Dynamical Research Of Two Classes Of Biological Models

HIV virus dynamics model and prey-predator model are two kinds of the important biological models, both of which are resource-consumer model in a broader sense. The difference is that the former describes micro organisms, but the latter describes macro organisms. The studies on HIV virus dynamics models can understand the quantitative relations of HIV virus and host cells, discover the mechanism of HIV infection and provide the estimation of viral load. Meanwhile, there exists the lag phenomenon for models in the process of interaction between HIV virus and host cells. Therefore,to research the delayed HIV virus dynamics models on dynamics behavior is the issue of theoretical and practical significance. The prey and predator populations exist the dependent and restrainted interactions. Doing the analysis and prediction on quantity relation of prey and predator is very important. This doctoral dissertation mainly researches resourceconsumer biological dynamics models based on different scales and includes the following parts: the HIV-1 virus dynamics model with nonlinear incidence rate and two classes of target cells, a delayed HIV-1 virus dynamics model with virus waning term and CTL immune response, and a prey-predator model with switched-harvest rate on predator.First, we study the HIV-1 virus dynamics model with nonlinear incidence rate and two classes of target cells: CD4+T-cells and macrophages. By using uniform persistence theory and constructing Lyapunov functional and applying La Salle invariant principle,we obtain that uninfected equilibrium and infected equilibrium are global attractive. Numerical simulations show that in the initial infection stage, the infected CD4+T-cells have larger contribution to the viral load, while in the late stage of infection, the infected macrophages seem to have larger contribution to the viral load. Therefore, this find indicates if we ignore the contributions of macrophages to HIV-1 infection and production,the risk of HIV-1 infection and the viral load will be underestimated.Second, we consider a delayed HIV-1 virus dynamics model with virus waning term and CTL immune response. Our aim is to discuss the variation of dynamics property for adding virus waning term to model. We construct Lyapunov functional to state global stability of the infection-free steady state. By applying uniform persistence theory, we obtain that system is permanent. And if the immune delay is equal to zero, the infected steady state is locally asymptotically stable. For the intercellular delay equalling zero, as virus waning rate increases, we obtain the bifurcation result which the stability of infected steady state changes. Numerical simulations show that the roles of the immune delay and virus waning rate are opposite on the stability of model: as virus waning rate increases,the stability of model strengthens, while as the immune delay increases, the stability of model weakens. We also discover that the immune delay does not affect the magnitude of the viral peak and the time to reach the peak. But as the intracellular delay increases, the time to achieve the viral peak is postponed and the magnitude of the viral peak becomes small.Finally, we discuss a predator-prey model with switched-harvest rate on predator.We assume the harvest rate on predator is the continuous and non-smooth piecewise function which depends on quantity of predator, i.e., the harvest rate is assumed to be proportional to the number of predator given that the harvest rate on predator is no larger than threshold value, otherwise it is a positive constant. We obtain the existent condition of multiple positive equilibria, respectively. By using Bendixson Theorem and constructing Lyapunov function, we demonstrate respectively that predator-extinct equilibrium and a coexistent equilibrium are globally asymptotically stable. And we obtain the sufficient condition of existence of backward bifurcation and inexistence of limit cycle. Based on the analysis results and numerical simulations above, model with the above-mentioned harvest rate possesses more abundant dynamical properties.