Long Term Behaviour of Two Interacting Birth-and-Death Processes

M.V. Menshikov, V.V. Scherbakov

2018, v.24, №1, 85-106

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 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.

Keywords: birth-and-death process, Lyapunov function method, recurrence, transience, stochastic population model

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