Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T07:04:01.121Z Has data issue: false hasContentIssue false
  • English
  • Français

The Stability and Slow Dynamics of Two-Spike Patterns for aClass of Reaction-Diffusion System

Published online by Cambridge University Press:  18 September 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: ward@math.ubc.ca
Get access

Abstract

The slow dynamics and linearized stability of a two-spike quasi-equilibrium solution to ageneral class of reaction-diffusion (RD) system with and without sub-diffusion isanalyzed. For both the case of regular and sub-diffusion, the method of matched asymptoticexpansions is used to derive an ODE characterizing the spike locations in the absence ofany 𝒪(1) time-scale instabilities of the two-spike quasi-equilibrium profile. These fastinstabilities result from unstable eigenvalues of a certain nonlocal eigenvalue problem(NLEP) that is derived by linearizing the RD system around the two-spike quasi-equilibriumsolution. For a particular sub-class of the reaction kinetics, it is shown that thediscrete spectrum of this NLEP is determined by the roots of some simple transcendentalequations. From a rigorous analysis of these transcendental equations, explicit sufficientconditions are given to predict the occurrence of either Hopf bifurcations or competitioninstabilities of the two-spike quasi-equilibrium solution. The theory is illustrated forseveral specific choices of the reaction kinetics.

Keywords

matched asymptotic expansionsbifurcationspikesnonlocal eigenvalue problemHopf bifurcationsub-diffusion
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 5: Bifurcations , 2013 , pp. 206 - 232
Copyright
© EDP Sciences, 2013

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

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
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., 1 (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
J. Ehrt, J. D. Rademacher, M. Wolfrum, First and second order semi-strong interaction of pulses in the Schnakenburg model. preprint, (2012).
Gierer, A., Meinhardt, H.. A theory of biological pattern formation. Kybernetik, 12, (1972), pp. 30–39. CrossRefGoogle ScholarPubMed
A. Golovin, A., J. Matkowsky, B., A. Volpert, V., Turing pattern formation in the Brusselator model with superdiffusion. SIAM J. Appl. Math. 69 (1), (2008), pp. 251–272. CrossRefGoogle Scholar
Gray, P., S. K., Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system A+2B → 3B, B → C. Chem. Eng. Sci. 39, (1984), pp. 1087–1097. 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., 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., 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
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).
C. S. Lin, W. M. Ni, I. Takagi. Large amplitude stationary solutions to a chemotaxis system. J. Diff. Eq. 72 (1), (1988), 1-27.
H. Meinhardt. The Algorithmic Beauty of Sea Shells. Springer-Verlag, Berlin, (1995).
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
I. Moyls, 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).
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., A. A., Nepomnyashchy. Front propagation problems with sub-diffusion. Discr. Cont. Dyn. Sys. Series A. 27 (2), (2010), pp. 827–846. CrossRefGoogle Scholar
Nec, Y., A. A., Nepomnyashchy, Linear stability of fractional reaction-diffusion systems. Math. Model. Nat. Phenom. 2 (2), (2007), pp. 77–105. CrossRefGoogle Scholar
Nec, Y., A. A., Nepomnyashchy, Turing instability in sub-diffusive reaction-diffusion systems. J. Physics A: Math. Theor. 40 (49), (2007), pp. 14687–14702. CrossRefGoogle Scholar
Nec, Y., M. J., Ward, The dynamics and stability of spike-type solutions to the Gierer-Meinhardt model with subdiffusion. Physica D. 241 (10), (2012), pp. 947–963. CrossRefGoogle Scholar
Nec, Y., Ward, M. J., An explicitly solvable nonlocal eigenvalue problem and the stability of a spike for a class of reaction-diffusion systems. Math. Model. of Nat. Phenom. 8 (2), (2013), pp. 55–87. CrossRefGoogle Scholar
Y., Nec Spike-type solutions to the one-dimensional Gierer-Meinhardt model with Lévy flights. Studies Appl. Math. 129 (3), (2012), pp. 272-299 Google Scholar
K. B. Oldham, J. Spanier. The fractional calculus. Academic Press, New York, 1974.
I. Podlubny. Fractional differential equations. Academic Press, San Diego, 1999.
Rademacher, J. D., First and second order semi-strong interface interaction in reaction-diffusion systems. SIAM J. App. Dyn. Syst. 12 (1), (2013), pp. 175–203. CrossRefGoogle Scholar
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., 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
I. Moyls, 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).
Turing, A.. The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. B. 327 (1952), pp. 37-72. CrossRefGoogle Scholar
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
Tzou, J. C., Nec, Y., Ward, M. J., The Stability of Localized Spikes for the 1-D Brusselator Reaction-Diffusion Model. Europ. J. Appl. Math. 24 (4), (2013), pp. 515–564. CrossRefGoogle Scholar
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, 3 (2), (2003), pp. 209-264. 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), 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