Hopf bifurcation analysis of a modified Brusselator model with sub-interval distributed delay

Shouzong Liu; Tingting Lu; Jianing Qin; Yaxin Niu; Shouzong Liu; Tingting Lu; Jianing Qin; Yaxin Niu

doi:10.3934/math.2026272

AIMS Mathematics

2026, Volume 11, Issue 3: 6569-6591. doi: 10.3934/math.2026272

Previous Article Next Article

Research article

Hopf bifurcation analysis of a modified Brusselator model with sub-interval distributed delay

1.
School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China
2.
Henan Provincial Center for Applied Mathematics, Xinyang Normal University, Xinyang 464000, China

Received: 28 January 2026 Revised: 02 March 2026 Accepted: 06 March 2026 Published: 13 March 2026
MSC : 34K18, 34K20, 37G15

In this paper, a modified Brusselator model incorporating a moderated autocatalytic term and a sub-interval distributed delay is investigated. The existence and uniqueness of a positive equilibrium are first established, together with explicit conditions for its local asymptotic stability. By deriving and analyzing the associated characteristic equation, two distinct Hopf bifurcation mechanisms are rigorously identified: one induced by variations in the delay center $ \tau $ and the other triggered by changes in the delay distribution width $ \varepsilon $. Sufficient conditions for the existence of two Hopf bifurcation branches in the $ (\tau, \varepsilon) $-parameter plane are obtained, and the corresponding transversality conditions are verified. Finally, numerical simulations are carried out to validate the theoretical results. The results reveal a clear destabilizing role of the delay center and a stabilizing, smoothing effect of the delay width, which highlights the important role of distributed delays in shaping the dynamical behavior of autocatalytic reaction systems.
- Brusselator model,
- sub-interval distributed delay,
- Hopf bifurcation,
- stability
Citation: Shouzong Liu, Tingting Lu, Jianing Qin, Yaxin Niu. Hopf bifurcation analysis of a modified Brusselator model with sub-interval distributed delay[J]. AIMS Mathematics, 2026, 11(3): 6569-6591. doi: 10.3934/math.2026272

Related Papers:

Abstract

In this paper, a modified Brusselator model incorporating a moderated autocatalytic term and a sub-interval distributed delay is investigated. The existence and uniqueness of a positive equilibrium are first established, together with explicit conditions for its local asymptotic stability. By deriving and analyzing the associated characteristic equation, two distinct Hopf bifurcation mechanisms are rigorously identified: one induced by variations in the delay center $ \tau $ and the other triggered by changes in the delay distribution width $ \varepsilon $. Sufficient conditions for the existence of two Hopf bifurcation branches in the $ (\tau, \varepsilon) $-parameter plane are obtained, and the corresponding transversality conditions are verified. Finally, numerical simulations are carried out to validate the theoretical results. The results reveal a clear destabilizing role of the delay center and a stabilizing, smoothing effect of the delay width, which highlights the important role of distributed delays in shaping the dynamical behavior of autocatalytic reaction systems.

