Abstract: Biomathematics is a brink subject between biology and mathematics, which studies and solves biological problem by means of mathematical method, and proceed with theoretical study to mathematical method that relates to biology. The population dynamics and the epidemic dynamics are two important branches of it. Two models for population dynamics and one model for epidemic dynamics are studied in the thesis, the study for them are great theoretical and practical significance.The bifurcation phenomena can occur in the parameter dependent systems. When the parameters are varied, changes may occur in the qualitative structure of the solutions for certain parameter value. These changes are called bifurcation and the parameter values are called bifurcation values. The type of bifurcation that connects equilibria with periodic solution is called Hopf bifurcation. In chapter 2, a predator-prey model with discrete and distributed delay is investigated. The associated characteristic equation is a special third degree exponential polynomials equation, taking discrete time delay r as a parameter, under some conditions, it is show that small time delay does not affect asymptotic stability of positive equilibrium, however, Hopf bifurcation occurs when r crosses some critical value, a family of periodic solutions bifurcate from positive equilibrium. Further, using the normal form theory and center manifold argument, the explicit formulae which determine the stability, direction and other properties of bifurcating periodic solutions are derived. Numerical simulations are also given to illustrate theoretical results.In chapter 3, a stage-structured predator-prey model with Beddington-DeAngelis functional response is investigated. By analyzing the locations of the roots of the associated characteristic equation, sufficient conditions are obtained for the local stability of a positive equilibrium, under some conditions, it is show that system undergoes Hopf bifurcation at the positive equilibrium when time delay r crosses some critical value, a family of periodic solutions bifurcate from positive equilibrium. By using an iteration technique and comparison argument, sufficient conditions are derived for the global stability of the positive equilibrium. Numerical simulations are also given to illustrate main results.In chapter 4, a delayed epidemic model with non-monotonic incidence rate is investigated. The basic reproductive number R_0 is derived. When R_0 = 1, by using Lyapunov-LaSalle invariance principle, the disease-free equilibrium is globally attractive. When R_0 < 1, by constructing Lyapunov functional, the disease-free equilibrium is globally asymptotically stable, then as long as (?) > (?)~*, time delay (?) does not affect the global stability of disease-free equilibrium. When R_0 > 1, by constructing Lyapunov functional, the endemic equilibrium is locally asymptotically stable, by using permanent theorem, the endemic equilibrium is permanent , then as long as 0≤(?) < (?)*, time delay (?) does not affect the permanence of the endemic equilibrium. Numerical simulations are also given to illustrate main results…

Key words: Time delay; Hopf bifurcation; Stability; epidemic model; Lyapunov functional; Permanence

# Studies on the Dynamic Behavior of Several Nonlinear Biomathematics Models

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