Time Lotka - On The Qualitative Analysis Of The Model

In this paper, the stability and bifurcation about Lotka-Volterra model with discrete and distributed delay and the difference equation with the delay are investigated. The questions about the balance state local asymptotic stability, the existence of the Hopf bifurcation, the stability of the periodic solutions of a bifurcation, the global asymptotic stability of solution, uniform persistence and stability of solution and the bifurcation about the difference equation are discussed.In the second chapter and third chapter, we study the following Lotka-Volterra model with the discrete and the distributed delays.In the second chapter, we discuss the local asymptotic stability of positive equilib-rium the model of, the existence Hopf bifurcation, the stability of periodic solutions of the bifurcation. According to the theory of characteristic equation and regarding the delay as parameter, necessary and sufficient condition for the local asymptotical stabi-lity of the steady state and the sufficient condition the existence of Hopf bifurcation are obtained. In term of the center manifold and normal form theory, the direction of Hopf bifurcation and the stability of periodic bifurcation solution are discussed. Finally, several numerical examples are given by Maltab to support the theoretical conclusions, besides the influence of every parameter for the bifurcation periodic solutions are discussed by compare the pictures.In the third chapter, the stability of positive equilibrium and the global asympto-tical stability of the solution of the model. In the term of the comparison principle,we obtain the sufficient conditions of the uniform persistence of the solution; through the construction Lyapunov functional. we give the sufficient condition of the globally asymptotic stability of the system; Finally, several numerical examples are given by Matlab to support the theoretical conclusions are correct. The properties of the continuous mode are studied in many literatures, such as the stability of the positive equilibrium in the model and the existence of Hopf bifurcation.But there is a few of literature in the study of discrete differential system, in view of this, in chapter4of the paper, the differential system with time delay are introduced as followingBy the theory about the characteristic value and Jury criterion, we research the local asymptotic stability and bifurcation problem in this part. According to the theory of characteristic equation and regarding the delay as parameter, necessary and sufficient condition for the local asymptotical stability of the steady state and the sufficient condition the existence of Flip bifurcation are obtained. In term of the center manifold and normal form theory, the direction of f bifurcation and the stability of periodic bifurcation solution are discussed. Finally, several numerical examples are given by Matlab to support the theoretical conclusions.