The Linear Stability of Symmetric Spike Patterns for a Bulk-Membrane Coupled Gierer-Meinhardt Model

We analyze a coupled bulk-membrane PDE model in which a scalar linear 2-D bulk diffusion process is coupled through a linear Robin boundary condition to a two-component 1-D reaction-diffusion (RD) system with Gierer--Meinhardt (nonlinear) reaction kinetics defined on the domain boundary. For this coupled model, in the singularly perturbed limit of a long-range inhibition and short-range activation for the membrane-bound species, asymptotic methods are used to analyze the existence of localized steady-state multispike membrane-bound patterns and to derive a nonlocal eigenvalue problem (NLEP) characterizing the O (1) time-scale instabilities of these patterns. A central, and novel, feature of this NLEP is that it involves a membrane Green's function that is coupled nonlocally to a bulk Green's function. When the domain is a disk, or in the well-mixed shadow-system limit corresponding to an infinite bulk diffusivity, this Green's function problem is analytically tractable, and as a result we will use a hybrid analytical-numerical approach to determine unstable spectra of this NLEP. This analysis characterizes how the 2-D bulk diffusion process and the bulk-membrane coupling modify the well-known linear stability properties of steady-state spike patterns for the 1-D Gierer-Meinhardt model in the absence of coupling. In particular, phase diagrams in parameter space for our coupled model characterizing either oscillatory instabilities due to Hopf bifurcations or competition instabilities due to zero-eigenvalue crossings are constructed. Finally, linear stability predictions from the NLEP analysis are confirmed with full numerical finite element simulations of the coupled PDE system. We remark that our approach is valid in more general settings than the disk or the well-mixed shadow system, with the key hurdle being the computation of the relevant Green's functions. By restricting our detailed analysis to these two specialized cases, we can bypass the computational challenges of calculating the Green's functions and therefore focus instead on the novel effects of coupling on the construction and stability of multispike solutions.