Convergence to steady state for a degenerate parabolic system of relevance to marine ecology

This paper deals with the degenerate parabolic system ut=uΔu+u(a1−b1u+c1v)ut=uΔu+u(a1−b1u+c1v) and vt=vΔv+v(a2−b2v+c2u)vt=vΔv+v(a2−b2v+c2u) 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.