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Abstract 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 strategies seem to require declining populations, as we show in our discussion.
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Higher-order dynamics in the replicator equation produce a limit cycle in rock-paper-scissors
Abstract Recent work has shown that pairwise interactions may not be sufficient to fully model ecological dynamics in the wild. In this letter, we consider a replicator dynamic that takes both pairwise and triadic interactions into consideration using a rank-three tensor. We study these new nonlinear dynamics using a generalized rock-paper-scissors game whose dynamics are well understood in the standard replicator sense. We show that the addition of higher-order dynamics leads to the creation of a subcritical Hopf bifurcation and consequently an unstable limit cycle. It is known that this kind of behaviour cannot occur in the pairwise replicator in any three-strategy games, showing the effect higher-order interactions can have on the resulting dynamics of the system. We numerically characterize parameter regimes in which limit cycles exist and discuss possible ways to generalize this approach to studying higher-order interactions.
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- Award ID(s):
- 1814876
- PAR ID:
- 10411022
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Europhysics Letters
- Volume:
- 142
- Issue:
- 3
- ISSN:
- 0295-5075
- Page Range / eLocation ID:
- Article No. 33001
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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