The influence of a piston on the diffusion spreading phenomenon
The work described here was carried out in collaboration with
Mingmin Zhang
and led to the writing of
this article.
We introduce a new diffusion model posed on a moving half-line
\( \{ z \geq b(t) \} \), whose boundary
\( z=b(t) \)
advances over time and acts as an impermeable wall repelling the individuals it encounters.
In other words, the boundary plays the role of a piston: far from the boundary, individuals spread by diffusion, while near the boundary they are blocked and then carried along by its motion.
From a mathematical point of view, the boundary condition imposed on the moving frontier is a Robin condition that exactly compensates the flux created by the motion of the domain.
This choice preserves the total mass of the population while revealing a new mechanism: the moving boundary acts against the natural spreading induced by diffusion.
The main objective of the article is therefore to understand how the speed of this piston influences the long-time behavior of the solution.
For this, we consider boundaries of the form
\( b(t)=c[(1+t)^\beta-1] \),
whose asymptotic behavior is
\( b(t)\sim ct^\beta \).
This model, though very elementary in appearance, already reveals a surprisingly rich asymptotic structure for the fundamental diffusion phenomenon.
In particular, we show the existence of a critical threshold
\( \beta=1/2 \),
which corresponds to the natural diffusive scale,
\( \sqrt{t} \).
When
\( \beta<1/2 \),
the boundary advances too slowly to deeply alter the diffusive spreading: the asymptotic profiles remain Gaussian, as for the heat equation with Neumann boundary condition.
In contrast, when
\( \beta>1/2 \),
the piston interacts strongly with the population and gives rise to exponential asymptotic profiles, reflecting an increased concentration near the moving boundary.
The case
\( \beta=1 \)
is particularly remarkable: diffusion and the advection induced by the moving boundary exactly balance, and the solution converges exponentially fast to a non-trivial stationary profile.
These results suggest that the simple displacement of a boundary can qualitatively transform the spreading of a population.
This naturally opens the way to the study of more complex models arising in population dynamics, in which the piston would be combined with growth, extinction, or interaction mechanisms:
Allee effect, intraspecific competition, predator-prey interactions, or density-dependent dynamics.
The general idea is that such a mobile confinement mechanism could give rise to new behaviors, where the speed of the boundary would no longer be only a geometric parameter, but a genuine driver of ecological dynamics.
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,
and has resulted in the writing of
this article.
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.
Stability of the trivial equilibrium in degenerate monostable reaction-diffusion equations
(presentation with movies)
This talk addresses the long-term behavior of reaction-diffusion equations \( \partial_{t} u = \Delta u
+ f(u) \) in \(
\mathbb{R}^{N} \), where the growth function \( f \) behaves as \( u^{1+p} \) when \( u \) is near the
origin.
Specifically, we are interested in the persistence versus extinction phenomena in a population
dynamics
context, where the function \( u \) represents a density of individuals distributed in space.
The degenerated behavior \( f(u) \sim u^{1+p} \) near the null equilibrium models the so-called Allee
effect, which
penalizes the growth of the population when the density is low. This effect simulates factors such as
inbreeding,
mating difficulties, or reduced resistance to extreme climatic events.
We will begin the presentation by discussing a result linking the questions of persistence and
extinction with the
dimension \( N \) and the intensity of the Allee effect \( p \), as established in the classical paper
by Aronson
and Weinberger (1978).
This result is closely related to the seminal work of Fujita (1966) on blow-up versus global
existence of
solutions to the superlinear equation \( \partial_{t} u = \Delta u + u^{1+p} \).
Following these preliminary results, we will focus on a reaction-diffusion system involving a heat
exchanger, where
the unknowns are coupled through the diffusion process, integrating super-linear and non-coupling
reactions.
An analysis of the solution frequencies for the purely diffusive heat exchanger will allow us to
estimate its
dispersal intensity, which is a key information for addressing blow-up versus global existence in
such
semi-linear problems.
This work represents a first step toward Fujita-type results for systems coupled by diffusion and raises
several
open questions, particularly regarding the exploration of more intricate diffusion mechanisms.
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 envi-
ronments, 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 sim-
ple 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
[22] and provide a uniform long-term control. Such control is useful to quantify the “dis-
persion intensity” of the diffusive process, enabling to demonstrate results related to per-
sistence 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.
Bridging Bulk and Surface: An Interacting Particle System Journey towards the Field-Road Diffusion Model
(presentation with movies)
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.
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...
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.
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.
