  In this work, we consider a system of differential equations modeling the
dynamics of some populations of preys and predators, moving in space according
to rapidly oscillating time-dependent transport terms, and interacting with
each other through a Lotka-Volterra term. These two contributions naturally
induce two separated time-scales in the problem. A generalized center manifold
theorem is derived to handle the situation where the linear terms are depending
on the fast time in a periodic way. The resulting equations are then amenable
to averaging methods. As a product of these combined techniques, one obtains an
autonomous differential system in reduced dimension whose dynamics can be
analyzed in a much simpler way as compared to original equations. Strikingly
enough, this system is of Lotka-Volterra form with modified coefficients.
Besides, a higher order perturbation analysis allows to show that the
oscillations on the original model destabilize the cycles of the averaged
Volterra system in a way that can be explicitely computed.
