Long term behaviour of locally interacting birth-and-death processes

TL;DR

This paper analyzes the long-term dynamics of a finite system of locally interacting birth-and-death processes on graphs, providing detailed asymptotic behavior for specific graph structures, with applications to population interactions and related models.

Contribution

It offers a comprehensive analysis of the asymptotic behavior of interacting birth-and-death processes on graphs, extending understanding of such systems in population modeling and related areas.

Findings

Asymptotic behavior characterized for constant degree graphs.

Detailed analysis for star graph configurations.

Connections established with particle systems and urn models.

Abstract

In this paper we study long-term evolution of a finite system of locally interacting birth-and-death processes labelled by vertices of a finite connected graph. A detailed description of the asymptotic behaviour is obtained in the case of both constant vertex degree graphs and star graphs. The model is motivated by modelling interactions between populations and is related to interacting particle systems, Gibbs models with unbounded spins, as well as urn models with interaction.