{"id":"https://openalex.org/W2080755434","doi":"https://doi.org/10.1137/100808381","title":"Stability of Spiky Solutions in a Competition Model with Cross-Diffusion","display_name":"Stability of Spiky Solutions in a Competition Model with Cross-Diffusion","publication_year":2011,"publication_date":"2011-01-01","ids":{"openalex":"https://openalex.org/W2080755434","doi":"https://doi.org/10.1137/100808381","mag":"2080755434"},"language":"en","primary_location":{"id":"doi:10.1137/100808381","is_oa":false,"landing_page_url":"https://doi.org/10.1137/100808381","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","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/A5015774352","display_name":"Th\u00e9odore Kolokolnikov","orcid":"https://orcid.org/0000-0002-7179-4325"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Theodore Kolokolnikov","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5057351789","display_name":"Juncheng Wei","orcid":"https://orcid.org/0000-0001-5262-477X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Juncheng Wei","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.2429,"has_fulltext":false,"cited_by_count":21,"citation_normalized_percentile":{"value":0.8766768,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"71","issue":"4","first_page":"1428","last_page":"1457"},"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":0.9998000264167786,"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":0.9998000264167786,"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/T11829","display_name":"Mathematical Biology Tumor Growth","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/T11764","display_name":"Evolution and Genetic Dynamics","score":0.9977999925613403,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.7230908274650574},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.6068850755691528},{"id":"https://openalex.org/keywords/domain","display_name":"Domain (mathematical analysis)","score":0.5856277346611023},{"id":"https://openalex.org/keywords/diffusion","display_name":"Diffusion","score":0.5742765665054321},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.5281313061714172},{"id":"https://openalex.org/keywords/hypergeometric-function","display_name":"Hypergeometric function","score":0.512582004070282},{"id":"https://openalex.org/keywords/instability","display_name":"Instability","score":0.49256235361099243},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.45763447880744934},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.4564386308193207},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.44275015592575073},{"id":"https://openalex.org/keywords/eigenfunction","display_name":"Eigenfunction","score":0.4372822940349579},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.41084885597229004},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.39272841811180115},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.37078481912612915},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.334162175655365},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.14371979236602783},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.08490756154060364}],"concepts":[{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.7230908274650574},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.6068850755691528},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.5856277346611023},{"id":"https://openalex.org/C69357855","wikidata":"https://www.wikidata.org/wiki/Q163214","display_name":"Diffusion","level":2,"score":0.5742765665054321},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5281313061714172},{"id":"https://openalex.org/C197320386","wikidata":"https://www.wikidata.org/wiki/Q21028472","display_name":"Hypergeometric function","level":2,"score":0.512582004070282},{"id":"https://openalex.org/C207821765","wikidata":"https://www.wikidata.org/wiki/Q405372","display_name":"Instability","level":2,"score":0.49256235361099243},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.45763447880744934},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.4564386308193207},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.44275015592575073},{"id":"https://openalex.org/C128803854","wikidata":"https://www.wikidata.org/wiki/Q1307821","display_name":"Eigenfunction","level":3,"score":0.4372822940349579},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.41084885597229004},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.39272841811180115},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.37078481912612915},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.334162175655365},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.14371979236602783},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.08490756154060364},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/100808381","is_oa":false,"landing_page_url":"https://doi.org/10.1137/100808381","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Life