{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T17:10:46Z","timestamp":1785431446655,"version":"3.56.0"},"reference-count":36,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100015808","name":"Student Innovation and Entrepreneurship Training Program of Shijiazhuang Tiedao University, \u201cResearch on Vibration Control of Complex Systems\u201d","doi-asserted-by":"publisher","award":["202510107356"],"award-info":[{"award-number":["202510107356"]}],"id":[{"id":"10.13039\/100015808","id-type":"DOI","asserted-by":"publisher"}]},{"award":["202510107356"],"award-info":[{"award-number":["202510107356"]}],"id":[{"id":"https:\/\/ror.org\/022e9e065","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This study proposes a stochastic time-delay pheromone feedback model to describe ant foraging dynamics, and investigates how response delays and environmental noise jointly induce stochastic oscillations and reorganize the system\u2019s probabilistic structure. By employing near-Hopf center-mode projection and stochastic averaging, we derive the first-order stochastic amplitude equation and analyze the stochastic dynamical properties near the deterministic delay-induced Hopf bifurcation. Subsequently, normalized Shannon entropy and Jensen\u2013Shannon divergence, computed relative to a pre-Hopf stochastic stationary reference distribution, are used to quantify uncertainty expansion and distributional reorganization in the stationary amplitude distribution and reconstructed state-variable distributions. The analytical results are supported by numerical simulations, which indicate that response delay primarily determines the transition from stable foraging to oscillatory behavior, while noise intensity mainly affects the dispersion and uncertainty of the amplitude distribution. Information-theoretic metrics further reveal noise-induced uncertainty growth and delay-induced probabilistic restructuring. This study elucidates the stability regulation mechanisms of ant foraging systems under stochastic conditions from a combined dynamical and information-theoretic perspective, and provides a theoretical reference for the design of delayed feedback in swarm intelligence systems.<\/jats:p>","DOI":"10.3390\/e28070751","type":"journal-article","created":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T12:41:38Z","timestamp":1782909698000},"page":"751","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Delay-Induced Hopf Bifurcation and Entropy-Based Distributional Uncertainty in a Stochastic Time-Delay Pheromone Feedback Model of Ant Foraging Dynamics"],"prefix":"10.3390","volume":"28","author":[{"given":"Jiaxin","family":"Zhu","sequence":"first","affiliation":[{"name":"Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luyan","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8279-4128","authenticated-orcid":false,"given":"Qiubao","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,7,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1146\/annurev-ento-011118-111923","article-title":"The Ecology of Collective Behavior in Ants","volume":"64","author":"Gordon","year":"2019","journal-title":"Annu. 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