{"id":"https://openalex.org/W3113092246","doi":"https://doi.org/10.1142/s0218127420502247","title":"A Diffusive One-Prey and Two-Cooperative-Predators Model with C\u2013M Functional Response","display_name":"A Diffusive One-Prey and Two-Cooperative-Predators Model with C\u2013M Functional Response","publication_year":2020,"publication_date":"2020-12-09","ids":{"openalex":"https://openalex.org/W3113092246","doi":"https://doi.org/10.1142/s0218127420502247","mag":"3113092246"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127420502247","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127420502247","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","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/A5015136648","display_name":"Xiao Yan","orcid":"https://orcid.org/0000-0003-0464-0200"},"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":"Xiao Yan","raw_affiliation_strings":["School of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an, Shaanxi 710119, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an, Shaanxi 710119, P. R. China","institution_ids":["https://openalex.org/I88830068"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101944007","display_name":"LI Yan-ling","orcid":"https://orcid.org/0000-0002-9614-4927"},"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":true,"raw_author_name":"Yanling Li","raw_affiliation_strings":["School of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an, Shaanxi 710119, P. R. China"],"raw_orcid":"https://orcid.org/0000-0002-9614-4927","affiliations":[{"raw_affiliation_string":"School of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an, Shaanxi 710119, P. R. China","institution_ids":["https://openalex.org/I88830068"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5010410185","display_name":"Yane Wang","orcid":"https://orcid.org/0000-0003-0381-0482"},"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":"Yan\u2019e Wang","raw_affiliation_strings":["School of Computer Science, Shaanxi Normal University, Xi\u2019an, Shaanxi 710119, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Computer Science, Shaanxi Normal University, Xi\u2019an, Shaanxi 710119, P. R. China","institution_ids":["https://openalex.org/I88830068"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5101944007"],"corresponding_institution_ids":["https://openalex.org/I88830068"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.23791279,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"30","issue":"15","first_page":"2050224","last_page":"2050224"},"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":0.991100013256073,"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9857000112533569,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/mathematics","display_name":"Mathematics","score":0.7412270307540894},{"id":"https://openalex.org/keywords/functional-response","display_name":"Functional response","score":0.6761213541030884},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.6474647521972656},{"id":"https://openalex.org/keywords/monotone-polygon","display_name":"Monotone polygon","score":0.578000009059906},{"id":"https://openalex.org/keywords/steady-state","display_name":"Steady state (chemistry)","score":0.5678137540817261},{"id":"https://openalex.org/keywords/dirichlet-boundary-condition","display_name":"Dirichlet boundary condition","score":0.5251112580299377},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4934455454349518},{"id":"https://openalex.org/keywords/fixed-point-index","display_name":"Fixed-point index","score":0.47367459535598755},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.4578482210636139},{"id":"https://openalex.org/keywords/maximum-principle","display_name":"Maximum principle","score":0.45556971430778503},{"id":"https://openalex.org/keywords/extinction","display_name":"Extinction (optical mineralogy)","score":0.4212135076522827},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4035136103630066},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.4012622535228729},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.3852904140949249},{"id":"https://openalex.org/keywords/predation","display_name":"Predation","score":0.3766963481903076},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.2290005087852478},{"id":"https://openalex.org/keywords/predator","display_name":"Predator","score":0.14294219017028809},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.11252027750015259},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10495331883430481},{"id":"https://openalex.org/keywords/optimal-control","display_name":"Optimal control","score":0.08541327714920044}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7412270307540894},{"id":"https://openalex.org/C110201527","wikidata":"https://www.wikidata.org/wiki/Q5508845","display_name":"Functional response","level":4,"score":0.6761213541030884},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.6474647521972656},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.578000009059906},{"id":"https://openalex.org/C8171440","wikidata":"https://www.wikidata.org/wiki/Q903414","display_name":"Steady state (chemistry)","level":2,"score":0.5678137540817261},{"id":"https://openalex.org/C110167270","wikidata":"https://www.wikidata.org/wiki/Q1193699","display_name":"Dirichlet boundary condition","level":3,"score":0.5251112580299377},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4934455454349518},{"id":"https://openalex.org/C2780668686","wikidata":"https://www.wikidata.org/wiki/Q5456324","display_name":"Fixed-point index","level":3,"score":0.47367459535598755},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.4578482210636139},{"id":"https://openalex.org/C31010330","wikidata":"https://www.wikidata.org/wiki/Q1914255","display_name":"Maximum principle","level":3,"score":0.45556971430778503},{"id":"https://openalex.org/C101991246","wikidata":"https://www.wikidata.org/wiki/Q5422078","display_name":"Extinction (optical