# American Institute of Mathematical Sciences

December  2019, 24(12): 6783-6815. doi: 10.3934/dcdsb.2019167

## Bifurcation analysis of an enzyme-catalyzed reaction system with branched sink

 1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China 2 Department of Mathematics, Chengdu Normal University, Chengdu, Sichuan 611130, China

* Corresponding author: Lan Zou, Email: lanzou@163.com

Received  September 2018 Revised  February 2019 Published  July 2019

Fund Project: This work is supported by NSFC grant 11831012 and 11771168

In this paper, we study the local bifurcations of an enzyme-catalyzed reaction system with positive parameters $\alpha$, $\beta$, $\gamma$ and integer $n\geq 2$. This system is orbitally equivalent to a polynomial differential system with order $n+2$. Although not all coordinates of equilibria can be computed because of the high degree of polynomial, parameter conditions for the coexistence of equilibria and their qualitative properties are obtained. Furthermore, it is proved that this system has various bifurcations, including saddle-node bifurcation, transcritical bifurcation, pitchfork bifurcation and Hopf bifurcation. Based on Lyapunov quantities, the order of weak focus is proved to be at most 3. Furthermore, parameter conditions of the exact order of weak focus are obtained. Finally, numerical simulations are employed to illustrate our results.

Citation: Juan Su, Bing Xu, Lan Zou. Bifurcation analysis of an enzyme-catalyzed reaction system with branched sink. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6783-6815. doi: 10.3934/dcdsb.2019167
##### References:
 [1] B. Aguda, L. Frisch and L. Olsen, Experimental evidence for the coexistence of oscillatory and steady states in the peroxidase-oxidase reaction, J. Amer. Chem. Soc., 112 (1990), 6652-6656. doi: 10.1021/ja00174a030. Google Scholar [2] X. Chen and W. Zhang, Decomposition of algebraic sets and applications to weak centers of cubic systems, J. Comput. Appl. Math., 232 (2009), 565-581. doi: 10.1016/j.cam.2009.06.029. Google Scholar [3] G. Collins and A. Akritas, Polynomial real root isolation using Descartes rule of signs, in Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation, ACM Press, 1976, 272–275.Google Scholar [4] F. 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. Google Scholar [5] D. Erle, K. Mayer and T. Plesser, The existence of stable limite cycles for enzyme catalyzed reactions with positive feedback, Math. Biosci., 44 (1979), 191-208. doi: 10.1016/0025-5564(79)90081-6. Google Scholar [6] T. Erneux and E. Reiss, Brussellator isolas, SIAM J. Appl. Math., 43 (1983), 1240-1246. doi: 10.1137/0143082. Google Scholar [7] I. Gelfand, M. Kapranov and A. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, Boston, 1994. doi: 10.1007/978-0-8176-4771-1. Google Scholar [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. Google Scholar [9] A. Goldbeter, Oscillatory enzyme reactions and Michaelis-Menten kinetics, FEBS Letters, 587 (2013), 2778-2784. doi: 10.1016/j.febslet.2013.07.031. Google Scholar [10] P. Gray and S. Scott, Chemical Oscillations and Instabilities: Non-linear Chemical Kinetics, Clarendon Press, Oxford, 1990. Google Scholar [11] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. doi: 10.1007/978-1-4612-1140-2. Google Scholar [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. Google Scholar [13] D. Huang, Y. Gong, Y. Tang and W. Zhang, Degenerate equilibria at infinity in the generalized Brusselator, Math. Comput. Model., 42 (2005), 167-179. doi: 10.1016/j.mcm.2004.02.041. Google Scholar [14] W. Ko, Bifurcations and asymptotic behavior of positive stead-state of an enzyme-catalyzed reaction-diffusion system, Nonlinearity, 29 (2016), 3777-3809. doi: 10.1088/0951-7715/29/12/3777. Google Scholar [15] K. Kwek and W. Zhang, Periodic solutions and dynamics of a multimolecular reaction system, Math. Comput. Model., 36 (2002), 189-201. doi: 10.1016/S0895-7177(02)00115-2. Google Scholar [16] R. Lefever and G. Nicolis., Chemical instabilities and sustained oscillations, J. Theor. Biol., 30 (1971), 267-284. doi: 10.1016/0022-5193(71)90054-3. Google Scholar [17] Z. Leng, B. Gao and Z. Wang, Qualitative analysis of a generalized system of saturated enzyme reactions, Math. Comput. Model., 49 (2009), 556-562. doi: 10.1016/j.mcm.2008.03.006. Google Scholar [18] 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. Google Scholar [19] J. Merkin, R. Satnoianu and S. Scott., Travelling waves in a differential flow reactor with simple autocatalytic kinetics, J. Eng. Math., 33 (1998), 157-174. doi: 10.1023/A:1004292023428. Google Scholar [20] M. Metcalf, J. Merkin and S. Scott, Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis, Proc. R. Soc. Lond. A, 447 (1994), 155-174. doi: 10.1098/rspa.1994.0133. Google Scholar [21] I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems. Ⅱ, J. Chem. Phys., 48 (1968), 1695–1700. doi: 10.1063/1.1668896. Google Scholar [22] J. Ritt, Differential Algebra, Amer. Math. Soc., Providence, 1950. doi: 10.1090/coll/033. Google Scholar [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. Google Scholar [24] Q. Zhang, L. Liu and W. Zhang, Local bifurcations of the enzyme-catalyzed reaction comprising a branched network, Int. J. Bifurcat. Chaos, 25 (2015), 1550081, 26pp. doi: 10.1142/S0218127415500819. Google Scholar [25] Q. Zhang, L. Liu and W. Zhang, Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network, Math. Biosci. Eng., 14 (2017), 1499-1514. doi: 10.3934/mbe.2017078. Google Scholar [26] Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Amer. Math. Soc., Providence, RI, 1992. Google Scholar

