KIBORD

This proposal consists in gathering three teams of mathematicians (ENS Cachan-CMLA, Univ. Paris-Dauphine-CEREMADE, Univ. Paris 6-LJLL) specialized in PDEs and their numerical simulation, and already having an experience in the collaboration with teams working in the sector of biology or medical studies. What we propose here is a detailed study of the modeling and the mathematical/numerical analysis of problems arising from different areas of biology, including cell biology, biological fluids, population dynamics and animal collective behavior. The common feature of those problems is the need to examine the qualitative properties and the numerical approximations of systems of PDEs which are specially designed to model them, and which are distinct from any of the systems appearing in physics. Most often, they have features which are specific to the application, like an infinite number of equations for coagulation-fragmentation, or cross diffusions terms for the spatial evolution of intelligent species. The methods to be used come from the most recent developments in PDE theory, they include existence of nontrivial remarkable states, entropy/entropy dissipation estimates, linear and nonlinear asymptotic stability, perturbative methods, duality lemmas, computer-aided proofs, time-dependent rescaling, renormalized solutions, etc. Our goal is in particular to extract informations from systems which are far from known steady states. Our study will be structured around three main themes : - Large multi-agent systems and their spatial structure. Those situations appear when a large number of individuals following a simple law gives rise to a complex behavior, as in statistical mechanics. Various levels of modeling lead to different classes of equations (kinetic, parabolic, etc.). The biological issues addressed here include swarming, chemotaxis, sprays, and neuron networks. - Growth, coalescence and fragmentation. Those phenomena occur in biology and (bio)chemistry at various scales (molecules, cells, etc.). They are modeled by infinite (or even continuous) numbers of PDEs leading to a wide variety of behaviors. The blow-up phenomenon known as gelation is especially interesting, and its study in the spatially inhomogeneous context is a recent and very promising issue. - New paradigms in reaction-diffusion models. Those models have been used in the modeling of living organisms for a very long time. New types of reaction-diffusion equations have recently drawn a lot of attention, among which reaction-cross diffusion systems, equations including nonlocal terms. Also recently devised methods (such as computer-aided existence proofs) shed a new light on classical problems (such as Turing instability). The program of research which is proposed will enable to propose several M2 internships for students having acquired skills in PDEs applied to biology/medicine. They will for example come out of the master 2 « spécialités » of Univ. Paris 6 and campus Paris-Saclay devoted to biomathematics with which the participants of the project have strong links, but also from other french or international high level graduate courses in biomathematics, PDEs and numerical analysis.