Convergence in competition models with small diffusion coefficients (Q1772319)

It is well known that for reaction-diffusion \(2\)-species Lotka-Volterra competition models with spatially independent reaction terms, global stability of an equilibrium for the reaction system implies global stability for the reaction-diffusion system. But this is not in general true for spatially inhomogeneous system. The authors consider this problem and show that for small enough diffusion coefficients, global convergence to an equilibrium holds for spatially inhomogeneous reaction-diffusion system, if for each point in space the reaction system has a globally attracting hyperbolic equilibrium. Their results are initial for understanding the connection between the asymptotic of reaction-diffusion systems with small diffusion coefficients and that of the corresponding reaction systems.