Harmless Delays for Permanence in a Class of

This paper studies a class of time-delay reaction]diffusion systems modeling the dynamics of single or interacting populations. In the logistic equation, we prove that when the magnitude of the instantaneous term is larger than that of the delay terms, the population growth u has the same asymptotic limit as in the case of no delay. For the predator]prey model, a condition on the interaction rates is given to ensure the permanence effect in the ecosystem regardless of the length of delay intervals. A permanence condition is also obtained in the N-species competition system with time delays. It is shown that when the natural growth rate a1, a2, . . . , aN.is in an unbounded parameter set L, the reaction]diffusion system has a positive global attractor. Finally, long-term behavior of the solutions for those time-delay systems is numerically demonstrated through finite-difference approximations and compared with the corresponding systems without delays.