EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 8 / No 5 (2013) Math. Model. Nat. Phenom., 8 5 (2013) 206-232 References
Free Access
Issue
Math. Model. Nat. Phenom.
Volume 8, Number 5, 2013
Bifurcations
Page(s) 206 - 232
DOI https://doi.org/10.1051/mmnp/20138513
Published online 18 September 2013
  1. W. Chen, M. J. Ward. Oscillatory instabilities of multi-spike patterns for the one-dimensional Gray-Scott model. Europ. J. Appl. Math., 20 (2), (2009), pp. 187–214. [CrossRef] [Google Scholar]
  2. A. Doelman, W. Eckhaus, T. J. Kaper. 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. [CrossRef] [Google Scholar]
  3. A. Doelman, W. Eckhaus, T. J. Kaper. 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. [CrossRef] [Google Scholar]
  4. A. Doelman, R. A. Gardner, T. Kaper. Large stable pulse solutions in reaction-diffusion equations. Indiana U. Math. Journ., 50 (1), (2001), pp. 443-507. [CrossRef] [Google Scholar]
  5. A. Doelman, T. Kaper. Semistrong pulse interactions in a class of coupled reaction-diffusion equations. SIAM J. Appl. Dyn. Sys., 2 (1), (2003), pp. 53-96. [CrossRef] [Google Scholar]
  6. A. Doelman, T. Kaper, K. Promislow, 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. [CrossRef] [MathSciNet] [Google Scholar]
  7. J. Ehrt, J. D. Rademacher, M. Wolfrum, First and second order semi-strong interaction of pulses in the Schnakenburg model. preprint, (2012). [Google Scholar]
  8. A. Gierer, H. Meinhardt. A theory of biological pattern formation. Kybernetik, 12, (1972), pp. 30–39. [CrossRef] [PubMed] [Google Scholar]
  9. A. A. Golovin, B. J. Matkowsky, V. A. Volpert, Turing pattern formation in the Brusselator model with superdiffusion. SIAM J. Appl. Math. 69 (1), (2008), pp. 251–272. [CrossRef] [MathSciNet] [Google Scholar]
  10. P. Gray, 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. [Google Scholar]
  11. B. I. Henry, S. L. Wearne. Existence of Turing instabilities in a two-species fractional reaction-diffusion system. SIAM J. Appl. Math., 62(3), (2002), pp. 870–887. [CrossRef] [Google Scholar]
  12. D. Iron, M. J. Ward, J. Wei. The stability of spike solutions to the one-dimensional Gierer-Meinhardt model. Physica D, 150 (1-2), (2001), pp. 25–62. [CrossRef] [Google Scholar]
  13. D. Iron, M. J. Ward, The dynamics of multi-spike solutions to the one-dimensional Gierer-Meinhardt model. SIAM J. Appl. Math. 62 (6), (2002), pp. 1924-1951. [CrossRef] [Google Scholar]
  14. T. Kolokolnikov, M. Ward, J. Wei. 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 [CrossRef] [MathSciNet] [Google Scholar]
  15. 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). [Google Scholar]
  16. C. S. Lin, W. M. Ni, I. Takagi. Large amplitude stationary solutions to a chemotaxis system. J. Diff. Eq. 72 (1), (1988), 1-27. [Google Scholar]
  17. H. Meinhardt. The Algorithmic Beauty of Sea Shells. Springer-Verlag, Berlin, (1995). [Google Scholar]
  18. R. Metzler, J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), pp. 1–77. [NASA ADS] [CrossRef] [Google Scholar]
  19. 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). [Google Scholar]
  20. C. Muratov, V. V. Osipov. Stability of the static spike autosolitons in the Gray-Scott model. SIAM J. Appl. Math. 62 (5), (2002), pp. 1463-1487. [CrossRef] [Google Scholar]
  21. Y. Nec, V. A. Volpert, A. A., Nepomnyashchy. Front propagation problems with sub-diffusion. Discr. Cont. Dyn. Sys. Series A. 27 (2), (2010), pp. 827–846. [Google Scholar]
  22. Y. Nec, A. A., Nepomnyashchy, Linear stability of fractional reaction-diffusion systems. Math. Model. Nat. Phenom. 2 (2), (2007), pp. 77–105. [CrossRef] [EDP Sciences] [Google Scholar]
  23. Y. Nec, A. A., Nepomnyashchy, Turing instability in sub-diffusive reaction-diffusion systems. J. Physics A: Math. Theor. 40 (49), (2007), pp. 14687–14702. [Google Scholar]
  24. Y. Nec, 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. [CrossRef] [Google Scholar]
  25. Y. Nec, M. J. Ward, 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. [Google Scholar]
  26. 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]
  27. K. B. Oldham, J. Spanier. The fractional calculus. Academic Press, New York, 1974. [Google Scholar]
  28. I. Podlubny. Fractional differential equations. Academic Press, San Diego, 1999. [Google Scholar]
  29. J. D. Rademacher, First and second order semi-strong interface interaction in reaction-diffusion systems. SIAM J. App. Dyn. Syst. 12 (1), (2013), pp. 175–203. [CrossRef] [Google Scholar]
  30. R. K. Saxena, A. M. Mathai, H. J. Haubold, Fractional reaction-diffusion equations. Astrophys. Space Sci. 305 (3), (2006), pp. 289–296. [Google Scholar]
  31. W. Sun, M. J. Ward, R. Russell. 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. [Google Scholar]
  32. 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). [Google Scholar]
  33. A. Turing. The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. B. 327 (1952), pp. 37-72. [Google Scholar]
  34. J. C. Tzou, A. Bayliss, B. J. Matkowsky, V. A. Volpert, 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. [CrossRef] [EDP Sciences] [Google Scholar]
  35. J. C. Tzou, A. Bayliss, B. J. Matkowsky, V. A. Volpert, Stationary and slowly moving localized pulses in a singularly perturbed Brusselator model. Europ. J. Appl. Math. 22 (5), (2011), pp. 423–453. [Google Scholar]
  36. 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. 24 (4), (2013), pp. 515–564. [CrossRef] [Google Scholar]
  37. H. van der Ploeg, A. Doelman. 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. [CrossRef] [Google Scholar]
  38. M. J. Ward, J. Wei. Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model. J. Nonlinear Science, 3 (2), (2003), pp. 209-264. [Google Scholar]
  39. J. Wei. On single interior spike solutions for the Gierer-Meinhardt system: uniqueness and stability estimates. Europ. J. Appl. Math. 10 (4), (1999), pp. 353-378. [CrossRef] [Google Scholar]
  40. 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. [Google Scholar]
  41. M. Wolfrum, J. Ehrt. Slow motion of quasi-stationary multi-pulse solutions by semistrong interaction in reaction-diffusion systems. WIAS Preprint 1233 (2007). [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.