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

Abstract

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. IntroductionMathematical models for swarming, schooling, and other aggregative behavior in biology have given us many tools to understand the fundamental behavior of collective motion and pattern formation that occurs in nature [10,6,2,26,25,14,7,13,27,19,33,32,23,11,17,37,38,34,36,9,15,29,21,20,24,8]. One of the key features of many of these models is that the social communication between individuals (sound, chemical detection, sight, etc...) is performed over different scales and are inherently nonlocal [11,22,2]. In the case of swarming, these nonlocal interactions between individuals usually consist of a shorter range repulsion to avoid collisions and medium to long range attraction to keep the swarm cohesive. While some models include anisotropy in this communication (e.g. an organism's eyes may have a limited field of vision) simplified isotropic interactions have been shown to capture many important swarming behaviors including milling [20,10]. More recently it has been shown [17,38,37] that the competition between the desire to avoid collisions and the desire to remain in a cohesive swarm can sometimes result in simple radially symmetric patterns such as rings, annuli and uniform circular patches and other times result in exceedingly complex patterns. Moreover how modelers select the strength and form of the repulsion near the origin

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