PDEs under pressure and applications in life science

12 oct. 2026, 02:00 → 16 oct. 2026, 23:59 Europe/Paris

Amphithéâtre L. Schwartz - RdC - Bâtiment 1R3 (Institut de Mathématiques de Toulouse)

Scope of the conference

The aim of the conference PDEs under pressure and applications in life science is to bring together experts and enthusiasts on PDEs modelling pressure effects in sciences - and especially in living systems. While pressure refers primarily to mechanical pressure, whose impact is crucial for example in cell and developmental biology, it has to be understood here in a wider sense, including ecological or evolutionary pressure, social or psychological pressure, etc. Topics include degenerate or nonlinear diffusion, cross-diffusion, fluid-mechanical PDEs, kinetic theory, nonlocal and aggregation models, and stochastic approaches. A particular focus will be given to the analysis of congestion models, which arise, for example, in granular media, crowds, cell and developmental biology, ecological populations, and swarm behavior. Talks will combine theoretical analysis, modeling, and interdisciplinary applications. We expect the participation of a vast audience, including early career researchers.

This conference is part of the CIMI thematic semester Interacting particles, PDEs and applications (IMT, spring-autumn 2026) including the MINT summer school and the conference Particle systems and PDEs XIV.

Content of the conference

The conference will be centered around three mini-courses and invited talks. A poster session will also be organized for contributed posters.

The courses will be given by:

• Maria Bruna (University of Oxford)
• Sophie Hecht (Sorbonne Université)

• Bertrand Maury (Université Paris-Sud)

 

The talks will be given by:

• Laurent Desvillettes (Université Paris Cité)

• Antoine Diez (RIKEN)

• Maxime Estavoyer (Aix-Marseille Université)

• Amic Frouvelle (Université Paris Dauphine)

• Thierry Goudon (INRIA Sophia Antipolis)

• Anastasiia Hraivoronska (Université Claude Bernard – Lyon 1 )

• Noureddine Igbida (Université de Limoges)

• Aline Lefebvre-Lepot (ENS Paris-Saclay)

• Nadia Loy (Politecnico di Torino)

• Marta Menci (Università Campus Bio-Medico di Roma)

• Kevin Painter (Politecnico di Torino)

• Steffen Plunder (Kyoto University)

• Cinzia Soresina (Università di Trento)

• Marie-Thérèse Wolfram (University of Warwick)

• Ewelina Zatorska (University of Warwick)

Registration

Registration is free but mandatory for organisational puprposes. Please fill the registration form by September 15th. A few students and early career researchers may ask for financial support (accommodation), they will need to fill in the registration form by July 3rd (notification by July 24th).

Schedule

The conference will start on Monday after lunch and will end on Friday around lunch time.

Conference dinner

We will meet on Thursday 15th of october at 18h45 for the conference dinner at : Aux Pieds Sous la Table, 4/8 Rue Arnaud Bernard, 31000 Toulouse.

Organisation

The organising committee is composed of:

• Marina Ferreira (CNRS and Université de Toulouse)
• Pierre Gervais (Université de Toulouse)

• Michèle Romanos (CNRS and Université de Lyon)

• Ariane Trescases (CNRS and Université de Toulouse)

Coordinators of the semester: Marina Ferreira, Francis Filbet, Pierre Gervais and Ariane Trescases

This conference is organised thanks to the support of the International Center of Mathematics and Computer Science in Toulouse (CIMI), Institut de Mathématiques de Toulouse, Institut Camille Jordan, the PEPR grant PREMMOVE and the RT Math Bio Santé.

lun. 12 octobre

14:00 Welcoming

1 Micro- and macroscopic modelling of crowding and pushing in corridors

Experiments with pedestrians show that both the geometry of a corridor and individuals' motivation to reach their target quickly have a strong effect on the overall dynamic in a crowd. I will present micro- and macroscopic models capturing these two effects, discuss their analysis and how they can be calibrated against experimental data. The resulting simulations reproduce key qualitative features observed experimentally, and I will discuss the dynamics and analytical properties of the underlying equations.

Orateur: Marie-Therese Wolfram (University of Warwick)

2 Hypo(coerciv-elliptic)ity in a model of aligning self-propelled particles

We consider a toy model of self-propelled particles (velocities constrained on the sphere) with alignment interaction (each individual tends to align with its close neighbors) and angular noise (to explore different directions).

