{"id":"https://openalex.org/W2019823335","doi":"https://doi.org/10.2478/v10006-011-0007-0","title":"Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays","display_name":"Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays","publication_year":2011,"publication_date":"2011-03-01","ids":{"openalex":"https://openalex.org/W2019823335","doi":"https://doi.org/10.2478/v10006-011-0007-0","mag":"2019823335"},"language":"en","primary_location":{"id":"doi:10.2478/v10006-011-0007-0","is_oa":true,"landing_page_url":"https://doi.org/10.2478/v10006-011-0007-0","pdf_url":"https://sciendo.com/pdf/10.2478/v10006-011-0007-0","source":{"id":"https://openalex.org/S117679522","display_name":"International Journal of Applied Mathematics and Computer Science","issn_l":"1641-876X","issn":["1641-876X","2083-8492"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Applied Mathematics and Computer Science","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://sciendo.com/pdf/10.2478/v10006-011-0007-0","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5003408413","display_name":"Changjin Xu","orcid":"https://orcid.org/0000-0001-5844-2985"},"institutions":[{"id":"https://openalex.org/I170172562","display_name":"Guizhou University of Finance and Economics","ror":"https://ror.org/02sw6yz40","country_code":"CN","type":"education","lineage":["https://openalex.org/I170172562"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Changjin Xu","raw_affiliation_strings":["(i) School of Mathematics and Statistics, Guizhou College of Finance and Economics, Luchongguan Rd 269, Guiyang 550004, PR China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"(i) School of Mathematics and Statistics, Guizhou College of Finance and Economics, Luchongguan Rd 269, Guiyang 550004, PR China","institution_ids":["https://openalex.org/I170172562"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5002021613","display_name":"Maoxin Liao","orcid":"https://orcid.org/0000-0001-8728-6977"},"institutions":[{"id":"https://openalex.org/I170172562","display_name":"Guizhou University of Finance and Economics","ror":"https://ror.org/02sw6yz40","country_code":"CN","type":"education","lineage":["https://openalex.org/I170172562"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Maoxin Liao","raw_affiliation_strings":["(i) School of Mathematics and Statistics, Guizhou College of Finance and Economics, Luchongguan Rd 269, Guiyang 550004, PR China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"(i) School of Mathematics and Statistics, Guizhou College of Finance and Economics, Luchongguan Rd 269, Guiyang 550004, PR China","institution_ids":["https://openalex.org/I170172562"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5102013345","display_name":"Xiaofei He","orcid":"https://orcid.org/0009-0001-9107-2354"},"institutions":[{"id":"https://openalex.org/I170172562","display_name":"Guizhou University of Finance and Economics","ror":"https://ror.org/02sw6yz40","country_code":"CN","type":"education","lineage":["https://openalex.org/I170172562"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiaofei He","raw_affiliation_strings":["(i) School of Mathematics and Statistics, Guizhou College of Finance and Economics, Luchongguan Rd 269, Guiyang 550004, PR China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"(i) School of Mathematics and Statistics, Guizhou College of Finance and Economics, Luchongguan Rd 269, Guiyang 550004, PR China","institution_ids":["https://openalex.org/I170172562"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5003408413"],"corresponding_institution_ids":["https://openalex.org/I170172562"],"apc_list":{"value":4080,"currency":"PLN","value_usd":1100},"apc_paid":{"value":4080,"currency":"PLN","value_usd":1100},"fwci":6.4065,"has_fulltext":true,"cited_by_count":53,"citation_normalized_percentile":{"value":0.96253401,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"21","issue":"1","first_page":"97","last_page":"107"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9998999834060669,"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"}},"topics":[{"id":"https://openalex.org/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9998999834060669,"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":0.9620000123977661,"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"}},{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.929099977016449,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.8869917392730713},{"id":"https://openalex.org/keywords/center-manifold","display_name":"Center manifold","score":0.8009846806526184},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7921940088272095},{"id":"https://openalex.org/keywords/transcritical-bifurcation","display_name":"Transcritical bifurcation","score":0.5350514650344849},{"id":"https://openalex.org/keywords/transcendental-equation","display_name":"Transcendental equation","score":0.5306940078735352},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5034546256065369},{"id":"https://openalex.org/keywords/saddle-node-bifurcation","display_name":"Saddle-node bifurcation","score":0.4978630542755127},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.4959042966365814},{"id":"https://openalex.org/keywords/bifurcation-diagram","display_name":"Bifurcation diagram","score":0.46297380328178406},{"id":"https://openalex.org/keywords/pitchfork-bifurcation","display_name":"Pitchfork bifurcation","score":0.45855942368507385},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4069015681743622},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.365476131439209},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.21142920851707458},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.10700011253356934},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09412223100662231},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.05839213728904724}],"concepts":[{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.8869917392730713},{"id":"https://openalex.org/C74774103","wikidata":"https://www.wikidata.org/wiki/Q4504202","display_name":"Center