Cyclic dominance in evolutionary games: A review (1408.6828v1)

Abstract: Rock is wrapped by paper, paper is cut by scissors, and scissors are crushed by rock. This simple game is popular among children and adults to decide on trivial disputes that have no obvious winner, but cyclic dominance is also at the heart of predator-prey interactions, the mating strategy of side-blotched lizards, the overgrowth of marine sessile organisms, and the competition in microbial populations. Cyclical interactions also emerge spontaneously in evolutionary games entailing volunteering, reward, punishment, and in fact are common when the competing strategies are three or more regardless of the particularities of the game. Here we review recent advances on the rock-paper-scissors and related evolutionary games, focusing in particular on pattern formation, the impact of mobility, and the spontaneous emergence of cyclic dominance. We also review mean-field and zero-dimensional rock-paper-scissors models and the application of the complex Ginzburg-Landau equation, and we highlight the importance and usefulness of statistical physics for the successful study of large-scale ecological systems. Directions for future research, related for example to dynamical effects of coevolutionary rules and invasion reversals due to multi-point interactions, are outlined as well.

• The paper details how cyclic dominance in mean-field environments often fails to ensure coexistence without spatial or networked interactions.
• It reveals that structured populations generate spatial patterns like spirals and waves, where mobility thresholds crucially affect biodiversity.
• The paper explores extended models beyond RPS, showing how alliances and nonlinear dynamics stabilize complex ecological and social systems.

Cyclic Dominance in Evolutionary Games: An Overview

The paper of cyclic dominance in evolutionary games offers insightful perspectives into the complex dynamics that underpin natural and social ecosystems. The paper "Cyclic dominance in evolutionary games: A review," provides a thorough examination of the conceptual frameworks and empirical findings associated with cyclic interactions, particularly in the context of the classic rock-paper-scissors (RPS) model as well as its extensions to more complex systems.

The foundational RPS game exemplifies cyclic dominance through interactions where each of the three strategies, rock, paper, and scissors, cyclically outcompetes one another. This mechanism is mirrored in various biological and ecological systems, notably in predator-prey dynamics, microbial competition, and even human social interactions. The paper underscores that cyclic dominance is not limited to three-strategy systems, highlighting the spontaneous emergence of such dynamics in multi-strategy evolutionary games under certain conditions.

Main Findings and Methodological Approaches

1. Mean-Field and Well-Mixed Populations: In the context of mean-field system analyses, the paper discusses the destabilizing effect of fluctuations in cycling competitions. The presence of an intrinsic cyclic loop is often insufficient to guarantee coexistence without spatial structure or networked interactions. The authors outline the conditions under which stable limit cycles and heteroclinic orbits occur, as well as the role of mutations in extending species coexistence.
2. Structured Populations and Pattern Formation: In structured populations, spatial patterning like spirals and traveling waves emerge as robust features supporting coexistence. The paper emphasizes how mobility, topology of interaction networks, and the inherent spatial structure affect these patterns significantly. Increased mobility has a threshold beyond which it jeopardizes biodiversity, illustrating the delicate balance between movement and spatial structure.
3. Extensions Beyond the Classic RPS Model: Moving beyond the simplicity of the RPS model, the review explores games where cyclic dominance emerges spontaneously. Particularly, the authors explore social dilemmas and public goods games where volunteering, reward, and punishment introduce cyclical interactions. These findings portray how strategy complexity increases with added layers of decision-making behaviors, resulting in cyclic dynamics that stabilize cooperation or competition in a structured population.
4. Formation and Impact of Alliances: When more than three strategies are involved, alliances become pivotal. Defensive alliances, where groups of strategies collaborate to counteract external invaders, highlight how subsystem solutions can determine the evolutionary outcomes. The authors argue that these alliances can alter power dynamics, sometimes favoring alliance-driven coexistences over individual strategies.
5. Nonlinear Dynamics and Ginzburg-Landau Framework: The paper introduces the use of the complex Ginzburg-Landau equation (CGLE) as a model for understanding spatiotemporal dynamics in cyclic systems. The CGLE offers a lens into how pattern formation and oscillatory behaviors emerge from underlying competitive interactions.

Implications and Future Directions

The implications of this research span ecological management, where understanding cyclic dominance could inform biodiversity preservation strategies. Furthermore, in the field of complex networks and social systems, recognizing patterns of cyclic dominance can provide insights into maintaining diversity and cooperation in human societies.

Future research directions include expanding the paper of cyclic dominance to encompass group interactions in more diverse and realistic spatial structures. The ongoing challenge remains to refine these theoretical models to incorporate empirical data, thereby solidifying the link between observed ecological patterns and theoretical predictions.

In conclusion, cyclic dominance offers a profound framework for understanding the persistence and stability of complex systems through non-linear and often counterintuitive dynamics. The evolved discussions in this paper pave the way for more sophisticated analyses that could unravel the intricate balance of interactions in biological and social ecosystems.