Global periodicity in a class of reaction-diffusion systems with time delays

In this paper we study a class of reaction-diffusion systems modelling the dynamics of "food-limited" populations with, periodic environmental data and time delays. The existence of a global attracting positive periodic solution is first established in the model without time delay. It is further shown that as long as the magnitude of the instantaneous self-limitation effects is larger than that of the time-delay effects, the positive periodic solution is also the global attractor in the time-delay system. Numerical simulations for both cases (with or without time delays) demonstrate the same asymptotic behavior (extinction or converging to the positive T-periodic solution, depending on the growth rate of the species).