With Constant Input And The Incidence Of Non-linear Dynamic System Model

Species dynamics and Epidemiology dynamics are both of great significance for mathematical bio-science. The questions regarding species dynamics concern the change law of the quantity and structure for each species, and how to protect, exploit as utilize species by human intervening; Epidemiology dynamics through the epidemiological models of quantitative and qualitative analysis as well as numeric stimulations , the process of the disease and the epidemic law can be revealed, the change and develop trend can be anticipated, and then theory basis for disease control and prevention can be provided. This paper reads as follows:First, establish two types of epidemic model with constant immigrant and non-linear incidence rate. To the epidemic model with constant immigrant and monotonic incidence rate, By carrying out the bifurcation analysis of the model, show that there exist some values of the model parameters such that numerous kinds of bifurcation occur for the model, such as Hopf bifurcation and Bogdanov-Takens bifurcation. To the epidemic model with constant immigrant and nonmonotonic incidence rate, By carrying out a global analysis of the model and studying the stability of the equilibrium, show that either the number of infective individuals tends to zero as time evolves or the disease persists.Second, consider the susceptible prey population has a constant immigrant, set up the spread of the disease in the prey of the predator - prey model, Study of solutions of the system is invariant, the existence of the equilibrium, and applying the theory of differential inequality study the boundedness of solutions, obtained the sufficient conditions of locally asymptotically stable of the equilibrium by the Routh-Hurwitz criterion. Moreover analyzed the global stability of the equilibrium by constructing appropriate Lyapunov functions, that is the conditions of the disease existent or not.