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Journal of Mathematical Biology

, Volume 33, Issue 2, pp 193–210 | Cite as

Travelling waves in a tissue interaction model for skin pattern formation

  • G. C. Cruywagen
  • P. K. Maini
  • J. D. Murray
Article

Abstract

Tissue interaction plays a major role in many morphogenetic processes, particularly those associated with skin organ primordia. We examine travelling wave solutions in a tissue interaction model for skin pattern formation which is firmly based on the known biology. From a phase space analysis we conjecture the existence of travelling waves with specific wave speeds. Subsequently, analytical approximations to the wave profiles are derived using perturbation methods. We then show numerically that such travelling wave solutions do exist and that they are in good agreement with our analytical results. Finally, the biological implications of our analysis are discussed.

Key words

Travelling waves Tissue interaction Perturbation procedure Wavespeed 

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References

  1. Cheer, A., Nuccitelli, R., Oster, G. F., Vincent, J.-P.: Cortical activity in vertebrate eggs. I. The activation waves. J. Theor. Biol. 124, 377–404 (1987)Google Scholar
  2. Cruywagen, G. C., Maini, P. K., Murray, J. D.: Sequential pattern formation in a model for skin morphogenesis. IMA J. Math. Appl Medic. Biol. 9, 227–248 (1992)Google Scholar
  3. Cruywagen, G. C., Maim, P. K., Murray, J. D.: Sequential and synchronous skin pattern formation. In: Othmer, H. G., Maim, P. K., Mrray, J. D. (eds.) Proceedings of the NATO ARW on Biological Pattern Formation, NATO Conference Series. New York: Plenum Press (in press)Google Scholar
  4. Cruywagen, G. C., Murray, J. D.: On a tissue interaction model for skin pattern formation. J. Nonlin. Sci. 2, 217–240 (1992)Google Scholar
  5. Cruywagen, G. C., Maini, P. K., Murray, J. D.: Bifurcating spatial patterns arising from travelling waves in a tissue interaction model. Appl. Math. Lett. 7(3), 63–66 (1994)Google Scholar
  6. Fife, P. C.: Mathematical aspects of reacting and diffusing systems. Lect Notes in Biomathematics 28. Berlin Heidelberg New York: Springer 1979Google Scholar
  7. Kolmogoroff, A., Petrovsky, I., Piscounoff, N.: Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Moscow Univ. Bull. Math. 1, 1–25 (1937)Google Scholar
  8. Lane, D. C., Murray, J. D., Manoranjan, V. S.: Analysis of wave phenomena in a morphogenetic mechanochemical model and an application to post-fertilisation waves on eggs. IMA J. Math. Appl. Medic. Biol. 4, 309–331 (1937)Google Scholar
  9. Mollison, D.: Spatial contact models for ecological and epidemic spread. J. Roy. Stat. Soc. B39, 283–326 (1977)Google Scholar
  10. Murray, J. D.: Nonlinear differential equation models in biology. Oxford: Clarendon Press 1977Google Scholar
  11. Murray, J. D.: Mathematical biology. Berlin Heidelberg New York: Springer 1989Google Scholar
  12. Murray, J. D., Cruywagen, G. C.: Threshold bifurcation in tissue interaction models for spatial pattern generation. Trans. R. Soc. Lond. A (in press)Google Scholar
  13. Murray, J. D., Cruywagen, G. C., Maini, P. K.: Pattern formation in tissue interaction models. In: Lecture Notes in Biomathematics 100 Levin S. (ed.) Berlin Heidelberg New York: Springer (in press)Google Scholar
  14. Murray, J. D., Oster, G. F.: Generation of biological pattern and form. IMA J. Maths. Appl. Med. Biol. 1, 51–75 (1984)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • G. C. Cruywagen
    • 1
  • P. K. Maini
    • 2
  • J. D. Murray
    • 1
  1. 1.Department of Applied Mathematics FS-20University of WashingtonSeattleUSA
  2. 2.Centre for Mathematical BiologyMathematical InstituteOxfordUK

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