Stabilizing a homoclinic stripe

Theodore Kolokolnikov, Michael Ward, Cheng Tzou, Juncheng Wei

    Research output: Contribution to journalArticleResearchpeer-review

    Abstract

    For a large class of reaction-diffusion systems with large diffusivity ratio, it is well known that a two-dimensional stripe (whose cross-section is a one-dimensional homoclinic spike) is unstable and breaks up into spots. Here, we study two effects that can stabilize such a homoclinic stripe. First, we consider the addition of anisotropy to the model. For the Schnakenberg model, we show that (an infinite) stripe can be stabilized if the fast-diffusing variable (substrate) is sufficiently anisotropic. Two types of instability thresholds are derived: zigzag (or bending) and break-up instabilities. The instability boundaries subdivide parameter space into three distinct zones: stable stripe, unstable stripe due to bending and unstable due to break-up instability. Numerical experiments indicate that the break-up instability is supercritical leading to a 'spotted-stripe' solution. Finally, we perform a similar analysis for the Klausmeier model of vegetation patterns on a steep hill, and examine transition from spots to stripes.
    LanguageEnglish
    Article number20180110
    Pages1-13
    Number of pages13
    JournalPhilosophical Transactions of the Royal Society A
    Volume376
    Issue number2135
    Early online date12 Nov 2018
    DOIs
    Publication statusPublished - 15 Nov 2018

    Fingerprint

    vegetation
    spikes
    diffusivity
    anisotropy
    thresholds
    cross sections

    Keywords

    • reaction–diffusion systems
    • pattern formation
    • stability of patterns

    Cite this

    Kolokolnikov, Theodore ; Ward, Michael ; Tzou, Cheng ; Wei, Juncheng. / Stabilizing a homoclinic stripe. In: Philosophical Transactions of the Royal Society A. 2018 ; Vol. 376, No. 2135. pp. 1-13.
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    abstract = "For a large class of reaction-diffusion systems with large diffusivity ratio, it is well known that a two-dimensional stripe (whose cross-section is a one-dimensional homoclinic spike) is unstable and breaks up into spots. Here, we study two effects that can stabilize such a homoclinic stripe. First, we consider the addition of anisotropy to the model. For the Schnakenberg model, we show that (an infinite) stripe can be stabilized if the fast-diffusing variable (substrate) is sufficiently anisotropic. Two types of instability thresholds are derived: zigzag (or bending) and break-up instabilities. The instability boundaries subdivide parameter space into three distinct zones: stable stripe, unstable stripe due to bending and unstable due to break-up instability. Numerical experiments indicate that the break-up instability is supercritical leading to a 'spotted-stripe' solution. Finally, we perform a similar analysis for the Klausmeier model of vegetation patterns on a steep hill, and examine transition from spots to stripes.",
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    Kolokolnikov, T, Ward, M, Tzou, C & Wei, J 2018, 'Stabilizing a homoclinic stripe', Philosophical Transactions of the Royal Society A, vol. 376, no. 2135, 20180110, pp. 1-13. https://doi.org/10.1098/rsta.2018.0110

    Stabilizing a homoclinic stripe. / Kolokolnikov, Theodore; Ward, Michael; Tzou, Cheng; Wei, Juncheng.

    In: Philosophical Transactions of the Royal Society A, Vol. 376, No. 2135, 20180110, 15.11.2018, p. 1-13.

    Research output: Contribution to journalArticleResearchpeer-review

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    AU - Kolokolnikov, Theodore

    AU - Ward, Michael

    AU - Tzou, Cheng

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