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

  • Yan Liu
  • Wenxue LiEmail author
  • Jiqiang Feng


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.


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



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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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of Technology (Weihai)WeihaiPeople’s Republic of China
  2. 2.College of mathematics and statisticsShenzhen UniversityShenzhenPeople’s Republic of China

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