Skip to main content

Advertisement

SpringerLink
Log in

Time Delay-Induced Instabilities and Hopf Bifurcations in General Reaction–Diffusion Systems

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

The distribution of the roots of a second order transcendental polynomial is analyzed, and it is used for solving the purely imaginary eigenvalue of a transcendental characteristic equation with two transcendental terms. The results are applied to the stability and associated Hopf bifurcation of a constant equilibrium of a general reaction–diffusion system or a system of ordinary differential equations with delay effects. Examples from biochemical reaction and predator–prey models are analyzed using the new techniques.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  • Bélair, J., Campbell, S.A.: Stability and bifurcations of equilibria in a multiple-delayed differential equation. SIAM J. Appl. Math. 54(5), 1402–1424 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Beretta, E., Kuang, Y.: Global analyses in some delayed ratio-dependent predator–prey systems. Nonlinear Anal. 32(3), 381–408 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Bodnar, M., Foryś, U., Poleszczuk, J.: Analysis of biochemical reactions models with delays. J. Math. Anal. Appl. 376(1), 74–83 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, S., Shi, J.: Global attractivity of equilibrium in Gierer–Meinhardt system with saturation and gene expression time delays (2012, submitted)

  • Chen, S., Wei, J., Shi, J.: Global stability and Hopf bifurcation in a delayed diffusive Leslie–Gower predator–prey system. Int. J. Bifurc. Chaos 22(3), 1250061 (2012)

    Article  MathSciNet  Google Scholar 

  • Cooke, K.L., Grossman, Z.: Discrete delay, distributed delay and stability switches. J. Math. Anal. Appl. 86(2), 592–627 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  • Crauste, F., Hbid, M.L., Kacha, A.: A delay reaction–diffusion model of the dynamics of botulinum in fish. Math. Biosci. 216(1), 17–29 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Culshaw, R.V., Ruan, S.: A delay-differential equation model of HIV infection of CD4(+) T-cells. Math. Biosci. 165(1), 27–39 (2000)

    Article  MATH  Google Scholar 

  • Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.-O.: Delay Equations: Functional-, Complex-, and Nonlinear Analysis. Applied Mathematical Sciences, vol. 110. Springer, New York (1995)

    MATH  Google Scholar 

  • Dutta, S., Ray, D.S.: Effects of delay in a reaction–diffusion system under the influence of an electric field. Phys. Rev. E 77(3), 036202 (2008)

    Article  Google Scholar 

  • Erneux, T.: Applied Delay Differential Equations. Surveys and Tutorials in the Applied Mathematical Sciences, vol. 3. Springer, New York (2009)

    MATH  Google Scholar 

  • Fan, D., Hong, L., Wei, J.: Hopf bifurcation analysis in synaptically coupled HR neurons with two time delays. Nonlinear Dyn. 62(1–2), 305–319 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Faria, T.: Stability and bifurcation for a delayed predator–prey model and the effect of diffusion. J. Math. Anal. Appl. 254(2), 433–463 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Ghosh, P.: Control of the Hopf–Turing transition by time-delayed global feedback in a reaction–diffusion system. Phys. Rev. E 84, 016222 (2011)

    Article  Google Scholar 

  • Ghosh, P., Sen, S., Ray, D.S.: Reaction-Cattaneo systems with fluctuating relaxation time. Phys. Rev. E 81, 026205 (2010)

    Article  Google Scholar 

  • Hadeler, K.P., Ruan, S.: Interaction of diffusion and delay. Discrete Contin. Dyn. Syst., Ser. B 8(1), 95–105 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Hale, J.K., Huang, W.Z.: Global geometry of the stable regions for two delay differential equations. J. Math. Anal. Appl. 178(2), 344–362 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional-Differential Equations. Applied Mathematical Sciences, vol. 99. Springer, New York (1993)

    MATH  Google Scholar 

  • Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. London Mathematical Society Lecture Note Series, vol. 41. Cambridge University Press, Cambridge (1981)

    MATH  Google Scholar 

  • Herz, A.V.M., Bonhoeffer, S., Anderson, R.M., May, R.M., Nowak, M.A.: Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay. Proc. Natl. Acad. Sci. USA 93(14), 7247–7251 (1996)

    Article  Google Scholar 

  • Hu, G.-P., Li, W.-T.: Hopf bifurcation analysis for a delayed predator–prey system with diffusion effects. Nonlinear Anal., Real World Appl. 11(2), 819–826 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Hu, G.-P., Li, W.-T., Yan, X.-P.: Hopf bifurcations in a predator–prey system with multiple delays. Chaos Solitons Fractals 42(2), 1273–1285 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Hutchinson, G.E.: Circular causal systems in ecology. Ann. N.Y. Acad. Sci. 50(4), 221–246 (1948)

    Article  Google Scholar 

  • Kolmanovskii, V., Myshkis, A.: Introduction to the Theory and Applications of Functional-Differential Equations. Mathematics and Its Applications, vol. 463. Kluwer Academic, Dordrecht (1999)

