Qualitative Analysis Of Reaction-diffusion Equations With Free Boundaries In Heterogeneous Environments

With the accelerating pace of globalization,the particularly increased trade and tourism,have resulted in the alien species arriving.Generally,the exotic species es-tablished cause significant damage to native biodiversity,which pose risks to valued e-cosystem,economic or human health.Simultaneously,the consequent climatic changes affect the geographical distribution of species.Moreover,it is possible to affect the inva-sion process of the species.In order to have a deep understanding of the heterogeneous environment,the simple reaction-diffusion model is constructed by the biomathemati-cian to investigate the invasive species in the new habitat.The diffusion profile and the spreading speed are two important sides studied by the biomathematician.In this paper,we use the reaction-diffusion model with the free boundary to pre-dict the diffusion process of the alien species in the heterogeneous habitat.With the reaction-diffusion model,we use the zero argument,the comparison principle and other methods to analyse the spreading profile and spreading speed of the alien species.In this doctoral thesis,We consider two different free boundary problems in the heterogeneous environment.Firstly,we define the heterogeneous habitat as favorable habitat or unfavorable habitat in radially symmetric environment.Besides,we study how the favorable habi-tat or unfavorable habitat,species diffusion rate and other factors influence on the in-vasive species.And the asymptotic behavior of the alien species is investigated.To better understand how the heterogeneous environment affects the long time behavior of the solution,we first give the uniqueness and existence of the solution with the free boundary.Then the definition and properties of the threshold value are given;Next,the sufficient condition of the spreading or vanishing of the invasive species is deter-mined by the threshold value.Furthermore,the "spreading-vanishing" dichotomy is followed by considering the diffusion rate,the initial habitat and the initial density of the species.Finally,the spreading speed of the exotic species is given by constructing the suitable upper and lower solution.Secondly,we study the sharp estimate of the solution of the free boundary problem when the spreading happens under the shifting speed of the climate less than a critical value in one dimension.In other words,as the spreading happens,the final profile of the alien species can be obtained as the shifting speed is less than some constant.Thus we only consider the spreading case in this thesis.We first use the suitable upper and lower solution to show that the spreading front has a uniform bound with a line.Then we illustrate that the spreading front tends to this line by using the zero argument.That is to say,when spreading happens,the species spread as this line in the end and the spreading speed is a constant.Furthermore,the final profile of the invasive species is given by applying the comparison principle.