Modeling Emergent Collective Behavior

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Three Behavioral Zones

•Zone of repulsion

•Zone of orientation

•Zone of attraction

•Blind Regiona

Adapted from Inada, 2002

The Logic of the ModelIf any neighbors in zone of repulsion (“Dude, 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

Desired direction for individual i. i = 1, 2, …, N

Repulsion Rule Orientation RuleAttraction Rule

The Three “Laws of Orientation”

Model Implementation: Compute Desired Heading

If any neighbors in zone of repulsion (repulsion has highest priority!):

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, turn 180 degrees-

Desired turning angle:

If f(t) < wt (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 Current Heading

Fraction defines how much of desired turn is actually achieved.

Quantifying group coherencePolarization

Angular Momentum

Moment Arm

Centroid

Average heading

Couzin Model ResultsCirculating 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

Rpred

Pure predator escape rule

Predator escape with obedience to the school

Obedience parameter (range = 0-1)h = 1: Fully obedient to the schoolh = 0: Ignore the school, move only to get away from predator

If predator is within the ‘danger zone’ (defined by Rpred)

Predator Attack: Further Considerations

What about multiple predators? And what about ratio of school sizeto number of predators?

Simulation Software (FishSim)

ReferencesI.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, 371-387

Adi Shklarsh, Gil Ariel, Elad Schneidman, Eshel Ben-Jacob. Smart Swarms of Bacteria-Inspired Agents with Performance Adaptable Interactions. PLoS Computational Biology, 2011; 7 (9): e1002177 DOI: 10.1371/journal.pcbi.1002177

y

Definition of Coordinate System