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Bifurcation analysis and chaos control in discrete-time glycolysis models

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Abstract

In this paper, the qualitative behavior of two discrete-time glycolysis models is discussed. The discrete-time models are obtained by implementing forward Euler’s scheme and nonstandard finite difference method. The parametric conditions for local asymptotic stability of positive steady-states are investigated. Moreover, we discuss the existence and directions of period-doubling and Neimark–Sacker bifurcations with the help of center manifold theorem and bifurcation theory. OGY feedback control and hybrid control methods are implemented in order to control chaos in discrete-time glycolysis model due to emergence of period-doubling and Neimark–Sacker bifurcations. Numerical simulations are provided to illustrate theoretical discussion.

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References

  1. E.E. Sel’kov, Self-oscillations in glycolysis. A simple model. Eur. J. Biochem. 4, 79–86 (1968)

    Article  Google Scholar 

  2. S.H. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering (Addison-Wesley, New York, 1994)

    Google Scholar 

  3. G.C. Layek, An Introduction to Dynamical Systems and Chaos (Springer, New Delhi, 2015)

    Book  Google Scholar 

  4. A. Goldbeter, R. Lefever, Dissipative structures for an allosteric model: application to glycolytic oscillations. Biophys. J. 12, 1302–1315 (1972)

    Article  CAS  Google Scholar 

  5. O. Decroly, A. Goldbeter, Birhythmicity, chaos and other patterns of temporal self-organization in a multiply regulated biochemical system. Proc. Natl. Acad. Sci. USA 79, 6917–6921 (1982)

    Article  CAS  Google Scholar 

  6. J. Wolf, J. Passarge, O.J. Somsen, J.L. Snoep, R. Heinrich, H.V. Westerhoff, Transduction of intracellular and intercellular dynamics in yeast glycolytic oscillations. Biophys. J. 78, 1145–1153 (2000)

    Article  CAS  Google Scholar 

  7. A. Goldbeter, Biochemical Oscillations and Biological Rhythms (Cambridge University Press, Cambridge, 1996)

    Book  Google Scholar 

  8. F.A. Davidson, B.P. Rynne, A priori bounds and global existence of solutions of the steady-state Sel’kov model. Proc. R. Soc. Edinb. Sect. A 130, 507–516 (2000)

    Article  Google Scholar 

  9. M.X. Wang, Non-constant positive steady-states of the Sel’kov model. J. Differ. Equ. 190(2), 600–620 (2003)

    Article  Google Scholar 

  10. R. Peng, Qualitative analysis of steady states to the Sel’kov model. J. Differ. Equ. 241, 386–398 (2007)

    Article  Google Scholar 

  11. M. Wei, J. Wu, G. Guo, Steady state bifurcations for a glycolysis model in biochemical reaction. Nonlinear Anal. RWA 22, 155–175 (2015)

    Article  Google Scholar 

  12. C.C. Felicio, P.C. Rech, Arnold tongues and the Devil’s Staircase in a discrete-time Hindmarsh–Rose neuron model. Phys. Lett. A 379, 2845–2847 (2015)

    Article  CAS  Google Scholar 

  13. A.D. Silva, P.C. Rech, Chaos and periodicity in a discrete-time Baier–Sahle model. Asian J. Math. Comput. Res.arch 15(2), 123–130 (2017)

    Google Scholar 

  14. R. E. Mickens, Positivity preserving discrete model for the coupled ODES modeling glycolysis, in Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, May 24–27, Wilmington, NC (2002), pp. 623–629

  15. Q. Din, Global stability and Neimark–Sacker bifurcation of a host-parasitoid model. Int. J. Syst. Sci. 48(6), 1194–1202 (2017)

    Article  Google Scholar 

  16. Q. Din, Neimark–Sacker bifurcation and chaos control in Hassell–Varley model. J. Differ. Equ. Appl. 23(4), 741–762 (2017)

    Article  Google Scholar 

  17. Q. Din, Ö.A. Gümüş, H. Khalil, Neimark–Sacker bifurcation and chaotic behaviour of a modified Host–Parasitoid model. Z. Naturforsch. A 72(1), 25–37 (2017)

