Description du projet
De nouveaux outils mathématiques pour révéler la dynamique complexe des systèmes humains
Les équations aux dérivées partielles (EDP) sont le langage de base dans lequel la plupart des lois de la physique ou de l’ingénierie peuvent être écrites et l’un des outils mathématiques les plus importants pour la modélisation en sciences sociales. Parmi les nombreux domaines d’application de la modélisation cinétique en biologie mathématique, le projet Nonlocal-CPD, financé par l’UE, se concentrera sur les EDP pour étudier les systèmes impliquant un grand nombre «d’individus» qui présentent un «comportement collectif» et tenter d’en tirer des informations «moyennées». Les études porteront également sur l’asymptotique de longue durée, les propriétés qualitatives et les schémas numériques pour la diffusion non linéaire, l’hydrodynamique et les équations cinétiques de modélisation du comportement collectif de systèmes à N corps.
Objectif
"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."
Champ scientifique
- natural sciencesbiological sciencesdevelopmental biology
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
- natural sciencesbiological scienceszoologymammalogy
- natural sciencescomputer and information sciencesartificial intelligencecomputational intelligence
Mots‑clés
Programme(s)
Thème(s)
Régime de financement
ERC-ADG - Advanced GrantInstitution d’accueil
OX1 2JD Oxford
Royaume-Uni