Study On Delay Predator-prey Systems With Plaque Diffusion

Population ecology is one of the most widely applied and mature branches in ecology mathematics and system development. The established model and method have a positive role for the development in the population ecology and other areas in biological mathematics. With the continuous development of economic society, the influence about human activities on the natural world is more and more prominent. So the research of human activities on the sustainable development of population ecology and itself has a realistic significance.This paper will mainly consider the prey system with time delay and diffusion and establish a predator-prey system according to the practice of ecological system model, the three kinds of different predator-prey systems with delay and diffusion will be studied..A predator-prey system with Holling II functional response,diffusion delay and three species nonautonomous predator-prey system with shelter, diffusion, delay, Holling III functional response are constructed. The sufficient conditions for the uniform persistence of these system is discussed using the comparison theorem. By constructing suitable Lyapunov functions, the sufficient conditions for the global asymptotic stability of these system are discussed. Numerical simulations are given, which validate the results.Then, an autonomous predator-prey system with diffusion, delay and state structure is studied. Based on time delay parameters, the sufficient condition of existence and stability of positive equilibrium point is given by mean of qualitative theory of differential equation. Through the analysis of system characteristic equation, a sufficient condition for the existence of the system Hopf bifurcation is established.In this paper, three types of predator-prey system with time delay and diffusion are built, which enrich the theory of population ecology model. Moreover, if we apply the theory model into the ecological practice, we will describe the population status more better and forecast the development and changes of the population in the future. Furthermore, we can provide the theory basis for human to protect the population and maintain the survival conditions for the population.