Collective behaviour of active particles with mean-field interaction
type: CreativeWork
description: This thesis is concerned with providing mathematical foundations for the emergent collective behaviour of ants. That is, we rigorously derive a macroscopic partial differential equation (PDE) from a nonlinear interacting stochastic particle system used to model ants and use the PDE to provide a phase diagram for the collective behaviour of the particle model. The PDE model sustains ants forming bidirectional lanes and ants forming spherical aggregates, and the behaviour depends critically on the ant antenna. This thesis contributes to the nonlinear analysis of pattern formations and emergent collective behaviour by showing rigorously how a relatively simple phase-space extended non-gradient, nonlocal and nonlinear single species model can create multiple metastable higher-dimensional patterns. In the second chapter, we introduce a microscopic agent-based model for a collective of interacting ants. We model the ants as a stochastic interacting particle system, where each particle has a position and orientation, as an active Brownian particle. The interaction of the particles couples the spatial distribution of the particles by a torque that acts on the orientations, in contrast with the chemotactic drift in the parabolic-elliptic Keller--Segel model. The particles also have a look-ahead mechanism, which models the antenna of ants, and allows the particles to anticipate the chemical field. We show typical particle trajectories for the linear model with several background chemical fields. The behaviour includes (damped) oscillations that are predicted by the inviscid linearised theory. We also show the typical trajectories for the interacting nonlinear model for bounded radially symmetric interaction kernels. This behaviour features aggregation and the formation of travelling clusters. We also introduce a singular interaction kernel, such that the chemical field is a distributional solution of an elliptic equation. Motivated by the behaviour of the nonlinear microscopi…
abstract: This thesis is concerned with providing mathematical foundations for the emergent collective behaviour of ants. That is, we rigorously derive a macroscopic partial differential equation (PDE) from a nonlinear interacting stochastic particle system used to model ants and use the PDE to provide a phase diagram for the collective behaviour of the particle model. The PDE model sustains ants forming bidirectional lanes and ants forming spherical aggregates, and the behaviour depends critically on the ant antenna. This thesis contributes to the nonlinear analysis of pattern formations and emergent collective behaviour by showing rigorously how a relatively simple phase-space extended non-gradient, nonlocal and nonlinear single species model can create multiple metastable higher-dimensional patterns. In the second chapter, we introduce a microscopic agent-based model for a collective of interacting ants. We model the ants as a stochastic interacting particle system, where each particle has a position and orientation, as an active Brownian particle. The interaction of the particles couples the spatial distribution of the particles by a torque that acts on the orientations, in contrast with the chemotactic drift in the parabolic-elliptic Keller--Segel model. The particles also have a look-ahead mechanism, which models the antenna of ants, and allows the particles to anticipate the chemical field. We show typical particle trajectories for the linear model with several background chemical fields. The behaviour includes (damped) oscillations that are predicted by the inviscid linearised theory. We also show the typical trajectories for the interacting nonlinear model for bounded radially symmetric interaction kernels. This behaviour features aggregation and the formation of travelling clusters. We also introduce a singular interaction kernel, such that the chemical field is a distributional solution of an elliptic equation. Motivated by the behaviour of the nonlinear microscopi…
author: De Wit, Oscar
language: eng
keywords: active matter, PDE, SDE, analysis, numerical analysis, pattern formation, singular, phase transition, well-posedness, Alikakos, Sobolev, overdamped, asymptotic
license: https://www.repository.cam.ac.uk/bitstreams/4a7dc7ec-22dc-42f2-a7e5-2ee9baf5a43c/download
license: https://creativecommons.org/licenses/by/4.0/
date published: 2025-09-12
date modified: 2026-07-22T22:24:21Z
provider: Cambridge University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Cambridge

PropertyValue
property ID: DOI
value: 10.17863/CAM.124054