Stability And Traveling Wave Solutions Of A Diffusion Systems

This article mainly consider two diffusive predator-prey systems, one is a Leslie-Gower system incorporating a prey refuge and the other is a Holling-II system which has a nonlinear density dependent. According to their different characteristics, we consider the stability and existence of traveling wave solutions of the two system, respectively. By controlling the diffusion and spreading speed, the prey could effectively survive together with the predator. So our study has a positive effect on prevention and controlling of population-extinction.In the first chapter, related knowledge about background and significance of dynamic diffusion predator-prey model are introduced. Then some influence of prey refuge and our main work are stated. At last, some related mathematical definition and theorems are also introduced.In the second chapter, we study the dynamic behaviour of a predator-prey system with prey refuge. By Shooting argument, Wazewski Theorem, Lyapunov function and LaSalle’s Invariance Principle, we proved that there is a heteroclinic orbit connecting local equilibrium and positive equilibrium, which means there may exist a traveling wave solution. At the same time, we find there also exists a small traveling wave train solution by using the Hopf bifurcation theorem.In the third chapter, we consider a predator-prey system with nonlin-ear density dependent and study the existence of equilibrium of the model, obtain the sufficient conditions of local asymptotically stability and globally asymptotically of the equilibriums. We also find the existence of traveling wave solution of this model. At last, numerical simulations are given to verify the theoretical results.