A cross-diffusion system modeling rivaling gangs

Global existence of bounded solutions and FCT stabilization for numerical simulation

authored by: Mario Fuest, Shahin Heydari
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.
Organisation(s): Institute of Applied Mathematics
External Organisation(s): Charles University
Type: Article
Journal: Mathematical Models and Methods in Applied Sciences
Volume: 34
Pages: 1-41
No. of pages: 41
ISSN: 0218-2025
Publication date: 29.06.2024
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Modelling and Simulation, Applied Mathematics
Electronic version(s): https://doi.org/10.48550/arXiv.2312.08147 (Access: Open)
https://doi.org/10.1142/S0218202524500349 (Access: Closed)

BibTeX

@article{baa59d2c12ad40ef84f3bdca3fd7927f,
title = "A cross-diffusion system modeling rivaling gangs: Global existence of bounded solutions and FCT stabilization for numerical simulation",
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.",
keywords = "asymptotic behavior, cross-diffusion, FEM-FCT stabilization, Gang territoriality, global existence, positivity preservation, separation",
author = "Mario Fuest and Shahin Heydari",
note = "Publisher Copyright: {\textcopyright} 2024 World Scientific Publishing Company.",
year = "2024",
month = jun,
day = "29",
doi = "10.48550/arXiv.2312.08147",
language = "English",
volume = "34",
pages = "1--41",
journal = "Mathematical Models and Methods in Applied Sciences",
issn = "0218-2025",
publisher = "World Scientific Publishing Co. Pte Ltd",
number = "9",
}