On Several Classes Of Reaction-diffusion Models In Heterogeneous Environment

A fundamental goal of theoretical ecology is to understand how the interaction-s of individual organisms with each other and with the environment determine the distribution of populations and the structure of communities.In particular,hetero-geneous environment in ecology plays an important role in maintaining biodiversity.Reaction-diffusion models provide a good framework for studying heterogeneous en-vironment influence population dynamics.This thesis is devoted to applications of reaction-diffusion models in biology,and three reaction-diffusion models in heteroge-neous environment are studied,focusing on the effect of environmental heterogeneity on population dynamics,evolution and disease spread.In chapter 2,we investigate a nonlocal reaction-diffusion-advection system mod-eling the population dynamics of two competing phytoplankton species in a eu-trophic environment,where nutrients are in abundance and the species are limited by light only for their metabolism.We first demonstrate that the system does not preserve the competitive order in the pointwise sense,caused by the nonlocal nature of the nonlinearity.Then we introduce a special cone K involving the cumulative distributions of the population densities,and a generalized notion of super-and subsolutions of the nonlocal competition system where the differential inequalities hold in the sense of the cone K.A comparison principle is then established for such super-and subsolutions,which implies the monotonicity of the underlying semiflow with respect to the cone K.As applications,we study the global dynamics of the single species system and the competition system.For the single species model,we show the uniqueness and global asymptotic stability of positive periodic solution.For two species model,we study the linear eigenvalue problem associated with the stability of semitrivial steady states,and analyze the effects of diffusion and advec-tion on the outcome of competition for some special cases,which has implications for the evolution of movement for phytoplankton species.In chapter 3,we continue to study the global dynamics of a nonlocal two-species phytoplankton model under a general setting,and focus on the joint effects of d-iffusion and advection on the outcome of competition,based on the monotonicity results in the previous chapter.We perform sufficient analysis on the local stabili-ty of two semi-trivial steady states and establish the non-existence of co-existence steady state,and obtain the global dynamics with the theory of monotone dynami-cal systems.The results illustrate that the combination of diffusion and advection can result in complex dynamical behaviors including competition exclusion and co-existence.The transition between different competitive outcomes may occur with a change in advection rate,which implies the composition of phytoplankton commu-nities can change when advective velocity change.In chapter 4,a diffusive logistic equation on n-dimensional periodically and isotropically evolving domains is investigated.We first derive the model and present the eigenvalue problem on evolving domains.Then we prove that the species persists if the diffusion rate d is below the critical value D0,while the species become extinct if it is above the critical value D0.Finally,we analyze the effect of domain evolution on the persistence of a species.Precisely,it depends on the average value ?-2,where ?(t)is the domain evolution rate,and ?-2=1/T ?0 T 1/?2(t).at.If ?-2>1,the periodical domain evolution has a negative effect on the persistence of a species.If ?-2<1,the periodical domain evolution has a positive effect on the persistence of a species.If ?-2=1,the periodical domain evolution has no effect on the persistence of a species.Numerical simulations are also presented to illustrate the analytical results.In chapter 5,we study a reaction-diffusion-advection SIS epidemic model in a spatially and temporally heterogeneous environment.We first introduce the ba-sic reproduction number R0 and establish the threshold dynamics in terms of R0.Some general qualitative properties of R0 are presented,then the paper is devoted to studying how the advection and diffusion of the infected individuals affect the reproduction number R0 for the special case that ?(x,t)-?(x,t)=V(x,t)is mono-tone with respect to spatial variable x.Our results suggest that if Vx(x,t)?0,(?)0 and V(x,t)changes sign about x,the advection is beneficial to eliminate the disease,whereas if Vx(x,t)?0,(?)0 and V(x,t)changes sign about x,the advection is bad for the elimination of disease.The effects of diffusion of the infected individuals on the persistence of the disease depends on the habitat is high-risk or low risk.