  Swarming patterns that emerge from the interaction of many mobile agents are
a subject of great interest in fields ranging from biology to physics and
robotics. In some application areas, multiple swarms effectively interact and
collide, producing complex spatiotemporal patterns. Recent studies have begun
to address swarm-on-swarm dynamics, and in particular the scattering of two
large, colliding swarms with nonlinear interactions. To build on early
numerical insights, we develop a mean-field approach that can be used to
predict the parameters under which colliding swarms are expected to form a
milling state. Our analytical method relies on the assumption that, upon
collision, two swarms oscillate near a limit-cycle, where each swarm rotates
around the other while maintaining an approximately constant and uniform
density. Using this approach we are able to predict the critical swarm-on-swarm
interaction coupling, below which two colliding swarms merely scatter, for near
head-on collisions as a function of control parameters. We show that the
critical coupling corresponds to a saddle-node bifurcation of a stable limit
cycle in the uniform, constant density approximation. Our results are tested
and found to agree with large multi-agent simulations.
