Long term behaviour of interacting birth-and-death processes

A single birth-and-death process on the set of non-negative integers is a classical probabilistic model for the size of a population. This is a continuous time Markov chain (CTMC) which evolves as follows. When the process is at state k, it can jump either to state k+1 (interpreted as birth event), or to state k-1, if k>0 (interpreted as death event), with transition rates that are state-dependent.

The project is devoted to the long term behaviour of a class of Markov processes that can be interpreted as a system of birth-and-death processes, whose components evolve subject to a certain interaction (interacting birth-and-death processes). Originally interacting birth-and-death processes were motivated by modelling competition between populations. In this case they are known as competition processes, which is a class of population probabilistic models. Another interesting case of interacting birth-and-death processes is a growth process motivated by physical phenomenon known as cooperative sequential adsorption (CSA). In CSA diffusing particles can get adsorbed by a material surface, when they hit it. The main peculiarity of CSA is that the adsorbed particles can change the adsorption properties of the material in a sense that they either attract, or repulse other particles. The growth process is
a system of pure birth processes whose components evolve subject to an interaction which is similar to that of CSA. In other words, the components of a growth process can either accelerate, or slow down the growth of each other. In the discrete time setting a growth process can be regarded as an interacting urn model. The latter is a class of random processes with reinforcement closely related to the generalised Polya urn model (another classical probabilistic model).