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Topological phases emerge in an ecological model

28 January 2021

An exotic phenomenon in condensed-matter systems illuminates the behavior of a one-dimensional model akin to the game rock-paper-scissors.

Topology explains some of the major discoveries in condensed-matter physics of the past 50 years, including the quantum Hall effect (see the article by Joseph Avron, Daniel Osadchy, and Ruedi Seiler, Physics Today, August 2003, page 38) and topological insulators (see Physics Today, April 2009, page 12). In such topological states, the material’s behavior derives from the topology of the band structure rather than the material’s symmetries, which explain most states of matter. Although topological phases are complicated to understand in detail, two essential features of their excitations are localization at the system’s boundary and robustness to defects. Now Erwin Frey and his colleagues at the Ludwig-Maximilians University Munich in Germany have identified the characteristics of topological phases in a nonlinear ecological model. The work points to the potential application of topology to other dynamic biological and ecological systems.

Credit: Enzoklop/CC BY-SA 3.0

The model Frey and his group investigated describes an element of game theory known as a rock-paper-scissors cycle. If you picture the three moves in the game rock-paper-scissors as the points of a triangle, as shown in the first figure, you can draw arrows to indicate which move wins. In a rock-paper-scissors cycle, the arrows instead become the rate at which mass moves in that direction—that is, the species represented by that point outcompetes another and becomes more numerous. With many of those cycles in a row, as shown below for S points, you have a toy model for population dynamics in an ecosystem.

Rock-paper-scissors chain.
Credit: J. Knebel, P. M. Geiger, E. Frey, Phys. Rev. Lett. 125, 258301 (2020)

The researchers numerically modeled rock-paper-scissors chains with different values of the so-called skewness r = r2/r3, in terms of two of the rates. For r = 1, no one species dominated, and the average mass stayed evenly distributed, as shown below by the diameter of the gray circles. For r < 1, the average mass gathered on the right, and for r > 1, the average mass gathered on the left. In other words, one species dominated the system, and the average mass became localized, although the system still oscillates. That behavior occurred regardless of how the mass was distributed initially and even when the researchers applied random perturbations to the rates in each triangle. In short, the behavior was robust. Those characteristics indicate the topological nature of the states, an observation that Frey and his students confirmed mathematically by deriving a topologically nontrivial band structure for the system.

Rock-paper-scissors cycles.
Credit: J. Knebel, P. M. Geiger, E. Frey, Phys. Rev. Lett. 125, 258301 (2020)

Such topological analysis extends to 2D networks. In fact, papers extending topology to 2D networks appeared on the arXiv preprint server a couple months after Frey’s paper was posted. Those works found chiral edge states, similar to the modes in 2D cold-atom lattices (see Physics Today, September 2020, page 14). In the future, topological concepts could potentially be introduced into any model composed of appropriately coupled oscillators. (J. Knebel, P. M. Geiger, E. Frey, Phys. Rev. Lett. 125, 258301, 2020.)

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