Journal of Nonlinear Science

, Volume 23, Issue 1, pp 1–38 | Cite as

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



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.


Second order transcendental polynomial Characteristic equation Reaction–diffusion Stability Hopf bifurcation 

Mathematics Subject Classification (2010)

34K08 34K18 34K20 35R10 92E20 



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).


  1. 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) MathSciNetMATHCrossRefGoogle Scholar
  2. Beretta, E., Kuang, Y.: Global analyses in some delayed ratio-dependent predator–prey systems. Nonlinear Anal. 32(3), 381–408 (1998) MathSciNetMATHCrossRefGoogle Scholar
  3. Bodnar, M., Foryś, U., Poleszczuk, J.: Analysis of biochemical reactions models with delays. J. Math. Anal. Appl. 376(1), 74–83 (2011) MathSciNetMATHCrossRefGoogle Scholar
  4. Chen, S., Shi, J.: Global attractivity of equilibrium in Gierer–Meinhardt system with saturation and gene expression time delays (2012, submitted) Google Scholar
  5. 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) MathSciNetCrossRefGoogle Scholar
  6. Cooke, K.L., Grossman, Z.: Discrete delay, distributed delay and stability switches. J. Math. Anal. Appl. 86(2), 592–627 (1982) MathSciNetMATHCrossRefGoogle Scholar
  7. 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) MathSciNetMATHCrossRefGoogle Scholar
  8. Culshaw, R.V., Ruan, S.: A delay-differential equation model of HIV infection of CD4(+) T-cells. Math. Biosci. 165(1), 27–39 (2000) MATHCrossRefGoogle Scholar
  9. 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) MATHGoogle Scholar
  10. 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) CrossRefGoogle Scholar
  11. Erneux, T.: Applied Delay Differential Equations. Surveys and Tutorials in the Applied Mathematical Sciences, vol. 3. Springer, New York (2009) MATHGoogle Scholar
  12. 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) MathSciNetMATHCrossRefGoogle Scholar
  13. 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) MathSciNetMATHCrossRefGoogle Scholar
  14. Ghosh, P.: Control of the Hopf–Turing transition by time-delayed global feedback in a reaction–diffusion system. Phys. Rev. E 84, 016222 (2011) CrossRefGoogle Scholar
  15. Ghosh, P., Sen, S., Ray, D.S.: Reaction-Cattaneo systems with fluctuating relaxation time. Phys. Rev. E 81, 026205 (2010) CrossRefGoogle Scholar
  16. Hadeler, K.P., Ruan, S.: Interaction of diffusion and delay. Discrete Contin. Dyn. Syst., Ser. B 8(1), 95–105 (2007) MathSciNetMATHCrossRefGoogle Scholar
  17. 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) MathSciNetMATHCrossRefGoogle Scholar
  18. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional-Differential Equations. Applied Mathematical Sciences, vol. 99. Springer, New York (1993) MATHGoogle Scholar
  19. 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) MATHGoogle Scholar
  20. 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) CrossRefGoogle Scholar
  21. 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) MathSciNetMATHCrossRefGoogle Scholar
  22. 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) MathSciNetMATHCrossRefGoogle Scholar
  23. Hutchinson, G.E.: Circular causal systems in ecology. Ann. N.Y. Acad. Sci. 50(4), 221–246 (1948) CrossRefGoogle Scholar
  24. Kolmanovskii, V., Myshkis, A.: Introduction to the Theory and Applications of Functional-Differential Equations. Mathematics and Its Applications, vol. 463. Kluwer Academic, Dordrecht (1999) MATHGoogle Scholar
  25. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Mathematics in Science and Engineering, vol. 191. Academic Press, Boston (1993) MATHGoogle Scholar
  26. 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) CrossRefGoogle Scholar
  27. 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) MathSciNetMATHCrossRefGoogle Scholar
  28. 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) MathSciNetMATHCrossRefGoogle Scholar
  29. 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) MathSciNetMATHCrossRefGoogle Scholar
  30. May, R.M.: Time-delay versus stability in population models with two and three trophic levels. Ecology 54(2), 315–325 (1973) CrossRefGoogle Scholar
  31. Murray, J.D.: Mathematical Biology. I: An Introduction, 3rd edn. Interdisciplinary Applied Mathematics, vol. 17. Springer, New York (2002) Google Scholar
  32. Nelson, P.W., Perelson, A.S.: Mathematical analysis of delay differential equation models of HIV-1 infection. Math. Biosci. 179(1), 73–94 (2002) MathSciNetMATHCrossRefGoogle Scholar
  33. Perelson, A.S., Nelson, P.W.: Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev. 41(1), 3–44 (1999) (electronic) MathSciNetMATHCrossRefGoogle Scholar
  34. 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) MATHGoogle Scholar
  35. Ruan, S.: On nonlinear dynamics of predator–prey models with discrete delay. Math. Model. Nat. Phenom. 4(2), 140–188 (2009) MathSciNetMATHGoogle Scholar
  36. 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) MATHCrossRefGoogle Scholar
  37. 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) MathSciNetMATHGoogle Scholar
  38. 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) CrossRefGoogle Scholar
  39. 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) MathSciNetMATHCrossRefGoogle Scholar
  40. Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Texts in Applied Mathematics, vol. 57. Springer, New York (2011) MATHCrossRefGoogle Scholar
  41. 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) MathSciNetMATHCrossRefGoogle Scholar
  42. 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) MathSciNetMATHCrossRefGoogle Scholar
  43. Song, Y., Yuan, S., Zhang, J.: Bifurcation analysis in the delayed Leslie–Gower predator–prey system. Appl. Math. Model. 33(11), 4049–4061 (2009) MathSciNetMATHCrossRefGoogle Scholar
  44. Wei, J., Ruan, S.: Stability and bifurcation in a neural network model with two delays. Physica D 130(3–4), 255–272 (1999) MathSciNetMATHCrossRefGoogle Scholar
  45. Wei, J., Yuan, Y.: Synchronized Hopf bifurcation analysis in a neural network model with delays. J. Math. Anal. Appl. 312(1), 205–229 (2005) MathSciNetMATHCrossRefGoogle Scholar
  46. Wu, J.: Theory and Applications of Partial Functional-Differential Equations. Applied Mathematical Sciences, vol. 119. Springer, New York (1996) MATHCrossRefGoogle Scholar
  47. 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) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinP.R. China
  2. 2.Department of MathematicsCollege of William and MaryWilliamsburgUSA

Personalised recommendations