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

Critical transition for colliding swarms