Spatio-temporal oscillation for a singular predator-prey model

We study an initial boundary value problem for a reaction-diffusion system arising in the study of a singular predator-prey system. Under an assumption on the growth rates, we first prove that the unique co-existence state is a center for the kinetic system. Then we prove that solutions of the diffusion system with equal diffusivity become spatially homogeneous and are subject to the kinetic part asymptotically.