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Multitype Bienaymé–Galton–Watson processes escaping extinction

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Serik Sagitov  and
Maria Conceição Serra
Serik Sagitov*
Affiliation:
Chalmers University of Technology and Göteborg University
Maria Conceição Serra*
Affiliation:
Chalmers University of Technology and Göteborg University
*
∗ Postal address: School of Mathematical Sciences, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden. Email address: serik@math.chalmers.se
∗∗ Current address: Department of Mathematics, University of Minho, Campus de Gualtar, 4710-057 Braga, Portugal.
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Abstract

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In the framework of a multitype Bienaymé–Galton–Watson (BGW) process, the event that the daughter's type differs from the mother's type can be viewed as a mutation event. Assuming that mutations are rare, we study a situation where all types except one produce on average less than one offspring. We establish a neat asymptotic structure for the BGW process escaping extinction due to a sequence of mutations toward the supercritical type. Our asymptotic analysis is performed by letting mutation probabilities tend to 0. The limit process, conditional on escaping extinction, is another BGW process with an enriched set of types, allowing us to delineate a stem lineage of particles that leads toward the escape event. The stem lineage can be described by a simple Markov chain on the set of particle types. The total time to escape becomes a sum of a random number of independent, geometrically distributed times spent at intermediate types.

Keywords

Bienaymé–Galton–Watson processdecomposableescape from extinctionmultitypewild-type branching process

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 92D25: Population dynamics (general)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 1 , March 2009 , pp. 225 - 246
Copyright
Copyright © Applied Probability Trust 2009 

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