Pattern formation for a type of reaction diffusion system with cross diffusion

Rasheed, Shaker M.; Abdulahad, Joseph G.; Salih, Viyan A. Mohammed

International Journal of Advanced and Applied Sciences

Int. j. adv. appl. sci.

EISSN: 2313-3724

Print ISSN: 2313-626X

Volume 4, Issue 4 (April 2017), Pages: 22-26

Title: Pattern formation for a type of reaction diffusion system with cross diffusion

Author(s): Shaker M. Rasheed *, Joseph G. Abdulahad, Viyan A. Mohammed Salih

Affiliation(s):

Department of Mathematics, Faculty of Science, University of Zakho, Kurdistan Region, Iraq

https://doi.org/10.21833/ijaas.2017.04.004

Full Text - PDF XML

Abstract:

In this paper, pattern formation for a Schnakenberg model is studied in one and two dimensions. The model has been studied when the diffusion is nonlinear and so called cross diffusion. The conditions of diffusion driven instability are applied to this model and shown that this model can formulate patterns, and the existence of bifurcation for specific parameters are shown and for different values of wave number k. The use of COMSOL Multiphysics finite element package in simulation shows nice graphs of pattern formations in two dimensions.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Schnakenberg model, Pattern formation, Cross diffusion

Article History: Received 5 December 2016, Received in revised form 10 February 2017, Accepted 18 February 2017

Digital Object Identifier:

https://doi.org/10.21833/ijaas.2017.04.004

Citation:

Rasheed SM, Abdulahad JG, and Salih VAM (2017). Pattern formation for a type of reaction diffusion system with cross diffusion. International Journal of Advanced and Applied Sciences, 4(4): 22-26

http://www.science-gate.com/IJAAS/V4I4/Rasheed.html

References:

Andreianov B, Bendahmane M, and Ruiz-Baier R (2011). Analysis of a finite volume method for a cross-diffusion model in population dynamics. Mathematical Models and Methods in Applied Sciences, 21(02): 307-344. https://doi.org/10.1142/S0218202511005064

Barrett JW and Blowey JF (2004). Finite element approximation of a nonlinear cross-diffusion population model. Numerische Mathematik, 98(2): 195-221. https://doi.org/10.1007/s00211-004-0540-y

Barrio RA, Varea C, Aragón JL, and Maini PK (1999). A two-dimensional numerical study of spatial pattern formation in interacting Turing systems. Bulletin of Mathematical Biology, 61(3): 483-505. https://doi.org/10.1006/bulm.1998.0093 PMid:17883228

Gambino G, Lombardo MC, and Sammartino M (2012). Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion. Mathematics and Computers in Simulation, 82(6): 1112-1132. https://doi.org/10.1016/j.matcom.2011.11.004

Gambino G, Lupo S, and Sammartino M (2015). Effects of cross-diffusion on Turing patterns in a reaction-diffusion Schnakenberg model. arXiv preprint arXiv:1501.04890. Available online at: https://arxiv.org/pdf/1501.04890v1.pdf

Gambino G, Lombardo MC, and Sammartino M (2007). Cross-diffusion driven instability for a Lotka-Volterra competitive reaction-diffusion system. In the WASCOM 2007 – 14th Conference on Waves and Stability in Continuous Media, World Sci. Publ., Hackensack, USA: 297–302

Madzvamuse A, Ndakwo HS, and Barreira R (2015). Cross-diffusion-driven instability for reaction-diffusion systems: Analysis and simulations. Journal of Mathematical Biology, 70(4): 709-743. https://doi.org/10.1007/s00285-014-0779-6 PMid:24671430

Rasheed SM (2014). Pattern formations dynamics in a reaction-diffusion model. International Journal of Pure and Applied Sciences and Technology, 21(1): 53-60.

Shigesada N, Kawasaki K, and Teramoto E (1979). Spatial segregation of interacting species. Journal of Theoretical Biology, 79(1): 83-99. https://doi.org/10.1016/0022-5193(79)90258-3

Tory E, Schwandt H, Ruiz-Baier R, and Berres S (2011). An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks and Heterogeneous Media, 6(EPFL-ARTICLE-170240): 401-423.

Turing AM (1952). The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237(641): 37-72. https://doi.org/10.1098/rstb.1952.0012

Vanag VK and Epstein IR (2009). Cross-diffusion and pattern formation in reaction–diffusion systems. Physical Chemistry Chemical Physics, 11(6): 897-912. https://doi.org/10.1039/B813825G PMid:19177206