At the mesoscopic scale (in the limit of a large number of particles), it takes the form of a kinetic equation coupling free transport (at constant velocity) and a local alignment-diffusion operator on the sphere (the velocity variable). The homogeneous version is well understood with a phase transition phenomenon: below a certain noise (or density) threshold, the velocity distribution converges towards isotropic equilibrium, while above this threshold, stable equilibria form a family of profiles that are more concentrated in velocity (von Mises distributions).

I will begin with an overview of the macroscopic behaviors that we expect to observe (or that we do not really understand) in the spatially inhomogeneous model. For example, we can formally derive fluid-type equations for densities above the critical threshold.

I will then present some results on the behavior (in short and long time) of the spatially inhomogeneous equation below the threshold, using hypoellipticity and hypocoercivity techniques, which require specific adjustments due to the fact that the velocity variable is constrained on the unit sphere. We obtain local (nonlinear) and quantitative stability of the isotropic equilibrium, with an exponential convergence rate when the space variable is on a flat torus. These results come from a long-standing collaboration with Émeric Bouin (Ceremade, Université Paris Dauphine - PSL).

Orateur: Amic Frouvelle (Univ. Paris Dauphine & Vienne)

16:00 Coffee Break

3 TBA 1/2 SH

Orateur: Sophie Hecht (Sorbonne Université)

18:00 Reception & Poster session

mar. 13 octobre

4 TBA 2/2 SH

Orateur: Sophie Hecht (Sorbonne Université)

10:30 Coffee break

5 Symmetry breaking in bacterial colonies in a kinetic reaction–transport model

We study the emergence of symmetry breaking in expanding bacterial colonies in the presence of a localized prey. While colony expansion is typically radially symmetric, environmental interactions can induce anisotropic propagation patterns. We introduce a kinetic reaction–transport model for motile and proliferating bacteria, combining persistent motion, stochastic reorientation, and division at the microscopic scale. From this description, we derive macroscopic equations for the bacterial density, leading to a nonlinear system governing colony dynamics. We analyze front propagation through travelling wave solutions and associated spreading speeds. We show that the presence of a prey alters the effective transport and proliferation terms, resulting in changes in front speed and triggering a breakdown of radial symmetry. Our approach combines analytical results, numerical simulations, and experimental data at both the colony and single-cell scales.

This work is carried out in collaboration with Vincent Calvez.

Orateur: Maxime Estavoyer (Aix-Marseille Université)

6 Starvation-driven cell patterning: integrating lab experiments and mathematical modelling

We present a reaction–diffusion model describing the interactions among cells, nutrients, and growth factors, aimed at capturing the emergence of starvation-driven cell pattern formation, a phenomenon recently observed in laboratory experiments under nutrient-limited growth conditions. Experiments and modelling were developed in parallel, enabling progressively more targeted experimental design while enhancing the biological realism of the model. This interdisciplinary feedback loop led to the formulation of new hypotheses and enabled the estimation of several key parameters. Numerical simulations show that the model reproduces pattern formation in both one- and two-dimensional spatial domains. To provide theoretical support for these findings, we performed a Turing instability analysis to investigate the potential for diffusion-driven instability. The analysis indicates that the observed patterns are not driven by chemotaxis; rather, they arise naturally under starvation conditions and display structural similarities to the Klausmeier model for vegetation pattern formation in semi-arid environments, suggesting the robustness of the underlying mechanism across biological scales.

Orateur: Cinzia Soresina

12:30 Lunch

7 Aggregation solutions to nonlocal kinetic equations

Highly concentrated patterns have been observed in a spatially heterogeneous, nonlocal, kinetic model with BGK type operators implementing a velocity jump process for cell migration, directed by the nonlocal sensing of either an external signal or the cell population density itself. We describe, in an asymptotic regime, the precise profile of these concentrations which, at the macroscale, are Dirac masses. Because Dirac concentrations look like Gaussian potentials, we use the Hopf–Cole transform to calculate the potential adapted to the problem. This potential, as in other similar situations, is obtained
through the viscosity solutions of a Hamilton–Jacobi equation. We begin with the linear case, when the heterogeneous external signal is given, and we show that the concentration profile obtained after the diffusion approximation is not correct and is a simple eikonal approximation of the true Hamilton-Jacobi equation. Its heterogeneous nature leads us to develop a new analysis of the implicit equation defining the Hamiltonian and a new condition to circumvent the 'dimensionality problem'. In the nonlinear case, when the signal occurs from the cell density itself, it is shown that the already observed linear instability (pattern formation) occurs when the Hamiltonian is convex-concave, a striking new feature of our approach.

