Journal of Dynamics and Differential Equations

, Volume 30, Issue 2, pp 667–685

# Graph-Theoretical Method to the Existence of Stationary Distribution of Stochastic Coupled Systems

• Yan Liu
• Wenxue Li
• Jiqiang Feng
Article

## Abstract

In this paper, by linking Fokker–Planck equations with stochastic coupled systems, a new method is provided to investigate the existence of a stationary distribution of stochastic coupled systems. Based on the graph theory and the Lyapunov method, an appropriate Lyapunov function associated with stationary Fokker–Planck equations is constructed. Moreover, a Lyapunov-type theorem and a coefficients-type criterion are obtained to guarantee the existence of a stationary distribution. Furthermore, theoretical results are applied to explore the existence of a stationary distribution of stochastic predator–prey models with dispersal and a sufficient criterion is presented correspondingly. Finally, a numerical example is given to illustrate the effectiveness of our results.

## Keywords

Graph theory Stationary distribution Stochastic coupled systems Fokker–Planck equation Stochastic predator–prey model

## Notes

### Acknowledgements

The authors really appreciate the reviewer’s valuable comments. The second author was supported by the NNSF of China (Nos. 11301112 and 11401136), the NNSF of Shandong Province (Nos. ZR2013AQ003 and ZR2014AQ010) and China Postdoctoral Science Foundation funded Project (No. 2014T70313). The third author was supported by the National Natural Science Foundation of China (61401283) and Educational Commission of Guangdong Province, China (2014KTSCX113).

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