Permanence and extinction of a three-species ratio-dependent food chain model with delay and prey diffusion

In this paper, we study a diffusive three-species ratio-dependent food chain model, using differential inequality, to obtain sufficient conditions that ensure the permanence of the system and the extinction of predator species. Our results reinforce the main result of Sun Wen, Shihua Chen and Huihai Mei Positive periodic solution of a more realistic three-species Lotka-Volterra model with delay and density regulation, Chaos, Solitons and Fractals, in press].