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The sequential loss of allelic diversity

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  01 February 2019

Guillaume Achaz ,
Amaury Lambert  and
Emmanuel Schertzer
Guillaume Achaz*
Affiliation:
Sorbonne Université and Collège de France
Amaury Lambert*
Affiliation:
Sorbonne Université and Collège de France
Emmanuel Schertzer*
Affiliation:
Sorbonne Université and Collège de France
*
Institut de Systématique, Évolution, Biodiversité (ISYEB), Muséum National d′Histoire Naturelle (MNHN), Sorbonne Université, 75005 Paris, France.
Laboratoire de Probabilités, Statistique et Modélisation (LPSM), Sorbonne Université, Case courrier 158, 4 Place Jussieu, 75252 Paris, France. Email address: amaury.lambert@upmc.fr
Laboratoire de Probabilités, Statistique et Modélisation (LPSM), Sorbonne Université, Case courrier 158, 4 Place Jussieu, 75252 Paris, France.
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Abstract

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In this paper we give a new flavour to what Peter Jagers and his co-authors call `the path to extinction'. In a neutral population of constant size N, assume that each individual at time 0 carries a distinct type, or allele. Consider the joint dynamics of these N alleles, for example the dynamics of their respective frequencies and more plainly the nonincreasing process counting the number of alleles remaining by time t. Call this process the extinction process. We show that in the Moran model, the extinction process is distributed as the process counting (in backward time) the number of common ancestors to the whole population, also known as the block counting process of the N-Kingman coalescent. Stimulated by this result, we investigate whether it extends (i) to an identity between the frequencies of blocks in the Kingman coalescent and the frequencies of alleles in the extinction process, both evaluated at jump times, and (ii) to the general case of Λ-Fleming‒Viot processes.

Keywords

CoalescentΛ‐Fleming‒Viotlook‐downextinctionpopulation geneticsurn modelintertwining

MSC classification

Primary: 92D10: Genetics
Secondary: 60G09: Exchangeability 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue A: Branching and Applied Probability , December 2018 , pp. 13 - 29
Copyright
Copyright © Applied Probability Trust 2018 

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