Numerical continuation for a fast-reaction system and its cross-diffusion limit

Christian Kuehn () and Cinzia Soresina ()
Additional contact information
Christian Kuehn: Technische Universität München
Cinzia Soresina: Technische Universität München

Partial Differential Equations and Applications, 2020, vol. 1, issue 2, 1-26

Abstract: Abstract In this paper we investigate the bifurcation structure of the cross-diffusion Shigesada–Kawasaki–Teramoto model (SKT) in the triangular form and in the weak competition regime, and of a corresponding fast-reaction system in 1D and 2D domains via numerical continuation methods. We show that the software pde2path can be exploited to treat cross-diffusion systems, reproducing the already computed bifurcation diagrams on 1D domains. We show the convergence of the bifurcation structure obtained selecting the growth rate as bifurcation parameter. Then, we compute the bifurcation diagram on a 2D rectangular domain providing the shape of the solutions along the branches and linking the results with the linearized analysis. In 1D and 2D, we pay particular attention to the fast-reaction limit by always computing sequences of bifurcation diagrams as the time-scale separation parameter tends to zero. We show that the bifurcation diagram undergoes major deformations once the fast-reaction systems limits onto the cross-diffusion singular limit. Furthermore, we find evidence for time-periodic solutions by detecting Hopf bifurcations, we characterize several regions of multi-stability, and improve our understanding of the shape of patterns in 2D for the SKT model.

Keywords: Bifurcations; Cross-diffusion; Fast-reaction; SKT model; Numerical continuation; 35Q92; 70K70; 35K59; 65P30 (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s42985-020-0008-7 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:pardea:v:1:y:2020:i:2:d:10.1007_s42985-020-0008-7

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/42985/

DOI: 10.1007/s42985-020-0008-7

Access Statistics for this article

Partial Differential Equations and Applications is currently edited by Zhitao Zhang

More articles in Partial Differential Equations and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().