Global Attractor of Reaction–Diffusion Gene Regulatory Networks with S-Type Delay

  • Xiao LiangEmail author
  • Xianglai Zhuo
  • Ruili Wang


This paper studies the existence of the global attractor for the reaction–diffusion gene regulatory networks with s-type delay. Firstly, two Hanalay inequalities are proposed and proved, then the uniform boundedness theorem is proved by using these inequalities and semigroup theory and functional analysis technique, the operator splitting technique based on the superposition principle is also utilized to deal with the asymptotic compact of the semigroup. Then some sufficient conditions on existence of global attractor are established, which are easily verifiable and have a wider application range. At last, two examples are discussed to validate the effectiveness of our results. Moreover, the simulation is given by using Matlab.


Global attractor Improved Hanalay inequality Gene regulatory networks S-type delay MRNA Protein 



We feel specially honored to meet the serious editor and rigorous reviewers. The paper is significantly improved after following their suggestion and comments. We are grateful to them. This work is founded by Shandong Provincial Natural Science Foundation under Contract No. ZR2017BA014, National Natural Science Foundation of China under Contract No. 61573008, and the Development Program for Defense Department of China under Contract No. C1520110002. The first author thanks Shandong University of Science and Technology for funding his visiting career in University of Southern California, Los Angeles, CA, from Jan. 2018–Jan. 2019.

Compliance with Ethical Standards

Conflicts of interest

The authors have declared that no conflict of interest exists.


