Existence Of Stationary Distribution For Several Kinds Of Stochastic Network Systems

Stationary distribution has been widely used in various fields,including stochastic predator-prey models in biological mathematics,epidemic models in infectious diseases,population models in demography and so on.Scholars have devoted themselves to the study of the existence of a stationary distribution.On the other hand,stochastic network systems have a wide range of applications and they can be used to simulate mechanical systems,power systems and many other practical systems,so it is very important to study this kind of systems.However,the research on the existence of a stationary distribution of stochastic network systems is not sufficient,so it is of practical significance to consider the existence of a stationary distribution of stochastic network systems.Based on the graph theory,Lyapunov second method and M-matrix theory,in this thesis,the existence of a stationary distribution of several kinds of stochastic network systems with different environmental noises is considered.The specific research contents are as follows.Firstly,the existence of a stationary distribution in network systems with white noise is analyzed.Through connecting the stationary Fokker-Planck equation with a stochastic network system,a suitable Lyapunov function is constructed,and sufficient conditions for the existence of a stationary distribution are given.Then,theoretical results are applied to the stochastic predator-prey model,with a criterion for the existence of a stationary distribution obtained.Numerical simulations are also given to illustrate the effectiveness of theoretical results.Secondly,the existence of a stationary distribution for stochastic network systems with Markovian switching is studied.A suitable Lyapunov function is constructed for the network system with Markovian switching,and two theorems are given to guarantee the existence of a stationary distribution.After that,theoretical results are applied to stochastic coupled oscillator models with Markovian switching,and sufficient conditions are given to ensure the existence of a stationary distribution.Finally,a numerical example is given to illustrate the effectiveness of theoretical results.Thirdly,the existence of a stationary distribution of stochastic multi-group models with multiple dispersal and Markovian switching is investigated.Based on multi-digraph theory,a suitable Lyapunov function is constructed,and sufficient conditions to guarantee the existence of a stationary distribution are given.As an application of theoretical results,stochastic coupled oscillators models with dispersal and Markovian switching are given and a theorem is presented to illustrate theoretical results.Fourthly,the effects of continuous feedback control for the existence of a stationary distribution of stochastic network systems are studied.Here,feedback control is introduced into the problem of the existence of a stationary distribution.Two theorems to ensure the existence of a stationary distribution are obtained.As an application of theoretical results,the existence of a stationary distribution for stochastic coupled oscillators models is considered and sufficient conditions to ensure the existence of a stationary distribution are presented.Finally,the influences of aperiodically intermittent control on the existence of a stationary distribution of stochastic multi-weights network systems are studied.By constructing a suitable Lyapunov function,sufficient conditions are given to ensure the existence of a stationary distribution.At the same time,the theoretical results are applied to stochastic multi-weights coupled oscillators models,and the effects of aperiodically intermittent control on the existence of a stationary distribution are analyzed.