  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 multi-spike
membrane-bound patterns, and to derive a nonlocal eigenvalue problem (NLEP)
characterizing $\mathcal{O}(1)$ 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 modifies 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.
