{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T15:54:23Z","timestamp":1760889263175,"version":"3.37.3"},"reference-count":28,"publisher":"World Scientific Pub Co Pte Ltd","issue":"10","funder":[{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"crossref","award":["11422111"],"award-info":[{"award-number":["11422111"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"crossref","award":["11371047"],"award-info":[{"award-number":["11371047"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"crossref","award":["11290141"],"award-info":[{"award-number":["11290141"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Science and technology project of Henan Province","award":["152102310320"],"award-info":[{"award-number":["152102310320"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2016,9]]},"abstract":"<jats:p> In this paper, we investigate the stability and Hopf bifurcation of a delayed density-dependent predator\u2013prey system with Beddington\u2013DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered such that the studied predator\u2013prey system conforms to the realistically biological environment. We start with the geometric criterion introduced by Beretta and Kuang [2002] and then investigate the stability of the positive equilibrium and the stability switches of the system with respect to the delay parameter [Formula: see text]. Especially, we generalize the geometric criterion in [Beretta &amp; Kuang, 2002] by introducing the condition [Formula: see text] which can be assured by the condition [Formula: see text], and adopting the technique of lifting to define the function [Formula: see text] for alternatively determining stability switches at the zeroes of [Formula: see text]s. Afterwards, by the Poincar\u00e9 normal form for Hopf bifurcation in [Kuznetsov, 1998] and the bifurcation formulae in [Hassard et al., 1981], we qualitatively analyze the properties for the occurring Hopf bifurcations of the system (3). Finally, an example with numerical simulations is given to illustrate the obtained results. <\/jats:p>","DOI":"10.1142\/s0218127416501650","type":"journal-article","created":{"date-parts":[[2016,9,23]],"date-time":"2016-09-23T05:53:59Z","timestamp":1474610039000},"page":"1650165","source":"Crossref","is-referenced-by-count":13,"title":["Stability and Hopf Bifurcation of a Delayed Density-Dependent Predator\u2013Prey System with Beddington\u2013DeAngelis Functional Response"],"prefix":"10.1142","volume":"26","author":[{"given":"Haiyin","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics and Information, Henan University of Economics and Law, Jinshui East Road, Zhengzhou 450002, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gang","family":"Meng","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, University of Chinese Academy of Sciences, Yuquan Road No. 19, Beijing 100049, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhikun","family":"She","sequence":"additional","affiliation":[{"name":"LMIB and School of Mathematics and Systems Science, Beihang University, Xueyuan Road No. 37, Beijing 100191, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2016,9,23]]},"reference":[{"key":"S0218127416501650BIB001","doi-asserted-by":"publisher","DOI":"10.1016\/0025-5564(90)90019-U"},{"volume-title":"Systems with Impulse Effect: Stability Theory and Applications","year":"1989","author":"Bainov D. D.","key":"S0218127416501650BIB002"},{"volume-title":"Impulsive Differential Equations: Periodic Solutions and Applications","year":"1993","author":"Bainov D. 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