Free Boundary Problems For Species Models In Time-periodic Environments

Biological invasion is a common biological phenomenon,it immediately influences the ecological balance of the nature.To understand the invasion and spreading of new species in time-periodic environments,we consider in this thesis some free boundary problems for species models,and give the spreading-vanishing dichotomy or quartering,some sufficient conditions for spreading and vanishing,and the asymptotic spreading speeds for the free boundaries when the species spread successfully.This thesis is organized as follows.In Chapter 1,we outline the background and research status.In Chapter 2,we consider a single species model with free boundary in time-periodic envi-ronments,in which the intrinsic growth rate allows to change sign for the spatial variable.We first discuss a time-periodic eigenvalue problem,establish the dependence of principle eigen-value on the diffusion rate,the length of domain and weighted function.Then we obtain the spreading-vanishing theorem,and provide some sufficient conditions for spreading and vanish-ing in terms of the signs of principle eigenvalue.Since the principle eigenvalue is not monotone with respect to the diffusion rate,we here just give the spreading and vanishing for the cases of small and large diffusion rate.Finally,we estimate the asymptotic spreading speed for the free boundary when the species spreads successfully.In Chapter 3,we study a two-species competition model with free boundary in time-periodic environments,in which the native species distribute in the whole space and the intrinsic growth rates of them may change signs with respect to the spatial variable.We first establish the spreading-vanishing theorem of the invasive species,and then obtain some sufficient conditions for spreading and vanishing by properties of the principle eigenvalue given in the Chapter 2.We finally estimate the asymptotic spreading speed for the free boundary when the invasive species spreads successfully.In Chapter 4,we investigate a two-species competition model with two free boundaries in time-periodic environments,in which advection terms are considered,and two species spread along the same direction with their expanding fronts may intersect each other at some time.We first discuss a time-periodic eigenvalue problem with advection term,and give the dependence of principle eigenvalue on the length of domain.Then we discuss a free boundary problem for sin-gle species with advection term in time-periodic environments,and give the spreading-vanishing theorem and sufficient conditions for spreading and vanishing.Applying the conclusions of s-ingle equation,we finally obtain the spreading-vanishing quartering,some sufficient conditions for spreading and vanishing of two species,and the asymptotic spreading speeds for the free boundaries when two species spread successfully.