Samuel Tréton

Articles publiés

A piston to counteract diffusion: The influence of an inward-shifting boundary on the heat equation in half-space (2025)
(avec M.Zhang) (SIAM Journal on Applied Mathematics) (DOI) (lien vers le postprint)
To better understand how populations respond to dynamic external pressure, we propose a new diffusion model in the moving half-line \( \{ z \geq b(t) \} \), where the boundary position \( b(t) \) is a given nondecreasing function of time. A Robin boundary condition is imposed at \( z = b(t) \) to prevent individuals from leaving the domain, so that the shifting boundary acts as an impermeable wall—a "piston"—that sweeps the individuals it encounters. Our analysis focuses on the cases where \( b(t) \sim ct^\beta \) with \( \beta \in [0, 1] \). We prove quantitative convergence results characterized by attraction toward self-similar profiles, based on entropy techniques and Duhamel's principle. When \( \beta \) goes through the critical value \( 1/2 \), the shape of the self-similar asymptotic profile switches from Gaussian to exponential. In particular, this profile turns out to be stationary when \( \beta = 1 \), reflecting a delicate balance between diffusion and advection induced by the moving boundary.
```
@article{TretonPiston25,
title = {A Piston to Counteract Diffusion: The Influence of an Inward-Shifting Boundary on the Heat Equation in Half-Space},
shorttitle = {A Piston to Counteract Diffusion},
author = {Tr{\'é}ton, Samuel and Zhang, Mingmin},
year = {2025},
journal = {SIAM Journal on Applied Mathematics},
doi = {10.1137/25M1757435},
abstract = {To better understand how populations respond to dynamic external pressure, we propose a new diffusion model
in the moving half-line { z ≥ b(t) }, where the boundary position b(t) is a given nondecreasing function
of time. A Robin boundary condition is imposed at z = b(t) to prevent individuals from leaving the domain, so
that the shifting boundary acts as an impermeable wall—a "piston"—that sweeps the individuals it encounters.
Our analysis focuses on the cases where b(t) ∼ ct^β with β ∈ [0, 1]. We prove quantitative
convergence results characterized by attraction toward self-similar profiles, based on entropy techniques
and Duhamel's principle. When β goes through the critical value 1/2, the shape of the
self-similar asymptotic profile switches from Gaussian to exponential. In particular, this profile turns out to be
stationary when β = 1, reflecting a delicate balance between diffusion and advection induced by the moving boundary.}
}
}
```

Emergence of complexity in opinion propagation: A reaction-diffusion model (2025)
(avec R.Ducasse) (Mathematical Modelling of Natural Phenomena) (DOI) (lien vers le postprint)

We analyze a model designed to describe the spread and accumulation of opinions in a population. Inspired by the social contagion paradigm, our model is built on the classical SIR model of Kermack and McKendrick and consists in a system of reaction-diffusion equations. In the scenario we consider, individuals within the population can adopt new opinions via interactions with others, following some simple rules. The individuals can gradually adopt more complex opinions over time.

Our main result is the characterization of a maximal complexity of opinions that can persist and propagate. In addition, we show how the parameters of the model influence this maximal complexity. Notably, we show that it grows almost exponentially with the size of the population, suggesting that large communities can foster the emergence of more complex opinions.

@article{DucasseEmergence25,
  title = {Emergence of Complexity in Opinion Propagation: {{A}} Reaction-Diffusion Model},
  shorttitle = {Emergence of Complexity in Opinion Propagation},
  author = {Ducasse, Romain and Tr{\'e}ton, Samuel},
  year = {2025},
  journal = {Mathematical Modelling of Natural Phenomena},
  volume = {20},
  pages = {25},
  issn = {0973-5348},
  doi = {10.1051/mmnp/2025025},
  abstract = {We analyze a model designed to describe the spread and accumulation of opinions in a population. Inspired by the social contagion paradigm, our model is built on the classical SIR model of Kermack and McKendrick and consists in a system of reaction-diffusion equations. In the scenario we consider, individuals within the population can adopt new opinions via interactions with others, following some simple rules. The individuals can gradually adopt more complex opinions over time. Our main result is the characterization of a maximal complexity of opinions that can persist and propagate. In addition, we show how the parameters of the model influence this maximal complexity. Notably, we show that it grows almost exponentially with the size of the population, suggesting that large communities can foster the emergence of more complex opinions.},
  keywords = {opinion dynamics, social contagion, reaction-diffusion, complexity, SIR model}
}

