  In this paper, we study a subclass of piecewise-deterministic Markov
processes with a Polish state space, involving deterministic motion punctuated
by random jumps that occur at exponentially distributed time intervals. Over
each of these intervals, the process follows a flow, selected randomly among a
finite set of all possible ones. Our main goal is to provide a set of
verifiable conditions guaranteeing the exponential ergodicity for such
processes (in terms of the bounded Lipschitz distance), which would refer only
to properties of the flows and the transition law of the Markov chain given by
the post-jump locations. Moreover, we establish a simple criterion on the
exponential ergodicity for a particular instance of these processes, applicable
to certain biological models, where the jumps result from the action of an
iterated function system with place-dependent probabilities.
