arXiv subjects

Grégory Faye

Publications and source records attributed to Grégory Faye.

At least 19 recordsLinked to original sources

arXiv2026

Spreading Properties of a City-Road Reaction-Diffusion Model on One-Dimensional Lattice

We propose and study a new model to describe biological invasions constrained on infinite homogeneous one dimensional metric graphs. Our model consists of an infinite PDE-ODE system where, at each vertex of the one-dimensional lattice $\mathbb{Z}$, we have a logistic equation, and connections between vertices are given by diffusion equations on the edges supplemented with Robin like boundary conditions at the vertices. We establish the main properties of the system and study the long time behavior of the solutions, especially by characterizing an asymptotic spreading speed for the system. In the fast diffusion regime, we derive a novel asymptotic model which exhibits similar propagation properties as the classical discrete Fisher-KPP on the one-dimensional lattice $\mathbb{Z}$.

math.AP↗

arXiv2026

Sharp asymptotics for a transport model with a nonlocal condition of the Fisher-KPP type at the boundary

This paper is concerned with the precise asymptotics, as time goes to infinity, of a transport problem in a half plane coupled with a nonlinear nonlocal boundary condition. This system arises from a class of models for the spatial spread of epdemics, its space independent version being the classical Kermack-McKendrick model. Using ideas pertaining to the study of nonlocal equations of the Fisher-KPP type, and exploiting the particular structure of the model, we prove that any initially localized solution will lag behind the minimal traveling wave, with a delay that grows logarithmically in time.

math.AP↗

arXiv2025

Spreading properties in Kermack-McKendrick models with nonlocal spatial interactions -- A new look

In this paper, we revisit the famous Kermack-McKendrick model with nonlocal spatial interactions by shedding new lights on associated spreading properties and we also prove the existence and uniqueness of traveling fronts. Unlike previous studies that have focused on integrated versions of the model for susceptible population, we analyze the long time dynamics of the underlying age-structured model for the cumulative density of infected individuals and derive precise asymptotic behavior for the infected population. Our approach consists in studying the long time dynamics of an associated transport equation with nonlocal spatial interactions whose spreading properties are close to those of classical Fisher-KPP reaction-diffusion equations. Our study is self-contained and relies on comparison arguments.

math.AP↗

arXiv2025

Wave propagation phenomena in nonlinear hierarchical neural networks with predictive coding feedback dynamics

We propose a mathematical framework to systematically explore the propagation properties of a class of continuous in time nonlinear neural network models comprising a hierarchy of processing areas, mutually connected according to the principles of predictive coding. We precisely determine the conditions under which upward propagation, downward propagation or even propagation failure can occur in both bi-infinite and semi-infinite idealizations of the model. We also study the long-time behavior of the system when either a fixed external input is constantly presented at the first layer of the network or when this external input consists in the presentation of constant input with large amplitude for a fixed time window followed by a reset to a down state of the network for all later times. In both cases, we numerically demonstrate the existence of threshold behavior for the amplitude of the external input characterizing whether or not a full propagation within the network can occur. Our theoretical results are consistent with predictive coding theories and allow us to identify regions of parameters that could be associated with dysfunctional perceptions.

math.AP↗

arXiv2025

Amplification of numerical wave packets for transport equations with two boundaries

The purpose of this note is to investigate the coupling of Dirichlet and Neumann numerical boundary conditions for the transport equation set on an interval. When one starts with a stable finite difference scheme on the lattice $\mathbb{Z}$ and each numerical boundary condition is taken separately with the Neumann extrapolation condition at the outflow boundary, the corresponding numerical semigroup on a half-line is known to be bounded. It is also known that the coupling of such numerical boundary conditions on a compact interval yields a stable approximation, even though large time exponentially growing modes may occur. We review the different stability estimates associated with these numerical boundary conditions and give explicit examples of such exponential growth phenomena for finite difference schemes with ''small'' stencils. This provides numerical evidence for the optimality of some stability estimates on the interval.

math.NA↗

arXiv2024

Nonlinear orbital stability of stationary shock profiles for the Lax-Wendroff scheme

