{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,10]],"date-time":"2025-12-10T05:19:21Z","timestamp":1765343961460,"version":"3.46.0"},"reference-count":24,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,12,7]],"date-time":"2025-12-07T00:00:00Z","timestamp":1765065600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Innovation and Entrepreneurship Curriculum Construction in Hebei Province","award":["Ordinary Differential Equations"],"award-info":[{"award-number":["Ordinary Differential Equations"]}]},{"name":"the Key Project of Natural Science Foundation of Hebei Province","award":["A2023210064"],"award-info":[{"award-number":["A2023210064"]}]},{"DOI":"10.13039\/501100001809","name":"the National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["U23A2068"],"award-info":[{"award-number":["U23A2068"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"the Hebei Provincial Central Guidance Local Science and Technology Development Project","award":["226Z4901G"],"award-info":[{"award-number":["226Z4901G"]}]},{"name":"the University-level Student Innovation and Entrepreneurship Training Program","award":["202510107355"],"award-info":[{"award-number":["202510107355"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We analyze the population dynamics of a microbial cross-protection model and derive the exact conditions under which a Fold\u2013Hopf bifurcation emerges. By applying center-manifold reduction and normal-form theory, we reduce the infinite-dimensional delay differential system to a finite-dimensional ordinary differential system, enabling rigorous bifurcation analysis. Numerical simulations reveal a rich repertoire of dynamical behaviors, including stable equilibria, sustained oscillations, and noise-induced irregularities. Our findings identify time-delay-induced Fold\u2013Hopf bifurcation as a fundamental mechanism driving oscillatory coexistence in cross-protection mutualisms, for previously reported experimental observations.<\/jats:p>","DOI":"10.3390\/axioms14120903","type":"journal-article","created":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T16:41:41Z","timestamp":1765212101000},"page":"903","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bifurcation Analysis in a Cross-Protection Model"],"prefix":"10.3390","volume":"14","author":[{"given":"Yufei","family":"Wu","sequence":"first","affiliation":[{"name":"Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China"}]},{"given":"Zikun","family":"Han","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China"}]},{"given":"Weixiang","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Information Science and Technology, Shijiazhuang Tiedao University, Shijiazhuang 050043, China"}]},{"given":"Yingting","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Management, Shijiazhuang Tiedao University, Shijiazhuang 050043, China"}]},{"given":"Qiubao","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China"},{"name":"Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"782","DOI":"10.1038\/nrmicro1747","article-title":"Microbial structuring of marine ecosystems","volume":"5","author":"Azam","year":"2007","journal-title":"Nat. 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