Stability Of Two Types Of Stochastic Nonlinear Differential Equations With Regime Switching And Its Applications

This dissertation mainly investigates stochastic stability of nonlinear differ-ential equations,and its application to Lotka-Volterra ecosystems.It examines the influences of the stochastic factors(Brownian motion,Markov chain and sin-gularly perturbed Markov chains)on long-term dynamical property.The whole dissertation consists of the following five chapters:Chapter 1 briefly introduces some related research on stochastic stability the-ory of differential equations and ecosystems.The contribution of this dissertation is also listed in this chapter.Hybrid differential equations with singularly perturbed Markov chains has been used in many practical scenarios such as manufacturing systems,control and optimization of large-scale systems.Up to now,related research mainly focuses on linear systems.Chapter 2 studies dynamical properties of hybrid nonlinear differ-ential equations with singularly perturbed Markov chains.The small parameter is used to reflect the different rates of regime switching among a large number of states of the discrete events.The Markov chains with two rates of regime switch-ing are called two-time-scale Markov chains or singular perturbed Markov chains.Under sufficient conditions of average principle,Using the perturbed Lyapunov method and asymptotic analysis,we derive the moment boundedness,moment exponential stability and convergence of asymptotic measures.Our result don’t need linear growth condition.Some examples of nonlinear SDEs with singularly perturbed Markov chains are given to illustrate these result.Coupled systems of nonlinear differential equations belong to another cat-egories of large-scale complex systems and have been applied widely,especially in the neural network,ecosystems and engineering control.Chapter 3 considers the coupled systems of stochastic differential equations with variable delays on networks.We first analyze the existence and uniqueness of global solution by combining the method of graph theory with the Lyapunov analysis.Then we utilize the directed graph theory to construct suitable Lyapunov functions.By the nonnegative semimartingale convergence theorem,we obtain the almost sure stability of sample solutions and the sufficient principles to locate their limit sets,which correlate closely with the topology property of the system.Finally we illus-trate the main results by examples from Lotka-Volterra ecosystems and vibration systems.Focusing on stochastic disturbance of population system involve the inside and outside,Chapter 4 considers the application of nonlinear stochastic systems in ecosystem.Sufficient conditions of stochastic permanence for generic stochas-tic multi-group Lotka-Volterra model,which are much weaker than the existing results in the literature are obtained.Then stochastic strong permanence and ergodic property for the mutualistic systems are derived.Moreover,the almost surely asymptotic estimate of solutions are given.These can specify some realistic recurring phenomena and reveal the fact that regime switching can suppress the impermanence.Due to the diversity of environment states and different rates among the ran-dom circumstances,basing on the previous chapter,Chapter 5 further investigates multi-group Lotka-Volterra models with singularly perturbed Markov chains.We first derive the weak convergence in finite horizon via Martingale method.By the method of perturbed Lyapunov analysis,using the structure of weak limit system as a bridge,we yield the stochastic(strong)permanence and extinction in long time scale.We go a further step to give the sufficient condition under which the measures of the original system converge asymptotically to the invariant measure of the weak limit system in infinite horizon.And the conditions can be regarded as average principles for the nonlinear multi-group ecosystem.The results reveals an interesting fact that the singularly perturbed Markov chain can suppress the impermanence.