  An investigation is undertaken of coupled reaction-diffusion systems in one
spatial dimension that are able to support, in different regions of their
parameter space, either an isolated spike solution, or stable localized
patterns with an arbitrary number of peaks. The distinction between the two
cases is drawn through the behavior of the far field, where there is either an
oscillatory or a monotonic decay. Several examples are studied, including the
Lugiato-Lefever model, a generalized Schakenberg system that arises in
cellular-level morphogensis and a continuum model of urban crime spread. In
each, it is found that localized patterns connected via a so-called homoclinic
snaking curve in parameter space transition into a single spike solution as a
second parameter is varied, via a change in topology of the snake into a series
of disconnected branches. In each case, the transition is caused by a so-called
Belyakov-Devaney transition between complex and real spatial eigenvalues of the
fair field of the primary pulse. A codimension-two problem is studied in detail
where a non-transverse homoclinic orbit undergoes this transition. A
Shilnikov-style analysis is undertaken which reveals the asymptotics of how the
infinite family of folds of multi-pulse orbits are all destroyed at the same
parameter value. The results are shown to be consistent with numerical
experiments on two of the examples.
