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An Explicitly Solvable Nonlocal Eigenvalue Problem and the Stability of a Spike for a Sub-Diffusive Reaction-Diffusion System

Published online by Cambridge University Press:  24 April 2013

Y. Nec  and
M. J. Ward
Y. Nec
Affiliation:
Department of Mathematics, University of British Columbia 1984 Mathematics Road, Vancouver, V6T1Z2, BC, Canada
M. J. Ward*
Affiliation:
Department of Mathematics, University of British Columbia 1984 Mathematics Road, Vancouver, V6T1Z2, BC, Canada
*
⋆ Corresponding author. E-mail: [email protected]
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Abstract

The stability of a one-spike solution to a general class of reaction-diffusion (RD) system with both regular and anomalous diffusion is analyzed. The method of matched asymptotic expansions is used to construct a one-spike equilibrium solution and to derive a nonlocal eigenvalue problem (NLEP) that determines the stability of this solution on an O(1) time-scale. For a particular sub-class of the reaction kinetics, it is shown that the discrete spectrum of this NLEP is determined in terms of the roots of certain simple transcendental equations that involve two key parameters related to the choice of the nonlinear kinetics. From a rigorous analysis of these transcendental equations by using a winding number approach and explicit calculations, sufficient conditions are given to predict the occurrence of Hopf bifurcations of the one-spike solution. Our analysis determines explicitly the number of possible Hopf bifurcation points as well as providing analytical formulae for them. The analysis is implemented for the shadow limit of the RD system defined on a finite domain and for a one-spike solution of the RD system on the infinite line. The theory is illustrated for two specific RD systems. Finally, in parameter ranges for which the Hopf bifurcation is unique, it is shown that the effect of sub-diffusion is to delay the onset of the Hopf bifurcation.

Keywords

matched asymptotic expansionsnonlocal eigenvalue problemwinding numberHopf bifurcationsub–diffusion
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 2: Anomalous diffusion , 2013 , pp. 55 - 87
Copyright
© EDP Sciences, 2013

