October 2017, 14(5&6): 1499-1514. doi: 10.3934/mbe.2017078

Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network

1. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

2. 

College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225, China

3. 

School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China

* Corresponding author: Weinian Zhang

Received  July 02, 2016 Accepted  January 2017 Published  May 2017

Fund Project: Supported by NSFC # 11221101, # 11231001, # 11501475 and SPDEF 16ZB0080

There have been some results on bifurcations of codimension one (such as saddle-node, transcritical, pitchfork) and degenerate Hopf bifurcations for an enzyme-catalyzed reaction system comprising a branched network but no further discussion for bifurcations at its cusp. In this paper we give conditions for the existence of a cusp and compute the parameter curves for the Bogdanov-Takens bifurcation, which induces the appearance of homoclinic orbits and periodic orbits, indicating the tendency to steady-states or a rise of periodic oscillations for the concentrations of the substrate and the product.

Citation: Qiuyan Zhang, Lingling Liu, Weinian Zhang. Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1499-1514. doi: 10.3934/mbe.2017078
References:
[1]

A. Betz and E. Sel'kov, Control of phosphofructokinase [PFK] activity in conditions simulating those of glycolysing yeast extract, FEBS Lett., 3 (1969), 5-9. doi: 10.1016/0014-5793(69)80082-7.

[2]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, New York, 1982.

[3]

F. A. Davidson and J. Liu, Global stability of the attracting set of an enzyme-catalysed reaction system, Math. Comput. Model., 35 (2002), 1467-1481. doi: 10.1016/S0895-7177(02)00098-5.

[4]

F. A. DavidsonR. Xu and J. Liu, Existence and uniqueness of limit cycles in an enzyme-catalysed reaction system, Appl. Math. Comput., 127 (2002), 165-179. doi: 10.1016/S0096-3003(01)00065-0.

[5]

F. DumortierR. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergod. Theor. Dyn. Syst., 7 (1987), 375-413. doi: 10.1017/S0143385700004119.

[6]

D. Erle, Nonuniqueness of stable limit cycles in a class of enzyme catalyzed reactions, J. Math. Anal. Appl., 82 (1981), 386-391. doi: 10.1016/0022-247X(81)90203-1.

[7]

D. ErleK. H. Mayer and T. Plesser, The existence of stable limit cycles for enzyme catalyzed reactions with positive feedback, Math. Biosci., 44 (1979), 191-208. doi: 10.1016/0025-5564(79)90081-6.

[8]

A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511608193.

[9]

A. Goldbeter and G. Dupont, Allosteric regulation, cooperativity and biochemical oscillations, Biophy. Chem., 37 (1990), 341-353. doi: 10.1016/0301-4622(90)88033-O.

[10]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1990. doi: 10.1007/978-1-4612-1140-2.

[11]

B. Hassard and K. Jiang, Unfolding a point of degenerate Hopf bifurcation in an enzyme-catalyzed reaction model, SIAM J. Math. Anal., 23 (1992), 1291-1304. doi: 10.1137/0523072.

[12]

X. HouR. Yan and W. Zhang, Bifurcations of a polynomial differential system of degree n in biochemical reactions, Comput. Math. Appl., 43 (2002), 1407-1423. doi: 10.1016/S0898-1221(02)00108-6.

[13]

J. P. KernévezG. JolyM. C. DubanB. Bunow and D. Thomas, Hysteresis, oscillations, and pattern formation in realistic immmobilized enzyme systems, J. Math. Biol., 7 (1979), 41-56. doi: 10.1007/BF00276413.

[14]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Appl. Math. Sci., 112, Springer, New York, 1995. doi: 10.1007/978-1-4757-2421-9.

[15]

Z. LengB. Gao and Z. Wang, Qualitative analysis of a generalized system of saturated enzyme reaction, Math. Comput. Model., 49 (2009), 556-562. doi: 10.1016/j.mcm.2008.03.006.

[16]

J. Liu, Coordination restriction of enzyme-catalysed reaction systems as nonlinear dynamical systems, Proc. R. Soc. Lond. A, 455 (1999), 285-298. doi: 10.1098/rspa.1999.0313.

