Chad Topaz, Macalester College

Dynamics of Collective Motion: Experiment, Topology, and PDE

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.