Boundary effect in competition processes

TL;DR

This paper analyzes the long-term behavior of a two-component Markov chain with competitive interactions, revealing that one component tends to infinity while the other oscillates between small values, with implications for models like Lotka-Volterra and urn processes.

Contribution

It characterizes the asymptotic behavior of a class of competitive Markov chains, including special cases like Lotka-Volterra and urn models, showing a universal boundary effect.

Findings

One component tends to infinity almost surely.

The other component oscillates between 0 and 1.

Results apply to models like Lotka-Volterra and Friedman's urn.

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).