`a`
Mathematical Biosciences and Engineering (MBE)
 

Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network
Pages: 1499 - 1514, Issue 5/6, October/December 2017

doi:10.3934/mbe.2017078      Abstract        References        Full text (594.3K)           Related Articles

Qiuyan Zhang - Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China (email)
Lingling Liu - School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China (email)
Weinian Zhang - Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China (email)

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.
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.       
4 F. A. Davidson, R. Xu and J. Liu, Existence and uniqueness of limit cycles in an enzyme-catalysed reaction system, Appl. Math. Comput., 127 (2002), 165-179.       
5 F. Dumortier, R. 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.       
6 D. Erle, Nonuniqueness of stable limit cycles in a class of enzyme catalyzed reactions, J. Math. Anal. Appl., 82 (1981), 386-391.       
7 D. Erle, K. H. Mayer and T. Plesser, The existence of stable limit cycles for enzyme catalyzed reactions with positive feedback, Math. Biosci., 44 (1979), 191-208.       
8 A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour, Cambridge University Press, Cambridge, 1996.
9 A. Goldbeter and G. Dupont, Allosteric regulation, cooperativity and biochemical oscillations, Biophy. Chem., 37 (1990), 341-353.
10 J. Guckenheimer and P. Holmes, Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1990.       
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.       
12 X. Hou, R. Yan and W. Zhang, Bifurcations of a polynomial differential system of degree $n$ in biochemical reactions, Comput. Math. Appl., 43 (2002), 1407-1423.       
13 J. P. Kernévez, G. Joly, M. C. Duban, B. Bunow and D. Thomas, Hysteresis, oscillations, and pattern formation in realistic immmobilized enzyme systems, J. Math. Biol., 7 (1979), 41-56.       
14 Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Appl. Math. Sci., 112, Springer, New York, 1995.       
15 Z. Leng, B. Gao and Z. Wang, Qualitative analysis of a generalized system of saturated enzyme reaction, Math. Comput. Model., 49 (2009), 556-562.       
16 J. Liu, Coordination restriction of enzyme-catalysed reaction systems as nonlinear dynamical systems, Proc. R. Soc. Lond. A, 455 (1999), 285-298.       
17 A. G. Marangoni, Enzymes Kinetics: A Modern Approach, Wiley-Interscience, Hoboken, NJ, 2003.
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.       
21 I. Stoleriu, F. 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.       
22 Y. Tang, D. Huang and W. Zhang, Direct parametric analysis of an enzyme-catalyzed reaction model, IMA J. Appl. Math., 76 (2011), 876-898.       
23 Y. Tang and W. Zhang, Bogdanov-Takens bifurcation of a polynomial differential system in biochemical reaction, Comput. Math. Appl., 48 (2004), 869-883.       
24 R. Varón, M. García-Moreno, F. García-Molina, M. E. Fuentes, E. Arribas, J. M. Yago, M. 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.       
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).       

Go to top