The Stability and Slow Dynamics of Localized Spot Patterns for the 3-D Schnakenberg Reaction-Diffusion Model
J. C. Tzou, S. Xie, T. Kolokolnikov, M. J. Ward
SIAM Journal on Applied Dynamical Systems
On a bounded three-dimensional domain Ω, 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 ε 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
... der in ε, 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 O(ε) correction to the strengths is found to depend on the spatial configuration of the spots through a certain Neumann Green's matrix G. When e = (1, . . . , 1) T is not an eigenvector of 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 leadingorder global threshold in terms of |Ω| 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 G in the special case when Ge = ke. Additionally, a local self-replication threshold for the strength of each spot is derived numerically, above which a spot splits into two. By examining O(ε) 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 O(1) instabilities, it is shown that the locations of spots in a quasi-equilibrium configuration evolve on a long O(ε −3 ) time-scale according to an ODE system characterized by a gradient flow of a certain discrete energy 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(x) is spatially variable, and that finite-time pinning away from minima of H can occur when A(x) is localized. The theory for linear stability and slow dynamics when Ω is the unit ball are compared favorably to numerical solutions of the Schnakenberg PDE.