  We show bifurcation of localized spike solutions from spatially constant
states in systems of nonlocally coupled equations in the whole space. The main
assumptions are a generic bifurcation of saddle-node or transcritical type for
spatially constant profiles, and a symmetry and second moment condition on the
convolution kernel. The results extend well known results for spots, spikes,
and fronts, in locally coupled systems on the real line, and for radially
symmetric profiles in higher space dimensions. Rather than relying on center
manifolds, we pursue a more direct approach, deriving leading order asymptotics
and Newton corrections for error terms. The key ingredient is smoothness of
Fourier multipliers arising from discrepancies between nonlocal operators and
their local long-wavelength approximations.
