Analysis On Some Reaction-diffusion Models With Spatial Heterogeneity

In the problems of population ecology or the spread of infectious diseases,the diffusion of species?viruses,cells,etc.?and the spatial heterogeneity of their habi-tat are two important factors that affect their dynamics.It is an effective method to investigate the above problems by using reaction-diffusion models based on the actual background.The purpose of this thesis is to establish and analyze several types of reaction-diffusion models with spatial heterogeneity.In the analysis,we mainly focus on the threshold dynamics of the model,traveling wave solutions,minimum wave speed,the effects of diffusion or spatial heterogeneity,the long-time behaviors of the solutions,the steady states of the system and the influence of the model parameters on the species.This thesis is divided into five chapters.In the first chapter,the research background and significance of the problems concerned in this thesis are introduced.Meanwhile,the major and innovation of this thesis are briefly stated.In the second chapter,a reaction-diffusion within-host HIV model is pro-posed.It incorporates cell mobility,spatial heterogeneity and cell-to-cell trans-mission,which depends on the diffusion ability of the infected cells.In the case of a bounded domain,the basic reproduction number R₀is established and shown as a threshold:the virus-free steady state is globally asymptotically stable if R₀<1and the virus is uniformly persistent if R₀>1.The explicit formula for R₀and the global asymptotic stability of the constant positive steady state are obtained for the case of homogeneous space.In the case of an unbounded domain and R₀>1,the existence of the traveling wave solutions is proved and the minimum wave speed c^*is obtained,providing the mobility of infected cells does not ex-ceed that of the virus.These results are obtained by using Schauder fixed point theorem,limiting argument,LaSalle's invariance principle and one-side Laplace transform.It is found that the asymptotic spreading speed may be larger than the minimum wave speed via numerical simulations.However,our simulations show that it is possible either to underestimate or overestimate the spread risk R₀if the spatial averaged system is used rather than one that is spatially explicit.The spread risk may also be overestimated if we ignore the mobility of the cells.It turns out that the minimum wave speed could be either underestimated or overestimated as long as the mobility of infected cells is ignored.In the third chapter,a reaction-diffusion system of two bacteria species com-peting a single limiting nutrient with the consideration of virus infection is derived and analyzed.Firstly,the well-posedness of the system,the existence of the triv-ial and semi-trivial steady states and some prior estimations of the steady states are given.Secondly,a single species subsystem with virus is studied.The sta-bility of the trivial and semi-trivial steady states and the uniform persistence of the subsystem are obtained.Further,taking the infective ability of virus as a bifurcation parameter,the global structure of the positive steady states and the effect of virus on the positive steady states are established via bifurcation theory and limiting arguments.It is shows that the backward bifurcation may occur.Some sufficient conditions for the existence,uniqueness and stability of the positive steady state are also obtained.Finally,some sufficient conditions on the existence of the positive steady states for the full system are derived by using the fixed point index theory.Some results on persistence or extinction for the full system are also obtained.In the fourth chapter,in order to study the influence of advection speed on the competitive outcomes of two invading species,we propose and investigate a reaction-diffusion-advection weak competition system with four free boundaries in one dimensional space.In the case of small advection speed,the explicit classification of the competitive outcomes,the estimation of the spreading speed,the long time behaviors of the solutions and the minimal habitat size which determines whether the species can always spread or not are obtained.The results are similar to the case with no advection.In the case of large advection speed,both two species cannot spread successfully,but may virtual spread downstream.In the case of medium-sized advection speed,some competitive outcomes and the long time behaviors of the solutions are also obtained.Mathematical results suggest that the competitive outcomes,which depend on the advection speed and moving parameters,are very complicated.Some criteria for spreading,vanishing and virtual spreading are also established in all cases.At the end,the main contents of this thesis are briefly summarized and some worthwhile future works related to our results are put forward.