doi: 10.3934/dcdss.2022103
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Pattern formation of Brusselator in the reaction-diffusion system

1. 

Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 211100, China

2. 

School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China

* Corresponding author: Jianwei Shen

Received  December 2021 Revised  March 2022 Early access April 2022

Time delay profoundly impacts reaction-diffusion systems, which has been considered in many areas, especially infectious diseases, neurodynamics, and chemistry. This paper aims to investigate the pattern dynamics of the reaction-diffusion model with time delay. We obtain the condition in which the system induced the Hopf bifurcation and Turing instability as the parameter of the diffusion term and time delay changed. Meanwhile, the amplitude equation of the reaction-diffusion system with time delay is also derived based on the Friedholm solvability condition and the multi-scale analysis method near the critical point of phase transition. We discussed the stability of the amplitude equation. Theoretical results demonstrate that the delay can induce rich pattern dynamics in the Brusselator reaction-diffusion system, such as strip and hexagonal patterns. It is evident that time delay causes steady-state changes in the spatial pattern under certain conditions but does not cause changes in pattern selection under certain conditions. However, diffusion and delayed feedback affect pattern formation and pattern selection. This paper provides a feasible method to study reaction-diffusion systems with time delay and the development of the amplitude equation. The numerical simulation well verifies and supports the theoretical results.

Citation: Yansu Ji, Jianwei Shen, Xiaochen Mao. Pattern formation of Brusselator in the reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022103
References:
[1]

H. Y. Alfifi, Feedback control for a diffusive and delayed Brusselator model: Semi-analytical solutions, Symmetry, 13 (2021), 725, 13 pp. 

[2]

T. BiancalaniD. Fanelli and F. Di Patti, Stochastic Turing patterns in the Brusselator model, Phys. Rev. E, 8 (2010), 046215, 8 pp. 

[3]

P. ClusellaM. C. Miguel and R. Pastor-Satorras, Amplitude death and restoration in networks of oscillators with random-walk diffusion, Commun. Phys., 4 (2021), 1-11. 

[4]

G. GambinoM. C. LombardoM. Sammartino and V. Sciacca, Turing pattern formation in the Brusselator system with nonlinear diffusion, Phys. Rev. E, 88 (2013), 042925, 12 pp. 

[5]

A. A. GolovinB. J. Matkowsky and V. A. Volpert, Turing Pattern Formation in the Brusselator Model with Superdiffusion, SIAM J. Appl. Math., 69 (2008), 251-272.  doi: 10.1137/070703454.

[6]

S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Applied Mathematical Sciences, 184. Springer, New York, 2013. doi: 10.1007/978-1-4614-6992-6.

[7]

S. V. Gurevich, Dynamics of localized structures in reaction-diffusion systems induced by delayed feedback, Phys. Rev. E, 87 (2013), 052922, 9 pp. 

[8]

J. Hale and S. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[9]

W. HorsthemkeK. Lam and P. K. Moore, Network topology and Turing instabilities in small arrays of diffusively coupled reactors, Phys. Lett. A, 328 (2004), 444-451. 

[10]

Y. Ji and J. Shen, Turing instability of Brusselator in the reaction-diffusion network, Complexity, 2020 (2020), 1572743, 12 pp. 

[11]

Y. Jia, Bifurcation and pattern formation of a tumor-immune model with time-delay and diffusion, Math. Comput. Simul., 178 (2020), 92-108.  doi: 10.1016/j.matcom.2020.06.011.

[12]

B. KostetM. TlidiF. TabbertT. Frohoff-HülsmannS. V. GurevichE. AverlantR. RojasG. Sonnino and K. Panajotov, Stationary localized structures and the effect of the delayed feedback in the Brusselator model, Philos. Trans. Roy. Soc. A, 376 (2018), 20170385, 18 pp.  doi: 10.1098/rsta.2017.0385.

[13]

Z. Liu, J. Shen, S. Cai and F. Yan, MicroRNA Regulatory Network: Structure and Function, Springer, Dordrecht, 2018. doi: 10.1007/978-94-024-1577-3.

[14]

O. Mason and M. Verwoerd, Graph theory and networks in biology, IET Syst. Biol., 1 (2007), 89-119. 

[15]

H. G. Othmer and L. E. Scriven, Instability and dynamic pattern in cellular networks, J. Theor. Biol., 32 (1971), 507-537. 

[16]

Q. Ouyang, Nonlinear Science and Introduction to Pattern Dynamics, Peking University Press, 2010.

[17]

B. Peña and C. Pérez-García, Stability of turing patterns in the Brusselator model, Phys. Rev. E, 64 (2001), 056213, 9 pp.  doi: 10.1103/PhysRevE.64.056213.

