Linear competition processes and generalized Pólya urns with removals

A competition process is a continuous time Markov chain that can be interpreted as a system of interacting birth-and-death processes, the components of which evolve subject to a competitive interaction. This paper is devoted to the study of the long-term behaviour of such a competition process, wher...

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Bibliographic Details
Published inStochastic processes and their applications Vol. 144; pp. 125 - 152
Main Authors Popov, Serguei, Shcherbakov, Vadim, Volkov, Stanislav
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.02.2022
Subjects
Birth-and-death process
Branching process
Competition process
Generalized Pólya urn with removals
Martingale
Matematik
Mathematics
Natural Sciences
Naturvetenskap
Branching process
Generalized Pólya urn with removals
60K35
Birth-and-death process
Competition process
Martingale
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Summary:A competition process is a continuous time Markov chain that can be interpreted as a system of interacting birth-and-death processes, the components of which evolve subject to a competitive interaction. This paper is devoted to the study of the long-term behaviour of such a competition process, where a component of the process increases with a linear birth rate and decreases with a rate given by a linear function of other components. A zero is an absorbing state for each component, that is, when a component becomes zero, it stays zero forever (and we say that this component becomes extinct). We show that, with probability one, eventually only a random subset of non-interacting components of the process survives. A similar result also holds for the relevant generalized Pólya urn model with removals.
ISSN:0304-4149
1879-209X
DOI:10.1016/j.spa.2021.11.001