Zealots tame oscillations in the spatial rock-paper-scissors game

TL;DR

This paper demonstrates that the presence of zealots in the spatial rock-paper-scissors game effectively suppresses oscillations and promotes biodiversity, even under conditions that typically threaten coexistence due to mobility and randomness.

Contribution

It introduces the concept that zealots can stabilize biodiversity in spatial cyclic dominance models, regardless of the type and strength of mobility or randomness.

Findings

Zealots significantly reduce oscillations at 5% occupancy.

Even a tiny fraction of zealots stabilizes species coexistence.

Zealots help recover stable mean-field dynamics under mobility and randomness.

Abstract

The rock-paper-scissors game is a paradigmatic model for biodiversity, with applications ranging from microbial populations to human societies. Research has shown, however, that mobility jeopardizes biodiversity by promoting the formation of spiral waves, especially if there is no conservation law in place for the total number of competing players. Firstly, we show that even if such a conservation law applies, mobility still jeopardizes biodiversity in the spatial rock-paper-scissors game if only a small fraction of links of the square lattice is randomly rewired. Secondly, we show that zealots are very effective in taming the amplitude of oscillations that emerge due to mobility and/or interaction randomness, and this regardless of whether the later is quenched or annealed. While even a tiny fraction of zealots brings significant benefits, at 5\% occupancy zealots practically destroy all oscillations regardless of the intensity of mobility, and regardless of the type and strength of randomness in the interaction structure. Interestingly, by annealed randomness the impact of zealots is qualitatively the same as by mobility, which highlights that fast diffusion does not necessarily destroy the coexistence of species, and that zealotry thus helps to recover the stable mean-field solution. Our results strengthen the important role of zealots in models of cyclic dominance, and they reveal fascinating evolutionary outcomes in structured populations that are a unique consequence of such uncompromising behavior.