{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T05:29:34Z","timestamp":1761629374861,"version":"3.40.5"},"reference-count":37,"publisher":"Wiley","license":[{"start":{"date-parts":[[2020,10,9]],"date-time":"2020-10-09T00:00:00Z","timestamp":1602201600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2020,10,9]]},"abstract":"<jats:p>We present a metapopulation dynamic model for the diffusively-coupled rock-paper-scissors (RPS) game with mutation in scale-free hierarchical networks. We investigate how the RPS game changes by mutation in scale-free networks. Only the mutation from rock to scissors (<jats:italic>R<\/jats:italic>-to-<jats:italic>S<\/jats:italic>) occurs with rate <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>\u03bc<\/mml:mi><\/mml:math>. In the network, a node represents a patch where the RPS game is performed. RPS individuals migrate among nodes by diffusion. The dynamics are represented by the reaction-diffusion equations with the recursion formula. We study where and how species coexist or go extinct in the scale-free network. We numerically obtained the solutions for the metapopulation dynamics and derived the transition points. The results show that, with increasing mutation rate <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>\u03bc<\/mml:mi><\/mml:math>, the extinction of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mi>P<\/mml:mi><\/mml:math> species occurs and then the extinction of <jats:italic>R<\/jats:italic> species occurs, and finally only <jats:italic>S<\/jats:italic> species survives. Thus, the first and second dynamical phase transitions occur in the scale-free hierarchical network. We also show that the scaling law holds for the population dynamics which suggests that the transition points approach zero in the limit of infinite size.<\/jats:p>","DOI":"10.1155\/2020\/6976328","type":"journal-article","created":{"date-parts":[[2020,10,9]],"date-time":"2020-10-09T22:20:07Z","timestamp":1602282007000},"page":"1-8","source":"Crossref","is-referenced-by-count":8,"title":["Diffusively-Coupled Rock-Paper-Scissors Game with Mutation in Scale-Free Hierarchical Networks"],"prefix":"10.1155","volume":"2020","author":[{"given":"Takashi","family":"Nagatani","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, Shizuoka University, Hamamatsu 432-8561, Japan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3349-5869","authenticated-orcid":true,"given":"Genki","family":"Ichinose","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Systems Engineering, Shizuoka University, Hamamatsu 432-8561, 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