Dynamic Models Related With Tumor Growth And Vector Diseases And Their Stability Analysis

Biomathematies, as an interdiscipline of biology and mathematics, aiming to research biology issues with the method of mathematics, has had rapid development. Biodynamies is a branch of hiomathematics, mathematical models play a key role in describing the biological dynamic behaviors. Biodynainics with delay is a field which has abundant pratical background and extensive application. Analyzing their dynamics by investigating the stability and bifurcation problems is a very important subject in the field of differential equations and mathematical biology. This thesis mainly applies the theories and methods, such as Liapunov stability theory, LaSalle invariance principle, uniform persistence theorem, normal form method, local Hopf bifurcation and so on, to study the modelling, local and global stability, persistence, local Hopf bifurcation of system for several biological dynamic system. Details are as follows:Firstly, a mathematical model of tumour-induced angiogenesis is developed basing on the biological meaning of the tumor-induced angiogenesis, After that, we introduce the delay in ordinary differential equation system. At first, the global existence, nonnegativity and boundedness of the solutions are discussed. After that, by analyzing the corresponding characteristic equations, the local stability of three boundary equilibria and the angiogenesis equilibrium of the model is discussed, respectively. By constructing Liapunov functionals and applying LaSalle invariance principle, we further consider global asymptotic stability of the boundary equilibria and the angiogenesis equilibrium. Finally, some numerical simulations are given to support the theoretical results.Secondly, we consider an epidemic model of a vector-borne disease with double delay. The basic reproduction number Ro, which is a threshold quantity for sta-bility of equilibria, is calculated. If R0≤1, by constructing the proper Liapunov functional, then the infection-free equilibrium is globally asymptotically stable. On the contrary, if R0>1, by constructing the proper Liapunov functional, the infec-tion equilibrium is globally asymptotically stable when one time delay is absent. Applying a permanence theorem for infinite dimensional systems, we obtain that the disease is always present when R0>1. Finally, some numerical simulations are given to support the theoretical results.At last, we consider a population dynamics model with two stage structure and two time delays. Firstly. we give the basic reproduction number R0(τ1,τ2). After that, we obtain the sufficient condition ensuring the global asymptotical stability of the boundary equilibrium using the Liapunov method. Furthermore, for the positive equilibrium. using delay as parameter, its dynamics are studied in terms of local analysis and local Hopf bifurcation analysis. The stability and direction of the Hopf bifurcation are determined by applying the normal form method and the center manifold theory. Numerical simulation results are given to support the theoretical predictions.