This paper deals with the degenerate parabolic system u t = u Δ u + u ( a 1 − b 1 u + c 1 v ) and v t = v Δ v + v ( a 2 − b 2 v + c 2 u ) with homogeneous Dirichlet conditions in a bounded domain. We show that any positive solutions converge exponentially to the unique steady state if the coefficients satisfy certain conditions. The result is relevant to models of the spatial growth of interacting species populations in a continuous, bounded domain.
