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
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