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

In this paper we study the pattern formation of a kinematic aggregation model for biological swarming in two dimensions

Andrea L. Bertozzi; Theodore Kolokolnikov; Hui Sun; David Uminsky; James von Brecht

2015

Scholarcy highlights

  • Mathematical 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
  • 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
  • Second order model Until now we have only considered the ground state patterns of the kinematic model for particle interactions, in particular the particles have no independent means of self-motility
  • We have investigated the stability of a ring pattern in a two-dimensional aggregation model
  • The threshold case p = 0 is interesting: as we show in Proposition 6.1, in this case it is possible for a discrete ring of N particles to be stable for very large N , it is unstable in the continuum limit

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