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Synchronized stationary distribution of stochastic multi-group models with dispersal

  • Yan Liu
  • Anran Liu
  • Wenxue Li
Original Article
  • 15 Downloads

Abstract

This paper is concerned with a new stationary distribution named synchronized stationary distribution. It is the first time to apply such kind of distribution to stochastic multi-group models with dispersal. And the existing region of synchronized stationary distribution is closely related to the coupling structure, stochastic disturbance intensity as well as the coefficients of models. We propose two main theorems to ensure the existence of a synchronized stationary distribution by combining Kirchhoff’s Matrix Tree Theorem in the graph theory as well as the Lyapunov method. Additionally, the value of our results is shown by applying them to stochastic coupled oscillators and stochastic coupled Rössler-like circuits with multiple dispersal. Correspondingly, two numerical examples are given to illustrate that our results are feasible and effective.

Keywords

Synchronized stationary distribution Kirchhoff’s Matrix Tree Theorem Stochastic multi-group models Stochastic coupled oscillators 

Notes

Acknowledgements

The authors really appreciate the valuable comments of editors and reviewers. This work was supported by Shandong Province Natural Science Foundation (Nos. ZR2018MA005, ZR2018MA020, ZR2017MA008); the Key Project of Science and Technology of Weihai (No. 2014DXGJMS08), the Project of Shandong Province Higher Educational Science and Technology Program of China (No. J18KA218) and the Innovation Technology Funding Project in Harbin Institute of Technology (No. HIT.NSRIF.201703).

