  We introduce and study the spatial replicator equation with higher order
interactions and both infinite (spatially homogeneous) populations and finite
(spatially inhomogeneous) populations. We show that in the special case of
three strategies (rock-paper-scissors) higher order interaction terms allow
travelling waves to emerge in non-declining finite populations. We show that
these travelling waves arise from diffusion stabilisation of an unstable
interior equilibrium point that is present in the aspatial dynamics. Based on
these observations and prior results, we offer two conjectures whose proofs
would fully generalise our results to all odd cyclic games, both with and
without higher order interactions, assuming a spatial replicator dynamic.
Intriguingly, these generalisations for $N \geq 5$ strategies seem to require
declining populations, as we show in our discussion.
