Dynamic Analysis Of Vegetation Pattern Mode

Water resources in arid and semi-arid areas are very limited.The lack of water resources leads to the uneven distribution of vegetation in these areas,which is called vegetation pattern in mathematics.In the past two decades of mathematical modeling work,scholars have attempted to explain vegetation patterns in arid and semi-arid regions through the self-organizing interaction between vegetation and water resources.The water-plant model is one of the most important models to describe vegetation evolution in ecosystems.Therefore,this thesis uses water-plant model to study the formation mechanism of vegetation patterns,and provides a theoretical basis for soil protection and desertification control in arid and semi-arid areas.In this thesis,by using Turing instability theory,bifurcated theory,energy method,(69) functional,Leray-Schauder degree theory and some other reactiondiffusion equation theories,we study the dynamic behavior of local and non-local water-plant model.In order to verify the validity of theoretical results,a lot of numerical simulations are done by using numerical methods and Matlab software.The main work of this thesis is summarized as follows:Chapter 1 presents the mathematical description of the interaction between water and plants and the construction of general water-plant mathematical models.Next,the research background and research status of pattern dynamics and waterplant models are expounded.Then,we summarize some preparatory knowledge and the main work of this thesis.In chapter 2,we mainly discuss the Turing pattern and bifurcation solutions of an extended Klausmeier model with cross-diffusion and nonlocal sustained grazing.Firstly,we analyze the stability of equilibrium points and saddle node bifurcation of ordinary differential systems.Secondly,we study the local stability and Turing instability of reaction-diffusion systems with nonlocal or local sustained grazing.The result show that the system may produce pattern under the influences of selfdiffusion and cross-diffusion.Moreover,both the grazing parameter and rainfall rate can induce transitions among bare soil state,vegetation pattern state and homogenous vegetation state.Then,numerical examples have been illustrated to verify our theoretical findings.Finally,we address the nonlocal reaction-diffusion system as a bifurcation problem,and analyze the existence and stability of bifurcation solutions.Chapter 3 offers a detailed mathematical analysis of an extended water-plant model with power exponent plant growth and local grazing to describe vegetation pattern formation in arid and semi-arid ecosystems.Firstly,we present some fundamental analytic properties of the ordinary differential model and the local reactiondiffusion model.In addition,we use the maximum principle,Hanarck inequality,Poincar′0)inequality and Leray-Schauder degree theory to report some characterizations for the non-constant positive steady state solutions,including a priori estimate of the positive solutions and the non-existence and existence of non-constant positive solutions.Finally,the ordinary differential model and local reaction-diffusion model are numerically simulated.It is worth pointing out that numerical simulation not only verifies the validity of the theoretical results,but also obtains some conclusions that can not be obtained by mathematical analysis.In chapter 4,based on the work of the chapter 2 and chapter 3,we propose an extended water-plant model with power exponent plant growth and two kinds of nonlocal grazing.Firstly,we present local stability and Turing instability of these two kinds of nonlocal models.And the results indicate that the unique positive equilibrium is locally stable for sustained grazing model when the infiltration parameter is equal to 1.However,under certain parameter constraints,Turing instability can be observed in the natural grazing model.When > 1,we can obtain parameter constraints for Turing instability in the sustained grazing model.Nevertheless,when ≥ 2,the parameter conditions for Turing instability can be obtained.However,the stability cannot be judged in the natural grazing model when ∈(1,2).The influence of nonlocal term on system stability is further understood through the comparative study of the two nonlocal models in this chapter and the local model in chapter 3.Secondly,the basic properties of positive solutions of two kinds of nonlocal steady state equations are analyzed by using the same method as in chapter 3.Finally,some notes on numerical simulation are given,based on different diffusion coefficients,infiltration parameter and rain fall parameter.Numerical simulation not only verifies the validity of the theoretical results,but also obtains some conclusions that can not be obtained in mathematical analysis.Chapter 5 focuses on a general class of reaction-diffusion-advection(RDA)system with linear cross-diffusion and cross-advection.By investigating the linearized stability of the constant equilibrium solution,we prove that the self-diffusion and self-advection terms do not affect the linear stabilization of the constant steady state,while the linear cross terms favors the destabilization of the constant steady state and the mechanism of pattern formation.The theoretical results are applied to water-vegetation model with cross-diffusion and cross-advection.