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

Published online by Cambridge University Press:  15 August 2016

Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria emails:,
Faculty of Mathematics, Technical University of Munich, Boltzmannstraße 3, 85748 Garching, Germany email:
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria emails:,


A cross-diffusion system modelling the information herding of individuals is analysed 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.

Copyright © Cambridge University Press 2016 

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AJ and LT acknowledge partial support from the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement Number 304617, the Austrian Science Fund (FWF), grants P22108, P24304, W1245, and the Austrian-French Program of the Austrian Exchange Service (ÖAD). CK acknowledges partial support by an APART fellowship of the Austrian Academy of Sciences (ÖAW) and by a Marie-Curie International Reintegration Grant by the EU/REA (IRG 271086).


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