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

TL;DR

This paper analyzes the long-term behavior of two interacting birth-and-death processes modeled as a Markov chain, providing conditions for transience and recurrence, and describing their asymptotic trajectories in detail.

Contribution

It introduces new conditions for transience and recurrence of interacting birth-and-death processes and characterizes their asymptotic behavior in special transient cases.

Findings

Conditions for transience and recurrence are established.

In some cases, the processes escape to infinity in an unusual manner.

Trajectories in transient cases are described with high precision.

Abstract

In this paper we study the long term evolution of a continuous time Markov chain formed by two interacting birth-and-death processes. The interaction between the processes is modelled by transition rates which are functions with suitable monotonicity properties. This is in line with the approach proposed by Gauss G.F. and Kolmogorov A.N. for modelling interaction between species in ecology. We obtain conditions for transience/recurrence of the Markov chain and describe in detail its asymptotic behaviour in special transient cases. In particular, we find that in some of these cases the Markov chain escapes to infinity in an unusual way, and the corresponding trajectories can be rather precisely described.