Skip to main content
SpringerLink
Log in

Localized Spot Patterns on the Sphere for Reaction-Diffusion Systems: Theory and Open Problems

  • Conference paper
  • First Online:
Mathematical and Computational Approaches in Advancing Modern Science and Engineering

Abstract

A new class of point-interaction problem characterizing the time evolution of spatially localized spots for reaction-diffusion (RD) systems on the surface of the sphere is introduced and studied. This problem consists of a differential algebraic system (DAE) of ODEs for the locations of a collection of spots on the sphere, and is derived from an asymptotic analysis in the large diffusivity ratio limit of certain singularly perturbed two-component RD systems. In Trinh and Ward (The dynamics of localized spot patterns for reaction-diffusion systems on the sphere. Nonlinearity Nonlinearity 29 (3), 766–806 (2016)), this DAE system was derived for the Brusselator and Schnakenberg RD systems, and herein we extend this previous analysis to the Gray-Scott RD model. Results and open problems pertaining to the determination of equilibria of this DAE system, and its relation to elliptic Fekete point sets , are highlighted. The potential of deriving similar DAE systems for more complicated modeling scenarios is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Barreira, R., Elliott, C.A., Madzvamuse, A.: The surface finite element method for pattern formation on evolving biological surfaces. J. Math. Biol. 63, 1095–1109 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boatto, S., Cabral, H.E.: Nonlinear stability of a latitudinal ring of point vortices on a non-rotating sphere. SIAM J. Appl. Math. 64 (1), 216–230 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bogomolov, V.A.: Dynamics of vorticity at a sphere. Fluid Dyn. (USSR) 6, 863–870 (1977)

    Google Scholar 

  4. Callahan, T.K.: Turing patterns with O(3) symmetry. Physica D 188 (1), 65–91 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chaplain, M.A.J, Ganesh, M., Graham, I.G.: Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth. J. Math. Biol. 42 (5), 387–423 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, W., Ward, M.J.: The stability and dynamics of localized spot patterns in the two-dimensional Gray-Scott model. SIAM J. Appl. Dyn. Syst. 10 (2), 582–666 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chossat, P., Lauterbach, R., Melbourne, I.: Steady-sate bifurcation with O(3) symmetry. Arch. Rat. Mech. Anal. 113, 313–376 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Coombs, D., Straube, R., Ward, M.J.: Diffusion on a sphere with localized traps: mean first passage time, eigenvalue asymptotics, and Fekete points. SIAM J. Appl. Math. 70 (1), 302–332 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dritschel, D.G., Boatto, S.: The motion of point vortices on closed surfaces. Proc. R. Soc. A 471, 20140890 (2015)

    Article  MathSciNet  Google Scholar 

  10. Kidambi, R., Newton, P.K.: Motion of three vortices on the sphere. Physica D 116 (1–2), 143–175 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kolokolnikov, T., Ward, M.J., Wei, J.: Spot self-replication and dynamics for the Schnakenberg model in a two-dimensional domain. J. Nonlinear Sci. 19 (1), 1–56 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kondo, S., Asai, R.: A reaction–diffusion wave on the skin of the marine angelfish Pomacanthus. Nature 376, 765–768 (1995)

    Article  Google Scholar 

  13. Landsberg, C., Voigt, V.: A multigrid finite element method for reaction-diffusion systems on surfaces. Comput. Vis. Sci. 13, 177–185 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Macdonald, C.B., Merriman, B., Ruuth, S.J.: Simple computation of reaction-diffusion processes on point clouds. Proc. Natl. Acad. Sci. U.S.A. 110 (23), 9209–9214 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Madzvamuse, A., Chung, A.H.W., Venkataraman, V.: Stability analysis and simulations of coupled bulk-surface reaction diffusion systems. Proc. R. Soc. A 471, 20140546 (2015)

