Spatial birth and death processes

Abstract

We examine birth and death processes where the birth/death rates depend on the particular configuration of the population. The easiest case is when the configurations form the subsets of some finite lattice (thus the state space is finite). In this case we look at the relationships between time-reversibility, nearest neighbour interactions, and the equilibrium state being a Markov random field. A more interesting case is when the entities which can be born or die can do so at any point in some bounded region of space (or the plane). This gives us a pure jump process, whose general properties are well-known, (see for example, Feller (1966) Vol. II Chapter X.3). We examine some particular examples and compute equilibrium distributions.