{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,25]],"date-time":"2025-11-25T08:57:21Z","timestamp":1764061041494,"version":"3.41.2"},"reference-count":30,"publisher":"World Scientific Pub Co Pte Ltd","issue":"13","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2021,10]]},"abstract":"<jats:p> In this paper, we classify the phase portraits in the Poincar\u00e9 disc of all the Kolmogorov systems [Formula: see text] which depend on six parameters. We prove that these systems have [Formula: see text] topologically distinct phase portraits in the Poincar\u00e9 disc. These systems are provided by a general three-dimensional Lotka\u2013Volterra system with a rational first integral of degree two of the form [Formula: see text], restricted to each surface [Formula: see text] varying [Formula: see text], with the additional assumption that they have a Darboux invariant of the form [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s0218127421502011","type":"journal-article","created":{"date-parts":[[2021,10,25]],"date-time":"2021-10-25T08:39:21Z","timestamp":1635151161000},"source":"Crossref","is-referenced-by-count":3,"title":["Planar Kolmogorov Systems Coming from Spatial Lotka\u2013Volterra Systems"],"prefix":"10.1142","volume":"31","author":[{"given":"\u00c9rika","family":"Diz-Pita","sequence":"first","affiliation":[{"name":"Departamento de Estat\u00edstica, An\u00e1lise Matem\u00e1tica e Optimizaci\u00f3n, Universidade de Santiago de Compostela, 15705, Santiago de Compostela, Spain"}]},{"given":"Jaume","family":"Llibre","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Aut\u00f2noma de Barcelona, 08193, Bellaterra, Barcelona, Spain"}]},{"given":"M. Victoria","family":"Otero-Espinar","sequence":"additional","affiliation":[{"name":"Departamento de Estat\u00edstica, An\u00e1lise Matem\u00e1tica e Optimizaci\u00f3n, Universidade de Santiago de Compostela, 15705, Santiago de Compostela, Spain"}]}],"member":"219","published-online":{"date-parts":[[2021,10,25]]},"reference":[{"key":"S0218127421502011BIB001","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2012.03.002"},{"key":"S0218127421502011BIB002","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127411030416"},{"volume-title":"Qualitative Theory of Second Order Dynamic Systems","year":"1973","author":"Andronov A. A.","key":"S0218127421502011BIB003"},{"key":"S0218127421502011BIB004","doi-asserted-by":"publisher","DOI":"10.1016\/0375-9601(80)90342-4"},{"key":"S0218127421502011BIB005","doi-asserted-by":"publisher","DOI":"10.1007\/BF01832841"},{"key":"S0218127421502011BIB007","doi-asserted-by":"publisher","DOI":"10.1016\/S0550-3213(98)00177-1"},{"key":"S0218127421502011BIB008","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-68304-6_4"},{"key":"S0218127421502011BIB009","doi-asserted-by":"publisher","DOI":"10.1137\/0136039"},{"key":"S0218127421502011BIB011","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-020-03162-2","volume":"2021","author":"Diz-Pita Pita.","year":"2021","journal-title":"Electron. J. Diff. Eqs."},{"key":"S0218127421502011BIB012","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(77)90136-X"},{"key":"S0218127421502011BIB013","series-title":"UniversiText","volume-title":"Qualitative Theory of Planar Differential Systems","author":"Dumortier F.","year":"2006"},{"key":"S0218127421502011BIB014","doi-asserted-by":"publisher","DOI":"10.1142\/S0217595917400061"},{"key":"S0218127421502011BIB015","doi-asserted-by":"publisher","DOI":"10.1007\/s12210-008-0023-7"},{"key":"S0218127421502011BIB016","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-03871-6"},{"volume-title":"Essays in Economic Dynamics","year":"1967","author":"Goodwin R. M.","key":"S0218127421502011BIB017"},{"key":"S0218127421502011BIB018","doi-asserted-by":"publisher","DOI":"10.1007\/BF01166429"},{"key":"S0218127421502011BIB019","first-page":"74","volume":"7","author":"Kolmogorov A.","year":"1936","journal-title":"Gi. Inst. Ital. Attuari"},{"volume-title":"Proc. Summer School of Theoretical Physics","year":"1975","author":"Laval G.","key":"S0218127421502011BIB020"},{"key":"S0218127421502011BIB021","doi-asserted-by":"publisher","DOI":"10.1016\/j.bulsci.2009.06.002"},{"key":"S0218127421502011BIB022","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2008.07.020"},{"key":"S0218127421502011BIB023","doi-asserted-by":"publisher","DOI":"10.1016\/j.bulsci.2007.12.001"},{"key":"S0218127421502011BIB024","doi-asserted-by":"publisher","DOI":"10.1137\/130923907"},{"key":"S0218127421502011BIB025","doi-asserted-by":"publisher","DOI":"10.1007\/s10440-020-00346-6"},{"key":"S0218127421502011BIB026","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1954-0060657-0"},{"volume-title":"Stability and Complexity in Model Ecosystems","year":"1974","author":"May R. M.","key":"S0218127421502011BIB027"},{"key":"S0218127421502011BIB028","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1975-0356138-6"},{"key":"S0218127421502011BIB029","unstructured":"Peixoto, M. M.   [1973]  Proccedings of a Symposium,  University of Bahia  (Acad. Press,  NY),  pp. 349\u2013420."},{"key":"S0218127421502011BIB030","volume":"64","author":"Schlomiuk D.","year":"2012","journal-title":"Electron. J. Diff. Eqs."},{"key":"S0218127421502011BIB031","doi-asserted-by":"publisher","DOI":"10.1007\/BF00307854"},{"key":"S0218127421502011BIB032","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2007.08.043"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127421502011","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,25]],"date-time":"2021-10-25T08:40:33Z","timestamp":1635151233000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127421502011"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10]]},"references-count":30,"journal-issue":{"issue":"13","published-print":{"date-parts":[[2021,10]]}},"alternative-id":["10.1142\/S0218127421502011"],"URL":"https:\/\/doi.org\/10.1142\/s0218127421502011","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"type":"print","value":"0218-1274"},{"type":"electronic","value":"1793-6551"}],"subject":[],"published":{"date-parts":[[2021,10]]},"article-number":"2150201"}}