| This thesis considers the following delayed Lotka-Volterra prey-predator model for the adult population When k=c2 and α=0,and there is no diffusion effect,by using the linearization method and analyzing of the distribution of roots of the associated characteristic equation in the complex plane,the local asymptotic stability of the boundary equilibrium point of the system is studied.Secondly,the comparative principle and the upper and lower solution method are used to prove the global asymptotic stability of the semi-trivial equilibrium point and the coexisting equilibrium point.In addition,when k=-c2 and α=1,a time-delayed reaction diffusion model with a Holling-II type functional response is also considered.When there is no coexistence equilibrium point in the system,the existence of the traveling wave solution connecting the two boundary equilibrium points when the wave velocity is greater than the critical wave velocity is proved.The main research contents of this paper are as follows:The first chapter reviews the research background and research status of the LotkaVolterra system with time delay,and elaborates the main research content of this paper.Chapter 2,considers the stability of the equilibrium point of the Lotka-Volterra preypredator adult population model with time delay and without diffusion,the linearization method is used to analyze the local asymptotic stability of the zero equilibrium point and the boundary equilibrium point of the system(2.1).Then,the comparison principle and the upper and lower solution method are used to prove the global asymptotic stability of the boundary equilibrium point êv and the coexistence equilibrium point E of the system(2.1).In chapter 3,a reaction-diffusion delay of the model with holling-ll functional response is also studied for the case when only the adult members of each species can diffuse.We prove the existence of a traveling front solution connecting the two boundary equilibraia for the case when there is no coexistence equilibrium.The proof of the existence of such a front uses the upper and lower solution method and monotone iteration principle. |
