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The stability of localized spikes for the 1-D Brusselator reaction–diffusion model

Published online by Cambridge University Press:  10 April 2013

J. C. TZOU ,
Y. NEC  and
M. J WARD
J. C. TZOU
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3125, USA email: tzou.justin@gmail.com
Y. NEC
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada emails: cranberryana@gmail.com, ward@math.ubc.ca
M. J WARD
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada emails: cranberryana@gmail.com, ward@math.ubc.ca
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Abstract

In a one-dimensional domain, the stability of localized spike patterns is analysed for two closely related singularly perturbed reaction–diffusion (RD) systems with Brusselator kinetics. For the first system, where there is no influx of the inhibitor on the domain boundary, asymptotic analysis is used to derive a non-local eigenvalue problem (NLEP), whose spectrum determines the linear stability of a multi-spike steady-state solution. Similar to previous NLEP stability analyses of spike patterns for other RD systems, such as the Gierer–Meinhardt and Gray–Scott models, a multi-spike steady-state solution can become unstable to either a competition or an oscillatory instability depending on the parameter regime. An explicit result for the threshold value for the initiation of a competition instability, which triggers the annihilation of spikes in a multi-spike pattern, is derived. Alternatively, in the parameter regime when a Hopf bifurcation occurs, it is shown from a numerical study of the NLEP that an asynchronous, rather than synchronous, oscillatory instability of the spike amplitudes can be the dominant instability. The existence of robust asynchronous temporal oscillations of the spike amplitudes has not been predicted from NLEP stability studies of other RD systems. For the second system, where there is an influx of inhibitor from the domain boundaries, an NLEP stability analysis of a quasi-steady-state two-spike pattern reveals the possibility of dynamic bifurcations leading to either a competition or an oscillatory instability of the spike amplitudes depending on the parameter regime. It is shown that the novel asynchronous oscillatory instability mode can again be the dominant instability. For both Brusselator systems, the detailed stability results from NLEP theory are confirmed by rather extensive numerical computations of the full partial differential equations system.

Keywords

BrusselatorSingular perturbationsQuasi-equilibriaNon-local eigenvalue problemHopf bifurcationAsynchronous oscillatory instabilityDynamically triggered instability
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 4 , August 2013 , pp. 515 - 564
Copyright
Copyright © Cambridge University Press 2013 

