Dynamical Behavior Analysis Of Two Kinds Of Mathematical Biology Models

In the 21 st century, Mathematical biology is one of the most popular topics, which has the particular role in solving the significant problems that human have to face. The stability study of the differential dynamic system always has been the core status in the field of mathematical biology. And it is widely used to analyze the biological phenomena which include the spread of infectious diseases, the interaction of population. These studies help us to understand the internal mechanism of these phenomena, and to find out the countermeasures about how to solve these problems. In this paper, we study the global stability of SEIRS epidemic model with the nonlinear incidence rates and vertical transmission. Then we analyze the dynamical behaviors of commensalisms Lotka-Volterra model with impulsive effect and also give its practical significance. Finally, the numerical simulation is given to verify the validity of the results in this paper. This paper mainly contains the following two parts:Firstly, the SEIRS epidemic model with the nonlinear incidence rates and vertical transmission is given. Through the construction and analysis of the basic reproductive number0R( p, q), we get the global dynamics of the model. If 0R( p, q) ?1, we can prove that the disease-free equilibrium is globally asymptotically stable and the disease always dies out with the Lyapunov stability theorem; If 0R( p, q) >1, the model have a unique positive equilibrium point, and a geometric approach is used to study the uniform persistence and the global asymptotic stability of the equilibrium point.Then, we propose the commensalisms Lotka-Volterra model with impulsive effect, and discuss some properties of the continuous survivability and the periodic solution of the system. Basing on the relevant theories of periodic impulsive differential equations, we analyze the uniform persistence conditions of the system. We prove that the system has only one periodic solution and the periodic solution has global attraction under some certain conditions.