  Species diversity in ecosystems is often accompanied by the self-organisation
of the population into fascinating spatio-temporal patterns. Here, we consider
a two-dimensional three-species population model and study the spiralling
patterns arising from the combined effects of generic cyclic dominance,
mutation, pair-exchange and hopping of the individuals. The dynamics is
characterised by nonlinear mobility and a Hopf bifurcation around which the
system's phase diagram is inferred from the underlying complex Ginzburg-Landau
equation derived using a perturbative multiscale expansion. While the dynamics
is generally characterised by spiralling patterns, we show that spiral waves
are stable in only one of the four phases. Furthermore, we characterise a phase
where nonlinearity leads to the annihilation of spirals and to the spatially
uniform dominance of each species in turn. Away from the Hopf bifurcation, when
the coexistence fixed point is unstable, the spiralling patterns are also
affected by nonlinear diffusion.
