{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T17:58:09Z","timestamp":1761155889068},"reference-count":42,"publisher":"World Scientific Pub Co Pte Lt","issue":"12","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2014,12]]},"abstract":"<jats:p> We explore a replicator\u2013mutator model of the repeated Prisoner's Dilemma involving three strategies: always cooperate (ALLC), always defect (ALLD), and tit-for-tat (TFT). The dynamics resulting from single unidirectional mutations are considered, with detailed results presented for the mutations TFT \u2192 ALLC and ALLD \u2192 ALLC. For certain combinations of parameters, given by the mutation rate \u03bc and the complexity cost c of playing tit-for-tat, we find that the population settles into limit cycle oscillations, with the relative abundance of ALLC, ALLD, and TFT cycling periodically. Surprisingly, these oscillations can occur for unidirectional mutations between any two strategies. In each case, the limit cycles are created and destroyed by supercritical Hopf and homoclinic bifurcations, organized by a Bogdanov\u2013Takens bifurcation. Our results suggest that stable oscillations are a robust aspect of a world of ALLC, ALLD, and costly TFT; the existence of cycles does not depend on the details of assumptions of how mutation is implemented. <\/jats:p>","DOI":"10.1142\/s0218127414300353","type":"journal-article","created":{"date-parts":[[2015,1,5]],"date-time":"2015-01-05T06:43:47Z","timestamp":1420440227000},"page":"1430035","source":"Crossref","is-referenced-by-count":15,"title":["Limit Cycles Sparked by Mutation in the Repeated Prisoner's Dilemma"],"prefix":"10.1142","volume":"24","author":[{"given":"Danielle F. P.","family":"Toupo","sequence":"first","affiliation":[{"name":"Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David G.","family":"Rand","sequence":"additional","affiliation":[{"name":"Department of Psychology, Yale University, New Haven, CT 06511, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Steven H.","family":"Strogatz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Cornell University, Ithaca, NY 14853, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2015,1,4]]},"reference":[{"key":"rf1","volume-title":"The Evolution of Cooperation","author":"Axelrod R.","year":"1984"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.81.066122"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1038\/327058a0"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1016\/0162-3095(92)90032-Y"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0630443100"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0500938102"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.78.026117"},{"key":"rf8","first-page":"274","volume":"80","author":"Fudenberg D.","year":"1990","journal-title":"Amer. Econ. Rev."},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/2011\/08\/P08007"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1016\/0022-5193(64)90038-4"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1126\/science.1070582"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0811503106"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1016\/0022-5193(79)90058-4"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139173179"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0502589102"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jebo.2012.10.018"},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1038\/327015a0"},{"key":"rf19","doi-asserted-by":"publisher","DOI":"10.1038\/nature06455"},{"key":"rf20","doi-asserted-by":"publisher","DOI":"10.1098\/rspb.2003.2643"},{"key":"rf21","doi-asserted-by":"publisher","DOI":"10.1006\/jtbi.1996.0243"},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1038\/355250a0"},{"key":"rf23","doi-asserted-by":"publisher","DOI":"10.1038\/364056a0"},{"key":"rf24","doi-asserted-by":"publisher","DOI":"10.1038\/nature02414"},{"key":"rf25","doi-asserted-by":"crossref","DOI":"10.2307\/j.ctvjghw98","volume-title":"Evolutionary Dynamics: Exploring the Equations of Life","author":"Nowak M. A.","year":"2006"},{"key":"rf27","doi-asserted-by":"publisher","DOI":"10.1137\/120878537"},{"key":"rf28","doi-asserted-by":"publisher","DOI":"10.1038\/nature02978"},{"key":"rf29","doi-asserted-by":"publisher","DOI":"10.1016\/j.jtbi.2008.09.015"},{"key":"rf30","doi-asserted-by":"publisher","DOI":"10.1016\/j.jtbi.2012.01.011"},{"key":"rf31","doi-asserted-by":"publisher","DOI":"10.1016\/j.tics.2013.06.003"},{"key":"rf32","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1214167110"},{"key":"rf33","doi-asserted-by":"crossref","DOI":"10.3998\/mpub.20269","volume-title":"Prisoner's Dilemma: A Study in Conflict and Cooperation","author":"Rapoport A.","year":"1965"},{"key":"rf34","doi-asserted-by":"publisher","DOI":"10.1007\/BF00948894"},{"key":"rf35","volume-title":"Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering","author":"Strogatz S. H.","year":"1994"},{"key":"rf36","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.70.037102"},{"key":"rf37","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.80.056104"},{"key":"rf38","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0903019106"},{"key":"rf39","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0602530103"},{"key":"rf40","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0912515107"},{"key":"rf41","doi-asserted-by":"publisher","DOI":"10.1086\/406755"},{"key":"rf43","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1206694109"},{"key":"rf44","doi-asserted-by":"publisher","DOI":"10.1126\/science.288.5467.850"},{"key":"rf45","doi-asserted-by":"publisher","DOI":"10.5890\/JAND.2013.04.007"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127414300353","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T19:48:22Z","timestamp":1565120902000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127414300353"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,12]]},"references-count":42,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2015,1,4]]},"published-print":{"date-parts":[[2014,12]]}},"alternative-id":["10.1142\/S0218127414300353"],"URL":"https:\/\/doi.org\/10.1142\/s0218127414300353","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,12]]}}}