Existence and Stability of Periodic Quasisolutions in Nonlinear Parabolic Systems with Discrete Delays1

The existence of periodic quasisolutions and the dynamics for a couple system of semilinear parabolic equations with discrete time delays and periodic coefficients are investigated using the method of upper and lower solutions. It is shown that if the reaction function in the system possesses a mixed quasimonotone property and the periodic boundary value system has a pair of coupled T-upper and lower solutions then the periodic boundary value problem has a pair of periodic quasisolutions and the sector between the quasisolutions is an attractor of the delayed periodic parabolic system. Under some additional conditions the periodic quasisolutions are exactly true periodic solutions of the periodic boundary value system. Finally, three models arising from ecology are given to illustrate the obtained results.