Dynamics Analysis Of Reaction Diffusion Ecosystem With Time Delay

The phenomenon of reaction diffusion is closely related to people's lives,for example,the spread of diseases,migration of birds and animals,population movements and invasion of alien species are all familiar proliferation events.A large number of questions raised in physics,chemistry,biology,infectious dis-eases,medicine,economics,and various engineering problems can be described by reaction-diffusion equations.Therefore,the study of reaction-diffusion equa-tions has a strong practical background and significant theoretical significance.In recent years,research on reaction-diffusion equations has received increasing atten-tion and in many problems the rate of changing of state depends not only on the state at the current time,but also on the state and space of the history.Therefore,these problems can be described by reaction-diffusion equations with time-delay.Through the study of reaction-diffusion equations with time-delay,it is not only possible to explain the process mathematically,but also to help to predict the future development trend of the process.This dissertation is composed of six chapters.In the first chapter,the background,significance and evolving of bifurcation analysis are briefly reviewed.Furthermore,we concisely introduce the main work of this dissertation.In the second chapter,we consider a diffusive predator-prey model with non-monotonic functional response.The stability of the positive homogeneous steady states and bifurcations of spatially homogeneous/nonhomogeneous time-periodic solutions as well as steady state solutions are studied.In particular,the formulas determining the bifurcation direction and the stability of the bifurcating periodic solutions are obtained.These results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulations.In the third chapter a diffusive predator-prey model with time delay is in-vestigated.The stability of the positive spatially homogeneous steady states and bifurcations of spatially homogeneous/nonhomogeneous time-periodic solutions as well as steady state solutions are studied.Some formulas are obtained to determine the bifurcation direction and the stability of the bifurcating periodic solutions.The fourth chapter is devoted to a reaction-diffusion system for a SIR epidemic model with time delay incidence rate.Firstly,the nonnegativity and boundedness of solutions determined by nonnegative initial values are obtained.Secondly,the existence and local stability of the disease-free equilibrium as well as the endemic equilibrium are investigated by analyzing the characteristic equations.Finally,the global asymptotical stability are obtained via Lyapunov functionals.In the fifth chapter we deal with a reaction-diffusion system for a SIRS epi-demic model with time delay and nonlinear incidence rate.On one hand,the existence and stability of the disease-free equilibrium,endemic equilibria and Hopf bifurcation are investigated by analyzing the characteristic equations.On the oth-er hand,the formulas determining the direction and the stability of the bifurcating periodic solutions are also established.In the sixth chapter a reaction-diffusion model with delay effect and Dirichlet boundary condition is considered.Firstly,the existence,multiplicity,and pat-terns of spatially nonhomogeneous steady-state solution are obtained by using Lyapunov-Schmidt reduction.Secondly,by means of space decomposition,we sub-tly discuss the distribution of eigenvalues of the infinitesimal generator associated with the linearized system at a spatially nonhomogeneous synchronous steady-state solution,and then we derive some sufficient conditions to ensure that the nontrivial synchronous steady-state solution is asymptotically stable.Thirdly,by using the symmetric bifurcation theory of differential equations coupled with rep-resentation theory of standard dihedral groups,we not only investigate the effect of time delay on the pattern formation,but also obtain some important results about the spontaneous bifurcation of multiple branches of nonlinear wave solutions and their spatio-temporal patterns.Finally,our general results are illustrated by an application to a model with one-dimensional spatial domain.