{"id":"https://openalex.org/W7204170205","doi":"https://doi.org/10.48550/arxiv.2608.22047","title":"The trace field on the prey nullcline: criticality of planar Hopf bifurcations on the admissible branch","display_name":"The trace field on the prey nullcline: criticality of planar Hopf bifurcations on the admissible branch","publication_year":2026,"publication_date":"2026-08-22","ids":{"openalex":"https://openalex.org/W7204170205","doi":"https://doi.org/10.48550/arxiv.2608.22047"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.22047","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.22047","pdf_url":"https://arxiv.org/pdf/2608.22047","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.22047","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5038924834","display_name":"E. Chan-L\u00f3pez","orcid":"https://orcid.org/0009-0003-7712-7907","observed_orcids":["https://orcid.org/0009-0003-7712-7907"]},"institutions":[{"id":"https://openalex.org/I93173995","display_name":"Universidad Ju\u00e1rez Aut\u00f3noma de Tabasco","ror":"https://ror.org/04ee58018","country_code":"MX","type":"education","lineage":["https://openalex.org/I93173995"]}],"countries":["MX"],"is_corresponding":true,"raw_author_name":"Chan-L\u00f3pez, E.","raw_affiliation_strings":["Divisi\u00f3n Acad\u00e9mica de Ciencias B\u00e1sicas , Universidad Ju\u00e1rez Aut\u00f3noma de Tabasco , 86690 Cunduac\u00e1n , Tabasco , Mexico"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Divisi\u00f3n Acad\u00e9mica de Ciencias B\u00e1sicas , Universidad Ju\u00e1rez Aut\u00f3noma de Tabasco , 86690 Cunduac\u00e1n , Tabasco , Mexico","institution_ids":["https://openalex.org/I93173995"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5038924834"],"corresponding_institution_ids":["https://openalex.org/I93173995"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"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":0.8345000147819519,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.8345000147819519,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.1639000028371811,"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"}},{"id":"https://openalex.org/T13377","display_name":"Ecosystem dynamics and resilience","score":0.00039999998989515007,"subfield":{"id":"https://openalex.org/subfields/2306","display_name":"Global and Planetary Change"},"field":{"id":"https://openalex.org/fields/23","display_name":"Environmental Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/trace-field","display_name":"trace field","score":0.6909999847412109},{"id":"https://openalex.org/keywords/predator-prey-dynamics","display_name":"predator-prey dynamics","score":0.7699999809265137},{"id":"https://openalex.org/keywords/oscillatory-instability","display_name":"oscillatory instability","score":0.6859999895095825},{"id":"https://openalex.org/keywords/criticality","display_name":"criticality","score":0.3479999899864197},{"id":"https://openalex.org/keywords/bifurcation-unfolding","display_name":"bifurcation unfolding","score":0.3109999895095825}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7703999876976013},{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.6700000166893005},{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.6507999897003174},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5139999985694885},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.4878000020980835},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.48010000586509705},{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.46059998869895935},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.44290000200271606},{"id":"https://openalex.org/C125611927","wikidata":"https://www.wikidata.org/wiki/Q17008131","display_name":"Criticality","level":2,"score":0.43309998512268066},{"id":"https://openalex.org/C155092403","wikidata":"https://www.wikidata.org/wiki/Q1785924","display_name":"Pitchfork bifurcation","level":5,"score":0.4016000032424927},{"id":"https://openalex.org/C128803854","wikidata":"https://www.wikidata.org/wiki/Q1307821","display_name":"Eigenfunction","level":3,"score":0.39430001378059387},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.36399999260902405},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.3490000069141388},{"id":"https://openalex.org/C47556283","wikidata":"https://www.wikidata.org/wiki/Q1058314","display_name":"Diffeomorphism","level":2,"score":0.3222000002861023},{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.3181999921798706},{"id":"https://openalex.org/C101225446","wikidata":"https://www.wikidata.org/wiki/Q740190","display_name":"Saddle-node bifurcation","level":4,"score":0.30799999833106995},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.3005000054836273},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.28139999508857727},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.2784999907016754},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.2759999930858612},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.27300000190734863},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.26969999074935913},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.2694000005722046},{"id":"https://openalex.org/C106195933","wikidata":"https://www.wikidata.org/wiki/Q7847935","display_name":"Truncation (statistics)","level":2,"score":0.2685000002384186},{"id":"https://openalex.org/C50429861","wikidata":"https://www.wikidata.org/wiki/Q3554830","display_name":"Stable manifold","level":2,"score":0.26330000162124634}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.22047","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.22047","pdf_url":"https://arxiv.org/pdf/2608.22047","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.22047","doi":"https://doi.org/10.48550/arxiv.2608.22047","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.22047","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.22047","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.22047","pdf_url":"https://arxiv.org/pdf/2608.22047","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204170205.pdf","grobid_xml":"https://content.openalex.org/works/W7204170205.grobid-xml"},"referenced_works_count":12,"referenced_works":["https://openalex.org/W1827761481","https://openalex.org/W1983173496","https://openalex.org/W1997297480","https://openalex.org/W2004728046","https://openalex.org/W2041263000","https://openalex.org/W2052125279","https://openalex.org/W2060852353","https://openalex.org/W2061879675","https://openalex.org/W2081696580","https://openalex.org/W2086851860","https://openalex.org/W2323873405","https://openalex.org/W7213559850"],"related_works":["https://openalex.org/W7140791292","https://openalex.org/W2796919817","https://openalex.org/W2786248442","https://openalex.org/W4410599594","https://openalex.org/W4390278982","https://openalex.org/W3193667357","https://openalex.org/W2903323143","https://openalex.org/W2619873116","https://openalex.org/W4394983254","https://openalex.org/W2809303293"],"abstract_inverted_index":{"Along":[0],"the":[1,9,21,35,54,84,96,113,130,138,141,156,160,176,183,189,199,202],"prey":[2],"nullcline":[3,63,131],"$y=g(x)$":[4],"of":[5,29,79,86,112,159,201,211],"a":[6,91],"planar":[7,120],"system,":[8],"Jacobian":[10],"entry":[11],"satisfies":[12],"$J_{11}=P\\,g'$,":[13],"where":[14,46,168],"$P=-f_{1y}|_\u03b3$.":[15],"Under":[16],"$P>0$":[17],"and":[18,60,66,88,173,224],"predator":[19],"self-damping,":[20],"Hopf":[22,36,121,177],"trace":[23,204],"condition":[24],"forces":[25],"$g'(x^\\ast)>0$:":[26],"critical":[27],"points":[28],"$g$":[30],"are":[31,116,226],"spectral":[32],"barriers,":[33],"so":[34],"locus":[37],"lies":[38],"on":[39],"ascending":[40],"branches.":[41],"This":[42],"geometric":[43],"localization":[44],"determines":[45],"oscillatory":[47],"instability":[48],"can":[49],"occur;":[50],"we":[51],"determine":[52],"how":[53],"bifurcation":[55,190],"unfolds":[56],"there.":[57],"Let":[58],"$D=f_{1x}+f_{2y}$":[59],"$\u03c4=D|_\u03b3$.":[61],"Using":[62],"coordinates":[64],"$(u=x,v=y-g(x))$":[65],"Hadamard's":[67],"lemma":[68],"gives":[69,125,166],"$\\dot":[70,72],"u=-vH$,":[71],"v=W$,":[73],"with":[74,155],"$H(u,0)=P$.":[75],"The":[76,206],"mixed":[77],"jet":[78,115],"$W$":[80],"is":[81,209],"determined":[82],"by":[83,217],"jets":[85],"$\u03c4$":[87],"$\u03bd=f_2|_\u03b3$,":[89],"yielding":[90],"closed":[92],"five-term":[93],"formula":[94],"for":[95,105],"first":[97,192],"Lyapunov":[98],"coefficient":[99],"$\\ell_1$":[100],"in":[101],"trace-field":[102],"form,":[103],"valid":[104],"predator-dependent":[106,152],"functional":[107],"responses.":[108],"Three":[109],"distinct":[110],"reductions":[111],"cubic":[114],"established.":[117],"In":[118],"general":[119],"bifurcations,":[122],"eigenvector":[123,212],"pairing":[124],"$\\partial\\ell_1/\\partial":[126],"f_{1yyy}=\\partial\\ell_1/\\partial":[127],"f_{2xxx}=0$.":[128],"Straightening":[129],"removes":[132,145],"two":[133],"further":[134],"slots,":[135],"while":[136,214],"within":[137],"Gause":[139],"class":[140],"reparametrization":[142],"$\u03c6\\mapsto":[143],"g$":[144],"$p'''$.":[146],"They":[147],"have":[148],"different":[149],"scopes.":[150],"For":[151],"responses,":[153],"comparison":[154],"$y$-affine":[157],"truncation":[158],"$3$-jet":[161],"at":[162,191,195],"fixed":[163],"linear":[164],"part":[165],"$\\ell_1=\\ell_1^{\\mathrm{aff}}+\u0394$,":[167],"$\u0394=\u039b\\bigl(f_{1xyy}+\u03c9_0^{-2}\\langle\\partial_x":[169],"f,\\nabla":[170,180],"D\\rangle":[171],"f_{1yy}\\bigr)$":[172],"$\u039b>0$.":[174],"On":[175],"locus,":[178],"$\\langle\\partial_x":[179],"D\\rangle=J_{11}\u03c4'+(\u03c9_0^2/P)D_y$.":[181],"Thus":[182],"same":[184],"quantity":[185],"$J_{11}=Pg'$":[186],"that":[187],"localizes":[188],"order":[193,197],"reappears":[194],"third":[196],"as":[198],"weight":[200],"tangential":[203],"derivative.":[205],"bridge":[207],"identity":[208],"independent":[210],"normalization,":[213],"$\\ell_1,\\ell_1^{\\mathrm{aff}},\u0394,\u039b$":[215],"scale":[216],"$|c|^2$":[218],"under":[219],"$q\\mapsto":[220],"cq$;":[221],"their":[222],"signs":[223],"ratios":[225],"unchanged.":[227]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-26T00:00:00"}