{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T01:28:37Z","timestamp":1776216517163,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,2]],"date-time":"2024-09-02T00:00:00Z","timestamp":1725235200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we study the local, global, and bifurcation properties of a planar nonlinear asymmetric discrete model of Ricker type that is derived from a Darwinian evolution strategy based on evolutionary game theory. We make a change of variables to both reduce the number of parameters as well as bring symmetry to the isoclines of the mapping. With this new model, we demonstrate the existence of a forward invariant and globally attracting set where all the dynamics occur. In this set, the model possesses two symmetric fixed points: the origin, which is always a saddle fixed point, and an interior fixed point that may be globally asymptotically stable. Moreover, we observe the presence of a supercritical Neimark\u2013Sacker bifurcation, a phenomenon that is not present in the original non-evolutionary model.<\/jats:p>","DOI":"10.3390\/sym16091139","type":"journal-article","created":{"date-parts":[[2024,9,2]],"date-time":"2024-09-02T10:07:35Z","timestamp":1725271655000},"page":"1139","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Invariant Sets, Global Dynamics, and the Neimark\u2013Sacker Bifurcation in the Evolutionary Ricker Model"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5991-9164","authenticated-orcid":false,"given":"Rafael","family":"Lu\u00eds","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Madeira, Campus Universit\u00e1rio da Penteada, 9020-105 Funchal, Portugal"},{"name":"Center for Mathematical Analysis, Geometry, and Dynamical Systems, Instituto Superior Tecnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6027-5191","authenticated-orcid":false,"given":"Brian","family":"Ryals","sequence":"additional","affiliation":[{"name":"Department of Mathematics, California State University Bakersfield, Bakersfield, CA 93311-1022, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1139\/f54-039","article-title":"Stock and recruitment","volume":"11","author":"Ricker","year":"1954","journal-title":"J. Fish. Res. Board Can."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1086\/283092","article-title":"Bifurcations and Dynamic Complexity in Simple Ecological Models","volume":"110","author":"May","year":"1976","journal-title":"Am. 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