TALKS
Panagiota Birmpa (Heriot-Watt University) - Non-equilibrium fluctuations for the stirring process with births and deaths.
We consider the one-dimensional stirring process on the segment {−N , . . . , N }, coupled to boundary dynamics that inject particles from the right reservoir and remove particles from the left reservoir, each acting on a window of fixed and finite size. In this talk, I will present the non-equilibrium fluctuations of the system when the initial configuration is given by a product measure associated with a smooth macroscopic profile. In this regime, the fluctuations are described by an Ornstein–Uhlenbeck process driven by the Laplacian and gradient operators, with boundary conditions determined by the hydrodynamic profile. A central step in the analysis is the derivation of sharp bounds for space and space–time v-functions of arbitrary degree associated with the centered occupation variables. In particular, we prove that the v-functions of degree 2 and 3 are of order N^{−1}, while those of degree at least 4 are of order N^{ −1−\zeta} for some \zeta> 0.
Alain Blaustein (INRIA Lille) - Long time stability for large systems of interacting agents
We analyze a stochastic model for systems constituted of a large number of interacting particles. This model has concentrated a lot of attention since it encodes phenomena encountered across a wide range of applications (aggregation, synchronization, fluid-like behavior). We establish its long-time stability for a general class of singular interactions and as a consequence, we will derive quantitative estimates for propagation of chaos in strong topologies.
Gioia Carinci (Università di Modena e R. Emilia) - From Discrete Bidding to Continuous Flows: A Multi-Agent Auction Model
We present an auction model in which multiple autonomous bidders attempt to sell their goods with the goal of maximizing their profits. Bidders operate without knowledge of one another and adjust their bids solely based on their most recent performance behavior (myopic adaptation). More precisely, a bidder who is awarded in a given round will increase their selling price in the next round, while one who is not awarded will decrease it. In each round, the auctioneer purchases the lowest $p$-fraction of the total energy offered by the bidders.
We find a system of differential equations governing the macroscopic dynamics and derive it as a scaling limit of the microscopic model. We find an explicit solution for the max-price evolution $q_t$ and show that in the long run bidders coordinate, i.e. they tend to bid the same value only depending on their initial distribution and the value $p$.
For Poisson-distributed initial bids, we obtain hydrodynamic limits and a central limit theorem for $q_t$. Finally we prove that when bidders have heterogeneous update speeds, the max price velocity becomes proportional to the harmonic mean of the velocities of the bidders at the max price.
Joint work with P. Ferrari, C. Franceschini, N. Manelli.
Martina Conte (Politecnico di Torino) - Kinetic and hydrodynamic descriptions of tumor-immune system competition
The dynamic competition between growing tumors and immune cells represents a critical biological process characterized by complex cellular interactions and immune evasion mechanisms. In this talk, we present a multi-scale mathematical framework that bridges kinetic mesoscopic formulations with their corresponding hydrodynamic macroscopic descriptions to capture these non-linear dynamics.
Starting from the description of cell-cell encounters via Boltzmann-like integro-differential equations, where interacting populations are structured by an internal state variable representing their competitive fitness, we extend the framework to spatially distributed scenarios that incorporate cell motility via velocity jump processes. Through appropriate hydrodynamic scaling limits, we formally derive macroscopic systems characterized by different types of linear and non-linear diffusion operators. We investigate the impact of these distinct diffusive regimes through stability analysis and numerical simulations. Our results highlight the strict connection between microscopic interaction rules and emergent macroscopic behavior, underscoring the key role of spatial structures in shaping immune surveillance.
Théophile Dolmaire (Università degli Studi dell'Aquila) - Inelastic collapse and global well-posedness in dissipative particle systems
When studying systems of particles, the very first step before any qualitative analysis relying on kinetic equations is to establish the well-posedness of the dynamics of the system. In the case of inelastic hard spheres, the dissipative collisions lead to emergence of clusters, and to the occurrence of infinitely many collisions in finite time, a phenomenon known as the inelastic collapse. This phenomenon remains poorly understood, yet it represents a major obstruction to a rigorous derivation of the inelastic Boltzmann equation. We will present recent results on collapsing systems, including the identification of new families of singularities. Besides, we will consider a particular class of inelastic hard sphere systems, in which a fixed amount of kinetic energy is lost in each sufficiently energetic collision. For this class, we establish the global well-posedness of the particle dynamics.
The results were obtained in collaboration with Roberto Castorrini (Università degli Studi della Tuscia), Eleni Hübner-Rosenau (Universität Regensburg) and Juan J. L. Velázquez (Universität Bonn).
