A meeting point of entropy and bifurcations in cross-diffusion herding

TL;DR

This paper analyzes a cross-diffusion system modeling herding behavior, exploring stability, bifurcations, and stationary solutions, and introduces a novel approach comparing bifurcation points with entropy-based method validity.

Contribution

It provides a combined analytical and numerical investigation of bifurcations and entropy structures in a herding model, highlighting parameter regimes where solutions exist or bifurcate.

Findings

Global existence of weak solutions in certain regimes

Identification of bifurcation points leading to non-homogeneous states

Main boundaries determined by bifurcation, degeneracy, and entropy conditions

Abstract

A cross-diffusion system modeling the information herding of individuals is analyzed in a bounded domain with no-flux boundary conditions. The variables are the species' density and an influence function which modifies the information state of the individuals. The cross-diffusion term may stabilize or destabilize the system. Furthermore, it allows for a formal gradient-flow or entropy structure. Exploiting this structure, the global-in-time existence of weak solutions and the exponential decay to the constant steady state is proved in certain parameter regimes. This approach does not extend to all parameters. We investigate local bifurcations from homogeneous steady states analytically to determine whether this defines the validity boundary. This analysis shows that generically there is a gap in the parameter regime between the entropy approach validity and the first local bifurcation. Next, we use numerical continuation methods to track the bifurcating non-homogeneous steady states globally and to determine non-trivial stationary solutions related to herding behaviour. In summary, we find that the main boundaries in the parameter regime are given by the first local bifurcation point, the degeneracy of the diffusion matrix and a certain entropy decay validity condition. We study several parameter limits analytically as well as numerically, with a focus on the role of changing a linear damping parameter as well as a parameter controlling the cross-diffusion. We suggest that our paradigm of comparing bifurcation-generated obstructions to the parameter validity of global-functional methods could also be of relevance for many other models beyond the one studied here.