Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-24T21:44:14.864Z Has data issue: false hasContentIssue false
  • English
  • Français

Stationary and slowly moving localised pulses in a singularly perturbed Brusselator model

Published online by Cambridge University Press:  06 May 2011

J. C. TZOU ,
A. BAYLISS ,
B. J. MATKOWSKY  and
V. A. VOLPERT
J. C. TZOU
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208-3125, USA email: b-matkowsky@northwestern.edu
A. BAYLISS
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208-3125, USA email: b-matkowsky@northwestern.edu
B. J. MATKOWSKY
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208-3125, USA email: b-matkowsky@northwestern.edu
V. A. VOLPERT
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208-3125, USA email: b-matkowsky@northwestern.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Recent attention has focused on deriving localised pulse solutions to various systems of reaction–diffusion equations. In this paper, we consider the evolution of localised pulses in the Brusselator activator–inhibitor model, long considered a paradigm for the study of non-linear equations, in a finite one-dimensional domain with feed of the inhibitor through the boundary and global feed of the activator. We employ the method of matched asymptotic expansions in the limit of small activator diffusivity and small activator and inhibitor feeds. The disparity of diffusion lengths between the activator and inhibitor leads to pulse-type solutions in which the activator is localised while the inhibitor varies on an O(1) length scale. In the asymptotic limit considered, the pulses become spikes described by Dirac delta functions and evolve slowly in time until equilibrium is reached. Such quasi-equilibrium solutions with N activator pulses are constructed and a differential-algebraic system of equations (DAE) is derived, characterising the slow evolution of the locations and the amplitudes of the pulses. We find excellent agreement for the pulse evolution between the asymptotic theory and the results of numerical computations. An algebraic system for the equilibrium pulse amplitudes and locations is derived from the equilibrium points of the DAE system. Both symmetric equilibria, corresponding to a common pulse amplitude, and asymmetric pulse equilibria, for which the pulse amplitudes are different, are constructed. We find that for a positive boundary feed rate, pulse spacing of symmetric equilibria is no longer uniform, and that for sufficiently large boundary flux, pulses at the edges of the pattern may collide with and remain fixed at the boundary. Lastly, stability of the equilibrium solutions is analysed through linearisation of the DAE, which, in contrast to previous approaches, provides a quick way to calculate the small eigenvalues governing weak translation-type instabilities of equilibrium pulse patterns.

