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Spike pinning for the Gierer–Meinhardt model

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  • Iron, David
  • Ward, Michael J.

Abstract

The pinning effect induced by two different types of spatial inhomogeneities on the dynamics and equilibria of a one-spike solution to the one-dimensional Gierer–Meinhardt (GM) activator–inhibitor model of morphogenesis is studied. The first problem that is treated is the shadow problem that results from taking the infinite inhibitor diffusivity limit in the GM model. For this problem, we show that an exponentially weak spatially varying activator diffusivity can stabilize an equilibrium spike-layer solution that would necessarily be unstable when the activator diffusivity was spatially uniform. The second problem that is treated is the full GM model in the presence of a spatially varying inhibitor decay rate. For this problem, we show that the equilibrium location of a one-spike solution depends on certain global properties of the inhibitor decay rate over the domain.

Suggested Citation

  • Iron, David & Ward, Michael J., 2001. "Spike pinning for the Gierer–Meinhardt model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 55(4), pages 419-431.
    • Handle: RePEc:eee:matcom:v:55:y:2001:i:4:p:419-431
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    References listed on IDEAS

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    1. Holloway, David M. & Harrison, Lionel G., 1995. "Order and localization in reaction-diffusion pattern," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 222(1), pages 210-233.
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    Cited by:

    1. Mai, F.X. & Qin, L.J. & Zhang, G., 2012. "Turing instability for a semi-discrete Gierer–Meinhardt system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(5), pages 2014-2022.
    2. Wang, Jinliang & Li, You & Zhong, Shihong & Hou, Xiaojie, 2019. "Analysis of bifurcation, chaos and pattern formation in a discrete time and space Gierer Meinhardt system," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 1-17.

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