Some Kinds Of Free Boundary Problems In Ecology And Epidemiology

In this doctoral thesis,we investigate some reaction-diffusion systems with free boundaries which could be used to describe the spatial spreading of biological species or epidemics.We study four classical biological models including a Lotka-Volterra type competition models with different habitats,a Lotka-Volterra type prey-predator models in the heterogeneous time-periodic environment,two compartmental models in epidemiology.At first,we study a free boundary problem of the diffusive competition model with different habitats.It is assumed that two species existed in different habitats,respectively.Both species would spread across their boundaries of habitats and move into new environments.In their separate habitats,the two species obey the diffusive logistic equations.In their common habitat,they would compete with each other.We first prove the existence,uniqueness and regularity of the global solution.Then give some uniform estimates which is significant to study the long time behavior when vanishing occurs.For spreading case,we obtain the long time behaviors when parameters satisfying different conditions.Some sufficient conditions for the spreading and vanishing are also established.Then,we undertake a study of a free boundary problem for a diffusive prey-predator model in the heterogeneous time-periodic environment which could describe a biological control progress.In this model,the local growth rates of two species may change signs and be very “negative” in a “suitable large region”.We investigate the spreadingvanishing dichotomy,obtain the long-time dynamical behavior of the solution under different conditions,give criteria for spreading and vanishing,and estimate the asymptotic spreading speed of the free boundary when spreading occurs.As an off-shoot of our analysis,we also obtain the existence of positive solutions to a time-periodic boundary value problem on half line associated with our free boundary problem.Finally,two epidemic models with free boundaries are studied.One of them is a reaction-diffusion system for an SIR epidemic model and the other is a nonlocal SIS epidemic model.For the SIR model,we focus on the long time behavior when vanishing occurs and sufficient conditions for the disease vanishing.For the SIS model,The existence,uniqueness and some estimates of the global solution are discussed first.Then,the long time behavior of the solution to the SIS free boundary problem is obtained for the disease vanishing case.At last,some sufficient conditions for the disease vanishing are established.For both two problems,we study the long time behavior of the solution for the associated heat equation,respectively.