Dynamics Of Reaction-diffusion Models With Free Boundary And Chemotaxis Effect

With the deepening of reaction-diffusion models researches,as well as the industrial demand,more and more complicated reaction-diffusion models have been established.For better description of the spatial orientation movement of the species by reaction-diffusion equations,in this paper,we investigate a class of free boundary type reaction-diffusion systems and a class of chemotaxis-reaction-diffusion systems.Firstly,we focus on a SIS epidemic reaction-diffusion model with a free bound-ary condition and nonhomogeneous coefficients:(?)where S and I represent the density of susceptible and infected individuals,respec-tively,and the infected individuals I processes a varying domain(g(t),h(t)),satis-fying the well known Stefan's condition:g'(t)=-kIx(g(t),t),h'(t)=-kIx(h(t),t).We show that this system admits a unique positive global in time classical solu-tion.Then,we introduce a basic reproduction number and establish a spreading-vanishing dichotomy:if limt?+?|g(t)|=limt?+?|h(t)|=+?,then the infected individuals I persists eventually,and the boundaries of the domain spread to the infinity;if limt?+?|g(t)|<+? and limt?+?|h(t)|<+? then the size of the domain tends to a finite number,and the infected individuals die out in the long run with limt?+??I?=0.In addition,we give out some sufficient conditions related to the diffusion coefficient d,spreading speed k,initial values S0,I0 and h0 for whether the disease spreading eventually or not.Secondly,we consider the following two-species competitive Keller-Segel sys-tem with Lotka-Volterra reactions(?)By semigroup theory,we obtain the local existence and uniqueness of the classical solutions of our system.Then,the sufficient conditions for the existence of global in time solutions has been given.After that,we investigate the asymptotic behavior of the solutions.Both considering the system in either strong competition case or weak competition case,we show that under certain conditions,there exist global asymptotic stable constant steady states.The lower and upper bound of the spacial propagating speed has also been considered.Finally,we extend this Lotka-Volterra competition reaction-diffusion-chemotaxis model to the free boundary problem:(?)We first give the local existence and global existence of classical solutions.Then,we introduce some known results about principal eigenvalue of diffusion-advection elliptic operator.Some spreading-vanishing dichotomies have been obtained both for the strong competition case and weak competition case.And the sufficient conditions for whether the species spreading successfully or not are provided.At last,without any additional restrictions,we claim that the solutions can spread successfully and persist eventually only if the size of the initial habitat is larger than a constant,which the similar phenomenon has been showed in the study for the non-chemotaxis competitive system in a known literature.