{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:11:53Z","timestamp":1760145113500,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,20]],"date-time":"2024-06-20T00:00:00Z","timestamp":1718841600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Jiangsu Higher Education Institutions of China","award":["20KJB110005","2018M642130"],"award-info":[{"award-number":["20KJB110005","2018M642130"]}]},{"name":"China Postdoctoral Science Foundation","award":["20KJB110005","2018M642130"],"award-info":[{"award-number":["20KJB110005","2018M642130"]}]},{"name":"Qing Lan Project of the Jiangsu Higher Education Institutions","award":["20KJB110005","2018M642130"],"award-info":[{"award-number":["20KJB110005","2018M642130"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper investigates the Hopf-like bifurcation of a fractional tumor-lymphatic model with two time delays. The two time delays are considered as branching parameters, and we analyze their influences on the dynamic properties of the model. Through an examination of the root distribution of the characteristic equation, we derive the properties of the positive steady state and the conditions for the occurrence of Hopf-like bifurcation near the positive equilibrium point. Numerical simulations are demonstrated to support our theoretical results.<\/jats:p>","DOI":"10.3390\/sym16060775","type":"journal-article","created":{"date-parts":[[2024,6,21]],"date-time":"2024-06-21T05:33:28Z","timestamp":1718948008000},"page":"775","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Hopf-like Bifurcation Analysis of a Fractional-Order Tumor-Lymphatic Model Involving Two Time Delays"],"prefix":"10.3390","volume":"16","author":[{"given":"Xueying","family":"Shi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Taizhou University, Taizhou 225300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoping","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Taizhou University, Taizhou 225300, China"},{"name":"Department of Mathematics, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chengdai","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"An","family":"Luo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Taizhou University, Taizhou 225300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xin","family":"Yin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Taizhou University, Taizhou 225300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.biosystems.2013.12.001","article-title":"Model for tumor growth with treatment by continuous and pulsed chemotherapy","volume":"116","author":"Borges","year":"2014","journal-title":"Biosystems"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1847","DOI":"10.1137\/120887898","article-title":"Bifurcations in delay differential equations and applications to tumor and immune system interaction models","volume":"12","author":"Bi","year":"2013","journal-title":"SIAM J. 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