Diffusion and the Predator-Prey Interaction

We consider the system of partial differential equations which describe the predator-prey interaction, where diffusion and spatial dependence are taken into account. The equations reduce to the Kolmogorov equations, when spatial dependence is neglected. We find conditions under which solutions are uniformly bounded, and we show that all solutions asymptotically enter a fixed region in phase space. We also consider the question of stability of those constant states which correspond to the extinction of one or both species.