Global-in-time behavior of the solution to a Gierer-Meinhardt system

Georgia Karali, Takashi Suzuki, Yoshio Yamada

    Research output: Contribution to journalArticle

    12 Citations (Scopus)

    Abstract

    Gierer-Meinhardt system is a mathematical model to describe biological pattern formation due to activator and inhibitor. Turing pattern is expected in the presence of local self-enhancement and long-range inhibition. The long-time behavior of the solution, however, has not yet been clarified mathematically. In this paper, we study the case when its ODE part takes periodic-in-time solutions, that is, τ = s+1/p-1 . Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits. Thus spatial patterns eventually disappear if those parameters are in a region without local self-enhancement or long-range inhibition.

    Original languageEnglish
    Pages (from-to)2885-2900
    Number of pages16
    JournalDiscrete and Continuous Dynamical Systems- Series A
    Volume33
    Issue number7
    DOIs
    Publication statusPublished - 2013 Jul

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    Keywords

    • Asymptotic behavior of the solution
    • Gierer-Meinhardt system
    • Hamilton structure
    • Reaction-diffusion equation
    • Turing pattern

    ASJC Scopus subject areas

    • Discrete Mathematics and Combinatorics
    • Applied Mathematics
    • Analysis

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