    Article  MathSciNet  Google Scholar 

  16. Matthews, P.C.: Pattern formation on a sphere. Phys. Rev. E 67 (3), 036206 (2003)

    Article  MathSciNet  Google Scholar 

  17. Nagata, W., Harrison, L.G., Wehner, S.: Reaction-diffusion models of growing plant tips: Bifurcations on hemispheres. Bull. Math. Biol. 6 (4), 571–607 (2003)

    Article  MATH  Google Scholar 

  18. Newton, P.K.: The N-Vortex Problem: Analytical Techniques. Springer, New York (2001)

    Book  MATH  Google Scholar 

  19. Newton, P.K., Sakajo, T.: Point vortex equilibria and optimal packings of circles on a sphere. Proc. R. Soc. A 467, 1468–1490 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Painter, K.J.: Modelling of pigment patterns in fish. In: Maini, P.K., Othmer, H.G. (eds.) Mathematical Models for Biological Pattern Formation. IMA Volumes in Mathematics and Its Applications, vol. 121, pp. 58–82. Springer-Verlag, New York (2000)

    Google Scholar 

  21. Plaza, R.G., Sánchez-Garduño, F., Padilla, P., Barrio, R.A., Maini, P.K.: The effect of growth and curvature on pattern formation. J. Dyn. Diff. Equ. 16 (4), 1093–1121 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Prigogine, I., Lefever, R.: Symmetry breaking instabilities in dissipative systems II. J. Chem. Phys. 48 (4), 1695–1700 (1968)

    Article  Google Scholar 

  23. Roberts, G.: Stability of relative equilibria in the planar n-vortex problem. SIAM J. Appl. Dyn. Syst. 12 (2), 1114–1134 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rozada, I., Ruuth, S., Ward, M.J.: The stability of localized spot patterns for the Brusselator on the sphere. SIAM J. Appl. Dyn. Syst. 13 (1), 564–627 (2014),

    Article  MathSciNet  MATH  Google Scholar 

  25. Sewalt, L., Doelman, A., Meijer, H., Rottschafer, V., Zagarias, A.: Tracking pattern evolution through extended center manifold reduction and singular perturbations. Physica D 298, 48–67 (2015)

    Article  MathSciNet  Google Scholar 

  26. Stortelder, W., de Swart, J., Pintér, J.: Finding elliptic Fekete point sets: two numerical solution approaches. J. Comput. Appl. Math. 130 (1–2), 205–216 (1998)

    MATH  Google Scholar 

  27. Trinh, P., Ward, M.J.: The dynamics of localized spot patterns for reaction-diffusion systems on the sphere. Nonlinearity 29 (3), 766–806 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Varea, C., Aragón, J.L., Barrio, R.A.: Turing patterns on a sphere. Phys. Rev. E. 60 (4), 4588–4592 (1999)

    Article  Google Scholar 

  29. Wei, J., Winter, M.: Stationary multiple spots for reaction-diffusion systems. J. Math. Biol. 57 (1), 53–89 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

PHT thanks Lincoln College, Oxford and the Zilkha Trust for generous funding. MJW gratefully acknowledges grant support from NSERC.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

    Alastair Jamieson-Lane & Michael J. Ward

  2. OCIAM, Mathematical Institute, University of Oxford, Oxford, UK

    Philippe H. Trinh

Authors
  1. Alastair Jamieson-Lane
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Philippe H. Trinh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael J. Ward
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael J. Ward .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, University of Montreal, Montreal, Québec, Canada

    Jacques Bélair

  2. Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada

    Ian A. Frigaard

  3. Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada

    Herb Kunze

  4. Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Roman Makarov

  5. MS2Discovery Institute, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Roderick Melnik

  6. Department of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

    Raymond J. Spiteri

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Jamieson-Lane, A., Trinh, P.H., Ward, M.J. (2016). Localized Spot Patterns on the Sphere for Reaction-Diffusion Systems: Theory and Open Problems. In: Bélair, J., Frigaard, I., Kunze, H., Makarov, R., Melnik, R., Spiteri, R. (eds) Mathematical and Computational Approaches in Advancing Modern Science and Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-30379-6_58

Download citation

Publish with us

Policies and ethics

Search

Navigation