Coupled Networks, Patterns and Complexity - Abstract
Kolokolnikov, Theodore
Animal group behaviour is a fascinating natural phenomenon that is observed at all levels of the animal kingdom, from beautiful bacterial colonies, insect swarms, fish schools and flocks of birds, to complex human population patterns. The emergence of very complex behaviour is often a consequence of individuals following very simple rules, without any external coordination. This poses intriguing biological, mathematical and even philosophical questions. In this talk, we discuss a simple model of insect swarming based on pairwise particle interactions with short-range attraction and long-range repulsion, which can lead to very complex and intriguing patterns in two or three dimensions. Depending on the relative strengths of attraction and repulsion, a multitude of various patterns is observed, from nearly-constant density swarms to annular solutions, to complex spot patterns that look like ßoccer balls". We show that many of these patterns can be understood in terms of stability and perturbations of "lower-dimensional" patterns. For example, spots arise as bifurcations of point clusters [delta concentrations]; annulus is a perturbation of a ring. Asymptotic methods provide a powerful tool to describe the stability, shape and precise dimensions of these complex patterns. We also consider the inverse problem: given a target pattern, how to custom-design the interaction force to obtain said pattern as a steady state. Finally, we look at the effects of the underlying network connectivity on the cohesion of the swarm. We consider the simplest possible non-trivial scenario: a one-dimensional swarm consisting of two clusters, where each particle 'feels' every other particle with probability p [i.e. Erdös--Rényi random graph model]. Using basic probability and random matrix theory, we derive instability thresholds on p for when the swarm loses its cohesion.