Biological Applications

  • Juncheng Wei
  • Matthias Winter
Part of the Applied Mathematical Sciences book series (AMS, volume 189)


We discuss a number of biological, chemical and ecological applications of pattern formation in reaction-diffusion systems. These include the head development and regeneration in Hydra, embryology for newt and Drosophila, pigmentation patterns on sea shells, fish and mammals and, finally, patterns on growing domains for angelfish body patterns and alligator tooth formation.


Activator Peak Mutual Exclusion Turing Instability Exponential Attractor Pigmentation Pattern 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 6.
    Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978) MathSciNetCrossRefMATHGoogle Scholar
  2. 20.
    Bohn, H.: Interkalare Regeneration und segmentale Gradienten bei den Extremitäten von Leucophaea-Larven. Wilhelm Roux’ Arch. 165, 303–341 (1970) CrossRefGoogle Scholar
  3. 22.
    Britton, N.F.: Essential Mathematical Biology, 2nd edn. Springer Undergraduate Mathematics Series. Springer, London (2003) CrossRefMATHGoogle Scholar
  4. 24.
    Castets, V., Dulos, E., Boissonade, J., De Kepper, P.: Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64, 2953–2956 (1990) CrossRefGoogle Scholar
  5. 28.
    Crampin, E.J., Gaffney, E.A., Maini, P.K.: Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull. Math. Biol. 61, 1093–1120 (1999) CrossRefGoogle Scholar
  6. 29.
    Crampin, E.J., Gaffney, E.A., Maini, P.K.: Mode doubling and tripling in reaction-diffusion patterns on growing domains: a piece-wise linear model. J. Math. Biol. 44, 107–128 (2002) MathSciNetCrossRefMATHGoogle Scholar
  7. 38.
    de Kepper, P., Castets, V., Dulos, E., Boissonade, J.: Turing-type chemical pattern in the chlorite-iodide-malonic acid reaction. Physica D 49, 161–169 (1991) CrossRefGoogle Scholar
  8. 40.
    del Pino, M., Wei, J.: Collapsing steady states of the Keller-Segel system. Nonlinearity 19, 661–684 (2006) MathSciNetCrossRefMATHGoogle Scholar
  9. 53.
    Dufiet, V., Boissonade, J.: Conventional and unconventional Turing patterns. J. Chem. Phys. 96, 664–673 (1992) CrossRefGoogle Scholar
  10. 54.
    Edelstein-Keshet, L.: Mathematical Models in Biology. SIAM Classics in Applied Mathematics, vol. 46. Society for Industrial and Applied Mathematics, Philadelphia (2005) CrossRefMATHGoogle Scholar
  11. 55.
    Efendiev, M., Yagi, A.: Continuous dependence on a parameter of exponential attractors for chemotaxis-growth system. J. Math. Soc. Jpn. 57, 167–181 (2005) MathSciNetCrossRefMATHGoogle Scholar
  12. 65.
    Fife, P.C.: Stationary patterns for reaction-diffusion systems. In: Nonlinear Diffusion. Research Notes in Mathematics, vol. 14, pp. 81–121. Pitman, London (1977) Google Scholar
  13. 66.
    Fife, P.C.: Large time behaviour of solutions of bistable nonlinear diffusion equations. Arch. Ration. Mech. Anal. 70, 31–46 (1979) MathSciNetCrossRefMATHGoogle Scholar
  14. 72.
    Gierer, A.: The Hydra model—a model for what? Int. J. Dev. Biol. 56, 437–445 (2012) CrossRefGoogle Scholar
  15. 73.
    Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik (Berlin) 12, 30–39 (1972) CrossRefGoogle Scholar
  16. 82.
    Hadeler, K.P., Rothe, F.: Travelling fronts in nonlinear diffusion equations. J. Math. Biol. 2, 251–263 (1975) MathSciNetCrossRefMATHGoogle Scholar
  17. 83.
    Haken, H.: Synergetics, an Introduction: Nonequilibrium Phase Transitions and Self-organization in Physics, Chemistry, and Biology, 3rd rev. enl. edn. Springer, New York (1983) CrossRefMATHGoogle Scholar
  18. 85.
