Analysis Of Dynamic Properties Of Coral Reef Models With Time Delay

The development of the law of things not only depends on the state at the moment,but also depends on the state at a certain point or a certain period of time in the past.Therefore,delayed differential equations(DDEs)can describe the natural phenomenamore accurate and reasonable than ordinary differential equations.Nowadays,DDEs are widely used in many fields such as ecology,physics and astronomy.Meanwhile,Hopf bifurcation of DDE is an important topic of research.Hopf bifurcation means that a small amplitude periodic solution comes out as a certain parameter varies and passes through a critical value.Sowe mainly discuss Hopf bifurcation occurred at the internal equilibrium point of the coral reef DDE model.Firstly,we transform the coral reef DDE model into the abstract ordinary differential form by the infinitesimal generator.Then by the normal form theory and the center manifold theory,the explicit formulae are derived to determine the direction of Hopf bifurcation and the stability of periodic solutions.In order to get a better understanding of dynamic mechanism of coral reefs,another threedimensional coral reef system with time delay is also discussed in this dissertation.First,we calculate equilibrium points of the system and linearized the system at the equilibrium points.The characteristic equation of the linearized system is analyzed.Consequently,local stability of equilibrium points of the system is discussed,and bifurcation conditions for the existence of Hopf bifurcation are also obtained.The dissertation is divided into four chapters,as follows:1.We briefly introduce the development of history ofordinary differential equations and delay differential equations based on the ecological point of view.2.Some basic concepts and important conclusions which would be used later in this dissertation are given.3.We provide a detailed analysis of dynamic propertiesof coral reef models with time delay,which can provide guidance for the protection of coral reefs.4.We summarize our research work.And then,direction of further study is also indicated in this section.