Coexistence And Qualitative Analysis Of Two Types Of Reaction Diffusion Models

By means of the reaction diffusion theory,the study of the interacting effects among species has been an important branch of the dynamics of biological models,and it has gained a series of valuable and meaningful achievements.Based on pre-vious theories and techniques,this article studies two types of biological models:a cooperative system with saturation and a competition model with Ivlev response and cross-diffusion.By using the extremum principle and bifurcation theory,we firstly study a cooperative model with saturation under homogenous Robin boundary conditionsFurthermore,making use of the fixed points index theory and the local bifurca-tion theory,we discuss a competition model with Ivlev response and cross-diffusionThe chief contents in this article are as follows:In section 1,we mainly state backgrounds,developments of two biological mod-els and arrangements of this thesis.In section 2,we investigate a cooperative model with saturation under ho-mogenous Robin boundary conditions.Firstly,a prior estimates of positive solutions can be established by the extremum principle and upper-lower solutions;Secondly,treating a as a bifurcation parameter,the bifurcations from semi-trivial solutions（a*,ηa*,0）and（a’,0,ηb）by the local bifurcation theory;Finally,the local bifurcation solutions can be extended to infinite by the global bifurcation theory.In section 3,a competition with Ivlev response and cross-diffusion is considered.First,we state some known results and give some a prior estimates;Furthermore,sufficient conditions to ensure the existence of positive solutions are given by employ-ing the degree theory;Finally,regarding the growth of one species as a bifurcation parameter,we derive the existence of positive solutions bifurcating from semi-trivial solutions(r^*,0,θ_r*)and（r^*,θ_k,0）by using the local bifurcation theory.