Three Behavioral Zones Zone of repulsion Zone of orientation Zone of attraction Blind Region  Adapted from Inada, 2002 RoRo

The Logic of the Model If any neighbors in zone of repulsion (“Dude, quit riding my tail!”).quit riding my tail Else If all neighbors are in the zone of orientation: If neighbors exist in both orientation and attraction zones: If all neighbors are in the zone of attraction: Otherwise fish is lost Get away--repulsion has highest priority. Orient to their (average) heading Come closer to their (average) position Then orient and get closer, in proper proportion Turn around 180 degrees and start heading back

y Definition of Coordinate System Unit direction vector of fish i wrt the horizontal Headings Position

The Three “Laws of Motion” Image credit: Repulsion Rule Attraction Rule Orientation Rule

Model Implementation: Compute Desired Heading Else If any neighbors in zone of repulsion (repulsion has highest priority!): If all neighbors are in the zone of orientation: If all neighbors are in the zone of attraction: Otherwise fish is lost, turn 180 degrees If neighbors exist in both orientation and attraction zones:  : orient/attract priority parameter. Range = [0-1]

Desired turning angle: If  (t) <  (Desired turn can be achieved in one time step) Else (Desired turn cannot be achieved in one time step, turn as much as possible toward desired heading) Model Implementation: Update Heading Fraction defines how much of desired turn is actually achieved.

Add a little noise; update position Gaussian noise term: turning angle error v i (t) v i (t+  ) “cruising speed”, assumed constant over time

Quantifying Group Behavior Polarization Angular Momentum Moment Arm Centroid

Couzin Model Results Circulating fish school—rare! Random/disorganized group behavior Emergent collective behavior: all fish pointed the same direction

Predator Attack! What rules do you obey? Move away from predator? Or stay with the school? Obey your thirst!

Run (errr, swim) for Your Lives!: The Predator Escape Rule R pred Pure predator escape rule Predator escape with obedience to the school Obedience parameter (range =  )  =  : Fully obedient to the school  =  : Ignore the school, move only to get away from predator If predator is within the ‘danger zone’ (defined by R pred )

Predator Attack: Further Considerations What about multiple predators? And what about ratio of school size to number of predators?

Simulation Software (FishSim)

References I.D. Couzin et al. Collective Memory and Spatial Sorting in Animal Groups, J. theor. Biol (2002) 218, 1-11 Y. Inada and K. Kawachi. Order and Flexibility in Motion of Fish Schools, J. theor. Biol (2002), 214,