Journal of Nonlinear Science

, Volume 19, Issue 3, pp 301–339 | Cite as

Spikes for the Gierer–Meinhardt System with Discontinuous Diffusion Coefficients

  • Juncheng Wei
  • Matthias WinterEmail author


We rigorously prove results on spiky patterns for the Gierer–Meinhardt system (Kybernetik (Berlin) 12:30–39, 1972) with a jump discontinuity in the diffusion coefficient of the inhibitor. Using numerical computations in combination with a Turing-type instability analysis, this system has been investigated by Benson, Maini, and Sherratt (Math. Comput. Model. 17:29–34, 1993a; Bull. Math. Biol. 55:365–384, 1993b; IMA J. Math. Appl. Med. Biol. 9:197–213, 1992).

Firstly, we show the existence of an interior spike located away from the jump discontinuity, deriving a necessary condition for the position of the spike. In particular, we show that the spike is located in one-and-only-one of the two subintervals created by the jump discontinuity of the inhibitor diffusivity. This localization principle for a spike is a new effect which does not occur for homogeneous diffusion coefficients. Further, we show that this interior spike is stable.

Secondly, we establish the existence of a spike whose distance from the jump discontinuity is of the same order as its spatial extent. The existence of such a spike near the jump discontinuity is the second new effect presented in this paper.

To derive these new effects in a mathematically rigorous way, we use analytical tools like Liapunov–Schmidt reduction and nonlocal eigenvalue problems which have been developed in our previous work (J. Nonlinear Sci. 11:415–458, 2001).

Finally, we confirm our results by numerical computations for the dynamical behavior of the system. We observe a moving spike which converges to a stationary spike located in the interior of one of the subintervals or near the jump discontinuity.


Pattern formation Discontinuous diffusion coefficients Steady states Stability 

Mathematics Subject Classification (2000)

35B35 76E30 35B40 76E06 


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  1. Benson, D.L., Maini, P.K., Sherratt, J.A.: Analysis of pattern formation in reaction diffusion models with spatially inhomogeneous diffusion coefficients. Math. Comput. Model. 17, 29–34 (1993a) zbMATHCrossRefMathSciNetGoogle Scholar
  2. Benson, D.L., Sherratt, J.A., Maini, P.K.: Diffusion driven instability in an inhomogeneous domain. Bull. Math. Biol. 55, 365–384 (1993b) zbMATHGoogle Scholar
  3. Benson, D.L., Maini, P.K., Sherratt, J.A.: Unravelling the Turing bifurcation using spatially varying diffusion coefficients. J. Math. Biol. 37, 381–387 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  4. Blair, S.S.: Limb development: Marginal fringe benefits. Curr. Biol. 7, R686–R690 (1997) CrossRefGoogle Scholar
  5. Blair, S.S.: Developmental biology: Notching the hindbrain. Curr. Biol. 14, R570–R572 (2004) CrossRefGoogle Scholar
  6. Boozer, A.H.: Equations for studies of feedback stabilization. Phys. Plasmas 5, 3350 (1998) CrossRefMathSciNetGoogle Scholar
  7. Dancer, E.N.: On stability and Hopf bifurcations for chemotaxis systems. Methods Appl. Anal. 8, 245–256 (2001) zbMATHMathSciNetGoogle Scholar
  8. Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik (Berlin) 12, 30–39 (1972) CrossRefGoogle Scholar
  9. Iron, D., Ward, M.J., Wei, J.: The stability of spike solutions to the one-dimensional Gierer–Meinhardt model. Physica D 150, 25–62 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  10. Irvine, K.D., Rauskolb, C.: Boundaries in development: Formation and function. Annu. Rev. Cell Dev. Biol. 17, 189–214 (2001) CrossRefGoogle Scholar
  11. Maini, P.K., Benson, D.L., Sherratt, J.A.: Pattern formation in reaction-diffusion models with spatially inhomogeneous diffusion coefficients. IMA J. Math. Appl. Med. Biol. 9, 197–213 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  12. Meinhardt, H.: Cell determination boundaries as organizing regions for secondary embryonic fields. Dev. Biol. 96, 375–385 (1983) CrossRefGoogle Scholar
  13. Mercader, I., Net, M., Knobloch, E.: Binary fluid convection in a cylinder. Phys. Rev. E 51, 339–350 (1995) CrossRefGoogle Scholar
  14. Murray, J.: Mathematical Biology I/II. Springer, Berlin (2002) Google Scholar
  15. Nishiura, Y., Teramoto, T., Yuan, X., Ueda, K.-I.: Dynamics of traveling pulses in heterogeneous media. Chaos 17, 037104 (2007) CrossRefMathSciNetGoogle Scholar
  16. Page, K., Maini, P.K., Monk, N.A.M.: Pattern formation in spatially heterogeneous Turing reaction-diffusion models. Physica D 181, 80–101 (2003) zbMATHMathSciNetGoogle Scholar
  17. Page, K., Maini, P.K., Monk, N.A.M.: Complex pattern formation in reaction-diffusion systems with spatially varying parameters. Physica D 202, 95–115 (2005) zbMATHMathSciNetGoogle Scholar
  18. Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B 237, 37–72 (1952) CrossRefGoogle Scholar
  19. Ward, M.J., Wei, J.: Hopf bifurcations and oscillatory instabilities of solutions for the one-dimensional Gierer–Meinhardt model. J. Nonlinear Sci. 13, 209–264 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  20. Ward, M., McInerney, D., Houston, P., Gavaghan, D., Maini, P.: The dynamics and pinning of a spike for a reaction-diffusion system. SIAM J. Appl. Math. 62, 1297–1328 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  21. Wei, J.: On single interior spike solutions of Gierer–Meinhardt system: uniqueness and spectrum estimates. Eur. J. Appl. Math. 10, 353–378 (1999) zbMATHCrossRefGoogle Scholar
  22. Wei, J., Winter, M.: On the two-dimensional Gierer–Meinhardt system with strong coupling. SIAM J. Math. Anal. 30, 1241–1263 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  23. Wei, J., Winter, M.: Spikes for the two-dimensional Gierer–Meinhardt system: The strong coupling case. J. Differ. Equ. 178, 478–518 (2000) CrossRefMathSciNetGoogle Scholar
  24. Wei, J., Winter, M.: Spikes for the two-dimensional Gierer–Meinhardt system: The weak coupling case. J. Nonlinear Sci. 11, 415–458 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  25. Wei, J., Winter, M.: On the Gierer–Meinhardt system with precursors. Discrete Contin. Dyn. Syst. Ser. A (2009, to appear) Google Scholar
  26. Wolpert, L.: Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25, 1–47 (1969) CrossRefGoogle Scholar
  27. Wolpert, L., Hornbruch, A.: Double anterior chick limb buds and models for cartilage rudiment specification. Development 109, 961–966 (1990) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsThe Chinese University of Hong KongShatinHong Kong
  2. 2.Department of Mathematical SciencesBrunel UniversityUxbridgeUK

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