Steady-state Bifurcation For Two Types Of Vegetation Dynamical Models

In recent years,with the increasingly serious desertification,the protection of vege-tation is becoming more and more important.As we all know,the spatial distribution of vegetation can be described by vegetation patterns.Due to climatic change,landform,hu-man factors,etc,the vegetation pattern structures are diversified.Semi-arid ecosystems are the most vulnerable to climatic change.Climatic change affects the growth,distribution and species richness of vegetation.Meanwhile,the absorption of vegetation root depends on the density of soil-water in semi-arid region.Indeed,Turing patterns correspond to a special type of nonconstant steady-state solutions.The analysis structure of vegetation pattern is given by the branch theory in this paper,which provides a new idea for understanding veg-etation pattern from the branch aspect.The study of vegetation pattern is helpful to better understand the formation mechanism of vegetation pattern and its influence on ecological function,so as to provide theoretical guidance for vegetation protection and desertification prevention.The steady-state bifurcation for two types of vegetation dynamical models is studied in this paper.The research content includes stability analysis of constant steady-state solutions,prior estimate of positive steady-state solution,branch structure of nonconstant steay-state solution and direction of branch solution.The research method mainly includes stability theory,the maximum principle,the implicit function theorem,the bifurcation theorem and the branching direction theorem.The main research content is as follows:Chapter 1 introduces the research background,present situation and main contents of a vegetation dynamical model with climatic factors and soil-water diffusion feedback and a nonlinear cross-diffusive vegetation dynamical model.Chapter 2 constructs a vegetation dynamical model with climatic factors（temperature,rainfall,ambientCO₂concentration and illumination）and soil-water diffusion feedback.Firstly,based on the analysis of linear stability and prior estimate of positive solution to model,the conditions of steady-state bifurcation are given.Then according to the Crandall-Rabinowitz bifurcation theorem,the implicit function theorem and the Rabinowitz global bifurcation theorem,the local and global structure of nonconstant steady-state solutions are proved.Finally,the numerical results demonstrate the existence of nonconstant steady-state solutions.The results show that with the increase of soil-water diffusion feedback coef-ficient,ambientCO₂concentration or solar radiation,spatial heterogeneity will gradually enhance.Meanwhile,the spatial heterogeneity will also gradually increase as temperature or rainfall decreases.These results reveal the formation mechanism of vegetation pattern,which provide a new viewpoint for protecting vegetation and controlling desertification.Chapter 3 forces on a nonlinear cross-diffusive vegetation dynamical model.We obtain some theoretical results,including prior estimate,existence and structure of nonconstant steady-state solution and direction of branch solution.The results show that nonconstant steady-state solutions are accompanied by steady-state bifurcation.In addition,with non-linear cross-diffusion coefficient increases,the spatial heterogeneity is gradually enhancing,the average biomass of vegetation will increase and vegetation is more survivable.Chapter 4 summarizes the content of this paper and its existing problems and prospects.