Stabilizing a homoclinic stripe

Theodore Kolokolnikov, Michael Ward, Cheng Tzou, Juncheng Wei

    Research output: Contribution to journalArticle

    3 Citations (Scopus)

    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.
    Original languageEnglish
    Article number20180110
    Pages (from-to)1-13
    Number of pages13
    JournalPhilosophical Transactions of the Royal Society A
    Volume376
    Issue number2135
    Early online date12 Nov 2018
    DOIs
    Publication statusPublished - 15 Nov 2018

    Keywords

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

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