Periodic Pattern Formation Analysis Numerically in a Chemical Reaction-Diffusion System

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A. K. M. Nazimuddin 1,* Md. Showkat Ali 2

1. Department of Mathematical and Physical Sciences, East West University, Dhaka-1212, Bangladesh.

2. Department of Applied Mathematics, University of Dhaka, Dhaka-1000, Bangladesh.

* Corresponding author.


Received: 11 Feb. 2019 / Revised: 21 Feb. 2019 / Accepted: 6 Mar. 2019 / Published: 8 Jul. 2019

Index Terms

Reaction Diffusion System, Periodic Traveling wave, Pattern Formation, Brusselator Model


In this paper, we analyze the pattern formation in a chemical reaction-diffusion Brusselator model. Twocomponent Brusselator model in two spatial dimensions is studied numerically through direct partial differential equation simulation and we find a periodic pattern. In order to understand the periodic pattern, it is important to investigate our model in one-dimensional space. However, direct partial differential equation simulation in one dimension of the model is performed and we get periodic traveling wave solutions of the model. Then, the local dynamics of the model is investigated to show the existence of the limit cycle solutions. After that, we establish the existence of periodic traveling wave solutions of the model through the continuation method and finally, we get a good consistency among the results.

Cite This Paper

A. K. M. Nazimuddin, Md. Showkat Ali, "Periodic Pattern Formation Analysis Numerically in a Chemical Reaction-Diffusion System ", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.5, No.3, pp.17-26, 2019. DOI:10.5815/ijmsc.2019.03.02


[1]Kopell, N. and Howard, LN. (1973). Plane-wave solutions to reaction-diffusion equations. Stud Appl. Math, 52:291–328.

[2]Ranta, E. and Kaitala, V. (1997). Travelling waves in vole population dynamics, Geochim. Cosmochim. Acta, 61:3503–3512.

[3]Bierman, S. M., Fairbairn, J. P., Petty, S. J., Elston, D. A., Tidhar, D. and Lambin, X. (2006). Changes over time in the spatiotemporal dynamics of cyclic populations of field voles (Microtus agrestis L.), The American Naturalist, 167 (4):583–590.

[4]Sherratt, J. A., Smith, M. J.  (2008). Periodic travelling waves in cyclic populations: field studies and reaction–diffusion models, Journal of the Royal Society Interface, 5 (22):483–505.

[5]DeVille, R. E. L. and Eijnden, E. V. (2007). Wavetrain response of an excitable medium to local stochastic forcing, Nonlinearity, 20 (1):51–74.

[6]Sherratt, J. A., Lord, G. J. (2007).  Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments, Theoretical Population Biology, 71 (1):1–11.

[7]Steinberg, V., Fineberg, J., Moses, E. and Rehberg, I. (1989). Pattern selection and transition to turbulence in propagating waves, Physica D: Nonlinear Phenomena, 37:359–383.

[8]Van Hecke, M., (2003). Coherent and incoherent structures in systems described by the 1D CGLE: experiments and identification, Physica D: Nonlinear Phenomena, 174 (1):134–151.

[9]Van Saarloos, W. (2003).  Front propagation into unstable states, Physics Reports, 386 (2): 29–222.

[10]Epstein, I. R. and Showalter, K. (1996). Nonlinear chemical dynamics: oscillations, patterns, and chaos, The Journal of Physical Chemistry, 100(31):13132–13147.

[11]Vanag, V. K., Epstein, I. R. (2008). Design and control of patterns in reaction-diffusion systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 18 (2):(026107)1-11.

[12]Bordyugov, G., Fischer, N., Engel, H., Manz, N. and Steinbock, O. (2010). Anomalous dispersion in the Belousov–Zhabotinsky reaction: Experiments and modeling, Physica D: Nonlinear Phenomena, 239 (11): 766–775.

[13]Sherratt, J. A. (2012).  Numerical continuation methods for studying periodic travelling wave (wavetrain) solutions of partial differential equations, Applied Mathematics & Computation, 218(9): 4684–4694.

[14]Dutt, AK. (2010). Turing pattern amplitude equation for a model glycolytic reaction-diffusion system, Journal of mathematical chemistry, 48(4):841–855.

[15]Kolokolnikov, T., Erneux, T., and Wei, J. (2006). Mesa-type patterns in the one-dimensional brusselator and their stability,  Physica D: Nonlinear Phenomena, 214(1):63–77.

[16]Tzou, JC., Matkowsky, B. J.  and Volpert, V. A. (2009). Interaction of turing and hopf modes in the super diffusive brusselator model, Applied Mathematics Letters, 22(9):1432–1437.

[17]Yang, L., Zhabotinsky, A. M., and Epstein, I. R. (2004). Stable squares and other oscillatory turing patterns in a reaction-diffusion model, Physical review letters, 92(19):198-303.

[18]Feng, B. (1988). Periodic Travelling-wave Solution of Bruselator, Acta Math. Appl. Sinica, 4(4): 324–332.

[19]Anguelov, R. & Stoltz, S. M. (2017). Stationary and oscillatory patterns in a coupled Brusselator model, Mathematics and Computers in Simulation, 133(C): 39-46. 

[20]Alqahtani, AM. (2018). Numerical simulation to study the pattern formation of reaction–diffusion Brusselator model arising in triple collision and enzymatic, Journal of Mathematical Chemistry, 56(6):1543-1566.

[21]Prigogine, I., Lefever, R. (1968).  Symmetry breaking instabilities in dissipative systems, The Journal of  Chemical Physics, 48(4): 1695–1700.

[22]Hassard, B. D., Kazarinoff, N. D. and Wan, Y. H. (1981). Theory and Applications of Hopf Bifurcation, Cambridge University Press.

[23]Marsden, J. E.  and McCracken, M. (1976). The Hopf Bifurcation and its Applications, Applied Math. Series, 19, New-York, Springer-Verlag.