GLOBAL ATTRACTORS OF REACTION-DIFFUSION SYSTEMS MODELING FOOD CHAIN POPULATIONS WITH DELAYS

In this paper, we study a reaction-diffusion system modeling the population dynamics of a four-species food chain with time delays. Under Dirichlet and Neumann boundary conditions, we discuss the existence of a positive global attractor which demonstrates the presence of a positive steady state and the permanence effect in the ecological system. Sufficient conditions on the interaction rates are given to ensure the persistence of all species in the food chain. For the case of Neumann boundary condition, we further obtain the uniqueness of a positive steady state, and in such case the density functions converge uniformly to a constant solution. Numerical simulations of the food-chain models are also given to demonstrate and compare the asymptotic behavior of the time-dependent density functions.