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Discrete and Continuous Distributed Delays in Replicator Dynamics

Author

Listed:
  • Nesrine Ben-Khalifa

    (University of Avignon)

  • Rachid El-Azouzi

    (University of Avignon)

  • Yezekael Hayel

    (University of Avignon)

Abstract

In this paper, we study evolutionary games and we examine the stability of the evolutionarily stable strategy (ESS) in the continuous-time replicator dynamics with distributed time delays. In many examples, the interactions between individuals take place instantaneously, but their impacts are not immediate and take a certain amount of time which is usually random. In this paper, we study the consequences of distributed delays on the stability of the replicator dynamics. The main results are that (i) under the exponential delay distribution, the ESS is asymptotically stable for any value of the rate parameter; (ii) the necessary and sufficient conditions for the asymptotic stability of the ESS under uniform and Erlang distributions; and (iii) for the discrete distributed delays, we derive a necessary and sufficient delay-independent stability conditions. We illustrate our results with numerical examples from the Hawk–Dove game.

Suggested Citation

  • Nesrine Ben-Khalifa & Rachid El-Azouzi & Yezekael Hayel, 2018. "Discrete and Continuous Distributed Delays in Replicator Dynamics," Dynamic Games and Applications, Springer, vol. 8(4), pages 713-732, December.
    • Handle: RePEc:spr:dyngam:v:8:y:2018:i:4:d:10.1007_s13235-017-0225-7
    DOI: 10.1007/s13235-017-0225-7
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    References listed on IDEAS

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    1. Iijima, Ryota, 2011. "Heterogeneous information lags and evolutionary stability," Mathematical Social Sciences, Elsevier, vol. 61(2), pages 83-85, March.
    2. Hiroshi Oaku, 2002. "Evolution with Delay," The Japanese Economic Review, Japanese Economic Association, vol. 53(1), pages 114-133, March.
    3. Elizabeth Wesson & Richard Rand, 2016. "Hopf Bifurcations in Delayed Rock–Paper–Scissors Replicator Dynamics," Dynamic Games and Applications, Springer, vol. 6(1), pages 139-156, March.
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    Cited by:

    1. Takahara, Akihiro & Sakiyama, Tomoko, 2023. "Twisted strategy may enhance the evolution of cooperation in spatial prisoner’s dilemma," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 629(C).
    2. Marek Bodnar & Jacek Miȩkisz & Raffi Vardanyan, 2020. "Three-Player Games with Strategy-Dependent Time Delays," Dynamic Games and Applications, Springer, vol. 10(3), pages 664-675, September.
    3. Chongyi Zhong & Hui Yang & Zixin Liu & Juanyong Wu, 2020. "Stability of Replicator Dynamics with Bounded Continuously Distributed Time Delay," Mathematics, MDPI, vol. 8(3), pages 1-12, March.
    4. Hu, Limi & Qiu, Xiaoling, 2022. "Stability analysis of game models with fixed and stochastic delays," Applied Mathematics and Computation, Elsevier, vol. 435(C).

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