- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Asymmetric cyclic interactions lead to stable mean-field equilibria while demographic noise induces sustained oscillations

A. L. L. Middleton, J. H. P. Dawes, and A. A. Jarai

K. M. Briggs

  • BT Research, Adastral Park, Martlesham Heath IP5 3RE, United Kingdom

PDF

Phys. Rev. E 114, 044211 – Published 16 October, 2026

DOI: https://doi.org/10.1103/9tjx-cw7q

Abstract

The dynamics of symmetric cyclically competing subpopulations is well known to lead to oscillatory dynamics. However, in some situations the competition is fundamentally asymmetric in that it is better modeled by binary interactions all of which involve the same subpopulation. Here we analyze a conceptual model of this second kind, motivated by design principles for communications networks where binary interactions inform decisions made by individual transmitters according to their competing operating policies, which may be to reduce overall network energy consumption, to improve coverage, or to limit interference between transmitters. We show that the mean-field dynamics in the asymmetric case always converges to equilibrium and cannot generate periodic oscillations. However, for a large but finite population of interacting agents, the intrinsic noise generated by the stochastic dynamics contains a dominant positive frequency component. We also analyze the robustness of the system to cyclically competing switching.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (44)

  1. T. Rogers and T. Gross, Consensus time and conformity in the adaptive voter model, Phys. Rev. E 88, 030102(R) (2013).
  2. H. Silk, G. Demirel, M. Homer, and T. Gross, Exploring the adaptive voter model dynamics with a mathematical triple jump, New J. Phys. 16, 093051 (2014).
  3. S. Redner, Reality-inspired voter models: A mini-review, C. R. Phys. 20, 275 (2019).
  4. A. Vendeville, Voter model can accurately predict individual opinions in online populations, Phys. Rev. E 111, 064310 (2025).
  5. R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29, 243 (1975).
  6. B. Sinervo and C. M. Lively, The rock-paper-scissors game and the evolution of alternative male strategies, Nature (London) 380, 240 (1996).
  7. B. Sinervo, B. Heulin, Y. Surget-Groba, J. Clobert, D. B. Miles, A. Corl, A. Chaine, and A. Davis, Models of density-dependent genic selection and a new rock-paper-scissors social system, Am. Nat. 170, 663 (2007).
  8. F. H. Busse and K. E. Heikes, Convection in a rotating layer: A simple case of turbulence, Science 208, 173 (1980).
  9. M. R. E. Proctor and C. A. Jones, The interaction of two spatially resonant patterns in thermal convection. Part I. Exact 1:2 resonance, J. Fluid Mech. 188, 301 (1988).
  10. J. H. P. Dawes, C. M. Postlethwaite, and M. R. E. Proctor, Instabilities induced by a weak breaking of a strong spatial resonance, Phys. D 191, 1 (2004).
  11. J. Guckenheimer and P. Holmes, Structurally stable heteroclinic cycles, Math. Proc. Cambridge Philos. Soc. 103, 189 (1988).
  12. P. Chossat and R. Lauterbach, Methods in equivariant bifurcations and dynamical systems, Advanced Series in Nonlinear Dynamics (World Scientific, Singapore, 2000), Vol. 15.
  13. D. C. Groothuizen Dijkema, V. Kirk, and C. M. Postlethwaite, Analysis of dynamics near heteroclinic networks in R4 with a projected map, SIAM J. Appl. Dyn. Syst. 25, 588 (2026).
  14. C. M. Postlethwaite and A. M. Rucklidge, Stability of cycling behaviour near a heteroclinic network model of Rock–Paper–Scissors–Lizard–Spock, Nonlinearity 35, 1702 (2022).
  15. Q. Yang, T. Rogers, and J. H. P. Dawes, Demographic noise slows down cycles of dominance, J. Theor. Biol. 432, 157 (2017).
  16. C. M. Postlethwaite and J. H. P. Dawes, Regular and irregular cycling near a heteroclinic network, Nonlinearity 18, 1477 (2005).
  17. Q. Yang, Oscillations in stochastic multi-species interactions, Ph.D. thesis, University of Bath, 2018.
  18. A. Szolnoki, M. Mobilia, L.-L. Jiang, B. Szczesny, A. M. Rucklidge, and M. Perc, Cyclic dominance in evolutionary games: A review, J. R. Soc. Interface 11, 20140735 (2014).
