  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.
