Qualitative Research On The Reaction Diffusion Models With A Protection Zone In A River Network

This thesis mainly research the long-term asymptotic behavior of the solution of the reaction-diffusion models with a protection zone in a one-dimensional river system.In the protection zone,the reaction term of the equation is a classical nonlinearity of Logistic type u(1-u).Outside the protection zone,i.e.in the original living environment of the species,the evolution of the species is affected by the strong Allee effect.That is to say,the reaction term of the equation is a bistable nonlinearity u(1-u)(u-θ),whereθ∈(0,1/2).In the first chapter of this thesis,we introduce the research background and the practical significance of the reaction-diffusion models.Then we represent the research problems and main conclusions of the thesis.The second chapter is mainly preliminary,including phase plane analysis,comparison principle,a general convergence results,etc.These knowledge is crucial in the proof of the main conclusions.In chapter 3,we consider the long-term asymptotic behavior of the solution of the reaction-diffusion model.In our model,the river system has two branches,and the water flow speed on each branch is the same constantβ.By analysis of the model,we obtain two important critical speeds,which are c0 and 2 with 0<c0<2.When-c0≤β<2,then big spreading happens.When-2<β<-c0,then small spreading happens.When |β|≥2,then virtual vanishing happens.In chapter 4,we consider the reaction-diffusion model with small water flow speed in half-space.For the reaction-diffusion model in half space,the solution only happens spreading.