    Hale, J.K., Peletier, L.A., Troy, W.C.: Stability and instability of the Gray-Scott model: the case of equal diffusion constants. Appl. Math. Lett. 12, 59–65 (1999) MathSciNetCrossRefMATHGoogle Scholar
  19. 86.
    Hale, J.K., Peletier, L.A., Troy, W.C.: Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis. SIAM J. Appl. Math. 61, 102–130 (2000) MathSciNetCrossRefMATHGoogle Scholar
  20. 88.
    Harland, R., Gerhard, J.: Formation and function of Spemann’s organizer. Annu. Rev. Cell Dev. Biol. 13, 661–667 (1997) CrossRefGoogle Scholar
  21. 91.
    Hillen, T., Painter, K.J.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58, 183–217 (2009) MathSciNetCrossRefMATHGoogle Scholar
  22. 93.
    Horstmann, D., Stevens, A.: A constructive approach to traveling waves in chemotaxis. J. Nonlinear Sci. 14, 1–25 (2004) MathSciNetCrossRefMATHGoogle Scholar
  23. 94.
    Horstmann, D., Winkler, M.: Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equ. 215, 52–107 (2005) MathSciNetCrossRefMATHGoogle Scholar
  24. 97.
    Ingham, P.W.: The molecular genetics of embryonic pattern formation in Drosophila. Nature 335, 25–34 (1988) CrossRefGoogle Scholar
  25. 102.
    Jäger, W., Luckhaus, S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc. 329, 819–824 (1992) MATHGoogle Scholar
  26. 108.
    Keller, K.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970) CrossRefMATHGoogle Scholar
  27. 119.
    Kondo, S., Asai, R.: A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus. Nature 376, 765–768 (1995) CrossRefGoogle Scholar
  28. 120.
    Kulesa, P.M., Cruywagen, G.C., Lubkin, S.R., Maini, P.K., Sneyd, J.S., Murray, J.D.: Modelling the spatial patterning of the teeth primordia in the lower jaw of Alligator mississippiensis. J. Biol. Syst. 3, 975–985 (1995) CrossRefGoogle Scholar
  29. 121.
    Kulesa, P.M., Cruywagen, G.C., Lubkin, S.R., Maini, P.K., Sneyd, J., Ferguson, M.W.J., Murray, J.D.: On a model mechanism for the spatial patterning of teeth primordia in the alligator. J. Theor. Biol. 180, 287–296 (1996) CrossRefGoogle Scholar
  30. 128.
    Lee, K.J., McCormick, W.D., Ouyang, Q., Swinney, H.L.: Pattern formation by interacting chemical fronts. Science 261, 192–194 (1993) CrossRefGoogle Scholar
  31. 129.
    Lee, K.J., McCormick, W.D., Pearson, J.E., Swinney, H.L.: Experimental observation of self-replicating spots in a reaction-diffusion system. Nature 369, 215–218 (1994) CrossRefGoogle Scholar
  32. 130.
    Lengyel, I., Epstein, I.R.: Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system. Science 251, 650–652 (1991) CrossRefGoogle Scholar
  33. 131.
    Levin, S.A.: The problem of pattern and scale in ecology. Ecology 73, 1943–1967 (1992) CrossRefGoogle Scholar
  34. 138.
    Madzvamuse, A., Wathen, A.J., Maini, P.K.: A moving grid finite element method applied to a model biological pattern generator. J. Comput. Phys. 190, 478–500 (2003) MathSciNetCrossRefMATHGoogle Scholar
  35. 139.
    Madzvamuse, A., Maini, P.K., Wathen, A.J.: A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains. J. Sci. Comput. 24, 247–262 (2005) MathSciNetCrossRefMATHGoogle Scholar
  36. 140.
    Maini, P.K.: Turing patterns in fish skin? Nature 380, 678 (1996) Google Scholar
  37. 142.
    Maini, P.K., Painter, K.J., Chau, H.: Spatial pattern formation in chemical and biological systems. J. Chem. Soc. Faraday Trans. 93, 3601–3610 (1997) CrossRefGoogle Scholar
  38. 143.
    Maini, P.K., Baker, R.E., Chuong, C.M.: The Turing model comes of molecular age. Science 314, 1397–1398 (2006) CrossRefGoogle Scholar
  39. 145.
    Meinhardt, H.: Models of Biological Pattern Formation. Academic Press, London (1982) Google Scholar
  40. 146.