In this article we study the spectral, linear and nonlinear stability of stationary shock profile solutions to the Lax-Wendroff scheme for hyperbolic conservation laws. We first clarify the spectral stability of such solutions depending on the convexity of the flux for the underlying conservation law. The main contribution of this article is a detailed study of the so-called Green's function for the linearized numerical scheme. As evidenced on numerical simulations, the Green's function exhibits a highly oscillating behavior ahead of the leading wave before this wave reaches the shock location. One of our main results gives a quantitative description of this behavior. Because of the existence of a one-parameter family of stationary shock profiles, the linearized numerical scheme admits the eigenvalue 1 that is embedded in its continuous spectrum, which gives rise to several contributions in the Green's function. Our detailed analysis of the Green's function describes these contributions by means of a so-called activation function. For large times, the activation function describes how the mass of the initial condition accumulates along the eigenvector associated with the eigenvalue 1 of the linearized numerical scheme. We can then obtain sharp decay estimates for the linearized numerical scheme in polynomially weighted spaces, which in turn yield a nonlinear orbital stability result for spectrally stable stationary shock profiles. This nonlinear result is obtained despite the lack of uniform ${\ell}$ 1 estimates for the Green's function of the linearized numerical scheme, the lack of such estimates being linked with the dispersive nature of the numerical scheme. This dispersive feature is in sharp contrast with previous results on the orbital stability of traveling waves or discrete shock profiles for parabolic perturbations of conservation laws.

math.AP↗

arXiv2024

The local limit theorem for complex valued sequences: the parabolic case

We give a complete expansion, at any accuracy order, for the iterated convolution of a complex valued integrable sequence in one space dimension. The remainders are estimated sharply with generalized Gaussian bounds. The result applies in probability theory for random walks as well as in numerical analysis for studying the large time behavior of numerical schemes.

math.NA↗

arXiv2023

Existence and stability of nonmonotone hydraulic shocks for the Saint Venant equations of inclined thin-film flow

Extending work of Yang-Zumbrun for the hydrodynamically stable case of Froude number F < 2, we categorize completely the existence and convective stability of hydraulic shock profiles of the Saint Venant equations of inclined thin-film flow. Moreover, we confirm by numerical experiment that asymptotic dynamics for general Riemann data is given in the hydrodynamic instability regime by either stable hydraulic shock waves, or a pattern consisting of an invading roll wave front separated by a finite terminating Lax shock from a constant state at plus infinity. Notably, profiles, and existence and stability diagrams are all rigorously obtained by mathematical analysis and explicit calculation.

math.AP↗

arXiv2023

Mathematical derivation of wave propagation properties in hierarchical neural networks with predictive coding feedback dynamics

Sensory perception (e.g. vision) relies on a hierarchy of cortical areas, in which neural activity propagates in both directions, to convey information not only about sensory inputs but also about cognitive states, expectations and predictions. At the macroscopic scale, neurophysiological experiments have described the corresponding neural signals as both forward and backward-travelling waves, sometimes with characteristic oscillatory signatures. It remains unclear, however, how such activity patterns relate to specific functional properties of the perceptual apparatus. Here, we present a mathematical framework, inspired by neural network models of predictive coding, to systematically investigate neural dynamics in a hierarchical perceptual system. We show that stability of the system can be systematically derived from the values of hyper-parameters controlling the different signals (related to bottom-up inputs, top-down prediction and error correction). Similarly, it is possible to determine in which direction, and at what speed neural activity propagates in the system. Different neural assemblies (reflecting distinct eigenvectors of the connectivity matrices) can simultaneously and independently display different properties in terms of stability, propagation speed or direction. We also derive continuous-limit versions of the system, both in time and in neural space. Finally, we analyze the possible influence of transmission delays between layers, and reveal the emergence of oscillations at biologically plausible frequencies.

q-bio.NC↗

arXiv2023

The logarithmic Bramson correction for Fisher-KPP equations on the lattice $\mathbb{Z}$

We establish in this paper the logarithmic Bramson correction for Fisher-KPP equations on the lattice $\mathbb{Z}$. The level sets of solutions with step-like initial conditions are located at position $c_*t-\frac{3}{2λ_*}\ln t+\mathcal{O}(1)$ as $t\rightarrow+\infty$ for some explicit positive constants $c_*$ and $λ_*$. This extends a well-known result of Bramson in the continuous setting to the discrete case using only PDE arguments. A by-product of our analysis also gives that the solutions approach the family of logarithmically shifted traveling front solutions with minimal wave speed $c_*$ uniformly on the positive integers, and that the solutions converge along their level sets to the minimal traveling front for large times.

math.AP↗

arXiv2022

Exponential asymptotic stability of Riemann shocks of hyperbolic systems of balance laws

For strictly entropic Riemann shock solutions of strictly hyperbolic systems of balance laws, we prove that exponential spectral stability implies large-time asymptotic orbital stability. As a preparation, we also prove similar results for constant solutions of initial value and initial boundary value problems, that seem to be new in this generality. Main key technical ingredients include the design of a nonlinear change of variables providing a hypocoercive Kawashima-type structure with dissipative boundary conditions in the high-frequency regime and the explicit identification of most singular parts of the linearized evolution, both being deduced from the mere spectral assumption.

