Uniform Persistence, Global Attractivity And Existence Of Positive Periodic Solutions Of A Two-species Ratio-dependent Predator-prey Diffusion Model With Variable Time Delays

In this paper, a two-species ratio-dependent predator-prey diffusion model with variable time delays is investigated. By means of differential inequalities, using coincidence degree theory, constructing some suitable Lyapunov functionals and utilizing some analysis techniques, we obtain the uniform persistence, global attractivity and existence of positive periodic solutions of the system.The organization of this paper is as follows: In Chapter 1, we introduce the background of the system. In Chapter 2, By constructing some suitable Lyapunov functionals and utilizing some analysis techniques, we establish some sufficient criteria for uniform persistence of the system, which are not relevant to the diffusion coefficients. Our results generalize and improve the results of [7-8]. In Chapter 3, by means of a set of Lyapunov functionals, a set of verifiable sufficient conditions are derived for the global attractivity of the positive solutions of the system. In Chapter 4, by using coincidence degree theory and constructing a homotopy operator, we obtain some sufficient conditions for the existence of positive periodic solutions of the system, which are not relevant to the diffusion coefficients.