Mathematical models for aggregation and self-collective behaviour

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The focus of the proposed research program is a theoretical and numerical investigation of mathematical models for aggregation/swarming phenomena. While such phenomena arise in a variety of areas, we are particularly interested in applications of aggregation models to population biology (chemotaxis of cells, swarming or flocking of animals), robotics and opinion formation. Various mathematical models exist in the literature, ranging from particle-based (ODE) to kinetic and continuum (PDE) descriptions. Most of these formulations lead to nonlinear and nonlocal differential equations which are challenging to analyze and simulate. The proposed research will concern extensions and generalizations of previous models for self-collective behaviour, as well as development and analysis of genuinely new models. A particular emphasis will be placed on the long time behaviour of the solutions and the resulting equilibrium configurations. An important class of aggregation models is described by an integro-differential equation for the evolution of the macroscopic density. The equation represents the active transport of the density by a velocity field that has a functional dependence on it, given by a convolution with an interaction potential. The model also has a corresponding discrete/ODE formulation. The interaction potential typically incorporates short-range repulsion and long-range attraction, and its properties are essential in the analysis and numerics of this class of equations. In addition to such interactions, the model may also include an external potential and/or diffusion terms. Both the discrete and the continuum models can be formulated as gradient flows of certain energy functionals; consequently, variational methods can be used to investigate their equilibria. Part of the proposed research will address various aspects related to this very general class of models that have been mostly overlooked so far: i) the role of boundaries and boundary conditions, ii) anisotropy (e.g., a limited perception field) and its effects, iii) models for multiple species, iv) aggregation on surfaces and manifolds. All aspects are very important in applications, but received little attention in the literature so far. We also plan to investigate several specific applications of aggregations models (e.g., to opinion formation and protein adsorption), as well as develop and study mathematically new models to address recent experimental observations on schooling fish. The proposed projects offer training opportunities for students with a variety of interests ranging from numerics and formal methods (asymptotics) to rigorous analysis. In addition, the projects can accommodate a wide range of skill levels, from unexperienced undergraduates to advanced PhD students and PDFs. The proposed program aims to advance basic mathematical research as well as applications.