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  • Open Access

Non-Markovian rock-paper-scissors games

Ohad Vilk1,*, Mauro Mobilia2,†, and Michael Assaf1,‡

  • *Contact author: ohad.vilk@mail.huji.ac.il
  • †Contact author: m.mobilia@leeds.ac.uk
  • ‡Contact author: michael.assaf@mail.huji.ac.il
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Phys. Rev. Research 7, 023284 – Published 20 June, 2025

DOI: https://doi.org/10.1103/4mm1-943p

Abstract

Evidence is mounting that species interactions often involve long-term memory, with highly varying waiting times between successive events and long-range temporal correlations. Accounting for memory undermines the common Markovian assumption, and dramatically impacts key ingredients of population dynamics including birth, foraging, predation, and competition processes. Here we study a critical aspect of population dynamics, namely, non-Markovian multispecies competition. This is done in the realm of the zero-sum rock-paper-scissors (zRPS) model that is broadly used in the life sciences to metaphorically describe cyclic competition among three interacting species. We develop a general non-Markovian formalism for multispecies dynamics, allowing us to determine the regions of the parameter space where each species dominates. In particular, when the dynamics are Markovian, the waiting times are exponentially distributed and the fate of the zRPS model in large well-mixed populations is encoded in a remarkably simple condition, often referred to as the “law of the weakest” (LOW), stating that the species with the lowest growth rate is the most likely to prevail. We show that the survival behavior and LOW of the zRPS model are critically affected by nonexponential waiting times and, especially, by their coefficient of variation. Our findings provide key insight into the influence of long waiting times on non-Markovian evolutionary processes.

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Phys. Rev. Research 7, 023284– Published 20 June, 2025
  • Received 22 September 2024
  • Accepted 22 May 2025
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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.

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