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Travelling Wave Solutions in a Tissue Interaction Model for Skin Pattern Formation

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Abstract

We discuss the existence and the uniqueness of travelling wave solutions for a tissue interaction model on skin pattern formation proposed by Cruywagen and Murray. The geometric theory of singular perturbations is employed.

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References

  1. Ai, S. (2003). Existence of traveling solutions in a tissue interaction model for skin pattern formation. J. Nonlinear Sci., in press.

  2. Chow, S.-N., Liu, W., and Yi, Y. (2000). Center manifolds for invariant sets of flows. J. Differential Equations 168(2), 355–385.

    Google Scholar 

  3. Cruywagen, G. C., and Murray, J. D. (1992). On a tissue interaction model for skin pattern formation. J. Nonlinear Sci. 2, 217–240.

    Google Scholar 

  4. Cruywagen, G. C., Maini, P. K., Murray, J. D. (1994). Traveling waves in a tissue interaction model for skin pattern formation. J. Math. Biol. 33, 193–210.

    Google Scholar 

  5. Cruywagen, G. C., Maini, P. K., Murray, J. D. (2000). An envelope method for analyzing sequential pattern formation. SIAM J. Appl. Math. 61, 213–231.

    Google Scholar 

  6. Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31, 53–98.

    Google Scholar 

  7. Gardner, R. A. (1993). An invariant-manifold analysis of electrophoretic traveling waves, J. Dynam. Differential Equations 5, 599–606.

    Google Scholar 

  8. Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math, Vol. 840, Springer-Verlag, New York.

    Google Scholar 

  9. Jones, C. K. R. T. (1995). Geometric singular perturbation theory. In Johnson, R. (ed.), Dynamical Systems, Springer-Verlag, Berlin, Heidelberg.

    Google Scholar 

  10. Lin, X.-B. (1996). Asymptotic expansion for layer solutions of a singularly perturbed reaction-diffusion system. Trans. Amer. Math. Soc. 348(2), 713–753.

    Google Scholar 

  11. Murray, J. D. (1989). Mathematical Biology, Springer, Berlin/Heidelberg/New York.

    Google Scholar 

  12. Sakamoto, K. (1990). Invariant manifolds in singular perturbation problems for ordinary differential equations. Proc. Roy. Soc. Edinburgh Sect. A 116, 45–78.

    Google Scholar 

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Ai, S., Chow, SN. & Yi, Y. Travelling Wave Solutions in a Tissue Interaction Model for Skin Pattern Formation. Journal of Dynamics and Differential Equations 15, 517–534 (2003). https://doi.org/10.1023/B:JODY.0000009746.52357.28

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  • DOI: https://doi.org/10.1023/B:JODY.0000009746.52357.28

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