A convergence theorem for a two-species competition system with slow diffusion

This article concerns the effect of slow diffusion in two-species competition-diffusion problem with spatially homogeneous nearly identical reaction terms. In this case all (nonnegative) equilibria are spatially homogeneous, and the set of nontrivial equilibria is the graph of a C1-curve. This article shows convergence of positive solutions to an equilibria which is determined by the initial data. The proof relies on the existence of a Lyapunov function and is adapted from [6] which dealt with linear diffusion.