Around the Field-Road Diffusion Model

A significant part of my thesis focused on the field-road reaction-diffusion model in the half-space \( \mathbb{R}^{N-1} \times \mathbb{R}^{\ast}_{+} \) (\( N \geq 2 \)), introduced in 2012 by H.Berestycki, J.-M.Roquejoffre, and L.Rossi in this paper. The main objective of this model is to capture new propagative phenomena in the presence of a line with fast diffusion. Our interest lies in studying the solution \( (u,v) = ( u(t,x) , v(t,x,y) ) \) of the following system of partial differential equations:

starting from an initial data \( (u_{0},v_{0}) = ( u_{0}(t,x) , v_{0}(t,x,y) ) \) bounded and integrable (a step, for instance).

Microscopic and Heuristic Simulation of the Field-Road Model.
The term "field" denotes the domain \( \mathbb{R}^{N-1} \times \mathbb{R}^{\ast}_{+} \), while "road" refers to the boundary of the field, that is the hyperplane \( \mathbb{R}^{N-1} \).

In the context of population dynamics, the function \( v \) can represent the density of individuals in the field, and \( u \) that on the road.

The diffusion of individuals is characterized by a coefficient \( d \) in the field and a coefficient \( D \) on the road, with typically, \( D \! \gg \! d \).

The exchange condition, a central element of the model, is defined in the second line of the system. This Robin-type boundary condition with source plays a key role in regulating the behavior of individuals at the edge of the field, whether they bounce back or are extracted towards the road, but also manages immigration from the road.

By taking the reaction function \( f \equiv 0 \), it can be shown that the mass of individuals is preserved over time; thus, we obtain a purely diffusive system.

In this paper co-authored with Matthieu Alfaro and Romain Ducasse, we write explicitely the solution of the diffusive field-road system and provide a uniform control in space over long times.

This asymptotic control allows us to estimate the "dispersive speed" of the model's diffusive mechanism, paving the way for questions of finite-time blow-up, persistence, or extinction when the system is equipped with super-linear reactions \( ( f(v) = v^{1+p}) \) or monostable-degenerate type reactions \( ( f(v) = v^{1+p}(1-v)) \) — the latter aimed at modeling growth with Allee effect.

Blow-up vs. Global Existence for a Fujita-Type Heat Exchanger System

During my thesis, I also focused on analyzing Fujita-type blow-up phenomena in reaction-diffusion systems, particularly in a context of coupling unknowns through the diffusion mechanism.

To put the problem in context, in 1966, the mathematician H.Fujita considered the semi-linear equation \( \partial_t u = \Delta u + u^{1+p} \) in \( \mathbb{R}^N \), where he highlighted the existence of a critical exponent \( p_F = 2/\! N \) separating systematic blow-up (\( p < p_F \)) from possible global existence (\( p > p_F \)) of positive solutions. This threshold, identified as \( 1/\! p = N/\! 2 \), corresponds to the balance ratio between the uniform algebraic decay rate of the Heat Equation \[ \partial_t \mathbf{u} = \Delta \mathbf{u} \] in \( \Vert \mathbf{u} (t,\cdot) \Vert_{L^\infty(\mathbb{R}^N)} \sim C/t^{N/\! 2} \), and the algebraic blow-up rate of the underlying ODE \[ \frac{d}{dt}U = U^{1+p} \] in \( U(t) \sim C/(T_{boom}-t)^{1/\! p} \). My work, summarized in this article, focused on the following Fujita-type system:

where the constant \( \kappa \) is either \( 0 \) or \( 1 \), acting as an on/off switch for the second non-linearity. My first contribution was to analyze the associated linear problem, demonstrating that the solutions converge exponentially fast towards those of a certain decoupled parabolic system. Subsequently, I determined the critical exponents \( p \) and \( q \) that separate systematic explosion from possible global existence.

Modeling the Field-Road System from a Stochastic Particle System

The work described here is part of a collaboration with Matthieu Alfaro and Mustapha Mourragui. Our goal is to reach the diffusive field-road model from a stochastic system of interacting particles.

For a fixed \( N \in \mathbb{N} \) (here, \( N \) represents the system size and not the dimension), we consider the lattices \( \Lambda_N := \mathbb{Z}/ N \mathbb{Z} \times [\![ 1 ; N-1 ]\!] \) (field) and \( \Gamma_N := \mathbb{Z}/ N \mathbb{Z} \) (road).

Each point of these lattices is called a site and is occupied by at most one particle — this is known as simple exclusion.

We evolve the system in continuous time: each event in the system (movement or creation/deletion of a particle) is equipped with an exponential clock and occurs each time the clock rings.

The frequency of each clock is set using the parameter of its exponential law — one of the subtleties being how to adjust this setting...

