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The Stability of Two-Community Replicator Dynamics with Discrete Multi-Delays

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  • Jinxiu Pi

    (School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China)

  • Hui Yang

    (School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China)

  • Yadong Shu

    (School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China)

  • Chongyi Zhong

    (School of Mathematics and Big data, Guizhou Education University, Guiyang 550018, China)

  • Guanghui Yang

    (School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China)

Abstract

This article investigates the stability of evolutionarily stable strategy in replicator dynamics of two-community with multi-delays. In the real environment, players interact simultaneously while the return of their choices may not be observed immediately, which implies one or more time-delays exists. In addition to using the method of classic characteristic equations, we also apply linear matrix inequality (i.e., LMI) to discuss the stability of the mixed evolutionarily stable strategy in replicator dynamics of two-community with multi-delays. We derive a delay-dependent stability and a delay-independent stability sufficient conditions of the evolutionarily stable strategy in the two-community replicator dynamics with two delays, and manage to extend the sufficient condition to n time delays. Lastly, numerical trials of the Hawk–Dove game are given to verify the effectiveness of the theoretical consequences.

Suggested Citation

  • Jinxiu Pi & Hui Yang & Yadong Shu & Chongyi Zhong & Guanghui Yang, 2020. "The Stability of Two-Community Replicator Dynamics with Discrete Multi-Delays," Mathematics, MDPI, vol. 8(12), pages 1-17, November.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2120-:d:451840
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    References listed on IDEAS

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    2. Gau, R.S. & Lien, C.H. & Hsieh, J.G., 2007. "Global exponential stability for uncertain cellular neural networks with multiple time-varying delays via LMI approach," Chaos, Solitons & Fractals, Elsevier, vol. 32(4), pages 1258-1267.
    3. Friedman, Daniel, 1991. "Evolutionary Games in Economics," Econometrica, Econometric Society, vol. 59(3), pages 637-666, May.
    4. Hu, Wenjun & Zhang, Gang & Tian, Haiyan, 2019. "The stability of imitation dynamics with discrete distributed delays," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 521(C), pages 218-224.
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