{"id":"https://openalex.org/W2076844805","doi":"https://doi.org/10.1137/12089884x","title":"Traveling Wave Solutions in a Reaction-Diffusion Model for Criminal Activity","display_name":"Traveling Wave Solutions in a Reaction-Diffusion Model for Criminal Activity","publication_year":2013,"publication_date":"2013-01-01","ids":{"openalex":"https://openalex.org/W2076844805","doi":"https://doi.org/10.1137/12089884x","mag":"2076844805"},"language":"en","primary_location":{"id":"doi:10.1137/12089884x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/12089884x","pdf_url":null,"source":{"id":"https://openalex.org/S4357572","display_name":"Multiscale Modeling and Simulation","issn_l":"1540-3459","issn":["1540-3459","1540-3467"],"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":"Multiscale Modeling & Simulation","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/A5084149439","display_name":"Henri Berestycki","orcid":"https://orcid.org/0000-0003-1724-2279"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"H. Berestycki","raw_affiliation_strings":["Ecole des Hautes Etudes en Sciences Sociales"],"affiliations":[{"raw_affiliation_string":"Ecole des Hautes Etudes en Sciences Sociales","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5040293177","display_name":"Nancy Rodr\u00edguez","orcid":"https://orcid.org/0000-0003-1023-9659"},"institutions":[{"id":"https://openalex.org/I97018004","display_name":"Stanford University","ror":"https://ror.org/00f54p054","country_code":"US","type":"education","lineage":["https://openalex.org/I97018004"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"N. Rodr\u00edguez","raw_affiliation_strings":["Stanford University"],"affiliations":[{"raw_affiliation_string":"Stanford University","institution_ids":["https://openalex.org/I97018004"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5079159174","display_name":"Lenya Ryzhik","orcid":"https://orcid.org/0000-0003-1892-4861"},"institutions":[{"id":"https://openalex.org/I97018004","display_name":"Stanford University","ror":"https://ror.org/00f54p054","country_code":"US","type":"education","lineage":["https://openalex.org/I97018004"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"L. Ryzhik","raw_affiliation_strings":["Stanford University"],"affiliations":[{"raw_affiliation_string":"Stanford University","institution_ids":["https://openalex.org/I97018004"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5084149439"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.8772,"has_fulltext":false,"cited_by_count":63,"citation_normalized_percentile":{"value":0.95893058,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"11","issue":"4","first_page":"1097","last_page":"1126"},"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":0.9997000098228455,"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":0.9997000098228455,"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":0.992900013923645,"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/T11252","display_name":"Evolutionary Game Theory and Cooperation","score":0.9725000262260437,"subfield":{"id":"https://openalex.org/subfields/3312","display_name":"Sociology and Political Science"},"field":{"id":"https://openalex.org/fields/33","display_name":"Social Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/reaction\u2013diffusion-system","display_name":"Reaction\u2013diffusion system","score":0.6713560819625854},{"id":"https://openalex.org/keywords/traveling-wave","display_name":"Traveling wave","score":0.6712334156036377},{"id":"https://openalex.org/keywords/diffusion","display_name":"Diffusion","score":0.5132939219474792},{"id":"https://openalex.org/keywords/stochastic-game","display_name":"Stochastic game","score":0.5002965927124023},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.49295544624328613},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.4887807071208954},{"id":"https://openalex.org/keywords/population","display_name":"Population","score":0.48636868596076965},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.4617604911327362},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4605191946029663},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.41759198904037476},{"id":"https://openalex.org/keywords/infinity","display_name":"Infinity","score":0.41444307565689087},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3575296401977539},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3264138102531433},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2241920828819275},{"id":"https://openalex.org/keywords/sociology","display_name":"Sociology","score":0.15830251574516296}],"concepts":[{"id":"https://openalex.org/C121231716","wikidata":"https://www.wikidata.org/wiki/Q901210","display_name":"Reaction\u2013diffusion system","level":2,"score":0.6713560819625854},{"id":"https://openalex.org/C2988173416","wikidata":"https://www.wikidata.org/wiki/Q37172","display_name":"Traveling wave","level":2,"score":0.6712334156036377},{"id":"https://openalex.org/C69357855","wikidata":"https://www.wikidata.org/wiki/Q163214","display_name":"Diffusion","level":2,"score":0.5132939219474792},{"id":"https://openalex.org/C22171661","wikidata":"https://www.wikidata.org/wiki/Q1074380","display_name":"Stochastic