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References

[1]Chen, W. & Ward, M. J. (1999) Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional Gray-Scott model. Europ. J. Appl. Math. 20 (2), 187–214.Google Scholar
[2]Doelman, A., Eckhaus, W. & Kaper, T. J. (2000) Slowly-modulated two-pulse solutions in the Gray-Scott model I: Asymptotic construction and stability. SIAM J. Appl. Math. 61 (3), 1080–1102.Google Scholar
[3]Doelman, A., Eckhaus, W. & Kaper, T. J. (2001) Slowly modulated two-pulse solutions in the Gray-Scott model II: Geometric theory, bifurcations, and splitting dynamics. SIAM J. Appl. Math. 66 (6), 2036–2062.Google Scholar
[4]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, 1–36.Google Scholar
[5]Doelman, A., Gardner, R. A. & Kaper, T. J. (2001) Large stable pulse solutions in reaction-diffusion equations. Indiana Univ. Math. J. 50 (1), 443–507.Google Scholar
[6]Doelman, A. & Kaper, T. (2003) Semistrong pulse interactions in a class of coupled reaction-diffusion equations. SIAM J. Appl. Dyn. Sys. 2 (1), 53–96.Google 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), 1760–1787.Google Scholar
[8]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), 1924–1951.CrossRefGoogle Scholar
[9]Iron, D., Ward, M. J. & Wei, J. (2001) The stability of spike solutions to the one-dimensional Gierer-Meinhardt model. Physica D 150 (1–2), 25–62.Google Scholar
[10]Iron, D., Wei, J. & Winter, M. (2004) Stability analysis of turing patterns generated by the Schnakenberg model. J. Math. Biol. 49 (4), 358–390.CrossRefGoogle ScholarPubMed
[11]Kerner, B. S. & Osipov, V. V. (1994) Autosolitions: A New Approach to Problem of Self-Organization and Turbulence, Kluwer, Dordrecht, Netherlands.Google Scholar
[12]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. Stud. Appl. Math. 115 (1), 21–71.Google Scholar
[13]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), 258–293.CrossRefGoogle Scholar
[14]Kolokolnikov, T., Ward, M. J. & Wei, J. (accepted) The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. DCDS-B 34.Google Scholar
[15]Lin, C. S., Ni, W. M. & Takagi, I. (1988) Large amplitude stationary solutions to a chemotaxis system. J. Diff. Equ. 72, 1–27.Google Scholar
[16]Muratov, C. B. & Osipov, V. V. (2000) Static spike autosolitons in the Gray-Scott model. J. Phys. A: Math. Gen. 33, 8893–8916.Google Scholar
[17]Muratov, C. B. & Osipov, V. V. (2002) Stability of the static spike autosolitons in the Gray–Scott model. SIAM J. Appl. Math. 62 (5), 1463–1487.Google Scholar
[18]Nicolis, G. & Prigogine, I. (1977) Self-Organization in Nonequilibrium Systems, Wiley, New York.Google Scholar
[19]Nishiura, Y. (2002) Far-From Equilibrium Dynamics, Translations of Mathematical Monographs, Vol. 209, AMS, Providence, RI.Google Scholar
[20]Pena, B. & Perez-Garcia, C. (2001) Stability of turing patterns in the Brusselator model. Phys. Rev. E. 64, 056213Google Scholar
[21]Peng, R. & Wang, M. (2005) Pattern formation in the Brusselator system. J. Math. Anal. Appl. 309 (1), 151–166.CrossRefGoogle Scholar
[22]Peng, R. & Yang, M. (2009) On steady-state solutions of the Brusselator-type system. Nonlinear Anal. Theory, Methods Appl. 71 (3–4), 1389–1394.Google Scholar
[23]Schenk, C. P., Or-Guil, M., Bode, M. & Purwins, H. (1997) Interacting pulses in three-component reaction-diffusion systems on two-dimensional domains. Phys. Rev. Lett. 78 (19), 3781–3784.Google Scholar
[24]Short, M. B., Bertozzi, A. L. & Brantingham, P. J. (2010) Nonlinear patterns in urban crime – Hotpsots, bifurcations, and suppression. SIAM J. Appl. Dyn. Sys. 9 (2), 462–483.Google Scholar
[25]Sun, W., Ward, M. J. & Russell, R. (2005) The slow dynamics of two-spike solutions for the Gray-Scott and Gierer-Meinhardt systems: Competition and oscillatory instabilities. SIAM J. Appl. Dyn. Sys. 4 (4), 904–953.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 co-dimension two bifurcation point. Math. Model. Nat. Phenom. 6 (1), 87–118.Google Scholar
[27]Tzou, J. C., Bayliss, A., Matkowsky, B. J. & Volpert, V. A. (2011) Stationary and slowly moving localized pulses in a singularly perturbed Brusselator model. Eur. J. Appl. Math. 22 (5), 423–453.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, 923–941.Google Scholar
[29]Vanag, V. K. & Epstein, I. R. (2007) Localized patterns in reaction-diffusion systems. Chaos 17 (3), 037110.Google Scholar
[30]Van der Ploeg, H. & Doelman, A. (2005) Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations. Indiana Univ. Math. J. 54 (5), 1219–1301.Google Scholar
[31]Ward, M. J. & Wei, J. (2002) The existence and stability of asymmetric spike patterns for the Schnakenberg model. Stud. Appl. Math. 109 (3), 229–264.Google Scholar
[32]Ward, M. J. & Wei, J. (2003) Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model. J. Nonlinear Sci. 13 (2), 209–264.Google Scholar
[33]Ward, M. J. & Wei, J., (2003) Hopf bifurcation of spike solutions for the shadow Gierer-Meinhardt model. Europ. J. Appl. Math. 14 (6), 677–711.Google Scholar
[34]Wei, J. (1999) On single interior spike solutions for the Gierer-Meinhardt system: Uniqueness and spectrum estimates. Eur. J. Appl. Math. 10 (4), 353–378.Google Scholar
[35]Wei, J. (2008) Existence and stability of spikes for the Gierer-Meinhardt system. In: Chipot, M. (editor), Handbook of Differential Equations, Stationary Partial Differential Equations, Vol. 5, Elsevier, Waltham, MA, pp. 489–581.Google Scholar
[36]Wittenberg, R. & Holmes, P. (1997) The limited effectiveness of normal forms: A critical review and extension of local bifurcation studies of the Brusselator PDE. Physica D 100, 1–40.Google Scholar