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Linear Instability, Turing Instability and Pattern Formation

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Parabolic Equations in Biology

Abstract

This chapter treats of one of the most fundamental observation for biological systems: the Turing instability mechanism. In his seminal paper, A. Turing suggests that a system of chemical substances reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis, that is boundary formation. Surprisingly, he shows how diffusion can generate unstability and this effect is also called ‘diffusion driven instabilities’. We present the argument for Turing instability, that mean instabilities where unstable modes remain finite. We illustrate the linear theory with several famous nonlinear examples: the non-local Fisher/KPP equation, the CIMA reaction, the brusselator, the Gray-Scott system etc.

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References

  1. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations, II. Existence of infinitely many solutions. Arch. Ration. Mech. Anal. 82(4), 347–375 (1983)

    MATH  MathSciNet  Google Scholar 

  2. Berestycki, H., Nicolaenko, B., Scheurer, B.: Traveling wave solutions to combustion models and their singular limits. SIAM J. Math. Anal. 16(6), 1207–1242 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berestycki, H., Nadin, G., Perthame, B., Ryzhik, L.: The non-local Fisher-KPP equation: traveling waves and steady states. Nonlinearity 22, 2813–2844 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Britton, N.F.: Spatial structures and periodic traveling waves in an integro-differential reaction-diffusion population model. SIAM J. Appl. Math. 50(6), 1663–1688 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  5. Castets, V., Dulos, E., Boissonade, J., De Kepper, P.: Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64(24), 2953–2956 (1990)

    Article  Google Scholar 

  6. De Kepper, P., Castets, V., Dulos, E., Boissonade, J.: Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction. Physica D 49, 161–169 (1991)

    Article  Google Scholar 

  7. FreeFEM++: Software available at http://www.freefem.org (2015)

  8. Génieys, S., Volpert, V., Auger, P.: Pattern and waves for a model in population dynamic with non-local consumption of resources. Math. Model. Nat. Phenom. 1(1), 65–82 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik 12, 30–39 (1972)

    Article  Google Scholar 

  10. Gourley, S.A.: Travelling front solutions of a non-local Fisher equation. J. Math. Biol. 41, 272–284 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gray, P., Scott, S.K.: Autocatalytic reactions in the isothermal continuous stirred tank reactor: isolas and other forms of multistability. Chem. Eng. Sci. 38(1), 29–43 (1983)

    Article  Google Scholar 

  12. Hecht, F.: New development in FreeFEM++. J. Numer. Math. 20(3–4), 251–265 (2012)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  15. Lefever, R., Lejeune, O.: On the origin of tiger bush. Bull. Math. Biol. 59(2), 263–294 (1997)

    Article  MATH  Google Scholar 

  16. Lengyel, I., Epstein, I.R.: Modeling of Turing structure in the chlorite-iodide-malonic acid-starch reaction system. Science 251, 650–652 (1991)

    Article  Google Scholar 

  17. Logak, E., Loubeau, V.: Travelling wave solutions to a condensed phase combustion model. Asymptot. Anal. 12(4), 259–294 (1996)

    MATH  MathSciNet  Google Scholar 

  18. Maini, P.K.: How the mouse got its stripes. PNAS 100(17), 9656–9657 (2003). doi:10.1073/pnas.1734061100

    Article  Google Scholar 

  19. Malchiodi, A., Montenegro, M.: Multidimensional boundary layers for a singularly perturbed Neumann problem. Duke Math. J. 124, 105–143 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Marion, M.: Qualitative properties of a nonlinear system for laminar flames without ignition temperature. Nonlinear Anal. 9(11), 1269–1292 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  21. Meinhardt, H.: Models of Biological Pattern Formation. Academic, London (1982)

    Google Scholar 

  22. Meinhardt, H.: The Algorithmic Beauty of Sea Shells. Springer, Heidelberg (1995)

    Book  Google Scholar 

  23. Morita, Y., Ogawa, T.: Stability and bifurcation of nonconstant solutions of a reaction-diffusion system with conservation of mass. Nonlinearity 23, 1387–1411 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  24. Muratov, C.B., Osipov, V.V.: Traveling spike autosolitons in the Gray-Scott model. Physica D 155, 112–131 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Murray, J.D.: Mathematical Biology, vols. 1 and 2, 2nd edn. Springer, New York (2002)

    Google Scholar 

  26. Nadin, G., Perthame, B., Tang, M.: Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation. C. R. Acad. Sci. Paris Ser. I 349, 553–557 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  27. Ni, W., Tang, M.: Turing patterns in the Lengyel-Epstein system for the CIMA reaction. Trans. Am. Math. Soc. 357, 3953–3969 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Ni, W.-M., Wei, J.: On positive solutions concentrating on spheres for the Gierer–Meinhardt system. J. Diff. Equ. 221, 158–189 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  29. Nicolis, G., Prigogine, I.: Self-organization in Non-equilibrium Systems. Wiley Interscience, New-York (1977)

    Google Scholar 

  30. Oster, G.F., Shubin, N., Murray, J.D., Alberch, P.: Evolution and morphogenetic rules: the shape of the vertebrate limb in ontogeny and phylogeny. Evolution 45, 862–884 (1988)

    Article  Google Scholar 

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

    Article  Google Scholar 

  32. Schnakenberg, J., Simple chemical reactions with limit cycle behavior. J. Theor. Biol. 81, 389–400 (1979)

    Article  MathSciNet  Google Scholar 

  33. Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. B 237, 37–72 (1952)

    Article  Google Scholar 

  34. Wei, J., Winter, M.: Symmetric and asymmetric multiple clusters in a reaction-diffusion system. NoDEA 14(5–6), 787–823 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  35. Yi, F., Wei, J., Shi, J.: Diffusion-driven instability and bifurcation in the Lengyel-Epstein reaction-diffusion system. Nonlinear Anal. Real World Appl. 9, 1038–1051 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Université Pierre et Marie Curie, Paris, France

    Benoît Perthame

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  1. Benoît Perthame
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Perthame, B. (2015). Linear Instability, Turing Instability and Pattern Formation. In: Parabolic Equations in Biology. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-19500-1_7

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