Global stability of nonhomogeneous equilibrium solution for the diffusive Lotka-Volterra competition model

A diffusive Lotka-Volterra competition model is considered and the combined effect of spatial dispersal and spatial variations of resource on the population persistence and exclusion is studied. A new Lyapunov functional method and a new integral inequality are developed to prove the global stability of non-constant equilibrium solutions in heterogeneous environment. The general result is applied to show that in a two-species system in which the diffusion coefficients, resource functions and competition rates are all spatially heterogeneous, the positive equilibrium solution is globally asymptotically stable when it exists, and it can also be applied to the system with arbitrary number of species under the assumption of spatially heterogeneous resource distribution, for which the monotone dynamical system theory is not applicable.