Qualitative Analysis For A Leslie-gower Predator-prey Model With Prey-taxis

In this dissertation,using the theories of bifurcation,the influence of prey-taxis term on the equilibrium pattern for the following predator-prey model are investigated,such as the linear stability and linear instability induced by prey-taxis,the existence and stability of the local bifurcating solutions,the directions of the branches,and the existence of the global bifurcating solutions analysis for this model.The organization of this paper is follows:In first section,we analysis the linear stability for the model(1)carefully,and discuss the effect of chemotaxis attraction and chemotaxis rejection on the stability of positive constant steady state solution(stabilization or destabilization).when chemotaxis rejection induces instability,it is discussed that whether the prey-taxis term can lead to the pattern formation.In second section,Using the maximum principle,the Harnack inequality,the standard elliptic regularity theorem and the Sobolev embedding theorem,we obtain a priori estimates of positive steady state solutions.In third section,Using the Crandall-Rabinowitz branch theory,we study the existence of positive bifurcating solutions.Specifically,according to the implicit function theorem,we find all the possible branch points in the trivial solution curve.Next,using the bifurcation theory,we obtain the specific expression of the local bifurcating solutions.Furthermore,we focus on the stability and the bifurcation directions of the local bifurcating solutions.Finally,Using the global bifurcation theory,we prove that the local bifurcation curves can be extended to the global branches.