Example of a simple exclusion system producing in the limit the Heat Equation on the interval \( (0,1) \) with Neumann boundary conditions...
(cf. [Baldasso, Menezes, Neumann, Souza])
This evolution produces a Markovian jump process with values in the state spaces \( \{0, 1\}^{\Lambda_N} \) for the field and \( \{0, 1\}^{\Gamma_N} \) for the road.

To connect this jump process to the solution of the field-road problem, we associate each state with an empirical measure which is essentially a sum of Dirac masses positioned where the particles are located.

An example of a càdlàg trajectory of empirical measures associated with a simple exclusion process on the torus \( \mathbb{Z}/ 3 \, \mathbb{Z} \).
As we send \( N \) to infinity, the Dirac masses become more numerous and invade the macroscopic landscape so that in the limit, the process of empirical measures converges to a measure that is absolutely continuous with respect to the Lebesgue measure.

The density of this limiting measure is then the solution of the expected deterministic problem.

Co-authors

Matthieu Alfaro — Romain Ducasse — Mustapha Mourragui

Talks

• Bridging Bulk and Surface: An Interacting Particle System Journey towards the Field-Road Diffusion Model (presentation with movies)
  
  Abstract

  This presentation explores the field-road diffusion model developed in 2012 by Berestycki, Roquejoffre, and Rossi. This parabolic system aims to capture the significant dispersal effects induced by lines of fast diffusion, with wide-ranging applications in population dynamics, ecology, and epidemiology.
  
  Initially, we will introduce the model, emphasizing its ability to simulate accelerated spread phenomena. We will then concentrate on the explicit determination of the fundamental solution to the macroscopic system, achieved through the application of a double integral transform, namely Fourier and Laplace. This analytical framework offers clear insights into the model's dynamics and sets the stage for exploring non-linear issues such as "persistence vs. extinction" phenomena in the presence of reaction terms with the so-called Allee effect.
  
  The second part of the talk will be dedicated to provide a stochastic foundation for the deterministic framework by deriving the governing equations of the diffusive field-road model from an interacting particle system. To introduce this approach, we will go back to the origins of the Symmetric Simple Exclusion Process (SSEP) which enables the rigorous derivation of solutions to the Heat equation on the torus. After outlining the principles of this type of particle system, we will see how it can be used to generate certain boundary conditions. This will allow us to introduce a microscopic dynamics for the field-road diffusion model and present our recent result on its hydrodynamic limit.
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• Blow-up vs. global existence for a Fujita type Heat echanger system (PDF without movies) (presentation with movies)
  
  Abstract

  In this presentation, we'll discuss finite-time explosion phenomena arising for certain superlinear reaction-diffusion problems.
  
  We'll start with an introduction to the seminal results of Japanese mathematician H.Fujita concerning the altered Heat equation by adding the unknown raised to the power 1+p (p>0) in the second member. In his 1966 work, Fujita highlighted a critical exponent for p, marking the threshold between systematic explosion and possible global existence of solutions. This distinction is based on an equilibrium ratio between two remarkable algebraic quantities associated with the reactive (growth) and diffusive (mass scattering) parts of the equation.
  
  We will then broaden our perspective by introducing a "heat exchanger" system where the unknowns are coupled by a diffusion mechanism, while integrating over-linear and non-coupling reactions as previously stated. A frequency analysis of the purely diffusive heat exchanger will enable us to estimate its "scattering intensity", leading to the main results of the talk concerning the systematic explosion and possible global existence of solutions of such a semilinear system.
  
  This work is a first step towards extending Fujita-type problems to diffusion-coupled systems, and raises several open questions, including the exploration of more complex diffusive mechanisms...


• Diffusive Dialogues: Mathematical Models of Individual Exchanging Across Spatial Domains (PDF without movies) (HTML presentation with movies)
  
  Abstract

  In this talk, we examine mathematical models that describe the diffusion and exchange of individuals across spatial domains.
  
  We begin with the field-road model, emphasizing its biological foundations and its importance in understanding fast diffusion channels in population dynamics and ecology. Following this, we explain how to derive the explicit solutions for the field-road model and provide estimates on the asymptotic decay rate of these solutions.
  
  This analytical framework paves the way for exploring non-linear issues, including Fujita-type blow-up phenomena, which we explain by outlining the key concepts involved.
  
  We then turn to the Heat-exchanger model, which serves as a tractable first approach for deriving some coupled-by-diffusion Fujita-type systems. We proceed to characterize the purely diffusive version of this model. With this foundation, we tackle the question of blow-up vs. global existence that arises when incorporating super-linear Fujita-type reaction terms.
  
  The presentation concludes with insights into a stochastic simple exclusion process for the field-road diffusion model. We provide a brief overview of the mechanics of this particle system, drawing parallels with the canonical example of the Heat equation.


• The field-road diffusion model (beamer here)
  
  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^∞ norm of these solutions.

Posters