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Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations
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 |
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.
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. 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.
doi: 10.1016/S0096-3003(01)00065-0. |
[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.
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. 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.
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. 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.
doi: 10.1016/S0898-1221(02)00108-6. |
[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.
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. Leng, B. 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. 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.
doi: 10.1007/s00285-004-0282-6. |
[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.
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. 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.
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. 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.
doi: 10.1016/S0096-3003(01)00065-0. |
[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.
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. 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.
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. 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.
doi: 10.1016/S0898-1221(02)00108-6. |
[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.
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. Leng, B. 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. 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.
doi: 10.1007/s00285-004-0282-6. |
[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.
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. 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.
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. |



Parameters |
Equilibria | Limit cycles and homoclinic orbits | Region in bifurcation diagram | |||
saddle | unstable focus | saddle | ||||
saddle | unstable focus | saddle | one homoclinic rrbit | |||
saddle | unstable focus | saddle | one limit cycle | |||
saddle | stable focus | saddle | ||||
saddle | stable focus | saddle | ||||
saddle-node | ||||||
cusp | ||||||
saddle-node |
Parameters |
Equilibria | Limit cycles and homoclinic orbits | Region in bifurcation diagram | |||
saddle | unstable focus | saddle | ||||
saddle | unstable focus | saddle | one homoclinic rrbit | |||
saddle | unstable focus | saddle | one limit cycle | |||
saddle | stable focus | saddle | ||||
saddle | stable focus | saddle | ||||
saddle-node | ||||||
cusp | ||||||
saddle-node |
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