Bifurcation Analysis Of Several Vegetation Spatiotemporal Dynamical Models

Desertification is one of the most serious environmental problems in the world.It not only threatens the survival and development of human beings,but also restricts the development of global economy.Affected by human activities such as grazing and urbanization and climate change,the area of arid and semi-arid areas is accelerating,and the expansion of these areas will make developing countries suffer from further land degradation and face the risk of desertification.Vegetation,as an important structural unit of ecosystem,plays an important role in water conservation,climate regulation and environment improvement.Vegetation pattern is one of the typical characteristics of ecosystems in arid and semi-arid areas.It can qualitatively describe the spatial distribution characteristics of vegetation.These spatial structures have a certain indication of the transfer of ecosystems,which can be used as early indicators of ecosystem improvement and degradation.In fact,the ecosystems in arid and semi-arid regions are fragile,and small disturbances in the environment may cause the system to mutate.In addition,the pattern is actually the product of bifurcation,which is a special kind of nonconstant steady state solution.Based on the feedback between vegetation and water,the bifurcation analysis of several types of vegetation spatiotemporal dynamical models is carried out.We qualitatively analyze the influence of parameters on the formation of vegetation pattern through bifurcation theory and normal form theory,in order to study the formation mechanism of vegetation pattern and the evolution law of vegetation ecosystem,which will also provide theoretical guidance for desertification early warning and management in arid and semi-arid areas.The main research contents of this paper are summarized as follows :(1)The spatiotemporal dynamics of self-diffusion vegetation-water model in arid flat environment is studied by using the bifurcation theory of reaction-diffusion equation.Based on the normal form theory of the reaction-diffusion equation,the spatiotemporal dynamic behavior of the system near the Turing-Hopf bifurcation point is accurately analyzed.The results show that the slight change of parameters can cause the switching between four different states : uniform state,time periodic state,spatial inhomogeneous steady state and spatial inhomogeneous periodic state.In addition,there is the bistability between the desert state and other states,which will provide some insights into whether the ecosystem is fragile.(2)The dynamics of cross-diffusion vegetation model and the effect of soil-water diffusion on vegetation patterns is studied by steady-state bifurcation analysis.There is at least one nonconstant steady state solution for the spatial vegetation system when the soil-water diffusion coefficient is appropriately large.Moreover,by using the Crandall-Rabinowitz bifurcation theorem and the implicit function theorem,the local structure of nonconstant steady state solutions is obtained.Next,the global steadystate bifurcation theory is used to globally extend the local steady-state bifurcation,and the global structure of the nonconstant steady state solution is obtained.It is found that with the gradual increase of soil water diffusion intensity,the spatial heterogeneity of vegetation gradually increases.(3)The bifurcation problem of vegetation model with infiltration delay is studied.Considering the fact that water infiltration into soil takes time,the dynamics of vegetation model with infiltration delay is analyzed.By selecting time delay and diffusion ratio as bifurcation parameters,the conditions for Hopf bifurcation,Turing bifurcation and Turing-Hopf bifurcation are obtained.Although theoretical analysis shows that time delay does not affect the critical value of Turing bifurcation,numerical simulation shows that the additive effect of time delay and diffusion will lead to more complex spatiotemporal structure of the system.(4)The existence of traveling wave solutions for vegetation models with nonlocal effects is analyzed.Firstly,the corresponding traveling wave equation is obtained by substituting the form of traveling wave solution into the original system.Then,by using the upper and lower solution method and Leray-Schauder degree theory,the results show that the equation has a wavefront solution connecting two constant equilibria under appropriate wave speed.