  The ecological invasion problem in which a weaker exotic species invades an
ecosystem inhabited by two strongly competing native species is modelled by a
three-species competition-diffusion system. It is known that for a certain
range of parameter values competitor-mediated coexistence occurs and complex
spatio-temporal patterns are observed in two spatial dimensions. In this paper
we uncover the mechanism which generates such patterns. Under some assumptions
on the parameters the three-species competition-diffusion system admits two
planarly stable travelling waves. Their interaction in one spatial dimension
may result in either reflection or merging into a single homoclinic wave,
depending on the strength of the invading species. This transition can be
understood by studying the bifurcation structure of the homoclinic wave. In
particular, a time-periodic homoclinic wave (breathing wave) is born from a
Hopf bifurcation and its unstable branch acts as a separator between the
reflection and merging regimes. The same transition occurs in two spatial
dimensions: the stable regular spiral associated to the homoclinic wave
destabilizes, giving rise first to an oscillating breathing spiral and then
breaking up producing a dynamic pattern characterized by many spiral cores. We
find that these complex patterns are generated by the interaction of two
planarly stable travelling waves, in contrast with many other well known cases
of pattern formation where planar instability plays a central role.
