Dissecting the snake: Transition from localized patterns to spike solutions

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 characterized through the behavior of the far field, where there is either an oscillatory or a monotonic decay. This transition is illustrated with two examples: a generalized Schnakenberg system that arises in cellular-level morphogenesis 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. The transition is caused by a so-called Belyakov–Devaney transition between complex and real spatial eigenvalues of the far 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 the examples.