Periodic Solutions of Parabolic Systems with Time Delays

Existence of maximal and minimal periodic solutions of a coupled system of parabolic equations with time delays and with nonlinear boundary conditions is discussed. The proof of the existence theorem is based on the method of upper and lower solutions and its associated monotone iterations. This method is constructive and can be used to develop a computational algorithm for numerical solutions of the periodic-parabolic system. An application is given to a competitor-competitormutualist model which consists of a coupled system of three reaction-diffusion equations with time delays.