References

[1]	S. Maće$\check{s}$ić, Z. $\check{C}$upić, L. Kolar-Anić, Model of a nonlinear reaction system with autocatalysis and autoinhibition: Stability of dynamic states, Hem. Ind., 66 (2012), 637–646. https://doi.org/10.2298/HEMIND120210034M doi: 10.2298/HEMIND120210034M
[2]	D. V. Kriukov, J. Huskens, A. S. Y. Wong, Exploring the programmability of autocatalytic chemical reaction networks, Nat. Commun., 15 (2024), 8289. https://doi.org/10.1038/s41467-024-52649-z doi: 10.1038/s41467-024-52649-z
[3]	I. Bashkirtseva, M. Pavletsov, T. Perevalova, L. Ryashko, Mechanisms of stochastic excitement in a nonlinear thermochemical model of autocatalysis, Nonlinear Dyn., 113 (2025), 2199–2213. https://doi.org/10.1007/s11071-024-10335-1 doi: 10.1007/s11071-024-10335-1
[4]	T. Erneux, E. Reiss, Brusselator isolas, SIAM J. Appl. Math., 43 (1983), 1240–1246. https://doi.org/10.1137/0143082 doi: 10.1137/0143082
[5]	I. Prigogine, G. Nicolis, Self-organisation in nonequilibrium systems: Towards a dynamics of complexity, In: Bifurcation analysis, Dordrecht: Springer, 1985, 3–12. https://doi.org/10.1007/978-94-009-6239-2_1
[6]	I. R. Epstein, J. A. Pojman, An introduction to nonlinear chemical dynamics: Oscillations, waves, patterns, and chaos, Oxford: Oxford University Press, 1998. https://doi.org/10.1093/oso/9780195096705.001.0001
[7]	J. Zheng, P. Zhang, X. Liu, Global existence and boundedness for an N-dimensional parabolic-elliptic chemotaxis-fluid system with indirect pursuit-evasion, J. Differ. Equations, 367 (2023), 199–228. https://doi.org/10.1016/j.jde.2023.04.042 doi: 10.1016/j.jde.2023.04.042
[8]	K. J. Brown, F. A. Davidson, Global bifurcation in the Brusselator system, Nonlinear Anal., 24 (1995), 1713–1725. https://doi.org/10.1016/0362-546X(94)00218-7 doi: 10.1016/0362-546X(94)00218-7
[9]	H. Y. Alfifi, S. M. Almuaddi, Stability analysis and Hopf bifurcation for the Brusselator reaction-diffusion system with gene expression time delay, Mathematics, 12 (2024), 1170. https://doi.org/10.3390/math12081170 doi: 10.3390/math12081170
[10]	S. Li, Y. Yang, X. Song, The joint impacts of maturation delay and digestion delay on the dynamics of a predator-prey model, J. Xinyang Norm. Univ. (Natural Science Edition), 38 (2025), 421–427. https://doi.org/10.3969/j.issn.2097-583X.2025.04.007 doi: 10.3969/j.issn.2097-583X.2025.04.007
[11]	A. M. Alqahtani, Numerical simulation to study the pattern formation of reaction-diffusion Brusselator model arising in triple collision and enzymatic, J. Math. Chem., 56 (2018), 1543–1566. https://doi.org/10.1007/s10910-018-0859-8 doi: 10.1007/s10910-018-0859-8
[12]	R. M. Jena, S. Chakraverty, H. Rezazadeh, D. D. Ganji, On the solution of time-fractional dynamical model of Brusselator reaction-diffusion system arising in chemical reactions, Math. Method Appl. Sci., 43 (2020), 3903–3913. https://doi.org/10.1002/mma.6141 doi: 10.1002/mma.6141
[13]	S. Zhao, P. Yu, W. Jiang, H. Wang, A new mechanism revealed by cross-diffusion-driven instability and double-Hopf bifurcation in the Brusselator system, J. Nonlinear Sci., 35 (2025), 21. https://doi.org/10.1007/s00332-024-10107-6 doi: 10.1007/s00332-024-10107-6
[14]	I. Prigogine, R. Lefever, Symmetry breaking instabilities in dissipative systems. Ⅱ, J. Chem. Phys., 48 (1968), 1695–1700. https://doi.org/10.1063/1.1668896 doi: 10.1063/1.1668896
[15]	M. Ghergu, V. Radulescu, Turing patterns in general reaction-diffusion systems of Brusselator type, Commun. Contemp. Math., 12 (2010), 661–679. https://doi.org/10.1142/S0219199710003968 doi: 10.1142/S0219199710003968
[16]	G. Guo, T. Wei, F. Jia, K. A. Abbakar, Turing instability of periodic solutions for a general Brusselator model with cross-diffusion, J. Math. Anal. Appl., 541 (2025), 128683. https://doi.org/10.1016/j.jmaa.2024.128683 doi: 10.1016/j.jmaa.2024.128683
