Samuel Tréton

Published articles

A piston to counteract diffusion: The influence of an inward-shifting boundary on the heat equation in half-space (2025)
(with M.Zhang) (SIAM Journal on Applied Mathematics) (DOI) (postprint link)
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{\'e}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)
(with R.Ducasse) (Mathematical Modelling of Natural Phenomena) (DOI) (postprint link)

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)
(with M.Alfaro and M.Mourragui) (Stochastics and Partial Differential Equations: Analysis and Computations) (DOI) (postprint link)
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) (postprint link)

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)
(with M.Alfaro and R.Ducasse) (Journal of Differential Equations) (DOI) (postprint link)

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}
}

Thesis manuscript

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

This thesis deals with the derivation and the analysis of spatially structured population models, including both stochastic and deterministic approaches.

The main goal of this work is to deepen our understanding of the intricate connections between individual-based dynamics and the collective behavior of populations, as well as the long-term behavior of the latter.

By focusing on models that illustrate exchanges between heterogeneous environments, we particularly investigate the relationships between certain interacting particle systems (simple exclusion processes) and reaction-diffusion equations.

Special attention is also given to the long-term behavior of the solutions of these equations, especially the criteria for population persistence or extinction.

Chapter 1 lays the theoretical background for reaction-diffusion equations and simple exclusion processes. This section provides the necessary foundation for the following chapters.

Chapter 2 explores the microscopic derivation, via a simple exclusion process, of a reaction-diffusion system known as the "field-road model", which is used to model the impact of fast diffusion channels in ecology and epidemiology.

In Chapter 3, we explicitly derive the solutions of the original field-road diffusion model and provide a uniform long-term control. Such control is useful to quantify the "dispersion intensity" of the diffusive process, enabling to demonstrate results related to persistence and extinction when a growth function with an Allee effect is introduced.

Finally, Chapter 4 examines Fujita-type results concerning blow-up versus possible global existence of solutions to a superlinear reaction-diffusion "heat exchanger" system. This study characterizes the stability of the zero equilibrium when a monostable reaction degenerate at $0$ (penalizing low-density growth) is included. This aspect is crucial for understanding the persistence and extinction phenomena mentioned above.