- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Dynamics of reaction-diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection

Julien Siebert1, Sergio Alonso2, Markus Bär2, and Eckehard Schöll1

  • 1Technische Universität, Institut für Theoretische Physik, Hardenberstrasse 36, 10623 Berlin, Germany
  • 2Physikalisch-Technische Bundesanstalt, Abbestrasse 2-12, 10587 Berlin, Germany

PDF

Phys. Rev. E 89, 052909 – Published 14 May, 2014

DOI: https://doi.org/10.1103/PhysRevE.89.052909

Abstract

A one-component bistable reaction-diffusion system with asymmetric nonlocal coupling is derived as a limiting case of a two-component activator-inhibitor reaction-diffusion model with differential advection. The effects of asymmetric nonlocal couplings in such a bistable reaction-diffusion system are then compared to the previously studied case of a system with symmetric nonlocal coupling. We carry out a linear stability analysis of the spatially homogeneous steady states of the model and numerical simulations of the model to show how the asymmetric nonlocal coupling controls and alters the steady states and the front dynamics in the system. In a second step, a third fast reaction-diffusion equation is included which induces the formation of more complex patterns. A linear stability analysis predicts traveling waves for asymmetric nonlocal coupling, in contrast to a stationary Turing patterns for a system with symmetric nonlocal coupling. These findings are verified by direct numerical integration of the full equations with nonlocal coupling.

View figure in article

Article Text

References (73)

