An Integro-Differential Equation Model for Alignment and Orientational Aggregation
Kyungkeun Kang, Benoit Perthame, Angela Stevens, and Juan J.L. Velazquez
Contact the author: Please use for correspondence this email.
Submission date: 18. Jan. 2007
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
Keywords and phrases: kinetic equations, asymptotic behavior, alignment of cells and filaments
Download full preprint: PDF (1934 kB)
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.