Analysis On Stability And Bifurcation For Delay Dynamical Systems In Biology

In this paper, two kinds of integer order HIV-1 models and a kind of fractional HIV-1 model are investigated. The stability of these three kinds of models is studied. Meanwhile, for the integer order HIV-1 model with delay, the Hopf bifurcation is discussed.Firstly, for the integer order HIV-1 model with delay, the local asymptotic stability of disease-free equilibria and endemic equilibria is discussed by employing the corresponding characteristic equation. At the same time, the sufficient conditions are presented for the global stability of the disease-free equilibrium in the model by constructing the Lyapunov functional. The Hopf bifurcation properties are obtained as the delay passes across a critical value. By using the center manifold and normal form theory, some local bifurcation results are obtained and the formulas for determining the direction of bifurcation and stability of the bifurcation periodic solution are derived. Finally, numerical simulations are presented to illustrate the effectiveness of theoretical analysis.Secondly, a fractional HIV-1 infection model with cell mediated immunity is given, and the local asymptotic stability properties of two equilibria are analyzed. The sufficient conditions for the asymptotic stability of the corresponding equilibria are established. At the same time, the numerical solution of the model is obtained by using the Euler method. Furthermore, the obtained results are verified by the numerical simulation.