KInetic models in Biology Or Related Domains – KIBORD

The methods to be used come from the most recent developments in PDE theory, they include
- the existence of nontrivial remarkable steady states, thanks to topological methods (degree theory) but also thanks to computer-aided proofs, in which the inifinite dimensional part of the equation is treated theoretically, while a finite-dimensional part is treated at the level of the computer (with a rigorous control of all errors);
- entropy/entropy dissipation estimates, leading to quantitative results of convergence to steady states, even where one starts from initial data which are far from the equilibrium.
- perturbative methods, including the recent developments involving enlargment of functional spaces enabling to obtain spectral gaps.
- duality lemmas, which provide some amount of parabolic regularization even with non-continuous coefficients in parabolic equations,
- time-dependent rescalings, which allow the study of the large-time behavior of equations in which no nontrivial steady state exists, and lead to proofs of existence of asymptotic profiles.
- renormalized solutions, which naturally appear in very singular kinetic or parabolic problems, and provide a good setting for situations in which the regularity of the solutions (such as defined by the known a priori estimates) is not sufficient to define rigorously distributional solutions.
All those methods are used for all types of equations (kinetic equations, hyperbolic systems, infinite-dimensional parabolic equations, integrodifferential systems, etc.).

J. A. Carrillo, F. James, F. Lagoutière et N. Vauchelet have studied the so-called aggregation equation, which describes the dynamics of bacteria when an attractive chemical substance is acting. The study of such a system leads to the introduction of measure solutions, since a finite time blowup is possible. This study has enabled a rigorous definition of the trajectories of aggregates of bacteria.

L. Desvillettes, T. Lepoutre, A. Moussa et A. Trescases worked on the existence of weak solutions for general systems of cross diffusion. First, they showed that the ellipticity condition coincides with the admissibility of the coefficients of the system. Then they solved a problem which appears regularly in the theory of those equations, namely the building of a good approximating problem.

M. Burger, A. Lorz and M.-T. Wolfram analysed a model of mean field game of Boltzmann type for the growth of knowledge. The authors concentrated on the existence of special solutions related to an exponential growth, and which are called pathways of equilibriated growth.

S. Mischler, C. Quininao and J. Touboul looked at an evolution equation of kinetic McKean-Vlasov type, which describes the statistical behavior of a neurone network in interaction, each neurone
following its own dynamics of noised Fitzhugh-Nagumo type. They show the existence of at least one nontrivial steady solution. Simulations are presented in the case of a regime of strong neuronal connexions, which leads to a complex dynamics (oscillation between two stationary solutions).

L. Boudin and F. Salvarani have compared the kinetic modeling in social sciences on real data coming from polls for the referendum for the independance of Scotland. The results produced by the model are quite interesting from the asymptotic point of view.

Recently a collaboration on innovative methods to control dengue has started in laboratory Jacques-Louis Lions, in collaboration with FioCruz in Rio de Janeiro, Brazil. The idea of this study is to control the dengue vector (that is the mosquito Aedes Egypti) by a population of mosquitoes that cannot transmit the virus. This study then reduces to the study of a system modeled by reaction-diffusion equations.
In recent works, different neural network models were analyzed and the qualitative behavior of the solutions was given in the case of weak connectivity between neurons. Two questions are particularly important: what are the most relevant models and what is the qualitative behavior of the solutions beyond the weak connectivity regime. S. Mischler, C. Quininao and J. Touboul are working on both of them
New results on duality lemmas have enabled to improve the theory of reversible chemistry reaction-diffusion systems. We now try, in a collaboration between M. Breden, L. Desvillettes and K. Fellner, to exploit those new results in oreder to get new results on the coagulation-fragmentation-diffusion equations.
An issue which recently received a lot of attention is the extension to the case of reaction diffusion equations of results known in the theory of ordinary differential equations for networks of chemistry reactions. Results in this direction are currentky investigated in the framework of a collaboration between L. Desvillettes, K. Fellner and B. Tang..

Here are some of the publications on the results obtained in the framework of the ANR Kibord which already appeared. The complete list can be found on the web site of the ANR Kibord.

L. Desvillettes, Th. Lepoutre, A. Moussa; Entropy, duality and Cross Diffusion, SIAM Journal of Mathematical Analysis 46 (1) 820-853.

Pierre Degond, Amic Frouvelle, Gaël Raoul, Local stability of perfect alignment for a spatially homogeneous kinetic model, J. Stat. Phys., 2014, 157 (1), pp.84-112.

B. Perthame, M. Tang, N. Vauchelet, Traveling wave solution of the Hele-Shaw model of tumor growth with nutrient, Math. Meth. Mod. in Appl. Sc., Vol 24 No 13 (2014), 489-508.

L. Corrias, M. Escobedo, J. Matos, Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane, J. Differential Equations 257 (2014), 1840-1878.

P. Gabriel, F. Salvarani, Exponential relaxation to self-similarity for the superquadratic fragmentation equation, Appl. Math. Lett. 27 (2014), 74-78

L. Almeida, C. Emako, N. Vauchelet, Existence and diffusive limit of a two-species kinetic model of chemotaxis, Kin. Rel. Models. Vol 8 no 2 (2015), 359-380.

L. Desvillettes, A. Trescases, New results for triangular reaction cross diffusion system, Journal of Mathematical Analysis and Applications 430 (2015), 32-59.

A. Lorz, and B. Perthame, Long-term behaviour of phenotypically structured models, Proceedings of the Royal Society A, Vol. 470, no. 2167, 2014

Pierre Gabriel, Global stability for the prion equation with general incidence, Mathematical Bioscience and Engineering, Vol 12, no 4, 2015, 789-801.

F. James, N. Vauchelet, Numerical methods for one-dimensional aggregation equations, SIAM J. Numer. Anal., Vol 53 no 2 (2015), 895-916.

M. Breden, L. Desvillettes, J.-P. Lessard, Rigorous numerics for nonlinear operators with tridiagonal dominant linear parts, Discrete and Continuous Dynamical Systems - Series A (DCDS-A) Vol 35 no 10 (2015).

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.