Descripción del proyecto
Nuevas herramientas matemáticas para desvelar la dinámica compleja de sistemas de muchos cuerpos
Las ecuaciones diferenciales parciales (EDP) son el lenguaje básico con el cual se pueden describir la mayoría de las leyes de la física o la ingeniería y es uno de las herramientas matemáticas para la modelización en ciencias sociales. Entre los numerosos ámbitos de aplicación de la modelización cinética en biología matemática, el proyecto Nonlocal-CPD, financiado con fondos europeos, se centrará en las EDP para estudiar sistemas que implican un amplio número de «individuos» que muestran «comportamiento colectivo» e intentarán obtener información «promedio» de ellos. Los estudios también se centrarán en las propiedades cualitativas asintóticas a largo plazo y los esquemas numéricos para las ecuaciones no lineales de difusión, hidrodinámica y cinética en la modelización del «comportamiento colectivo» de sistemas de muchos cuerpos.
Objetivo
"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."
Ámbito científico
- natural sciencesbiological sciencesdevelopmental biology
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
- natural sciencesbiological scienceszoologymammalogy
- natural sciencescomputer and information sciencesartificial intelligencecomputational intelligence
Palabras clave
Programa(s)
Régimen de financiación
ERC-ADG - Advanced GrantInstitución de acogida
OX1 2JD Oxford
Reino Unido