This paper focuses on the existence of a stationary distribution of stochastic Markov jump coupled systems (SMJCSs) for the first time, in which the coupling effect is considered. A new technique that is combining the graph theory, M-matrix method with the Lyapunov method is used to study stationary distribution, and sufficient conditions are presented to ensure the existence of a stationary distribution, which are more applicable and suitable for various fields, such as neural networks, biomathematics, physics and so forth. Moreover, sufficient conditions presented indicate that the existing region of stationary distribution is related to stochastic disturbance and the dimension of a system closely. Also, theoretical results are applied to stochastic Markov jump coupled oscillators systems in physics and then a specific theorem is presented. Eventually, some simulations are given to verify the feasibility and availability of our theoretical results. (C) 2019 Elsevier Ltd. All rights reserved.

Graph-theoretic approach as a new technique is applied to analyze the issue of stationary distribution for stochastic multi-group models (MGMs) with Markovian switching and dispersal. The feature of our model is combining the white noise, the dispersal with Markovian switching, which can model systems in practice reasonably. By constructing a global Lyapunov function for stochastic MGMs with Markovian switching and dispersal, sufficient criteria, which guarantee the existence of a stationary distribution, are presented via the combination of the graph theory, the Lyapunov method and the M-matrix method. Moreover, stochastic coupled oscillators with Markovian switching and dispersal are presented as a practical application to illustrate our results in the end.