Periodic Solution And Permanence For A Non-autonomous Predator-Prey Diffusive System And The Periodic Solution On Time Scales

The persistence and existence of periodic solutions of predator-prey systems are an important direction of mathematical ecology research. It has been an important topic to give a definite criterion of the properties of a complicated ecological system. Many ecologist and mathematician have paid much attention in this field. The study of the quality property of the solution for the non-autonomous predator-prey systems has a wide application.The theory of time scales can unify continuous and discrete cases, which pioneers a new mathematical area. This theory unifies differential and difference equations, reveals the essence of continuous and discrete cases, avoids repeat study. Recently, the research of predator-prey systems on time scales has been more and more popular.In this paper, we study a non-autonomous predator-prey diffusion system with delay and ratio-dependent. In the second part, we will prove the persistence of the system and find the conditions for the existence of periodic solutions by using the Gaines and Mawhin's continuation theorem of coincidence degree theory. In the third part, we introduce the time scale, which unifies continuous and discrete system, and establish the existence result of positive periodic solution for the model when the time scale T is R or Z.