In this paper we study the pattern formation of a kinematic aggregation model for biological swarming in two dimensions. The swarm is represented by particles and the dynamics are driven by a gradient flow of a non-local interaction potential which has a local repulsion long range attraction structure. We leverage a co-dimension one formulation of the continuum gradient flow to characterize the stability of ring solutions for general interaction kernels. In the regime of long-wave instability we show that the resulting ground state is as a low mode bifurcation away from the ring and use weakly nonlinear analysis to provide conditions for when this bifurcation is a pitchfork. In the regime of short-wave instabilities we show that the rings break up into fully 2D ground states in the large particle limit. We analyze the dependence on the stability of a ring on the number of particles and provide examples of complex multi-ring bifurcation behavior as the number of particles increases. We are also able to provide a solution for the “designer potential” problem in 2D. Finally, we characterize the stability of the rotating rings in the second order kinetic swarming model.
Bertozzi, A.L., Kolokolnikov, T., Sun, H., Uminsky, D., & von Brecht, J. (2015). Ring patterns and their bifurcations in a nonlocal model of biological swarms. Communications in Mathematical Sciences, Volume 13 (4). Pages: 955-985. http://dx.doi.org/10.4310/CMS.2015.v13.n4.a6