Study Of Discrete And Reaction-diffusion Biological Dynamic System

This paper mainly investigates the discrete biological dynamic system and the contin-uous spatial epidemic model. And it is organized as follows.In chapter 2, we study the existence and global stability of periodic solution for dis-crete predator-prey system with the Beddington-DeAndelis functional response and predatorcannibalism. By using continuation theorem which was proposed by Gaines and Mawhin,the existence conditions of at least one periodic solution of system are obtained. And thesu?cient conditions which ensure the global stability of the positive periodic solution arederived by constructing a special Lyapunov function.In chapter 3, we investigate a discrete-time epidemic model with nonlinear incidencerate by qualitative analysis and numerical simulation. It is verified that there are phenomenaof the transcritical bifurcation, ?ip bifurcation, Hopf bifurcation types and chaos. Also thelargest Lyapunov exponents are numerically computed to confirm further the complexity ofthese dynamic behaviors. The obtained results show that discrete epidemic model can haverich dynamics behavior.In chapter 4, pattern formation of a spatial epidemic model with both self and crossdi?usion is investigated. From the Turing theory, it is well known that Turing patternformation can not occur for the equal self di?usion coe?cients. However, combined with crossdi?usion, the system will emerge isolated groups, i.e., stripe-like or spotted or coexistence ofboth, which we show by both mathematical analysis and numerical simulations. Our studyshows that the interaction of self and cross di?usion can be considered as an importantmechanism for the appearance of complex spatiotemporal dynamics in epidemic models.In chapter 5, we investigate an epidemic model with both di?usion and migration. In the previous work (Sun et al 2007 J. Stat. Mech. P11011), we studied the model only withdi?usion and obtained stationary Turing pattern. However, combined with migration, themodel will exhibit typical traveling pattern, which is shown by both mathematical analysisand numerical simulations. The results obtained well extend the finding of pattern formationin the epidemic model and may well explain the field observed in the real world.