Localized Pattern Formation and Oscillatory Instabilities in a Three-component Gierer Meinhardt Model
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Gai, Chunyi Saadi, Fahad Al |
| author_facet | Gai, Chunyi Saadi, Fahad Al |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_01401 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Localized Pattern Formation and Oscillatory Instabilities in a Three-component Gierer Meinhardt Model Gai, Chunyi Saadi, Fahad Al Analysis of PDEs Dynamical Systems 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. |
| title | Localized Pattern Formation and Oscillatory Instabilities in a Three-component Gierer Meinhardt Model |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2510.01401 |