Linear competition processes and generalized Pólya urns with removals

Serguei Popov, Vadim Shcherbakov and Stanislav Volkov

Stochastic Processes and their Applications, 2022, vol. 144, issue C, 125-152

Abstract: 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.

Keywords: Birth-and-death process; Competition process; Branching process; Generalized Pólya urn with removals; Martingale (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1016/j.spa.2021.11.001

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