Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Stochastic Patterns and the Role of Crowding

Discontinuity, Nonlinearity, and Complexity 2(4) (2013) 301--319 | DOI:10.5890/DNC.2013.11.001

Claudia Cianci$^{1}$; Duccio Fanelli$^{2}$

$^{1}$ Dipartimento di Sistemi e Informatica, University of Florence and INFN, Via S. Marta 3, 50139 Florence, Italy

$^{2}$ Dipartimento di Fisica e Astronomia, University of Florence and INFN, via Sansone 1, Sesto Fiorentino, 50019 Florence, Italy

Download Full Text PDF

 

Abstract

A stochastic variant of the Brusselator model is investigated. The model accounts for a long range coupling among constituents, as well as for the finite capacity of the embedding medium. The mean field limit of the model is studied and the condition for Turing and wave instability obtained. A degenerate, cusp like transition that separates the domains of Turing and wave order can take place. The point of transition is worked out analytically. Interestingly, the region of Turing instability, as delimited by such transition point, can set in also if the inhibitor diffuses slower then the activator. This is a consequence of the generalized diffusion scheme here analyzed and which originates from having imposed an effect of spatial competition. Beyond the deterministic, mean field picture, we elaborate on the role of stochastic corrections. Granularity, endogenous to the system, can eventually materialize in waves or Turing like patterns, that we here categorized in distinct classes.

References

  1. [1]  Murray, J. (2002), Mathematical Biology, Second Edition, Springer.
  2. [2]  Turing, A.M. (1952), The chemical basis of morphogenesis, Philos. Trans. R. Soc. London Ser. B, 273 (37), 37–72.
  3. [3]  Strogatz, S. (2001), Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering, Perseus Book Group.
  4. [4]  Belousov, B.P. (1959) A Periodic Reaction and itsMechanismin Collection of Short Papers on RadiationMedicine for 1958, Med. Publ. Moschow, 1959, A. M. Zhabotinsky, Biofizka, 9, 306–311.
  5. [5]  Butler, T., Goldenfeld, N. (2011), Fluctuation-driven Turing patterns, Physical Review E, 84, 011112.
  6. [6]  Biancalani, T., Galla, T., and McKane, A. (2011), Quasi-waves in a Brusselator model with nonlocal interaction, arXiv:1104.0567V1.
  7. [7]  Biancalani, T., Fanelli, D., Patti, F. Di (2010), Stochastic Turing patterns in the Brusselator model, Physical Review E, 81, 046215
  8. [8]  Fanelli, D., Cianci, C., and Patti, F.Di (2013), Turing instabilities in reaction-diffusion systems with cross diffusion, European Physical Journal B, to appear, DOI: 10.1140/epjb/e2013-30649-7.
  9. [9]  Kumar, N. and Horsthemke, W. (2011), Effects of cross term diffusion on Turing bifurcations in two-species reaction-transport system, Physical Review E, 83, 036105(2011).
  10. [10]  Fanelli, D. and McKane, A. (2010), Diffusion in a crowded environment, Physical Review E, 82, 021113.
  11. [11]  van Kampen, N.G. (2007), Stochastic Processes in Physics and Chemistry, Elsevier, Amsterdam, Third edition.
  12. [12]  Nicola, E.M. (2001), Ph.D. Thesis, University of Dresden, 2001), URL: http://nbn-resolving.de/urn:nbn:de:swb: 14-1036499969687-26395.
  13. [13]  Nicola, E.M., Baer, M., and Egel, H. (2006), Physical Review E, 73, 066225.
  14. [14]  Lugo, C., and McKane, A. (2008), Quasicycles in a spatial predator-prey model, Physical Review E, 78, 051911.
  15. [15]  de Anna, P., Patti, F. Di, Fanelli, D., McKane, A., and Dauxois, T. (2010), Spatial model of autocatalitic reactions, Physical Review E, 81, 056110.