Ecology - Epidemic Model With Constant Input Study

Eco-epidemiological model is an important aspect of the study of epidemiological model.Currently, a large number of epidemiological model only consider the spread of disease insingle species. In the natural world, a lot of the time delayed phenomenon exist in the world,which play an important role in the spread of diseases and the growth of population. So itis of more biological significance to study the systems of two or more interacting species andthe delayed systems.In this paper, according to the study of eco-epidemiological systems both at home andabroad in recent years, we mainly introduce the following questions:First, the regular eco-epidemiological model. There are many epidemiological modelstudy the susceptible prey with constant input, but in eco-epidemiological model it is almostnever seen. The main consideration here is the prey with constant input. For the spreadof the disease in the prey or the predator, the boundedness of solutions is obtained, andthe su?cient conditions of locally asymptotically stable of the equilibria is studied by theRouth ? Hurwitz criterion, by constructing appropriate Lyapunov functions the globalstability of the equilibria is investigated. Furthermore, we study the permanence of thesystem by comparison theorem.Second, the time delayed eco-epidemiological model. On the basis of the above, weconsider the prey with constant input and the disease spread in the predators also. Againsttwo time delayed phenomenon of ecological significance, by regarding the delayτas thebifurcation parameter, the stability of the positive equilibrium and the existence of Hopfbifurcation are investigated. Then, using the normal form theory and the center manifoldreduction, the direction of bifurcation and the stability of the bifurcating periodic solutionsare studied. Finally, numerical simulation results are given to support the theoretical results.