Elsevier

Journal of Differential Equations

Volume 246, Issue 4, 15 February 2009, Pages 1387-1421
Journal of Differential Equations

An integro-differential equation model for alignment and orientational aggregation

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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.

MSC

47G20
78M35
92C37

Keywords

Aggregation
Alignment
Integro-differential equation

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