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Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional Gray-Scott model

Published online by Cambridge University Press:  01 April 2009

WAN CHEN  and
MICHAEL J. WARD
WAN CHEN
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, CanadaV6T 1Z2 email: [email protected]
MICHAEL J. WARD
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, CanadaV6T 1Z2 email: [email protected]
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Abstract

The dynamics and oscillatory instabilities of multi-spike solutions to the one-dimensional Gray-Scott reaction–diffusion system on a finite domain are studied in a particular parameter regime. In this parameter regime, a formal singular perturbation method is used to derive a novel ODE–PDE Stefan problem, which determines the dynamics of a collection of spikes for a multi-spike pattern. This Stefan problem has moving Dirac source terms concentrated at the spike locations. For a certain subrange of the parameters, this Stefan problem is quasi-steady and an explicit set of differential-algebraic equations characterizing the spike dynamics is derived. By analysing a nonlocal eigenvalue problem, it is found that this multi-spike quasi-equilibrium solution can undergo a Hopf bifurcation leading to oscillations in the spike amplitudes on an O(1) time scale. In another subrange of the parameters, the spike motion is not quasi-steady and the full Stefan problem is solved numerically by using an appropriate discretization of the Dirac source terms. These numerical computations, together with a linearization of the Stefan problem, show that the spike layers can undergo a drift instability arising from a Hopf bifurcation. This instability leads to a time-dependent oscillatory behaviour in the spike locations.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 20 , Issue 2 , April 2009 , pp. 187 - 214
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Astrov, Y. A. & Purwins, H. G. (2006) Spontaneous division of dissipate solitions in a planar gas-discharge with high ohmic electrode. Phys. Lett. A 358 (5–6), 404–408.CrossRefGoogle Scholar
[2]Astrov, Y. A. & Purwins, H. G. (2001) Plasma spots in a gas discharge system: Birth, scattering and formation of molecules. Phys. Lett. A 283 (3–4), 349–354.Google Scholar
[3]Beyer, R. P. & Leveque, R. J. (1992) Analysis of a one-dimensional model for the immersed boundary method. SIAM J. Numer. Anal. 29 (2), 332–364.CrossRefGoogle Scholar
[4]Blom, J. G. & Zegeling, P. A. (1994) Algorithm 731: A moving-grid interface for systems of one-dimensional time-dependent partial differential equations. ACM Trans. Math. Software 20 (2), 194–214.Google Scholar
[5]Bode, M., Liehr, A. W., Schenk, C. P. & Purwins, H. G. (2002) Interactions of dissipative solitons: Particle-like behaviour of localized structures in a three component reaction–diffusion system. Physica D (1–2) 45–66.Google Scholar
[6]Davis, P. W., Blanchedeau, P., Dullos, E. & De Kepper, P. (1998) Dividing blobs, chemical flowers, and patterned islands in a reaction–diffusion system. J. Phys. Chem. A 102 (43), 8236–8244.Google Scholar
[7]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
[8]Doelman, A., Eckhaus, W. & Kaper, T. J. (2000) Slowly modulated two-pulse solutions in the Gray-Scott model II: Geometric theory, bifurcations, and splitting dynamics. SIAM J. Appl. Math. 61 (6), 2036–2061.Google Scholar
[9]Doelman, A., Gardner, R. A. & Kaper, T. J. (1998) Stability analysis of singular patterns in the 1D Gray-Scott model: A matched asymptotics approach. Physica D 122 (1–4), 1–36.Google Scholar
[10]Doelman, A., Gardner, R. A. & Kaper, T. J. (2002) A stability index analysis of 1D patterns of the Gray-Scott model. Memoirs of the AMS 155.CrossRefGoogle Scholar
[11]Doelman, A. & Kaper, T. (2003) Semi-strong pulse interactions in a class of coupled reaction–diffusion equations. SIAM J. Appl. Dyn. Sys. 2 (1), 53–96.Google Scholar
[12]Doelman, A., Kaper, T. & Promislow, K. (2007) Nonlinear asymptotic stability of the semi-strong pulse dynamics in a regularized Gierer-Meinhardt model. SIAM J. Math. Anal. 38 (6), 1760–1789.Google Scholar
[13]Ei, S. (2002) The motion of weakly interacting pulses in reaction–diffusion systems. J. Dynam. Differential Equations 14 (1), 85–137.Google Scholar
[14]Ei, S. & Ikeda, H. (2006) Dynamics of Front Solutions in Reaction–Diffusion Systems in One Dimension, Preprint.Google Scholar
