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Bimatrix Replicator Dynamics with Periodic Impulses

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  • Xinmiao An

    (Beijing Normal University)

  • Xiaomin Wang

    (Beijing Normal University)

  • Boyu Zhang

    (Beijing Normal University)

Abstract

This paper investigates the bimatrix replicator dynamics with periodic impulses. In biological system, impulsive perturbations may due to the occurrence of an unfavorable physical environment, or due to the seasonal life history effects in the physiological and reproductive mechanisms of the population. We show that impulsive perturbations can lead to complicated dynamical behaviors. On the one hand, the system can have multiple $$\tau $$ τ -periodic solutions, where the lower bound of the number of solutions is increasing linearly in the impulsive period $$\tau $$ τ . On the other hand, for shorter impulsive period, we provide a differential approximation for the impulsive dynamical system. By analyzing the resulting differential dynamics, we show that the interior equilibrium (which corresponds to a $$\tau $$ τ -periodic solution) must be globally stable if it exists. Furthermore, when the impulsive effect is weak, all interior trajectories of the impulsive dynamics evolve to a small neighborhood of the interior equilibrium of the bimatrix replicator dynamics. In summary, longer impulsive period causes an increase in the complexity of the evolutionary process and shorter period promotes evolutionary stability of the interior equilibrium where multiple strategies can coexist.

Suggested Citation

  • Xinmiao An & Xiaomin Wang & Boyu Zhang, 2020. "Bimatrix Replicator Dynamics with Periodic Impulses," Dynamic Games and Applications, Springer, vol. 10(3), pages 676-694, September.
    • Handle: RePEc:spr:dyngam:v:10:y:2020:i:3:d:10.1007_s13235-019-00344-w
    DOI: 10.1007/s13235-019-00344-w
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    References listed on IDEAS

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    1. Ralf Korn, 1999. "Some applications of impulse control in mathematical finance," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 50(3), pages 493-518, December.
    2. Ross Cressman, 2003. "Evolutionary Dynamics and Extensive Form Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262033054, December.
    3. Min Zhao & Yanzhen Wang & Lansun Chen, 2012. "Dynamic Analysis of a Predator-Prey (Pest) Model with Disease in Prey and Involving an Impulsive Control Strategy," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-18, August.
    4. Zhang, Yujuan & Xiu, Zhilong & Chen, Lansun, 2005. "Dynamic complexity of a two-prey one-predator system with impulsive effect," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 131-139.
    5. Bo-Yu Zhang & Ross Cressman & Yi Tao, 2010. "Cooperation and Stability through Periodic Impulses," PLOS ONE, Public Library of Science, vol. 5(3), pages 1-7, March.
    6. Sun, Mingjing & Liu, Yinli & Liu, Sujuan & Hu, Zuoliang & Chen, Lansun, 2016. "A novel method for analyzing the stability of periodic solution of impulsive state feedback model," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 425-434.
    7. Wang, Weiming & Wang, Hailing & Li, Zhenqing, 2007. "The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy," Chaos, Solitons & Fractals, Elsevier, vol. 32(5), pages 1772-1785.
    8. Li, Li, 2015. "Patch invasion in a spatial epidemic model," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 342-349.
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    1. Ilaria Colivicchi & Gianluca Iannucci, 2023. "The Environmental Responsibility of Firms and Insurance Coverage in an Evolutionary Game," Dynamic Games and Applications, Springer, vol. 13(3), pages 801-818, September.

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