  19. Th. Ruijgrok and M. Ruijgrok, A reaction-diffusion equation for a cyclic system with three components, J. Stat. Phys. 87, 1145 (1997).
  20. N. Marchetti, N. R. Prasad, J. Johansson, and T. Cai, Self-organizing networks: State-of-the-art, challenges and perspectives, in 2010 8th International Conference on Communications Bucharest, Romania (IEEE, Piscataway, NJ, 2010), pp. 503–508.
  21. J. Moysen and L. Giupponi, From 4G to 5G: Self-organized network management meets machine learning, Comput. Commun. 129, 248 (2018).
  22. A. L. L. Middleton, Distributed optimisation of LTE systems. Ph.D. thesis, University of Bath, 2019.
  23. A. Ghafourian, O. Georgiou, E. Barter, and T. Gross, Wireless localization with diffusion maps. Sci. Rep. 10, 20655 (2020).
  24. P. Bao, Q. Xu, and Y. Yang, Modeling and aggregated control of large-scale 5G base stations and backup energy storage systems towards secondary frequency support, Appl. Energy 357, 122498 (2024).
  25. A. Aggarwal, P. Kalita, and D. Selvamuthu, Stochastic modeling of a base station in 5G wireless networks for energy apsects using advanced sleep mechanism, Methodol. Comput. Appl. Probab. 27, 58 (2025).
  26. C. W. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences (Springer, Berlin, 2009).
  27. T. G. Kurtz, Strong approximation theorems for density dependent Markov chains, Stochastic Processes Appl. 6, 223 (1978).
  28. A. J. McKane and T. J. Newman, Predator-prey cycles from resonant amplification of demographic stochasticity, Phys. Rev. Lett. 94, 218102 (2005).
  29. P. H. Baxendale and P. E. Greenwood, Sustained oscillations for density dependent Markov processes, J. Math. Biol. 63, 433 (2011).
  30. J. H. P. Dawes and M. Souza, A derivation of Holling's type I, II and III functional responses in predator-prey systems, J. Theor. Biol. 327, 11 (2013).
  31. D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 81, 2340 (1977).
  32. N. Abdennur, Gillespie stochastic simulation algorithm, GitHub 2025, https://github.com/nvictus/Gillespie.
  33. F. Di Patti, R. Arbel-Goren, D. Fanelli, and J. Stavans, Demographic noise induced patterns in space and time, Phys. Rep. 1179, 1 (2026).
  34. L. S. Wang, J. Yu, Y. Liang, and J. Zhang, The breakdown of linear quasi-cycles: Demographic noise and absorbing boundaries in finite predator–prey system, Electon. Res. Arch. 34, 4248 (2026).
  35. N. G. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed. (Elsevier, Amsterdam, 2007).
  36. H. Risken and T. Frank, The Fokker-Planck Equation: Methods of Solutions and Applications, Springer Series in Synergetics (Springer, New York, 1996).
  37. M. A. Murphy, S. R. Gathmann, C. J. Bartel, O. A. Abdelrahman, and P. J. Dauenhauer, Catalytic resonance theory: Circumfluence of programmable catalytic loops, J. Catal. 430, 115343 (2024).
  38. M. A. Murphy, K. Noordhoek, S. R. Gathmann, P. J. Dauenhauer, and C. J. Bartel, Catalytic resonance theory: Forecasting the flow of programmable catalytic loops, Digital Discovery 4, 411 (2025).
  39. A. G. Fredrickson, Stochastic triangular reactions, Chem. Eng. Sci. 21, 687 (1966).
  40. L. Onsager, Reciprocal relations in irreversible processes. I., Phys. Rev. 37, 405 (1931).
  41. M. Alberich-Carramiñana, A. Ferragut, and J. Llibre, Quadratic planar differential systems with algebraic limit cycles via quadratic plane Cremona maps, Adv. Math. 389, 107924 (2021).
  42. W. A. Coppel, Some quadratic systems with at most one limit cycle, in Dynamics Reported: A Series in Dynamical Systems and their Applications (Springer Nature, Berlin, 1989), Vol. 2, p. 61.
  43. B. Coll and J. Llibre, Limit cycles for a quadratic system with an invariant straight line and some evolution of phase portraits, in Qualitative Theory of Differential Equations, Colloquia Mathematica Societatis Janos Bolyai Vol. 53 (Bolyai Institut, Szeged, Hungary, 1988), pp. 111–123.
  44. R. Bamón, Quadratic vector fields in the plane have a finite number of limit cycles, Publ. Math. Inst. Hautes Études Sci. 64, 111 (1986).

Outline

Information

Phys. Rev. E 114, 044211– Published 16 October, 2026
  • Received 21 January 2026
  • Revised 6 July 2026
  • Accepted 24 September 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation