A Cooperative System With A Free Boundary In Time-periodic And Space-inhomogeneous Environment

It is well known that the birth and death rates of many species depend on the seasons,so for the evolutionary of species.The geographical environment may also affect the distribution of individuals since it determines the distribution of resources.Due to the above reasons,it is necessary to consider the effect of inhomogeneous environment in population dynamics.In the current thesis,we study a Lotka-Volterra type cooperative system with a free boundary in time-periodic and space-inhomogeneous environment,in which a new introduced species and the aborigine are cooperative.The initial value of the new species admits a nonempty compact support,and the habitat of the new species will be larger and larger with the increase of time.First,we investigate the existence and uniqueness of solutions to the free boundary problem.By a change of variables,the free boundary becomes a fixed boundary,then the local existence is proved by the contracting mapping principle,during which the regularity of the solution is also obtained.By the estimation to the bounds of solutions,we confirm the global existence when the coefficients satisfy proper conditions.To describe the long time behavior of the free boundary problem,we also study the existence of positive solutions of the corresponding periodic boundary value problem.More precisely,with the help of upper and lower solutions and comparison principle,we obtain the existence by monotone iteration since it is a cooperative system.Moreover,we obtain the upper and lower bounds of these positive solutions.Finally,we study the long time behavior of such a system by presenting the criteria of spreading-vanishing dichotomy.By eigenvalue/eigenfunction theory and upper and lower solutions,some sufficient conditions on the persistence and extinction of new species are given.Our results imply that the persistence of new species requires proper conditions on the initial value,and the coupled nonlinearity plays a positive role on the spreading of the new species.