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

摘要

A cross-diffusion system modeling the information herding of individuals isanalyzed in a bounded domain with no-flux boundary conditions. The variablesare the species' density and an influence function which modifies theinformation state of the individuals. The cross-diffusion term may stabilize ordestabilize the system. Furthermore, it allows for a formal gradient-flow orentropy structure. Exploiting this structure, the global-in-time existence ofweak solutions and the exponential decay to the constant steady state is provedin certain parameter regimes. This approach does not extend to all parameters.We investigate local bifurcations from homogeneous steady states analyticallyto determine whether this defines the validity boundary. This analysis showsthat generically there is a gap in the parameter regime between the entropyapproach validity and the first local bifurcation. Next, we use numericalcontinuation methods to track the bifurcating non-homogeneous steady statesglobally and to determine non-trivial stationary solutions related to herdingbehaviour. In summary, we find that the main boundaries in the parameter regimeare given by the first local bifurcation point, the degeneracy of the diffusionmatrix and a certain entropy decay validity condition. We study severalparameter limits analytically as well as numerically, with a focus on the roleof changing a linear damping parameter as well as a parameter controlling thecross-diffusion. We suggest that our paradigm of comparingbifurcation-generated obstructions to the parameter validity ofglobal-functional methods could also be of relevance for many other modelsbeyond the one studied here.