    Meinhardt, H.: Hierarchical inductions of cell states: a model for segmentation in Drosophila. J. Cell Sci., Suppl. 4, 357–381 (1986) CrossRefGoogle Scholar
  41. 147.
    Meinhardt, H.: A model for pattern-formation of hypostome, tentacles, and foot in hydra: how to form structures close to each other, how to form them at a distance. Dev. Biol. 157, 321–333 (1993) CrossRefGoogle Scholar
  42. 148.
    Meinhardt, H.: Growth and patterning—dynamics of stripe formation. Nature 376, 722–723 (1995) CrossRefGoogle Scholar
  43. 149.
    Meinhardt, H.: Organizer and axes formation as a self-organizing process. Int. J. Dev. Biol. 45, 177–188 (2001) Google Scholar
  44. 150.
    Meinhardt, H.: Primary body axes of vertebrates: generation of a near-Cartesian coordinate system and the role of Spemann-type organizer. Dev. Dyn. 235, 2907–2919 (2006) CrossRefGoogle Scholar
  45. 151.
    Meinhardt, H.: Models of biological pattern formation: from elementary steps to the organization of embryonic axes. Curr. Top. Dev. Biol. 81, 1–63 (2008) CrossRefGoogle Scholar
  46. 152.
    Meinhardt, H.: The Algorithmic Beauty of Sea Shells, 4th edn. Springer, Berlin (2009) CrossRefGoogle Scholar
  47. 153.
    Meinhardt, H.: Modeling pattern formation in hydra—a route to understanding essential steps in development. Int. J. Dev. Biol. 56, 447–462 (2012) CrossRefGoogle Scholar
  48. 154.
    Meinhardt, H., Gierer, A.: Generation and regeneration of sequences of structures during morphogenesis. J. Theor. Biol. 85, 429–450 (1980) MathSciNetCrossRefGoogle Scholar
  49. 160.
    Mimura, M., Tsujikawa, T.: Aggregating pattern dynamics in a chemotaxis model including growth. Physica A 230, 499–543 (1996) CrossRefGoogle Scholar
  50. 161.
    Moos, M., Wang, S.W., Krinks, M.: Anti-dorsalizing morphogenetic protein is a novel tgf-beta homolog expressed in the Spemann organizer. Development 121, 4293–4301 (1995) Google Scholar
  51. 165.
    Murray, J.D.: Mathematical Biology I: An Introduction, 3rd edn. Interdisciplinary Applied Mathematics, vol. 17. Springer, Berlin (2002) Google Scholar
  52. 166.
    Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd edn. Interdisciplinary Applied Mathematics, vol. 18. Springer, Berlin (2003) Google Scholar
  53. 169.
    Ni, W.-M.: The Mathematics of Diffusion. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 82. Society for Industrial and Applied Mathematics, Philadelphia (2011) CrossRefMATHGoogle Scholar
  54. 180.
    Nicolis, G., Prigogine, I.: Self-organization in Non-equilibrium Systems. Wiley, New York (1977) Google Scholar
  55. 188.
    Nüsslein-Volhard, C.: Determination of the embryonic axes of Drosophila. Development 1(Suppl.), 1–10 (1991) Google Scholar
  56. 189.
    Nüsslein-Volhard, C.: Coming to Life: How Genes Drive Development. Yale University Press, New Haven (2006) Google Scholar
  57. 190.
    Nüsslein-Volhard, C., Wieschaus, E.: Mutations affecting segment number and polarity in Drosophila. Nature 287, 795–801 (1980) CrossRefGoogle Scholar
  58. 194.
    Ouyang, Q., Swinney, H.L.: Transition from a uniform state to hexagonal and striped Turing patterns. Nature 352, 610–612 (1991) CrossRefGoogle Scholar
  59. 195.
    Ouyang, Q., Swinney, H.L.: Transition to chemical turbulence. Chaos 1, 411–420 (1991) CrossRefMATHGoogle Scholar
  60. 198.
    Painter, K.J., Maini, P.K., Othmer, H.G.: Stripe formation in juvenile Pomacanthus explained by a generalised Turing mechanism with chemotaxis. Proc. Natl. Acad. Sci. USA 96, 5549–5554 (1999) CrossRefGoogle Scholar
  61. 202.
