  We investigate the large population dynamics of a family of stochastic
particle systems with three-state cyclic individual behaviour and
parameter-dependent transition rates. On short time scales, the dynamics turns
out to be approximated by an integrable Hamiltonian system whose phase space is
foliated by periodic trajectories. This feature suggests to consider the
effective dynamics of the long-term process that results from averaging over
the rapid oscillations. We establish the convergence of this process in the
large population limit to the solutions of an explicit stochastic differential
equation. Remarkably, this averaging phenomenon is complemented by the
convergence of stationary measures. The proof of averaging follows the
Stroock-Varadhan approach to martingale problems and relies on a fine analysis
of the system's dynamical features.
