  Dynamical systems containing heteroclinic cycles and networks can be invoked
as models of intransitive competition between three or more species. When
populations are assumed to be well-mixed, a system of ordinary differential
equations (ODEs) describes the interaction model. Spatially extending these
equations with diffusion terms creates a system of partial differential
equations which captures both the spatial distribution and mobility of species.
In one spatial dimension, travelling wave solutions can be observed, which
correspond to periodic orbits in ODEs that describe the system in a
steady-state travelling frame of reference. These new ODEs also contain a
heteroclinic structure. For three species in cyclic competition, the topology
of the heteroclinic cycle in the well-mixed model is preserved in the
steady-state travelling frame of reference. We demonstrate that with four
species, the heteroclinic cycle which exists in the well-mixed system becomes a
heteroclinic network in the travelling frame of reference, with additional
heteroclinic orbits connecting equilibria not connected in the original cycle.
We find new types of travelling waves which are created in symmetry-breaking
bifurcations and destroyed in an orbit flip bifurcation with a cycle between
only two species. These new cycles explain the existence of "defensive
alliances" observed in previous numerical experiments. We further describe the
structure of the heteroclitic network for any number of species, and we
conjecture how these results may generalise to systems of any arbitrary number
of species in cyclic competition.
