Invariant measures for contact processes with state dependent birth and death rates

In this paper, we consider contact processes on locally compact separable metric spaces with birth and death rates heterogeneous in space. Conditions on the rates that ensure the existence of invariant measures of contact processes are formulated. One of the crucial condition is the so-called critical regime condition. To prove the existence of invariant measures we used our approach proposed in \cite{PZh}. We discuss in details the multi-species contact model with a compact space of marks (species) in which both birth and death rates depend on the marks.