Amit Einav (Durham University) - Paths of order in a jungle of chaos
Systems that revolve around the interactions of many elements are a constant part of our day to day lives. Yet for all their prevalence, trying to explore mathematical models of such systems, theoretically or numerically, can often be a herculean task. To try and address this difficulty, it was realised early on (back in the late 19th century) that we do not need to understand how each and every element in the system behaves. Instead, it is often enough to understand how a typical or average element does.
A revolutionary idea that birthed a new way to investigate systems of many elements was formalised in the work of Mark Kac in 1956. Kac suggested to find how an average particle in dilute gas behaves by considering a “probabilistic model” of the gas, expressed via a PDE for the probability to find the system in various configurations, together with the idea that as the number of particles in the gas increases, they become more and more independent. The latter is often known as molecular chaos, or chaos. Combining these two ingredients, Kac was able to find an equation that describes how a “limiting average particle” evolves in his settings – which ended up being a one-dimensional variant of the celebrated Boltzmann equation.
These ideas are far more general and powerful than their application in Kac’s model, and they have formed the framework of what we now call the mean field limit approach. At its heart, the mean field limit approach has two ingredients:
An average model for the system, expressed via a PDE for its probability measure.
An asymptotic correlation relation that expresses the emerging phenomena we expect to get as the number of elements goes to infinity.
In recent decades the use of the mean field limit approach has expanded beyond physical models into the realm of biology, economy, and societal studies. Yet in almost all cases considered so far, the sole correlation relation used was chaos. This seems to be inappropriate in settings that have a tendency for adherence such as biological swarming.
In our talk we will briefly discuss the background to the mean field limit approach as well as Kac’s model and its limiting equation. We will then jump to 2013 and consider the Choose the Leader model, or CL model, which is a Kac-like animal swarming model introduced by Carlen, Degond, and Wennberg where chaoticity breaks. Motivated by the desire to understand this model better, we will introduce two new asymptotic correlation relations, order and partial order, and see how they arise naturally in the CL model.
Nicolas Fournier (Sorbonne University) - Stochastic particle systems for the Keller-Segel equation
The Keller-Segel equation describes the movement of cells via chemotaxis. The cells diffuse in the plane and release a chemical. This chemical, which also diffuses, attracts the cells. This leads to a singular interaction between the cells (via the chemical). This interaction is critical: depending on the values of the constants, or a cluster of cells may or not emerge in finite time.
We will discuss finite particle systems approximating this equation, both in the elliptic case where the chemical diffuses instantaneously and in the parabolic case where the product diffuses at a finite rate.
Based on joint works with B. Jourdain, Y. Tardy and M. Tomasevic
Sepideh Mirrahimi (CNRS and University of Toulouse) - Link between stochastic individual-based evolutionary models and Hamilton-Jacobi equations
An approach involving Hamilton-Jacobi equations has been developed during the last two decades to study models from evolutionary biology. Such Hamilton-Jacobi equations are derived, in the regime of small mutational variance, from integro-differential models, which are themselves derived from stochastic individual based models in the limit of large populations. These limiting procedures do not always commute, leading possibly to artefacts in the population dynamics. In this work, we derive such a Hamilton-Jacobi equation, directly from a stochastic individual based model. This derivation leads to a new limit corresponding to a free boundary Hamilton-Jacobi equation, taking into account possible extinctions of sub-populations. This is a joint work with N. Champagnat, S. Méléard and C. Tran.
Michel Pain (CNRS and University of Toulouse) - Polynomial slowdown in an angle-dependent 2d branching Brownian motion
I will present a particle system on the plane where each particle moves according to a 2d Brownian motion and splits into two new particles at a rate which depends on its position, but only through its angular coordinate. Let $b(\theta)$ denote the branching rate of a particle at a position of polar coordinates $(r,\theta)$. In a joint work with Julien Berestycki and David Geldbach, we investigated the maximal displacement at a large time for this model. Depending on the behavior of the function $b$ near its maximal value, several phases appear and we focus on a case where some polynomial slowdown occurs. The proof relies both on probabilistic tool, as well as PDE techniques to estimate the transition kernel of a Brownian motion weighted by integrals of its path.
