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The Coalescent in Peripatric Metapopulations

Part of: Special processes Markov processes Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Amaury Lambert  and
Chunhua Ma
Amaury Lambert*
Affiliation:
Université Pierre et Marie Curie and Collège de France
Chunhua Ma*
Affiliation:
Université Pierre et Marie Curie, Collège de France and Nankai University
*
∗ Postal address: Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, CNRS UMR 7599, Paris, France. Email address: amaury.lambert@upmc.fr
∗∗ Postal address: School of Mathematical Sciences and LPMC, Nankai University, Tianjin, P. R. China. Email address: mach@nankai.edu.cn
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Abstract

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We consider a dynamic metapopulation involving one large population of size N surrounded by colonies of size εNN, usually called peripheral isolates in ecology, where N → ∞ and εN → 0 in such a way that εNN → ∞. The main population, as well as the colonies, independently send propagules to found new colonies (emigration), and each colony independently, eventually merges with the main population (fusion). Our aim is to study the genealogical history of a finite number of lineages sampled at stationarity in such a metapopulation. We make assumptions on model parameters ensuring that the total outer population has size of the order of N and that each colony has a lifetime of the same order. We prove that under these assumptions, the scaling limit of the genealogical process of a finite sample is a censored coalescent where each lineage can be in one of two states: an inner lineage (belonging to the main population) or an outer lineage (belonging to some peripheral isolate). Lineages change state at constant rate and (only) inner lineages coalesce at constant rate per pair. This two-state censored coalescent is also shown to converge weakly, as the landscape dynamics accelerate, to a time-changed Kingman coalescent.

Keywords

Censored coalescentmetapopulationweak convergenceperipheral isolatepopulation geneticsperipatric speciation

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60G10: Stationary processes 92D10: Genetics 92D15: Problems related to evolution
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 2 , June 2015 , pp. 538 - 557
Copyright
© Applied Probability Trust 

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