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Perturbative Traveling-Wave Solution for a Flux-Limited Reaction-Diffusion Morphogenesis Equation

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Abstract

In this study, we investigate a porous medium-type flux-limited reaction—diffusion equation that arises in morphogenesis modeling. This nonlinear partial differential equation is an extension of the generalized Fisher—Kolmogorov—Petrovsky—Piskunov (Fisher-KPP) equation in one-dimensional space. The approximate analytical traveling-wave solution is found by using a perturbation method. We show that the morphogen concentration propagates as a sharp wave front where the wave speed has a saturated value. The numerical solutions of this equation are also provided to compare them with the analytical predictions. Finally, we qualitatively compare our theoretical results with those obtained in experimental studies.

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References

  1. J. Murray, Mathematical Biology (Springer-Verlag, New York, 1989).

    Book  Google Scholar 

  2. R. Fisher, Ann. Eugen. 7, 355 (1937).

    Article  Google Scholar 

  3. A. Kolmogorov, I. Petrovskii and N. Piscounov, Selected works of A. N. Kolmogorov I, edited by V. Tikhomirov (Kluwer, Dordrecht, 1991), pp. 242–270.

  4. E. Dessaud et al., Nature 450, 717 (2007).

    Article  ADS  Google Scholar 

  5. K. W. Rogers and A. F. Schier, Annu. Rev. Cell Dev. Biol. 27, 377 (2011).

    Article  Google Scholar 

  6. J. Briscoe and P. P. Thérond, Nat. Rev. Mol. Cell Biol. 14, 416 (2013).

    Article  Google Scholar 

  7. E. Simon, A. Aguirre-Tamaral, G. Aguilar and I. Guerrero, J. Dev. Biol. 4, 34 (2016), http://www.mdpi.com/2221-3759/4/4/34.

    Article  Google Scholar 

  8. A. D. Lander, Q. Nie and F. Y. Wan, Dev. Cell 2, 785 (2002), http://www.sciencedirect.com/science/article/pii/S153458070200179X.

    Article  Google Scholar 

  9. K. Saha and D. V. Schaffer, Development 133, 889 (2006).

    Article  Google Scholar 

  10. S. Kondo and T. Miura, Science 329, 1616 (2010), https://science.sciencemag.org/content/329/5999/1616.

    Article  ADS  MathSciNet  Google Scholar 

  11. A. M. Turing, Phil. Trans. Roy. Soc. B 237, 37 (1952).

    ADS  Google Scholar 

  12. F. Crick, Nature 225, 420 (1970).

    Article  ADS  Google Scholar 

  13. A. Gierer and H. Meinhardt, Kybernetik 12, 30 (1972), https://doi.org/10.1007/BF00289234.

    Article  Google Scholar 

  14. M. Verbeni et al., Phys. Life Rev. 10, 457 (2013), http://www.sciencedirect.com/science/article/pii/S1571064513000833.

    Article  ADS  Google Scholar 

  15. Ó. Sánchez et al., Hedgehog signaling protocols (Springer, New York, 2015), pp. 19–33.

    Book  Google Scholar 

  16. P. Rosenau, Phys. Rev. A 46, R7371 (1992), https://doi.org/10.1103/PhysRevA.46.R7371.

    Article  ADS  Google Scholar 

  17. Y. Brenier, Optimal transportation and applications (Springer, Berlin Heidelberg, 2003), pp. 91–121.

    Book  Google Scholar 

  18. A. Chertock, A. Kurganov and P. Rosenau, Non-linearity 16, 1875 (2003), https://doi.org/0951-7715/16/i=6/a=301.

    ADS  Google Scholar 

  19. V. Caselles, Publ. Mat. 57, 144 (2013).

    Article  Google Scholar 

  20. A. Kurganov and P. Rosenau, Nonlinearity 19, 171 (2006), https://doi.org/0951-7715/19/i=1/a=009.

    Article  ADS  MathSciNet  Google Scholar 

  21. F. Andreu, V. Caselles and J. Mazon, J. Differ. Equations 248, 2528 (2010), http://www.sciencedirect.com/science/article/pii/S0022039610000124.

    Article  ADS  Google Scholar 

  22. F. Andreu, J. Calvo, J. Mazon and J. Soler, J. Differ. Equations 252, 5763 (2012), http://www.sciencedirect.com/science/article/pii/S0022039612000411.

    Article  ADS  Google Scholar 

  23. M. Garrione and L. Sanchez, Bound. Value Probl. 2015, 45 (2015), https://doi.org/10.1186/s13661-015-0303-y.

    Article  Google Scholar 

  24. J. Campos, P. Guerrero, O. Sánchez and J. Soler, Ann. I. H. Poincaré: NA 30, 141 (2013), http://www.sciencedi-rect.com/science/article/pii/S0294144912000637.

  25. J. Campos and J. Soler, Nonlinear Anal. 137, 266 (2016), http://www.sciencedirect.com/science/article/pii/S0362546X1500437X.

    Article  MathSciNet  Google Scholar 

  26. J. Calvo et al., EMS Surv. Math. Sci 2, 131 (2015).

    Article  MathSciNet  Google Scholar 

  27. J. Calvo et al., Invent. Math. 206, 57 (2016), https://doi.org/10.1007/s00222-016-0649-5.

    Article  ADS  MathSciNet  Google Scholar 

  28. J. Calvo, Computational Mathematics, Numerical Analysis and Applications (Springer, Milan/Heidelberg, 2017), pp. 189–194.

    Book  Google Scholar 

  29. J. Calvo, SeMA J. 75, 173 (2018), https://doi.org/10.1007/s40324-017-0128-y.

    Article  MathSciNet  Google Scholar 

  30. M. Garrione and L. Sanchez, Bound. Value Probl. 2015, 45 (2015), https://doi.org/10.1186/s13661-015-0303-y.

    Article  Google Scholar 

  31. F. S. Garduño and P. Maini, Appl. Math. Lett. 7, 47 (1994), http://www.sciencedirect.com/science/article/pii/0893965994900515.

    Article  MathSciNet  Google Scholar 

  32. W. Ngamsaad and S. Suantai, Commun. Nonlinear Sci. Numer. Simul. 35, 88 (2016), http://www.sciencedi-rect.com/science/article/pii/S1007570415003718.

    Article  MathSciNet  Google Scholar 

  33. W. Newman, J. Theor. Biol. 85, 325 (1980).

    Article  Google Scholar 

  34. H. J. Eberland L. Demaret, Electron. J. Diff. Equ. Conf. 15, 77 (2007).

    Google Scholar 

  35. W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1988).

    MATH  Google Scholar 

  36. G. B. Arfken, Mathematical Methods for Physicists (Academic Press, San Diego, 1985).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Division of Physics, School of Science, University of Phayao, Phayao, 56000, Thailand

    Waipot Ngamsaad

  2. Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Suthep Suantai

Authors
  1. Waipot Ngamsaad
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  2. Suthep Suantai
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Correspondence to Waipot Ngamsaad.

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Ngamsaad, W., Suantai, S. Perturbative Traveling-Wave Solution for a Flux-Limited Reaction-Diffusion Morphogenesis Equation. J. Korean Phys. Soc. 76, 323–329 (2020). https://doi.org/10.3938/jkps.76.323

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  • DOI: https://doi.org/10.3938/jkps.76.323

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