Research On Stability And Bifurcation Control For Delayed DynamicalSystems In Biological

The evolvement of Ecological system is not only depended on the state of the present time but also depended on the bypast state, so the ecological system with time delays can be more close to ecological system. In this dissertation, stability, bifurcation and bifurcation control for the population dynamical systems with three delays are considered. The aim of bifurcation control is to design a controller to modify the bifurcation properties of a given dynamic systems in biology, and achieving some desirable dynamical behaviors. By empoloying theory of stability, bifurcation theorem and theory of bifurcation control, a systematic investigation into the population dynamical systemis with three delays has been studies, and the time delay hybrid controller has been designed. At the same time, the exponentional stability of the delayed epidemic model is also considered. The main contents in this dissertation can be stated as follows:Firstly, two species predator-prey system with three delays is investigated. The stability and Hopf bifurcations of the system are investigated. Moreover, a new hybrid strategy is proposed to control the Hopf bifurcation, in which parameter perturbation and time-delayed state feedback are used to delay the onset of an inherent bifurcation or let the bifurcation disapper.Secondly, the problem of Hopf bifurcation for a two species competition system with three delays is considered. In particular, the formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions are derived by using the normal form theory and center manifold theorem. Finally, numerical simulations are given to support the theoretical results.Thirdly, three species delayed Lotka-Volterra system is researched, the problem of Hopf bifurcation control for a three species delayed competitive system is researched, a new hybrid strategy is proposed to control the Hopf bifurcation, in which state feedback and parameter perturbation are used to delay the onset of an inherent bifurcation or let the bifurcation disapper. In particular, the formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions of controlled system are also derived by using the normal form theory and center manifold theorem. Finally, numerical simulation results confirm that the hybrid control is efficient in controlling Hopf bifurcation.Fourthly, a nonlinear red tide dynamic model to study the effect of a toxicant emitted into the environment is considered. Using the method of the modern nonlinear dynamics, the stability and bifurcation of the system are discussed, and the thresholds of persistence and extinction for each species have been obtained.It's showed that that a sequence of Hopf bifurcations occur at the interior equilibrium as the delay increase or the growth rates increase.Finally, numerical simulations are given to support the theoretical results.Fifthly, the stability of two delayed epidemic models are researched. an SIS epidemic model with distributed time delays is discussed by employing inequity technique, some new sufficient conditions on global exponential stability of the disease-free equilibrium and global exponential stability of the endemic equilibrium are obtained.Sixthly, by using the theory of differential equations, we discuss the propagation regularity of network viruses without the latent period and with the latent period.We also obtain the critical value R0 which determine whether the network viruses remove or not.It is proved that there is a free equilibrium (the network virus remove) if R0 < 1, and an epidemic equilibrium (the network virus prevalent) if R0 > 1, so that we put forward a method of removing the network viruses. Stability of the free equilibriums and the endemic equilibriums are discussed.This research can use manpower and material resources more sparingly than the method of statistics.