Journal of Mathematical Biology

, Volume 75, Issue 1, pp 145–198 | Cite as

Survival of a recessive allele in a Mendelian diploid model

  • Rebecca Neukirch
  • Anton Bovier


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 \(\alpha >0\).


Adaptive dynamics Population genetics Mendelian reproduction Diploid population Nonlinear birth-and-death process Genetic variability 

Mathematics Subject Classification

60K35 92D25 60J85 


  1. Athreya KB, Ney PE (2011) T. E. Harris and branching processes. Ann Probab 39(2):429–434Google Scholar
  2. 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) Google Scholar
  3. 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) Google Scholar
  4. Bovier A (2006) Metastability: a potential theoretic approach. International Congress of Mathematicians, vol III. Eur Math Soc, Zürich, pp 499–518Google Scholar
  5. Bovier A, den Hollander F (2015) Metastability: a potential-theoretic approach, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 351. Springer, ChamGoogle Scholar
  6. Bürger R (2000) The mathematical theory of selection, recombination, and mutation. In: Wiley series in mathematical computational biology. Wiley, ChichesterGoogle Scholar
  7. Champagnat N (2006) A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stoch Process Appl 116(8):1127–1160MathSciNetCrossRefzbMATHGoogle Scholar
  8. Champagnat N, Méléard S (2011) Polymorphic evolution sequence and evolutionary branching. Probab Theory Relat Fields 151(1–2):45–94MathSciNetCrossRefzbMATHGoogle Scholar
  9. Champagnat N, Ferričre R, Ben Arous G (2002) The canonical equation of adaptive dynamics: a mathematical view. Selection 2(1–2):73–83Google Scholar
  10. Champagnat N, Ferrière R, Méléard S (2008) From individual stochastic processes to macroscopic models in adaptive evolution. Stoch Models 24(suppl. 1):2–44MathSciNetCrossRefzbMATHGoogle Scholar
  11. Collet P, Méléard S, Metz JA (2013) A rigorous model study of the adaptive dynamics of mendelian diploids. J Math Biol 67(3):569–607MathSciNetCrossRefzbMATHGoogle Scholar
  12. Coron C (2014) Stochastic modeling of density-dependent diploid populations and the extinction vortex. Adv Appl Probab 46(2):446–477MathSciNetCrossRefzbMATHGoogle Scholar
  13. Coron C (2016) Slow-fast stochastic diffusion dynamics and quasi-stationarity for diploid populations with varying size. J Math Biol 72(1–2):171–202MathSciNetCrossRefzbMATHGoogle Scholar
  14. Coron C, Méléard S, Porcher E, Robert A (2013) Quantifying the mutational meltdown in diploid populations. Am Nat 181(5):623–636CrossRefGoogle Scholar
  15. Crow JF, Kimura M (1970) An introduction to population genetics theory. Harper & Row Publishers, New YorkzbMATHGoogle Scholar
  16. Dieckmann U, Law R (1996) The dynamical theory of coevolution: a derivation from stochastic ecological processes. J Math Biol 34(5–6):579–612MathSciNetCrossRefzbMATHGoogle Scholar
  17. Ewens WJ (2004) Mathematical population genetics. I. In: Interdisciplinary applied mathematics, vol 27, 2nd edn. Springer, New YorkGoogle Scholar
  18. Fisher R (1918) The correlation between relatives on the supposition of Mendelian inheritance. Trans R Soc Edinb 42:399–433Google Scholar
  19. Fournier N, Méléard S (2004) A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann Appl Probab 14(4):1880–1919MathSciNetCrossRefzbMATHGoogle Scholar
  20. Freidlin MI, Wentzell AD (1984) Random perturbations of dynamical systems, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 260. Springer, New YorkzbMATHGoogle Scholar
  21. Golubitsky M, Guillemin V (1973) Stable mappings and their singularities. In: Graduate texts in mathematics, vol 14. Springer, New YorkGoogle Scholar
  22. Haldane J (1924a) A mathematical theory of natural and artificial selection. Part I. Trans Camb Phil Soc 23:19–41Google Scholar
  23. Haldane J (1924b) A mathematical theory of natural and artificial selection. Part II. Trans Camb Phil Soc Biol Sci 1:158–163Google Scholar
  24. Hirsch MW, Pugh CC, Shub M (1977) Invariant manifolds. In: Lecture notes in mathematics, vol 583. Springer, BerlinGoogle Scholar
  25. Hofbauer J, Sigmund K (1990) Adaptive dynamics and evolutionary stability. Appl Math Lett 3(4):75–79MathSciNetCrossRefzbMATHGoogle Scholar
  26. Kisdi É, Geritz SA (1999) Adaptive dynamics in allele space: evolution of genetic polymorphism by small mutations in a heterogeneous environment. Evolution 60:993–1008CrossRefGoogle Scholar
  27. 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):133–141CrossRefGoogle Scholar
  28. Metz J, Nisbet R, Geritz S (1992) How should we define ’fitness’ for general ecological scenarios? Trends Ecol Evol 7(6):198–202CrossRefGoogle Scholar
  29. 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 183–231Google Scholar
  30. Nagylaki T (1992) Introduction to theoretical population genetics. In: Biomathematics, vol 21. Springer, BerlinGoogle Scholar
  31. Perko L (2001) Differential equations and dynamical systems. In: Texts in applied mathematics, vol 7, 3rd edn. Springer, New YorkGoogle Scholar
  32. Rouhani S, Barton N (1987) The probability of peak shifts in a founder population. J Theor Biol 126(1):51–62CrossRefGoogle Scholar
  33. Wright S (1931) Evolution in Mendelian populations. Genetics 16:97–157Google Scholar
  34. 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 140–142Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikRheinische Friedrich-Wilhelms-UniversitätBonnGermany

Personalised recommendations