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An Integro-Differential Equation Model for Alignment and Orientational Aggregation
Kyungkeun Kang, Benoit Perthame, Angela Stevens and Juan J.L. Velazquez
We study an integro-differential equation modeling angular alignment of interacting bundles of cells or filaments. A bifurcation analysis of the related stationary problem was done by Geigant and Stoll in [J. Math. Biol. 46 (2003), no. 6, 537--563].
Here we analyze the time dependent problem and prove that the type of alignment (one or multidirectional) depends on the initial distribution, the interaction potential, and the preferred optimal orientation of the bundles of cells or filaments. Our main technical tool is the analysis of the evolution of suitable functionals for the cell density, which allows to also specify the direction(s) where the final alignment takes place.