Vortices determine the dynamics of biodiversity in cyclical interactions with protection spillovers

TL;DR

This study explores how protection spillovers in a modified rock-paper-scissors game influence biodiversity dynamics in structured populations, revealing complex spatial patterns driven by vortex density and randomness.

Contribution

It introduces the impact of protection spillovers on spatial and temporal patterns in structured populations, extending classical cyclic dominance models.

Findings

Vortices determine the system's spatiotemporal dynamics.

Protection spillovers lead to unique spatial patterns not seen in classical models.

Randomness influences wave propagation and synchronization.

Abstract

If rock beats scissors and scissors beat paper, one might assume that rock beats paper too. But this is not the case for intransitive relationships that make up the famous rock-paper-scissors game. However, the sole presence of paper might prevent rock from beating scissors, simply because paper beats rock. This is the blueprint for the rock-paper-scissors game with protection spillovers, which has recently been introduced as a new paradigm for biodiversity in well-mixed microbial populations. Here we study the game in structured populations, demonstrating that protection spillovers give rise to spatial patterns that are impossible to observe in the classical rock-paper-scissors game. We show that the spatiotemporal dynamics of the system is determined by the density of stable vortices, which may ultimately transform to frozen states, to propagating waves, or to target waves with reversed propagation direction, depending further on the degree and type of randomness in the interactions among the species. If vortices are rare, the fixation to waves and complex oscillatory solutions is likelier. Moreover, annealed randomness in interactions favors the emergence of target waves, while quenched randomness favors collective synchronization. Our results demonstrate that protection spillovers may fundamentally change the dynamics of cyclic dominance in structured populations, and they outline the possibility of programming pattern formation in microbial populations.