[15]Frankel, M., Kovacic, G., Roytburd, V. & Timofeyev, I. (2000) Finite-dimensional dynamical system modeling thermal instabilities. Physica D 137 (3–4), 295–315.Google Scholar
[16]Ikeda, T. & Nishiura, Y. (1994) Pattern selection for two breathers. SIAM J. Appl. Math. 54 (1), 195–230.Google Scholar
[17]Iron, D. & Ward, M. J. (2002) The dynamics of multi-spike solutions to the one-dimensional Gierer-Meinhardt model. SIAM J. Appl. Math. 62 (6), 1924–1951.Google Scholar
[18]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
[19]Kirk, C. M. & Olmstead, W. E. (2005) Blow-up solutions of the two-dimensional heat equation due to a localized moving source. Anal. Appl. (Singapore) 3 (1), 1–16.CrossRefGoogle Scholar
[20]Kolokolnikov, T., Sun, W., Ward, M. J. & Wei, J. (2006) The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation. SIAM J. Appl. Dyn. Sys. 5 (2), 313–363.CrossRefGoogle Scholar
[21]Kolokolnikov, T., Erneux, T. & Wei, J. (2006) Mesa-type patterns in the one-dimensional brusselator and their stability. Physica D 214 (1), 63–77.Google Scholar
[22]Kolokolnikov, T., Ward, M. & Wei, J. (2006) The stability of spike equilibria in the one-dimensional Gray-Scott model: The low feed-rate regime. Studies Appl. Math. 115 (1), 21–71.Google Scholar
[23]Kolokolnikov, T., Ward, M. & Wei, J. (2005) The stability of spike equilibria in the one-dimensional Gray-Scott model: The pulse-splitting regime. Physica D 202 (3–4), 258–293.Google Scholar
[24]Kolokolnikov, T., Ward, M. & Wei, J. (2006) Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model. Interfaces Free Bound. 8 (2), 185–222.Google Scholar
[25]Lee, K. J., McCormick, W. D., Pearson, J. E. & Swinney, H. L. (1994) Experimental observation of self-replicating spots in a reaction–diffusion system. Nature 369, 215–218.Google Scholar
[26]Mimura, M., Nagayama, M. & Sakamoto, K. (1995) Pattern dynamics in an exothermic reaction. Physica D 84 (1–2), 58–71.Google Scholar
[27]Muratov, C. & Osipov, V. V. (2001) Traveling spike auto-solitons in the Gray-Scott model. Physica D 155 (1–2), 112–131.Google Scholar
[28]Muratov, C. & Osipov, V. V.Stability of the static spike autosolitons in the Gray-Scott model. SIAM J. Appl. Math. 62 (5), 1463–1487.Google Scholar
[29]Muratov, C. & Osipov, V. V. (2000) Static spike autosolitons in the Gray-Scott model. J. Phys. A: Math Gen. 33, 8893–8916.Google Scholar
[30]Muratov, C. & Osipov, V. V. (2001) Spike sutosolitons and pattern formation scenarios in the two-dimensional Gray-Scott model. Eur. Phys. J. B. 22, 213–221.Google Scholar
[31]Nishiura, Y. & Fujii, H. (1987) Stability of singularly perturbed solutions to systems of reaction–diffusion equations. SIAM J. Math. Anal. 18, 1726–1770.CrossRefGoogle Scholar
[32]Nishiura, Y. & Mimura, H. (1989) Layer oscillations in reaction–diffusion systems. SIAM J. Appl. Math. 49 (2), 481–514.Google Scholar
[33]Nishiura, Y. & Ueyama, D. (1999) A skeleton structure of self-replicating dynamics. Physica D 130 (1–2), 73–104.Google Scholar
[34]Nishiura, Y. & Ueyama, D. (2001) Spatio-temporal chaos for the Gray-Scott model. Physica D 150 (3–4), 137–162.Google Scholar
[35]Ockendon, J., Howison, S., Lacey, A. & Movchan, A. (2001) Applied Partial Differential Equations, Oxford University Press, Oxford.Google Scholar
[36]Park, J. H., Bayliss, A., Matkowsky, B. J. & Nepomnyashchy, A. A. (2006) On the route to extinction in nonadiabatic solid flames. SIAM J. Appl. Math. 66 (3), 873–895.Google Scholar
[37]Pearson, J. E. (1993) Complex patterns in a simple system. Science 216, 189–192.Google Scholar
[38]Reynolds, W. N., Ponce-Dawson, S. & Pearson, J. E. (1997) Dynamics of self-replicating spots in reaction–diffusion systems. Phys. Rev. E 56 (1), 185–198.Google Scholar
[39]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. App. Dyn. Syst. 4 (4), 904–953.Google Scholar
[40]Tornberg, A. K. & Engquist, B. (2004) Numerical approximations of singular source terms in differential equations. J. Comput. Phys. 200 (2), 462–488.Google Scholar
[41]Ueyama, D. (1999) Dynamics of self-replicating patterns in the one-dimensional Gray-Scott model. Hokkaido Math J. 28 (1), 175–210.Google Scholar
[42]Vanag, V. K. & Epstein, I. R. (2007) Localized patterns in reaction–diffusion systems. Chaos 17 (3), 037110.CrossRefGoogle ScholarPubMed
[43]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
[44]Wei, J. (1999) On single interior spike solutions for the Gierer-Meinhardt system: Uniqueness and stability estimates. Europ. J. Appl. Math. 10 (4), 353–378.Google Scholar