game","level":2,"score":0.5002965927124023},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.49295544624328613},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.4887807071208954},{"id":"https://openalex.org/C2908647359","wikidata":"https://www.wikidata.org/wiki/Q2625603","display_name":"Population","level":2,"score":0.48636868596076965},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.4617604911327362},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4605191946029663},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.41759198904037476},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.41444307565689087},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3575296401977539},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3264138102531433},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2241920828819275},{"id":"https://openalex.org/C144024400","wikidata":"https://www.wikidata.org/wiki/Q21201","display_name":"Sociology","level":0,"score":0.15830251574516296},{"id":"https://openalex.org/C149923435","wikidata":"https://www.wikidata.org/wiki/Q37732","display_name":"Demography","level":1,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/12089884x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/12089884x","pdf_url":null,"source":{"id":"https://openalex.org/S4357572","display_name":"Multiscale Modeling and Simulation","issn_l":"1540-3459","issn":["1540-3459","1540-3467"],"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":"Multiscale Modeling & Simulation","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/16","display_name":"Peace, Justice and strong institutions","score":0.75}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":21,"referenced_works":["https://openalex.org/W43144976","https://openalex.org/W356830591","https://openalex.org/W1583602735","https://openalex.org/W1866311589","https://openalex.org/W1967439982","https://openalex.org/W1984535955","https://openalex.org/W1985314283","https://openalex.org/W2020718248","https://openalex.org/W2027283010","https://openalex.org/W2053419780","https://openalex.org/W2056583384","https://openalex.org/W2061191702","https://openalex.org/W2066739572","https://openalex.org/W2075663353","https://openalex.org/W2079949023","https://openalex.org/W2080763235","https://openalex.org/W2088047416","https://openalex.org/W2095446265","https://openalex.org/W2123906784","https://openalex.org/W2155458379","https://openalex.org/W4249855545"],"related_works":["https://openalex.org/W1997022868","https://openalex.org/W2061818987","https://openalex.org/W2058973986","https://openalex.org/W4287780914","https://openalex.org/W3034862729","https://openalex.org/W3023599262","https://openalex.org/W2064520184","https://openalex.org/W1970842439","https://openalex.org/W2706310289","https://openalex.org/W2054370674"],"abstract_inverted_index":{"We":[0,34,126],"study":[1,100],"a":[2,15,40,109,120,123],"reaction-diffusion":[3],"system":[4],"of":[5,39,51,92,103,112,130,142,160],"partial":[6],"differential":[7],"equations,":[8],"which":[9],"can":[10],"be":[11,14],"taken":[12],"to":[13,89,163,179],"basic":[16],"model":[17],"for":[18,118],"criminal":[19,52,76,82,93],"activity,":[20,83],"first":[21,64],"introduced":[22],"in":[23,122,135,167,188,196],"[Berestycki":[24],"and":[25,78],"Nadal,":[26],"J.":[27],"Appl.":[28],"Math.,":[29],"21":[30],"(2010),":[31],"pp.":[32],"371--399].":[33],"show":[35,65],"that":[36,66,114,145],"the":[37,48,56,101,116,128,140,157,165,168,185,189,197],"assumption":[38],"population's":[41],"natural":[42,61],"tendency":[43],"towards":[44],"crime":[45,121],"significantly":[46],"changes":[47],"long-time":[49],"behavior":[50],"activity":[53,77,94],"patterns.":[54],"Under":[55],"right":[57],"assumptions":[58],"on":[59],"these":[60],"tendencies":[62],"we":[63,99,155,175],"there":[67],"exists":[68],"traveling":[69,147],"wave":[70,131],"solutions":[71,144],"connecting":[72],"zones":[73,79],"with":[74,80],"no":[75],"high":[81],"known":[84,183],"as":[85,149,184],"hotspots.":[86],"This":[87],"corresponds":[88],"an":[90],"invasion":[91,166],"onto":[95],"all":[96],"space.":[97],"Second,":[98],"problem":[102,187],"preventing":[104],"such":[105],"invasions":[106],"by":[107,138],"employing":[108],"finite":[110,124],"number":[111],"resources":[113,161],"reduce":[115],"payoff":[117],"committing":[119],"region.":[125],"make":[127],"concept":[129],"propagation":[132],"mathematically":[133],"rigorous":[134],"this":[136],"situation":[137],"proving":[139,193],"existence":[141],"entire":[143],"approach":[146],"waves":[148],"time":[150],"approaches":[151],"negative":[152],"infinity.":[153],"Furthermore,":[154],"characterize":[156],"minimum":[158],"amount":[159],"necessary":[162],"prevent":[164],"case":[169],"when":[170],"prevention":[171],"is":[172,181],"possible.":[173],"Finally,":[174],"apply":[176],"our":[177],"theory":[178],"what":[180],"commonly":[182],"gap":[186],"excitable":[190],"media":[191],"literature,":[192],"existing":[194],"conjectures":[195],"literature.":[198]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":8},{"year":2022,"cited_by_count":9},{"year":2021,"cited_by_count":8},{"year":2020,"cited_by_count":7},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":11}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}