    MATH  Google Scholar 

  • Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Mathematics in Science and Engineering, vol. 191. Academic Press, Boston (1993)

    MATH  Google Scholar 

  • Kyrychko, Y.N., Blyuss, K.B., Hogan, S.J., Schöll, E.: Control of spatiotemporal patterns in the Gray–Scott model. Chaos 19(4), 043126 (2009)

    Article  Google Scholar 

  • Lee, S.S., Gaffney, E.A., Monk, N.A.M.: The influence of gene expression time delays on Gierer–Meinhardt pattern formation systems. Bull. Math. Biol. 72(8), 2139–2160 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Li, X., Wei, J.: On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays. Chaos Solitons Fractals 26(2), 519–526 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Li, X., Ruan, S., Wei, J.: Stability and bifurcation in delay-differential equations with two delays. J. Math. Anal. Appl. 236(2), 254–280 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • May, R.M.: Time-delay versus stability in population models with two and three trophic levels. Ecology 54(2), 315–325 (1973)

    Article  Google Scholar 

  • Murray, J.D.: Mathematical Biology. I: An Introduction, 3rd edn. Interdisciplinary Applied Mathematics, vol. 17. Springer, New York (2002)

    Google Scholar 

  • Nelson, P.W., Perelson, A.S.: Mathematical analysis of delay differential equation models of HIV-1 infection. Math. Biosci. 179(1), 73–94 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Perelson, A.S., Nelson, P.W.: Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev. 41(1), 3–44 (1999) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  • Ruan, S.: Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator–prey systems with discrete delays. Q. Appl. Math. 59(1), 159–173 (2001)

    MATH  Google Scholar 

  • Ruan, S.: On nonlinear dynamics of predator–prey models with discrete delay. Math. Model. Nat. Phenom. 4(2), 140–188 (2009)

    MathSciNet  MATH  Google Scholar 

  • Ruan, S., Wei, J.: On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion. IMA J. Math. Appl. Med. Biol. 18(1), 41–52 (2001)

    Article  MATH  Google Scholar 

  • Ruan, S., Wei, J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 10(6), 863–874 (2003)

    MathSciNet  MATH  Google Scholar 

  • Sen, S., Ghosh, P., Riaz, S.S., Ray, D.S.: Time-delay-induced instabilities in reaction–diffusion systems. Phys. Rev. E 80(4), 046212 (2008)

    Article  Google Scholar 

  • Shayer, L.P., Campbell, S.A.: Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays. SIAM J. Appl. Math. 61(2), 673–700 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Texts in Applied Mathematics, vol. 57. Springer, New York (2011)

    Book  MATH  Google Scholar 

  • Song, Y., Wei, J.: Bifurcation analysis for Chen’s system with delayed feedback and its application to control of chaos. Chaos Solitons Fractals 22(1), 75–91 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Song, Y., Wei, J., Yuan, Y.: Stability switches and Hopf bifurcations in a pair of delay-coupled oscillators. J. Nonlinear Sci. 17(2), 145–166 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Song, Y., Yuan, S., Zhang, J.: Bifurcation analysis in the delayed Leslie–Gower predator–prey system. Appl. Math. Model. 33(11), 4049–4061 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Wei, J., Ruan, S.: Stability and bifurcation in a neural network model with two delays. Physica D 130(3–4), 255–272 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Wei, J., Yuan, Y.: Synchronized Hopf bifurcation analysis in a neural network model with delays. J. Math. Anal. Appl. 312(1), 205–229 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Wu, J.: Theory and Applications of Partial Functional-Differential Equations. Applied Mathematical Sciences, vol. 119. Springer, New York (1996)

    Book  MATH  Google Scholar 

  • Zuo, W., Wei, J.: Stability and Hopf bifurcation in a diffusive predatory–prey system with delay effect. Nonlinear Anal., Real World Appl. 12(4), 1998–2011 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank two anonymous referees for very helpful comments which greatly improved the manuscript. Parts of this work was done when SSC visited College of William and Mary in 2010–2011, and she would like to thank CWM for warm hospitality.

Partially supported by a grant from China Scholarship Council (Chen), NSF grant DMS-1022648 and Shanxi 100 talent program (Shi), China-NNSF grants 11031002 (Wei).

Author information

Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, P.R. China

    Shanshan Chen & Junjie Wei

  2. Department of Mathematics, College of William and Mary, Williamsburg, VA, 23187-8795, USA

    Junping Shi

Authors
  1. Shanshan Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junping Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Junjie Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shanshan Chen.

Additional information

Communicated by Sue Anne Campbell.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, S., Shi, J. & Wei, J. Time Delay-Induced Instabilities and Hopf Bifurcations in General Reaction–Diffusion Systems. J Nonlinear Sci 23, 1–38 (2013). https://doi.org/10.1007/s00332-012-9138-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00332-012-9138-1

Keywords

Mathematics Subject Classification (2010)

Search

Navigation