Journal of Mathematical Biology

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

Survival of a recessive allele in a Mendelian diploid model

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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 \(\alpha >0\).

Keywords

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

Mathematics Subject Classification

60K35 92D25 60J85 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

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

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