Herds
Modelling of gregarious behavior
Herds of gregarious mammals offers a natural laboratory for the observation of emergence phenomena.
Like in vehicular traffic or crowds, the spatial extension of the agents gives rise to congestion phenomena and a jamming phase transition. Indeed, the density of individuals cannot exceed a maximal density threshold at which they are in contact with each other. The morpho-dynamics of herd structures can be described in the frame of Self-Organized Hydrodynamics (flows with constant velocity norm) with maximal density constraint (or constrained SOH model), using the singular pressure model.
The left picture shows a typical structure taken by such a flow in the jammed region, where it is incompressible. The velocity is uniform along straight lines perpendicular to the velocity itself and the flow lines are parallel curves. This geometry accounts for the structuration of the flow in parallel lines, like in a herd. The pink region corresponds to the area where the congestion density is reached. Its boundary models the boundary of the herd.
The videos below show simulation results of the constrained SOH model. They show the collision of two herds in the midst of a lower density population spinning around the center. The density and velocity videos are shown. The particular structure of the velocity field as described above is clearly visible on the second video.
These two videos correspond to the constrained SOH model with no background pressure (in the uncongested region, the fluid is pressureless) and with information speed equal to fluid speed. The following links provide videos of other test cases:
With background pressure, pressure waves initiated by the collision of the two herds radiate into the unjammed region. When the information speed is smaller than the fluid speed (which is the standard case), the formation of congestions is enhanced.
Individual based models consisting of a two-particle species extension of the Vicsek model have also been proposed (see L. Navoret's web page). Their macroscopic limit leads to various extension of the Self-Organized Hydrodynamics.