Nonlocal Hydrodynamic Models of Interacting Agents

Agents in an interacting system typically organise their dynamics based on the behaviour of their neighbours. This is often observed in the animal kingdom - herds of mammals, schools of fish, flocks of birds - but also in a variety of economic, engineering, and social settings. Usually, due to huge number of agents in the system, detailed, microscopic description of such behaviour is impossible to analyse and to simulate numerically, and thus its applicability is rather limited.

Instead, in this project we propose to focus on the macroscopic models described by systems of Partial Differential Equations (PDEs). These equations allow us to capture interactions between multiple agents/species/phases in an elegant reformulation involving a small number of nonlocal hydrodynamic conservation laws. It often leads to completely unexplored classes of systems, but also opens the possibility of using a wide range of tools and techniques from the modern theory of PDEs to provide essential insight into the dynamics of large complex systems of interacting agents.

Solving the fundamental problems from the objectives of this proposal will provide a profound mathematical understanding of PDEs emerging in the modelling of collective behaviour. It will allow us to characterise the qualitative properties of the macroscopic models, and to asses whether they are fit to describe the achieving of consensus or emergence of complex patterns observable in nature. We are especially excited to gain this knowledge for the formally derived macroscopic models, as this could provide the arguments for their validity and applicability.

The long-term goal of the project is to use the multidisciplinary environment of the University College London to establish a new research group developing and analysing new PDE models, along with novel numerical techniques for emerging challenges in Mathematical Biology and Mathematical Physics. The core of this environment will be a team composed of the PI, two PDRAs and the UCL funded PhD student. Results will be widely disseminated in diverse environments, including the mathematical fluid mechanics and kinetic theory communities, but also within other disciplines such as computational science, or civil engineering.

This project has a potential to influence especially medicine, biology and social science, where collective behaviour and self organisation of interacting agents is ubiquitous. The important examples include growth of biological tissues, patterns in motions of animals, or road traffic. But surprisingly, very similar mechanisms are observable also in variety of economic, engineering, and social settings - the distribution of goods, spacecraft formation, sensor networks, digital media arts, and the emergence of languages in primitive societies. Mathematical modelling and analysis are indispensable in discovering conditions leading, for instance, to creation of congestions in a system. It is of key importance to understand these factors, as it might help to prevent, for example, diseases such as cancer, formation of traffic jams on road, or variety of financial and social crises.

Investigation of mathematical models of interacting agents triggered out substantial progress in many different fields of mathematics, especially in: measure theory, optimal transportation, probability theory, functional, harmonic, stochastic and numerical analysis, dynamical systems, and ordinary and partial differential equations. Enormous interest in this area can be confirmed by awarding highest honours to mathematicians who contributed most significantly to this progress, but also by exponential growth of special sessions, conferences, or even whole thematic programs devoted to emerging problems.

The focus of this project is on PDE models that can be, at least formally, derived from the microscopic agent-based systems. Such formulation opens the possibility of using a wide range of tools and techniques from the modern theory of PDEs in order to supply us with essential insight into dynamics of large complex systems of interacting agents. However, despite all the excellent progress that has been made already, some of the fundamental problems like existence and stability of solutions remain open even for the simplest systems. The techniques developed in this research will deepen our theoretical understanding of these PDEs models and to establish whether they are fit to describe arriving to consensus or emergence of complex patterns observable in nature. It is an important argument in favour of macroscopic PDEs models which are much more convenient for the purposes of further numerical analysis. Having a reliable model along with the numerical scheme is of key importance to forecast, for example, the most efficient drug, or the safest walkways in densely populated areas. This project aims at making significant advances not only for these two selected applications, but for a whole class of underlying PDE systems.