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

TL;DR

The paper investigates a three-component Gierer–Meinhardt reaction–diffusion system in the semi-strong regime with $ extcolor{purple}{ abla1}$, where a nontrivial background $a>0$ enables spike nucleation and rich spike dynamics. By combining matched asymptotic analysis with numerical path-following, it constructs localized spike equilibria, derives a nonlinear outer problem that governs spike nucleation thresholds (e.g., $D_{nuc}$), and reveals two distinct oscillatory instabilities: amplitude oscillations driven by large-eigenvalue instabilities and oscillatory spike motion driven by small-eigenvalue instabilities, both mediated by the additional component $w$ and by the time-scaling parameters $ heta$ and $ au$. The stability analysis via nonlocal eigenvalue problems shows that the upper and lower spike branches respond differently to parameter changes, with Hopf bifurcations occurring under certain regimes, and full simulations confirm the predicted transition scenarios, including boundary nucleation and interstitial spike creation. Overall, the work demonstrates that adding a third component to the GM model yields significantly richer spike dynamics and opens questions about the spectral properties of the resulting NLEP and its extension to more complex geometries and multi-spike configurations.

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.

Figures (15)

• Figure 1: Time-dependent PDE simulations of \ref{['Gm']} using FlexPDEflexpde2015 illustrating spike nucleation behavior as the domain half-length $L$ slowly increases as $D_v = 2-1.5*10^{-4}t$. The transitions where boundary spikes first emerge and then later when spikes are nucleated between the interior and boundary spikes. Parameters: $\delta_1 = 0.03^2, \delta_2=\delta_1^2, \theta = \tau =0$, $a =0.7, b=1, c=1$ and $l=4$.
• Figure 2: Full simulations by Flexpde flexpde2015 illustrating oscillatory behavior in spike amplitudes and spike motion. (a) Oscillations in spike amplitude triggered by a large-eigenvalue instability as $\theta=1.5$, which is above the critical value $\theta_h= 1.34$, the oscillations eventually destroy the spike structure. (b) The amplitude of $u(x)$ versus time for the interior spike centered at $x = 0$. Other parameters are: $\delta_1=0.01^2, a=0.01, b=1, c=1, l=1, D_v=1, \tau=0.$
• Figure 3: Full simulations by Flexpde illustrating spike motions for different values of $\tau>\tau_h=1.18$; (a) For $\tau=1.2$, the interior spike exhibits small oscillations. (b) For $\tau=1.5$, the spike undergoes larger excursions and eventually drifts toward the boundary. Other parameters are: $\delta_1=0.01^2, D_v=1, a=0.01, b=1, c=1, l=1,\theta=0.$
• Figure 4: Comparison between analytical and numerical results for $V_0$ as $D_v$ is varied. The solid curve is the analytical result by solving the coupled nonlinear system (\ref{['sec2_Newton_1']}) and (\ref{['sec2_implicit_mu']}) using Newton’s method. The red stars are obtained by full simulations of the GM model \ref{['Gm']} using pde2pathpde2path. Parameters: $\delta_1 = 0.01^2,a=1, b=3, c=1$ and $\ell = 3$.
• Figure 5: Comparison between asymptotic and numerical results for the nucleation threshold $D_{nuc}$ versus $a$. The solid curves are the asymptotic results given in \ref{['nuc_threshold']}. The red stars are the numerical results for the saddle-node bifurcation point as computed from \ref{['Gm']} using pde2pathpde2path. Parameters: $\delta_1 = 0.01^2$, $l=4$, $b=1$ and $c = 1$.

Theorems & Definitions (6)

• Lemma 1
• proof
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4