Stability And Bifurcation Problem Studies Of Several Biological Dynamic Systems

Biomathematics, as an interdiscipline, has had rapid development. Biodynam-ics is a branch of biomathematics, mathematical models play a key role in describing the biological dynamic behaviors. Biodynamics with delay is a field which has abun-dant practical background and extensive application. The study on stability and bifurcation problems of delayed biodynamic system plays a vital role in the de-velopment of the actual application fileds, among which the stability embodys the structured balance. The stability study to a system in an infinite-dimensional s-pace, especially, the global stability study, will show the dynamic property of system more comprehensively and in-depth. So-called bifurcation is that some characteris-tics of system occur mutation phenomenon when the parameters change and pass some threshold values. Hopf bifurcation is a common important branch, it main-ly studies that the stability of equilibrium changes and produces small amplitude periodic solutions near the equilibrium as parameters’ changes. This thesis main-ly applies the theories and methods, such as, Lyapunov stability theory, LaSalle invariance principle, topological degree theory, center manifold theorem, normal form method, global bifurcation theorem, and so on, to study the local and global stability, existence of periodic solutions, fixed point bifurcation, local and global Hopf bifurcation, persistence of system for several biological dynamic systems. The details are as follows:Firstly, an SIRS model is studied, using the delay as parameter, it is obtained that the globally asymptotical stability of disease-free equilibrium, locally asymptot-ical stability, existence of Hopf bifurcation of endemic equilibrium and persistence of system. After that, a non-autonomous SIR model is studied, using the coincidence degree theory, the sufficient conditions which ensure the global existence of positive periodic solution, uniqueness and global stability are also obtained. Considering the effect of incubation period, furthermore, an SEIRS model is investigated, it is obtained that the globally asymptotical stability of disease-free equilibrium, locally asymptotical stability of endemic equilibrium, existence of global Hopf bifurcation and the sufficient conditions of persistence of system.Secondly, complicated dynamics of a plankton ecosystem is investigated. At first, the stability analysis on equilibria of ordinary differential equation system is given. After that, introducing the delay in ordinary differential equation system and using delay as parameter, it is obtained that the sufficient conditions that the boundary equilibrium is globally asymptotically stable and unstable. Furthermore, under certain conditions, the stability switches phenomenon can occur near the pos- itive equilibrium and the periodic solutions may appear, as the increasing of delay, the periodic solutions still exist, which shows the existence of global Hopf bifurca-tion. At the same time, it is found that, as the increasing of toxin release rate, the unstable intervals of positive equilibrium narrow, which shows the toxin contributes to the stability of system. At last, introducing the diffusion term on the basis of the delayed system and investigating the effects both diffusion and delay, it obtains that the diffusion can not vary the stability of equilibrium, that is, the Turing unsta-bility cannot occur. The effects of big and small diffusions to Hopf bifurcation are investigated, under certain conditions, the space inhomogeneous periodic solutions may produce and the algorithm is given to determine the properties on bifurcation periodic solutions.At last, the chaos control strategies of two systems are studied. Firstly, a plank-ton ecosystem with two delays is investigated, using delays as parameters, it obtains that the system may appear stability switches phenomenon. As the changes of pa-rameters, chaos phenomenon may occur, furthermore, by numerical simulations, it can get that the increasing delay and toxin release rate can make chaos disappear. The increasing maximum conversion of zooplankton can make system from stable state, undergoing period-doubling bifurcation, lead to chaos finally. Secondly, a chaotic plankton ecosystem is investigated using delayed feedback control method, taking delay as parameter, the conditions that the controlled system occurs Hopf bifurcation are obtained, that is, under certain conditions, the unstable periodic solution of system can change into stable periodic solution or stable equilibrium when the delay takes some certain value, which shows the effectiveness of control.