{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T15:35:29Z","timestamp":1772292929328,"version":"3.50.1"},"reference-count":17,"publisher":"World Scientific Pub Co Pte Lt","issue":"07","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2014,7]]},"abstract":"<jats:p> In the paper, we study the existence of periodic solutions in an invariant manifold for a predator\u2013prey model with periodic harvesting rate. Without harvesting, the predator\u2013prey system has a positive equilibrium of center type, which undergoes Hopf bifurcation. Under the periodic harvesting, we first reduce this predator\u2013prey system to a complex normal form. Then, we define the Poincar\u00e9 mapping of this system by using the polar coordinates, and prove that there exists an integral manifold and periodic solution with the period of the harvesting rate. An example is provided to illustrate the result. <\/jats:p>","DOI":"10.1142\/s0218127414500965","type":"journal-article","created":{"date-parts":[[2014,7,30]],"date-time":"2014-07-30T05:32:54Z","timestamp":1406698374000},"page":"1450096","source":"Crossref","is-referenced-by-count":3,"title":["Periodic Solutions for a Predator\u2013Prey Model with Periodic Harvesting Rate"],"prefix":"10.1142","volume":"24","author":[{"given":"Yilei","family":"Tang","sequence":"first","affiliation":[{"name":"Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dongmei","family":"Xiao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2014,7,30]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1016\/S0169-5347(00)01908-X"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-5193(89)80211-5"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1007\/s002850000078"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1111\/j.1939-7445.2003.tb00113.x"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1137\/120895858"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511665639"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.2307\/1940008"},{"key":"rf9","volume-title":"Differential Equations, Dynamical Systems and An Introduction to Chaos","author":"Hirsch M.","year":"2004"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1007\/s002850100079"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1006\/bulm.1998.0072"},{"key":"rf12","first-page":"325","volume":"21","author":"Kuang Y.","year":"1999","journal-title":"Fields Instit. Commun."},{"key":"rf13","first-page":"37","volume":"26","author":"Qian M.","year":"1983","journal-title":"Acta Math. Sin."},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.2307\/3799196"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1007\/s002850100097"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139903428719"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2005.11.048"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127414500965","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T00:59:44Z","timestamp":1565139584000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127414500965"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,7]]},"references-count":17,"journal-issue":{"issue":"07","published-online":{"date-parts":[[2014,7,30]]},"published-print":{"date-parts":[[2014,7]]}},"alternative-id":["10.1142\/S0218127414500965"],"URL":"https:\/\/doi.org\/10.1142\/s0218127414500965","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,7]]}}}