    Article  CAS  Google Scholar 

  18. Q. Din, Complexity and chaos control in a discrete-time prey–predator model. Commun. Nonlinear Sci. Numer. Simul. 49, 113–134 (2017)

    Article  Google Scholar 

  19. Q. Din, Qualitative analysis and chaos control in a density-dependent host–parasitoid system. Int. J. Dyn. Control (2017). https://doi.org/10.1007/s40435-017-0341-7

  20. Q. Din, Global stability of Beddington model. Qual. Theor. Dyn. Syst. 16(2), 391–415 (2017)

    Article  Google Scholar 

  21. Q. Din, U. Saeed, Bifurcation analysis and chaos control in a host–parasitoid model. Math. Methods. Appl. Sci. (2017). https://doi.org/10.1002/mma.4395

  22. Q. Din, Controlling chaos in a discrete-time prey–predator model with Allee effects. Int. J. Dyn. Control (2017). https://doi.org/10.1007/s40435-017-0347-1

  23. Q. Din, A.A. Elsadany, H. Khalil, Neimark–Sacker bifurcation and chaos control in a fractional-order plant-herbivore model. Discrete Dyn. Nat. Soc. 2017, 1–15 (2017)

    Google Scholar 

  24. Z. He, X. Lai, Bifurcation and chaotic behavior of a discrete-time predator–prey system. Nonlinear Anal. RWA 12, 403–417 (2011)

    Article  Google Scholar 

  25. Z. Jing, J. Yang, Bifurcation and chaos in discrete-time predator–prey system. Chaos Soliton Fract. 27, 259–277 (2006)

    Article  Google Scholar 

  26. X. Liu, D. Xiao, Complex dynamic behaviors of a discrete-time predator–prey system. Chaos Soliton Fract. 32, 80–94 (2007)

    Article  Google Scholar 

  27. H.N. Agiza, E.M. ELabbasy, H. EL-Metwally, A.A. Elsadany, Chaotic dynamics of a discrete prey–predator model with Holling type II. Nonlinear Anal. RWA 10, 116–129 (2009)

    Article  Google Scholar 

  28. B. Li, Z. He, Bifurcations and chaos in a two-dimensional discrete Hindmarsh–Rose model. Nonlinear Dyn. 76(1), 697–715 (2014)

    Article  Google Scholar 

  29. L.-G. Yuan, Q.-G. Yang, Bifurcation, invariant curve and hybrid control in a discrete-time predator–prey system. Appl. Math. Model. 39(8), 2345–2362 (2015)

    Article  Google Scholar 

  30. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, 1983)

    Book  Google Scholar 

  31. C. Robinson, Dynamical Systems: Stability (Symbolic Dynamics and Chaos, Boca Raton, 1999)

    Google Scholar 

  32. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, New York, 2003)

    Google Scholar 

  33. Y.H. Wan, Computation of the stability condition for the Hopf bifurcation of diffeomorphism on \(R^2\). SIAM. J. Appl. Math. 34, 167–175 (1978)

    Article  Google Scholar 

  34. Y.A. Kuznetsov, Elements of Applied Bifurcation Theory (Springer, New York, 1997)

    Google Scholar 

  35. E. Ott, C. Grebogi, J.A. Yorke, Controlling chaos. Phys. Rev. Lett. 64(11), 1196–1199 (1990)

    Article  CAS  Google Scholar 

  36. S. Lynch, Dynamical Systems with Applications Using Mathematica (Birkhäuser, Boston, 2007)

    Google Scholar 

  37. X.S. Luo, G.R. Chen, B.H. Wang et al., Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems. Chaos Solitons Fract. 18, 775–783 (2004)

    Article  Google Scholar 

Download references

Acknowledgements

The author is grateful to the referees for their excellent suggestions which greatly improve the presentation of the paper.

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  1. Department of Mathematics, The University of Poonch Rawalakot, Rawalakot, Pakistan

    Qamar Din

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  1. Qamar Din
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Correspondence to Qamar Din.

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Din, Q. Bifurcation analysis and chaos control in discrete-time glycolysis models. J Math Chem 56, 904–931 (2018). https://doi.org/10.1007/s10910-017-0839-4

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  • DOI: https://doi.org/10.1007/s10910-017-0839-4

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