[18]

R. Peng and M. Wang, Pattern formation in the Brusselator system, J. Math. Anal. Appl., 309 (2005), 151-166.  doi: 10.1016/j.jmaa.2004.12.026.

[19]

I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems. Ⅱ, J. Chem. Phys., 48 (1968), 1695-1700. 

[20]

S. Ruan and J. Wei, On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, IMA J. Math. Appl. Medicine Biol., 18 (2001), 41-52. 

[21]

J. ShenZ. LiuW. ZhengF. Xu and L. Chen, Oscillatory dynamics in a simple gene regulatory network mediated by small RNAs, Phys. A, 388 (2009), 2995-3000.  doi: 10.1016/j.physa.2009.03.032.

[22]

R. SunJ. LiH. Zhang and Y. Yang, Kinetic Pattern Formation with Intermolecular Interactions: A Modified Brusselator Model, Chinese J. Polym. Sci., 39 (2021), 1673-1679. 

[23]

M. TlidiY. GandicaG. SonninoE. Averlant and K. Panajotov, Self-replicating spots in the Brusselator model and extreme events in the one-dimensional case with delay, Entropy, 8 (2016), 64, 10 pp. 

[24]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[25]

X. WangZ. SongZ. LiL. Chang and Z. Wang, Delay-induced patterns in a reaction-diffusion system on complex networks, New J. Phys., 23 (2021), 073022, 12 pp.  doi: 10.1088/1367-2630/ac0ebc.

[26]

W.-X. XieS.-P. CaoL. Cai and X.-X. Zhang, Study on Turing patterns of Gray-Scott model via amplitude equation, Int. J. Bifurcation Chaos, 30 (2020), 2050121, 19 pp.  doi: 10.1142/S0218127420501217.

[27]

K. ZhangH. WangC. Qiao and Q. Ouyang, Hexagonal standing-wave patterns in periodically forced reaction–diffusion systems, Chin. Phys. Lett., 23 (2006), 1414-1417. 

[28]

Q. Zheng and J. Shen, Pattern formation in the FitzHugh-Nagumo model, Comput. Math. Appl., 70 (2015), 1082-1097.  doi: 10.1016/j.camwa.2015.06.031.

[29]

Q. Zheng and J. Shen, Turing instability and amplitude equation of reaction-diffusion system with multivariable, Math. Probl. Eng., 2020 (2020), 1381095, 7 pp.  doi: 10.1155/2020/1381095.

[30]

Q. Zheng and J. Shen, Turing instability induced by random network in FitzHugh-Nagumo model, Appl. Math. Comput., 381 (2020), 125304, 13 pp.  doi: 10.1016/j.amc.2020.125304.

[31]

Q. ZhengJ. Shen and Z. Wang, Pattern formation and oscillations in reaction-diffusion model with p53-Mdm2 feedback loop, Int. J. Bifurcation Chaos, 29 (2019), 1930040, 13 pp.  doi: 10.1142/S0218127419300404.

[32]

Q. ZhengJ. Shen and Y. Xu, Turing instability in the reaction-diffusion network, Phys. Rev. E, 102 (2020), 062215, 9 pp.  doi: 10.1103/PhysRevE.102.062215.

[33]

Q. ZhuJ. ShenF. Han and W. Lu, Bifurcation analysis and probabilistic energy landscapes of two-component genetic network, IEEE Access, 8 (2020), 150696-150708. 

show all references

References:
[1]

H. Y. Alfifi, Feedback control for a diffusive and delayed Brusselator model: Semi-analytical solutions, Symmetry, 13 (2021), 725, 13 pp. 

[2]

T. BiancalaniD. Fanelli and F. Di Patti, Stochastic Turing patterns in the Brusselator model, Phys. Rev. E, 8 (2010), 046215, 8 pp. 

[3]

P. ClusellaM. C. Miguel and R. Pastor-Satorras, Amplitude death and restoration in networks of oscillators with random-walk diffusion, Commun. Phys., 4 (2021), 1-11. 

[4]

G. GambinoM. C. LombardoM. Sammartino and V. Sciacca, Turing pattern formation in the Brusselator system with nonlinear diffusion, Phys. Rev. E, 88 (2013), 042925, 12 pp. 

[5]

A. A. GolovinB. J. Matkowsky and V. A. Volpert, Turing Pattern Formation in the Brusselator Model with Superdiffusion, SIAM J. Appl. Math., 69 (2008), 251-272.  doi: 10.1137/070703454.

[6]

S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Applied Mathematical Sciences, 184. Springer, New York, 2013. doi: 10.1007/978-1-4614-6992-6.

[7]

S. V. Gurevich, Dynamics of localized structures in reaction-diffusion systems induced by delayed feedback, Phys. Rev. E, 87 (2013), 052922, 9 pp. 

