Qualitative Analysis On A Nonlinear Differential Dynamic System

The main research of this text is about the models of nonlinear differential dynamic system. The full text can be divided into three parts.In the first part, we formulated and analyzed the predator-prey model of prey with epidemic and vertical transmission. The boundness of solutions are studied, and the sufficient condition of locally asymptotically stability of the equilibrium are studied by latent root method and Hurwitz method. Furthermore, the global stability of the equilibrium, and the sufficient condition of global stability of positive equilibrium are also obtained. We look the model with vertical transmissionrate q = 1 as an especial example. We do the same studying. Finally, we get the conclusion:When the vertical transmission rate equal to 1, the disease will be to local infection. When the vertical transmission is less than 1, we can predominate the global stability of the local equilibrium by controlling predatory rate of infective prey and susceptible prey. Then we can prevent prevalence of disease.In the second part, we study an impulse predator-prey system with nonmonotonic predatory rate. We discuss the extinction and permanent existence by the impulse comparison theorem, Floquet multiplier and Liapunov function. Finally, we get the relevant sufficient condition. In the third part, The SIRS epidemical model with impulsive vaccinations is discussed, andthe infective rate is function β(N). We proved the existence and global stability of the disease-free periodic solution by impulsive comparison theorem and get the conclusion: we can control the threshold R|-₂ by adjusting impulsive vaccinations, then we can prevent thespreading of disease.Wei Zou (Application mathematics) Directed by: Zuoliang Xiong Professor...