[17]

A. G. Marangoni, Enzymes Kinetics: A Modern Approach, Wiley-Interscience, Hoboken, NJ, 2003. doi: 10.1002/0471267295.

[18]

L. Michaelis and M. L. Menten, Die kinetik der invertinwirkung, Biochem. Z., 49 (1913), 333-369.

[19]

J. D. Murray, Mathematical Biology Ⅰ: An Introduction, Interdisciplinary Applied Mathematics 17, Springer, Berlin, 2002.

[20]

H. G. Othmer and J. A. Aldridge, The effects of cell density and metabolite flux on cellular dynamics, J. Math. Biol., 5 (1978), 169-200. doi: 10.1007/BF00275897.

[21]

I. StoleriuF. A. Davidson and J. Liu, Effects of priodic input on the quasi-steady state assumptions for enzyme-catalyzed reactions, J. Math. Biol., 50 (2005), 115-132. doi: 10.1007/s00285-004-0282-6.

[22]

Y. TangD. Huang and W. Zhang, Direct parametric analysis of an enzyme-catalyzed reaction model, IMA J. Appl. Math., 76 (2011), 876-898. doi: 10.1093/imamat/hxr005.

[23]

Y. Tang and W. Zhang, Bogdanov-Takens bifurcation of a polynomial differential system in biochemical reaction, Comput. Math. Appl., 48 (2004), 869-883. doi: 10.1016/j.camwa.2003.05.012.

[24]

R. VarónM. García-MorenoF. García-MolinaM. E. FuentesE. ArribasJ. M. YagoM. Ll. Amo-Saus and E. Valero, Two new regulatory properties arising from the transient phase kinetics of monocyclic enzyme cascades, J. Math. Chem., 38 (2005), 437-450. doi: 10.1007/s10910-004-6895-6.

[25]

Y. -Q. Ye et al., Theory of Limit Cycles, Transl. Math. Monogr. 66 American Mathematical Society, Providence, RI, 1986.

[26]

Z. -F. Zhang, T. -R. Ding, W. -Z. Huang and Z. -X. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monogr., 101 Amer. Math. Soc., Providence, RI, 1992.

[27]

Q. Zhang, L. Liu and W. Zhang, Local bifurcations of the enzyme-catalyzed reaction comprising a branched network, Int. J. Bifur. Chaos, 25 (2015), 155081 (26 pages). doi: 10.1142/S0218127415500819.

show all references

References:
[1]

A. Betz and E. Sel'kov, Control of phosphofructokinase [PFK] activity in conditions simulating those of glycolysing yeast extract, FEBS Lett., 3 (1969), 5-9. doi: 10.1016/0014-5793(69)80082-7.

[2]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, New York, 1982.

[3]

F. A. Davidson and J. Liu, Global stability of the attracting set of an enzyme-catalysed reaction system, Math. Comput. Model., 35 (2002), 1467-1481. doi: 10.1016/S0895-7177(02)00098-5.

[4]

F. A. DavidsonR. Xu and J. Liu, Existence and uniqueness of limit cycles in an enzyme-catalysed reaction system, Appl. Math. Comput., 127 (2002), 165-179. doi: 10.1016/S0096-3003(01)00065-0.

[5]

F. DumortierR. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergod. Theor. Dyn. Syst., 7 (1987), 375-413. doi: 10.1017/S0143385700004119.

[6]

D. Erle, Nonuniqueness of stable limit cycles in a class of enzyme catalyzed reactions, J. Math. Anal. Appl., 82 (1981), 386-391. doi: 10.1016/0022-247X(81)90203-1.

[7]

D. ErleK. H. Mayer and T. Plesser, The existence of stable limit cycles for enzyme catalyzed reactions with positive feedback, Math. Biosci., 44 (1979), 191-208. doi: 10.1016/0025-5564(79)90081-6.

[8]

A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511608193.

[9]

A. Goldbeter and G. Dupont, Allosteric regulation, cooperativity and biochemical oscillations, Biophy. Chem., 37 (1990), 341-353. doi: 10.1016/0301-4622(90)88033-O.