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References

Chen, W., Ward, M. J.. Oscillatory instabilities of multi-spike patterns for the one-dimensional Gray-Scott model. Europ. J. Appl. Math., 20 (2), (2009), pp. 187–214. CrossRefGoogle Scholar
Chen, W., Ward, M. J.. The stability and dynamics of localized spot patterns in the two-dimensional Gray-Scott model. SIAM J. Appl. Dyn. Sys., 10 (2), (2011), pp. 582–666. CrossRefGoogle Scholar
Doelman, A., Eckhaus, W., Kaper, T. J.. Slowly modulated two-pulse solutions in the Gray-Scott model i: asymptotic construction and stability. SIAM J. Appl. Math., 61 (3), (2000), pp. 1080–1102. CrossRefGoogle Scholar
Doelman, A., Eckhaus, W., Kaper, T. J.. Slowly modulated two-pulse solutions in the Gray-Scott model ii: geometric theory, bifurcations, and splitting dynamics. SIAM J. Appl. Math., 61 (6), (2000), pp. 2036–2061. CrossRefGoogle Scholar
Doelman, A., Gardner, R. A., Kaper, T.. Large stable pulse solutions in reaction-diffusion equations. Indiana U. Math. Journ., 50 (1), (2001), pp. 443–507. CrossRefGoogle Scholar
Doelman, A., Kaper, T.. Semistrong pulse interactions in a class of coupled reaction-diffusion equations. SIAM J. Appl. Dyn. Sys., 2 (1), (2003), pp. 53–96. CrossRefGoogle Scholar
Doelman, A., Kaper, T., Promislow, K.. Nonlinear asymptotic stability of the semi-strong pulse dynamics in a regularized Gierer-Meinhardt model. SIAM J. Math. Anal., 38 (6), (2007), pp. 1760–1789. CrossRefGoogle Scholar
Doelman, A., van der Ploeg, H.. Homoclinic stripe patterns. SIAM J. Appl. Dyn. Sys., 1 (1), (2002), pp. 65–104. CrossRefGoogle Scholar
J. Ehrt, J. D. Rademacher, M. Wolfrum. First and second order semi-strong interaction of pulses in the Schnakenberg model. preprint, (2012).
Golovin, A. A., Matkowsky, B. J., Volpert, V. A.. Turing pattern formation in the Brusselator model with superdiffusion. SIAM J. Appl. Math., 69 (1), (2008), pp. 251–272. CrossRefGoogle Scholar
Henry, B. I., Wearne, S. L.. Existence of Turing instabilities in a two-species fractional reaction-diffusion system. SIAM J. Appl. Math., 62 (3), (2002), pp. 870–887. CrossRefGoogle Scholar
Iron, D., Ward, M. J.. A metastable spike solution for a non-local reaction-diffusion model. SIAM J. Appl. Math., 60 (3), (2000), pp. 778-802. CrossRefGoogle Scholar
Iron, D., Ward, M. J., Wei, J.. The stability of spike solutions to the one-dimensional Gierer-Meinhardt model. Physica D, 150 (1-2), (2001), pp. 25–62. CrossRefGoogle Scholar
Iron, D., Ward, M. J.. The dynamics of multi-spike solutions to the one-dimensional Gierer-Meinhardt model. SIAM J. Appl. Math., 62 (6), (2002), pp. 1924-1951. CrossRefGoogle Scholar
Kolokolnikov, T., Ward, M. J., Wei, J.. The stability of spike equilibria in the one-dimensional Gray-Scott model: the low feed-rate regime. Studies in Appl. Math., 115 (1), (2005), pp. 21–71 CrossRefGoogle Scholar
Koloklonikov, T., Ward, M. J., Wei, J.. Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model. Interfaces and Free Boundaries, 8 (2), (2006), pp. 185–222. CrossRefGoogle Scholar
T. Kolokolnikov, M. Ward, J. Wei. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Disc. Cont. Dyn. Sys Series B., to appear, (2013), (34 pages).
Lin, C. S., Ni, W. M., Takagi, I.. Large amplitude stationary solutions to a chemotaxis system. J. Diff. Eq., 72 (1), (1988), pp. 1-27. CrossRefGoogle Scholar
Metzler, R., Klafter, J.. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep., 339. (2000), pp. 1–77. CrossRefGoogle Scholar
Muratov, C., Osipov, V. V.. Stability of the static spike autosolitons in the Gray-Scott model. SIAM J. Appl. Math., 62 (5), (2002), pp. 1463-1487. CrossRefGoogle Scholar
Nec, Y., Volpert, V. A., Nepomnyashchy, A. A.. Front propagation problems with sub-diffusion. Discr. Cont. Dyn. Sys. Series A, 27 (2), (2010), pp. 827–846. CrossRefGoogle Scholar
Nec, Y., Nepomnyashchy, A. A.. Linear stability of fractional reaction-diffusion systems. Math. Model. Nat. Phenom., 2 (2), (2007), pp. 77–105. CrossRefGoogle Scholar
Nec, Y., Nepomnyashchy, A. A.. Turing instability in sub-diffusive reaction-diffusion systems. J. Physics A: Math. Theor., 40 (49), (2007), pp. 14687–14702. CrossRefGoogle Scholar
Nec, Y., Ward, M. J.. The dynamics and stability of spike-type solutions to the Gierer-Meinhardt model with subdiffusion. Physica D, 241 (9), (2012), pp. 947–963. CrossRefGoogle Scholar
Y. Nec, M. J. Ward. The stability and slow dynamics of two-spike patterns for a class of reaction-diffusion system. submitted, (2013), (28 pages)
K. B. Oldham, J. Spanier. The fractional calculus. Academic Press, New York, 1974.
I. Podlubny. Fractional differential equations. Academic Press, San Diego, 1999.
J. D. Rademacher. First and second order semi-strong interface interaction in reaction-diffusion systems. SIAM J. App. Dyn. Syst., (2012), to appear.
Saxena, R. K., Mathai, A. M., Haubold, H. J., Fractional reaction-diffusion equations. Astrophys. Space Sci., 305 (3), (2006), pp. 289–296. CrossRefGoogle Scholar
Sun, W., Tang, T., Ward, M. J., Wei, J.. Numerical challenges for resolving spike dynamics for two reaction-diffusion systems. Studies in Appl. Math., 111 (1), (2003), pp. 41-84. CrossRefGoogle Scholar
Sun, W., Ward, M. J., Russell, R.. The slow dynamics of two-spike solutions for the Gray-Scott and Gierer-Meinhardt systems: competition and oscillatory instabilities. SIAM J. App. Dyn. Sys., 4 (4), (2005), pp. 904–953. CrossRefGoogle Scholar
W. H. Tse, M. J. Ward. On explicitly solvable nonlocal eigenvalue problems and the stability of localized pulses. to be submitted, Applied Math Letters, (2013).
Tzou, J. C., Bayliss, A., Matkowsky, B. J., Volpert, V. A., Interaction of Turing and Hopf models in the superdiffusive Brusselator model near a codimension two bifurcation point. Math. Model. Nat. Phenom. 6 (1), (2011), pp. 87–118. CrossRefGoogle Scholar
Tzou, J. C., Bayliss, A., Matkowsky, B. J., Volpert, V. A., Stationary and slowly moving localized pulses in a singularly perturbed Brusselator model. Europ. J. Appl. Math., 22 (5), (2011), pp. 423–453. CrossRefGoogle Scholar
J. C. Tzou, Y. Nec, M. J. Ward, The Stability of Localized Spikes for the 1-D Brusselator Reaction-Diffusion Model. Europ. J. Appl. Math., (2012), under review.
van der Ploeg, H., Doelman, A.. Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations. Indiana U. Math. J., 54 (5), (2005), pp. 1219–1301. Google Scholar
Ward, M. J., Wei, J.. Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model. J. Nonlinear Science, 13 (2), (2003), pp. 209–264. CrossRefGoogle Scholar
Ward, M. J., Wei, J.. Hopf bifurcations of spike solutions for the shadow Gierer-Meinhardt model. Europ. J. Appl. Math., 14 (6), (2003), pp. 677–711. CrossRefGoogle Scholar
Wei, J.. On single interior spike solutions for the Gierer-Meinhardt system: uniqueness and stability estimates. Europ. J. Appl. Math., 10 (4), (1999), pp. 353–378. CrossRefGoogle Scholar
J. Wei. Existence and stability of spikes for the Gierer-Meinhardt system. book chapter in Handbook of Differential Equations, Stationary Partial Differential Equations. Vol. 5 (M. Chipot ed.), Elsevier, (2008), pp. 489–581.
Wolfrum, M., Ehrt, J.. Slow motion of quasi-stationary multi-pulse solutions by semistrong interaction in reaction-diffusion systems. WIAS Preprint 1233 (2007). Google Scholar