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Mathematical Biosciences and Engineering

2011, Volume 8, Issue 3: 659-676. doi: 10.3934/mbe.2011.8.659
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A note on the replicator equation with explicit space and global regulation

  • 1.
    Applied Mathematics–1, Moscow State University of Railway Engineering, Obraztsova 9, Moscow, 127994
  • Received: 01 July 2010 Accepted: 29 June 2018 Published: 01 June 2011

  • MSC : Primary: 35K57, 35B35, 91A22; Secondary: 92D25.

  • A replicator equation with explicit space and global regulation is considered. This model provides a natural framework to follow frequencies of species that are distributed in the space. For this model, analogues to classical notions of the Nash equilibrium and evolutionary stable state are provided. A sufficient condition for a uniform stationary state to be a spatially distributed evolutionary stable state is presented and illustrated with examples.

    Citation: Alexander S. Bratus, Vladimir P. Posvyanskii, Artem S. Novozhilov. A note on the replicator equation with explicit space and global regulation[J]. Mathematical Biosciences and Engineering, 2011, 8(3): 659-676. doi: 10.3934/mbe.2011.8.659

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  • A replicator equation with explicit space and global regulation is considered. This model provides a natural framework to follow frequencies of species that are distributed in the space. For this model, analogues to classical notions of the Nash equilibrium and evolutionary stable state are provided. A sufficient condition for a uniform stationary state to be a spatially distributed evolutionary stable state is presented and illustrated with examples.


  • This article has been cited by:

    1. A. S. Bratus, V. P. Posvyanskii, A. S. Novozhilov, A. Morozov, Replicator Equations and Space, 2014, 9, 0973-5348, 47, 10.1051/mmnp/20149304
    2. Alexander S. Bratus, Vladimir P. Posvyanskii, Artem S. Novozhilov, Solutions with a bounded support promote permanence of a distributed replicator equation, 2017, 96, 0003-6811, 2652, 10.1080/00036811.2016.1236921
    3. Alexander S. Bratus, Chin-Kun Hu, Mikhail V. Safro, Artem S. Novozhilov, On Diffusive Stability of Eigen’s Quasispecies Model, 2016, 22, 1079-2724, 1, 10.1007/s10883-014-9237-4
    4. O. Kuzenkov, E. Ryabova, A. Morozov, Variational Principle for Self-replicating Systems, 2015, 10, 0973-5348, 115, 10.1051/mmnp/201510208
    5. T. S. Yakushkina, A distributed replicator system corresponding to a bimatrix game, 2016, 40, 0278-6419, 19, 10.3103/S0278641916010064
    6. E. N. Pavlovich, A. S. Bratus’, Exclusion of dominated species in open replicator systems, 2013, 24, 1046-283X, 136, 10.1007/s10598-013-9165-2
    7. N. K. Volosova, A. K. Volosova, K. A. Volosov, S. P. Vakulenko, Graphs for the Replicator Equations and the “Tragedy of the Exhaustion of the Common Resource”, 2020, 12, 2070-0482, 246, 10.1134/S2070048220020167
    8. Наталья Константиновна Волосова, Natal'ya Konstantinovna Volosova, Александра Константиновна Волосова, Aleksandra Konstantinovna Volosova, Константин Александрович Волосов, Konstantin Aleksandrovich Volosov, Сергей Петрович Вакуленко, Sergei Petrovich Vakulenko, Графы задач для репликаторных уравнений и "трагедия исчерпания общего ресурса", 2019, 31, 0234-0879, 101, 10.1134/S0234087919080069
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