[10]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1990. doi: 10.1007/978-1-4612-1140-2.

[11]

B. Hassard and K. Jiang, Unfolding a point of degenerate Hopf bifurcation in an enzyme-catalyzed reaction model, SIAM J. Math. Anal., 23 (1992), 1291-1304. doi: 10.1137/0523072.

[12]

X. HouR. Yan and W. Zhang, Bifurcations of a polynomial differential system of degree n in biochemical reactions, Comput. Math. Appl., 43 (2002), 1407-1423. doi: 10.1016/S0898-1221(02)00108-6.

[13]

J. P. KernévezG. JolyM. C. DubanB. Bunow and D. Thomas, Hysteresis, oscillations, and pattern formation in realistic immmobilized enzyme systems, J. Math. Biol., 7 (1979), 41-56. doi: 10.1007/BF00276413.

[14]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Appl. Math. Sci., 112, Springer, New York, 1995. doi: 10.1007/978-1-4757-2421-9.

[15]

Z. LengB. Gao and Z. Wang, Qualitative analysis of a generalized system of saturated enzyme reaction, Math. Comput. Model., 49 (2009), 556-562. doi: 10.1016/j.mcm.2008.03.006.

[16]

J. Liu, Coordination restriction of enzyme-catalysed reaction systems as nonlinear dynamical systems, Proc. R. Soc. Lond. A, 455 (1999), 285-298. doi: 10.1098/rspa.1999.0313.

[17]

A. G. Marangoni, Enzymes Kinetics: A Modern Approach, Wiley-Interscience, Hoboken, NJ, 2003. doi: 10.1002/0471267295.

[18]

L. Michaelis and M. L. Menten, Die kinetik der invertinwirkung, Biochem. Z., 49 (1913), 333-369.

[19]

J. D. Murray, Mathematical Biology Ⅰ: An Introduction, Interdisciplinary Applied Mathematics 17, Springer, Berlin, 2002.

[20]

H. G. Othmer and J. A. Aldridge, The effects of cell density and metabolite flux on cellular dynamics, J. Math. Biol., 5 (1978), 169-200. doi: 10.1007/BF00275897.

[21]

I. StoleriuF. A. Davidson and J. Liu, Effects of priodic input on the quasi-steady state assumptions for enzyme-catalyzed reactions, J. Math. Biol., 50 (2005), 115-132. doi: 10.1007/s00285-004-0282-6.

[22]

Y. TangD. Huang and W. Zhang, Direct parametric analysis of an enzyme-catalyzed reaction model, IMA J. Appl. Math., 76 (2011), 876-898. doi: 10.1093/imamat/hxr005.

[23]

Y. Tang and W. Zhang, Bogdanov-Takens bifurcation of a polynomial differential system in biochemical reaction, Comput. Math. Appl., 48 (2004), 869-883. doi: 10.1016/j.camwa.2003.05.012.

[24]

R. VarónM. García-MorenoF. García-MolinaM. E. FuentesE. ArribasJ. M. YagoM. Ll. Amo-Saus and E. Valero, Two new regulatory properties arising from the transient phase kinetics of monocyclic enzyme cascades, J. Math. Chem., 38 (2005), 437-450. doi: 10.1007/s10910-004-6895-6.

[25]

Y. -Q. Ye et al., Theory of Limit Cycles, Transl. Math. Monogr. 66 American Mathematical Society, Providence, RI, 1986.

[26]

Z. -F. Zhang, T. -R. Ding, W. -Z. Huang and Z. -X. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monogr., 101 Amer. Math. Soc., Providence, RI, 1992.

[27]

Q. Zhang, L. Liu and W. Zhang, Local bifurcations of the enzyme-catalyzed reaction comprising a branched network, Int. J. Bifur. Chaos, 25 (2015), 155081 (26 pages). doi: 10.1142/S0218127415500819.

