Nonlocal Hydrodynamic Models of Interacting Agents

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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.