Analysis Of Dynamics For Reaction-Diffusion Systems In Advective Environment

Reaction-diffusion population models in advective environment have been widely concerned by many scholars in recent years.Time delay,time-periodic environment and nonlinear boundary response are common phenomena in the natural world.Based on the spectral theory of linear operators and the abstract branch theory of operator equations,this dissertation investigates the dynamical properties of the reaction-diffusion-advection population models under the influence of time delay,time period and nonlinear boundary response.The main contents are as follows:Firstly,the stability and Hopf bifurcation of a classical two-species LotkaVolterra competition-diffusion-advection model with discrete delay are considered.The existence of the global solution of the system is proved by the step-by-step method of the delayed differential equation,and the implicit function theorem is used to obtain that there is at least one spatially nonhomogeneous positive steady state under some parameter conditions.By analyzing the corresponding characteristic equations and taking the time delay as the bifurcation parameter,the local stability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are obtained.We also study the globally asymptotical stability of positive steady state when there is no delay.Inspired by the work of Chen et al.(2018 J.Differ.Equ.264 5333-5359),the stability and direction of the Hopf bifurcation are deduced by introducing a weighted inner product associated with advection term.Numerical simulations are carried out to verify the theoretical analysis results.Then,the stability and Hopf bifurcation of a general reaction-diffusionadvection two species model with nonlocal delay and Dirichlet boundary condition are studied.Under certain parameter conditions,the existence of at least one spatially nonhomogeneous positive steady state is obtained by using the implicit function theorem.By analyzing the corresponding characteristic equations and taking the time delay as the bifurcation parameter,the local stability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are obtained.It is proved that Hopf bifurcation near the critical value is forward and the bifurcating periodic solution is stable on the central manifold by using the center manifold theory and the normal form method.Moreover,the results are applied to a Lotka-Volterra competitiondiffusion-advection model with nonlocal delay.In addition,we investigate the spatiotemporal dynamics of a time-periodic Lotka-Volterra competition-diffusion-advection system.Previous results by Huston et al.(2001 J.Math.Biol.43 501-533)and recent work by Bai et al.(2023 J.Eur.Math.Soc.,online)showed the dynamics of a time-periodic Lotka-Volterra competition-diffusion system with identical growth rate and without advection,in which coexistence and competition exclusion can both occur.Now,our attention will be paid on the effects of the interspecific competition coefficients on the local and global stability of semi-trivial periodic solutions,and the structure of bifurcating coexistence periodic solutions arising from these semi-trivial periodic solutions.Furthermore,we show a relatively complete understanding on the global dynamics of a special system for the weak competition,strong-weak competition and strong competition cases.Finally,the stability and bifurcation of a reaction-diffusion-advection single species model with nonlinear boundary condition is considered.The stability of the trivial steady state is investigated by studying the corresponding eigenvalue problem.The existence and stability of nontrivial steady states are proved by applying the Crandall-Rabinowitz bifurcation Theorem,the Lyapunov-Schmidt reduction method and perturbation analysis,in which bifurcation from simple eigenvalue and that from degenerate simple eigenvalue are both possible.The general results are applied to a parabolic equation with monostable nonlinear boundary condition,and to a parabolic equation with sublinear interior growth and superlinear boundary condition.