Multi-spike Patterns in the Gierer–Meinhardt System with a Nonzero Activator Boundary Flux

The structure, linear stability, and dynamics of localized solutions to singularly perturbed reaction–diffusion equations have been the focus of numerous rigorous, asymptotic, and numerical studies in the last few decades. However, with a few exceptions, these studies have often assumed homogeneous boundary conditions. Motivated by the recent focus on the analysis of bulk-surface coupled problems, we consider the effect of inhomogeneous Neumann boundary conditions for the activator in the singularly perturbed one-dimensional Gierer–Meinhardt reaction–diffusion system. We show that these boundary conditions necessitate the formation of spikes that concentrate in a boundary layer near the domain boundaries. Using the method of matched asymptotic expansions, we construct boundary layer spikes and derive a class of shifted nonlocal eigenvalue problems analogous to those studied in Maini et al. (Chaos 17(3):037106, 2007) for which we rigorously prove partial stability results. Moreover, by using a combination of asymptotic, rigorous, and numerical methods we investigate the structure and linear stability of selected one- and two-spike patterns. In particular, we find that inhomogeneous Neumann boundary conditions increase both the range of parameter values over which asymmetric two-spike patterns exist and are stable.