Permanence and Positive Periodic Solutions of n-Species Competition Reaction]Diffusion Systems with Spatial Inhomogeneity

This paper is mainly devoted to the study of permanence and existence of positive periodic solutions of n-species competition reaction]diffusion systems with spatial inhomogeneity, which are defined on a bounded domain and subjected to Neumann boundary conditions. We first discuss in detail the single-species periodic reaction]diffusion equations. Then by using comparison and maximum principle arguments, we get the sufficient conditions in terms of the principal eigenvalue of periodic-parabolic problem for the n-species competition reaction]diffusion systems to be permanent. We further show by using the Schauder fixed point theorem that under the same condition the system admits at least one positive periodic solution. The special case of Lotka]Volterra systems with diffusion and periodic coefficients is also discussed. Q 1996 Academic Press, Inc.