The Dynamics Study Of A Lotka-Volterra Type Reaction-diffusion-convection-competitive System

Reaction-diffusion equation(s)are frequently used to describe some real problems arising in physics,chemistry and ecology.The existence and qualitative properties of solutions of such equations are important research topics in the theoretical study of PDE and its application.This thesis mainly investigated a Lotka-Volterra type competition-diffusion-advection system from river ecology.By using some standard PDE theory,monotone dynamical systems theory as well as some nonlinear analytic skills,we discussed the ef-fect of some important parameters,e.g.,advection rate,resource function and boundary conditions,on the potential dynamical behaviors,and revealed the essential influence of different environment(indicated by different values of parameters)on the real problem.Theoretically,it is expected that we can further develop some methods and techniques to deal with such nonlinear problems,and from the application point of view,we can provide some evidence to understand the mechanism of some real problems.The specific contents are arranged as follows:To explore the evolution of movement strategy in advective environments,chapter 2 of this thesis studied the following model:ut = duxx-?1ux + u[r-u-v],0<x<L,t>0,vt = dvxx-?2vx + v[r-u-v],0<x<L,t>0,(3)dux(0,t)-?1u(0,t)= 0,ux(L,t)= 0,0<x<L,dvx(0,t)-?2v(0,t)= 0,vx(L,t)=0 0<x<L,where u and v denote the population density of two aquatic species,respectively;d>0 is the diffusion rate caused by turbulence or self-propelling;?1,?2>0 measure the effective advection speeds incurred by unidirectional water flow of two species;r,a positive constant,stands for the local growth rate.We assume the upstream end is closed,so the no-flux type condition is considered;while at the downstream end,we suppose it is connected with a big lake,and so the free-flow condition is imposed.Furthermore,we assume that both populations are competing for the same resources and have the same competition ability and diffusion rate.The only difference of two competitors lies in their advection rates.Under these hypothesis,we aim to figure out whether strong or weak advection strength is more helpful for species to win the competition.By employing the maximum principle,the theory of principle eigenvalue,monotone dynamical systems theory and some PDE analytic skills,we proved that the semi-trivial steady state corresponding to the smaller advection rate is globally asymptotically stable,that is,the species with slower advection rate will wipe out the other one completely,and thus win the competition finally.Chapter 3 of this thesis aims to generalize the above conclusion to a more general situation,where the boundary condition at the downstream end becomes more flexible,including the well-known Neumann,No-flux and Robin type condition as special case,and the resource function now may depend nontrivially on the spatial variable x(which reflects a more reasonable environment).The model under consideration is as below:ut = duxx-?ux+ u[r(x)-u-v],0<x<L,t>0,vt = dvxx-?vx + v[r(x)-u-v],0<x<L,t>0,dux(0,t)-?u(0,t)= dvx(0,t)-?v(0,t)= 0,t>0,dux(L,t)-?u(L,t)=-b1?u(L,t),t>0,(4)dvx(L,t)-?v(L,t)-b2?v(L,t),=-b2?v(L,t)t>,u(x,0)= u0(x)?,(?)0,0<x L,v(x,0)= v0(x)?,(?)0,0<x<L,where the parameters b1,b2 ? 0 are used to measure the loss rate of population at the downstream end,which was firstly proposed by the famous biological mathematician M.Lewis et al.[19].When the resource function r is decreasing in x,we can establish a similar conclusion to that in chapter 2;while for the increasing case,under certain conditions,we can illustrate that the competitive exclusion principle may not hold;instead,two populations can co-exist eventually.