[8]

J. Hale and S. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[9]

W. HorsthemkeK. Lam and P. K. Moore, Network topology and Turing instabilities in small arrays of diffusively coupled reactors, Phys. Lett. A, 328 (2004), 444-451. 

[10]

Y. Ji and J. Shen, Turing instability of Brusselator in the reaction-diffusion network, Complexity, 2020 (2020), 1572743, 12 pp. 

[11]

Y. Jia, Bifurcation and pattern formation of a tumor-immune model with time-delay and diffusion, Math. Comput. Simul., 178 (2020), 92-108.  doi: 10.1016/j.matcom.2020.06.011.

[12]

B. KostetM. TlidiF. TabbertT. Frohoff-HülsmannS. V. GurevichE. AverlantR. RojasG. Sonnino and K. Panajotov, Stationary localized structures and the effect of the delayed feedback in the Brusselator model, Philos. Trans. Roy. Soc. A, 376 (2018), 20170385, 18 pp.  doi: 10.1098/rsta.2017.0385.

[13]

Z. Liu, J. Shen, S. Cai and F. Yan, MicroRNA Regulatory Network: Structure and Function, Springer, Dordrecht, 2018. doi: 10.1007/978-94-024-1577-3.

[14]

O. Mason and M. Verwoerd, Graph theory and networks in biology, IET Syst. Biol., 1 (2007), 89-119. 

[15]

H. G. Othmer and L. E. Scriven, Instability and dynamic pattern in cellular networks, J. Theor. Biol., 32 (1971), 507-537. 

[16]

Q. Ouyang, Nonlinear Science and Introduction to Pattern Dynamics, Peking University Press, 2010.

[17]

B. Peña and C. Pérez-García, Stability of turing patterns in the Brusselator model, Phys. Rev. E, 64 (2001), 056213, 9 pp.  doi: 10.1103/PhysRevE.64.056213.

[18]

R. Peng and M. Wang, Pattern formation in the Brusselator system, J. Math. Anal. Appl., 309 (2005), 151-166.  doi: 10.1016/j.jmaa.2004.12.026.

[19]

I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems. Ⅱ, J. Chem. Phys., 48 (1968), 1695-1700. 

[20]

S. Ruan and J. Wei, On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, IMA J. Math. Appl. Medicine Biol., 18 (2001), 41-52. 

[21]

J. ShenZ. LiuW. ZhengF. Xu and L. Chen, Oscillatory dynamics in a simple gene regulatory network mediated by small RNAs, Phys. A, 388 (2009), 2995-3000.  doi: 10.1016/j.physa.2009.03.032.

[22]

R. SunJ. LiH. Zhang and Y. Yang, Kinetic Pattern Formation with Intermolecular Interactions: A Modified Brusselator Model, Chinese J. Polym. Sci., 39 (2021), 1673-1679. 

[23]

M. TlidiY. GandicaG. SonninoE. Averlant and K. Panajotov, Self-replicating spots in the Brusselator model and extreme events in the one-dimensional case with delay, Entropy, 8 (2016), 64, 10 pp. 

[24]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[25]

X. WangZ. SongZ. LiL. Chang and Z. Wang, Delay-induced patterns in a reaction-diffusion system on complex networks, New J. Phys., 23 (2021), 073022, 12 pp.  doi: 10.1088/1367-2630/ac0ebc.

[26]

W.-X. XieS.-P. CaoL. Cai and X.-X. Zhang, Study on Turing patterns of Gray-Scott model via amplitude equation, Int. J. Bifurcation Chaos, 30 (2020), 2050121, 19 pp.  doi: 10.1142/S0218127420501217.

[27]

K. ZhangH. WangC. Qiao and Q. Ouyang, Hexagonal standing-wave patterns in periodically forced reaction–diffusion systems, Chin. Phys. Lett., 23 (2006), 1414-1417. 

[28]

Q. Zheng and J. Shen, Pattern formation in the FitzHugh-Nagumo model, Comput. Math. Appl., 70 (2015), 1082-1097.  doi: 10.1016/j.camwa.2015.06.031.

[29]

Q. Zheng and J. Shen, Turing instability and amplitude equation of reaction-diffusion system with multivariable, Math. Probl. Eng., 2020 (2020), 1381095, 7 pp.  doi: 10.1155/2020/1381095.

[30]

Q. Zheng and J. Shen, Turing instability induced by random network in FitzHugh-Nagumo model, Appl. Math. Comput., 381 (2020), 125304, 13 pp.  doi: 10.1016/j.amc.2020.125304.

[31]

Q. ZhengJ. Shen and Z. Wang, Pattern formation and oscillations in reaction-diffusion model with p53-Mdm2 feedback loop, Int. J. Bifurcation Chaos, 29 (2019), 1930040, 13 pp.  doi: 10.1142/S0218127419300404.

