Numerical simulations of Turing patterns in a reaction-diffusion model with the Chebyshev spectral method

  • Maliha Tehseen Saleem
  • Ishtiaq AliEmail author
Regular Article


Multi-species models play an important role in both ecology and mathematical ecology due to their practical relevance and universal existence. Some phenomena include but are not limited to osculating solutions behavior, multiple steady states and spatial patterns formation. In this article we study the numerical approximation of Turing patterns corresponding to the steady state solutions of systems of reaction-diffusion equations subject to zero-flux boundary conditions. We apply Chebyshev spectral methods which proved to be numerical methods that can significantly speed up the computation of systems of reaction-diffusion equations in the spatial part, while the temporal part is discretized using the Euler scheme in one dimension. For the evaluation of Turing instabilities and bifurcation of the steady state problem, we used the eigenvalues of the Jacobian matrix. The proposed scheme is then extended to the two-dimensional problem. We found that our numerical scheme is in very good agreement with other schemes available in the literature.


  1. 1.
    J.D. Murray, Mathematical Biology, 2nd edition (Springer-Verlag, Berlin, 1993)Google Scholar
  2. 2.
    Darcy Wentworth Thompson, On Growth and Form (1942)Google Scholar
  3. 3.
    A.M. Turing, Philos. Trans. R. Soc. London B 237, 37 (1952)CrossRefGoogle Scholar
  4. 4.
    P.K. Maini, K.J. Painter, H.N.P. Chau, Faraday Trans. 93, 3601 (1997)CrossRefGoogle Scholar
  5. 5.
    A. Gierer, H. Meinhardt, Kybernetik 12, 30 (1972)CrossRefGoogle Scholar
  6. 6.
    A.J. Koch, H. Meinhardt, Rev. Mod. Phys. 66, 1481 (1994)CrossRefGoogle Scholar
  7. 7.
    H. Meinhardt, Models of Biological Pattern Formation (Academic, London, 1982)Google Scholar
  8. 8.
    Shigeru Kondo, Rihito Asai, Nature 376, 765 (1995)CrossRefGoogle Scholar
  9. 9.
    K.J. Painter, H.G. Othmer, P.K. Maini, Proc. Natl. Acad. Sci. U.S.A. 96, 5549 (1999)CrossRefGoogle Scholar
  10. 10.
    D. Calhoun, C. Helzel, SIAM J. Sci. Comput. 31, 4066 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Allen R. Sanderson, Mike Kirby, Chris R. Johnson, Lingfa Yang, J. Graphics Tools 11, 47 (2006)CrossRefGoogle Scholar
  12. 12.
    John E. Pearson, Science 261, 189 (1993)CrossRefGoogle Scholar
  13. 13.
    Qianqian Zheng, Jianwei Shen, Zhijie Wang, PLoS ONE 13, e0190176 (2018)CrossRefGoogle Scholar
  14. 14.
    Gendai Gu, Hongxiao Peng, Int. J. Mod. Nonlinear Theory Appl. 4, 215 (2015)CrossRefGoogle Scholar
  15. 15.
    Tatiana T. Marquez-Lago, Pablo Padilla, Theor. Biol. Med. Modell. 11, 7 (2014)CrossRefGoogle Scholar
  16. 16.
    Il-Hie Lee, Ung-In Cho, Bull. Kor. Chem. Soc. 21, 1213 (2000)Google Scholar
  17. 17.
    J. Aragon, M. Torres, D. Gil, R.A. Barrio, P.K. Maini, Phys. Rev. E 65, 051913 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    M.A.J. Chaplain, M. Ganesh, I.G. Graham, J. Math. Biol. 42, 387 (2001)MathSciNetCrossRefGoogle Scholar
  19. 19.
    R. Barrio, C. Varea, J. Aragon, P. Maini, Bull. Math. Biol. 61, 483 (1999)CrossRefGoogle Scholar
  20. 20.
    Teemu Leppanen, Computational Studies of Pattern Formation in Turing systems, Doctoral dissertation (Helsinki University of Technology, 2004)Google Scholar
  21. 21.
    Jean Tyson Schneider, Perfect stripes from a general Turing model in different geometries (Boise State University, August 2012)Google Scholar
  22. 22.
    R.A. Barrio, J.L. Aragon, M. Torres, P.K. Maini, Physica D 61, 168 (2002)Google Scholar
  23. 23.
    C. Varea, J.L. Aragon, R.A. Barrio, Phys. Rev. E 60, 4588 (1999)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsCOMSATS UniversityIslamabadPakistan

Personalised recommendations