Analysis On Stability Of Stochastic Biological Systems With Time Delays

There always exist stochastic perturbations in the real biological systems. In or-der to describe the systems, and show their developments and changes, it is necessaryto take stochastic perturbations fully into account during systems modeling. Besides,time delays are commonly encountered in practice. If considering the efects of stochas-tic perturbations on biological systems with time delays, one should naturally resort tostochastic diferential equations with delays or, more general, stochastic functional dif-ferential equations. In this dissertation, based on the Lyapunov stability theory andstochastic functional diferential equations theory, stability problems of stochastic bio-logical systems with time delays are investigated by employing Lyapunov functions, thecomparison principle, inequalities technique as well as nonsingular M-matrix. The maincontributions of this dissertation are summarized as follows:1. First, the background, research signifcance and current situation of the selectedtopic are reviewed. Then, some preliminaries are presented.2. A delayed prey-predator system with random perturbation is investigated, whichis also based on a modifed version of the Lesile-Gower and Holling II schemes. Byapplying the comparison principle and Ito's formula, a unique positive global solution andits pth moment's upper bound are obtained respectively. And then its global asymptoticstability is proved by constructing Lyapunov functional. Finally, two types of numericalsimulations, state portrait and phase portrait, are presented to illustrate the correctnessof the proposed theory.3. A stochastic SIR epidemic system with distributed time delay is investigated.First, the existence and uniqueness of global positive solution is proved. Furthermore,the asymptotic properties of the positive solution are investigated. There is neitherdisease-free equilibrium nor endemic equilibrium for the stochastic delayed system afteradding stochastic perturbations to the associated deterministic system. In order to showthe stability to some extent, asymptotic properties around the disease-free equilibriumand endemic equilibrium of the deterministic system are studied. Finally, numericalsimulations are presented to illustrate the mathematical fndings.4. Stochastic delayed epidemic systems with one Markovian switching parameterincluding stochastic switched SIR epidemic system with discrete or distributed time de-lay and stochastic switched SIRS with time delay are investigated. The existence anduniqueness of the global positive solution of every system is proved respectively. Us- ing diferential Lyapunov functions, stochastic stability of the disease-free equilibrium ofthe associated stochastic switched system with time delay is proved. In addition, suf-cient stability conditions are obtained for the stochastic subsystems. Finally, numericalsimulations are illustrate to support the mathematical fndings.5. A stochastic delayed SEIRS epidemic model with some Markovian switchingparameters is constructed and investigated. It is shown that this model has a uniqueglobal positive solution. Using Lyapunov methods, exponential mean square stability ofthe disease-free equilibrium of the linearization of the system is proved under suitableconditions, and then stochastic stability of the disease-free equilibrium of the system isalso studied. Moreover, the sufcient stability condition of the disease-free equilibrium ofthe subsystem is obtained. Numerical simulations are presented to illustrate the results.Finally, the main results of the dissertation are concluded and some issues for futureresearch are proposed.