[32]

Q. ZhengJ. Shen and Y. Xu, Turing instability in the reaction-diffusion network, Phys. Rev. E, 102 (2020), 062215, 9 pp.  doi: 10.1103/PhysRevE.102.062215.

[33]

Q. ZhuJ. ShenF. Han and W. Lu, Bifurcation analysis and probabilistic energy landscapes of two-component genetic network, IEEE Access, 8 (2020), 150696-150708. 

Figure 1.  (a) Hopf bifurcation of $ u $ about $ b $ when $ \theta = 0.05 $ and $ \tau = 0 $. The time evolution corresponding to system (2) when $ b = 1.5 $(b), $ b = 2.5 $(c) and $ b = 7.5 $(d)
Figure 2.  The curve of the bifurcation critical point $ b_{H} $ of the system (2) with the feedback strength $ \theta $ when without diffusion and $ \tau = 0 $
Figure 3.  Concentration-time profiles of variables $ u $ and $ v $ in the system (2) for different $ \tau $ when $ d_1 = d_2 = 0 $, $ a = 1 $, $ \theta = 0.05 $ and $ b = 1.5 $. The time delays are (a) $ \tau = 1 $, (b) $ \tau = 1.5 $ and (c) $ \tau = 2 $ respectively. (d) Bifurcation of the concentration $ u $ versus $ \tau $
Figure 4.  The pattern formation of $ u $ and $ v $ in the system(2) when $ a = 1 $, $ b = 1.5 $, $ d_1 = 0.004 $, $ \tau = 0 $, $ \theta = 0 $, (a)(b) $ d_2 = 0.3 $; (c)(d) $ d_2 = 0.4 $; (e)(f) $ d_2 = 0.5 $
Figure 5.  The pattern formation of $ u $ and $ v $ in the system(2) when $ a = 1 $, $ b = 1.5 $, $ d_1 = 0.15 $, $ d_2 = 2.1 $, (a)(b) $ \tau = 0 $, $ \theta = 0 $; (c)(d) $ \tau = 0 $, $ \theta = 0.45 $; (e)(f) $ \tau = 0.1 $, $ \theta = 0.45 $
Figure 6.  The pattern formation of $ u $ and $ v $ in the system(2) when $ a = 1 $, $ b = 2.5 $, $ d_1 = 0.04 $, $ d_2 = 0.2 $, $ \tau = 0 $, $ \theta = 0.35 $
Figure 7.  The pattern formation of $ u $ and $ v $ in the system(2) when $ a = 1 $, $ b = 2.5 $, $ d_1 = 0.01 $, $ d_2 = 0.2 $, $ \tau = 0 $, $ \theta = 0.35 $
Figure 8.  The pattern formation of $ u $ and $ v $ in the system(2) when $ a = 1 $, $ b = 2.5 $, $ d_1 = 0.01 $, $ d_2 = 0.8 $, $ \tau = 0, \theta = 0.35 $
Figure 9.  The pattern formation of $ u $ and $ v $ in the system(2) when $ a = 1 $, $ b = 2.5 $, $ d_1 = 0.01 $, $ d_2 = 0.2 $, $ \tau = 0.05, \theta = 0.5 $
Figure 10.  The pattern formation of $ u $ and $ v $ in the system(2) when $ a = 1 $, $ b = 2.5 $, $ d_1 = 0.04 $, $ d_2 = 0.2 $, $ \tau = 0.05, \theta = 0.5 $
Figure 11.  The pattern formation of $ u $ and $ v $ in the system(2) when $ a = 1 $, $ b = 2.5 $, $ d_1 = 0.01 $, $ d_2 = 0.2 $, $ \tau = 5, \theta = 0.25 $
Figure 12.  The pattern formation of $ u $ and $ v $ in the system(2) when $ a = 1 $, $ b = 2.5 $, $ d_1 = 0.04 $, $ d_2 = 0.2 $, $ \tau = 0.5, \theta = 0.25 $
Figure 13.  The pattern formation of $ u $ and $ v $ in the system(2) when $ a = 1 $, $ b = 2.5 $, $ d_1 = 0.5 $, $ d_2 = 2 $, $ \tau = 0 $ (a)(b) $ \theta = 0.15 $, (c)(d) $ \theta = 0.25 $, (e)(f) $ \theta = 0.45 $
Figure 14.  The pattern formation of $ u $ and $ v $ in the system(2) when $ a = 1 $, $ b = 2.5 $, $ d_1 = 0.5 $, $ d_2 = 2 $ and $ \theta = 0.35 $, (a)(b) $ \tau = 0 $; (c)(d) $ \tau = 0.1 $; (e)(f) $ \tau = 1 $
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