  We are interested in prey-predator communities where the predator population
evolves much faster than the prey's (e.g. insect-tree communities). We
introduce a piecewise deterministic model for these prey-predator communities
that arises as a limit of a microscopic model when the number of predators goes
to infinity. We prove that the process has a unique invariant probability
measure and that it is exponentially ergodic. Further on, we rescale the
predator dynamics in order to model predators of smaller size. This slow-fast
system converges to a community process in which the prey dynamics is averaged
on the predator equilibria. This averaged process admits an invariant
probability measure which can be computed explicitly. We use numerical
simulations to study the convergence of the invariant probability measures of
the rescaled processes.
