Dynamic Analysis Of Some Competitive Systems In Advective Environments

This thesis studies the global dynamics of two species competition system in advective environment.By qualitative analysis,maximum principle,spectral theory,upper and lower solution method and the theory of monotone dynamical systems,we investigate how the parameters(diffusion rate,advection rate,inter-specific competition intensities,loss rate of individuals at the upstream and downstream ends,resource function)appearing in the systems affect the dynamics of the system,and then obtain different dynamics in different ranges,which reflect the essential effects of different environments on the practical problem.In Chapter 1,we introduce the background of a two-species competition model in advective environment and its research status,and then present the main contents of our thesis.In Chapter 2,we are concerned with the global dynamics of a Lotka-Volterra competitiondiffusion-advection system for small diffusion rates in heterogenous environment.Supposing that the resource functions of the two species are different,we firstly prove the stability of semi-trivial steady-state solutions when one of the diffusion rates is sufficiently small.Then it is proved that the coexistence steady state exists and the coexistence steady state is globally asymptotically stable when both diffusion rates are sufficiently small.In Chapter 3,the global dynamics of competitive diffusion-convection systems with general boundary conditions are considered.First,a complete classification of all possible longtime behaviors of steady state is given.In addition,Using spectral theory and the theory of monotone dynamical systems,we study the effects of diffusion rates,advection rates,interspecific competition coefficients and boundary conditions on contents of competition when two species compete for the same resource.In Chapter 4,dynamical behaviors of a Lotka-Volterra competitive system from river ecology are considered.Here Neumann boundary condition is assumed at the upstream end.At the downstream end,we suppose that the population may be exposed to differing magnitudes of loss of individuals.With the help of spectrum theory and the theory of monotone dynamical systems,we study the effects of diffusion rates,downstream loss rates and interspecific competition coefficients on the dynamical behaviors of the system.We obtain the following contents of competition: either two competitors survive,or two species coexist,or both species become extinction.In Chapter 5,global dynamics of the competition model of two populations in a homogeneous environment is discussed.It is assumed that the diffusion rates,advection rates and downstream loss rates of these two species are different.It is worth noting that we assume the upstream end is Neumann boundary condition,and there is always a net loss of individuals at the downstream end caused by water flow or both water flow and diffusion.The results showed that when the downstream loss is large(both water flow and diffusion cause downstream individual loss)or when the downstream loss was small(only water flow causes downstream individual loss),two species coexist,or go to extinction or only one species survives.In Chapter 6,we consider the global dynamics of a two-species Lotka-Volterra competitiondiffusion-advection system.It is assumed that the total resources of the two competitors are fixed at the same level and that one species has two modes of diffusion : diffusion and biased movement,while the other species only diffuse.With the help of spectral theory and the theory of monotone dynamical systems,the value range of advection rates,diffusion rates,inter-specific competition coefficients and resource functions for coexistence of two species are obtained.In the final Chapter 7,a summary on the main content of this dissertation and future research prospects are presented.