References

  1. 1.
    Wang Y, Gao J (2014) Global dynamics of multi-group SEI animal disease models with indirect transmission. Chaos Solitons Fractals 69:81–89MathSciNetCrossRefGoogle Scholar
  2. 2.
    Li H, Huang C, Chen G, Liao X, Huang T (2017) Distributed consensus optimization in multiagent networks with time-varying directed topologies and quantized communication. IEEE Trans Cybern 47:2044–2057CrossRefGoogle Scholar
  3. 3.
    Yang Q, Mao X (2013) Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations. Nonlinear Anal Real World Appl 14:1434–1456MathSciNetCrossRefGoogle Scholar
  4. 4.
    Zhu W, Li H, Jiang Z (2017) Consensus of multi-agent systems with time-varying topology: an event-based dynamic feedback scheme. Int J Robust Nonlinear Control 27:1339–1350MathSciNetCrossRefGoogle Scholar
  5. 5.
    Feng X, Wang K, Zhang F, Teng Z (2017) Threshold dynamics of a nonlinear multi-group epidemic model with two infinite distributed delays. Math Methods Appl Sci 40:2762–2771MathSciNetCrossRefGoogle Scholar
  6. 6.
    Xu J, Geng Y, Zhou Y (2017) Global stability of a multi-group model with distributed delay and vaccination. Math Methods Appl Sci 40:1475–1486MathSciNetCrossRefGoogle Scholar
  7. 7.
    Guo Y, Ding X, Li Y (2016) Stochastic stability for pantograph multi-group models with dispersal and stochastic perturbation. J Frankl Inst Eng Appl Math 353:2980–2998MathSciNetCrossRefGoogle Scholar
  8. 8.
    Zhang C, Li W, Wang K (2015) Graph-theoretic approach to stability of multi-group models with dispersal. Discrete Contin Dyn Syst Ser B 20:259–280MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen W, Zhang B, Ma Q (2018) Decay-rate-dependent conditions for exponential stability of stochastic neutral systems with Markovian jumping parameters. Appl Math Comput 321:93–105MathSciNetGoogle Scholar
  10. 10.
    Feng J, Xu C (2018) A graph-theoretic approach to exponential stability of stochastic complex networks with time-varying delays. Neurocomputing 272:453–460CrossRefGoogle Scholar
  11. 11.
    Wang P, Jin W, Su H (2018) Synchronization of coupled stochastic complex-valued dynamical networks with time-varying delays via aperiodically intermittent adaptive control. Chaos 28:043114MathSciNetCrossRefGoogle Scholar
  12. 12.
    Zong X, Wu F, Huang C (2016) The moment exponential stability criterion of nonlinear hybrid stochastic differential equations and its discrete approximations. Proc R Soc Edinb Sect A Math 146:1303–1328MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lu J, Ho D, Wu L (2009) Exponential stabilization of switched stochastic dynamical networks. Nonlinearity 22:889–911MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lu J, Ho D (2011) Stabilization of complex dynamical networks with noise disturbance under performance constraint. Nonlinear Anal Real World Appl 12:1974–1984MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zhang Y, Sun J (2010) Stability of impulsive linear hybrid systems with time delay. J Syst Sci Complex 23:738–747MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zhang Y (2013) Stability of discrete-time delay Markovian jump systems with stochastic non-linearity and impulses. IET Control Theory Appl 7:2178–2187MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zhu E, Yin G, Yuan Q (2016) Stability in distribution of stochastic delay recurrent neural networks with Markovian switching. Neural Comput Appl 27:2141–2151CrossRefGoogle Scholar
  18. 18.
    Ma S, Kang Y (2018) Exponential synchronization of delayed neutral-type neural networks with Lévy noise under non-Lipschitz condition. Commun Nonlinear Sci Numer Simul 57:372–387MathSciNetCrossRefGoogle Scholar
  19. 19.
    Xiao Y, Tang S, Sun Z, Song X (2018) Positive role of multiplication noise in attaining complete synchronization on large complex networks of dynamical systems. Appl Math Model 54:803–816MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wu Y, Yan S, Fan M, Li W (2018) Stabilization of stochastic coupled systems with Markovian switching via feedback control based on discrete-time state observations. Int J Robust Nonlinear Control 1:247–265MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wang J, Feng J, Xu C, Zhao Y (2013) Exponential synchronization of stochastic perturbed complex networks with time-varying delays via periodically intermittent pinning. Commun Nonlinear Sci Numer Simul 18:3146–3157MathSciNetCrossRefGoogle Scholar
  22. 22.
    Feng J, Yang P, Zhao Y (2016) Cluster synchronization for nonlinearly time-varying delayed coupling complex networks with stochastic perturbation via periodically intermittent pinning control. Appl Math Comput 291:52–68MathSciNetGoogle Scholar
  23. 23.
    Zhang W, Li C, Huang T, Qi J (2014) Global exponential synchronization for coupled switched delayed recurrent neural networks with stochastic perturbation and impulsive effects. Neural Comput Appl 25:1275–1283CrossRefGoogle Scholar
