Convergence in almost periodic competition diffusion systems

The paper deals with the convergence of positive solutions for almost-perio dic competition diffusion systems. Tlc asymptotic almost periodicity of a p ositive solution for such a system is described by the almost periodicity o f the omega -limit set of the corresponding positive motion in the associat ed skew-product flow. In the framework of the skew-product flow, it will be proved that the omega -limit set of any spatially homogeneous positive mot ion contains at most two minimal sets which are both almost automorphic. It will also be proved that if each spatially homogeneous positive solution i s asymptotically almost periodic and each spatially homogeneous positive al most periodic solution is lower (upper) asymptotically Lyapunov stable, the n every positive solution converges to a spatially homogeneous almost perio dic solution. Several important special cases are described where every pos itive solution converges to a spatially homogeneous almost-periodic solutio n. (C) 2001 Academic Press.