  We study the linear stability properties of spatially localized single- and
multi-peak states generated in a subcritical Turing bifurcation in the
Meinhardt model of branching. In one spatial dimension, these states are
organized in a foliated snaking structure owing to peak-peak repulsion but are
shown to be all linearly unstable, with the number of unstable modes increasing
with the number of peaks present. Despite this, in two spatial dimensions
direct numerical simulations reveal the presence of stable single- and
multi-spot states whose properties depend on the repulsion from nearby spots as
well as the shape of the domain and the boundary conditions imposed thereon.
Front propagation is shown to trigger the growth of new spots while
destabilizing others. The results indicate that multi-variable models may
support new types of behavior that are absent from typical two-variable models.