Orateur: Nadia Loy (Politecnico di Torino)

8 Analytical and numerical advances on pressureless Euler systems with non-local alignment and chemotaxis

In this talk I will present recent analytical and numerical results on pressureless Euler systems which combine non-local interactions and chemotaxis. The system under investigation arises as the macroscopic counterpart of a class of hybrid discrete-continuum models for collective cell migration, in which cells interact mechanically through alignment and attraction-repulsion effects, and respond to the gradient of a chemical signal they themselves produce. The absence of a pressure term, replaced by non-local integral terms retaining information about the microscopic dynamics, represents the main issue relating to loss of regularity in finite time.
I will present analytical results concerning local well-posedness of the solution together with a continuation criterion showing that breakdown, if it occurs, can only be caused by the blow-up of the velocity gradient. A comparison with the chemotaxis-free case, already investigated in the literature of the field, shows that the subcritical or supercritical character of the dynamics is no longer determined by the initial configuration alone, but also by the cumulative, memory-type contribution of the chemotactic coupling.
Results are tested numerically in one space dimension, through a study of the transition between regular and singular regimes, of the robustness of the detected breakdown time with respect to discretization parameters, and of the role of the chemotactic strength in the onset of the singularity. In a two-dimensional setting, numerical simulations highlight the ability of the model to reproduce network-like patterns and immune-cancer interactions observed in cancer-on-chip experiments.

Orateur: Marta Menci (Università Campus Bio-Medico di Roma)

16:00 Coffee break

9 Strong solutions to the triangular SKT cross diffusion system

We present a work in collaboration with Hector Bouton and Helge Dietert, in which we prove that the solutions to the triangular SKT (Shigesada-Kawasaki-Teramoto) cross diffusion system (a system appearing in the dynamics of populations in competition) are strong in dimension 3 (it was proven in dimension 2 by Lou, Ni and Wu in the 90s). The proof uses a new approach to the Krylov-Safonov theory, in the special case when the solution is increasing with time.

Orateur: Laurent Desvillettes

mer. 14 octobre

10 Microscopic and macroscopic crowd motion models of the granular type : is it possible to incorporate some sort of "politeness" in such mechanical models ? 1/2

Orateur: Bertrand Maury

10:30 Coffee break

11 The fully discrete JKO scheme for constrained gradient flows

We introduce a fully discrete version of the Jordan–Kinderlehrer–Otto (JKO) scheme for gradient flows subject to hard constraints. We discuss the construction of the scheme and establish its convergence toward the corresponding evolution equations. The fully discrete formulation naturally leads to computational methods based on discrete optimal transport, for which we use the network simplex algorithm. We illustrate the approach through numerical simulations and applications to crowd motion, tumor growth, and specific cross-diffusion systems.

Orateur: Anastasia Hraivoronska

12 TBA

TBA

Orateur: Noureddine Igbida

12:30 Lunch

13 Modelling Interacting Particle Systems 1/2 MB

In this mini-course, I will introduce some of the main ideas and techniques for modelling systems of many interacting particles. Starting from stochastic particle-based descriptions, I will discuss different scalings of the interactions, from mean-field regimes to strongly interacting systems, and show how they can be used to capture different effects at the macroscopic level. I will then discuss methods for deriving continuum PDE models and how microscopic interactions are reflected in the resulting equations and collective behaviour. In the second part, I will illustrate these ideas in the context of active matter and self-propelled particles, with examples including bacteria and ants.

Orateur: Maria Bruna (University of Oxford)

16:00 Coffee break

jeu. 15 octobre

14 Microscopic and macroscopic crowd motion models of the granular type : is it possible to incorporate some sort of "politeness" in such mechanical models ? 2/2

Orateur: Bertrand Maury

10:30 Coffee break

15 Interactive web simulations of cell mobility with contacts

During my PhD with Sara Merino-Aceituno, I fell in love with the Gauss-Seidel principle, which has been speeding up computations for over 200 years. In my case, using early terminated Gauss-Seidel iterations turned out useful for approximating contact forces between hard spheres in overdamped models. Using properties of prox-regular sets, we also proved convergence of the corresponding numerical method.