Bridging bulk and surface: An interacting particle system towards the field-road diffusion model (2025)
(avec M.Alfaro et M.Mourragui) (Stochastics and Partial Differential Equations: Analysis and Computations) (DOI) (lien vers le postprint)
We recover the so-called field-road diffusion model as the hydrodynamic limit of an interacting particle system. The former consists of two parabolic PDEs posed on two sets of different dimensions (a "field" and a "road" in a population dynamics context), and coupled through exchange terms between the field's boundary and the road. The latter stands as a Symmetric Simple Exclusion Process (SSEP): particles evolve on two microscopic lattices following a Markov jump process, with the constraint that each site cannot host more than one particle at the same time. The system is in contact with reservoirs that allow to create or remove particles at the boundary sites. The dynamics of these reservoirs are slowed down compared to the diffusive dynamics, to reach the reactions and the boundary conditions awaited at the macroscopic scale. This issue of bridging two spaces of different dimensions is, as far as we know, new in the hydrodynamic limit context, and raises perspectives towards future related works.
```
@article{AlfaroBridging25,
title = {Bridging Bulk and Surface: An Interacting Particle System towards the Field-Road Diffusion Model},
shorttitle = {Bridging Bulk and Surface},
author = {Alfaro, Matthieu and Mourragui, Mustapha and Tr{\'e}ton, Samuel},
year = {2025},
month = may,
journal = {Stochastics and Partial Differential Equations: Analysis and Computations},
issn = {2194-041X},
doi = {10.1007/s40072-025-00367-y},
abstract = {We recover the so-called field-road diffusion model as the hydrodynamic limit of an interacting particle system. The former consists of two parabolic PDEs posed on two sets of different dimensions (a ``field'' and a ``road'' in a population dynamics context), and coupled through exchange terms between the field's boundary and the road. The latter stands as a Symmetric Simple Exclusion Process (SSEP): particles evolve on two microscopic lattices following a Markov jump process, with the constraint that each site cannot host more than one particle at the same time. The system is in contact with reservoirs that allow to create or remove particles at the boundary sites. The dynamics of these reservoirs are slowed down compared to the diffusive dynamics, to reach the reactions and the boundary conditions awaited at the macroscopic scale. This issue of bridging two spaces of different dimensions is, as far as we know, new in the hydrodynamic limit context, and raises perspectives towards future related works.}
}
```

Blow-up vs. global existence for a Fujita-type Heat exchanger system (2024)
(SIAM Journal on Mathematical Analysis (SIMA)) (DOI) (lien vers le postprint)

We analyze a reaction-diffusion system on ℝ^N which models the dispersal of individuals between two exchanging environments for its diffusive component and incorporates a Fujita-type growth for its reactive component. The originality of this model lies in the coupling of the equations through diffusion, which, to the best of our knowledge, has not been studied in Fujita-type problems. We first consider the underlying diffusive problem, demonstrating that the solutions converge exponentially fast towards those of two uncoupled equations, featuring a dispersive operator that is somehow a combination of Laplacians. By subsequently adding Fujita-type reaction terms to recover the entire system, we identify the critical exponent that separates systematic blow-up from possible global existence.

@article{TretonBlowUp24,
  title = {Blow-{{Up}} vs. {{Global Existence}} for a {{Fujita-Type Heat Exchanger System}}},
  author = {Tr{\'e}ton, Samuel},
  year = {2024},
  journal = {SIAM Journal on Mathematical Analysis},
  pages = {2191--2212},
  publisher = {{Society for Industrial and Applied Mathematics}},
  issn = {0036-1410},
  doi = {10.1137/23M1587440},
  abstract = {We analyze a reaction-diffusion system on \R^N which models the dispersal of individuals between two exchanging environments for its diffusive component and incorporates a Fujita-type growth for its reactive component. The originality of this model lies in the coupling of the equations through diffusion, which, to the best of our knowledge, has not been studied in Fujita-type problems. We first consider the underlying diffusive problem, demonstrating that the solutions converge exponentially fast towards those of two uncoupled equations, featuring a dispersive operator that is somehow a combination of Laplacians. By subsequently adding Fujita-type reaction terms to recover the entire system, we identify the critical exponent that separates systematic blow-up from possible global existence.},
  keywords = {reaction-diffusion system, Fujita blow-up phenomena, critical exponent, global solutions, heat exchanger system}
}

