Bifurcations And Periodic Solutions For Delayed Lotka-Volterra Diffusion Systems

In recent years, the study of bifurcation problems has been one of important subjects in dynamical systems and has been applied extensively in many fields such as mechanics, physics, chemistry, biology, ecology, control, numerical calculations, engineering technology and economics and social sciences etc. In this thesis we consider mainly periodic solutions and bifurcation problems of the two-dimensional Lotka-Volterra system described by the following partial functional differential equationsIn the absence of diffusion effects, by regarding the delayτas the bifurcation parameter and analyzing the characteristic equations of the linearized system of the original system at the positive equilibrium, the sufficient conditions ensuring that the positive equilibrium is asymptotically stable and the conditions guaranteeing that the system can bifurcate periodic solutions from the positive equilibrium are established. In addition, by applying the normal form theory and the center manifold reduction for delayed differential equations, an explicit algorithm determining the direction of Hopf bifurcation, the stability and period of bifurcated periodic solutions is given. Finally, by using a topological global existence result, we give the global existence of bifurcated periodic solutions and it is found that the system exists always a nonconstant periodic solution after the second critical value of the delay.In the presence of diffusion effects and Neumann boundary conditions, by linearizing the system at the spatial homogeneous positive equilibrium and analyzing the corresponding characteristic equation, the stability of positive equilibrium is studied and the conditions under which the system undergoes a Hopf bifurcation of periodic solutions are obtained. Furthermore, by using the normal form theory and the center manifold reduction for partial functional differential equations, an explicit algorithm determining the direction of Hopf bifurcation of spatial homogeneous periodic solutions, the stability and period of bifurcated periodic solutions is given. In view of this algorithm, the sufficient conditions ensuring that the bifurcated periodic solutions are orbitally asymptotically stable and unstable on the center manifold are obtained. Meanwhile, to verify the theoretical conclusions obtained in this part, some numerical simulations are also included.When there are diffusion effects and Dirichlet boundary conditions, we firstly obtain the conditions such that the system exists positive equilibrium solutions and give the asymptotic expression of these positive equilibrium solutions. Secondly, by linearizing the system at the unique positive equilibrium solution and analyzing the associated characteristic equation, it is found that the positive equilibrium solution is asymptotically stable when the delay equals to zero and when it is increased to certain critical value, the positive equilibrium solution will loss the stability and a spatially heterohomogeneous periodic solution will bifurcate from this positive equilibrium solution. With the further increase of delay, at another sequence of critical values of delay, although the positive equilibrium solution is always unstable, a spatially heterohomogeneous periodic solution can also bifurcate from this positive equilibrium solution. For the direction of Hopf bifurcation and the stability and period of bifurcated periodic solutions, we give an explicit algorithm for determining these properties by using the normal form theory and the center manifold reduction for partial functional differential equations. According to this algorithm, it is shown that the bifurcated periodic solutions through Hopf bifurcations at the first critical value of delay are orbitally asymptotically stable on the center manifold and the bifurcated periodic solutions through Hopf bifurcations at other critical values of delay are unstable. Some numerical simulations are also given to verify our theoretical results.