An Integro-Differential Equation Model for Alignment and Orientational Aggregation

Kyungkeun Kang, Benoit Perthame, Angela Stevens, and Juan J.L. Velazquez

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Submission date: 18. Jan. 2007
Pages: 42
published in: Journal of differential equations, 246 (2009) 4, p. 1387-1421 
DOI number (of the published article): 10.1016/j.jde.2008.11.006
Bibtex
Keywords and phrases: kinetic equations, asymptotic behavior, alignment of cells and filaments
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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 [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.