Dynamics Of Reaction-diffusion Models With Nonlocal Delay And Chemotaxis Effect

The development of natural science and technology largely depends on the progress and achievements of biology,chemistry and physics,while the precision of these subjects themselves provides an important guarantee for their progress and achievements.The precision of disciplines is usually implemented by developing mathematical models,while most of the mathematical models can be summarized up as reaction-diffusion models.There has been growing attention and signifi-cant progress on the study of reaction-diffusion models because it involve a lot of problems arising from numerous mathematical models in biology,chemistry and physics,and has strong practical background and application value.As reaction-diffusion models were applied to the broader fields of natural science,however,it was found that many physical,chemical and biological phenomena can not be explained by simple reaction-diffusion mechanisms.Instead they need to be inter-preted by chemotaxis and delay.The reaction-diffusion models used to describe the phenomena of chemotaxis and delay are referred to as reaction-diffusion models with delay and chemotaxis effect.In view of reaction-diffusion models with delay and chemotaxis effect reflects practical problems and phenomena much better than simple reaction-diffusion models,they now becomes more and more attention of scholars since the last decade or so.In the study of reaction-diffusion models with delay and chemotaxis effect,dynamics is an active field full of rich practical back-ground and wide applications.In this dissertation,we investigate the dynamics of reaction-diffusion models with nonlocal delay and chemotaxis effect,with empha-sis on the existence,stability and bifurcation of steady-state solutions as well the existence of traveling wave solutions.This dissertation is organized as follows:Firstly,we investigate the local and global existence of solutions of a reaction-diffusion model with nonlocal delay and chemotaxis effect under Dirichlet boundary condition by using Amann's existence theory for quasilinear parabolic systems.Next,we study the existence and multiplicity of nontrivial steady-states nearby origin by means of Lyapunov-Schmidt reduction and the implicit function theorem.Moreover,we investigate the stability and Hopf bifurcation of nontrivial steady-states by analyzing the characteristic equation.Whereafter,we study the stability and bifurcation direction of Hopf bifurcating periodic orbits by using Lyapunov-Schmidt reduction,the implicit function theorem and S~1-equivariant theory.In particular,we illustrate our general results by applications to a chemotaxis model with Logistic source,nonlocal delay and one-dimensional spatial domain.Secondly,we study the existence and multiplicity of nontrivial steady-states nearby non-zero equilibrium of a reaction-diffusion model with nonlocal delay and chemotaxis effect under Neumann boundary condition by using Lyapunov-Schmidt reduction and the implicit function theorem.Next,we investigate the stability and Hopf bifurcation of nontrivial steady-states by analyzing the characteristic equation.Moreover,we study the stability and bifurcation direction of Hopf bifur-cating periodic orbits by using Lyapunov-Schmidt reduction,the implicit function theorem and S~1-equivariant theory.Finally,we illustrate our general results by applications to a chemotaxis model with local delay and one-dimensional spatial domain.Thirdly,we investigate the existence of traveling wavefronts with large wave speed for a reaction-diffusion model with nonlocal delay and chemotaxis effect by applying the perturbation method.Firstly,we study the existence of heteroclinic solutions by analyzing the characteristic equation of the reaction equation at two equilibria.Moreover,we convert the existence of traveling wavefronts into an equivalent operator equation for the existence of solutions in a Banach space by using traveling wave transformation and method of variation of constants.Finally,we prove the existence of traveling wavefronts by applying Banach fixed-point theorem.Finally,we study the existence of periodic traveling waves with large wave speed for a reaction-diffusion model with nonlocal delay and chemotaxis effect by applying the perturbation method.Firstly,we investigate the existence of period-ic solutions by analyzing the characteristic equation of the reaction equation at a non-zero equilibrium.Moreover,we study the stability and bifurcation direction of periodic solutions by using normal form theory and center manifold theorem.Next,we convert the existence of periodic traveling waves into an equivalent op-erator equation for the existence of solutions in a Banach space by using traveling wave transformation and method of variation of constants.Finally,we prove the existence of periodic traveling waves by applying Lyapunov-Schmidt reduction and generalized implicit function theorem.