Skip to main content
SpringerLink
Log in

Survival of a recessive allele in a Mendelian diploid model

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

Abstract

In this paper we analyse the genetic evolution of a diploid hermaphroditic population, which is modelled by a three-type nonlinear birth-and-death process with competition and Mendelian reproduction. In a recent paper, Collet et al. (J Math Biol 67(3):569–607, 2013) have shown that, on the mutation time-scale, the process converges to the Trait-Substitution Sequence of adaptive dynamics, stepping from one homozygotic state to another with higher fitness. We prove that, under the assumption that a dominant allele is also the fittest one, the recessive allele survives for a time of order at least \(K^{1/4-\alpha }\), where K is the size of the population and \(\alpha >0\).

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Athreya KB, Ney PE (2011) T. E. Harris and branching processes. Ann Probab 39(2):429–434

  • Baar M, Bovier A, Champagnat N (2016) From stochastic individual-based models to the canonical equation of adaptive dynamics—in one step. Ann Appl Probab (online first)

  • Billiard S, Smadi C (2016) The interplay of two mutations in a population of varying size: a stochastic eco-evolutionary model for clonal interference. Stoch Process Appl (in press)

  • Bovier A (2006) Metastability: a potential theoretic approach. International Congress of Mathematicians, vol III. Eur Math Soc, Zürich, pp 499–518

  • Bovier A, den Hollander F (2015) Metastability: a potential-theoretic approach, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 351. Springer, Cham

  • Bürger R (2000) The mathematical theory of selection, recombination, and mutation. In: Wiley series in mathematical computational biology. Wiley, Chichester

  • Champagnat N (2006) A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stoch Process Appl 116(8):1127–1160

    Article  MathSciNet  MATH  Google Scholar 

  • Champagnat N, Méléard S (2011) Polymorphic evolution sequence and evolutionary branching. Probab Theory Relat Fields 151(1–2):45–94

    Article  MathSciNet  MATH  Google Scholar 

  • Champagnat N, Ferričre R, Ben Arous G (2002) The canonical equation of adaptive dynamics: a mathematical view. Selection 2(1–2):73–83

  • Champagnat N, Ferrière R, Méléard S (2008) From individual stochastic processes to macroscopic models in adaptive evolution. Stoch Models 24(suppl. 1):2–44

    Article  MathSciNet  MATH  Google Scholar 

  • Collet P, Méléard S, Metz JA (2013) A rigorous model study of the adaptive dynamics of mendelian diploids. J Math Biol 67(3):569–607

    Article  MathSciNet  MATH  Google Scholar 

  • Coron C (2014) Stochastic modeling of density-dependent diploid populations and the extinction vortex. Adv Appl Probab 46(2):446–477

    Article  MathSciNet  MATH  Google Scholar 

  • Coron C (2016) Slow-fast stochastic diffusion dynamics and quasi-stationarity for diploid populations with varying size. J Math Biol 72(1–2):171–202

    Article  MathSciNet  MATH  Google Scholar 

  • Coron C, Méléard S, Porcher E, Robert A (2013) Quantifying the mutational meltdown in diploid populations. Am Nat 181(5):623–636

    Article  Google Scholar 

  • Crow JF, Kimura M (1970) An introduction to population genetics theory. Harper & Row Publishers, New York

    MATH  Google Scholar 

  • Dieckmann U, Law R (1996) The dynamical theory of coevolution: a derivation from stochastic ecological processes. J Math Biol 34(5–6):579–612

    Article  MathSciNet  MATH  Google Scholar 

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

  • Fisher R (1918) The correlation between relatives on the supposition of Mendelian inheritance. Trans R Soc Edinb 42:399–433

    Google Scholar 

  • Fournier N, Méléard S (2004) A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann Appl Probab 14(4):1880–1919

    Article  MathSciNet  MATH  Google Scholar 

  • Freidlin MI, Wentzell AD (1984) Random perturbations of dynamical systems, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 260. Springer, New York

    MATH  Google Scholar 

  • Golubitsky M, Guillemin V (1973) Stable mappings and their singularities. In: Graduate texts in mathematics, vol 14. Springer, New York

  • Haldane J (1924a) A mathematical theory of natural and artificial selection. Part I. Trans Camb Phil Soc 23:19–41

  • Haldane J (1924b) A mathematical theory of natural and artificial selection. Part II. Trans Camb Phil Soc Biol Sci 1:158–163

  • Hirsch MW, Pugh CC, Shub M (1977) Invariant manifolds. In: Lecture notes in mathematics, vol 583. Springer, Berlin

  • Hofbauer J, Sigmund K (1990) Adaptive dynamics and evolutionary stability. Appl Math Lett 3(4):75–79

    Article  MathSciNet  MATH  Google Scholar 

  • Kisdi É, Geritz SA (1999) Adaptive dynamics in allele space: evolution of genetic polymorphism by small mutations in a heterogeneous environment. Evolution 60:993–1008

    Article  Google Scholar 

  • Marrow P, Law R, Cannings C (1992) The coevolution of predator–prey interactions: Esss and red queen dynamics. Proc R Soc Lond B Biol Sci 250(1328):133–141

    Article  Google Scholar 

  • Metz J, Nisbet R, Geritz S (1992) How should we define ’fitness’ for general ecological scenarios? Trends Ecol Evol 7(6):198–202

    Article  Google Scholar 

  • Metz JAJ, Geritz SAH, Meszéna G, Jacobs FJA, van Heerwaarden JS (1996) Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In: Stochastic and spatial structures of dynamical systems (Amsterdam, 1995), Konink. Nederl. Akad. Wetensch. Verh. Afd. Natuurk. Eerste Reeks, vol 45. North-Holland, Amsterdam, pp 183–231

  • Nagylaki T (1992) Introduction to theoretical population genetics. In: Biomathematics, vol 21. Springer, Berlin

  • Perko L (2001) Differential equations and dynamical systems. In: Texts in applied mathematics, vol 7, 3rd edn. Springer, New York

  • Rouhani S, Barton N (1987) The probability of peak shifts in a founder population. J Theor Biol 126(1):51–62

    Article  Google Scholar 

  • Wright S (1931) Evolution in Mendelian populations. Genetics 16:97–157

    Google Scholar 

  • Yule G (1907) On the theory of inheritance of quantitative compound characters on the basis of Mendel’s laws—a preliminary note. In: Reports of the 3rd international congress on genetics. Spottiswoode, London, pp 140–142

Download references

Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität, Endenicher Allee 60, 53115, Bonn, Germany

    Rebecca Neukirch & Anton Bovier

Authors
  1. Rebecca Neukirch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anton Bovier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rebecca Neukirch.

Additional information

We acknowledge financial support from the German Research Foundation (DFG) through the Hausdorff Center for Mathematics, the Cluster of Excellence ImmunoSensation, and the Priority Programme SPP1590 Probabilistic Structures in Evolution. We thank Loren Coquille for help with the numerical simulations and for fruitful discussions.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Neukirch, R., Bovier, A. Survival of a recessive allele in a Mendelian diploid model. J. Math. Biol. 75, 145–198 (2017). https://doi.org/10.1007/s00285-016-1081-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-016-1081-6

Keywords

Mathematics Subject Classification

Search

Navigation