Figure 1.  Reaction scheme
Figure 2.  Bifurcation surfaces projection on the $(a, \kappa)$-plane
Figure 3.  Bifurcation diagrams of system (4) for the case that $c>1$ and $b=(c+1)^2/4$
Figure 4.  An attracting limit cycle
Table 4.  Dynamics of system (4) in various cases of parameter $(a, \kappa)$
Parameters $(a, \kappa)$ Equilibria Limit cycles and homoclinic orbits Region in bifurcation diagram
$E_0$ $E_1$ $E_2$ $E_*$
$ \mathcal{D}_{I}$ saddle unstable focus saddle $\mathcal{D}_{I}$
$ \mathcal{L}$ saddle unstable focus saddle one homoclinic rrbit $\mathcal{L}$
$\mathcal{D}_{II}$ saddle unstable focus saddle one limit cycle $\mathcal{D}_{II}$
$\mathcal{H}$ saddle stable focus saddle $\mathcal{H}$
$\mathcal{D}_{III}$ saddle stable focus saddle $\mathcal{D}_{III}$
$ \mathcal{SN}^+$ saddle-node $\mathcal{SN}^+$
$ \mathcal{D}_{IV}$ $\mathcal{D}_{IV}$
$(a_*, \kappa_*)$ cusp $(a_*, \kappa_*)$
$ \mathcal{SN}^-$ saddle-node $\mathcal{SN}^-$
Parameters $(a, \kappa)$ Equilibria Limit cycles and homoclinic orbits Region in bifurcation diagram
$E_0$ $E_1$ $E_2$ $E_*$
$ \mathcal{D}_{I}$ saddle unstable focus saddle $\mathcal{D}_{I}$
$ \mathcal{L}$ saddle unstable focus saddle one homoclinic rrbit $\mathcal{L}$
$\mathcal{D}_{II}$ saddle unstable focus saddle one limit cycle $\mathcal{D}_{II}$
$\mathcal{H}$ saddle stable focus saddle $\mathcal{H}$
$\mathcal{D}_{III}$ saddle stable focus saddle $\mathcal{D}_{III}$
$ \mathcal{SN}^+$ saddle-node $\mathcal{SN}^+$
$ \mathcal{D}_{IV}$ $\mathcal{D}_{IV}$
$(a_*, \kappa_*)$ cusp $(a_*, \kappa_*)$
$ \mathcal{SN}^-$ saddle-node $\mathcal{SN}^-$
[1]

Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041

[2]

Hebai Chen, Xingwu Chen, Jianhua Xie. Global phase portrait of a degenerate Bogdanov-Takens system with symmetry. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1273-1293. doi: 10.3934/dcdsb.2017062

[3]

Hebai Chen, Xingwu Chen. Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ). Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4141-4170. doi: 10.3934/dcdsb.2018130

[4]

Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205

[5]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363

[6]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643

[7]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109

[8]

Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362

[9]

Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561

[10]

John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170

[11]

Chikahiro Egami. Mechanism for the color transition of the Belousov-Zhabotinsky reaction catalyzed by cerium ions and ferroin. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2527-2544. doi: 10.3934/dcdsb.2018061

[12]

Stefan Siegmund. Normal form of Duffing-van der Pol oscillator under nonautonomous parametric perturbations. Conference Publications, 2001, 2001 (Special) : 357-361. doi: 10.3934/proc.2001.2001.357

[13]

Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529

[14]

Anke D. Pohl. A dynamical approach to Maass cusp forms. Journal of Modern Dynamics, 2012, 6 (4) : 563-596. doi: 10.3934/jmd.2012.6.563

[15]

Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523

[16]

Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503

[17]

Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093

[18]

Tatsuki Mori, Kousuke Kuto, Masaharu Nagayama, Tohru Tsujikawa, Shoji Yotsutani. Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization. Conference Publications, 2015, 2015 (special) : 861-877. doi: 10.3934/proc.2015.0861

[19]

Chih-Yuan Chen, Shin-Hwa Wang, Kuo-Chih Hung. S-shaped bifurcation curves for a combustion problem with general arrhenius reaction-rate laws. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2589-2608. doi: 10.3934/cpaa.2014.13.2589

[20]

Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183

2017 Impact Factor: 1.23

Metrics

  • PDF downloads (23)
  • HTML views (55)
  • Cited by (0)

Other articles
by authors

[Back to Top]