  We study the stochastic dynamics of a system of interacting species in a
stochastic environment by means of a continuous-time Markov chain with
transition rates depending on the state of the environment. Models of gene
regulation in systems biology take this form. We characterise the finite-time
distribution of the Markov chain, provide conditions for ergodicity, and
characterise the stationary distribution (when it exists) as a mixture of
Poisson distributions. The mixture measure is uniquely identified as the law of
a fixed point of a stochastic recurrence equation. This recursion is crucial
for statistical computation of moments and other distributional features.
