Linear competition processes and generalised Pólya urns with removals

Serguei Popov, Vadim Shcherbakov, Stanislav Volkov

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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 Polya urn model with removals.
Original languageEnglish
Article numberSPA3886
Pages (from-to)125-152
Number of pages28
JournalStochastic Processes and Their Applications
Early online date11 Nov 2021
Publication statusPublished - Feb 2022

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