Mathematical investigation of two dimensional pattern formation

Authors

  • Seyed Ali Madani Tonekaboni

    School of Mathematics, University of Waterloo, Ontario, Canada

  • Ali Shademani

    School of Mechanical Engineering, University of Tehran, Tehran, Iran

How to Cite

Madani Tonekaboni, S. A., & Shademani, A. (2013). Mathematical investigation of two dimensional pattern formation. International Journal of Biological Research, 2(1), 1-5. https://doi.org/10.14419/ijbr.v2i1.1502

Received date: November 11, 2013

Accepted date: December 5, 2013

Published date: December 24, 2013

DOI:

https://doi.org/10.14419/ijbr.v2i1.1502

Abstract

In this paper, one of the significant effects on two dimensional pattern formations of chemical reactions concerned with diffusion of species is investigated. Gray-Scott model is employed to study the effect of diffusion on reaction rate and distribution of the reactants. Nonlinear dimensionless partial differential equations of the problem are solved using explicit finite difference method. Contours of one agent are obtained for different parameter values and time dependencies of the patterns are investigated. Different time scales of the problem are also took into consideration.

 

Keywords: Diffusion, finite difference method, Gray-Scott model, pattern formation, time scale.

References

  1. Aris, R. (1975). The mathematical theory of diffusion and reaction in permeable catalysts. Vol. 1: The theory of the steady state. Clarendon, Oxford.
  2. Tonekaboni, S. A. M., Abad, A. S. M., Karimi, S., & Shabani, M. (2013). New Solution of Substrate Concentration in the Biosensor Response by Discrete Homotopy Analysis Method. World Journal of Engineering and Technology, 1, 27.
  3. Siepmann, J., & Göpferich, A. (2001). Mathematical modeling of bioerodible, polymeric drug delivery systems. Advanced Drug Delivery Reviews, 48(2), 229-247.
  4. Turing, A. (1952). The Chemical Basis Of Morphogenesis. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237(641), 37-72.
  5. FRS, J. D. M. (1993). Animal Coat Patterns and Other Practical Applications of Reaction Diffusion Mechanisms. In Mathematical Biology (pp. 435-480). Springer Berlin Heidelberg.
  6. Castets, V., Dulos, E., Boissonade, J., & De Kepper, P. (1990). Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Physical Review Letters, 64, 2953-2956.
  7. Maini, P., Painter, K., & Chau, H. P. (1997). Spatial pattern formation in chemical and biological systems. Journal of the Chemical Society, Faraday Transactions, 93(20), 3601-3610.
  8. R.A. Barrio, L. Zhang, (1999) Proc. IV Congreso Nacional de Biolog a del Desarrollo, Vol. 18, Bol. Ed. Bioquim., p. 32;
  9. D.W. McLaughlin, (2002) Modelling the primary visual cortex of the macaquemonkey, Physica D 168-169, 35-44.
  10. Leppänen, T., Karttunen, M., Kaski, K., Barrio, R. A., & Zhang, L. (2002). A new dimension to Turing patterns. Physica D: Nonlinear Phenomena, 168, 35-44.
  11. Pearson, J. E. (1993). Complex patterns in a simple system. arXiv preprint patt-sol/9304003.

Downloads

How to Cite

Madani Tonekaboni, S. A., & Shademani, A. (2013). Mathematical investigation of two dimensional pattern formation. International Journal of Biological Research, 2(1), 1-5. https://doi.org/10.14419/ijbr.v2i1.1502

Received date: November 11, 2013

Accepted date: December 5, 2013

Published date: December 24, 2013