Skip to main content Accessibility help

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



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.



Hide All

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).



Hide All
[1] Arnold, A., Abdallah, N. B. & Negulescu, C. (1996) Liapunov functionals and large-time-asymptotics of mean-field nonlinear Fokker-Planck equations. Transp. Theory Stat. Phys. 25 (7), 733751.
[2] Achleitner, F. & Kuehn, C. (2015) On bounded positive stationary solutions for a nonlocal Fisher-KPP equation. Nonl. Anal. A: Theor. Meth. Appl. 112, 1529.
[3] Amann, H. (1989) Dynamic theory of quasilinear parabolic systems. III. Global existence. Math. Z. 202, 219250.
[4] Arnold, A., Markowich, P., Toscani, G. & Unterreiter, A. (2001) On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Commun. Partial Differ. Equ. 26 (1–2), 43100.
[5] Agranovich, M. S. & Vishik, M. I. (1964) Elliptic problems with a parameter and parabolic problems of general type. Russ. Math. Surv. 19 (3), 53157.
[6] Bakry, D., Gentil, I. & Ledoux, M. (2014) Analysis and Geometry of Markov Diffusion Operators, Springer.
[7] Burger, M., Markowich, P. & Pietschmann, J.-F. (2011) Continuous limit of a crowd motion and herding model: Analysis and numerical simulations. Kinet. Relat. Mod. 4, 10251047.
[8] Carrillo, J. A., Jüngel, A., Markowich, P. A., Toscani, G. & Unterreiter, A. (2001) Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatshefte für Math. 133 (1), 182.
[9] Chertock, A., Kurganov, A., Wang, X. & Wu, Y. (2012) On a chemotaxis model with saturated chemotactic flux. Kinet. Relat. Mod. 5, 5195.
[10] Crandall, M. G. & Rabinowitz, P. H. (1971) Bifurcation from simple eigenvalues. J. Funct. Anal. 8 (2), 321340.
[11] Crandall, M. G. & Rabinowitz, P. H. (1973) Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Ration. Mech. Anal. 52, 161180.
[12] Doedel, E. J., Champneys, A., Dercole, F., Fairgrieve, T., Kuznetsov, Y., Oldeman, B., Paffenroth, R., Sandstede, B., Wang, X. & Zhang, C. (2007) Auto 2007p: Continuation and bifurcation software for ordinary differential equations (with homcont). URL:, accessed July 1st, 2016.
[13] Desvillettes, L. & Fellner, K. (2006) Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations. J. Math. Anal. Appl. 319 (1), 157176.
[14] Desvillettes, L. & Fellner, K. (2007) Entropy methods for reaction-diffusion systems. Discrete Cont. Dyn. Sys. (suppl.) 304312.
[15] Dreher, M. & Jüngel, A. (2012) Compact families of piecewise constant functions in L p (0,T;B). Nonlin. Anal. 75, 30723077.
[16] Delitala, M. & Lorenzo, T. (2014) A mathematical model for value estimation with public information and herding. Kinet. Relat. Mod. 7, 2944.
[17] Dankowicz, H. & Schilder, F. (2013) Recipes for Continuation. SIAM.
[18] Evans, L. C. (2002) Partial Differential Equations, AMS.
[19] Fife, P. C. (1973) Semilinear elliptic boundary value problems with small parameters. Arch. Ration. Mech. Anal. 52 (3), 205232.
[20] Gabriel, P. (2012) Long-time asymptotics for nonlinear growth-fragmentation equations. Commun. Math. Sci. 10, 787820.
[21] Govaerts, W. F. (1987) Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, PA.
[22] Galiano, G. & Selgas, V. (2014) On a cross-diffusion segregation problem arising from a model of interacting particles. Nonlin. Anal.: Real World Appl. 18, 3449.
[23] Henderson, M. E. (2002) Multiple parameter continuation: Computing implicitly defined k-manifolds. Int. J. Bif. Chaos 12 (3), 451476.
[24] Hittmeir, S. & Jüngel, A. (2011) Cross diffusion preventing blow up in the two-dimensional Keller-Segel model. SIAM J. Math. Anal. 43, 9971022.
[25] Horstmann, D. (2011) Generalizing the keller-segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species. J. Nonlin. Sci. 21, 231270.
[26] Hillen, T. & Painter, K. (2002) Volume filling and quorum sensing in models for chemosensitive movement. Canad. Appl. Math. Quart. 10, 501543.
[27] Januauskas, A. (1998) Classification of second-order partial differential equation systems elliptic in the petrovskii sense. Lithuanian Math. J. 38, 5963.
[28] Jüngel, A. (2015) The boundedness-by-emtropy method for cross-diffusion systems. Nonlinearity 28, 19632001.
[29] Jiang, J. & Zhang, Y. (2009) On convergence to equilibria for a chemotaxis model with volume-filling effect. Asympt. Anal. 65, 79102.
[30] Keller, H. (1977) Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Rabinowitz, P. (editor), Applications of Bifurcation Theory, Academic Press, pp. 359384.
[31] Kielhoefer, H. (2004) Bifurcation Theory: An Introduction with Applications to PDEs, Springer.
[32] Krauskopf, B., Osinga, H. M. & Galán-Vique, J. (editors) (2007) Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems, Springer.
[33] Keller, E. & Segel, S. (1970) Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399415.
[34] Kuehn, C. (2015). Efficient gluing of numerical continuation and a multiple solution method for elliptic PDEs. Applied Mathematics and Computation, 266, 656674.
[35] Lions, P.-L. (2015) Some new classes of nonlinear Kolmogorov equations. Talk at the 16th Pauli Colloquium, Wolfgang-Pauli Institute.
[36] Liero, M. & Mielke, A. (2013) Gradient structures and geodesic convexity for reaction-diffusion systems. Phil. Trans. Roy. Soc. A 371, 20120346.
[37] Lambda, H. & Seaman, T. (2008) Market statistics of a psychology-based heterogeneous agent model. Intern. J. Theor. Appl. Finance 11, 717737.
[38] Ni, W.-M. (1998) Diffusion, cross-diffusion and their spike-layer steady states. Not. Amer. Math. Soc. 45 (1), 918.
[39] Shi, J. & Wang, X. (2009) On the global bifurcation for quasilinear elliptic systems on bounded domains. J. Differ. Equ. 246, 27882812.
[40] Temam, R. (1997) Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer.
[41] Uhlenbeck, K. (1972) Eigenfunctions of Laplace operators. Bull. Amer. Math. Soc. 78, 10731076.
[42] Uecker, H., Wetzel, D. & Rademacher, J. D. M. (2014) pde2path - A Matlab package for continuation and bifurcation in 2D elliptic systems. Num. Math.: Th. Meth. Appl. 7, 58106.
[43] Wolansky, G. (2002) Multi-components chemotactic system in the absence of conflicts. Europ. J. Appl. Math. 13, 641661.
[44] Wrzosek, D. (2004) Global attractor for a chemotaxis model with prevention of overcrowding. Nonlin. Anal. 59, 12931310.
[45] Wang, X. & Xu, Q. (2013) Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem. J. Math. Biol. 66 (6), 12411266.
[46] Zinsl, J., & Matthes, D. (2015). Transport distances and geodesic convexity for systems of degenerate diffusion equations. Calculus of Variations and Partial Differential Equations, 54 (4), 33973438.


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



Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.