Dynamics of a Continuous Model for Flocking

Ed Ottin 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, v0

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

),(v

),(

tx

tx

Conservation of Boids:

Velocity Equation:

Upt

)(1

vvv

4

0v t

1 2

3

Governing Equations:

v)v

v1(

1]v[

20

2

W

(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.

p 1

*

*

1)( *

Tp

Here -1 represents a ‘screening length’ past which the

interaction between boids at and becomes ineffective.

In this case, satisfies:

and U satisfies:

)'(22 xxuu

)(022 xuUU

These equations for u and U apply in 1D, 2D, and 3D.

'x x

(2). Long-range attraction: U

')',(),'(0 xdxxutxuU

|'|4

|)'|exp()',(

xx

xxxxu

00 u

)',( xxu

Our choice for satisfies:)',( xxw

)'(22 xxww v

(3). Velocity orientation relaxation term:

')],(v),'(v[),'()'(]v[ 0 xdtxtxtxxxwwW

(4). Speed regulation term:

This term brings all boids to a common speed v0. If |v| > v0 (|v| < v0 ), then this term decreases (increases) |v|. If , the speed |v| is clamped to v0.

v)v

v1(

12

0

2

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 (v0):

Equilibrium

00vv x

Equilibrium Equations:

0)()(1

xUdx

dxp

dx

d

)()()(

02

2

2

xuxUdx

xUd

0)(2

12

dx

d

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

0*u

T

Equilibrium Solutions

045.0,2.0 and the density at x = 0 is determined to be

5604.00

An Example

Solving the potential equation, we get

The profile is symmetric about 0x

Waves and Stability

Equilibrium:

Perturbations:

000

00

vv

)(

x

x

)exp()(~ zikyikstxf zy

...ˆ

)(1ˆ1

222

2

00

0

0

W

kdx

du

d

dpH

and the notation signifies the operator

2)'(')(

'1

22

2 xxexfdxxf

dx

d

1

22

2

dx

d

dx

dxizkykK zy 000

0

ˆ

v

Basic Equation:

000

10

2 2ˆˆˆˆ xxWsKHKs

where:

Long Wavelength Expansion

Ordering Scheme:

102

0

22

10

2

ˆˆˆ ,ˆˆ

~ ,1~~

KKKUHH

kk zy

wkk

π

2 ,

2

slab ofwidth w

Analysis:

)( equation )(

:in equations Expand

00

OO

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

0

042 BksCsA

.dˆ

,d,d

dd

dd

2

dd11

2

dd1

00

00

0

0

xWC

xBxA

xx

xp

ux

Comment:

The eigenfunction from the analysis represents a small rigid x-displacement whose amplitude varies as exp(ikyy + ikzz).

xx

xxd

)(d00

0~)()(

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.

VBVA

Linearized equations are a coupled system for , vx, and vy. 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

... where

,d)exp()()(),(0

(k)A

kinrkJkAr

n

nnn

.and22 yxr

Solution using Linearized System

Add the repulsive potential, linearize the original equations 0

t

Black = Lower Density,

White = Higher Density

10 0 100.15

0.075

0

0.075

0.15

Re()

Im()

x

10 0 100.01

0.005

0

0.005

0.01

Revx Imvx Vx

x

10 0 100.3

0.15

0

0.15

0.3

Revy Imvy Vy

x

10 0 100.15

0.075

0

0.075

0.15

Re()

Im()

x

10 0 100.01

0.005

0

0.005

0.01

Revx Imvx Vx

x

10 0 100.3

0.15

0

0.15

0.3

Revy Imvy Vy

x

10 0 100.15

0.075

0

0.075

0.15

Re()

Im()

x

10 0 100.01

0.005

0

0.005

0.01

Revx Imvx Vx

x

10 0 100.3

0.15

0

0.15

0.3

Revy Imvy Vy

x

10 0 100.15

0.075

0

0.075

0.15

Re()

Im()

x

10 0 100.01

0.005

0

0.005

0.01

Revx Imvx Vx

x

10 0 100.3

0.15

0

0.15

0.3

Revy Imvy Vy

x

10 0 100.15

0.075

0

0.075

0.15

Re()

Im()

x

10 0 100.015

0.0075

0

0.0075

0.015

Revx Imvx Vx

x

10 0 100.3

0.15

0

0.15

0.3

Revy Imvy Vy

x

r0.967 i 0.122

10 0 100.15

0.075

0

0.075

0.15

Re()

Im()

x

10 0 100.015

0.0075

0

0.0075

0.015

Revx Imvx Vx

x

10 0 100.3

0.15

0

0.15

0.3

Revy Imvy Vy

x

r=-1.072, i=-0.025 r=1.072, i=-0.025 r=-1.025, i=-0.075 r=1.025, i=-0.075 r=-0.978, i=-0.122 r=0.978, r=-0.122

First Three Lowest Eigenmodesk = 0.4, = 0.3, 0/* = 0.8, = 1, = 2, = 100

r0 i 0.123

10 0 100.3

0.15

0

0.15

0.3

Re()

Im()

x

10 0 100.12

0.06

0

0.06

0.12

Revx Imvx

x

10 0 100.2

0.1

0

0.1

0.2

Revy Imvy

x

r=0, r=-0.123 r0 i 0.091

10 0 100.3

0.15

0

0.15

0.3

Re()

Im()

x

10 0 100.12

0.06

0

0.06

0.12

Revx Imvx

x

10 0 100.2

0.1

0

0.1

0.2

Revy Imvy

x

r=0, r=-0.091r0 i 0.158

10 0 100.3

0.1

0.1

0.3

0.5

Re()

Im()

x

10 0 100.05

0.025

0

0.025

0.05

Revx Imvx

x

10 0 100.1

0.05

0

0.05

0.1

Revy Imvy

x

r=0, r=-0.158

ZERO REAL FREQUENCY EIGENMODES

k=0.01, =0.3, =0.8, =1, =2, =100

0 y 0 x

0 0x 0 x x2

0 0 0

0y 0 x

0 0

2

2

2 20 2 2

dk V i ( V )

dx

dP dPd dV i i i [ ] i [ V ] i V

dx d dx dx

dPd dV k k [ ] [ V ]

dx d dx

1[ ] dx (x )exp( k 1 x x )

k 1

[ V] dx exp( k xk

��������������0 02

1/ 22y0 * 0

0 1/ 2 2 1/ 2* 0 * 0 * 0

V(x)x ) (x ) V(x ) dx exp( x x ) (x )

k(0)2T 4 (0)= k= =

u u ( u )

NORMALIZED EQUATIONS

Dimensionless parameters

0 .03 0 .02 0 .01 0 0 .01 0 .02 0 .032

1 .5

1

0 .5

0

0 .5

1

Im z( )

Re z( )

4 2 0 2 410 1

10 0 .5

10 0

99 .5

99

98 .5

98

Im z( )

Re z( )

k=0.01, =0.3, =0.8, =1, =2, =100