  1. 1.
    Jong H (2002) Modeling and simulation of genetic regulatory systems: a literature review. J Comput Biol 9:67–103MathSciNetCrossRefGoogle Scholar
  2. 2.
    Karlebach G, Shamir R (2008) Modelling and analysis of gene regulatory networks. Nat Rev Mol Cell Biol 9:770–780CrossRefGoogle Scholar
  3. 3.
    Thomas R (1973) Boolean formalization of genetic control circuits. J Theor Biol 42:563–585CrossRefGoogle Scholar
  4. 4.
    Klipp E (2005) Systems biology in practice: concepts, implementation and application. Wiley, WeinheimCrossRefGoogle Scholar
  5. 5.
    Li C, Chen L, Aihara K (2007) Stochastic synchronization of genetic oscillator networks. BMC Syst Biol 11:6–12CrossRefGoogle Scholar
  6. 6.
    Chen L, Aihara K (2002) Stability of genetic regulatory networks with time delays. IEEE Trans Circuits Syst Fundam Theory Appl 49:602–608MathSciNetCrossRefGoogle Scholar
  7. 7.
    Goodwin B (1965) Oscillatory behavior of enzymatic control processes. Adv Enzyme Reg 3:425–439CrossRefGoogle Scholar
  8. 8.
    van Zon J, Morelli M, Tanase-Nicola S, Wolde P (2006) Diffusion of transcription factors can drastically enhance the noise in gene expression. Biophys J 91:4350–4367CrossRefGoogle Scholar
  9. 9.
    Busenberg S, Mahaffy J (1985) Interaction of spatial diffusion and delays in models of genetic control by repression. J Mol Biol 22:313–333MathSciNetzbMATHGoogle Scholar
  10. 10.
    Knierim J, Zhang K (2012) Attractor dynamics of spatially correlated neural activity in the limbic system. Annu Rev Neurosci 35:267–285CrossRefGoogle Scholar
  11. 11.
    Zhang X, Han YY, Wu L, Zou JH (2016) M-matrix-based globally asymptotic stability criteria for genetic regulatory networks with time-varying discrete and unbounded distributed delays. Neurocomputing 174:1060–1069CrossRefGoogle Scholar
  12. 12.
    Ma Q, Shi G, Xu S, Zou Y (2011) Stability analysis for delayed genetic regulatory networks with reaction–diffusion terms. Neural Comput Appl 20:507–516CrossRefGoogle Scholar
  13. 13.
    Fan X, Xue Y, Zhang X, Ma J (2017) Finite-time state observer for delayed reaction–diffusion genetic regulatory networks. Neurocomputing 227:18–28CrossRefGoogle Scholar
  14. 14.
    Zhou J, Xu S, Shen H (2011) Finite-time robust stochastic stability of uncertain stochastic delayed reaction–diffusion genetic regulatory networks. Neurocomputing 74:2790–2796CrossRefGoogle Scholar
  15. 15.
    Huang S, Eichler G, Bar-Yam Y, Ingber D (2005) Cell fates as high-dimensional attractor states of a complex gene regulatory network. Phys Rev Lett 94:1–4Google Scholar
  16. 16.
    Hale J, Lunel Verduyn SM (1993) Introduction to functional differential equations. Springer, New YorkCrossRefGoogle Scholar
  17. 17.
    Liang X, Wang L, Wang R (2018) Random attractor of reaction diffusion Hopfield neural networks driven by Wiener processes. Math Probl Eng 2018:2538658MathSciNetGoogle Scholar
  18. 18.
    Temam R (1997) Infinite-dimensional dynamical systems in mechanics and physics. Springer, New YorkCrossRefGoogle Scholar
  19. 19.
    Liang X, Wang L, Wang Y, Wang R (2016) Dynamical behavior of delayed reaction–diffusion Hopfield neural networks driven by infinite dimensional Wiener processes. IEEE Trans Neural Netw Learn Syst 27:1231–1242MathSciNetGoogle Scholar
  20. 20.
    Wang L, Wang Y (2008) Stochastic exponential stability of the delayed reaction diffusion interval neural networks with Markovian jumpling parameters. Phys Lett A 356:346–352CrossRefGoogle Scholar
  21. 21.
    Sun J, Wan L (2005) Convergence dynamics of stochastic reaction–diffusion recurrent neural networks with delays. Int J Bifurc Chaos Appl Sci Eng 15:2131–2144MathSciNetCrossRefGoogle Scholar
  22. 22.
    Babin A, Vishik M (1983) Attractors of partial differential evolution equations and estimates of their dimension. Rus Math Surv 38:151–213CrossRefGoogle Scholar
  23. 23.
    Chueshov ID (1993) Global attractors for non-linear problems of mathematical physics. Rus Math Surv 48:133–161CrossRefGoogle Scholar
  24. 24.
    Da Prato G, Zabczyk J (1992) Stochastic equations in infinite dimensions. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  25. 25.
    Liang X, Wang R (2018) Global well-posedness and dynamical behavior of delayed reaction–diffusion BAM neural networks driven by Wiener processes. IEEE Access 6:69265–69278CrossRefGoogle Scholar
  26. 26.
    Guo R, Zhang Z, Gao M (2018) State estimation for complex-valued memristive neural networks with time-varying delays. Adv Differ Equ 118:210–219MathSciNetzbMATHGoogle Scholar
  27. 27.
    Meng X, Zhao S, Feng T et al (2016) Dynamics of a novel nonlinear stochastic sis epidemic model with fouble epidemic hypothesis. J Math Anal Appl 433:227–242MathSciNetCrossRefGoogle Scholar
  28. 28.
    Zhuo X, Zhang F (2018) Stability for a new discrete ratio-dependent predator-prey system. Qual Theory Dyn Syst 17:189–202MathSciNetCrossRefGoogle Scholar
  29. 29.
    Zhao W, Li J, Meng X (2015) Dynamical analysis of sir epidemic model with nonlinear pulse vaccination and lifelong immunity. Discrete Dyn Nat Soc 2015:1–10 Article ID 848623MathSciNetzbMATHGoogle Scholar
  30. 30.
    Meng X, Zhang L (2016) Evolutionary dynamics in a Lotka–Volterra competition model with impulsive periodic disturbance. Math Methods Appl Sci 39:177–188MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ma H, Jia Y (2016) Stability analysis for stochastic differential equations with infinite Markovian switchings. J Math Anal Appl 435:593–605MathSciNetCrossRefGoogle Scholar
  32. 32.
    Meng X, Zhao S, Zhang W (2015) Adaptive dynamics analysis of a predator-prey model with selective disturbance. J Math Anal Appl 266:946–958MathSciNetzbMATHGoogle Scholar
  33. 33.
    Zhuo X (2018) Global attractability and permanence for a new stage-structured delay impulsive ecosystem. J Appl Anal Comput 8:457–457MathSciNetGoogle Scholar
  34. 34.
    Bian F, Zhao W (2017) Dynamical analysis of a class of prey-predator model with Beddington–DeAngelis functional response, stochastic perturbation, and impulsive toxicant input. Complexity 2017:1–18CrossRefGoogle Scholar
  35. 35.
    Zhang H, Hu J, Zou L, Yu X, Wu Z (2018) Event-based state estimation for time-varying stochastic coupling networks with missing measurements under uncertain occurrence probabilities. Int J Gen Syst 47:506–521MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of Mathematics and System ScienceShandong University of Science and TechnologyQingdaoChina
  2. 2.Institute of Applied Physics and Computational MathematicsBeijingChina

Personalised recommendations