Stochastic Stability and Bifurcation for the Selkov Model with Noise

Document Type : Research Paper


Department of Mathematics, Yazd University, 89195-741 Yazd, Iran


In this paper, we consider a stochastic Selkov model for the glycolysis process. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic mathematical modeling. First, we employ polar coordinate transformation and stochastic averaging method to transform the original system into an Itô averaging diffusion system. Next, we investigate the stochastic dynamical bifurcation of the Itô averaging amplitude equation by studying the qualitative changes of invariant measures and explore the phenomenological bifurcation (P-bifurcation) by using the counterpart Fokker-Planck equation. Finally, some numerical simulations are presented to verify our analytic arguments.


1. A. Dutta, D. Das, D. Banerjee and J. K. Bhattacharjee, Estimating the boundaries of a limit cycle in a 2d dynamical system using renormalization group, Commun. Nonlinear Sci. Numer. Simulat. 57 (2018) 47–57.
2. J. Murray,
Mathematical Biology, Springer, New York, USA, 1989.
3. J. Higgins, A chemical mechanism for oscillation of glycolytic intermediates in yeast cells,
Proc Natl Acad Sci USA. 51 (1964) 989–994.
4. E. E. Selkov, Self-oscillations in glycolysis,
Uropean J. Biochemistry 4 (1968) 79–86.
5. K. I. Papadimitriou and E. M. Drakakis, Cmos weak-inversion log-domain glycolytic oscillator: a cytomimetic circuit example,
Int. J. Circuit Theory Appl. (2012) 1–22.
6. A. Goldbeter,
Biochemical oscillations and biological rhythms, Cambridge University Press, UK, 1996.
7. L. Klitzing and A. Betz, Metabolic control in flow systems
, Archiv. Microbio. 71 (1970) 220–225.
8. S. H. Strogatz,
Nonlinear Dynamics and Chaos, Addison Wesley, Reading, MA, 1994.
9. J. C. Artés, J. Llibre and C.Valls, Dynamics of the higgins-selkov and selkov systems,
Chaos, Solitons and Fractals 114 (2018) 145–150.
10. P. Sarkar, The linear response of a glycolytic oscillator, driven by a multiplicative colored noise
, J. Stat. Mech: Theory Exp. 2016 (12) (2016) 123202.
11. C. Li and J. Zhang, Stochastic bifurcation analysis in brusselator system with white noise,
Adv. Differ. Equ. 2019 (1) (2019) 1–16.
12. Y. Y. Ma and L. J. Ning, Stochastic p-bifurcation of fractional derivative van der pol system excited by gaussian white noise,
Indian J. Phys. 93 (1) (2019) 61–66.
13. C. Xu, Phenomenological bifurcation in a stochastic logistic model with 
correlated colored noises, Appl. Math. Letters 101 (2020) 106064.
14. M. Fatehi Nia and M. H. Akrami, Stability and bifurcation in a stochastic vocal folds model, Commun. Nonlinear Sci. Numer. Simulat. 79 (2019) 104898.
15. C. Kong, Z. Chen and X.-B. Liu, On the stochastic dynamical behaviors of a nonlinear oscillator under combined real noise and harmonic excitations,
J. Comput. Nonlinear Dynam. 12 (3) (2017) 031015.
16. A. Rounak and S. Gupta, Stochastic p-bifurcation in a nonlinear impact oscillator with soft barrier under ornstein–uhlenbeck process,
Nonlinear Dyn. 99 (2020) 2657–2674.
17. Y. Li, Z. Wu, G. Zhang, F. Wang and Y. Wang, Stochastic p-bifurcation in a bistable van der pol oscillator with fractional time-delay feedback under gaussian white noise excitation,
Adv. Differ. Equ. 2019 (1) (2019) 448. DOI:10.1186/s13662-019-2356-1
18. G. J. Fezeu, I. S. Mokem Fokou, C. Nono Dueyou Buckjohn, M. Siewe Siewe and C. Tchawoua, Resistance induced p-bifurcation and ghost-stochastic resonance of a hybrid energy harvester under colored noise,
Phys. A: Stat. Mech. Appl. 557 (2020) 124857.
19. Z. Huang, Q. Yang, and J. Cao, Stochastic stability and bifurcation for the chronic state in marchuk’s model with noise,
Appl. Math. Model. 35 (2011) 5842–5855.
20. X. Mao,
Stochastic differential equations and applications, 2nd ed., Woodhead Publishing, Chichester, England, 2007.
21. C. Luo and S. Guo, Stability and bifurcation of two-dimensional stochastic differential equations with multiplicative excitations,
Bull. Malaysian Math. Sci. Soc. 40 (2) (2017) 795–817.
22. L. Arnold,
Random Dynamical Systems, In: Johnson R. (eds) Dynamical Systems.
Lecture Notes in Mathematics, vol 1609
. Springer, Berlin, Heidelberg, 1995.
23. R. Z. Khas’minskii, On the principle of averaging for itô’s stochastic differential equations
Kybernetika (Prague) 4 (1968) 260–279. (in Russian)
24. U. Wagner and W. Weding, On the calculation of stationary solutions of multidimensional fokker-planck equation by orthogonal function,
Nonlinear Dyn. 29 (2000) 283–306.
25. P. E. Kloeden and E. Platen,
Numerical solution of stochastic differential equations, corrected ed., Stochastic Modelling and Applied Probability, SpringerVerlag, New York, 1995.
26. O. S. Fard, Linearization and nonlinear stochastic differential equations with locally lipschitz condition,
Appl. Math. Sci. 1 (2007) 2553–2563.