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.
Bibliographical noteFunding Information:
K. Kang’s work was supported by the Max-Planck-Institute for Mathematics in the Sciences (MPI MIS) while staying in Leipzig and by the Korean Government via a Korean Research Foundation Grant (MOEHRD, Basic Research Promotion Fund, KRF-2006-311-C00007) and the Korea Science and Engineering Foundation (KOSEF) grant (MEST, No. R01-2008-000-11008-0). Angela Stevens was employed at the MPI MIS in Leipzig when working on this paper. J.J.L. Velázquez work was supported by the Humboldt foundation during a stay at the MPI MIS in Leipzig and by the DGES Research Grant MTM2007-61755.
All Science Journal Classification (ASJC) codes
- Applied Mathematics