  24. 24.
    Gan Q (2013) Synchronization of unknown chaotic neural networks with stochastic perturbation and time delay in the leakage term based on adaptive control and parameter identification. Neural Comput Appl 22:1095–1104CrossRefGoogle Scholar
  25. 25.
    Acebrón J, Perales A, Spigler R (2001) Bifurcations and global stability of synchronized stationary states in the Kuramoto model for oscillator populations. Phys Rev E 64:016218CrossRefGoogle Scholar
  26. 26.
    Hasminskii R (1980) Stochastic stability of differential equations. In: Mechanics and analysis. Sijthoff and Noordhoff, NetherlandsGoogle Scholar
  27. 27.
    Huang W, Ji M, Liu Z, Yi Y (2015) Steady states of Fokker–Planck equations: I. Existence. J Dyn Differ Equ 27:721–742MathSciNetCrossRefGoogle Scholar
  28. 28.
    Li M, Shuai Z (2010) Global-stability problem for coupled systems of differential equations on networks. J Differ Equ 248:1–20MathSciNetCrossRefGoogle Scholar
  29. 29.
    Guo H, Li M, Shuai Z (2008) A graph-theoretic approach to the method of global Lyapunov functions. Proc Am Math Soc 136:2793–2802MathSciNetCrossRefGoogle Scholar
  30. 30.
    Liu Y, Li W, Feng J (2018) Graph-theoretical method to the existence of stationary distribution of stochastic coupled systems. J Dyn Differ Equ 30:667–685MathSciNetCrossRefGoogle Scholar
  31. 31.
    Zhao Y, Yuan S, Ma J (2015) Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment. Bull Math Biol 77:1285–1326MathSciNetCrossRefGoogle Scholar
  32. 32.
    He Y, Zhang D, Zhang H (2006) A novel atomic force microscope with high stability and scan speed. Instrum Sci Technol 34:547–554CrossRefGoogle Scholar
  33. 33.
    Zhao Y, Yuan S, Zhang T (2016) The stationary distribution and ergodicity of a stochastic phytoplankton allelopathy model under regime switching. Commun Nonlinear Sci Numer Simul 37:131–142MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zhu C, Zhu W, Yang Y (2012) Design of feedback control of a nonlinear stochastic system for targeting a pre-specified stationary probability distribution. Probab Eng Mech 30:20–26CrossRefGoogle Scholar
  35. 35.
    Kinnally M, Williams R (2010) On existence and uniqueness of stationary distributions for stochastic delay differential equations with positivity constraints. Electron J Probab 15:409–451MathSciNetCrossRefGoogle Scholar
  36. 36.
    West D (1996) Introduction to graph theory. Prentice Hall, Upper Saddle RiverzbMATHGoogle Scholar
  37. 37.
    Lajmanovich A, Yorke J (1976) A deterministic model for gonorrhea in a nonhomogeneous population. Math Biosci 28:221–236MathSciNetCrossRefGoogle Scholar
  38. 38.
    Li M, Jin Z, Sun G, Zhang J (2017) Modeling direct and indirect disease transmission using multi-group model. J Math Anal Appl 446:1292–1309MathSciNetCrossRefGoogle Scholar
  39. 39.
    Mao X (2007) Stochastic differential equations and applications. Horwood Publishing, ChichesterzbMATHGoogle Scholar
  40. 40.
    Huang W, Ji M, Liu Z, Yi Y (2015) Steady states of Fokker–Planck equations: II. Non-existence. J Dyn Differ Equ 27:743–762MathSciNetCrossRefGoogle Scholar
  41. 41.
    Huang W, Ji M, Liu Z, Yi Y (2016) Steady states of Fokker–Planck equations: III. Degenerate diffusion. J Dyn Differ Equ 28:127–141MathSciNetCrossRefGoogle Scholar
  42. 42.
    Zou X, Fan D, Wang K (2013) Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete Contin Dyn Syst Ser B 18:1507–1519MathSciNetCrossRefGoogle Scholar
  43. 43.
    Zou X, Lv J (2017) A new idea on almost sure permanence and uniform boundedness for a stochastic predator-prey model. J Frankl Inst Eng Appl Math 354:6119–6137MathSciNetCrossRefGoogle Scholar
  44. 44.
    Guo W, Cai Y, Zhang Q, Wang W (2018) Stochastic persistence and stationary distribution in an SIS epidemic model with media coverage. Physica A 492:2220–2236MathSciNetCrossRefGoogle Scholar
  45. 45.
    Ha S, Lee J, Li Z (2017) Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators. Netw Heterog Media 12:1–24MathSciNetCrossRefGoogle Scholar
  46. 46.
    Tôrres L, Hespanha J, Moehlis J (2015) Synchronization of identical oscillators coupled through a symmetric network with dynamics: a constructive approach with applications to parallel operation of inverters. IEEE Trans Autom Control 60:3226–3241MathSciNetCrossRefGoogle Scholar
  47. 47.
    Heisler I, Braun T, Zhang Y (2003) Experimental investigation of partial synchronization in coupled chaotic oscillators. Chaos 13:185–194CrossRefGoogle Scholar
  48. 48.
    Xiao Y, Tang S, Xu S (2012) Theoretical analysis of multiplicative-noise-induced complete synchronization in global coupled dynamical network. Chaos 22:013110MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of Technology (Weihai)WeihaiPeople’s Republic of China

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