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Zoology of a non-local cross-diffusion model for two species
File(s)
17m1128782.pdf (956.57 KB)
Published version
Author(s)
Carrillo de la Plata, J
Huang, Yanghong
Schmidtchen, MArkus
Type
Journal Article
Abstract
We study a nonlocal two species cross-interaction model with cross-diffusion. We propose a positivity preserving finite volume scheme based on the numerical method introduced in [J. A. Carrillo, A. Chertock, and Y. Huang, Commun. Comput. Phys., 17 (2015), pp. 233--258] and explore this new model numerically in terms of its long-time behaviors. Using the so-gained insights, we compute analytical stationary states and travelling pulse solutions for a particular model in the case of attractive-attractive/attractive-repulsive cross-interactions. We show that, as the strength of the cross-diffusivity decreases, there is a transition from adjacent solutions to completely segregated densities, and we compute the threshold analytically for attractive-repulsive cross-interactions. Other bifurcating stationary states with various coexistence components of the support are analyzed in the attractive-attractive case. We find a strong agreement between the numerically and the analytically computed steady states in these particular cases, whose main qualitative features are also present for more general potentials.
Date Issued
2018-04-05
Date Acceptance
2018-01-25
Citation
SIAM Journal on Applied Mathematics, 2018, 78 (2), pp.1078-1104
URI
DOI
ISSN
0036-1399
Publisher
Society for Industrial and Applied Mathematics
Start Page
1078
End Page
1104
Journal / Book Title
SIAM Journal on Applied Mathematics
Volume
78
Issue
2
Copyright Statement
© 2018 SIAM. Published by SIAM under the terms
of the Creative Commons 4.0 license (https://creativecommons.org/licenses/by/4.0/)
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Grant Number
EP/P031587/1
Subjects
Science & Technology
Physical Sciences
Mathematics, Applied
Mathematics
cross-diffusion
nonlocal aggregation-diffusion systems
volume exclusion
AGGREGATION MODEL
INTERACTING POPULATIONS
INTERACTION ENERGY
EMERGENT BEHAVIOR
LOCAL MINIMIZERS
CELL-ADHESION
BLOW-UP
EQUATIONS
DYNAMICS
PARTICLES
0102 Applied Mathematics
Applied Mathematics
Publication Status
Published