{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T10:57:29Z","timestamp":1760785049008,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,29]],"date-time":"2022-01-29T00:00:00Z","timestamp":1643414400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we studied a nutrient\u2013phytoplankton model with time delay and diffusion term. We studied the Turing instability, local stability, and the existence of Hopf bifurcation. Some formulas are obtained to determine the direction of the bifurcation and the stability of periodic solutions by the central manifold theory and normal form method. Finally, we verify the above conclusion through numerical simulation.<\/jats:p>","DOI":"10.3390\/axioms11020056","type":"journal-article","created":{"date-parts":[[2022,1,29]],"date-time":"2022-01-29T23:02:06Z","timestamp":1643497326000},"page":"56","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Hopf Bifurcation Analysis of a Diffusive Nutrient\u2013Phytoplankton Model with Time Delay"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5052-1416","authenticated-orcid":false,"given":"Ruizhi","family":"Yang","sequence":"first","affiliation":[{"name":"Department of Mathematics, Northeast Forestry University, Harbin 150040, China"}]},{"given":"Liye","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Northeast Forestry University, Harbin 150040, China"}]},{"given":"Dan","family":"Jin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Northeast Forestry University, Harbin 150040, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.mbs.2013.06.002","article-title":"Phytoplankton-zooplankton dynamics in periodic environments taking into account eutrophication","volume":"245","author":"Luo","year":"2013","journal-title":"Math. 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