    Pankratz, M.J., Jäckle, H.: Making stripes in the Drosophila embryo. Trends Genet. 6, 287–292 (1990) CrossRefGoogle Scholar
  62. 204.
    Pearson, J.E., Horsthemke, W.: Turing instabilities with nearly equal diffusion constants. J. Chem. Phys. 90, 1588–1599 (1989) MathSciNetCrossRefGoogle Scholar
  63. 207.
    Reynolds, J., Pearson, J., Ponce-Dawson, S.: Dynamics of self-replicating patterns in reaction diffusion systems. Phys. Rev. Lett. 72, 2797–2800 (1994) CrossRefGoogle Scholar
  64. 208.
    Reynolds, J., Pearson, J., Ponce-Dawson, S.: Dynamics of self-replicating spots in reaction-diffusion systems. Phys. Rev. E 56, 185–198 (1997) MathSciNetCrossRefGoogle Scholar
  65. 215.
    Segel, L.A., Levin, S.A.: Applications of nonlinear stability theory to the study of the effects of dispersion on predator-prey interactions. In: Piccirelli, R. (ed.) Selected Topics in Statistical Mechanics and Biophysics. Conference Proceedings, vol. 27. American Inst. Physics, New York (1976) Google Scholar
  66. 216.
    Sick, S., Reinker, S., Timmer, J., Schlake, T.: WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism. Science 314, 1447–1450 (2006) CrossRefGoogle Scholar
  67. 218.
    Sleeman, B.D., Ward, M.J., Wei, J.: The existence and stability of spike patterns in a chemotaxis model. SIAM J. Appl. Math. 65, 790–817 (2005) MathSciNetCrossRefMATHGoogle Scholar
  68. 219.
    Spemann, H.: Embryonic Development and Induction. Yale University Press, New Haven (1938) Google Scholar
  69. 220.
    Spemann, H., Mangold, H.: Über Induktion von Embryonalanlagen durch Implantation artfremder Organisatoren. Wilhelm Roux’ Arch. Entwicklungsmech. Org. 100, 599–638 (1924) Google Scholar
  70. 221.
    Stevens, A.: The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J. Appl. Math. 61, 183–212 (2000) MathSciNetCrossRefMATHGoogle Scholar
  71. 229.
    Trembley, A.: Mémoires pour servir à l’histoire d’un genre de polypes d’eau douce, à bras en forme de cornes. Jean & Herman Verbeek, Leiden (1744) Google Scholar
  72. 232.
    Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 237, 37–72 (1952) CrossRefGoogle Scholar
  73. 234.
    Vastano, J.A., Pearson, J.E., Horsthemke, W., Swinney, H.L.: Chemical pattern formation with equal diffusion coefficients. Phys. Lett. A 124, 320–324 (1987) CrossRefGoogle Scholar
  74. 235.
    Vastano, J.A., Pearson, J.E., Horsthemke, W., Swinney, H.L.: Turing patterns in an open reactor. J. Chem. Phys. 88, 6175–6181 (1988) CrossRefGoogle Scholar
  75. 237.
    Wang, Z.-A., Hillen, T.: Classical solutions and pattern formation for a volume filling chemotaxis model. Chaos 17, 037108 (2007) MathSciNetCrossRefGoogle Scholar
  76. 239.
    Ward, M.J.: Asymptotic methods for reaction-diffusion systems: past and present. Bull. Math. Biol. 68, 1151–1167 (2006) CrossRefGoogle Scholar
  77. 284.
    Wilby, O.K., Webster, G.: Studies on the transmission of hypostome inhibition in hydra. J. Embryol. Exp. Morphol. 24, 583–593 (1970) Google Scholar
  78. 285.
    Wilby, O.K., Webster, G.: Experimental studies on axial polarity in hydra. J. Embryol. Exp. Morphol. 24, 595–613 (1970) Google Scholar
  79. 288.
    Wolpert, L.: Positional information and pattern regulation in regeneration of hydra. Symp. Soc. Exp. Biol. 25, 391–415 (1971) Google Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  • Juncheng Wei
    • 1
  • Matthias Winter
    • 2
  1. 1.Department of MathematicsThe Chinese University of Hong KongHong KongChina
  2. 2.Department of Mathematical SciencesBrunel UniversityUxbridgeUK

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