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

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.