Dynamics in 30species preadtor-prey models with time delays

We study a differential equation system with diffusion and time delays which models the dynamics of predator-prey interactions within three biological species. Our main focus is on the persistence (non-extinction) of u-species which is at the bottom of the nutrient hierarchy, and the permanence effect (long-term survival of all the predators and prey) in this model. When u-species persists in the absence of its predators, we generate a condition on the interaction rates to ensure that it does not go extinction under the predation of the v- and w-species. With certain additional conditions, we can further obtain the permanence effect (long-term survival of all three species) in the ecological system. Our proven results also explicitly present the effects of all the environmental data (growth rates and interaction rates) on the ultimate bounds of the three biological species. Numerical simulations of the model are also given to demonstrate the pattern of dynamics (extinction, persistence, and permanence)in the ecological model.