{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T12:46:39Z","timestamp":1766407599789,"version":"3.41.2"},"reference-count":36,"publisher":"World Scientific Pub Co Pte Ltd","issue":"10","funder":[{"name":"Team Building Project for Graduate Tutors in Chongqing","award":["JDDSTD201802"],"award-info":[{"award-number":["JDDSTD201802"]}]},{"name":"National Natural Science Foundation of China under Grants","award":["11801047"],"award-info":[{"award-number":["11801047"]}]},{"name":"National Natural Science Foundation of China under Grants","award":["11961024"],"award-info":[{"award-number":["11961024"]}]},{"name":"Joint Training Base Construction Project for Graduate Students in Chongqing","award":["JDLHPYJD2021016"],"award-info":[{"award-number":["JDLHPYJD2021016"]}]},{"name":"the Program of Chongqing Municipal Education Com- mission","award":["KJQN201900707"],"award-info":[{"award-number":["KJQN201900707"]}]},{"name":"Group Building Scientific Innovation Project for universi- ties in Chongqing","award":["CXQT21021"],"award-info":[{"award-number":["CXQT21021"]}]},{"name":"Natural Science Foundation of Chongqing under Grant","award":["cstc2019jcyj-msxmX0755"],"award-info":[{"award-number":["cstc2019jcyj-msxmX0755"]}]},{"name":"Graduate Research and Innovation Project of Chongqing","award":["CYS21370"],"award-info":[{"award-number":["CYS21370"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2022,8]]},"abstract":"<jats:p> Impulsive control strategies have been widely used in cancer treatment and linear impulsive control has always been considered in previous studies. We propose a novel tumour-immune model with nonlinear killing rate as state-dependent feedback control, which can better reflect the saturation effects of the tumour and immune cell mortalities due to chemotherapy, and its dynamic behaviors are investigated. The paper aims to discuss the transcritical and subcritical bifurcations of the model. To begin with, the threshold conditions for tumour eradication and tumour persistence in the model without pulse interventions are provided. We define the Poincar\u00e9 map of the proposed model and then address the existence and orbital asymptotically stability of the model\u2019s tumour-free periodic solution. Furthermore, by using the bifurcation theory of the discrete one-parameter family of maps, which is determined by the Poincar\u00e9 mapping, we investigate the model\u2019s transcritical and subcritical pitchfork bifurcations with respect to the key parameter. <\/jats:p>","DOI":"10.1142\/s0218127422501553","type":"journal-article","created":{"date-parts":[[2022,8,29]],"date-time":"2022-08-29T03:49:35Z","timestamp":1661744975000},"source":"Crossref","is-referenced-by-count":2,"title":["Bifurcation Analysis of a Tumour-Immune Model with Nonlinear Killing Rate as State-Dependent Feedback Control"],"prefix":"10.1142","volume":"32","author":[{"given":"Likun","family":"Guan","sequence":"first","affiliation":[{"name":"College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, P. R. China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7529-5649","authenticated-orcid":false,"given":"Jin","family":"Yang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, P. R. China"}]},{"given":"Yuanshun","family":"Tan","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, P. R. China"}]},{"given":"Zijian","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, P. R. China"}]},{"given":"Robert A.","family":"Cheke","sequence":"additional","affiliation":[{"name":"Natural Resources Institute, University of Greenwich at Medway, Central Avenue, Chatham Maritime, Chatham, Kent, ME4 4TB, UK"}]}],"member":"219","published-online":{"date-parts":[[2022,8,29]]},"reference":[{"doi-asserted-by":"publisher","key":"S0218127422501553BIB001","DOI":"10.1016\/S0140-6736(06)69153-8"},{"key":"S0218127422501553BIB002","volume-title":"Impulsive Differential Equations: Periodic Solutions and Applications","volume":"66","author":"Bainov D.","year":"1993"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB003","DOI":"10.1016\/j.jde.2008.02.007"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB004","DOI":"10.1158\/0008-5472.CAN-05-0328"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB005","DOI":"10.1115\/1.4043001"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB006","DOI":"10.4064\/ap84-1-1"},{"volume-title":"Cancer Immunotherapy","year":"2013","author":"Couzin-Frankel J.","key":"S0218127422501553BIB007"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB008","DOI":"10.1016\/S0895-7177(03)00125-0"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB009","DOI":"10.1016\/S0895-7177(03)00131-6"},{"key":"S0218127422501553BIB010","first-page":"395","volume":"13","author":"Ga\u0142ach M.","year":"2003","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB011","DOI":"10.1038\/nature13988"},{"volume-title":"Ordinary Differential Equations","year":"1969","author":"Hale J. K.","key":"S0218127422501553BIB012"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB013","DOI":"10.1164\/rccm.200501-057OC"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB014","DOI":"10.1016\/0022-247X(90)90199-P"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB015","DOI":"10.1016\/S0092-8240(05)80260-5"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB016","DOI":"10.1142\/S0218127417502236"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB017","DOI":"10.1016\/j.jaci.2004.03.020"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB018","DOI":"10.1038\/nature13904"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB019","DOI":"10.1200\/JCO.2003.06.041"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB020","DOI":"10.1080\/00036811.2016.1153630"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB021","DOI":"10.1007\/s40435-015-0204-z"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB022","DOI":"10.1101\/gad.169029.111"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB023","DOI":"10.1080\/00207728808547133"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB024","DOI":"10.1007\/s00285-004-0290-6"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB025","DOI":"10.1016\/j.mbs.2008.06.008"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB026","DOI":"10.1016\/j.jtbi.2012.05.022"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB027","DOI":"10.1142\/S0218127416500395"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB028","DOI":"10.1016\/j.apm.2017.03.023"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB029","DOI":"10.1137\/080742762"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB030","DOI":"10.1142\/S0218127413500685"},{"volume-title":"World Cancer Report: Cancer Research for Cancer Prevention","year":"2020","author":"Wild C.","key":"S0218127422501553BIB031"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB032","DOI":"10.1002\/bem.20177"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB033","DOI":"10.1016\/j.matcom.2014.09.001"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB034","DOI":"10.1016\/j.cnsns.2019.02.025"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB035","DOI":"10.1142\/S0218127420501047"},{"doi-asserted-by":"publisher","key":"S0218127422501553BIB036","DOI":"10.1137\/19M1279320"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127422501553","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,29]],"date-time":"2022-08-29T03:51:33Z","timestamp":1661745093000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S0218127422501553"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8]]},"references-count":36,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2022,8]]}},"alternative-id":["10.1142\/S0218127422501553"],"URL":"https:\/\/doi.org\/10.1142\/s0218127422501553","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"type":"print","value":"0218-1274"},{"type":"electronic","value":"1793-6551"}],"subject":[],"published":{"date-parts":[[2022,8]]},"article-number":"2250155"}}