Mean-field analysis of a dynamical phase transition in a cellular automaton model for collective motion

A cellular automaton model is presented for random walkers with biologically motivated interactions favoring local alignment and leading to collective motion or swarming behavior. The degree of alignment is controlled by a sensitivity parameter, and a dynamical phase transition exhibiting spontaneous breaking of rotational symmetry occurs at a critical parameter value. The model is analyzed using nonequilibrium mean field theory: Dispersion relations for the critical modes are derived, and a phase diagram is constructed. Mean field predictions for the two critical exponents describing the phase transition as a function of sensitivity and density are obtained analytically.Comment: 4 pages, 4 figures, final version as publishe

STABILITY OF PEAK SOLUTIONS OF A NON-LINEAR TRANSPORT EQUATION ON THE CIRCLE

Abstract. We study solutions of a transport-diffusion equation on the circle. The velocity of turning is given by a non-local term that models attraction and repulsion between elongated particles. Having mentioned basics like invariances, instability criteria and non-existence of time-periodic solutions, we prove that the constant steady state is stable at large diffusion. We show that without diffusion localized initial distributions and attraction lead to formation of several peaks. For peak-like steady states two kinds of peak stability are analyzed: first spatially discretized with respect to the relative position of the peaks, then stability with respect to nonlocalized perturbations. We prove that more than two peaks may be stable up to translation and slight rearrangements of the peaks. Our fast numerical scheme which is based on the Fouriertransformed system allows to study the long-time behaviour of the equation. Numerical examples show backward bifurcation, mixed-mode solutions, peaks with unequal distances, coexistence of one-peak and two-peak solutions and peak formation in a case of purely repulsive interaction. 1

An Integro-Differential Model For Orientational Distributions Of F-Actin In Cells

. Angular self-organization of the actin cytoskeleton is modeled as a process of instant changing of filament orientation in the course of specific actin-actin interactions. These interactions are modified by cross-linking actin-binding proteins. This problem was raised first in (Civelekoglu and EdelsteinKeshet, 1994). When restricted to a two-dimensional configuration, the mathematical model consists of a single Boltzmann-like integro-differential equation for the one-dimensional angular distribution. Linear stability analysis, asymptotic analysis and numerical results reveal that at certain parameter values of actin-actin interactions spontaneous alignment of filaments in the form of unipolar or bipolar bundles or orthogonal networks can be expected. Key words. actin cytoskeleton, master equation, Boltzmann equation, integrodifferential equation, peak solution AMS subject classifications. 45K05, 92C 1. Introduction. The formation of orientational order has been of great scientific i..