{"id":"https://openalex.org/W4416200174","doi":"https://doi.org/10.1155/cplx/9715552","title":"Bifurcation Dynamics and Complex Behavior in a Discrete\u2010Time Predator\u2013Prey Model With Cross\u2010Species Interaction Incorporating Holling Type\u2010II Response","display_name":"Bifurcation Dynamics and Complex Behavior in a Discrete\u2010Time Predator\u2013Prey Model With Cross\u2010Species Interaction Incorporating Holling Type\u2010II Response","publication_year":2025,"publication_date":"2025-01-01","ids":{"openalex":"https://openalex.org/W4416200174","doi":"https://doi.org/10.1155/cplx/9715552"},"language":"en","primary_location":{"id":"doi:10.1155/cplx/9715552","is_oa":true,"landing_page_url":"https://doi.org/10.1155/cplx/9715552","pdf_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1155/cplx/9715552","source":{"id":"https://openalex.org/S207319443","display_name":"Complexity","issn_l":"1076-2787","issn":["1076-2787","1099-0526"],"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":"Complexity","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1155/cplx/9715552","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5028967774","display_name":"Muhammad Rafaqat","orcid":"https://orcid.org/0000-0002-7437-5994"},"institutions":[{"id":"https://openalex.org/I142204992","display_name":"University of Lahore","ror":"https://ror.org/051jrjw38","country_code":"PK","type":"education","lineage":["https://openalex.org/I142204992"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Muhammad Rafaqat","raw_affiliation_strings":["Department of Mathematics and Statistics ,  The University of Lahore ,  Lahore ,  Pakistan ,  uol.edu.pk"],"raw_orcid":"https://orcid.org/0000-0002-7437-5994","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics ,  The University of Lahore ,  Lahore ,  Pakistan ,  uol.edu.pk","institution_ids":["https://openalex.org/I142204992"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103214864","display_name":"Syed Tauseef Saeed","orcid":"https://orcid.org/0009-0001-4221-052X"},"institutions":[{"id":"https://openalex.org/I142204992","display_name":"University of Lahore","ror":"https://ror.org/051jrjw38","country_code":"PK","type":"education","lineage":["https://openalex.org/I142204992"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Syed Tauseef Saeed","raw_affiliation_strings":["Department of Mathematics and Statistics ,  The University of Lahore ,  Lahore ,  Pakistan ,  uol.edu.pk"],"raw_orcid":"https://orcid.org/0009-0001-4221-052X","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics ,  The University of Lahore ,  Lahore ,  Pakistan ,  uol.edu.pk","institution_ids":["https://openalex.org/I142204992"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014356417","display_name":"S. Saleem","orcid":"https://orcid.org/0000-0001-6217-3626"},"institutions":[{"id":"https://openalex.org/I82952536","display_name":"King Khalid University","ror":"https://ror.org/052kwzs30","country_code":"SA","type":"education","lineage":["https://openalex.org/I82952536"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Salman Saleem","raw_affiliation_strings":["Center for Artificial Intelligence (CAI) ,  King Khalid University ,  Abha ,  61421 ,  Saudi Arabia ,  kku.edu.sa","Department of Mathematics ,  College of Science ,  King Khalid University ,  Abha ,  61413 ,  Saudi Arabia ,  kku.edu.sa"],"raw_orcid":"https://orcid.org/0000-0001-6217-3626","affiliations":[{"raw_affiliation_string":"Center for Artificial Intelligence (CAI) ,  King Khalid University ,  Abha ,  61421 ,  Saudi Arabia ,  kku.edu.sa","institution_ids":["https://openalex.org/I82952536"]},{"raw_affiliation_string":"Department of Mathematics ,  College of Science ,  King Khalid University ,  Abha ,  61413 ,  Saudi Arabia ,  kku.edu.sa","institution_ids":["https://openalex.org/I82952536"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5056254181","display_name":"Feyisa Edosa Merga","orcid":"https://orcid.org/0000-0002-8737-8684"},"institutions":[{"id":"https://openalex.org/I1443707","display_name":"Jimma University","ror":"https://ror.org/05eer8g02","country_code":"ET","type":"education","lineage":["https://openalex.org/I1443707"]}],"countries":["ET"],"is_corresponding":true,"raw_author_name":"Feyisa Edosa Merga","raw_affiliation_strings":["Department of Mathematics ,  Jimma University ,  Jimma ,  Ethiopia ,  ju.edu.et"],"raw_orcid":"https://orcid.org/0000-0002-8737-8684","affiliations":[{"raw_affiliation_string":"Department of Mathematics ,  Jimma University ,  Jimma ,  Ethiopia ,  ju.edu.et","institution_ids":["https://openalex.org/I1443707"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5056254181"],"corresponding_institution_ids":["https://openalex.org/I1443707"],"apc_list":{"value":2300,"currency":"USD","value_usd":2300},"apc_paid":{"value":2300,"currency":"USD","value_usd":2300},"fwci":3.2389,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.92961404,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":98,"max":99},"biblio":{"volume":"2025","issue":"1","first_page":null,"last_page":null},"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":0.9487000107765198,"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":0.9487000107765198,"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/T10244","display_name":"Chaos