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Stable spike clusters for the precursor Gierer-Meinhardt system in .

Juncheng Wei1, Matthias Winter2, Wen Yang3

  • 11Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2 Canada.

Calculus of Variations and Partial Differential Equations
|February 4, 2020
PubMed
Summary

Researchers constructed stable Gierer-Meinhardt system spike clusters. These clusters, with varying numbers of spikes, demonstrate linear stability by balancing repulsive interactions and attraction to inhomogeneity minimums.

Keywords:
35B2535B3535B4092C15

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Area of Science:

  • Mathematical Biology
  • Pattern Formation
  • Reaction-Diffusion Systems

Background:

  • The Gierer-Meinhardt system is a fundamental model for pattern formation, particularly in biological contexts.
  • Understanding the stability of localized patterns (spikes) is crucial for explaining biological structures.
  • Previous studies have explored spike formation but stability analysis in specific parameter regimes remains key.

Purpose of the Study:

  • To construct and analyze the linear stability of spike clusters in the Gierer-Meinhardt system.
  • To investigate the influence of small inhibitor and activator diffusivities on spike cluster behavior.
  • To determine the conditions under which k-spike clusters can achieve linear stability.

Main Methods:

  • Analytical construction of k-spike cluster solutions for the Gierer-Meinhardt system.
  • Linear stability analysis of the constructed spike cluster equilibria.
  • Numerical exploration of spike configurations, including polygonal arrangements with and without a central spike.

Main Results:

  • Demonstrated the existence of linearly stable k-spike clusters for any positive integer k.
  • Showcased stable spike clusters located at vertices of polygons, with and without a central spike.
  • Identified stability limits: up to 3 spikes for clusters without a center, and up to 6 spikes for clusters with a center.

Conclusions:

  • Stable spike clusters can form in the Gierer-Meinhardt system under specific conditions of small diffusivities.
  • The balance between repulsive spike interactions (due to low inhibitor diffusivity) and attraction to precursor inhomogeneity minima drives stability.
  • These findings provide insights into the mechanisms underlying complex pattern formation in reaction-diffusion systems.