Spike density distribution for the Gierer–Meinhardt model with precursor

We consider the Gierer–Meinhardt model in one dimension with a spatially-dependent precursor μ(x) which was proposed as a model of cell differentiation with control over pattern distribution in space. Assuming that the activator diffuses much slower than the inhibitor, such a system is well-known to admit solutions where the activator concentrates at N “spikes”. In the large-N limit, we derive the effective spike density for an arbitrary μ(x). We show that this density satisfies a first-order separable ODE. As a consequence, we derive instability thresholds for N spikes that correspond to a singularity in the ODE for the density. We recover, as a special case, the well-known stability thresholds for constant μ first derived in Iron et al. (2001), as well as cluster solutions that concentrate near the minimum of μ(x) that were recently discovered in Wei and Winter (2017). The main trick is applying Taylor expansions and geometric series to the equations of effective spike dynamics.