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An integro-differential equation model for alignment and orientational aggregation |
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| Authors: | Kyungkeun Kang Benoit Perthame |
| Affiliation: | a Department of Mathematics, Sungkyunkwan University and Institute of Basic Science, Suwon 440-746, Republic of Korea b Départment de Mathématiques appliquées, CNRS UMR 8553, Ecole Normale Supérieure, 45, rue d'Ulm, F-75230 Paris cedex 05, France c University of Heidelberg, Applied Mathematics and Bioquant, BQ 0021, INF 267, D-69120 Heidelberg, Germany d Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Facultad de Ciencias Matemáticas, Universidad Complutense, 28040 Madrid, Spain |
| Abstract: | 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 [E. Geigant, M. Stoll, Bifurcation analysis of an orientational aggregation model, J. Math. Biol. 46 (6) (2003) 537-563]. Here we analyze the time-dependent problem and prove that the type of alignment (one- or multi-directional) 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. |
| Keywords: | 47G20 78M35 92C37 |
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