Global Dynamics Of A Population Model From River Ecology

This thesis is devoted to the study of the global dynamics of a Lotka-Volterra type competition-diffusion-advection system arising in river ecology.The feature of this model-ing system lies in the general boundary conditions imposed at the downstream end includ-ing the classical Dirichlet,Neumann and Robin type boundary conditions.Biologically.different type of boundary conditions reflect different environmental situation at the river end.By the virtue of the basic theory of differential equations and dynamical systems,the spectrum theory as well as some skills of nonlinear analysis,the global dynamics of this competition system is finally revealed.Mathematically,we will develop further the method and skills to deal with such nonlinear problems.Biologically,our theoretical results may help people better understand the mechanism behind the evolution of dispersal strategy in advective environmentsThe main body of this thesis is divided into three chapters.In the first chapter,we mainly introduce the biological background,the recent development in this direction,and our main problem and results.The second chapter is devoted to a brief statement on several fundamental theories such as the theory of monotone dynamical systems and the theory of principal eigenvalue.In chapter 3,we aim to give the qualitative analysis on the modeling system.Specifically,in subsection 3.1,the eigenvalue problem of a single equation is analyzed and three properties of the principal eigenvalue are exhibited;in subsection 3.2,some a priori estimates on the semi-trivial steady states and positive steady states are given;and in subsection 3.3,based on the establishment of no positive steady states,a clear picture on the local stability of semi-trivial steady states,and the existing theory of monotone dynamical systems,the proof of our main result concerning the global dynamics is displayed.