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

In this paper we study the long term evolution of a continuous time Markov chain formed by two interacting birth-and-death processes and motivated by modelling interaction between populations. We show transience/recurrence of the Markov chain under fairly general assumptions on transition rates and describe in more detail its asymptotic behaviour in some transient cases. 1 The model and results A birth-and-death process on Z+ = {0, 1, 2, . . .} is a continuous time Markov chain that evolves as follows. Given a current state k it jumps either to k+1, or to k−1 (if k > 0) at certain state dependent rates. The long term behaviour of such Markov chains is well known. Namely, given a set of transition rates one can, in principle, determine whether the corresponding birth-and-death process is recurrent/positive recurrent, or transient/explosive (e.g., see [9], [11] and [13]). In this paper we study the long term behaviour of a continuous time Markov chain formed by two interacting birth-and-death processes. A birth-and-death process on Z+ is a classic probabilistic model for the size of a population, therefore the Markov chain can be interpreted in terms of two interacting populations. In recent years a range of probabilistic models motivated by modelling evolution of interacting populations has been proposed (e.g., see [1], [2], [3], [5] , [7], [12] and references therein). These models differ in both technical frameworks and research methods. Our model is motivated by modelling competition and collaboration between populations (discussed later) and can be thought of as a non-homogeneous random walk in the quarter plane. Non-homogeneous random walks appear in modelling population evolution in biology (e.g. [4] and [15]), as in many other applications, and are of interest in their own right (e.g. see [14] and references therein). Department of Mathematical Sciences, Durham University, UK. Email address: [email protected] Department of Mathematics, Royal Holloway, University of London, UK. Email address: [email protected]