A cross-diffusion system modeling rivaling gangs: Global existence of bounded solutions and FCT stabilization for numerical simulation

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Autoren

  • Mario Fuest
  • Shahin Heydari

Organisationseinheiten

Externe Organisationen

  • Charles University
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Details

OriginalspracheEnglisch
Seiten (von - bis)1-41
Seitenumfang41
FachzeitschriftMathematical Models and Methods in Applied Sciences
Jahrgang34
Ausgabenummer9
PublikationsstatusVeröffentlicht - 29 Juni 2024

Abstract

In this paper, we study a gang territorial model consisting of two parabolic and two ordinary differential equations, where a taxis-type mechanism models that the two rivaling gangs are repelled by each other's graffiti. Our main analytical finding shows the existence of global, bounded classical solutions. By making use of quantitative global estimates, we prove that these solutions converge to homogeneous steady states if the initial data are sufficiently small. Moreover, we perform numerical experiments which show that for different choices of parameters, the system may become diffusion- or convection-dominated, where in the former case the solutions converge toward constant steady states while in the latter case nontrivial asymptotic behavior such as segregation is observed. In order to perform these experiments, we apply a nonlinear finite element flux-corrected transport method (FEM-FCT) which is positivity-preserving. Then we treat the nonlinearities in both the system and the proposed nonlinear scheme simultaneously using fixed-point iteration.

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A cross-diffusion system modeling rivaling gangs: Global existence of bounded solutions and FCT stabilization for numerical simulation. / Fuest, Mario; Heydari, Shahin.
in: Mathematical Models and Methods in Applied Sciences, Jahrgang 34, Nr. 9, 29.06.2024, S. 1-41.

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

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keywords = "asymptotic behavior, cross-diffusion, FEM-FCT stabilization, Gang territoriality, global existence, positivity preservation, separation",
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T1 - A cross-diffusion system modeling rivaling gangs

T2 - Global existence of bounded solutions and FCT stabilization for numerical simulation

AU - Fuest, Mario

AU - Heydari, Shahin

N1 - Publisher Copyright: © 2024 World Scientific Publishing Company.

PY - 2024/6/29

Y1 - 2024/6/29

N2 - In this paper, we study a gang territorial model consisting of two parabolic and two ordinary differential equations, where a taxis-type mechanism models that the two rivaling gangs are repelled by each other's graffiti. Our main analytical finding shows the existence of global, bounded classical solutions. By making use of quantitative global estimates, we prove that these solutions converge to homogeneous steady states if the initial data are sufficiently small. Moreover, we perform numerical experiments which show that for different choices of parameters, the system may become diffusion- or convection-dominated, where in the former case the solutions converge toward constant steady states while in the latter case nontrivial asymptotic behavior such as segregation is observed. In order to perform these experiments, we apply a nonlinear finite element flux-corrected transport method (FEM-FCT) which is positivity-preserving. Then we treat the nonlinearities in both the system and the proposed nonlinear scheme simultaneously using fixed-point iteration.

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KW - asymptotic behavior

KW - cross-diffusion

KW - FEM-FCT stabilization

KW - Gang territoriality

KW - global existence

KW - positivity preservation

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