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

TL;DR

This paper explores the bifurcation structures of a cross-diffusion system and its fast-reaction limit using numerical continuation, revealing how bifurcations evolve and identifying complex solution behaviors in 1D and 2D domains.

Contribution

It adapts pde2path for cross-diffusion systems, demonstrates convergence of bifurcation structures in the fast-reaction limit, and links bifurcation analysis with pattern formation and Turing instability.

Findings

Bifurcation diagrams undergo major deformations in the fast-reaction limit.

Evidence of time-periodic solutions via Hopf bifurcations.

Identification of multi-stability regions and pattern shapes in 2D.

Abstract

In this paper we investigate the bifurcation structure of the triangular SKT model in the weak competition regime and of the corresponding fast-reaction system in 1D and 2D domains via numerical continuation methods. We show that the software pde2path can be adapted 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 Turing instability 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.