The Stability and Slow Dynamics of Localized Spot Patterns for the 3-D Schnakenberg Reaction-Diffusion Model

Abstract

On a bounded three-dimensional domain $\Omega$, a hybrid asymptotic-numerical method is employed to analyze the existence, linear stability, and slow dynamics of localized quasi-equilibrium multispot patterns of the Schnakenberg activator-inhibitor model with bulk feed-rate $A$ in the singularly perturbed limit of small diffusivity $\varepsilon^2$ of the activator component. By approximating each spot as a Coulomb singularity, a nonlinear system of equations is formulated for the strength of each spot. To leading order in $\varepsilon$, two types of solutions are identified: symmetric patterns for which all strengths are identical, and asymmetric patterns for which each strength takes on one of two distinct values. The $\mathcal{O}(\varepsilon)$ correction to the strengths is found to depend on the spatial configuration of the spots through a certain Neumann Green's matrix $\mathcal{G}$. When $\mathbf{e} = (1,\dots,1)^T$ is not an eigenvector of $\mathcal{G}$, a detailed numerical and (in the case of two spots) asymptotic characterization is performed for the resulting imperfection-sensitive bifurcation structure. For symmetric multispot patterns, a leading-order global threshold in terms of $|\Omega|$ and parameters of the Schnakenberg model is obtained, below which a competition instability is triggered leading to the annihilation of one or more spots. A corresponding refined threshold is established in terms of eigenvalues of $\mathcal{G}$ in the special case when $\mathcal{G}\mathbf{e} = k\mathbf{e}$. Additionally, a local self-replication threshold for the strength of each spot is derived numerically, above which a spot splits into two. By examining $\mathcal{O}(\varepsilon)$ corrections to spot strengths, a prediction is made as to which spot will be next to split as $A$ is slowly tuned. When the pattern is stable to $\mathcal{O}(1)$ instabilities, it is shown that the locations of spots in a quasi-equilibrium configuration evolve on a long $\mathcal{O}(\varepsilon^{-3})$ time-scale according to an ODE system characterized by a gradient flow of a certain discrete energy $\mathcal{H}$, the minima of which define stable equilibrium points of the ODE. The theory also illustrates that new equilibrium points can be created when $A = A(\mathbf{x})$ is spatially variable, and that finite-time pinning away from minima of $\mathcal{H}$ can occur when $A(\mathbf{x})$ is localized. The theory for linear stability and slow dynamics when $\Omega$ is the unit ball are compared favorably to numerical solutions of the Schnakenberg PDE.