by Dirk Brockmann

This explorable illustrates one of the most famous and most fundamental models for the emergence of flocking, swarming and synchronized behavior in animal groups. The model was originally published in a 1995 paper by Tamás Vicsek and co-workers and is therefore called the Vicsek-Model. The model can explain why transitions to flocking behavior in groups of animals are often not gradual. Instead, one can expect a sudden emergence of flocking and synchronized movements if a critical density is crossed.

Press Play and keep on reading….

This is how it works

In the model a group of $N$ particles move around in space, a square arena with periodic boundary conditions (when particles hit one of the walls, they reenter on the other side, like ghosts). The particles move at a constant speed $v$ which is the same for all particles (you can change it with the speed slider).

However, the particles change the heading $\theta$ as time goes on. There are two factors that shape how particles change their heading:

  1. A random force that makes them wiggle around
  2. An alignment force that nudges a particle to align with others around it

So, given that the heading of each particle $n$ at time $t$ is denoted by $\theta_ n (t)$, the random force changes the heading in a time interval $\Delta t$ like so:

$$ \theta_n (t+\Delta t) = \theta_n (t)+\eta_ n(t), $$

where $\eta_ n(t)$ is a random number with zero mean and some variance. You can change the magnitude of the random heading changes with the wiggle slider in the control panel.

The alignment force works like this: At every time step a particle evaluates the mean heading $\left < \theta(t) \right > _n$ of all other particles around it in an interaction radius $R$ and aligns its own heading to match this mean heading. So with this and the wiggle the complete update of the heading is given by:

$$ \theta_n (t+\Delta t) = \left< \theta(t) \right>_ n +\eta_ n(t). $$

The size of the interaction radius can be adjusted with the interaction radius slider.

You can now explore under which conditions all particles will equilibrate to a state where the whole flock will eventually move into one direction (approximately). For a large flock, it helps increasing the speed of the particles with the speed slider.

The optional color by heading toggle paints the particles according to their heading.

You may want to check out the explorables “Flock’n Roll”, “Into the Dark”, and “Thrilling Milling Schelling Herings”, all of which implement a model for collective behavior and swarming, very similar (but slightly more complicated) to the Vicsek-Model. Also the model for pedestrian dynamics in explorable “The Walking Head”, is quite similar to the Vicsek-Model.

Further Reading

Related Explorables:

Horde of the Flies

The Vicsek-Model


Oscillators that sync and swarm


Oscillators that sync and swarm

Come Together

Chemotaxis in Dictyostelium discoideum

Janus Bunch

Dynamics of two-phase coupled oscillators

Critically Inflammatory

A forrest fire model