A Class Of Pulse Born And Infectious Diseases Predator Prey Model,

Since the pioneering work of Kermack and Mckendrick (1927) on SIR, epidemiologicalmodels have received much attention from scientists. In standard epidemiological models,only single-species models are considered, and some threshold results are obtained. However,the actual situation is not always like this. In natural world, species never exist alonefrom each other. They may spread diseases, compete with others for space and food, or ispreyed by other species. Therefore, it is more biological significance to consider the e?ectof interacting species when we study the dynamical behaviors of epidemiological models. Sofar, little attention has been paid to merging these two areas of research. In the paper, weconsider the predator-prey models with the disease in the prey and the predator, respectively.In chapter 2, we analyze and formulate the predator-prey model with the disease inthe prey, considering density-dependent of two species. The boundness of solutions and theexistence of the equilibria are studied, and the su?cient conditions of locally asymptoticallystable of the equilibria are obtained by the Routh-Hurwitz criterion. Furthermore, we analyzethe global stability of the equilibria by using Lyapunov functions, and obtain the conditionsof the disease extincting and the disease persistence.In chapter 3, we investigate an predator-prey system with the disease in the prey andimpulsive birth. The conditions for the stability of infection-free periodic solution are givenby applying Floquet theory of linear periodic impulsive equation. And we give the conditionsof persistence by constructing a consequence of some abstract monotone iterative schemes.By using the method of coincidence degree, a set of su?cient conditions are derived for theexistence of at least one strictly positive periodic solution.In chapter 4, we consider an predator-prey model with the disease in the predator andimpulsive birth. By using the coincidence degree theorem, a set of easily verifiable su?cientconditions are obtained for the existence of at least one strictly positive periodic solutions.