manifold","level":5,"score":0.8009846806526184},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7921940088272095},{"id":"https://openalex.org/C55610538","wikidata":"https://www.wikidata.org/wiki/Q1502581","display_name":"Transcritical bifurcation","level":5,"score":0.5350514650344849},{"id":"https://openalex.org/C139609574","wikidata":"https://www.wikidata.org/wiki/Q2273509","display_name":"Transcendental equation","level":3,"score":0.5306940078735352},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5034546256065369},{"id":"https://openalex.org/C101225446","wikidata":"https://www.wikidata.org/wiki/Q740190","display_name":"Saddle-node bifurcation","level":4,"score":0.4978630542755127},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.4959042966365814},{"id":"https://openalex.org/C86243350","wikidata":"https://www.wikidata.org/wiki/Q2595915","display_name":"Bifurcation diagram","level":4,"score":0.46297380328178406},{"id":"https://openalex.org/C155092403","wikidata":"https://www.wikidata.org/wiki/Q1785924","display_name":"Pitchfork bifurcation","level":5,"score":0.45855942368507385},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4069015681743622},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.365476131439209},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.21142920851707458},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.10700011253356934},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09412223100662231},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.05839213728904724},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","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":3,"locations":[{"id":"doi:10.2478/v10006-011-0007-0","is_oa":true,"landing_page_url":"https://doi.org/10.2478/v10006-011-0007-0","pdf_url":"https://sciendo.com/pdf/10.2478/v10006-011-0007-0","source":{"id":"https://openalex.org/S117679522","display_name":"International Journal of Applied Mathematics and Computer Science","issn_l":"1641-876X","issn":["1641-876X","2083-8492"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Applied Mathematics and Computer Science","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.390.6957","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.390.6957","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://matwbn.icm.edu.pl/ksiazki/amc/amc21/amc2117.pdf","raw_type":"text"},{"id":"pmh:oai:zbc.uz.zgora.pl:46901","is_oa":false,"landing_page_url":"https://zbc.uz.zgora.pl/dlibra/docmetadata?showContent=true&id=46901","pdf_url":null,"source":{"id":"https://openalex.org/S4306400400","display_name":"Digital library of Zielona Gora (University of Zielona G\u00f3ra)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I46305939","host_organization_name":"University of Zielona G\u00f3ra","host_organization_lineage":["https://openalex.org/I46305939"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"https://www.amcs.uz.zgora.pl/?action=paper&paper=537","raw_type":"artyku\u0142"}],"best_oa_location":{"id":"doi:10.2478/v10006-011-0007-0","is_oa":true,"landing_page_url":"https://doi.org/10.2478/v10006-011-0007-0","pdf_url":"https://sciendo.com/pdf/10.2478/v10006-011-0007-0","source":{"id":"https://openalex.org/S117679522","display_name":"International Journal of Applied Mathematics and Computer Science","issn_l":"1641-876X","issn":["1641-876X","2083-8492"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Applied Mathematics and Computer Science","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2019823335.pdf","grobid_xml":"https://content.openalex.org/works/W2019823335.grobid-xml"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W1531715726","https://openalex.org/W1535516791","https://openalex.org/W1586658033","https://openalex.org/W1973000709","https://openalex.org/W1981100186","https://openalex.org/W1982962620","https://openalex.org/W1991330350","https://openalex.org/W1991632067","https://openalex.org/W2004216864","https://openalex.org/W2014613110","https://openalex.org/W2021320112","https://openalex.org/W2028932921","https://openalex.org/W2031133895","https://openalex.org/W2043960238","https://openalex.org/W2056233524","https://openalex.org/W2061491569","https://openalex.org/W2067791524","https://openalex.org/W2079871064","https://openalex.org/W2086604080","https://openalex.org/W2090347294","https://openalex.org/W2625711953","https://openalex.org/W4285719527","https://openalex.org/W4289708816"],"related_works":["https://openalex.org/W2546487931","https://openalex.org/W56905705","https://openalex.org/W2019027784","https://openalex.org/W3042510364","https://openalex.org/W2078078632","https://openalex.org/W2325048728","https://openalex.org/W2356715721","https://openalex.org/W2387129305","https://openalex.org/W2024878967","https://openalex.org/W2385123893"],"abstract_inverted_index":{"Stability":[0],"and":[1,42,54,72],"Hopf":[2,43,57,63],"bifurcation":[3,44,58],"analysis":[4],"for":[5,50,79],"a":[6,16],"Lotka-Volterra":[7,18],"predator-prey":[8,19],"model":[9,20],"with":[10,21],"two":[11,22],"delays":[12,23],"In":[13],"this":[14],"paper,":[15],"two-species":[17],"is":[24,40,45],"considered.":[25],"By":[26],"analyzing":[27],"the":[28,33,37,52,81],"associated":[29],"characteristic":[30],"transcendental":[31],"equation,":[32],"linear":[34],"stability":[35,53],"of":[36,56],"positive":[38],"equilibrium":[39],"investigated":[41],"demonstrated.":[46],"Some":[47,76],"explicit":[48],"formulae":[49],"determining":[51],"direction":[55],"periodic":[59],"solutions":[60],"bifurcating":[61],"from":[62],"bifurcations":[64],"are":[65,84],"obtained":[66],"by":[67],"using":[68],"normal":[69],"form":[70],"theory":[71],"center":[73],"manifold":[74],"theory.":[75],"numerical":[77],"simulations":[78],"supporting":[80],"theoretical":[82],"results":[83],"also":[85],"included.":[86]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":7},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":6},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":11}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}