Gaël Raoul (Ecole Polytechnique) - Growing random planar network with oriented branching and fusion
The formation and spatial organization of biological networks play a central role in many contexts, such as organ vascularization or mycelial growth. While Partial Differential Equation models can accurately describe the advance of network fronts, much less is known about the structure of the network once it is fully developed. In particular, characterizing the statistical properties of mature networks remains a challenging problem. In this talk, we introduce and analyze a simple model for network formation. In this model, branches grow along straight lines at constant speed and are allowed to branch only on their right-hand side. We show that this system can be related to a branching process on rectangles, which provides a valuable basis to analyze the network formation process. Through this connection, we derive structural and statistical properties of the resulting networks. Although the model is deliberately simplified, we believe that the methods and insights developed here may be useful for studying more general network formation processes. This work is a collaboration with Vincent Bansaye and Miliça Tomasevic.
Josué Tchouanti (University of Toulouse) - Evolutionary dynamics of a fragmented population in a mean-field network of interconnected demes
In this talk, we study migration as a key driver of the evolution and phenotypic heterogeneity of spatially structured populations. We
consider a metapopulation framework where the evolutionary dynamics within each patch is modeled by a Moran process describing the evolution of a quantitative trait in a population of fixed size through resampling and mutation. Migration between patches is introduced to account for interactions between populations. The main question is how migration influences long-term evolution both within a single patch and at the scale of the whole metapopulation. To address this, we study several scaling limits of the model. Under the assumption of rare mutations and migrations, we adapt a technique from Champagnat & Lambert (2007) to
derive a mean-field network of Trait Substitution Sequences (TSS) describing successive dominant traits in each patch. As the number of patches grows, we obtain a propagation of chaos: patches become i.i.d., and their TSS converges to a McKean–Vlasov pure jump stochastic process. In the limit of small mutational effects with slower migration, the TSS converges, on a new migration timescale, to a stochastic differential equation representing a new canonical equation of adaptive dynamics with selection, genetic drift, and migration-driven jumps.
Milica Tomasevic (CNRS and Ecole polytechnique) - On the Go or Grow particles
In this talk we study a branching particle system of diffusion processes on the real line interacting through their rank in the system. Namely, each particle follows an independent Brownian motion, but only $K\geq 1$ particles on the far right are allowed to branch with constant rate, whilst the remaining particles have an additional positive drift of intensity $\chi>0$. This is the so called Go or Grow hypothesis, which serves as an elementary hypothesis to model cells in a capillary tube moving upwards a chemical gradient. Despite the discontinuous character of the coefficients for the movement of particles and their demographic events, we first obtain the limit behavior of the population as $K\to \infty$ by weighting the individuals by $1/K$. Then, on the microscopic level when $K$ is fixed, we investigate numerically the speed of propagation of the particles and recover a threshold behavior according to the parameter χ consistent with the already known behavior of the limit. Finally, by studying numerically the ancestral lineages we will see how one could categorize the traveling fronts as pushed or pulled according to the critical parameter $\chi$.
This is a joint work with M. Demircigil (Univ. of Arizona).
Oliver Tough (University of Durham) - Stationary solutions of the FKPP equation are unique in dimension up to 6, but not necessarily in dimension at least 7.
We consider stationary solutions of the FKPP equation, i.e. solutions of Delta u+f(x,u)=0, with Dirichlet boundary conditions. We show that under a general KPP-type condition on f, stationary solutions are unique in dimensions d<=6 and exist if and only if the generalised principal eigenvalue of Delta+f_s(x,0) is strictly positive, but counterexamples for both exist in dimension d>=7. We generalise this to a general condition on f including for instance f:=c(x)(u-u^{1+\gamma}), where we show d<=2+4/\gamma is both a sufficient and sharp condition. Our conditions on the boundary of Omega are very general.
We apply this to branching Brownian motion as follows. A branching process survives globally if it never dies out, and survives locally if it returns to an arbitrarily chosen ball at arbitrarily large times. We show that branching Brownian motion cannot survive globally without surviving locally (i.e. ``drifting off to infinity'') in dimensions d<=6, for very general domains with killing on the boundary, but we exhibit a simple counterexample on the whole space in dimensions d>=7.
Oscar de Wit (Université Claude Bernard Lyon 1) - Ants and mean-field interactions: singular interactions and multiple invariant measures
In this talk we discuss mean-field particle systems on continuous periodic domains by the example of a simple model for ants. The model shares features with the Vicsek model and the active Ising model. We cover well-posedness, propagation of chaos and the mean-field limit for non-singular and singular interactions. We also cover the existence of multiple invariant measures via the Lyapunov-Schmidt method and a super- to subcritical phase transition.
The slides of the talks will be uploaded here