math.AP↗

arXiv2022

Asymptotic behavior of nonlocal bistable reaction-diffusion equations

In this paper, we study the asymptotic behavior of the solutions of nonlocal bistable reaction-diffusion equations starting from compactly supported initial conditions. Depending on the relationship between the nonlinearity, the interaction kernel and the diffusion coefficient, we show that the solutions can either: propagate, go extinct or remain pinned. We especially focus on the latter regime where solutions are pinned by thoroughly studying discontinuous ground state solutions of the problem for a specific interaction kernel serving as a case study. We also present a detailed numerical analysis of the problem.

math.AP↗

arXiv2021

Generalized gaussian bounds for discrete convolution powers

We prove a uniform generalized gaussian bound for the powers of a discrete convolution operator in one space dimension. Our bound is derived under the assumption that the Fourier transform of the coefficients of the convolution operator is a trigonometric rational function, which generalizes previous results that were restricted to trigonometric polynomials. We also allow the modulus of the Fourier transform to attain its maximum at finitely many points over a period.

math.NA↗

arXiv2021

Spreading properties for SIR models on homogeneous trees

We consider an epidemic model of SIR type set on a homogeneous tree and investigate the spreading properties of the epidemic as a function of the degree of the tree, the intrinsic basic reproduction number and the strength of the interactions within the population of infected individuals. When the degree is one, the homogeneous tree is nothing but the standard lattice on the integers and our model reduces to a SIR model with discrete diffusion for which the spreading properties are very similar to the continuous case. On the other hand, when the degree is larger than two, we observe some new features in the spreading properties. Most notably, there exists a critical value of the strength of interactions above which spreading of the epidemic in the tree is no longer possible.

math.AP↗

arXiv2021

Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity

We determine the asymptotic spreading speed of the solutions of a Fisher-KPP reaction-diffusion equation, starting from compactly supported initial data, when the diffusion coefficient is a fixed bounded monotone profile that is shifted at a given forcing speed and satisfies a general uniform ellipticity condition. Depending on the monotony of the profile, we are able to characterize this spreading speed as a function of the forcing speed and the two linear spreading speeds associated to the asymptotic problems. Most notably, when the profile of the coefficient diffusion is increasing we show that there is an intermediate range for the forcing speed where spreading actually occurs at a speed which is larger than the linear speed associated with the homogeneous state around the position of the front. We complement our study with the construction of strictly monotone traveling front solutions with strong exponential decay near the unstable state when the profile of the coefficient diffusion is decreasing and in the regime where the forcing speed is precisely the selected spreading speed.

math.AP↗

arXiv2021

Sharp stability for finite difference approximations of hyperbolic equations with boundary conditions

In this article, we consider a class of finite rank perturbations of Toeplitz operators that have simple eigenvalues on the unit circle. Under a suitable assumption on the behavior of the essential spectrum, we show that such operators are power bounded. The problem originates in the approximation of hyperbolic partial differential equations with boundary conditions by means of finite difference schemes. Our result gives a positive answer to a conjecture by Trefethen, Kreiss and Wu that only a weak form of the so-called Uniform Kreiss-Lopatinskii Condition is sufficient to imply power boundedness.

math.AP↗

arXiv2020

Dynamics of epidemic spreading on connected graphs

We propose a new model that describes the dynamics of epidemic spreading on connected graphs. Our model consists in a PDE-ODE system where at each vertex of the graph we have a standard SIR model and connexions between vertices are given by heat equations on the edges supplemented with Robin like boundary conditions at the vertices modeling exchanges between incident edges and the associated vertex. We describe the main properties of the system, and also derive the final total population of infected individuals. We present a semi-implicit in time numerical scheme based on finite differences in space which preserves the main properties of the continuous model such as the uniqueness and positivity of solutions and the conservation of the total population. We also illustrate our results with a selection of numerical simulations for a selection of connected graphs.

math.AP↗

arXiv2018

Rigorous derivation of the nonlocal reaction-diffusion FitzHugh-Nagumo system

We introduce a spatially extended transport kinetic FitzHugh-Nagumo model with forced local interactions and prove that its hydrodynamic limit converges towards the classical nonlocal reaction-diffusion FitzHugh-Nagumo system. Our approach is based on a relative entropy method, where the macroscopic quantities of the kinetic model are compared with the solution to the nonlocal reaction-diffusion system. This approach allows to make the rigorous link between kinetic and reaction-diffusion models.

math.AP↗