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Kingman’s model with random mutation probabilities: convergence and condensation I

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  25 February 2022

Linglong Yuan
Linglong Yuan*
Affiliation:
University of Liverpool and Xi’an Jiaotong-Liverpool University
*
*Postal address: University of Liverpool, Department of Mathematical Sciences, Peach Street, L69 7ZL, Liverpool, UK. Email address: yuanlinglongcn@gmail.com
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Abstract

For a one-locus haploid infinite population with discrete generations, the celebrated model of Kingman describes the evolution of fitness distributions under the competition of selection and mutation, with a constant mutation probability. This paper generalises Kingman’s model by using independent and identically distributed random mutation probabilities, to reflect the influence of a random environment. The weak convergence of fitness distributions to the globally stable equilibrium is proved. Condensation occurs when almost surely a positive proportion of the population travels to and condenses at the largest fitness value. Condensation may occur when selection is favoured over mutation. A criterion for the occurrence of condensation is given.

Keywords

Population dynamicsmutation–selection balancehouse of cardsfitness distributionsize-biased distributiondistributional equation

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G10: Stationary processes 92D15: Problems related to evolution 92D25: Population dynamics (general)
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 1 , March 2022 , pp. 311 - 335
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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