Global Dynamics Of Models Related With HIV Infection And Microorganism Flocculation

The time-delayed biological dynamic system is an important research field in biomathematics,which is of great theoretical significance as well as has widespread actual background and practical value.In time-delayed biological dynamic system,the dynamic properties of stability and permanence are important research topics.In this dissertation,by mainly employing Lyapunov functional method coupled with Lyapunov-LaSalle theorem or its variant,and permanence theoretical techniques,the global dynamics of the proposed HIV infection model and that of the related microorganism flocculation models are investigated,Based on the research methods of these models,a generalization of the Lyapunov-LaSalle theorem and its related applications are given.Details are as follows:In chapter 3,we establish a class of delay differential equations model of HIV infection dynamics with nonlinear transmissions and apoptosis induced by infected cells,and then consider the global properties of this model system.By employing the Lyapunov\s second method or constructing suitable Lyapunov functional,we prove the infection-free equilibrium of the system is globally asymptotically stable if the basic reproduction number R0<1,and globally attractive iF R0=1.If R.O>1.the positive equilibrium of the system is locally asymptotically stable and this system is permanent are proved,an analysis method for exploring some explicit expressions of the eventual lower bounds of positive solutions for such systems is proposed.In chapter 4,we consider the global dynamics of a microorganism flocculation model with time delay.In this model system,there may exist a forward bifurcation or backward bifurcation.However,it is difficult to use the research methods that the aforementioned virus model system used to study the dynamics of' such system.Hence,we put up a new idea,that is,by considering a Lyapunov functional L on the orbit through a given initial data(?)after some time T = T(?),we identify ?-limit set ?(?)of(?).Therefore,by using this idea,we study the global stability of the equilibria of the system under certain conditions Furthermore,we also investigate the permanence of the system,and an explicit,expression of the eventual lower bound of microorganism concentration is given.In chapter 5,a time-delayed model of microorganism flocculation with satu-rated functional responses is presented We first analyse the local dynamics of this model system with bifurcations in parameter fields,and then when threshold pa-rameter R0>1,we prove the collection of microorganisms is sustainable as well as propose an analysis method for exploring an explicit eventual lower bound of such microorganism concentration in a large phase space.This system has a back-ward bifurcation if ?-<R0<1 under an additional condition,which implies that the microorganism-free equilibrium coexists with a microorganism equilibrium.In these cases,we introduce an idea for variant,of the Lyapunov-LaSalle theorem.In other words,for a given initial dat(?),we identify its ?-limit set ?((?)by consid-ering a Lyapunov functional L on ?(?),which implies a Lyapunov functional L on the orbit through(?)after some T = T((?)).Therefore,we establish some sufficient conditions for the global stabilities of the microorganism-free equilibrium and the microorganism equilibrium under the corresponding conditions.In chapter 6,in view of the research methods on the global dynamics of the former systems,we give a generalization for the Lyapunov-LaSalle theorem and its modified version,besides,establish some methods for global stability of an au-tonomous differential system with time delays.