Stability Of Semitrivial Solutions And Existence Of Positive Solutions For A Predator-prey Model In Heterogeneous Environments

In this paper,we mainly consider the dynamic behavior of the predator-prey model with Holling-Tanner functional response function in heterogeneous environments,as follows:that is,stability of semi-trivial solutions and existence of positive solutions are studied in the equilibrium state.In this paper involved in upper and lower solutions?extremal principle?Fredholm theory?bifurcation theory?fixed point index theory.This article has four chapters:In chapter one,we briefly introduce the domestic and international research situa-tion of the predator-prey model,the content of this paper and the structural arrangement of this paper.In chapter two,we mainly introduce the related concepts,basic theorems and con-clusions.In chapter three,we use the eigenvalue theory to study the stability of semi-trivial solution of the model.In chapter four,the existence of positive solutions of the model will be studied from two aspects.Firstly,a priori estimates of positive solutions of the model and the nec-essary conditions for the existence of positive solutions are obtained,using the extreme principle and structure of the upper and lower solution method;Secondly,by means of local bifurcation theory,it is given that the sufficient conditions for the existence of positive solutions;Finally,using the global bifurcation theory and the fixed point index theory shows that local bifurcation generated by the model in the semi-trivial solution can be extended to global bifurcation.