Keywords

Brusselator modelPulse evolutionQuasi-equilibriaDifferential-algebraic equationsMatched asymptotic expansionsSingular perturbations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 5 , October 2011 , pp. 423 - 453
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Chen, W. & Ward, M. J. (2011) The stability and dynamics of localized spot patterns in the two-dimensional Gray-Scott model. SIAM J. Appl. Dyn. Sys. (to appear).CrossRefGoogle Scholar
[2]De Wit, A., Lima, D., Dewel, G. & Borckmans, P. (1996) Spatiotemporal dynamics near a codimension-two point. Phys. Rev. E 54, pp. 261–271.CrossRefGoogle Scholar
[3]Doelman, A., Eckhaus, W. & Kaper, T. J. (2000) Slowly-modulated two-pulse solutions in the Gray-Scott model : Part I. Asymptotic construction and stability. SIAM J. Appl. Math. 61 (3), pp. 1080–1102.CrossRefGoogle Scholar
[4]Doelman, A., Eckhaus, W. & Kaper, T. J. (2001) Slowly modulated two-pulse solutions in the Gray–Scott model: Part II. Geometric theory, bifurcations, and splitting dynamics. SIAM J. Appl. Math. 66 (6), pp. 2036–2062.CrossRefGoogle Scholar
[5]Doelman, A., Gardner, R. A. & Kaper, T. J. (1998) Stability analysis of singular patterns in the 1D Gray-Scott model: A matched asymptotic approach. Physica D 122, pp. 1–36.CrossRefGoogle Scholar
[6]Doelman, A. & Kaper, T. (2003) Semistrong pulse interactions in a class of coupled reaction-diffusion equations. SIAM J. Appl. Dyn. Syst. 2 (1), pp. 53–96.CrossRefGoogle Scholar
[7]Doelman, A., Kaper, T. J. & Promislow, K. (2007) Nonlinear asymptotic stability of the semistrong pulse dynamics in a regularized Gierer–Meinhardt model. SIAM J. Math. Anal. 38 (6), pp. 1760–1787.CrossRefGoogle Scholar
[8]Gierer, A. & Meinhardt, H. (1972) A theory of biological pattern Formation. Kybernetic 12, pp. 30–39.CrossRefGoogle ScholarPubMed
[9]Iron, D. & Ward, M. J. (2000) A metastable spike solution for a nonlocal reaction-diffusion model. SIAM J. Appl. Math. 60 (3), pp. 778–802.CrossRefGoogle Scholar
[10]Iron, D., Ward, M. J. & Wei, J. (2001) The stability of spike solutions to the one-dimensional Gierer-Meinhardt model. Physica D 150, pp. 25–62.CrossRefGoogle Scholar
[11]Iron, D. & Ward, M. J. (2002) The dynamics of multi-spike solutions for the one-dimensional Gierer-Meinhardt Model. SIAM J. Appl. Math. 62 (6), pp. 1924–1951.CrossRefGoogle Scholar
[12]Iron, D., Wei, J. & Winter, M. (2004) Stability analysis of turing patterns generated by the Schnakenberg model. J. Math. Biol. 49, pp. 358–390.CrossRefGoogle ScholarPubMed
[13]Kerner, B. S. & Osipov, V. V. (1989) Autosolitons. Sov. Phys.-Usp. 32, pp. 101–138.CrossRefGoogle Scholar
[14]Kerner, B. S. & Osipov, V. V. (1990) Self-organization in active distributed media: Scenarios for the spontaneous formation and evolution of dissipative structures. Sov. Phys.-Usp. 33, pp. 679–719.CrossRefGoogle Scholar
[15]Kolokolnikov, T., Erneux, T. & Wei, J. (2006) Mesa-type patterns in the one-dimensional Brusselator and their stability. Physica D 214, pp. 63–77.CrossRefGoogle Scholar
[16]Kolokolnikov, T., Ward, M. J. & Wei, J. (2005) The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The low feed rate regime. Studies in Appl. Math. 115 (1), pp. 21–71.CrossRefGoogle Scholar
[17]Kolokolnikov, T., Ward, M. J. & Wei, J. (2005) The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The pulse-splitting regime. Physica D 202 (3–4), pp. 258–293.CrossRefGoogle Scholar
[18]Kolokolnikov, T., Ward, M. J. & Wei, J. (2006) Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model. Interfaces and Free Boundaries 8 (2), pp. 185–222.Google Scholar
[19]Kolokolnikov, T., Ward, M. J. & Wei, J. (2009) Spot self-replication and dynamics for the Schnakenburg model in a two-dimensional domain. J. Nonlinear Sci. 19 (1), pp. 1–56.CrossRefGoogle Scholar
[20]Muratov, C. B. & Osipov, V. V. (2000) Static spike autosolitons in the Gray-Scott model. J. Phys. A: Math. Gen. 33, pp. 8893–8916.CrossRefGoogle Scholar
[21]Muratov, C. B. & Osipov, V. V. (2002) Stability of the static spike autosolitons in the Gray–Scott model. SIAM J. Appl. Math. 62 (5), pp. 1463–1487.CrossRefGoogle Scholar
[22]Nicolis, G. & Prigogine, I. (1977) Self-Organization in Nonequilibrium Systems, Wiley, New York.Google Scholar
[23]Nishiura, Y., Teramoto, T. & Ueda, K. (2003) Scattering and separators in dissipative systems. Phys. Rev. E 67, 056210.CrossRefGoogle ScholarPubMed
[24]Nishiura, Y. & Ueyama, D. (1999) A skeleton structure of self-replicating dynamics. Physica D 130, pp. 73–104.CrossRefGoogle Scholar
[25]Pearson, J. E. (1993) Complex patterns in a simple system. Science 216, pp. 189–192.CrossRefGoogle Scholar
[26]Tzou, J. C., Bayliss, A., Matkowsky, B. J. & Volpert, V. A. (2011) Interaction of turing and Hopf modes in the superdiffusive Brusselator model near a codimension two bifurcation point. Math. Model. Nat. Phenom. 6 (1), pp. 87–118.CrossRefGoogle Scholar
[27]Turing, A. M. (1952) The Chemical Basis of Morphogenesis. Philos. Trans. R. Soc. London Ser. B 237, pp. 37–72.Google Scholar
[28]Vanag, V. K. (2004) Waves and patterns in reaction diffusion systems. Belousov-Zhabotinsky reaction in water-in oil microemulsions. Phys.-Usp. 47, pp. 923–941.CrossRefGoogle Scholar
[29]Ward, M. J. & Wei, J. (2002) Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: Equilibria and stability. European J. Appl. Math. 13 (3), pp. 283–320.CrossRefGoogle Scholar
[30]Ward, M. J. & Wei, J. (2002) The existence and stability of asymmetric spike patterns for the Schnakenberg model. Studies in Appl. Math. 109, pp. 229–264.CrossRefGoogle Scholar
[31]Ward, M. J. & Wei, J. (2008) Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model. J. Nonlinear Sci. 13 (2), pp. 209–264.CrossRefGoogle Scholar