A note on scaling limits for truncated birth-and-death processes with interaction

In this note we consider a Markov chain formed by a finite system of truncated birth-death processes with local interaction. We study an asymptotic behaviour of the Markov chain as the range of possible values of its components becomes large. In particular, we show that appropriately scaled Markov chain converges to a diffusion process, and derive conditions for existence of the stationary distribution of the limit diffusion process in some special cases.