show all references

##### References:
 [1] B. Aguda, L. Frisch and L. Olsen, Experimental evidence for the coexistence of oscillatory and steady states in the peroxidase-oxidase reaction, J. Amer. Chem. Soc., 112 (1990), 6652-6656. doi: 10.1021/ja00174a030. Google Scholar [2] X. Chen and W. Zhang, Decomposition of algebraic sets and applications to weak centers of cubic systems, J. Comput. Appl. Math., 232 (2009), 565-581. doi: 10.1016/j.cam.2009.06.029. Google Scholar [3] G. Collins and A. Akritas, Polynomial real root isolation using Descartes rule of signs, in Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation, ACM Press, 1976, 272–275.Google Scholar [4] F. 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. Google Scholar [5] D. Erle, K. Mayer and T. Plesser, The existence of stable limite cycles for enzyme catalyzed reactions with positive feedback, Math. Biosci., 44 (1979), 191-208. doi: 10.1016/0025-5564(79)90081-6. Google Scholar [6] T. Erneux and E. Reiss, Brussellator isolas, SIAM J. Appl. Math., 43 (1983), 1240-1246. doi: 10.1137/0143082. Google Scholar [7] I. Gelfand, M. Kapranov and A. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, Boston, 1994. doi: 10.1007/978-0-8176-4771-1. Google Scholar [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. Google Scholar [9] A. Goldbeter, Oscillatory enzyme reactions and Michaelis-Menten kinetics, FEBS Letters, 587 (2013), 2778-2784. doi: 10.1016/j.febslet.2013.07.031. Google Scholar [10] P. Gray and S. Scott, Chemical Oscillations and Instabilities: Non-linear Chemical Kinetics, Clarendon Press, Oxford, 1990. Google Scholar [11] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. doi: 10.1007/978-1-4612-1140-2. Google Scholar [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. Google Scholar [13] D. Huang, Y. Gong, Y. Tang and W. Zhang, Degenerate equilibria at infinity in the generalized Brusselator, Math. Comput. Model., 42 (2005), 167-179. doi: 10.1016/j.mcm.2004.02.041. Google Scholar [14] W. Ko, Bifurcations and asymptotic behavior of positive stead-state of an enzyme-catalyzed reaction-diffusion system, Nonlinearity, 29 (2016), 3777-3809. doi: 10.1088/0951-7715/29/12/3777. Google Scholar [15] K. Kwek and W. Zhang, Periodic solutions and dynamics of a multimolecular reaction system, Math. Comput. Model., 36 (2002), 189-201. doi: 10.1016/S0895-7177(02)00115-2. Google Scholar [16] R. Lefever and G. Nicolis., Chemical instabilities and sustained oscillations, J. Theor. Biol., 30 (1971), 267-284. doi: 10.1016/0022-5193(71)90054-3. Google Scholar [17] Z. Leng, B. Gao and Z. Wang, Qualitative analysis of a generalized system of saturated enzyme reactions, Math. Comput. Model., 49 (2009), 556-562. doi: 10.1016/j.mcm.2008.03.006. Google Scholar [18] 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. Google Scholar [19] J. Merkin, R. Satnoianu and S. Scott., Travelling waves in a differential flow reactor with simple autocatalytic kinetics, J. Eng. Math., 33 (1998), 157-174. doi: 10.1023/A:1004292023428. Google Scholar [20] M. Metcalf, J. Merkin and S. Scott, Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis, Proc. R. Soc. Lond. A, 447 (1994), 155-174. doi: 10.1098/rspa.1994.0133. Google Scholar [21] I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems. Ⅱ, J. Chem. Phys., 48 (1968), 1695–1700. doi: 10.1063/1.1668896. Google Scholar [22] J. Ritt, Differential Algebra, Amer. Math. Soc., Providence, 1950. doi: 10.1090/coll/033. Google Scholar [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. Google Scholar [24] Q. Zhang, L. Liu and W. Zhang, Local bifurcations of the enzyme-catalyzed reaction comprising a branched network, Int. J. Bifurcat. Chaos, 25 (2015), 1550081, 26pp. doi: 10.1142/S0218127415500819. Google Scholar [25] Q. Zhang, L. Liu and W. Zhang, Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network, Math. Biosci. Eng., 14 (2017), 1499-1514. doi: 10.3934/mbe.2017078. Google Scholar [26] Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Amer. Math. Soc., Providence, RI, 1992. Google Scholar