control and synchronization","score":0.012900000438094139,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.005400000140070915,"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/phase-portrait","display_name":"Phase portrait","score":0.7311000227928162},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6639000177383423},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.6093999743461609},{"id":"https://openalex.org/keywords/chaotic","display_name":"Chaotic","score":0.5425999760627747},{"id":"https://openalex.org/keywords/center-manifold","display_name":"Center manifold","score":0.5123999714851379},{"id":"https://openalex.org/keywords/invariant-manifold","display_name":"Invariant manifold","score":0.503000020980835},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.4934999942779541},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.47600001096725464},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.4620000123977661},{"id":"https://openalex.org/keywords/exponential-stability","display_name":"Exponential stability","score":0.41290000081062317}],"concepts":[{"id":"https://openalex.org/C122283620","wikidata":"https://www.wikidata.org/wiki/Q2532943","display_name":"Phase portrait","level":4,"score":0.7311000227928162},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6639000177383423},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.6093999743461609},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5559999942779541},{"id":"https://openalex.org/C2777052490","wikidata":"https://www.wikidata.org/wiki/Q5072826","display_name":"Chaotic","level":2,"score":0.5425999760627747},{"id":"https://openalex.org/C74774103","wikidata":"https://www.wikidata.org/wiki/Q4504202","display_name":"Center manifold","level":5,"score":0.5123999714851379},{"id":"https://openalex.org/C20483540","wikidata":"https://www.wikidata.org/wiki/Q4200516","display_name":"Invariant manifold","level":2,"score":0.503000020980835},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.4934999942779541},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.47600001096725464},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.4620000123977661},{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.41290000081062317},{"id":"https://openalex.org/C145912823","wikidata":"https://www.wikidata.org/wiki/Q113558","display_name":"Dynamics (music)","level":2,"score":0.41100001335144043},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.3856000006198883},{"id":"https://openalex.org/C126701199","wikidata":"https://www.wikidata.org/wiki/Q264224","display_name":"Complex dynamics","level":2,"score":0.3709000051021576},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3698999881744385},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.3650999963283539},{"id":"https://openalex.org/C191544260","wikidata":"https://www.wikidata.org/wiki/Q1238630","display_name":"Lyapunov exponent","level":3,"score":0.3386000096797943},{"id":"https://openalex.org/C207821765","wikidata":"https://www.wikidata.org/wiki/Q405372","display_name":"Instability","level":2,"score":0.33160001039505005},{"id":"https://openalex.org/C2778671503","wikidata":"https://www.wikidata.org/wiki/Q7542110","display_name":"Slow manifold","level":3,"score":0.3221000134944916},{"id":"https://openalex.org/C174914120","wikidata":"https://www.wikidata.org/wiki/Q4914973","display_name":"Biological applications of bifurcation theory","level":5,"score":0.3199999928474426},{"id":"https://openalex.org/C2776829284","wikidata":"https://www.wikidata.org/wiki/Q1341651","display_name":"Lyapunov stability","level":3,"score":0.3066999912261963},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.3018999993801117},{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.30079999566078186},{"id":"https://openalex.org/C75380026","wikidata":"https://www.wikidata.org/wiki/Q868454","display_name":"Euler method","level":3,"score":0.29760000109672546},{"id":"https://openalex.org/C101225446","wikidata":"https://www.wikidata.org/wiki/Q740190","display_name":"Saddle-node bifurcation","level":4,"score":0.29589998722076416},{"id":"https://openalex.org/C33962884","wikidata":"https://www.wikidata.org/wiki/Q378637","display_name":"Dynamical system (definition)","level":3,"score":0.2946999967098236},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2791999876499176},{"id":"https://openalex.org/C205606062","wikidata":"https://www.wikidata.org/wiki/Q5249645","display_name":"Decoupling (probability)","level":2,"score":0.2770000100135803},{"id":"https://openalex.org/C77405623","wikidata":"https://www.wikidata.org/wiki/Q598451","display_name":"System dynamics","level":2,"score":0.27469998598098755},{"id":"https://openalex.org/C55689738","wikidata":"https://www.wikidata.org/wiki/Q15963867","display_name":"Discrete time and continuous time","level":2,"score":0.2687999904155731},{"id":"https://openalex.org/C50429861","wikidata":"https://www.wikidata.org/wiki/Q3554830","display_name":"Stable manifold","level":2,"score":0.2648000121116638},{"id":"https://openalex.org/C86243350","wikidata":"https://www.wikidata.org/wiki/Q2595915","display_name":"Bifurcation