The field-road diffusion model: fundamental solution and asymptotic behavior (2023)
(avec M.Alfaro et R.Ducasse) (Journal of Differential Equations) (DOI) (lien vers le postprint)

We consider the linear field-road system, a model for fast diffusion channels in population dynamics and ecology. This system takes the form of a system of PDEs set on domains of different dimensions, with exchange boundary conditions. Despite the intricate geometry of the problem, we provide an explicit expression for its fundamental solution and for the solution to the associated Cauchy problem. The main tool is a Fourier (on the road variable)/Laplace (on time) transform. In addition, we derive estimates for the decay rate of the L^∞ norm of these solutions.

@article{AlfaroFieldRoad23,
  title = {The Field-Road Diffusion Model: {{Fundamental}} Solution and Asymptotic Behavior},
  shorttitle = {The Field-Road Diffusion Model},
  author = {Alfaro, Matthieu and Ducasse, Romain and Tr{\'e}ton, Samuel},
  year = {2023},
  journal = {Journal of Differential Equations},
  volume = {367},
  pages = {332--365},
  issn = {0022-0396},
  doi = {10.1016/j.jde.2023.05.002},
  abstract = {We consider the linear field-road system, a model for fast diffusion channels in population dynamics and ecology. This system takes the form of a system of PDEs set on domains of different dimensions, with exchange boundary conditions. Despite the intricate geometry of the problem, we provide an explicit expression for its fundamental solution and for the solution to the associated Cauchy problem. The main tool is a Fourier (on the road variable)/Laplace (on time) transform. In addition, we derive estimates for the decay rate of the $L^{\infty}$ norm of these solutions.},
  keywords = {Decay rate,Diffusion, Exchange boundary conditions, Field-road model, Fundamental solution}
}

Manuscrit de thèse

Analyse de Dynamiques d’Échanges Microscopiques et Macroscopiques pour l’Écologie et l’Épidémiologie (2024)

Cette thèse porte sur la dérivation et l’analyse de modèles de populations structurées en espace, de nature stochastique et déterministe. L’objectif principal de ce travail est d’améliorer notre compréhension des liens complexes entre les dynamiques individu- centrées et le comportement global des populations, ainsi que l’évolution en temps long de ces dernières. En mettant l’accent sur certains modèles présentant des dynamiques d’échanges entre milieux hétérogènes, on explore les relations entre certains systèmes de particules en interaction (processus d’exclusion simple) et les équations de réaction- diffusion. Une attention particulière est également portée à l’analyse du comportement en temps long des solutions de ces dernières, notamment aux critères de persistance ou d’extinction des populations.

On commence par introduire dans le Chapitre 1 les principaux fondements théoriques des équations de réaction-diffusion et des processus d’exclusion simple. Cette partie établit les prérequis essentiels pour les chapitres qui suivent.

Le Chapitre 2 est consacré à la dérivation microscopique, à partir d’un processus d’exclusion simple, d’un système de réaction-diffusion connu sous le nom de “champ-route”, utilisé pour modéliser l’impact des lignes de diffusion rapide en écologie et épidémiologie.

Dans le Chapitre 3, on rend explicite les solutions du système champ-route diffusif original et on en fournit un contrôle uniforme en temps long. Ce type de contrôle s’avère utile pour quantifier “l’intensité de dispersion” du processus diffusif et permet notamment de montrer des résultats de persistance et d’extinction lorsqu’une fonction de croissance avec effet Allee est introduite.

Enfin, le Chapitre 4 concerne des résultats de type Fujita sur l’explosion en temps fini, par opposition à la possible existence globale des solutions, d’un système de réaction- diffusion sur-linéaire “échangeur de chaleur”. Cette étude permet de caractériser la stabilité de l’équilibre nul lorsqu’on ajoute une réaction monostable dégénérée en 0, pénalisant la croissance des faibles densités. Ce point représente la clé de voûte de la caractérisation des phénomènes de persistance et d’extinction mentionnés plus haut.