Clment Sire Laboratoire de Physique Thorique CNRS & Universit - - PowerPoint PPT Presentation

Analytical and numerical study

• f a realistic model for

fish schools

Clément Sire

Laboratoire de Physique Théorique

CNRS & Université Paul Sabatier Toulouse, France

www.lpt.ups-tlse.fr

Guy Theraulaz, Daniel Calovi, Ugo Lopez, J. Gautrais (CRCA, Toulouse) Hugues Chaté, Sandrine Ngo (CEA, Saclay)

Collective motion in fish schools

Swarming, schooling, milling

Introduction

• Several models reproduce qualitatively the

collective behaviors in fish schools, wild insect swarms, flocks of birds…

• The Vicsek Model (1995)

, ( 1) ( )

( 1) arg ( ) ( ) ( , , ) , 1 1 are the noise intensity interaction radius, and velocity Fixing and , the control parameters are or the densi y e t

 

    

  

      



Order parame

j R i

N t t i i i i j

R t t t t v v R v

i i

r r e

1

1











ter

i

N i

N e

Vicsek Model

Second order nature of the transition challenged by kinetic theory: mode instability – stripes – destabilizes the long-range order just below the

• nset of flocking (Grégoire et al. 2004, Bertin

et al. 2006, Chaté et al. 2007, Ihle 2010)

* / *

| | ( ) ,

 

        

c

L L

( ) ~| | 0.45(3), 1.6(3), 2.3(4)

c 

           (Baglietto 2012) et al.

Experiments by the CRCA team

• Need for realistic models based on constraining

and validating experiments

• These experiments (1, 2, 5… up to 30 fish) permit

to identify “individual laws”, “elementary interactions”, and “microscopic parameters”

Some important aspects

• Forces mediated by vision are in general not

conservative (no law of action-reaction)

• Are forces really additive (a finite amount of

information can be treated)? Instead, forces may be an average over the local environment

• Do fish (or birds) interact through metric
• r topologic (Voronoï diagram) forces?

(not crucial in a tank)

Long-range attractive interactions?

• Additive (?) attractive force are mediated by

vision and should be a linear function of the (solid) angle spread of the (group of) fish

• Long-range attractive (~gravitation) force…

but screened by obstacles

1

~ ~ ( )

d

r a r F



      

a  r

Basic model validated by CRCA experiments

• n Barred Flagtail (Kuhlia Mugil), and more recently, Hemigrammus
• J. Gautrais et al., J. Math. Biol. (2009); Plos Comput. Biology (2012)
• Constant velocity
• Individual (2D) angular velocity evolving

according to an Ornstein-Uhlenbeck process

• The target angular velocity includes the effect of alignment

and attraction (metric/topological) forces

~ 0.1 0.6 m/s  v

2 * 2

1 ( ) ( ), ˆ ( ~ / ; ~ )

i

i i i i i

d d d t v dt dt dt v v



                 r e

Basic model validated by CRCA experiments

• Topological alignment force and attraction force
• Phenomenological effect of vision angle
• + repulsive interaction with the wall
• Averaging

*

sin( sin( ( ) ) 1 cos )      



            

P i j j j i i i j ij

k v k r

~ v ~

ij

r

* * ,

( 1 ~ 6)

i j i i j i i

N N   

 





( )

W

k

V 

k k v

Basic model validated by CRCA experiments

• Experiments vs model simulations

Swarming to schooling transition as the velocity (and hence, alignment) is increased

• Mean fish distance and magnetization vs velocity
• Mean square displacement in a tank

Basic model validated by CRCA experiments

12

r v v P

2( )

x t   t

1

1 (Polarization)

i

N i

P N

 



e

Order parameter

12

r P v

2 Fish 5 Fish 10 Fish 15 Fish 30 Fish

Mean inter-individual distance r12 Alignment P Experiment Model No interaction

Empirical investigation of fish schooling

Comparison between model predictions and experimental data

Dimensionless equations of motion

• Inertial effects on the angle dynamics are negligible
• in numerical simulations (no milling phase for

)

2 * 2 , *

angle of fish velocity position of f with respect to the horizontal a ish ; ngle

1 2 , cos ,sin ) sin( ) sin( ) 1 c s ) (

• (



 

              

   

  

                 



i i ij

i

i i i i j i i i i j i j i j i ij ij ij

i i

d d d dt dt N dt r

r

r e

2 1 2

view of fish looking at fish ; distance between fish and

ˆ For 0.24 m/s, / 0.1s, 2 / ( ) 0.48 0.024, 2.7, s 1.7             





          

