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Noise-Induced Quasi-Heteroclinic Cycle in a Rock–Paper–Scissors Game with Random Payoffs

Author

Listed:
  • Tian-Jiao Feng

    (Chinese Academy of Sciences
    University of Chinese Academy of Sciences)

  • Jie Mei

    (Chinese Academy of Sciences
    University of Chinese Academy of Sciences)

  • Rui-Wu Wang

    (Northwestern Polytechnical University)

  • Sabin Lessard

    (University of Montreal)

  • Yi Tao

    (Chinese Academy of Sciences
    Northwestern Polytechnical University
    Yunnan University)

  • Xiu-Deng Zheng

    (Chinese Academy of Sciences)

Abstract

The rock–paper–scissors game is one of the main theoretical models in evolutionary game theory and has been used successfully to explain some observed phenomena in biology, economics and social science. In order to explore the influence of environmental noise on cyclic dominance in the rock–paper–scissors game, stochastic stability in a continuous-time dynamics of the game with random payoffs in pairwise interactions is investigated by using the stochastic stability theory of Itô’s stochastic differential equations. After deducing a stochastic replicator equation for the strategy frequencies in a symmetric version of the game, stochastic stability conditions for constant equilibrium states are obtained and stochastic simulations of the global dynamics are performed. The main results are that (1) none of the fixation states of the system can be stochastically stable; and (2) an increase in the noise level (or stochastic fluctuation intensity) can result in the loss of stochastic stability of the constant interior equilibrium. More importantly, the simulation results not only match the theoretical predictions but also show the appearance of a noise-induced quasi-heteroclinic cycle when the constant interior equilibrium loses its stochastic stability as the noise level increases.

Suggested Citation

  • Tian-Jiao Feng & Jie Mei & Rui-Wu Wang & Sabin Lessard & Yi Tao & Xiu-Deng Zheng, 2022. "Noise-Induced Quasi-Heteroclinic Cycle in a Rock–Paper–Scissors Game with Random Payoffs," Dynamic Games and Applications, Springer, vol. 12(4), pages 1280-1292, December.
    • Handle: RePEc:spr:dyngam:v:12:y:2022:i:4:d:10.1007_s13235-021-00414-y
    DOI: 10.1007/s13235-021-00414-y
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    References listed on IDEAS

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    1. Benjamin C. Kirkup & Margaret A. Riley, 2004. "Antibiotic-mediated antagonism leads to a bacterial game of rock–paper–scissors in vivo," Nature, Nature, vol. 428(6981), pages 412-414, March.
    2. Fudenberg, D. & Harris, C., 1992. "Evolutionary dynamics with aggregate shocks," Journal of Economic Theory, Elsevier, vol. 57(2), pages 420-441, August.
    3. Helbing, Dirk, 1992. "Interrelations between stochastic equations for systems with pair interactions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 181(1), pages 29-52.
    4. Hommes, Cars H. & Ochea, Marius I., 2012. "Multiple equilibria and limit cycles in evolutionary games with Logit Dynamics," Games and Economic Behavior, Elsevier, vol. 74(1), pages 434-441.
    5. Tobias Reichenbach & Mauro Mobilia & Erwin Frey, 2007. "Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games," Nature, Nature, vol. 448(7157), pages 1046-1049, August.
    6. Michael B. Elowitz & Stanislas Leibler, 2000. "A synthetic oscillatory network of transcriptional regulators," Nature, Nature, vol. 403(6767), pages 335-338, January.
    7. Andrew R. Tilman & Joshua B. Plotkin & Erol Akçay, 2020. "Evolutionary games with environmental feedbacks," Nature Communications, Nature, vol. 11(1), pages 1-11, December.
    8. Mauro Mobilia & Alastair M. Rucklidge & Bartosz Szczesny, 2016. "The Influence of Mobility Rate on Spiral Waves in Spatial Rock-Paper-Scissors Games," Games, MDPI, vol. 7(3), pages 1, September.
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