The Research On Traveling Wave Solutions Of Some Classes Of Holling Type Predator-prey Systems

In recent years, the predator-prey relation has become a very important part in mathematics and ecology, it has attracted more and more attention from Mathematics and ecologists. Especially, since entering the 21st century, our earth continuously suffered all kinds of natural disasters, under such circumstances, we should pay attention to the predator-prey relation, in some sense, humans is a natural predator. So the predator-prey theory has a great value in both theory and application. One of the most important ques-tions is to prove the existence of traveling wave solutions and to find the exact traveling solutions, which has received a great deal of attention of many mathematicians and biologists. Base on the ex-isted Lotka-Volterra population models, we consider some classes of predator-prey systems with Holling functional response. We make much investigation in the existence of traveling wave solutions of these systems, we also consider the exact traveling solution to the diffusing Lotka-Volterra equations. This thesis is composed of four chapters.In the first chapter, we introduce the history of mathematical ecology's development, the existed related work and the origin of the problems we discussed. The main work of this paper is also simply introduced in this chapter. In the second chapter, we consider the exact traveling solu-tion to the diffusing Lotka-Volterra equations. By using the Riccati equation rational expansion method, we obtain a series of new trav-eling wave solutions.In the third chapter, we make investigation in the existence of traveling wave solutions in a diffusive predator-prey system with Holling type-Ⅲfunctional response. By using the shooting argu-ment and the invariant manifold theory, we obtain sufficient condi-tions for the existence of traveling wave solution.In the last chapter, we make investigation in the existence of traveling wave solutions in a diffusive predator-prey system with Holling type-Ⅳfunctional response. By using the shooting argu-ment and the invariant manifold theory, we obtain sufficient condi-tions for the existence of traveling wave solution.