  A diffusive Lotka-Volterra competition model is considered for the combined
effect of spatial dispersal and spatial variations of resource on the
population persistence and exclusion. First it is shown 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. Secondly the existence and
global asymptotic stability of the positive and semi-trivial equilibrium
solutions are obtained for the model with arbitrary number of species under the
assumption of spatially heterogeneous resource distribution. A new Lyapunov
functional method is developed to prove the global stability of a non-constant
equilibrium solution in heterogeneous environment.
