An Asymptotic Analysis of Localized Three-Dimensional Spot Patterns for the Gierer--Meinhardt Model: Existence, Linear Stability, and Slow Dynamics

Localized spot patterns, where one or more solution components concentrate at certain points in the domain, are a common class of localized pattern for reaction-diffusion systems, and they arise in a wide range of modeling scenarios. Although there is a rather well-developed theoretical understanding for this class of localized pattern in one and two space dimensions, a theoretical study of such patterns in a three-dimensional setting is, largely, a new frontier. In an arbitrary bounded three-dimensional domain, the existence, linear stability, and slow dynamics of localized multispot patterns are analyzed for the well-known singularly perturbed Gierer--Meinhardt activator-inhibitor system in the limit of a small activator diffusivity $\varepsilon^2\ll 1$. Our main focus is to classify the different types of multispot patterns and predict their linear stability properties for different asymptotic ranges of the inhibitor diffusivity $D$. For the range $D={\mathcal O}(\varepsilon^{-1})\gg 1$, although both symmetric and asymmetric quasi-equilibrium spot patterns can be constructed, the asymmetric patterns are shown to be always unstable. On this range of $D$, it is shown that symmetric spot patterns can undergo either competition instabilities or a Hopf bifurcation, leading to spot annihilation or temporal spot amplitude oscillations, respectively. For $D={\mathcal O}(1)$, only symmetric spot quasi-equilibria exist and they are linearly stable on ${\mathcal O}(1)$ time intervals. On this range, it is shown that the spot locations evolve slowly on an ${\mathcal O}(\varepsilon^{-3})$ time scale toward their equilibrium locations according to an ODE gradient flow, which is determined by a discrete energy involving the reduced-wave Green's function. The central role of the far-field behavior of a certain core problem, which characterizes the profile of a localized spot, for the construction of quasi-equilibria in the $D={\mathcal O}(1)$ and $D={\mathcal O}(\varepsilon^{-1})$ regimes, and in establishing some of their linear stability properties, is emphasized. Finally, for the range $D={\mathcal O}(\varepsilon^{2})$, it is shown that spot quasi-equilibria can undergo a peanut-splitting instability, which leads to a cascade of spot self-replication events. Predictions of the linear stability theory are all illustrated with full PDE numerical simulations of the Gierer--Meinhardt model.

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