Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games

Abstract: Biodiversity is essential to the viability of ecological systems. Species diversity in ecosystems is promoted by cyclic, non-hierarchical interactions among competing populations. Such non-transitive relations lead to an evolution with central features represented by the `rock-paper-scissors' game, where rock crushes scissors, scissors cut paper, and paper wraps rock. In combination with spatial dispersal of static populations, this type of competition results in the stable coexistence of all species and the long-term maintenance of biodiversity. However, population mobility is a central feature of real ecosystems: animals migrate, bacteria run and tumble. Here, we observe a critical influence of mobility on species diversity. When mobility exceeds a certain value, biodiversity is jeopardized and lost. In contrast, below this critical threshold all subpopulations coexist and an entanglement of travelling spiral waves forms in the course of temporal evolution. We establish that this phenomenon is robust, it does not depend on the details of cyclic competition or spatial environment. These findings have important implications for maintenance and evolution of ecological systems and are relevant for the formation and propagation of patterns in excitable media, such as chemical kinetics or epidemic outbreaks.

• The paper establishes that species coexistence in cyclic games hinges on a critical mobility threshold derived via spatially explicit simulations.
• The authors employ evolutionary game theory and stochastic modeling to reveal how spiral wave patterns emerge at low mobility, preserving biodiversity.
• Exceeding the critical mobility value disrupts spatial heterogeneity, leading to the extinction of two species and a collapse in biodiversity.

Mobility and Biodiversity in Cyclic Dominance Games

The paper by Reichenbach, Mobilia, and Frey explores the intricate dynamics of species coexistence within ecosystems governed by cyclic interactions, akin to the rock-paper-scissors game. The study foregrounds the role of spatial mobility in maintaining or disrupting biodiversity among competing populations. Specifically, it analyzes how mobility influences the stability of species coexistence, providing a robust framework for understanding biodiversity through a combination of evolutionary game theory and stochastic modeling.

Key Findings and Methodology

The central premise of this study is the critical role of mobility in ecological systems exhibiting cyclic dominance. The authors employ a spatially explicit model where three species—denoted as $A$, $B$, and $C$—interact following the rules of a modified rock-paper-scissors dynamic. Each species can outcompete another while being vulnerable to the third, forming a closed loop of competitive interactions. The model incorporates both local interactions (such as selection and reproduction) and global processes mediated by mobility, operationalized through the ability of species to move and colonize new territories.

Through extensive computational simulations and analytical derivations, the authors identify a critical threshold of mobility, denoted as $M_c$. Below this threshold, subpopulations coexist in dynamic patterns characterized by spiral waves, which enhance species diversity. Conversely, if mobility exceeds this critical value, the spatial heterogeneity diminishes, often leading to the extinction of two species and the dominance of one, thereby annihilating biodiversity.

Numerical and Empirical Insights

The authors report that for low mobility, the dynamic interactions form stable patterns that extend the temporal persistence of biodiversity. These patterns, described as traveling spiral waves, are indicative of stable coexistence and are robust against stochastic disturbances, a common feature in ecosystems. The work effectively bridges theoretical constructs with empirical observations, suggesting that similar patterns are observable in microbial communities, such as in cyclic dominance among strains of E.coli.

The model’s robustness is further validated through its alignment with complex systems theory, particularly by employing frameworks such as the complex Ginzburg-Landau equation to describe the spatiotemporal behavior of the system. This mathematical approach allows the derivation of the critical mobility threshold and provides insight into the scaling relations that govern pattern formation within these systems.

Implications and Future Directions

This research has profound implications for understanding biodiversity's resilience against environmental changes. It suggests that spatial structure and mobility are crucial factors in sustaining diverse ecological communities. The identification of a critical mobility threshold highlights the precarious balance within ecosystems—where both too little and too much mobility can significantly alter community dynamics.

The findings open avenues for further empirical validation and experimental design, such as manipulating mobility in controlled laboratory setups to observe microbial pattern formations, potentially informing conservation strategies and the management of biodiversity hotspots. Moreover, the theoretical framework can be extended to explore the broader implications of spatial dynamics in social and economic systems, where similar cyclic interactions may occur.

In sum, this paper provides a nuanced perspective on the interplay between movement and diversity in cyclic ecological systems, laying the groundwork for interdisciplinary applications in studying complex adaptive systems.