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 that are given by functions with suitable monotonicity properties. This is in line with the approach proposed by Gause and Kolmogorov 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.
Menshikov, M., & Shcherbakov, V. (2018). Long term behaviour of two interacting birth-and-death processes. Markov processes and related fields, 24(1), 85-106