Project description
New mathematical tools to uncover the complex dynamics of many-body systems
Partial differential equations (PDEs) are the basic language in which most of the laws in physics or engineering can be written and one of the most important mathematical tools for modelling in social sciences. Amongst the numerous application areas of kinetic modelling in mathematical biology, the EU-funded Nonlocal-CPD project will focus on PDEs to study systems involving a large number of 'individuals' that display 'collective behaviour' and try to obtain 'averaged' information from them. Studies will also focus on long-time asymptotics, qualitative properties and numerical schemes for nonlinear diffusion, hydrodynamic and kinetic equations in the modelling of many-body systems' collective behaviour.
Objective
"This proposal focuses on the development of new mathematical tools to analyse theoretical, numerical and
modelling aspects of novel applications of nonlinear nonlocal aggregation-diffusion equations in Mathematical Biology and in classical problems of kinetic theory. Among the numerous areas of applications of kinetic modelling in Mathematical Biology, we will concentrate on phenomena identified, at the modelling stage, as systems involving a large number of ""individuals"" showing ""collective behaviour"" and how to obtain ""averaged"" information from them. Individuals behavior can be modelled via stochastic/deterministic ODEs from which one obtains mesoscopic/macroscopic descriptions based on mean-field PDEs leading to continuum mechanics, hydrodynamic and/or kinetic systems. Understanding the interplay between the interaction behaviour (nonlocal, nonlinear), the diffusion (nonlinear), the transport phenomena, and the synchronization is my main mathematical goal.
The proposed research is centred on developing tools underpinning the analysis of long time asymptotics, phase transitions, stability of patterns, consensus and clustering, and qualitative properties of these models. On the other hand, designing numerical schemes to accurately solve these models is key not only to understand theoretical issues but also crucial in applications. We will focus on the important case of the Landau equation with applications in weakly nonlinear plasmas by means of the gradient flow techniques. On the other hand, we showcase our tools in patterns and consensus by focusing on zebra fish patterning formation, as example of spontaneous self-organisation processes in developmental biology, and grid cells for navigation in mammals, as prototype for the synchronization of neural networks. This project connects with other areas of current interest in science and technology such as agent-based models in engineering: global optimization, clustering, and social sciences."
Fields of science
- natural sciencesbiological sciencesdevelopmental biology
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
- natural sciencesbiological scienceszoologymammalogy
- natural sciencescomputer and information sciencesartificial intelligencecomputational intelligence
Keywords
Programme(s)
Topic(s)
Funding Scheme
ERC-ADG - Advanced GrantHost institution
OX1 2JD Oxford
United Kingdom