In this talk, I want to present the basic numerical analysis and share my experiences including both strengths and weaknesses of the method. With view on interdisciplinary applications, I will present examples of interactive web simulations, comment on implementation tips for GPUs and share thoughts on using (open-weight) LLM coding assistants for modelling.

Orateur: Steffen Plunder

16 Strong attraction-repulsion interactions and shape selection in flocking models

We study the collective dynamics of a population subject to self-consistent attraction-repulsion interactions and an external velocity field. We start form a Vlasov-like description of the population in the regime of strong interaction forces. We show that the population asymptotically concentrates within a domain $\Omega(t)=\Omega_0+X(t)$ whose shape $\Omega_0$ is determined by the minimization of the interaction energy while the evolution of the domain’s center of mass $X(t)$ is determined by the external force field.
In addition, we show that inside the domain the motion of the organisms is governed by the lake equation, a variant of the incompressible Euler system. The analysis uses variational approaches and modulated energy methods.

Orateur: Thierry Goudon (INRIA)

12:30 Lunch

17 Modelling Interacting Particle Systems 2/2 MB

In this mini-course, I will introduce some of the main ideas and techniques for modelling systems of many interacting particles. Starting from stochastic particle-based descriptions, I will discuss different scalings of the interactions, from mean-field regimes to strongly interacting systems, and show how they can be used to capture different effects at the macroscopic level. I will then discuss methods for deriving continuum PDE models and how microscopic interactions are reflected in the resulting equations and collective behaviour. In the second part, I will illustrate these ideas in the context of active matter and self-propelled particles, with examples including bacteria and ants.

Orateur: Maria Bruna (University of Oxford)

16:00 Coffee break

18:45 Conference dinner

We will meet at 18h45 at : Aux Pieds Sous la Table, 4/8 Rue Arnaud Bernard, 31000 Toulouse.

ven. 16 octobre

18 Close interactions in immersed granular media

This presentation focuses on the simulation of dense granular media composed of macroscopic particles suspended in a viscous fluid, such as mud. These systems exhibit extremely complex macroscopic behavior. At the heart of these behaviors lie the microscopic physical interactions between neighboring particles, in particular lubrication effects induced by the interstitial fluid, as well as solid contacts. These interactions are essential not only for understanding immersed granular flows, but also in the broader context of particle suspensions. Despite their importance, the influence of these microscopic mechanisms on the overall dynamics of the system remains poorly understood.

A major difficulty stems from the singular nature of these effects as the distance between particles goes to zero, giving rise to time singularities. In this talk, we will present new contact-dynamics-like models that account for both lubrication and solid contact, with or without friction. We will show that these models can be used to develop stable and robust numerical schemes. At each time step, the incremental problem to be solved is a constrained optimization problem. The presentation will be illustrated with numerical results. We will conclude with a brief discussion of how this type of microscopic model can inspire the construction of one-dimensional macroscopic models, in which the non-overlapping constraint at the particle scale naturally translates into a maximum-density constraint at the continuous level.

Orateur: Aline Lefebre lepot

19 TBA

TBA

Orateur: Antoine Diez (RIKEN)

10:30 Coffee break

20 Stability and weak strong uniqueness of weak solutions to two-component hydrodynamic systems

Orateur: Ewelina Zatorska (Univ. Warwick)

21 Phenotype-structured PDE models for chemotaxis self-organisation.

Chemotaxis is a powerful mechanism for driving the self-organisation of populations that range from cells to organisms. Recently there has been significant interest in the impact of phenotype structuring on populations, where in the context of chemotaxis this structuring could introduce variability in traits that range from their sensitivity to gradients, proliferation, or secretion of autoattractants. In this talk I will describe a nonlocal PDE framework that incorporates a continuous phenotype structuring into a chemotactic population, and show how this structuring can impact on two classic phenomena of chemotactic system: waves and aggregation behaviour. Work with Tommaso Lorenzi, Fiona Macfarlane, and Chiara Villa.

Orateur: Kevin Painter (Politecnico di Torino)

12:30 Closing