Dynamically Generated Hierarchies in Games of Competition and the Role of Stochastic Fluctuations

• Applicant: Professorin Dr. Hildegard Meyer-Ortmanns
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: from 2015 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 290307074

• Final Report Year: 2019

Final Report Abstract

We succeeded in the design of predation rules between nine species such that the species organize themselves into three clusters of three species each. Rock-paper-scissors is then played on two scales, between the clusters and within the clusters. This provides an example of nested, self-similar dynamics of winnerless competition. Assigned to a two-dimensional grid this leads to the formation of nested spirals. Of particular interest are the interfaces between the large spiral arms of different clusters. We numerically measured the growth of these spirals, the width of these interfaces and their fluctuations. We obtain critical exponents for the growth and its saturation, which correspond to values obtained for curvature driven coarsening processes that belong to the Edwards-Wilkinson universality class as long as the fluctuations are small. For large fluctuations, new exponents are found, not known from other universality classes. The characteristics and conditions for observing this type of universal behavior deserve further analysis. For a simpler version of a single 3-species rock-paper-scissors game we focused on rare large stochastic fluctuations which can kick the system out of a stable 3-species coexistence-fixed point towards 1-species saddles, so that the rare events amount to the extinction of two out of three species. The analytical approach is based on a WKB-ansatz for the solution of the master equation. It allows an estimate of the extinction time in units of the relaxation time. The value of these analytical estimates lies in the assessment of whether rare events are so rare that they can be safely ignored in a given period of time.

Publications

• Rock-paper-scissors played within competing domains in predator-prey games, J. Stat. Mech., 113402~1-21 (2016)
  D. Labavic and H. Meyer-Ortmanns
  
• Breaking of time-translation invariance in Kuramoto dynamics with multiple time scales, Europhys. Lett.118, 40006 (2017)
  S. Esmaeili, D. Labavic, M. Pleimling and H. Meyer-Ortmanns
  
• Phys. Rev. E 96, 012147 (2017)
  B. L. Brown and M. Pleimling
  
• Dynamically generated hierarchies in games of competition, Phys. Rev. E99, 062116 (2019)
  B. L. Brown, H. Meyer-Ortmanns and M. Pleimling