Study On The Stochastic Dynamic Systems

The paper goes deep into studying the dynamic behaviors of several kinds of nonlin-ear stochastic population systems. Considering the effect of stochastic perturbation, time delay and diffusion on the stability of systems, and applying Lyapunov function, Ito for-mula and stochastic processes theory and methods, we investigate the dynamic behaviors of stochastic population systems, including existence and uniqueness of solutions, stabil-ity, stochastic persistence and extinction, boundedness and so on. Numerical simulations showing the complex dynamics of stochastic population systems are given. This paper is composed of six chapters and main results are described as follows:In Chapter1, we introduce the historical background and the recent development of stochastic population dynamic systems. Moreover, the main purposes of this paper are also briefly introduced.In Chapter2, we investigate the dynamic behaviors of an epidemic system with non-linear incidence rate and diffusion or stochastic perturbation. By analyzing characteristic equation and Lyapunov function, the stabilities of the endemic equilibrium of the deter-ministic system are studied. The sufficient conditions of global asymptotical stability of the equilibrium for the diffusion system are also obtained, i.e., the conditions of the dis-ease extinction or persistence. We carry out the analytical study for the stochastic system in details by constructing Lyapunov function and using stochastic differential equation theory and Ito formula, also find out the conditions for stochastically ultimate bounded-ness, permanence and asymptotic stability of the endemic equilibrium.In Chapter3, we investigate the dynamic behaviors of a Harrison-type predator-prey system involving time delay within stochastic environment. By using Routh-Hurwitz criterion and Lyapunov function, the stabilities of the deterministic and delay systems are showed. The stability and direction of Hopf bifurcation are studied by applying the center manifold theorem and the normal form theory, numerical simulations have been used to demonstrate the change of time-series plot and phase plot with respect to the time delay. We construct the stochastic delay differential equation system to discuss the effect of environmental noise on the dynamical behavior. For example, existence and uniqueness of the positive solutions, stochastically ultimately boundness and so on.In Chapter4, we investigate the dynamic behaviors of a Holling-type II phytoplankton-zooplankton system with stochastic perturbations. By constructing Lyapunov function, some sufficient conditions to ensure that the equilibrium of system is asymptotically sta-ble and Hopf bifurcation exists near the equilibrium are derived. Under the perturbation of environmental noise, we show that there is a globally positive solution to the stochastic system, which is stochastically ultimately bounded and permanent. Also we find out some sufficient conditions for stochastically asymptotically stable of the solution.In Chapter5, we investigate a spatial version of a Michaelis-Menten-type predator-prey system incorporating a prey refuge, which contains some important factors, such as environmental noise, external periodic forces and diffusion processes. We give a general survey of the linear stability analysis and determine the condition of Turing instability, and present numerical evidence of time evolution of patterns in the system. Also the ef-fect of refuge, environmental noise and external periodic forces on the reaction-diffusion predation system are discussed.In Chapter6, the studying contents and the main results of this paper are briefly summarized, some research prospects and the directions are also proposed.Our obtained results explain many biodynamic behaviors of stochastic population systems, such as populations permanence and extinction, etc., which helps people to un-derstand biodynamics scientifically so that some interactions of population and interac-tions of between population and environment can be intend to control.