  This paper studies birth and death processes in interactive random
environments where the birth and death rates and the dynamics of the state of
the environment are dependent on each other. Two models of a random environment
are considered: a continuous-time Markov chain (finite or countably infinite)
and a reflected (jump) diffusion process. The background is determined by a
joint Markov process carrying a specific interactive mechanism, with an
explicit invariant measure whose structure is similar to a product form. We
discuss a number of queueing and population-growth models and establish
conditions under which the above-mentioned invariant measure can be derived.
  Next, an analysis of the rate of convergence to stationarity is performed for
the models under consideration. We consider two settings leading to either an
exponential or a polynomial convergence rate. In both cases we assume that the
underlying environmental Markov process has an exponential rate of convergence,
but the convergence rate of the joint Markov process is determined by certain
conditions on the birth and death rates. To prove these results a coupling
method turns out to be useful.
