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Boundary effect in compretition processes

Shcherbakov, Vadim and Volkov, Stanislav LU orcid (2019) In Journal of Applied Probability
Abstract
This paper studies the long-term behaviour of a continuous time Markov chain formed by two non-negative integer valued components that evolve subject to a competitive interaction. In the absence of interaction the Markov chain is just a pair of independent linear birth processes with immigration. Interactions of interest include, as a special case, the famous Lotka-Volterra interaction. The Markov chain with another special case of interaction can be interpreted as an urn model with ball removals and is reminiscent, in a sense, of Friedman's urn model. We show that, with probability one, eventually one of the components of the process tends to infinity, while the other component oscillates between values 0 and 1 (between values 0 and 2 in... (More)
This paper studies the long-term behaviour of a continuous time Markov chain formed by two non-negative integer valued components that evolve subject to a competitive interaction. In the absence of interaction the Markov chain is just a pair of independent linear birth processes with immigration. Interactions of interest include, as a special case, the famous Lotka-Volterra interaction. The Markov chain with another special case of interaction can be interpreted as an urn model with ball removals and is reminiscent, in a sense, of Friedman's urn model. We show that, with probability one, eventually one of the components of the process tends to infinity, while the other component oscillates between values 0 and 1 (between values 0 and 2 in a special case).
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Contribution to journal
publication status
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in
Journal of Applied Probability
publisher
Applied Probability Trust
ISSN
1475-6072
language
English
LU publication?
yes
id
93cc5174-3319-468a-a3a9-158ba13c4033
date added to LUP
2019-05-07 09:13:58
date last changed
2021-04-16 10:29:35
@article{93cc5174-3319-468a-a3a9-158ba13c4033,
  abstract     = {{This paper studies the long-term behaviour of a continuous time Markov chain formed by two non-negative integer valued components that evolve subject to a competitive interaction. In the absence of interaction the Markov chain is just a pair of independent linear birth processes with immigration. Interactions of interest include, as a special case, the famous Lotka-Volterra interaction. The Markov chain with another special case of interaction can be interpreted as an urn model with ball removals and is reminiscent, in a sense, of Friedman's urn model. We show that, with probability one, eventually one of the components of the process tends to infinity, while the other component oscillates between values 0 and 1 (between values 0 and 2 in a special case).<br/>}},
  author       = {{Shcherbakov, Vadim and Volkov, Stanislav}},
  issn         = {{1475-6072}},
  language     = {{eng}},
  publisher    = {{Applied Probability Trust}},
  series       = {{Journal of Applied Probability}},
  title        = {{Boundary effect in compretition processes}},
  year         = {{2019}},
}