Reaction scheme with branched sink
Partition of parameter quadrant for $(\alpha, \gamma)\in \mathbb{R}_+^2$
Phase portraits of system (7) with $(n, \alpha, \beta, \gamma) = (4, 0.55, 50, 0.1)$ in (A) and $(n, \alpha, \beta, \gamma) = (4, 0.57, 50, 0.1)$ in (B)
Phase portraits of system (7) with $(n, \alpha, \beta, \gamma) = (4, 0.67247, 10, 0.2)$ in (A) and $(n, \alpha, \beta, \gamma) = (4, 0.672, 10, 0.2)$ in (B)
Oscillation of substrate and product in the reaction. Solutions of system (43) with initial value $(x(0), y(0)) = (0.569, 0.255)$
Two limit cycles bifurcate from Hopf bifurcation
Parameter conditions of equilibria for system (7)
 Possibility of parameters Equilibria $(\alpha, \gamma)\in \mathcal{D}_0\cup \mathcal{D}_4 \cup\mathcal{L}_1 \cup\mathcal{L}_3 \cup \mathcal{P}_0$ $E_b$ $(\alpha, \gamma)\in \mathcal{L}_4$ $E_b$ $E_0$ $(\alpha, \gamma)\in \mathcal{D}_{1}\cup \mathcal{L}_{2}$ $E_b$ $E_1$ $(\alpha, \gamma)\in \mathcal{D}_{2}$ $E_b$ $E_2$ $(\alpha, \gamma)\in \mathcal{D}_{3}$ $E_b$ $E_1$ $E_2$
 Possibility of parameters Equilibria $(\alpha, \gamma)\in \mathcal{D}_0\cup \mathcal{D}_4 \cup\mathcal{L}_1 \cup\mathcal{L}_3 \cup \mathcal{P}_0$ $E_b$ $(\alpha, \gamma)\in \mathcal{L}_4$ $E_b$ $E_0$ $(\alpha, \gamma)\in \mathcal{D}_{1}\cup \mathcal{L}_{2}$ $E_b$ $E_1$ $(\alpha, \gamma)\in \mathcal{D}_{2}$ $E_b$ $E_2$ $(\alpha, \gamma)\in \mathcal{D}_{3}$ $E_b$ $E_1$ $E_2$
 [1] Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419 [2] Ping Liu, Junping Shi, Yuwen Wang. A double saddle-node bifurcation theorem. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2923-2933. doi: 10.3934/cpaa.2013.12.2923 [3] Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233 [4] Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203 [5] Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026 [6] Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 [7] Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 [8] John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 [9] Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152 [10] 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 [11] Russell Johnson, Francesca Mantellini. A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 209-224. doi: 10.3934/dcds.2003.9.209 [12] Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71 [13] R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147 [14] Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197 [15] Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247 [16] Patrick M. Fitzpatrick, Jacobo Pejsachowicz. Branching and bifurcation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1955-1975. doi: 10.3934/dcdss.2019127 [17] Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098 [18] 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 [19] 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 [20] Eric Benoît. Bifurcation delay - the case of the sequence: Stable focus - unstable focus - unstable node. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 911-929. doi: 10.3934/dcdss.2009.2.911

2018 Impact Factor: 1.008