Existence Of Traveling Wave Solutions For Some Predator-prey Systems

The interaction of predators and preys is a very important and common phenomenon in biology.And it plays a crucial role in the process of biological evolution.In fact,the interaction of predators and preys not only restricts preys,but also has a great impact on predators.In recent years,it has been one of the hot topics to study the interaction of predator and prey by establishing the reaction-diffusion equation in mathematical biology.Traveling wave solution is an important solution of reaction-diffusion equation,which can describe the invasion of species and the spread of the disease.Therefore,it is very important to study the traveling wave solutions of reaction-diffusion equations.This paper mainly studied the traveling wave solutions of several predator-prey models.First of all,a diffusive predator-prey model with Bazykin functional response is proposed.Under the condition that the diffusion coefficient of the prey is ignored,the existence of traveling wave solutions connecting the boundary equilibrium point and the coexistence equilibrium point and connecting the zero equilibrium point and the coexistence equilibrium point are proved on some given conditions,respectively.Our main arguments are based on a simplified shooting method,a sandwich method and the construction of an appropriate Liapunov function.Then,the nonexistence of the traveling wave solutions is proved by using the method of contradiction and the stable manifold theorem.The oscillatory of both two types of the traveling wave solutions are investigated when they approach the positive equilibrium.Secondly,a reaction-diffusion system which contains two predators-one prey is studied.We first derive the local stability of all equilibrium points and the global asymptotic stability of the positive equilibrium by constructing a Liapunov function.Then,the existence of traveling wave solutions is proved by constructing a pair of upper and lower solutions and using Schauder's fixed point theorem.Finally,under certain conditions the nonexistence of traveling wave solutions is established by the theory of the spreading speed.Finally,we study a predator-prey model with spatial diffusion and time delay.Firstly,we give the sufficient conditions for the existence of positive equilibrium points.Secondly,by using the Schauder's fixed point theorem and constructing a pair of explicit upper and lower solutions,we proved the existence of the traveling wave solution connecting the zero equilibrium point and the positive equilibrium point.