{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T20:17:08Z","timestamp":1774901828763,"version":"3.50.1"},"reference-count":49,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,12]],"date-time":"2022-07-12T00:00:00Z","timestamp":1657584000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"The scientific research project of Ningde Normal University","award":["2022ZX415"],"award-info":[{"award-number":["2022ZX415"]}]},{"name":"The scientific research project of Ningde Normal University","award":["2021J011155"],"award-info":[{"award-number":["2021J011155"]}]},{"name":"The Natural Science Foundation of Fujian Province","award":["2022ZX415"],"award-info":[{"award-number":["2022ZX415"]}]},{"name":"The Natural Science Foundation of Fujian Province","award":["2021J011155"],"award-info":[{"award-number":["2021J011155"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This article revisits the stability property of a symbiotic model of commensalism with Michaelis\u2013Menten type harvesting in the first commensal populations. By constructing some suitable Lyapunov functions, we provide a thorough analysis of the dynamic behaviors of the subsystem composed of the second and third species. After that, by applying the stability results of this subsystem and combining with the differential inequality theory, sufficient conditions which ensure the global attractivity of the equilibria are obtained. The results obtained here essentially improve and generalize some known results.<\/jats:p>","DOI":"10.3390\/axioms11070337","type":"journal-article","created":{"date-parts":[[2022,7,12]],"date-time":"2022-07-12T20:52:41Z","timestamp":1657659161000},"page":"337","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Global Attractivity of Symbiotic Model of Commensalism in Four Populations with Michaelis\u2013Menten Type Harvesting in the First Commensal Populations"],"prefix":"10.3390","volume":"11","author":[{"given":"Lili","family":"Xu","sequence":"first","affiliation":[{"name":"College of Mathematics and Physics, Ningde Normal University, Ningde 352000, China"}]},{"given":"Yalong","family":"Xue","sequence":"additional","affiliation":[{"name":"College of Mathematics and Physics, Ningde Normal University, Ningde 352000, China"}]},{"given":"Qifa","family":"Lin","sequence":"additional","affiliation":[{"name":"College of Mathematics and Physics, Ningde Normal University, Ningde 352000, China"}]},{"given":"Chaoquan","family":"Lei","sequence":"additional","affiliation":[{"name":"College of Mathematics and Physics, Ningde Normal University, Ningde 352000, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"121061","DOI":"10.1016\/j.physa.2019.121061","article-title":"Stability of a stochastic delay commensalism model with Lvy jumps","volume":"527","author":"Deng","year":"2019","journal-title":"Phys. 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