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

Analytical and numerical study of 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) N        ( ) i t ( 1) arg ( ) e t j t i i   , i j R 0     ( 1 ) ( 1 ) r t r t v e   0 ( 1) i i t i  , , are the noise intensity interaction radius, and velocity , R v 0 0   Fixing and , the control parameters are or the densi y t R v 0 0 N 1    Order parame ter e  N i  i 1

Vicsek Model       ( ) ~| | (Baglietto et al. 2012)    c | |        * / * c ( ) , L L       0.45(3), 1.6(3), 2.3(4)  Second order nature of the transition challenged by kinetic theory: mode instability – stripes – destabilizes the long-range order just below the onset of flocking (Grégoire et al. 2004, Bertin et al. 2006, Chaté et al. 2007, Ihle 2010)

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 or 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  1 d   a   ( ) ~ ~   F r   r a r  Long-range attractive (~gravitation) force… but screened by obstacles

Basic model validated by CRCA experiments on Barred Flagtail (Kuhlia Mugil), and more recently, Hemigrammus J. Gautrais et al., J. Math. Biol. (2009); Plos Comput. Biology (2012)   Constant velocity ~ 0.1 0.6 m/s v  Individual (2D) angular velocity evolving according to an Ornstein-Uhlenbeck process   2 1 ( d d d r           * i i i ) ( ), e t v   i i 2 dt dt dt i     ˆ ( ~ / ; ~ ) v v  The target angular velocity includes the effect of alignment and attraction (metric/topological) forces

Basic model validated by CRCA experiments  Topological alignment force and attraction force ~ v ~ r ij  Phenomenological effect of vision angle                * sin( ) sin( ) 1 cos ( ) k v k r      j i j i P i j ij i j  + repulsive interaction with the wall ( ) k W V  k k v  Averaging 1      * * ( ~ 6) N i j i i N   , j i i

Basic model validated by CRCA experiments  Experiments vs model simulations Swarming to schooling transition as the velocity (and hence, alignment ) is increased

Basic model validated by CRCA experiments  Mean fish distance and magnetization vs velocity r P v 12 r P 12 N 1  e  Order parameter P  N i  1 i v (Polarization) v  Mean square displacement in a tank   2 ( ) x t t

Empirical investigation of fish schooling Comparison between model predictions and experimental data Mean inter-individual Alignment P distance r 12 Experiment 2 Fish Model No interaction 5 Fish 10 Fish 15 Fish 30 Fish

Dimensionless equations of motion   2 1 d d d r            *  i i i 2 , ( cos ,sin ) e   i j i i i 2 dt dt N dt i   , i j i                  * sin( ) sin( ) 1 c s o ( ) r      j i j i ij ij ij    position of f ish ; angle of fish velocity with respect to the horizontal r i i i i    a ngle view of fish looking at fish ; distance between fish and i j r i j ij ij           ˆ 2 1 For 0.24 m/s, / 0.1s, 2 / ( ) 0.48 s v v 0    k k          0.024, 2.7, P 1.7    2 0  V             * Alignment ( ) , with ( ) cos V    j i i j i  (XY model, in-between 2 and mean- field ) d  Inertial effects on the angle dynamics are negligible      in numerical simulations ( no milling phase for 1 0 )

Phase diagram without a tank       plane  ( 0.024; 1; ) DC, UL, SN, CS, HC & GT, New J. Phys. (2014) N 1    Order parameters : Po lari za tion P e  N i  1 i N 1    Mill in g M e e  r N i i  1 i I II I I-II Alignment II I II III   2 0 c Attraction

Existence of a third narrow elongated phase   for ; observed in some fish schools III III School of Atlantic herring ( Clupea harengus ) Photo courtesy of P. Brehmer - IRD

Phase diagram without a tank       plane  ( 0.024; 1; )      Experimental parameters lie not far from the 2.5, 1.7 transition line : real fishes can slightly modify their velocity to go from swarming to schooling (notably in the presence of a predator )  Divergence of the polarization susceptibility near the transition line With P. Schumacher (CRCA)  Swarming transition near the mean-field transition line     (see hereafter) 2 ( 0) c

Mean-field theory  Variables:   Coordinates ( , ) the center of mass of the school r r vs  Velocity angle    Continuous density distribution of fish ( , , ) r  Local order parameter        ( , ) ( ( , ), ( , )) (cos ,sin ) M r M r M r M 0 0 0 x y            ( , ) cos , ( , ) sin (averages at fixed and ) M r M r r x y     ( 0): ( , ) ( ,0) Unifo rm schooling pha se M r M 0 0           ( / 2): ( , ) ( s i n ,cos ) Isotropi c milling phas e M r M 0 0

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   ( 0) of mass a              sin( ) ( ') r r r r dr d     * ( ) r a r  A       ( ') r r r dr d  r a              2 such that ( ') 6 N r r r dr d a 0 i  r a r    Expanding the top integral and assuming | | / ~ , 2 r 0 3       * ( ) sin( ) r r  A 2 2 r 0 0    which tends to a lign the velocity to the direction

Mean-field theory equations of motion (  =0)   2 d d                 * 2 sin( ) ( )sin( ) M r 0 0 2 dt dt r d     , with = e M e M e e      0 0 , r 0 0 dt  dr d                 cos( ) cos( ), sin( ) si n ( ) M r M 0 0 0 0 dt dt         0.024, 2. 7, ( ) / ~ 1.7 ( wall of the tan k) r r . . . Exp Exp Exp   0 Fokker - Planck equation ( )       2 1                    * sin( ) si n ( ) M            0 0 2 t r               cos( ) c o ( s ) M    0 0 r