[17]	Y. Li, Hopf bifurcations in general systems of Brusselator type, Nonlinear Anal. Real World Appl., 28 (2016), 32–47. https://doi.org/10.1016/j.nonrwa.2015.09.004 doi: 10.1016/j.nonrwa.2015.09.004
[18]	Y. Ji, J. Shen, X. Mao, Pattern formation of Brusselator in the reaction-diffusion system, Discrete Contin. Dyn. Syst. Ser. S, 16 (2023), 434–459. https://doi.org/10.3934/dcdss.2022103 doi: 10.3934/dcdss.2022103
[19]	M. Liao, Q. R. Wang, Stability and bifurcation analysis in a diffusive Brusselator-type system, Int. J. Bifurcat. Chaos, 26 (2016), 1650119. https://doi.org/10.1142/S0218127416501194 doi: 10.1142/S0218127416501194
[20]	T. Zheng, L. Zhang, Y. Luo, X. Zhou, H. Li, Z. Teng, Stability and Hopf bifurcation of a stage-structured cannibalism model with two delays, Int. J. Bifurcat. Chaos, 31 (2021), 2150242. https://doi.org/10.1142/S0218127421502424 doi: 10.1142/S0218127421502424
[21]	W. Chen, L. Zhang, N. Wang, Z. Teng, Bifurcation analysis and chaos for a double-strains HIV coinfection model with intracellular delays, saturated incidence and logistic growth, Math. Comput. Simulation, 223 (2024), 617–641. https://doi.org/10.1016/j.matcom.2024.04.025 doi: 10.1016/j.matcom.2024.04.025
[22]	B. Song, Y. Zhang, Y. Sang, L. Zhang, Stability and Hopf bifurcation on an immunity delayed HBV/HCV model with intra- and extra-hepatic coinfection and saturation incidence, Nonlinear Dyn., 111 (2023), 14485–14511. https://doi.org/10.1007/s11071-023-08580-x doi: 10.1007/s11071-023-08580-x
[23]	X. Yang, G. Zhang, Turing-Hopf bifurcation of Brusselator model with cross-diffusion and Gene expression time delay, Int. J. Bifurcat. Chaos, 35 (2025), 2550159. https://doi.org/10.1142/S0218127425501597 doi: 10.1142/S0218127425501597
[24]	Q. Li, B. Liu, Traveling wave fronts to a Mackey-Glass model involving distinct delays in diffusion term and birth function, Math. Method Appl. Sci., 48 (2025), 15549–15558. https://doi.org/10.1002/mma.11109 doi: 10.1002/mma.11109
[25]	Y. Lv, Z. Liu, Turing-Hopf bifurcation analysis and normal form of a diffusive Brusselator model with gene expression time delay, Chaos Soliton Fract., 152 (2021), 111478. https://doi.org/10.1016/j.chaos.2021.111478 doi: 10.1016/j.chaos.2021.111478
[26]	S. Ruan, G. S. K. Wolkowicz, Bifurcation analysis of a chemostat model with a distributed delay, J. Math. Anal. Appl., 204 (1996), 786–812. https://doi.org/10.1006/jmaa.1996.0468 doi: 10.1006/jmaa.1996.0468
[27]	Z. Wang, B. Hu, L. Zhu, J. Lin, M. Xu, D. Wang, Hopf bifurcation analysis for Parkinson oscillation with heterogeneous delays: A theoretical derivation and simulation analysis, Commun. Nonlinear Sci. Numer. Simul., 114 (2022), 106614. https://doi.org/10.1016/j.cnsns.2022.106614 doi: 10.1016/j.cnsns.2022.106614
[28]	Z. Wang, B. Hu, W. Zhou, M. Xu, D. Wang, Hopf bifurcation mechanism analysis in an improved cortex-basal ganglia network with distributed delays: An application to Parkinsons disease, Chaos Soliton Fract., 166 (2023), 113022. https://doi.org/10.1016/j.chaos.2022.113022 doi: 10.1016/j.chaos.2022.113022
[29]	T. Yu, S. Yuan, T. Zhang, The effect of delay interval on the feedback control for a turbidostat model, J. Frankl. Inst., 358 (2021), 7628–7649. https://doi.org/10.1016/j.jfranklin.2021.08.003 doi: 10.1016/j.jfranklin.2021.08.003
[30]	X. Shi, Y. Kuang, A. Makroglou, S. Mokshagundam, J. Li, Oscillatory dynamics of an intravenous glucose tolerance test model with delay interval, Chaos, 27 (2017), 114324. https://doi.org/10.1063/1.5008384 doi: 10.1063/1.5008384

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)