Dynamics of a Continuous Model for Flocking Ed Ott in collaboration with Tom Antonsen Parvez Guzdar Nicholas Mecholsky

Dynamical Behavior in Observed Bird Flocks and Fish Schools

1.Flock equilibria 2.Relaxation to equilibrium 3.Stability of the flock 4.Response to an external stimulus, e.g. flight around a small obstacle: poster of Nick Mecholsky Our Objectives- Introduce a model and use it to investigate:

Characteristics of Common Microscopic Models of Flocks 1.Nearby repulsion (to avoid collisions) 2.Large scale attraction (to form a flock) 3.Local relaxation of velocity orientations to a common direction 4.Nearly constant speed, v 0

Continuum Model Many models evolve the individual positions and velocities of a large number of discrete boids. Another approach (the one used here) considers the limit in which the number of boids is large and a continuum description is applicable. Let The number density of boids The macroscopic (locally averaged) boid velocity field

Conservation of Boids: Velocity Equation: Governing Equations:

(1). Short range repulsion: This is a pressure type interaction that models the short range repulsive force between boids. The denominator prevents  from exceeding  * so that the boids do not get too close together.

Here   -1 represents a ‘screening length’ past which the interaction between boids at and becomes ineffective. In this case, satisfies: and U satisfies: These equations for u and U apply in 1D, 2D, and 3D. (2). Long-range attraction:

Our choice for satisfies: (3). Velocity orientation relaxation term:

(4). Speed regulation term: This term brings all boids to a common speed v 0. If |v| > v 0 (|v| < v 0 ), then this term decreases (increases) |v|. If, the speed |v| is clamped to v 0.

We consider a one dimensional flock in which the flock density, in the frame moving with the flock, only depends on x. Additionally, v is independent of x and is constant in time (v 0 ): Equilibrium

Equilibrium Equations:

The equilibrium equations combine to give an energy like form where  depends on a dimensionless parameter   defined below) and both  and x are made dimensionless by their respective physical parameters  * and   Equilibrium Solutions

and the density at x = 0 is determined to be An Example

Solving the potential equation, we get The profile is symmetric about

Waves and Stability Equilibrium : Perturbations :

and the notation signifies the operator Basic Equation: where:

Long Wavelength Expansion Ordering Scheme:

Analysis: Inner product of equation for with annihilates higher order terms to give:

Comment: The eigenfunction from the analysis represents a small rigid x-displacement whose amplitude varies as exp(ik y y + ik z z).

We have also done a similar analysis for a cylindrical flock with a long wavelength perturbation along the cylinder axis. Cylindrical Flock

Numerical Analysis of Waves and Stability Use a standard algorithm to determine eigenvalues and eigenvectors. The solutions give all three branches of eigenvalues and their respective eigenfunctions. Linearized equations are a coupled system for ,  v x, and  v y. Discretize these functions of position, and arrange as one large vector.

Preliminary Conclusions From Numerical Stability Code:  All eigenmodes are stable (damped).  For small k (wavelength >> layer width), the damping rate is much larger than the real frequency.  For higher k (wavelength ~ layer width), the real frequency becomes bigger than the damping rate.

Flock Obstacle Avoidance We consider the middle of a very large flock moving at a constant velocity in the positive x direction. The density of the boids is uniform in all directions. The obstacle is represented by a repulsive Gaussian hill

Fourier-Bessel transform in Solution using Linearized System Add the repulsive potential, linearize the original equations

Black = Lower Density, White = Higher Density