Dynamics Of Models With Nonlocal Effect And Chemotaxis Effect

There exist some objective phenomena in the field of life sciences and social sciences,such as chemotaxis and nonlocal effect.The ecological environment becomes more rich and colorful due to these phenomena.Therefore,the investigation of these effects can help us to understand the formation mechanism of some complex behavior.One way to study these effects is to establish a mathematical model,mainly reaction-diffusion model,which has attracted many researchers recently.However,the model with chemotaxis may admit a finite time blow-up solution,and the standard comparison principle is invalid in models with nonlocal interaction.Therefore,the study of these effects can promote the development of mathematical theory.This dissertation is devoted to investigate the dynamic behaviours of several models with chemotaxis and nonlocal effects and the main content will be divided into the following four parts:(1)We investigate a class of predator-prey model with prey-taxis,where the prey-taxis has volume-filling effect.The main contribution is to describe the effects of diffusion coefficients and the prey-taxis on the existence of nonconstant steady states.First,we give the local stability analysis of constant steady states and construct the Lyapunov function to obtain the global asymptotical stability of boundary and interior constant steady states.Then,we give some priori estimates.Finally,we employ some inequality techniques to obtain the nonexistence conditions of nonconstant steady states,and we use the fixed-point index theory to obtain the existence conditions of nonconstant steady states.(2)We study a chemotaxis-competition population model with two chemicals,where the population diffusions are functions of the corresponding chemical and maybe degenerate.Our goal is to investigate the global existence of solutions and their long-time behaviors.First,treating diffusion functions as weighted functions and using the method of energy estimation,we obtain the global existence of solutions.Then under different strengths of competition,we construct Lyapunov function to obtain long-time behavior of solutions.Finally,using linear stability analysis and numerical simulations,we find that the system will exhibit different complex spatio-temporal patterns for different chemicals production mechanisms.(3)We investigate a predator-prey model with nonlocal competition and give bifurcation analysis by applying Lyapunov-Schmidt method.First,we give a stability analysis of constant steady states and present the effects of nonlocal on their stability.Then,applying the Lyapunov-Schmidt reduction,we study steady-state bifurcation,where we obtain the existence,multiplicity and stability of spatially nonhomogeneous steady-state solutions.Meanwhile,using the Lyapunov-Schmidt reduction,the implicit function theorem and S1-equivariant theory,we obtain the stability and bifurcation direction of Hopf bifurcating periodic orbit.Furthermore,we give a simple description of fold-Hopf singularity by using the LyapunovSchmidt reduction.Finally,we give a numerical example.(4)We investigate a nonlocal Fisher-KPP model in time-periodic and spaceheterogeneous media under three different boundary conditions,obtain uniform persistence of solutions and the existence,uniqueness and stability of positive time periodic solutions.First,using the semi-group theory and sup-and sub-solution method,we give the global existence and boundedness of solutions.Then,using the iteration method and the comparison principle for nonlocal dispersal equations,we obtain uniform persistence of solutions.Finally,under three different conditions,we present the existence,uniqueness and stability of positive time periodic solutions.