Document Type

Article

Publication Date

2015

Abstract

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.

Comments

This work is a pre-print version of an article published by International Press

Published version: http://dx.doi.org/10.4310/CMS.2015.v13.n4.a6

DOI

10.4310/CMS.2015.v13.n4.a6

Download

Included in

Mathematics Commons

Share

COinS