  1. H. Haken, Synergetics, An Introduction, 3rd ed. (Springer, Berlin, 1983).
  2. A. S. Mikhailov, Foundations of Synergetics, 2nd ed. (Springer, Berlin, 1994), Vol. I.
  3. Chemical Waves and Patterns, edited by R. Kapral and K. Showalter (Kluwer, Dordrecht, 1995).
  4. J. P. Keener and J. Sneyd, Mathematical Physiology (Springer, New York, 1998).
  5. S. Kondo and T. Miura, Science 329, 1616 (2010).
  6. A. T. Winfree, When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias (Princeton University Press, Princeton, NJ, 1987); includes indexes.
  7. J. M. Davidenko, A. M. Pertsov, R. Salomonsz, W. Baxter, and J. Jalife, Nature (London) 355, 349 (1992).
  8. N. A. Gorelova and J. Bures, J. Neurobiol. 14, 353 (1983).
  9. M. A. Dahlem and S. C. Müller, Exp. Brain Res. 115, 319 (1997).
  10. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer-Verlag, Berlin, 1984).
  11. M. Falcke and H. Engel, Phys. Rev. E 50, 1353 (1994).
  12. M. Bär, M. Falcke, M. Hildebrand, M. Neufeld, H. Engel, and M. Eiswirth, Int. J. Bifurcation Chaos Appl. Sci. Eng. 4, 499 (1994).
  13. M. Falcke, H. Engel, and M. Neufeld, Phys. Rev. E 52, 763 (1995).
  14. M. Bertram, C. Beta, M. Pollmann, A. S. Mikhailov, H. H. Rotermund, and G. Ertl, Phys. Rev. E 67, 036208 (2003).
  15. C. Beta, M. Bertram, A. S. Mikhailov, H. H. Rotermund, and G. Ertl, Phys. Rev. E 67, 046224 (2003).
  16. C. Beta, M. G. Moula, A. S. Mikhailov, H. H. Rotermund, and G. Ertl, Phys. Rev. Lett. 93, 188302 (2004).
  17. U. Middya, M. Sheintuch, M. D. Graham, and D. Luss, Physica D 63, 393 (1992).
  18. M. Sheintuch and S. Shvartsman, AIChE J. 42, 1041 (1996).
  19. F. Plenge, P. Rodin, E. Schöll, and K. Krischer, Phys. Rev. E 64, 056229 (2001).
  20. M. Meixner, P. Rodin, E. Schöll, and A. Wacker, Eur. Phys. J. B 13, 157 (2000).
  21. E. Schöll, Nonlinear Spatio-Temporal Dynamics and Chaos in Semiconductors, Nonlinear Science Series Vol. 10 (Cambridge University Press, Cambridge, 2001).
  22. A. S. Mikhailov and K. Showalter, Phys. Rep. 425, 79 (2006).
  23. M. Sheintuch, Physica D 102, 125 (1996).
  24. A. Yochelis and M. Sheintuch, Phys. Rev. E 80, 056201 (2009).
  25. A. Yochelis and M. Sheintuch, Phys. Chem. Chem. Phys. 11, 9210 (2009).
  26. A. Yochelis and M. Sheintuch, Phys. Chem. Chem. Phys. 12, 3957 (2010).
  27. A. Yochelis and M. Sheintuch, Phys. Rev. E 81, 025203 (2010).
  28. A. B. Rovinsky and M. Menzinger, Phys. Rev. Lett. 69, 1193 (1992).
  29. A. B. Rovinsky and M. Menzinger, Phys. Rev. Lett. 70, 778 (1993).
  30. Y. Khazan and L. M. Pismen, Phys. Rev. Lett. 75, 4318 (1995).
  31. Y. Balinsky (Khazan) and L. M. Pismen, Phys. Rev. E 58, 4524 (1998).
  32. R. A. Satnoianu, J. H. Merkin, and S. K. Scott, Physica D 124, 345 (1998).
  33. A. Tóth, D. Horváth, and W. van Saarloos, J. Chem. Phys. 111, 10964 (1999).
  34. H. Malchow, J. Marine Syst. 7, 193 (1996).
  35. C. A. Klausmeier, Science 284, 1826 (1999).
  36. J. von Hardenberg, E. Meron, M. Shachak, and Y. Zarmi, Phys. Rev. Lett. 87, 198101 (2001).
  37. F. Borgogno, P. D'Odorico, F. Laio, and L. Ridolfi, Rev. Geophys. 47, 1 (2009).
  38. D. M. Petrich and R. E. Goldstein, Phys. Rev. Lett. 72, 1120 (1994).
  39. L. M. Pismen, Patterns and Interfaces in Dissipative Dynamics, Springer Series in Synergetics (Springer, Berlin, 2006).
  40. E. M. Nicola, M. Or-Guil, W. Wolf, and M. Bär, Phys. Rev. E 65, 055101(R) (2002).
  41. D. Tanaka and Y. Kuramoto, Phys. Rev. E 68, 026219 (2003).
  42. S.-i. Shima and Y. Kuramoto, Phys. Rev. E 69, 036213 (2004).
  43. A. Das and C. D. Gilbert, Nature (London) 399, 655 (1999).
  44. K. Kang, M. Shelley, and H. Sompolinsky, Proc. Natl. Acad. Sci. U.S.A. 100, 2848 (2003).
  45. X. Xu, W. Bosking, G. Sáry, J. Stefansic, D. Shima, and V. Casagrande, J. Neurosci. 24, 6237 (2004).
  46. J. Christoph and M. Eiswirth, Chaos 12, 215 (2002).
  47. M. Sheintuch and O. Nekhamkina, J. Chem. Phys. 107, 8165 (1997).
  48. G. Bordyugov and H. Engel, Phys. Rev. E 74, 016205 (2006).
  49. P. Colet, M. A. Matias, L. Gelens, and D. Gomila, Phys. Rev. E 89, 012914 (2014).
  50. L. Gelens, M. A. Matias, D. Gomila, T. Dorissen, and P. Colet, Phys. Rev. E 89, 012915 (2014).
  51. N. Mazouz, G. Flätgen, and K. Krischer, Phys. Rev. E 55, 2260 (1997).
  52. M. Hildebrand, H. Skødt, and K. Showalter, Phys. Rev. Lett. 87, 088303 (2001).
  53. E. M. Nicola, M. Bär, and H. Engel, Phys. Rev. E 73, 066225 (2006).
  54. R. Friedrichs and A. Engel, Europhys. Lett. 63, 826 (2003).
  55. J. Christoph, R. D. Otterstedt, M. Eiswirth, N. I. Jaeger, and J. L. Hudson, J. Chem. Phys. 110, 8614 (1999).
  56. Y. J. Li, J. Oslonovitch, N. Mazouz, F. Plenge, K. Krischer, and G. Ertl, Science 291, 2395 (2001).
  57. H. Varela, C. Beta, A. Bonnefont, and K. Krischer, Phys. Rev. Lett. 94, 174104 (2005).
  58. B. Belintsev, M. Livshits, and M. Volkenstein, Z. Phys. B 44, 345 (1981).
  59. M. A. Dahlem, F. M. Schneider, and E. Schöll, Chaos 18, 026110 (2008).
  60. F. M. Schneider, E. Schöll, and M. A. Dahlem, Chaos 19, 015110 (2009).
  61. M. A. Dahlem, R. Graf, A. J. Strong, J. P. Dreier, Y. A. Dahlem, M. Sieber, W. Hanke, K. Podoll, and E. Schöll, Physica D 239, 889 (2010).
  62. Y. Kuramoto and D. Battogtokh, Nonlinear Phenom. Complex Syst. (Minsk, Belarus) 5, 380 (2002).
  63. D. M. Abrams and S. H. Strogatz, Phys. Rev. Lett. 93, 174102 (2004).
  64. A. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, Nat. Phys. 8, 658 (2012).
  65. M. R. Tinsley, S. Nkomo, and K. Showalter, Nat. Phys. 8, 662 (2012).
  66. E. A. Martens, S. Thutupalli, A. Fourrière, and O. Hallatschek, Proc. Natl. Acad. Sci. U.S.A. 110, 10563 (2013).
  67. I. Omelchenko, O. E. Omel'chenko, P. Hövel, and E. Schöll, Phys. Rev. Lett. 110, 224101 (2013).
  68. M. Kim, M. Bertram, M. Pollmann, A. von Oertzen, A. S. Mikhailov, H. H. Rotermund, and G. Ertl, Science 292, 1357 (2001).
  69. Y. N. Kyrychko, K. B. Blyuss, S. J. Hogan, and E. Schöll, Chaos 19, 043126 (2009).
  70. H. Malchow and L. Schimansky-Geier, Noise and Diffusion in Bistable Nonequilibrium Systems (B. G. Teubner, Berlin, 1985).
  71. F. Schlögl, Z. Phys. 253, 147 (1972).
  72. F. Schlögl, C. Escher, and R. S. Berry, Phys. Rev. A 27, 2698 (1983).
  73. Note that the Fourier transform of the function [ab/(a+b)]exp[−(1/2)(a+b)|x|−(1/2)(a−b)x] is 1/{2π[1+ik(b−a)/ab+k2/ab]}.

Outline

Information

Phys. Rev. E 89, 052909– Published 14 May, 2014
  • Received 14 January 2014
  • Revised 10 April 2014

©2014 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation