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Moran models and Wright–Fisher diffusions with selection and mutation in a one-sided random environment

Part of: Markov processes Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  09 March 2023

Fernando Cordero Open the ORCID record for Fernando Cordero [Opens in a new window]  and
Grégoire Véchambre Open the ORCID record for Grégoire Véchambre [Opens in a new window]
Fernando Cordero*
Affiliation:
Bielefeld University
Grégoire Véchambre*
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences
*
*Postal address: Faculty of Technology, Bielefeld University, Box 100131, 33501 Bielefeld, Germany. Email address: fcordero@techfak.uni-bielefeld.de
**Postal address: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55, Zhongguancun East Road, Haidian District, Beijing, China. Email address: vechambre@amss.ac.cn
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Abstract

Consider a two-type Moran population of size N with selection and mutation, where the selective advantage of the fit individuals is amplified at extreme environmental conditions. Assume selection and mutation are weak with respect to N, and extreme environmental conditions rarely occur. We show that, as $N\to\infty$, the type frequency process with time sped up by N converges to the solution to a Wright–Fisher-type SDE with a jump term modeling the effect of the environment. We use an extension of the ancestral selection graph (ASG) to describe the genealogical picture of the model. Next, we show that the type frequency process and the line-counting process of a pruned version of the ASG satisfy a moment duality. This relation yields a characterization of the asymptotic type distribution. We characterize the ancestral type distribution using an alternative pruning of the ASG. Most of our results are stated in annealed and quenched form.

Keywords

Wright–Fisher diffusionMoran modelrandom environmentancestral selection graphduality

MSC classification

Primary: 82C22: Interacting particle systems 92D15: Problems related to evolution
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J27: Continuous-time Markov processes on discrete state spaces
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 3 , September 2023 , pp. 701 - 767
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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