Details
| Original language | English |
|---|---|
| Pages (from-to) | 1-41 |
| Number of pages | 41 |
| Journal | Mathematical Models and Methods in Applied Sciences |
| Volume | 34 |
| Issue number | 9 |
| Publication status | Published - 29 Jun 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.
Keywords
- asymptotic behavior, cross-diffusion, FEM-FCT stabilization, Gang territoriality, global existence, positivity preservation, separation
ASJC Scopus subject areas
- Mathematics(all)
- Modelling and Simulation
- Mathematics(all)
- Applied Mathematics
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In: Mathematical Models and Methods in Applied Sciences, Vol. 34, No. 9, 29.06.2024, p. 1-41.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
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.
AB - 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.
KW - asymptotic behavior
KW - cross-diffusion
KW - FEM-FCT stabilization
KW - Gang territoriality
KW - global existence
KW - positivity preservation
KW - separation
UR - http://www.scopus.com/inward/record.url?scp=85197907690&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2312.08147
DO - 10.48550/arXiv.2312.08147
M3 - Article
AN - SCOPUS:85197907690
VL - 34
SP - 1
EP - 41
JO - Mathematical Models and Methods in Applied Sciences
JF - Mathematical Models and Methods in Applied Sciences
SN - 0218-2025
IS - 9
ER -