Individual based and mean-field modeling of direct aggregation

Handle URI:
http://hdl.handle.net/10754/334573
Title:
Individual based and mean-field modeling of direct aggregation
Authors:
Burger, Martin; Haskovec, Jan; Wolfram, Marie-Therese
Abstract:
We introduce two models of biological aggregation, based on randomly moving particles with individual stochasticity depending on the perceived average population density in their neighborhood. In the firstorder model the location of each individual is subject to a density-dependent random walk, while in the second-order model the density-dependent random walk acts on the velocity variable, together with a density-dependent damping term. The main novelty of our models is that we do not assume any explicit aggregative force acting on the individuals; instead, aggregation is obtained exclusively by reducing the individual stochasticity in response to higher perceived density. We formally derive the corresponding mean-field limits, leading to nonlocal degenerate diffusions. Then, we carry out the mathematical analysis of the first-order model, in particular, we prove the existence of weak solutions and show that it allows for measure-valued steady states. We also perform linear stability analysis and identify conditions for pattern formation. Moreover, we discuss the role of the nonlocality for well-posedness of the first-order model. Finally, we present results of numerical simulations for both the first- and second-order model on the individual-based and continuum levels of description. 2012 Elsevier B.V. All rights reserved.
KAUST Department:
King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
Citation:
Burger M, Ha-kovec J, Wolfram M-T (2013) Individual based and mean-field modeling of direct aggregation. Physica D: Nonlinear Phenomena 260: 145-158. doi:10.1016/j.physd.2012.11.003.
Publisher:
North-Holland
Journal:
Physica D: Nonlinear Phenomena
Issue Date:
1-Oct-2013
DOI:
10.1016/j.physd.2012.11.003
PubMed ID:
24926113
PubMed Central ID:
PMC4047626
Type:
Article
ISSN:
01672789
Appears in Collections:
Articles

Full metadata record

DC FieldValue Language
dc.contributor.authorBurger, Martinen
dc.contributor.authorHaskovec, Janen
dc.contributor.authorWolfram, Marie-Thereseen
dc.date.accessioned2014-11-11T14:30:14Z-
dc.date.available2014-11-11T14:30:14Z-
dc.date.issued2013-10-01en
dc.identifier.citationBurger M, Ha-kovec J, Wolfram M-T (2013) Individual based and mean-field modeling of direct aggregation. Physica D: Nonlinear Phenomena 260: 145-158. doi:10.1016/j.physd.2012.11.003.en
dc.identifier.issn01672789en
dc.identifier.pmid24926113en
dc.identifier.doi10.1016/j.physd.2012.11.003en
dc.identifier.urihttp://hdl.handle.net/10754/334573en
dc.description.abstractWe introduce two models of biological aggregation, based on randomly moving particles with individual stochasticity depending on the perceived average population density in their neighborhood. In the firstorder model the location of each individual is subject to a density-dependent random walk, while in the second-order model the density-dependent random walk acts on the velocity variable, together with a density-dependent damping term. The main novelty of our models is that we do not assume any explicit aggregative force acting on the individuals; instead, aggregation is obtained exclusively by reducing the individual stochasticity in response to higher perceived density. We formally derive the corresponding mean-field limits, leading to nonlocal degenerate diffusions. Then, we carry out the mathematical analysis of the first-order model, in particular, we prove the existence of weak solutions and show that it allows for measure-valued steady states. We also perform linear stability analysis and identify conditions for pattern formation. Moreover, we discuss the role of the nonlocality for well-posedness of the first-order model. Finally, we present results of numerical simulations for both the first- and second-order model on the individual-based and continuum levels of description. 2012 Elsevier B.V. All rights reserved.en
dc.language.isoenen
dc.publisherNorth-Hollanden
dc.rightshttp://creativecommons.org/licenses/by-nc-nd/3.0/en
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/3.0/en
dc.subjectDegenerate parabolic equationen
dc.subjectDensity dependent random walken
dc.subjectDirect aggregationen
dc.subjectMean field limiten
dc.subjectBiological aggregationsen
dc.subjectDegenerate diffusionsen
dc.subjectExistence of weak solutionsen
dc.subjectMathematical analysisen
dc.subjectMean field limitsen
dc.subjectPopulation densitiesen
dc.subjectRandom Walken
dc.subjectPopulation statisticsen
dc.subjectRandom processesen
dc.subjectAgglomerationen
dc.titleIndividual based and mean-field modeling of direct aggregationen
dc.typeArticleen
dc.contributor.departmentKing Abdullah University of Science and Technology, Thuwal, Saudi Arabiaen
dc.identifier.journalPhysica D: Nonlinear Phenomenaen
dc.identifier.pmcidPMC4047626en
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionInstitut fr Numerische und Angewandte Mathematik, Westflische Wilhelms-Universitt Mnster, Einsteinstr. 62, 48149 Mnster, Germanyen
dc.contributor.institutionFaculty of Mathematics, Universitt Wien, Nordbergstrasse 15, A-1090 Wien, Austriaen
dc.contributor.institutionDAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdomen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
kaust.authorHaskovec, Janen

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