Abstract :
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.
