{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T14:26:12Z","timestamp":1740147972795,"version":"3.37.3"},"reference-count":15,"publisher":"Wiley","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11061016"],"award-info":[{"award-number":["11061016"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computational and Mathematical Methods in Medicine"],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:p>A modified Leslie-Gower predator-prey system with two delays is investigated. By choosing<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:msub><mml:mi>\u03c4<\/mml:mi><mml:mn>1<\/mml:mn><\/mml:msub><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:msub><mml:mi>\u03c4<\/mml:mi><mml:mn>2<\/mml:mn><\/mml:msub><\/mml:mrow><\/mml:math>as bifurcation parameters, we show that the Hopf bifurcations occur when time delay crosses some critical values. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and chaotic behaviors are observed. Finally, using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of the periodic solutions.<\/jats:p>","DOI":"10.1155\/2014\/619132","type":"journal-article","created":{"date-parts":[[2014,4,7]],"date-time":"2014-04-07T19:09:39Z","timestamp":1396897779000},"page":"1-12","source":"Crossref","is-referenced-by-count":4,"title":["Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge"],"prefix":"10.1155","volume":"2014","author":[{"given":"Qingsong","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yiping","family":"Lin","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jingnan","family":"Cao","sequence":"additional","affiliation":[{"name":"School of Computer Science, Beijing University of Post Telecommunication, Beijing 100876, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"issue":"1-2","key":"1","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1093\/biomet\/45.1-2.16","volume":"45","year":"1958","journal-title":"Biometrika"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/j.apm.2009.02.008"},{"issue":"3","key":"3","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/S0362-546X(97)00491-4","volume":"32","year":"1998","journal-title":"Nonlinear Analysis: Theory, Methods and Applications"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-247X(02)00718-7"},{"year":"1977","key":"5"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1007\/s002850050080"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/j.physleta.2005.10.056"},{"key":"8","first-page":"37","volume":"40","year":"2008","journal-title":"Punjab University Journal of Mathematics"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2007.03.015"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2009.01.052"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1093\/imamat\/hxp013"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2005.10.003"},{"year":"1981","key":"13"},{"issue":"12","key":"14","doi-asserted-by":"crossref","first-page":"4799","DOI":"10.1090\/S0002-9947-98-02083-2","volume":"350","year":"1998","journal-title":"Transactions of the American Mathematical Society"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2006.12.032"}],"container-title":["Computational and Mathematical Methods in Medicine"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/cmmm\/2014\/619132.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/cmmm\/2014\/619132.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/cmmm\/2014\/619132.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,22]],"date-time":"2017-06-22T07:10:24Z","timestamp":1498115424000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/cmmm\/2014\/619132\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":15,"alternative-id":["619132","619132"],"URL":"https:\/\/doi.org\/10.1155\/2014\/619132","relation":{},"ISSN":["1748-670X","1748-6718"],"issn-type":[{"type":"print","value":"1748-670X"},{"type":"electronic","value":"1748-6718"}],"subject":[],"published":{"date-parts":[[2014]]}}}