Effects of competition on pattern formation in the rock-paper-scissors game

TL;DR

This study explores how competition rate influences pattern formation and biodiversity in the rock-paper-scissors game, revealing critical thresholds for stable spirals and the role of noise and mobility in ecological dynamics.

Contribution

It provides a detailed analysis of the effects of competition rate on pattern stability and biodiversity, introducing a deterministic PDE model to reproduce observed phenomena.

Findings

Higher competition rates promote biodiversity but can disrupt spatial patterns.

A critical competition rate exists for stable spiral formation, increasing with system size and mobility.

Pattern disintegration is linked to percolation of vacant sites and noise effects.

Abstract

We investigate the impact of cyclic competition on pattern formation in the rock-paper-scissors game. By separately considering random and prepared initial conditions, we observe a critical influence of the competition rate $p$ on the stability of spiral waves and on the emergence of biodiversity. In particular, while increasing values of $p$ promote biodiversity, they may act detrimental on spatial pattern formation. For random initial conditions, we observe a phase transition from biodiversity to an absorbing phase, whereby the critical value of mobility grows linearly with increasing values of $p$ on a log-log scale, but then saturates as $p$ becomes large. For prepared initial conditions, we observe the formation of single-armed spirals, but only for values of $p$ that are below a critical value. Once above, the spirals break up and form disordered spatial structures, mainly because of the percolation of vacant sites. Thus, there exists a critical value of the competition rate $p_{c}$ for stable single-armed spirals in finite populations. Importantly though, $p_{c}$ increases with increasing system size, because noise reinforces the disintegration of ordered patterns. In addition, we also find that $p_{c}$ increases with the mobility. These phenomena are reproduced by a deterministic model that is based on nonlinear partial differential equations. Our findings indicate that competition is vital for the sustenance of biodiversity and emergence of pattern formation in ecosystems governed by cyclical interactions.