The Qualitative Analysis For The Solution Of A Predator-prey And Cooperation Model With A Free Boundary

In this paper we study a predator-prey and cooperation model with a free bound-ary in a high dimensional and heterogenous environment.The parameters of the environment and model are radial symmetry.We study the long time behaviour of the solutions of the reaction-diffusion equation corresponding this model in the weak and strong heterogenous environment respectively.Firstly,in this paper we give the existence and uniqueness of the global solutions and the estimation of uniformly boundedness for the solutions.For the weak heterogenous environment,we get the spreading-vanishing dichotomy of the solutions for the problem,which implies that the biological populations either obtain development as the time goes to infinity such that they successfully survive in the new environment or they fail to spread and die out in the long run.Then we construct the suitable supper-solution and give the criteria for spreading-vanishing.Finally,an estimate for the spreading speed is given when the species are successfully developed.In the last part of this paper,we study the same subject in the strong heterogenous en-vironment.Considering the diffusion coefficients and the initial habitat radius as parameters respectively,we study the properties of the principal eigenvalue for the corresponding eigenvalue problems,by which we give the spreading-vanishing dichotomy and the criteria.