mineralogy)","level":2,"score":0.4212135076522827},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4035136103630066},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4012622535228729},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.3852904140949249},{"id":"https://openalex.org/C188382862","wikidata":"https://www.wikidata.org/wiki/Q170430","display_name":"Predation","level":2,"score":0.3766963481903076},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2290005087852478},{"id":"https://openalex.org/C96857902","wikidata":"https://www.wikidata.org/wiki/Q2138108","display_name":"Predator","level":3,"score":0.14294219017028809},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.11252027750015259},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10495331883430481},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.08541327714920044},{"id":"https://openalex.org/C147789679","wikidata":"https://www.wikidata.org/wiki/Q11372","display_name":"Physical chemistry","level":1,"score":0.0},{"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/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","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/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127420502247","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127420502247","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Life in Land","score":0.5600000023841858,"id":"https://metadata.un.org/sdg/15"}],"awards":[{"id":"https://openalex.org/G1811438643","display_name":null,"funder_award_id":"61907030","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2913164961","display_name":null,"funder_award_id":"GK201903088","funder_id":"https://openalex.org/F4320335787","funder_display_name":"Fundamental Research Funds for the Central Universities"},{"id":"https://openalex.org/G7025122447","display_name":null,"funder_award_id":"1171262","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7169471730","display_name":"\u591a\u5bc6\u5ea6\u53cd\u5e94\u6269\u6563\u6a21\u578b\u7684\u8ba1\u7b97\u673a\u8f85\u52a9\u5206\u6790\u7814\u7a76\u4e0e\u5e94\u7528","funder_award_id":"61672021","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8388672120","display_name":null,"funder_award_id":"2018JQ1021","funder_id":"https://openalex.org/F4320324173","funder_display_name":"Natural Science Foundation of Shaanxi Province"},{"id":"https://openalex.org/G8539411763","display_name":"\u53cd\u5e94\u6269\u6563\u6a21\u578b\u65f6\u7a7a\u6591\u56fe\u7684\u8ba1\u7b97\u673a\u8f85\u52a9\u5206\u6790\u7814\u7a76\u4e0e\u5e94\u7528","funder_award_id":"61872227","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320324173","display_name":"Natural Science Foundation of Shaanxi Province","ror":null},{"id":"https://openalex.org/F4320335787","display_name":"Fundamental Research Funds for the Central Universities","ror":null}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W997355000","https://openalex.org/W1175360599","https://openalex.org/W1588745929","https://openalex.org/W1970373918","https://openalex.org/W1974891663","https://openalex.org/W1987315552","https://openalex.org/W1999852355","https://openalex.org/W2012721015","https://openalex.org/W2013350755","https://openalex.org/W2025516574","https://openalex.org/W2027857246","https://openalex.org/W2027871831","https://openalex.org/W2042129337","https://openalex.org/W2043619255","https://openalex.org/W2046724040","https://openalex.org/W2059517472","https://openalex.org/W2073862313","https://openalex.org/W2084216026","https://openalex.org/W2092696532","https://openalex.org/W2096356504","https://openalex.org/W2125914782","https://openalex.org/W2252696324","https://openalex.org/W2323873405","https://openalex.org/W2325470240","https://openalex.org/W2955711053","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2129001831","https://openalex.org/W4234536609","https://openalex.org/W2365962591","https://openalex.org/W2392461588","https://openalex.org/W2037096150","https://openalex.org/W1970373918","https://openalex.org/W2392011893","https://openalex.org/W2883225148","https://openalex.org/W2530403616","https://openalex.org/W2058947045"],"abstract_inverted_index":{"This":[0],"paper":[1],"is":[2,61,72],"dedicated":[3],"to":[4,19,151,158],"a":[5,8],"study":[6,120],"of":[7,28,51,66,69,109,129,163],"diffusive":[9],"one-prey":[10],"and":[11,38,49,63,91,112,127,140],"two-cooperative-predators":[12],"model":[13],"with":[14],"C\u2013M":[15],"functional":[16],"response":[17],"subject":[18],"Dirichlet":[20],"boundary":[21],"conditions.":[22],"We":[23],"first":[24],"discuss":[25],"the":[26,33,39,43,47,64,103,107,113,121,125,130,134,153,161,166],"existence":[27],"positive":[29],"steady":[30,81,96],"states":[31,53,82],"by":[32,102,133],"fixed":[34],"point":[35],"index":[36],"theory":[37],"degree":[40],"theory.":[41],"In":[42,117],"meantime,":[44],"we":[45,119],"analyze":[46],"uniqueness":[48],"stability":[50],"coexistence":[52],"under":[54,88],"conditions":[55],"that":[56,84],"one":[57,93],"predator\u2019s":[58],"consumer":[59],"rate":[60],"small":[62],"effect":[65],"interference":[67],"intensity":[68],"another":[70],"predator":[71],"large.":[73],"Then,":[74],"steady-state":[75],"bifurcations":[76],"from":[77,92],"two":[78],"strong":[79],"semi-trivial":[80,95],"(provided":[83],"they":[85],"uniquely":[86],"exist":[87],"some":[89],"conditions)":[90],"weak":[94],"state":[97],"are":[98,147],"investigated":[99],"in":[100],"detail":[101],"Crandall\u2013Rabinowitz":[104],"bifurcation":[105],"theorem,":[106],"technique":[108],"space":[110],"decomposition":[111],"implicit":[114],"function":[115],"theorem.":[116],"addition,":[118],"asymptotic":[122],"behaviors":[123],"including":[124],"extinction":[126],"permanence":[128],"time-dependent":[131],"system":[132],"comparison":[135],"principle,":[136],"upper-lower":[137],"solution":[138],"method":[139],"monotone":[141],"iteration":[142],"scheme.":[143],"Finally,":[144],"numerical":[145],"simulations":[146],"done":[148],"not":[149],"only":[150],"validate":[152],"theoretical":[154],"conclusions,":[155],"but":[156],"also":[157],"further":[159],"clarify":[160],"impacts":[162],"parameters":[164],"on":[165],"three":[167],"species.":[168]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}