Communications in Mathematical Sciences

Volume 13 (2015)

Number 4

Special Issue in Honor of George Papanicolaou’s 70th Birthday

Guest Editors: Liliana Borcea, Jean-Pierre Fouque, Shi Jin, Lenya Ryzhik, and Jack Xin

Ring patterns and their bifurcations in a nonlocal model of biological swarms

Pages: 955 – 985

DOI: https://dx.doi.org/10.4310/CMS.2015.v13.n4.a6

Authors

Andrea L. Bertozzi (Department of Mathematics, University of California at Los Angeles)

Theodore Kolokolnikov (Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada)

Hui Sun (Department of Mathematics, University of California at San Diego)

David Uminsky (Department of Mathematics, University of San Francisco, California, U.S.A.)

James von Brecht (Department of Mathematics, University of California at Los Angeles)

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 review and expand upon recent developments of this class of problems as well as present new results. As in previous work, 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 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 of 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.

Keywords

aggregation swarming, pattern formation, dynamical systems

2010 Mathematics Subject Classification

35B36, 35Q82, 70F45, 70H33, 82C22

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Published 12 March 2015