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arXiv · 2510.01401

Localized Pattern Formation and Oscillatory Instabilities in a Three-component Gierer Meinhardt Model

Chunyi Gai·Fahad Al Saadi
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Abstract

In this paper, we introduce a three-component Gierer-Meinhardt model in the semi-strong interaction regime, characterized by an asymptotically large diffusivity ratio. A key feature of this model is that the interior spike can undergo Hopf bifurcations in both amplitude and position, leading to rich oscillatory dynamics not present in classical two-component systems. Using asymptotic analysis and numerical path-following, we construct localized spike equilibria and analyze spike nucleation that occurs through slow passage beyond a saddle-node bifurcation. Moreover, stability of spike equilibrium is analyzed by introducing time-scaling parameters, which reveal two distinct mechanisms: amplitude oscillations triggered by large-eigenvalue instabilities and oscillatory spike motion associated with small eigenvalues. Numerical simulations illustrate these dynamics and their transition regimes. This dual mechanism highlights richer spike behavior in three-component systems and suggests several open problems for future study.

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Chunyi Gai, Fahad Al Saadi. 2025-10-01. Localized Pattern Formation and Oscillatory Instabilities in a Three-component Gierer Meinhardt Model. https://arxiv.org/abs/2510.01401

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