A Diffusive Species-Interactive Epidemiological Model

Since the pioneering work of Kermack-Mckendrick(1927) on SIR, epidemiological models 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 the natural world, species never exist alone from each other. They may spread diseases, compete with others for space or food, or is predated by other species. Therefore, it is of more biological significance to consider the effect of interacting species when we study the dynamical behaviors of epidemiological models. So far little attention has been paid to merging these two areas of research. In this paper, emphasis is put on such an eco-epidemiological system in which only the predator-prey model with disease in the prey is considered. The whole paper is consisted of four parts:In Chapter 1, the topic is put forward after introducing the origin, background and history of current research situation about the related work.In Chapter 2, the prior estimates for the solutions of parabolic system are derived.In Chapter 3, a sufficient condition for the existence of each steady solution is given. And the local asymptotic stability around each of the equilibria is discussed by using the characteristic decomposition and linearization.In Chapter 4, a sufficient condition for the global asymptotic stability around theboundary equilibrium E₁ is firstly given by upper and lower solutions. It shows that ifthe infection rate of susceptible prey is small, the capturing rate of susceptible prey by the predator is small, and the death rate of predator is big, the infected prey and thepredator will extinct. A sufficient condition for the global asymptotic stability around the positive equilibrium is also given by constructing a suitable Liapunov function. It shows that if the intra-specific competition of susceptible prey is weak enough, the infection rate of susceptible prey is big, the death rate of infected prey is small, and the capturing rate of susceptible prey by the predator is small, then the positive equilibrium is globally asymptotically stable.