Hopf Bifurcation And Stability Of A Competition Diffusion System With Delay

In the study of ecology, to consider the population development of species is a very important subject. The species is a fairly complex system. On one hand, we will consider various spatially inhomogeneous predator-prey models, in this kind of model the density of the two species is not well-distributed and the species can move from the places with higher density to the places with lower density. On the other hand, in these systems there always exists the affection of time-delay. In common, the models with time-delay are of two types. One is the Logistic model with discrete delay, and the other is the Volterra model with infinite delay.In this paper we consider a spatially inhomogeneous competition diffusion system with distributed delay.The researchers in the past only considered the existence conditions of the spatially homogeneous periodic solutions of the kind of models, and regarding the delay or the diffusion rate as a bifurcation parameter, the existence conditions of Hopf bifurcation, and the direction of Hopf bifurcation with Neumann boundary condition. What differ from that of the researchers in the past is the consideration of the direction of Hopf bifurcation with Dirichlet boundary condition , that is, the conditions when the Hopf bifurcation periodic solutions are orbitally asymptotically stable with asymptotic phase.Our basic idea come from the results given by Tang[30]. We adopt the method used by Hassard in the Hopf bifurcation theorem in [24], which is to judge the existence and the direction of Hopf bifurcation, and the stability of the Hopf bifurcation periodic solutions. Firstly, we rewrite the original problem in a form of operator differential equation. In order to determine the Poincare normal form of linear operator in the operator differetial equation, we need to reduce the vector-value function to the two dimensional case on center manifold. Secondly, in order to determine the Floquent index of the Poincare normal form, we need to compute the coefficients of the expansions of each functions in the operator differetial equation. When we computed these coefficients, considering the complexity of the model, we have to compute the director of Hopf bifurcation when the parameters are in some special case.we arrange this paper as follows:In Chapter 1, we introduce the biology background, talk about the history of the study in Logistic and Volterra models and give the new subjects for further study. Moreover, we also line out the useful fundamental theorems.In Chapter 2, we will study the stability of the Hopf bifurcation periodic solutions of the competition-diffusion model with time-delay. Firstly, we introduce the problem considered in this paper. Secondly, regarding the diffusion rate as a bifurcation parameter, we show that the one-dimensional competition diffusion system with infinite delay and Dirichlet boundary condition exhibits the spatiotemporal structure near the steady state of the system. Thirdly, by the method in Hassard [24] and Zhou [27]when they study the unbounded functional differential equations with time-delay, we study the stability of the Hopf bifurcation solutions near the steady state of the system. Finally, we give some examples.In Chapter 3, The existence of a bounded global attactor for Olmstead model with homogeneous Dirichlet boundary condition is proved under some condition on the parameters.