Propagation And Asymptotic Behavior Of Several Reaction-Diffusion Models

Free boundaries problems are more realistic and meaningful than fixed boundary parabolic system in ecology,where the free boundary represents expanding fronts of the species.In this thesis we first investigate several free boundary problems of the diffusive population models.The main concerns are the dynamical properties: the existence,uniqueness and regularity of global solution,long time behaviors,the criteria governing spreading and vanishing,asymptotic spreading speeds of species and asymptotic speeds of free boundaries when spreading happens,and so on.At last,we study a reaction-diffusion competition model with seasonal succession,mainly concern the existence of traveling waves,and establish the spreading properties for a large class of solutions,and then compare it to the corresponding free boundaries problem.Moreover,combined with the real life,the biological significance of those systems and conclusions can be analyzed.Firstly,we study the diffusive Leslie-Gower prey-predator model with double free boundaries in one space dimension.We assume that the prey distributes in R,the predator(invasive species)exists initially in a bounded interval,and then invades into the new environment from two sides as time increases.We first obtain the existence,uniqueness and regularity estimates of global solution.Then we provide the spreading-vanishing dichotomy and the criteria for spreading and vanishing.Notice that the term v/u(which appears in the second equation)may be unbounded when u tends to zero,we need to adopt some new techniques and carefully estimates to overcome these difficulties.Secondly,we are devoted to investigate a free boundary problem for the diffusive mutualist model with advection in one space dimension.Species u and v are mutually beneficial,and have the different free boundary on the right side.We first prove the existence and uniqueness of global solution,and then obtain the regularity and uniform estimates.Next,we discuss the sufficient conditions for the spreading and vanishing of two species.Finally,we provide more accurate estimates of the long time behaviors of solution component(u,v),and asymptotic speeds of two free boundaries when spreading happens.Thirdly,the diffusive competition model with seasonal succession and different free boundaries in one space dimension is studied.Two species compete the common resources in the good season,while they hibernate in the bad one.There exist many difficulties due to the reaction terms are discontinuous caused by the seasonal succession.So we need to adopt some new methods for this time-periodic free boundaries problem.We first prove the existence and uniqueness of global solutions.Under the weak competition assumption(i.e.,coexistence case),we then study the long time behaviors and the sharp criteria for spreading and vanishing.Moreover,when spreading occurs,sharper estimates of asymptotic spreading speed of two species and asymptotic speeds of two free boundaries are also given.Finally,we investigate the propagation dynamics of a reaction-diffusion competition model with seasonal succession in R.Under the weak competition condition,the corresponding kinetic system admits a globally asymptotically stable positive periodic solution((?)(t),(?)(t)).By the method of upper and lower solutions and the Schauder fixed point theorem,we prove the existence of traveling wave solutions connecting(0,0)to((?)(t),(?)(t)).At last,we use the comparison arguments to establish the spreading properties for a large class of solutions.