Stability and attractivity of periodic solutions of parabolic systems with time delays

This paper is concerned with the existence, stability, and global attractivity of time-periodic solutions for a class of coupled parabolic equations in a bounded domain. The problem under consideration includes coupled system of parabolic and ordinary differential equations, and time delays may appear in the nonlinear reaction functions. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement using Schauder fixed point theorem, while the stability and attractivity analysis is for quasimonotone nondecreasing and mixed quasimonotone reaction functions using the monotone iterative scheme. The results for the general system are applied to the standard parabolic equations without time delay and to the corresponding ordinary differential system. Applications are also given to three Lotka-Volterra reaction diffusion model problems, and in each problem a sufficient condition on the reaction rates is obtained to ensure the stability and global attractivity of positive periodic solutions.