Analysis of a Brusselator reaction-diffusion model with nonlinear inhibition

Shouzong Liu; Yaxin Niu; Peiyang Chai; Shouzong Liu; Yaxin Niu; Peiyang Chai

doi:10.3934/math.2025855

AIMS Mathematics

2025, Volume 10, Issue 8: 19126-19146. doi: 10.3934/math.2025855

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Research article

Analysis of a Brusselator reaction-diffusion model with nonlinear inhibition

College of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China

Received: 07 June 2025 Revised: 09 August 2025 Accepted: 19 August 2025 Published: 22 August 2025
MSC : 35J25, 35K50, 47H11, 58C15

In this paper, the pattern formation of a Brusselator reaction-diffusion model with nonlinear inhibition was investigated. Through linear stability analysis and spectral theory, mathematical criteria for the existence and stability of spatial patterns were established. The main results included explicit stability conditions for homogeneous steady states and spectral criteria linking eigenvalue multiplicities to pattern existence. A series of numerical simulations were performed to validate theoretical predictions, revealing clear transitions between uniform and spatially heterogeneous solutions. The work provides fundamental insights into pattern-forming mechanisms for controlling spatial organization in nonlinear reaction-diffusion systems.
- Brusselator model,
- reaction-diffusion equations,
- steady-state solution,
- stability,
- pattern formation
Citation: Shouzong Liu, Yaxin Niu, Peiyang Chai. Analysis of a Brusselator reaction-diffusion model with nonlinear inhibition[J]. AIMS Mathematics, 2025, 10(8): 19126-19146. doi: 10.3934/math.2025855

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Abstract

In this paper, the pattern formation of a Brusselator reaction-diffusion model with nonlinear inhibition was investigated. Through linear stability analysis and spectral theory, mathematical criteria for the existence and stability of spatial patterns were established. The main results included explicit stability conditions for homogeneous steady states and spectral criteria linking eigenvalue multiplicities to pattern existence. A series of numerical simulations were performed to validate theoretical predictions, revealing clear transitions between uniform and spatially heterogeneous solutions. The work provides fundamental insights into pattern-forming mechanisms for controlling spatial organization in nonlinear reaction-diffusion systems.

References

[1]	I. R. Epstein, J. A. Pojman, An introduction to nonlinear chemical dynamics: Oscillations, waves, patterns, and chaos, New York: Oxford Academic, 1998. https://doi.org/10.1093/oso/9780195096705.001.0001
[2]	M. C. Cross, P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), 851–1112. https://doi.org/10.1103/RevModPhys.65.851 doi: 10.1103/RevModPhys.65.851
[3]	P. Gray, S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability, Chem. Eng. Sci., 38 (1983), 29–43. https://doi.org/10.1016/0009-2509(83)80132-8 doi: 10.1016/0009-2509(83)80132-8
[4]	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
[5]	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
[6]	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
[7]	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), 1–40. https://doi.org/10.1007/s00332-024-10107-6 doi: 10.1007/s00332-024-10107-6
[8]	P. Gray, S. K. Scott, J. H. Merkin, The Brusselator model of oscillatory reactions, J. Chem. Soc. Farandy Trans. 1, 84 (1988), 993–1012. https://doi.org/10.1039/F19888400993 doi: 10.1039/F19888400993
[9]	J. J. Tyson, What everyone should know about the Belousov-Zhabotinsky reaction, In: Frontiers in Mathematical Biology, Lecture Notes in Biomathematics, Berlin: Springer, 100 (1994), 569–587. https://doi.org/10.1007/978-3-642-50124-1_33
[10]	B. Li, G. Guo, Diffusion-driven instability and Hopf bifurcation in spatial homogeneous Brusselator model, In: Advances in Future Computer and Control Systems, Advances in Intelligent and Soft Computing, Berlin: Springer, 160 (2012), 249–255. https://doi.org/10.1007/978-3-642-29390-0_41
[11]	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
[12]	M. Chen, R. Wu, B. Liu, L. Chen, Turing-Turing and Turing-Hopf bifurcations in a general diffusive Brusselator model, J. Appl. Math. Mec., 103 (2023), e201900111. https://doi.org/10.1002/zamm.201900111 doi: 10.1002/zamm.201900111
[13]	H. Liu, B. Ge, Spatial Turing patterns of periodic solutions for the Brusselator system with cross-diffusion-like coupling, Int. J. Bifurcat. Chaos, 33 (2023), 2350148. https://doi.org/10.1142/S0218127423501481 doi: 10.1142/S0218127423501481
[14]	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
[15]	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
[16]	X. Zhu, M. Huang, H. Bao, X. Zhang, Mechanistic insights into nonlinear effects in copper-catalyzed asymmetric esterification, Nat. Commun., 16 (2025), 2183. https://doi.org/10.1038/s41467-025-57380-x doi: 10.1038/s41467-025-57380-x
[17]	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
[18]	L. Kong, C. Zhu, Diffusion-driven codimension-2 Turing-Hopf bifurcation in the general Brusselator model, Math. Method. Appl. Sci., 44 (2021), 11456–11468. https://doi.org/10.1002/mma.7504 doi: 10.1002/mma.7504
[19]	H. Y. Alfifi, Feedback control for a diffusive and delayed Brusselator model: Semi-analytical solutions, Symmetry, 13 (2021), 725. https://doi.org/10.3390/sym13040725 doi: 10.3390/sym13040725
[20]	M. Winkler, A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization, J. Funct. Anal., 276 (2019), 1339–1401. https://doi.org/10.1016/j.jfa.2018.12.009 doi: 10.1016/j.jfa.2018.12.009
[21]	J. Zheng, Y. Ke, Eventual smoothness and stabilization in a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization, J. Differ. Equations, 328 (2022), 228–260. https://doi.org/10.1016/j.jde.2022.04.042 doi: 10.1016/j.jde.2022.04.042
[22]	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
[23]	A. Turing, The chemical basis of morphogenesis, Philos. T. Roy. Soc. B, 237 (1952), 37–72. https://doi.org/10.1098/rstb.1952.0012 doi: 10.1098/rstb.1952.0012
[24]	Y. Lou, W. M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differ. Equations, 154 (1999), 157–190. https://doi.org/10.1006/jdeq.1998.3559 doi: 10.1006/jdeq.1998.3559
[25]	L. Nirenberg, Topics in nonlinear functional analysis, New York: New York University-Courant Institute of Mathematical Sciences, 6 (2001). https://doi.org/10.1090/cln/006
[26]	R. Peng, J. Shi, M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471–1488. https://doi.org/10.1088/0951-7715/21/7/006 doi: 10.1088/0951-7715/21/7/006

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