Active Brownian Particle and Random Walk Theories of the Motions of   Zooplankton: Application to Experiments with Swarms of Daphnia
  Active Brownian Particles are self-propelled particles that move in a dissipative medium subject to random forces, or noise . Additionally, they can be confined by an external field and/or they can interact with one another. The external field may actually be an attractive marker, for example a light field (as in the experiment) or an energy potential or a chemical gradient (as in the theory). The potential energy can also be the result of interparticle attractive and/or repulsive forces summed over all particles (a mean field potential). Four, qualitatively different motions of the particles are possible: at small particle density their motions are approximately independent of one another subject only to the external field and the noise, which results in moving randomly through or performing rotational motions about a central point in space. At increasing densities interactions play an important role and individuals form a swarm performing several types of self-organized collective motion. We apply this model for the description of zooplankton Daphnia swarms. In the case of the zooplankton Daphnia (and probably many other aquatic animals that form similar motions as well) this vortex is hydrodynamical but motivated by the self-propelled motion of the individuals. Similar vortex-type motions have been observed for other creatures ranging in size from bacteria to flocks of birds and schools of fish. However, our experiment with Daphnia is unique in that all four motions can be observed in controlled laboratory conditions with the same animal. Moreover, the theory, presented in both continuous differential equation and random walk forms, offers a quantitative, physically based explanation of the four motions. 
Udo Erdmann, Werner Ebeling, Lutz Schimansky-Geier, Anke Ordemann,   Frank Moss