The Study Of A Ratio-dependent Predator-prey Model With Mixed Boundary Conditions

This paper deals with a ratio-dependent predator-prey model with diffusion in which the predator is subject to the homogeneous Robin boundary condition and the prey to the homogeneous Neumann condition. In the model, the predator, in addition to the prey in the system considered, has other natural food sources. By mainly using Leray-Schauder degree theory, this paper proves that under certain conditions there exists a positive steady-state solution to the model . Furthermore, according to the bifurcation theory and the linear stability theory, this paper studies local or global bifurcation, local stability and uniqueness of the positive steady-state solution, and the asymptotic behavior of the non-negative solution to the model.The contents of the article are as follows.Chapter 1 introduces the research significance of biomathematical model,current research situation of this field and main tasks of this article.Chapter 2 lists some basic concepts, terms and theories, such as the eigenvalue problems, the maximum principle, Sobolev imbedding theorem, fixed point index theory, the bifurcation theory and linear stability theory and so on.In Chapter 3, the existence of a positive steady-state solution to the model is mainly investigated. It firstly discusses the necessary condition for existence of a positive steady-state solution to the model. In succession, by using the index theory, topologic degree theory and the approximation method, it has been proved that the model at least possesses a positive steady-state solution under certain conditions.Then it discusses the asymptotic behavior of the non-negative solution to the model.In Chapter 4, in terms of the bifurcation theory and the linear stability theory, and taking the birth rate of one species as the bifurcation parameter, the bifurcation of its positive solutions which emanates from the semi-trivial solution is analyzed. The stability and uniqueness of the semi-trivial solution and the bifurcation solution are studied and global bifurcation is discussed.In conclusion some problems which are to be resolved in the future are discussed.