  • Yakushkina T, Saakian DB, Bratus A, Hu C-K (2015) Evolutionary games with randomly changing payoff matrices. J Phys Soc Jpn 84(6):064802

    Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Departamento de Matemática and Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516, Caparica, Portugal

    Fabio A. C. C. Chalub

  2. Departamento de Matemática Aplicada, Universidade Federal Fluminense, R. Mário Santos Braga, s/n, Niterói, RJ, 22240-920, Brasil

    Max O. Souza

Authors
  1. Fabio A. C. C. Chalub
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Max O. Souza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fabio A. C. C. Chalub.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chalub, F.A.C.C., Souza, M.O. On the stochastic evolution of finite populations. J. Math. Biol. 75, 1735–1774 (2017). https://doi.org/10.1007/s00285-017-1135-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-017-1135-4

Keywords

Mathematics Subject Classification

Search

Navigation