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Mean-field models for segregation dynamics

Part of: Partial differential equations Partial differential equations, boundary value problems General higher-order equations and systems

Published online by Cambridge University Press:  23 December 2020

MARTIN BURGER ,
JAN-FREDERIK PIETSCHMANN ,
HELENE RANETBAUER ,
CHRISTIAN SCHMEISER  and
MARIE-THERESE WOLFRAM
MARTIN BURGER
Affiliation:
Department Mathrmatik Cauerstr, 11 91058 Erlangen, Germany email: martin.burger@fau.de
JAN-FREDERIK PIETSCHMANN
Affiliation:
Technische Universität Chemnitz, Reichenhainer Straße 41, Chemnitz, Germany email: jfpietschmann@math.tu-chemnitz.de
HELENE RANETBAUER
Affiliation:
University of Vienna, Oskar-Morgenstern-Platz 1, Vienna, Austria emails: helene.ranetbauer@univie.ac.at; christian.schmeiser@univie.ac.at
CHRISTIAN SCHMEISER
Affiliation:
University of Vienna, Oskar-Morgenstern-Platz 1, Vienna, Austria emails: helene.ranetbauer@univie.ac.at; christian.schmeiser@univie.ac.at
MARIE-THERESE WOLFRAM
Affiliation:
University of Warwick, Coventry CV4 7AL, UK RICAM, Altenbergerstr. 69, 4040 Linz, Austria email: m.wolfram@warwick.ac.uk
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Abstract

In this paper, we derive and analyse mean-field models for the dynamics of groups of individuals undergoing a random walk. The random motion of individuals is only influenced by the perceived densities of the different groups present as well as the available space. All individuals have the tendency to stay within their own group and avoid the others. These interactions lead to the formation of aggregates in case of a single species and to segregation in the case of multiple species. We derive two different mean-field models, which are based on these interactions and weigh local and non-local effects differently. We discuss existence and stability properties of solutions for both models and illustrate the rich dynamics with numerical simulations.

Keywords

Mean-field modesnon-local partial differential equationsintegro partial differential equationslinear stabilityasymptotic behaviour

MSC classification

Primary: 35G61: Initial-boundary value problems for nonlinear higher-order systems 45K05: Integro-partial differential equations 35B35: Stability
Secondary: 65N08: Finite volume methods 35B40: Asymptotic behavior of solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 1 , February 2022 , pp. 111 - 132
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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