{"id":"https://openalex.org/W1974652010","doi":"https://doi.org/10.1155/2013/201936","title":"Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays","display_name":"Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays","publication_year":2013,"publication_date":"2013-01-01","ids":{"openalex":"https://openalex.org/W1974652010","doi":"https://doi.org/10.1155/2013/201936","mag":"1974652010"},"language":"en","primary_location":{"id":"doi:10.1155/2013/201936","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/201936","pdf_url":"https://downloads.hindawi.com/journals/jam/2013/201936.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://downloads.hindawi.com/journals/jam/2013/201936.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5032880028","display_name":"Ruiqing Shi","orcid":"https://orcid.org/0000-0002-7012-7651"},"institutions":[{"id":"https://openalex.org/I88830068","display_name":"Shaanxi Normal University","ror":"https://ror.org/0170z8493","country_code":"CN","type":"education","lineage":["https://openalex.org/I88830068"]},{"id":"https://openalex.org/I94310126","display_name":"Shanxi Normal University","ror":"https://ror.org/03zd3ta61","country_code":"CN","type":"education","lineage":["https://openalex.org/I94310126"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Ruiqing Shi","raw_affiliation_strings":["College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China","School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China"],"raw_orcid":"https://orcid.org/0000-0002-7012-7651","affiliations":[{"raw_affiliation_string":"College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China","institution_ids":["https://openalex.org/I88830068"]},{"raw_affiliation_string":"School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China","institution_ids":["https://openalex.org/I94310126"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5050506510","display_name":"Junmei Qi","orcid":null},"institutions":[{"id":"https://openalex.org/I94310126","display_name":"Shanxi Normal University","ror":"https://ror.org/03zd3ta61","country_code":"CN","type":"education","lineage":["https://openalex.org/I94310126"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Junmei Qi","raw_affiliation_strings":["School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China","institution_ids":["https://openalex.org/I94310126"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5041520845","display_name":"Sanyi Tang","orcid":"https://orcid.org/0000-0002-3324-746X"},"institutions":[{"id":"https://openalex.org/I88830068","display_name":"Shaanxi Normal University","ror":"https://ror.org/0170z8493","country_code":"CN","type":"education","lineage":["https://openalex.org/I88830068"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Sanyi Tang","raw_affiliation_strings":["College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China","institution_ids":["https://openalex.org/I88830068"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5032880028"],"corresponding_institution_ids":["https://openalex.org/I88830068","https://openalex.org/I94310126"],"apc_list":{"value":1050,"currency":"USD","value_usd":1050},"apc_paid":{"value":1050,"currency":"USD","value_usd":1050},"fwci":0.0,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.10686689,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"2013","issue":null,"first_page":"1","last_page":"10"},"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":1.0,"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":1.0,"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":0.9958999752998352,"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/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9786999821662903,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.7417593598365784},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.6568008065223694},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.5475763082504272},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4904880225658417},{"id":"https://openalex.org/keywords/center-manifold","display_name":"Center manifold","score":0.47832775115966797},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4646650552749634},{"id":"https://openalex.org/keywords/stage","display_name":"Stage (stratigraphy)","score":0.4271664023399353},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.42572298645973206},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3703625798225403},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09797179698944092},{"id":"https://openalex.org/keywords/machine-learning","display_name":"Machine learning","score":0.09447234869003296},{"id":"https://openalex.org/keywords/biology","display_name":"Biology","score":0.055443763732910156}],"concepts":[{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.7417593598365784},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.6568008065223694},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.5475763082504272},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4904880225658417},{"id":"https://openalex.org/C74774103","wikidata":"https://www.wikidata.org/wiki/Q4504202","display_name":"Center manifold","level":5,"score":0.47832775115966797},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4646650552749634},{"id":"https://openalex.org/C146357865","wikidata":"https://www.wikidata.org/wiki/Q1123245","display_name":"Stage (stratigraphy)","level":2,"score":0.4271664023399353},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.42572298645973206},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3703625798225403},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09797179698944092},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.09447234869003296},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.055443763732910156},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","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":4,"locations":[{"id":"doi:10.1155/2013/201936","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/201936","pdf_url":"https://downloads.hindawi.com/journals/jam/2013/201936.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.jam/1394808185","is_oa":false,"landing_page_url":"https://projecteuclid.org/euclid.jam/1394808185","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},{"id":"pmh:oai:RePEc:hin:jnljam:201936","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/JAM/2013/201936.xml","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"pmh:oai:doaj.org/article:da59eaf6533946fc9cc28f16e1029ea2","is_oa":false,"landing_page_url":"https://doaj.org/article/da59eaf6533946fc9cc28f16e1029ea2","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Journal of Applied Mathematics, Vol 2013 (2013)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2013/201936","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/201936","pdf_url":"https://downloads.hindawi.com/journals/jam/2013/201936.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2919516586","display_name":null,"funder_award_id":"2011M501428","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3886440964","display_name":null,"funder_award_id":"2013021002-2","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G606003498","display_name":"\u7efc\u5408\u5bb3\u866b\u6cbb\u7406\u4e0eBt\u4f5c\u7269\u6297\u6027\u7ba1\u7406\u7684\u6570\u5b66\u6a21\u578b\u7814\u7a76","funder_award_id":"11171199","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1974652010.pdf","grobid_xml":"https://content.openalex.org/works/W1974652010.grobid-xml"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W1531715726","https://openalex.org/W1586658033","https://openalex.org/W1734689372","https://openalex.org/W1977833600","https://openalex.org/W1979616160","https://openalex.org/W1986093545","https://openalex.org/W1997592679","https://openalex.org/W2003392944","https://openalex.org/W2003858003","https://openalex.org/W2004797631","https://openalex.org/W2013156080","https://openalex.org/W2014462152","https://openalex.org/W2015250294","https://openalex.org/W2016373708","https://openalex.org/W2018564440","https://openalex.org/W2025904721","https://openalex.org/W2028932921","https://openalex.org/W2032487189","https://openalex.org/W2034167224","https://openalex.org/W2039470073","https://openalex.org/W2040019394","https://openalex.org/W2041735995","https://openalex.org/W2041823824","https://openalex.org/W2049822021","https://openalex.org/W2052131048","https://openalex.org/W2055523903","https://openalex.org/W2063441776","https://openalex.org/W2074768841","https://openalex.org/W2076045511","https://openalex.org/W2084741964","https://openalex.org/W2087099377","https://openalex.org/W2327295268","https://openalex.org/W2331625639","https://openalex.org/W2474479551"],"related_works":["https://openalex.org/W1987790493","https://openalex.org/W2153388945","https://openalex.org/W2038701038","https://openalex.org/W1975783007","https://openalex.org/W1996991127","https://openalex.org/W2382752082","https://openalex.org/W1988489950","https://openalex.org/W1986426368","https://openalex.org/W2363522146","https://openalex.org/W2061491569"],"abstract_inverted_index":{"We":[0],"propose":[1],"a":[2,15,51],"three-dimensional":[3,20],"stage-structured":[4],"predatory-prey":[5],"model":[6],"with":[7],"discrete":[8],"and":[9,33,78,86],"distributed":[10],"delays.":[11],"By":[12],"use":[13,73],"of":[14,36,39,74,88],"new":[16,41],"variable,":[17],"the":[18,31,40,46,62,84],"original":[19],"system":[21],"transforms":[22],"into":[23],"an":[24],"equivalent":[25],"four-dimensional":[26],"system.":[27,42],"Firstly,":[28],"we":[29,54,82],"study":[30],"existence":[32],"local":[34],"stability":[35,87],"positive":[37],"equilibrium":[38],"And,":[43],"by":[44,72],"choosing":[45],"time":[47,63],"delay<mml:math":[48,64],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[49,65],"id=\"M1\"><mml:mrow><mml:mi>\u03c4</mml:mi></mml:mrow></mml:math>as":[50],"bifurcation":[52,58],"parameter,":[53],"show":[55],"that":[56],"Hopf":[57,89],"may":[59],"occur":[60],"as":[61],"id=\"M2\"><mml:mrow><mml:mi>\u03c4</mml:mi></mml:mrow></mml:math>passes":[66],"through":[67],"some":[68,93],"critical":[69],"values.":[70],"Secondly,":[71],"normal":[75],"form":[76],"theory":[77],"central":[79],"manifold":[80],"argument,":[81],"establish":[83],"direction":[85],"bifurcation.":[90],"At":[91],"last,":[92],"simple":[94],"discussion":[95],"is":[96],"presented.":[97]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2018,"cited_by_count":1}],"updated_date":"2026-07-31T08:31:51.225901","created_date":"2025-10-10T00:00:00"}