{"id":"https://openalex.org/W2056160862","doi":"https://doi.org/10.1137/060657947","title":"Hopf Bifurcation in Differential Equations with Delay for Tumor\u2013Immune System Competition Model","display_name":"Hopf Bifurcation in Differential Equations with Delay for Tumor\u2013Immune System Competition Model","publication_year":2007,"publication_date":"2007-01-01","ids":{"openalex":"https://openalex.org/W2056160862","doi":"https://doi.org/10.1137/060657947","mag":"2056160862"},"language":"en","primary_location":{"id":"doi:10.1137/060657947","is_oa":false,"landing_page_url":"https://doi.org/10.1137/060657947","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5016073934","display_name":"Radouane Yafia","orcid":"https://orcid.org/0000-0002-9824-9036"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Radouane Yafia","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5016073934"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.6832,"has_fulltext":false,"cited_by_count":57,"citation_normalized_percentile":{"value":0.7509049,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"67","issue":"6","first_page":"1693","last_page":"1703"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11829","display_name":"Mathematical Biology Tumor Growth","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11829","display_name":"Mathematical Biology Tumor Growth","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/2739","display_name":"Public Health, Environmental and Occupational Health"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},{"id":"https://openalex.org/T11764","display_name":"Evolution and Genetic Dynamics","score":0.9951000213623047,"subfield":{"id":"https://openalex.org/subfields/1311","display_name":"Genetics"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.8206796646118164},{"id":"https://openalex.org/keywords/limit-cycle","display_name":"Limit cycle","score":0.7507936954498291},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6898773908615112},{"id":"https://openalex.org/keywords/delay-differential-equation","display_name":"Delay differential equation","score":0.6383416056632996},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.598884105682373},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.5857811570167542},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5093759298324585},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.4897947311401367},{"id":"https://openalex.org/keywords/competition","display_name":"Competition (biology)","score":0.4701578617095947},{"id":"https://openalex.org/keywords/biological-applications-of-bifurcation-theory","display_name":"Biological applications of bifurcation theory","score":0.42220380902290344},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.42016321420669556},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.3904472589492798},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.15704834461212158},{"id":"https://openalex.org/keywords/control","display_name":"Control (management)","score":0.1566891074180603},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.13273528218269348},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.12724432349205017}],"concepts":[{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.8206796646118164},{"id":"https://openalex.org/C93357160","wikidata":"https://www.wikidata.org/wiki/Q1545916","display_name":"Limit cycle","level":3,"score":0.7507936954498291},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6898773908615112},{"id":"https://openalex.org/C147504518","wikidata":"https://www.wikidata.org/wiki/Q2146125","display_name":"Delay differential equation","level":3,"score":0.6383416056632996},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.598884105682373},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.5857811570167542},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5093759298324585},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.4897947311401367},{"id":"https://openalex.org/C91306197","wikidata":"https://www.wikidata.org/wiki/Q45767","display_name":"Competition (biology)","level":2,"score":0.4701578617095947},{"id":"https://openalex.org/C174914120","wikidata":"https://www.wikidata.org/wiki/Q4914973","display_name":"Biological applications of bifurcation theory","level":5,"score":0.42220380902290344},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.42016321420669556},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.3904472589492798},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.15704834461212158},{"id":"https://openalex.org/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"score":0.1566891074180603},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.13273528218269348},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.12724432349205017},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/060657947","is_oa":false,"landing_page_url":"https://doi.org/10.1137/060657947","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":25,"referenced_works":["https://openalex.org/W157111480","https://openalex.org/W1029467092","https://openalex.org/W1522687244","https://openalex.org/W1607633954","https://openalex.org/W1929982142","https://openalex.org/W1969273190","https://openalex.org/W1971201122","https://openalex.org/W1979575182","https://openalex.org/W1981608920","https://openalex.org/W1986837879","https://openalex.org/W2007175366","https://openalex.org/W2011158867","https://openalex.org/W2017687145","https://openalex.org/W2040837794","https://openalex.org/W2045728287","https://openalex.org/W2048631223","https://openalex.org/W2062210472","https://openalex.org/W2066340345","https://openalex.org/W2087207248","https://openalex.org/W2112900204","https://openalex.org/W2116148182","https://openalex.org/W2154508073","https://openalex.org/W2333206676","https://openalex.org/W2400039264","https://openalex.org/W3152125479"],"related_works":["https://openalex.org/W2115351049","https://openalex.org/W2353430195","https://openalex.org/W2094768441","https://openalex.org/W2001548992","https://openalex.org/W178781247","https://openalex.org/W2543553286","https://openalex.org/W2045360568","https://openalex.org/W323081086","https://openalex.org/W2255858401","https://openalex.org/W2001119836"],"abstract_inverted_index":{"This":[0],"paper":[1],"deals":[2],"with":[3,44],"the":[4,8,16,19,51,56,62,70,75,81,89,97],"qualitative":[5],"analysis":[6,71],"of":[7,41,68,83,149],"solutions":[9],"to":[10,15,79,137,143],"a":[11,28,39,66,84],"model":[12,31],"that":[13,50],"refers":[14],"competition":[17,35],"between":[18],"immune":[20],"system":[21,40],"and":[22,123],"an":[23],"aggressive":[24],"host":[25],"such":[26],"as":[27,65],"tumor.":[29],"The":[30,93],"which":[32,115],"describes":[33],"this":[34],"is":[36,48,72,135],"governed":[37],"by":[38,100,139],"differential":[42],"equations":[43],"one":[45],"delay.":[46],"It":[47,134],"shown":[49],"dynamics":[52],"depends":[53],"crucially":[54],"on":[55,74],"time":[57,63],"delay":[58,64],"parameter.":[59],"By":[60],"using":[61],"parameter":[67],"bifurcation,":[69],"focused":[73],"Hopf":[76],"bifurcation":[77],"problem":[78],"predict":[80],"occurrence":[82],"limit":[85],"cycle":[86],"bifurcating":[87],"from":[88],"nontrivial":[90],"steady":[91],"state.":[92],"obtained":[94],"results":[95,146],"depict":[96],"oscillations,":[98],"given":[99],"simulations":[101],"(see":[102,120],"[M.":[103],"Galach,":[104],"Int.":[105],"J.":[106,124,127],"Appl.":[107],"Math.":[108,128],"Comput.":[109],"Sci.,":[110],"13":[111],"(2003),":[112],"pp.":[113,132],"395\u2013406]),":[114],"are":[116],"observed":[117],"in":[118],"reality":[119],"[D.":[121],"Kirschner":[122],"C.":[125],"Panetta,":[126],"Biol.,":[129],"37":[130],"(1998),":[131],"235\u2013252]).":[133],"suggested":[136],"examine":[138],"laboratory":[140],"experiments":[141],"how":[142],"employ":[144],"these":[145],"for":[147],"control":[148],"tumor":[150],"growth.":[151]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":7},{"year":2021,"cited_by_count":8},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":3}],"updated_date":"2026-03-29T08:15:47.926485","created_date":"2025-10-10T00:00:00"}
