Outline • Motivation - The midges and the swarm - The adaptive gravity model • Stable Swarming with Adaptivity

- Adaptivity as a self-stabilization mechanism

- Jeans Instability

• Conclusions

The midges and the swarm

In the lab (Stanford U.) :

Nick Ouellette - PI

Rui Ni

James Puckett

The midges and the swarm

The midges (Chironomidae)

• Non-biting midges

• Only male swarm (mating ritual)

• How many ?

• Where ?

• When ?

10−104

stream edges

dawn and dusk

1−100

Black felt “swarm markers”

Overhead light source – ON/OFF

nature lab

The midges and the swarm

Method: - High-speed stereo-

imaging using three synchronized cameras

(100 fps) - Automated motion

tracking algorithm

Trajectories of midges vs. time

Measurement: Kinematics – r (𝑡), v(𝑡), 𝑎 (𝑡)

In the lab (Stanford U.) :

The midges and the swarm

In the lab (Stanford U.) :

Nick Ouellette - PI

Rui Ni (Post-doc)

James Puckett (Post-doc)

• Long-range Interaction (“force”) • Swarm in the dark • Not influenced by chemical signals

The midges and the swarm

x (mm)

𝑎𝑥 (mm/𝑠𝑒𝑐2)

50 100 −50 −100

500

−500

−1000

1000

𝑁 = 10

Assumptions: • Long range interaction • Pairwise interaction • uniform density • spherical symmetry

2

1F

r

3

2ˆ

d rF r r

r

F Kr

Linear restoring force – effective spring constant

Isotropic Harmonic Oscillator

𝐾

𝐾

𝐾

The only possible force:

The Adaptive Gravity Model

The Model • Acoustic attraction – Johnston’s organ

The Adaptive Gravity Model

𝟏

𝒓𝟐

Acceleration towards the source 𝒂~𝟏

𝒓𝟐

• “Acoustic Gravity”

• Flight sound intensity decays as

Another feature (in the lab):

The linear force decreases for larger swarms

𝑥 (𝑚𝑚)

What is missing ?

A model of the acoustic interactions

sR

• A typical feature of sensory systems

Adaptivity (as a part of the Fold Change Detection Mechanism)

The Adaptive Gravity Model

Scalar symmetry For any two stimuli and the output is the same

S ( 0)p S p

Sensitive to directionality but not to the overall amplitude !

Acoustic input Output

2

1ˆiijeff

j i j

I C rr r

i ijj

s s ii

effij

j

sF

s

Motsch & Tadmor 2011

logS effFShoval et al. 2010

effFSStimulus -

response, effective force

11,..., ,... 11,..., ,...( ) ( )eff ij eff ijF p s p s F s s

𝑅𝑠

3

2

sR

i j

d r

r r

3

2 2

1sR

sj i j i j

d rR

r r r r

1

s

KR

with adaptivity Isotropic Harmonic Oscillator

F Kr

effective spring constant

The Adaptive Gravity Model

Uniform density & spherical symmetry

ˆ

1

ij

j i jieff

j i j

r

r rF C

r r

Adaptive Gravity – Evidence Supported by data – 122 swarms !

Black – raw data Red – Binned average Blue – (-1) slope (spherical) /(-2) slope (cylindrical)

2

1

s

KR

1

s

KR

Large swarms are elongated along the vertical axis

The Adaptive Gravity Model

Dependence of The Effective Force on The Density (Uniform)

density

Regular gravity 2

ijij

ij

i ijj

rs C

r

s s

2

1

r

At the center: 4( )

3

CS r r

Adaptive gravity ii

effij

j

sF

s

At the center: 2

2( ) ( )

3eff

s s

C rF r r O

R R

At the center: 2

6 2 2

2( ) ( )

ln( )eff

s s s

C rF r r O

R R R

2

3 233

3( ) ( )

3 ( 1)

eff nsn

s s

C n rF r r O

RR R

3n

3n

Adaptive forces

2

1

r

1nr

( 2)n

ijij n

ij

iieff

ijj

rs C

r

sF

s

Increasing density

Stronger pull

Increasing density

Weaker pull

Marginal

density

density

Jeans Instability (Gravity)

sR

• Balance: gravitational pull random velocities

• collapse (minimal density for collapse)

• If

GJeans

esc colt t

Escape time (random

velocities) 2v

sesc

Rt

Time for collapse

F kr

2colt

k

2

2

4 3 v

3 16GJeans

s

Ck

R C

Jeans Instability ( Adaptive Gravity)

sR

• No critical density !

• If

GJeans

esc colt t

Escape time (random

velcoities) 2v

sesc

Rt

Time for collapse

F kr

2colt

k

22

4v

3 3s

s

R CCk

R

Jeans Instability ( Adaptive Forces )

sR

• Stabilization at a particular density

• If

AJeans

esc colt t

Escape time (random

velcoities) 2v

sesc

Rt

Time for collapse

F kr

2colt

k

1nr

( 2)n

6 2 3 2 2

3

3

3 3 2 233

42 13 ( )

ln( )

3 4 313 1

33 ( 1)

sAJeans

s s s

nsA

Jeansnsn

s s

CRCn k Exp

R R R

C n C n Rn k

RR R

v

v

AJeans

t

stability

• Midge swarm dynamics is dominated by long range acoustic interactions

• The interactions are adaptive - weaker when the background intensity is higher.

• Adaptivity, for general power-law interactions, stabilizes the swarm against collapse

• A prediction: A Selection of a particular density for higher power law interactions

Conclusions

( 2)n

Acknowledgements

Experimental group (Yale U.):

Nick Ouellette - PI Rui Ni James Puckett

Reuven Ianconescu (Research Assistant)

Nir Gov – PI

Some Results...

(mm)

2rExtended Virial Theorem

2 0T W pr ds Surface Pressure

– keeps the swarm together

Density profile

Theoretical Data

Poisson-Boltzmann equation w/ cut-off

Ellipsoidal Approximation

Boundary closer to the center – Stiffer effective spring