P

ij

i j r i j

v v k k 1    

*

Alignment ) ) cos (XY model, ( , with ) ( 2

j i i j i

d V V       



        in-between and mean- field

• Phase diagram without a tank

DC, UL, SN, CS, HC & GT, New J. Phys. (2014)

1 1

1 1

i i i

N i N i

P N M N

   

  

 

r

e e e Mill Po in lari za g tion Order parameters :

( 0.024; 1; )        plane

I II

Attraction

I

Alignment

I-II III II

  2

c

I II

Existence of a third narrow elongated phase for ; observed in some fish schools

School

• f

Atlantic herring (Clupea harengus) Photo courtesy of P. Brehmer - IRD

 

III III

• Experimental parameters lie not far from the

transition line: real fishes can slightly modify their velocity to go from swarming to schooling (notably in the presence

• f a predator)
• Divergence of the polarization

susceptibility near the transition line

• Swarming transition near the mean-field transition line

(see hereafter) 2.5, 1.7    

With P. Schumacher (CRCA)

Phase diagram without a tank

( 0.024; 1; )        plane

2 ( 0)

c

   

Mean-field theory

• Variables:
• Local order parameter

( , ) Velocity angle Coordinates the center of mass of the school Continuous density distribution of fish ( , , ) r vs r       r ) ( ), )) (cos ,sin ) ) cos ) sin (averages at fixed and ) ( 0): ) ( ,0) ( / 2): ) ( s ( , i ( , ( , ( , , ( , ( , ( n ,                               

x y x y

M M M M M r M r r r r r r r M Unifo Isotropi rm schooling pha c milling phas M e se M M ,cos ) 

Mean-field theory (attraction force)

• If the density is smooth enough, the attractive force between

fishes acts as an effective attraction force toward the center

• f mass

* A 2 2 * A 2 0 0

sin( ) ( ') ( ) ( ') such that ( ') 6 Expanding the top integral and assuming | | / ~ , 3 ( ) sin( ) 2 which tends to

r a r a i r a

r r dr d r dr d N r dr d a r r r r                   

  

                       

  

r r r a r r r r r lign    the velocity to the direction

a

( 0)  

Mean-field theory equations of motion (=0)

2 * 2 , . . .

, with 2 sin( ) ( )sin( ) cos( ) cos( ), sin( ) ( ) 0.024, 2. = si 7, ( ) / ~ 1.7 ( wall of the tan n k)

    

                                              

r Exp Exp Exp

d d M r dt dt d dt dr d M r M dt dt r M r M r e e e e

   

2 * 2

si 1 sin( ) ( ) cos( ) c n s )

• (

                                                   t M r M r Fokker - Planck equation ( )

Diffusion coefficient of a single fish

   

2 2 2 2

2 0, 1 C( ) [ ( ) (0)] 1 exp( / ) 2 ( ) lim [ ( ) (0)] / ex 2 ( ) ~ ~ p 2 2

  

         

   

                 



t

d d d dt dt dt t t t t D t t C t dt r e r r

2 2 1 ˆ

ˆ ( ) Expressing length and time in the original units with ( ) ˆ ) ( ~ phase noise) ~ ~ 2 2 ( 2

  

               

    

         D v D v D D v v Weak phase noise ⇒ large

1/2 ˆ  

 D Strong phase noise ⇒ small

Mean-field theory

• 1

( , , ) exp ( )

r

M r r r dr Z               



Exact solution for (swarming)

 

           ( ) 1 ( ) exp cos , cos Complete analogy with the (Antoni & Ruffo 1995) the XY model with all spins interacting wit h ea r M M Z i.e. Exact solution for (space irrelevant; schooling / swarming) HMF model      ch other ~ ( ), with 2

c c

M

Conclusion

• Realistic model for fish schools validated by experiments
• The general issue of topologic/metric force: relevance of

long-range interactions? (~self-gravitating Brownian particles → school cohesion at low noise; Chavanis & CS)

• The milling phase is present when vision effects are taken

into account, along with a narrow elongated phase

• Biologically relevant parameters are close to the

swarming/schooling transition line, where fish can quickly adjust to their environment

• Introduction of a mean-field theory
• Including the effect of vision (non conservative attractive force) to

reproduce the milling phase

• Allowing for non uniform/non isotropic order parameter
• Time evolution (instability modes, dynamical transitions…)

• Fish i

Fish j

• Fish j

Fish i

ij

r