diagram","level":4,"score":0.26190000772476196},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.2614000141620636},{"id":"https://openalex.org/C85075877","wikidata":"https://www.wikidata.org/wiki/Q1154357","display_name":"Bifurcation theory","level":4,"score":0.2596000134944916},{"id":"https://openalex.org/C203479927","wikidata":"https://www.wikidata.org/wiki/Q5165939","display_name":"Controller (irrigation)","level":2,"score":0.25209999084472656}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1155/cplx/9715552","is_oa":true,"landing_page_url":"https://doi.org/10.1155/cplx/9715552","pdf_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1155/cplx/9715552","source":{"id":"https://openalex.org/S207319443","display_name":"Complexity","issn_l":"1076-2787","issn":["1076-2787","1099-0526"],"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":"Complexity","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:d267b3f3c1824a72957527855662d93d","is_oa":true,"landing_page_url":"https://doaj.org/article/d267b3f3c1824a72957527855662d93d","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":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Complexity, Vol 2025 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/cplx/9715552","is_oa":true,"landing_page_url":"https://doi.org/10.1155/cplx/9715552","pdf_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1155/cplx/9715552","source":{"id":"https://openalex.org/S207319443","display_name":"Complexity","issn_l":"1076-2787","issn":["1076-2787","1099-0526"],"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":"Complexity","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5298213512","display_name":null,"funder_award_id":"RGP.2/339/46","funder_id":"https://openalex.org/F33954902","funder_display_name":"Deanship of Scientific Research, King Khalid University"}],"funders":[{"id":"https://openalex.org/F33954902","display_name":"Deanship of Scientific Research, King Khalid University","ror":null},{"id":"https://openalex.org/F4320324433","display_name":"King Khalid University","ror":"https://ror.org/052kwzs30"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4416200174.pdf","grobid_xml":"https://content.openalex.org/works/W4416200174.grobid-xml"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W1827761481","https://openalex.org/W1972726492","https://openalex.org/W1997297480","https://openalex.org/W2035379472","https://openalex.org/W2089876099","https://openalex.org/W2102892532","https://openalex.org/W2103948957","https://openalex.org/W2129782605","https://openalex.org/W2131915744","https://openalex.org/W3037283348","https://openalex.org/W3103408287","https://openalex.org/W4206744643","https://openalex.org/W4220949616","https://openalex.org/W4236990395","https://openalex.org/W4251015067","https://openalex.org/W4385446777","https://openalex.org/W4403707351","https://openalex.org/W4406255483","https://openalex.org/W4409169707"],"related_works":[],"abstract_inverted_index":{"We":[0],"investigate":[1],"the":[2,30,39,51,64,73,99,105,133,148],"nonlinear":[3,46,167],"dynamics":[4,116],"of":[5,41,56,66,109],"a":[6,12,19,122,156],"discrete\u2010time":[7],"predator\u2013prey":[8],"model":[9],"governed":[10],"by":[11,118],"Holling":[13],"Type\u2010II":[14],"functional":[15],"response.":[16],"Starting":[17],"from":[18,75],"biologically":[20],"motivated":[21],"continuous\u2010time":[22],"system,":[23],"we":[24,95,120],"derive":[25,96],"its":[26],"discrete":[27,163],"analogue":[28],"via":[29],"explicit":[31],"Euler":[32],"method":[33],"and":[34,53,69,80,91,107,127,144,161],"employ":[35],"nondimensionalization":[36],"to":[37,78,103],"reduce":[38],"number":[40],"parameters.":[42],"The":[43,151],"resulting":[44],"two\u2010dimensional":[45],"system":[47,84,135],"is":[48],"analyzed":[49],"for":[50,63,98,159],"existence":[52],"local":[54,137],"stability":[55,108],"fixed":[57],"points.":[58],"Analytical":[59],"conditions":[60,130],"are":[61],"established":[62],"occurrence":[65],"flip":[67],"(period\u2010doubling)":[68],"Neimark\u2013Sacker":[70],"bifurcations,":[71,119],"characterizing":[72],"transition":[74],"steady":[76],"states":[77],"periodic":[79],"quasi\u2010periodic":[81],"behavior":[82],"as":[83],"parameters":[85],"vary.":[86],"Employing":[87],"center":[88],"manifold":[89],"theory":[90],"normal":[92],"form":[93],"computations,":[94],"expressions":[97],"first":[100],"Lyapunov":[101],"coefficient":[102],"determine":[104],"direction":[106],"bifurcating":[110],"invariant":[111],"curves.":[112],"To":[113],"suppress":[114],"chaotic":[115],"induced":[117],"implement":[121],"hybrid":[123],"feedback":[124],"control":[125],"mechanism":[126],"establish":[128],"sufficient":[129],"under":[131],"which":[132],"controlled":[134],"regains":[136],"asymptotic":[138],"stability.":[139],"Numerical":[140],"results,":[141],"bifurcation":[142],"diagrams,":[143],"phase":[145],"portraits":[146],"corroborate":[147],"theoretical":[149],"results.":[150],"framework":[152],"developed":[153],"herein":[154],"provides":[155],"rigorous":[157],"foundation":[158],"analyzing":[160],"stabilizing":[162],"ecological":[164],"models":[165],"with":[166],"interaction":[168],"terms.":[169]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-11-14T00:00:00"}