in Land","id":"https://metadata.un.org/sdg/15","score":0.6899999976158142}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":25,"referenced_works":["https://openalex.org/W1511279087","https://openalex.org/W1605268299","https://openalex.org/W1895763814","https://openalex.org/W1965638376","https://openalex.org/W1967724962","https://openalex.org/W1973118029","https://openalex.org/W1983932758","https://openalex.org/W1986187534","https://openalex.org/W1987347396","https://openalex.org/W2015939010","https://openalex.org/W2017324017","https://openalex.org/W2026570637","https://openalex.org/W2029615251","https://openalex.org/W2038975311","https://openalex.org/W2040731886","https://openalex.org/W2050434873","https://openalex.org/W2066496593","https://openalex.org/W2076644075","https://openalex.org/W2092051148","https://openalex.org/W2098607588","https://openalex.org/W2113257692","https://openalex.org/W2121979536","https://openalex.org/W2131610689","https://openalex.org/W2267029276","https://openalex.org/W2316491317"],"related_works":["https://openalex.org/W2053448926","https://openalex.org/W2899860737","https://openalex.org/W1990842824","https://openalex.org/W2006122712","https://openalex.org/W2991529292","https://openalex.org/W128178425","https://openalex.org/W2767914060","https://openalex.org/W3214805258","https://openalex.org/W1575765233","https://openalex.org/W2989723623"],"abstract_inverted_index":{"We":[0,82,173],"consider":[1,74],"the":[2,9,28,75,91,111,125,141,150,154,176,189,203,215,232,240,245],"Shigesada\u2013Kawasaki\u2013Teramoto":[3],"model":[4],"of":[5,11,15,22,30,78,103,113,127,140,147,168,178,191,217,239],"species":[6],"segregation":[7],"in":[8,27,36,39,135,164,223],"limit":[10,38],"high":[12],"cross-diffusion":[13],"rate":[14,21],"one":[16,40,148,169,195,224],"species,":[17],"and":[18,46,59,119,225,235],"small":[19,129,208],"diffusion":[20],"another.":[23],"Recently,":[24],"steady":[25,180],"states":[26,181],"shape":[29],"an":[31],"inverted":[32],"spike":[33,187],"were":[34],"constructed":[35],"this":[37,71],"dimension":[41],"[Y.":[42],"Lou,":[43],"W.-M.":[44],"Ni,":[45],"S.":[47],"Yotsutani,":[48],"Discrete":[49,62],"Contin.":[50,63],"Dyn.":[51,64],"Syst.,":[52,65],"10":[53],"(2004),":[54],"pp.":[55,68],"435\u2013458;":[56],"Y.":[57],"Wu":[58],"Q.":[60],"Xu,":[61],"29":[66],"(2011),":[67],"367\u2013385].":[69],"In":[70,194],"paper":[72],"we":[73,99],"stability":[76],"properties":[77],"such":[79,106],"spiky":[80],"states.":[81],"show":[83],"that":[84,107],"K":[85,108],"symmetric":[86,185],"spikes":[87,109],"are":[88,116,199,210,228],"stable":[89,117],"if":[90,118,121],"domain":[92,112],"length":[93],"is":[94,131,157],"sufficiently":[95],"large.":[96],"More":[97],"precisely,":[98],"derive":[100],"a":[101,128,136,160,165,183,192],"sequence":[102],"thresholds":[104,206],"$0=L_1<L_2<L_3<L_4<L_5\\ldots$":[105],"on":[110],"size":[114],"$2KL$":[115],"only":[120],"$L>L_K$.":[122],"When":[123,152],"$K=2$,":[124],"instability":[126,156],"eigenvalue":[130],"triggered":[132],"first,":[133],"resulting":[134,163],"very":[137],"slow":[138],"drift":[139],"two":[142,226],"spikes,":[143],"with":[144],"eventual":[145],"absorption":[146],"by":[149,213],"other.":[151],"$K\\geq3$,":[153],"primary":[155],"due":[158],"to":[159,182,201,230,236],"large":[161,204],"eigenvalue,":[162],"quick":[166],"death":[167],"or":[170],"more":[171],"spikes.":[172],"also":[174],"extend":[175],"construction":[177],"one-dimensional":[179],"radially":[184],"two-dimensional":[186],"at":[188],"center":[190],"disk.":[193],"dimension,":[196],"hypergeometric":[197],"functions":[198],"utilized":[200],"study":[202],"eigenvalues;":[205],"for":[207],"eigenvalues":[209],"derived":[211],"indirectly":[212],"classifying":[214],"bifurcations":[216],"asymmetric":[218],"patterns.":[219],"Full":[220],"numerical":[221],"simulations":[222],"dimensions":[227],"performed":[229],"confirm":[231],"asymptotic":[233],"results":[234],"explore":[237],"some":[238],"dynamical":[241],"scenarios":[242],"away":[243],"from":[244],"equilibrium":[246],"state.":[247]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":5},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
