Diffusion-mediated persistence in two-species competition Lotka-Volterra model

We consider a system composed of two Lotka-Volterra patches connected by diffusion. Each patch has two competitors. Conditions for persistence of the system are given. It is proved that the system can be made persistent under appropriate diffusion coefficients ensuring the instability of boundary equilibria, even if each species is not persistent within each patch. The choice of the coefficients depends closely on the patch dynamics without diffusion.