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Spatial patterns for an interaction-diffusion equation in morphogenesis.

M A Mimura, Y Nishiura

    Journal of Mathematical Biology
    |April 18, 1979
    PubMed
    Summary
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    Slight tissue asymmetry in a Gierer-Meinhardt model generates stable patterns from uniform morphogen distribution. This study uses perturbed bifurcation theory to analyze pattern formation in morphogenesis.

    Area of Science:

    • Developmental Biology
    • Mathematical Biology
    • Theoretical Physics

    Background:

    • Morphogenesis involves pattern formation driven by interactions and diffusion.
    • The Gierer-Meinhardt model describes activator-inhibitor systems in biological development.
    • Turing's theory of diffusion-driven instability is foundational to understanding pattern formation.

    Purpose of the Study:

    • To analyze pattern formation in a Gierer-Meinhardt interaction-diffusion model.
    • To investigate the impact of initial asymmetry on pattern stability.
    • To explore the mathematical conditions for steady-state solutions.

    Main Methods:

    • Perturbed bifurcation theory was employed.
    • Analysis of an interaction-diffusion equation from morphogenesis.

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  • Investigation of steady-state solutions for parameter ranges.

    • Main Results:

    • Slightly asymmetric initial conditions lead to stable, striking patterns.
    • Pattern formation is dependent on the degree of asymmetry.
    • The study discusses global existence of steady-state solutions.

    Conclusions:

    • Asymmetry is a key factor in generating robust patterns in morphogenesis.
    • The Gierer-Meinhardt model, analyzed via bifurcation theory, captures essential aspects of pattern emergence.
    • Mathematical analysis provides insights into the conditions for stable biological patterns.