Dynamics of Collective Motion: Experiment, Topology, and PDE

Chad Topaz
Department of Mathematics, Statistics, and Computer Science, Macalester College, Saint Paul

Biological aggregations such as bird flocks, fish schools, and insect swarms are striking examples of self-organized collective motion. The challenges of collective motion research include determining individual-level behaviors, assessing macroscopic group properties, and elucidating the connection between the two. In this talk, I present representative work addressing each challenge. First, to determine individual-level behaviors, I perform motion tracking experiments on pea aphids and use the data to develop an unbiased correlated random walk model. Second, to assess group-level dynamics, I apply topological data analysis to the influential interacting particle model of Vicsek et al. (1995). This analysis assigns a topological signature to a set of aggregation data and detects dynamical events that are undetected by standard methods. Third, I investigate a nonlocal PDE model for aggregation. In the PDE modeling framework, one can specify individual-level rules and determine the corresponding group behavior. The nonlocal model is well-approximated by a local, degenerate Cahn-Hilliard model that is more amenable to analysis and computation. Using the Cahn-Hilliard model, I demonstrate how environmental factors can suppress the aggregation's peak population density, which is essential for controlling locust outbreaks.