- 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

Analysis of stationary droplets in a generic Turing reaction-diffusion system

Thomas E. Woolley* and Ruth E. Baker

Philip K. Maini

José Luis Aragón

Rafael A. Barrio

  • Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St. Giles’, Oxford OX1 3LB, United Kingdom

  • Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St. Giles’, Oxford OX1 3LB, United Kingdom and Oxford Centre for Integrative Systems Biology, Department of Biochemistry, University of Oxford, South Parks Road, Oxford OX1 3QU, United Kingdom

  • Centro de Física Aplicada y Tecnologia Avanzada, UNAM, Apartado Postal 1-1010, 76000 Queretaro, Mexico

  • Instituto de Física, UNAM, Apartado Postal 20-364, 01000 México, D.F., Mexico

  • *woolley@maths.ox.ac.uk
PDF

Phys. Rev. E 82, 051929 – Published 23 November, 2010

DOI: https://doi.org/10.1103/PhysRevE.82.051929

Abstract

Solitonlike structures called “droplets” are found to exist within a paradigm reaction-diffusion model that can be used to describe patterning in a number of biological systems, for example, on the skin of various fish species. They have also been found in many other systems that can be modeled with a complex Ginzburg-Landau system. These droplets can be analyzed in the biological paradigm model because the system has two nonzero stable steady states that are symmetric; however, the asymmetric case is more challenging. We first review the properties of the paradigm system and then extend a recently developed perturbation technique [D. Gomila et al., J. Opt. B: Quantum Semiclassical Opt. 6, S265 (2004)] to investigate the weakly asymmetric case. We compare the results of our mathematical analysis with numerical simulations and show good agreement in the region where the assumptions hold.

View figure in article

Article Text

References (36)

  1. P. DeKepper, V. Castets, E. Dulos, and J. Boissonade, Physica D 49, 161 (1991).
  2. D. G. Míguez, M. Dolnik, A. P. Muñuzuri, and L. Kramer, Phys. Rev. Lett. 96, 048304 (2006).
  3. A. M. Turing, Philos. Trans. R. Soc. London, Ser. B 237, 37 (1952).
  4. A. Gierer and H. Meinhardt, Biol. Cybern. 12, 30 (1972).
  5. S. Kondo and R. Asai, Nature (London) 376, 765 (1995).
  6. S. Sick, S. Reinker, J. Timmer, and T. Schlake, Science 314, 1447 (2006).
  7. S. S. Liaw, C. C. Yang, R. T. Liu, and J. T. Hong, Phys. Rev. E 64, 041909 (2001).
  8. H. Meinhardt, P. Prusinkiewicz, and D. R. Fowler, The Algorithmic Beauty of Sea Shells (Springer Verlag, Berlin, 2003).
  9. R. A. Barrio, C. Varea, J. L. Aragón, and P. K. Maini, Bull. Math. Biol. 61, 483 (1999).
  10. R. A. Barrio, R. E. Baker, B. Vaughan, Jr., K. Tribuzy, M. R. de Carvalho, R. Bassanezi, and P. K. Maini, Phys. Rev. E 79, 031908 (2009).
  11. W. J. Firth and G. K. Harkness, Asian J. Phys. 7, 665 (1998).
  12. W. J. Firth and A. J. Scroggie, Phys. Rev. Lett. 76, 1623 (1996).
  13. D. Gomila, P. Colet, G. L. Oppo, and M. S. Miguel., J. Opt. B: Quantum Semiclassical Opt. 6, S265 (2004).
  14. C. Varea, D. Hernández, and R. A. Barrio, J. Math. Biol. 54, 797 (2007).
  15. D. Michaelis, U. Peschel, F. Lederer, D. V. Skryabin, and W. J. Firth, Phys. Rev. E 63, 066602 (2001).
  16. D. Gomila, P. Colet, M. San Miguel, A. J. Scroggie, and G. L. Oppo, IEEE J. Quantum Electron. 39, 238 (2003).
  17. J. D. Gunton, M. San Miguel, and P. S. Sahni, Phase Transitions and Critical Phenomena (Academic, London, 1983).
  18. G. L. Oppo, A. J. Scroggie, and W. J. Firth, J. Opt. B: Quantum Semiclassical Opt. 1, 133 (1999).
  19. D. Gomila, Ph.D. thesis, Universitat de les Illes Balears, 2003.
  20. D. Gomila, P. Colet, G. L. Oppo, and M. S. Miguel, Phys. Rev. Lett. 87, 194101 (2001).
  21. J. D. Murray, Mathematical Biology I: An Introduction, 3rd ed. (Springer, New York, 2003), Vol. 1.
  22. B. Ermentrout, Proc. R. Soc. Lond. A 434, 413 (1991).
  23. P. Coullet, C. Elphick, and D. Repaux, Phys. Rev. Lett. 58, 431 (1987).
  24. D. W. Jordan and P. Smith, Mathematical Techniques: An Introduction for the Engineering, Physical, and Mathematical Sciences (Oxford University Press, Oxford, 2002).
  25. T. Leppänen, Ph.D. thesis, Helsinki University of Technology, 2004.
  26. M. R. de Carvalho, N. R. Lovejoy, and R. S. Rosa, Check List of the Freshwater Fishes of South and Central America, Chapter Family Potamotrygonidae (River Stingrays) (EDIPUCRS, Porto Alegre, 2003), p. 22.
  27. J. P. Keener, Principles of Applied Mathematics (Westview Press, Boulder, CO, 2000).
  28. D. V. Skryabin, A. Yulin, D. Michaelis, W. J. Firth, G. L. Oppo, U. Peschel, and F. Lederer, Phys. Rev. E 64, 056618 (2001).
  29. E. Süli and D. F. Mayers, An Introduction to Numerical Analysis (Cambridge University Press, Cambridge, 2003).
  30. J. M. McSloy, Ph.D. thesis, University of Strathclyde, 2002.
  31. C. B. Moler and G. W. Stewart, SIAM Journal on Numerical Analysis 10, 241 (1973).
  32. C. F. Van Loan, SIAM Journal on Numerical Analysis 12, 819 (1975).
  33. T. B. Thorson, J. K. Langhammer, and M. I. Oetinger, Environ. Biol. Fishes 9, 3 (1983).
  34. E. A. Dawes, Quantitative Problems in Biology (Longman, London, 1956).
  35. R. Dillon, C. Gadgil, and H. G. Othmer, Proc. Natl. Acad. Sci. U.S.A. 100, 10152 (2003).
  36. P. Charvet-Almeida, M. L. G. Araújo, and M. P. Almeida, J. Northwest Atl. Fish. Sci. 35, 165 (2005).

Outline

Information

Phys. Rev. E 82, 051929– Published 23 November, 2010
  • Received 21 July 2010

© 2010 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation