arXiv subjects

Quentin Griette

Publications and source records attributed to Quentin Griette.

At least 19 recordsLinked to original sources

arXiv2026

Propagation phenomena in KPP-bistable periodic patchy environments

This paper first investigates the propagation dynamics of solutions to the Cauchy problem for a one-dimensional reaction-diffusion equation in a spatially periodic environment consisting of two distinct patch types. The novelty of this work lies in the systematic analysis of a KPP-bistable heterogeneous framework. In this setting, the respective patch lengths, the linear stability of the zero solution and the positive periodic steady state, and the magnitude of the initial data play crucial roles in the long-time dynamics. We first establish persistence properties of the species, showing that uniform persistence holds when the zero steady state of the associated periodic patch model is unstable, while local persistence is obtained under additional suitable conditions. Using a dynamical systems approach, we further establish spreading properties and demonstrate the existence of pulsating traveling waves in two different cases, depending on whether the trivial solution is unstable or stable. Finally, we present two sets of sufficient conditions characterizing species extinction.

math.AP↗

arXiv2026

On advective nonlocal operators: multiplicity of principal eigenpairs

We study the existence and multiplicity of principal eigenvalues and eigenfunctions for a periodically heterogeneous nonlocal dispersal model with advection. The operator we consider is resolvent-positive but not resolvent-compact; therefore, the classical Krein-Rutman theory cannot be applied directly. When the advection coefficient has a constant sign, we prove the existence and uniqueness of the principal eigenvalue and the corresponding normalized eigenfunction. In sharp contrast, when the advection does not have a constant sign, the problem is more involved and leads to surprising results. Depending on the coefficients of the equation, the principal eigenproblem can either have a unique normalized solution or a continuum of solutions, at the boundary of which there exists a principal eigenvector with a singular measure component. In the latter situation, all the constructed eigenvalues are embedded in the continuous spectrum of our operator. We completely characterize the eigenvalues associated with positive eigenvectors, even when the eigenvector is a Radon measure. Finally, we discuss an application to a nonlinear KPP-type equation with nonlocal dispersal, which possesses a continuum of nontrivial stationary solutions, a different behavior from the classical KPP equation with local diffusion.

math.AP↗

arXiv2026

Propagation dynamics of an acid-mediated invasion model with degenerate tumor diffusion

We investigate traveling wave fronts for an acid-mediated tumor invasion model with density-dependent degenerate diffusion. The model is a partially diffusive PDE--ODE system of Gatenby--Gawlinski type, in which the tumor diffusion coefficient $D(U)$ is allowed to be a general decreasing function satisfying $D(1)=0$. This degeneracy causes the traveling wave equation for the tumor component to lose uniform ellipticity near the healthy state, and hence standard arguments for nondegenerate reaction diffusion systems are not directly applicable. To overcome this difficulty, we introduce a nonlinear change of variables which removes the degeneracy from the highest-order term of the tumor equation. For each fixed admissible tumor profile, the acid profile is represented by a Green kernel, while the healthy-tissue profile is obtained from an explicit integral formula. The transformed tumor profile is then constructed as the stationary limit of a uniformly parabolic auxiliary problem. By combining comparison principles, local Schauder estimates, carefully chosen super- and sub-solutions, and the Schauder fixed-point theorem, we prove the existence of traveling wave fronts for every wave speed $\theta\ge 2\sqrt{rD(0)}$. The resulting wave connects the tumor-dominant state $(0,1,1)$ at $z=-\infty$ to the healthy state $(1,0,0)$ at $z=+\infty$. We further establish strict pointwise bounds, monotonicity of all wave components, and one-sided exponential asymptotic estimates in both the transformed variable and the original traveling-wave variable.

math.AP↗

arXiv2026

Front propagation in hybrid reaction-diffusion epidemic models with spatial heterogeneity. Part II: Pulsating traveling waves

We consider a two-species reaction-diffusion system in one space dimension that is derived from an epidemiological model in a spatially periodic environment with two types of pathogens: the wild type and the mutant. The system is of a hybrid nature, partly cooperative and partly competitive, but neither of these entirely. As a result, the comparison principle does not hold. In the previous work, we studied the propagation properties of the solutions to the Cauchy problem for this system and showed, among other things, that the spreading speeds of the fronts to the right and to the left directions, denoted by $ c^*_R$ and $ c^*_L$, can be characterized by using certain principal eigenvalues, and studied the homogenization limit as the spatial period $L$ tends to $0$, and also discussed the long-time behavior of solutions behind the fronts. In the present paper we prove the existence of pulsating traveling waves in the right direction (resp. left direction) with speed $c$ for any $c\geq c^*_R$ (resp. $c\geq c^*_L$), where $c^*_R$ and $c^*_L$ denote the aforementioned spreading speeds in the right and left directions. We also prove that the leading edge of any traveling wave has the exponential decay rate that is anticipated from formal linear analysis, thus extending part of the results of Hamel 2008 to systems of equations. Finally, we present an example in which the two speeds $c^*_R$ and $c^*_L$ are different. This is done by considering a multi-scale singular limit problem. This result highlights a marked difference between our system and scalar KPP type equations.

math.AP↗

arXiv2026

Sharp threshold dynamics for a bistable age-structured population model

This paper is devoted to the long-term dynamics of solutions to the Gurtin-MacCamy population model with a bistable birth function. We consider a one-parameter monotone family of initial distributions for the population such that for small values of the parameter, the corresponding population density gets extinct as time passes, whereas for large values of them, the solutions exhibit a different behavior. We are interested in the intermediate set of values for the parameters, which are called threshold parameters. We prove the existence of a sharp transition between these two asymptotic dynamics; that is, there exists exactly one threshold value when the age-dependent birth rate of the population has compact support, utilizing the theory of monotone dynamical systems. The case when the birth rate is non-compactly supported is more intricate to deal with, as has been observed in several works, even if the nonlinear birth function is monostable. Nevertheless, the approach used in the present work turns out to be effective to handle a particular birth rate with noncompact support by translating the dynamics of the age-structured model into an integro-differential system.

math.AP↗

arXiv2025

Slowly oscillating periodic solutions in a nonlinear Volterra equation with non-symmetric feedback

In this work we study a nonlinear Volterra equation with non-symmetric feedback that arises as a particular case of the Gurtin-MacCamy model in population dynamics. We are particularly interested in the existence of slowly oscillating periodic solutions when the trivial stationary state is unstable. Here the absence of symmetry of the nonlinearity prevents the use of many traditional strategies to obtain a priori estimates on the solution. Without a precise knowledge of the period of the solution, we manage to prove the forward invariance of a carefully constructed set of initial data whose properties imply the slowly oscillating character of all continuations. We prove the existence of periodic solutions by constructing a homeomorphism between our set and a convex subset of a different Banach space, thereby showing that it possesses the fixed-point property. Finally, in a singular limit of a parameter, we show that this periodic solution converges to the solution of a wellknown discrete difference equation. We conclude the paper with some numerical simulations to illustrate the existence of the periodic orbit as well as the singular limit behavior.

math.AP↗

arXiv2025

Front propagation into unstable states for periodic monotone reaction-diffusion systems

In this paper we study the invasion fronts of spatially periodic monotone reaction-diffusion systems in a multi-dimensional setting. We study the pulsating traveling waves that connect the trivial equilibrium, for which all components of the state variable are identically equal to zero, to a uniformly persistent stationary state, for which all components are uniformly positive. When the trivial equilibrium is linearly unstable, we show that all pulsating traveling waves have a speed that is greater than the speed of the linearized system at the equilibrium, in any given direction. If moreover the nonlinearity is sublinear, then we can construct a pulsating traveling wave that travels at any super-linear speed in any given direction (i.e. the minimal speed is linearly determined). We also show that pulsating traveling waves are monotonic in time as soon as the nonlinearity is sub-homogeneous. Beyond these general qualitative properties, the main focus of the paper is to derive sufficient conditions for the existence and nonexistence of pulsating waves propagating in any given direction. Our proof of the existence part relies upon a new level of understanding of the multi-dimensional pulsating waves observed from a direction-dependent coordinate system.

math.AP↗

arXiv2024

Front propagation in hybrid reaction-diffusion epidemic models with spatial heterogeneity. Part I: Spreading speed and asymptotic behavior

We consider a two-species reaction-diffusion system in one space dimension that is derived from an epidemiological model in a spatially periodic environment with two types of pathogens: the wild type and the mutant. The system is of a hybrid nature, partly cooperative and partly competitive, but neither of these entirely. As a result, the comparison principle does not hold for the whole system. We study spreading properties of solution fronts when the infection is localized initially. We show that there is a well-defined spreading speed both in the right and left directions and that it can be computed from the linearized equation at the leading edge of the propagation front. Next we study the case where the coefficients are spatially homogeneous and show that, when spreading occurs, every solution to the Cauchy problem converges to the unique positive stationary solution as $t\to\infty$. Finally we consider the case of rapidly oscillating coefficients, that is, when the spatial period of the coefficients, denoted by $\varepsilon$, is very small. We show that there exists a unique positive stationary solution, and that every positive solution to the Cauchy problem converges to this stationary solution as $t\to\infty$. We then discuss the homogenization limit as $\varepsilon\to 0$.

math.AP↗

arXiv2024

Traveling waves with continuous profile for hyperbolic Keller-Segel equation

This work describes a hyperbolic model for cell-cell repulsion with population dynamics. We consider the pressure produced by a population of cells to describe their motion. We assume that cells try to avoid crowded areas and prefer locally empty spaces far away from the carrying capacity. Here, our main goal is to prove the existence of traveling waves with continuous profiles. This article complements our previous results about sharp traveling waves. We conclude the paper with numerical simulations of the PDE problem, illustrating such a result.

math.AP↗

arXiv2023

Robin Hood model versus Sheriff of Nottingham model: transfers in population dynamics

We study the problem of transfers in a population structured by a continuous variable corresponding to the quantity being transferred. The model takes the form of an integro-differential equations with kernels corresponding to the specific rules of the transfer process. We focus our interest on the well-posedness of the Cauchy problem in the space of measures. We characterize transfer kernels that give a continuous semiflow in the space of measures and derive a necessary and sufficient condition for the stability of the space $L^1$ of integrable functions. We construct some examples of kernels that may be particularly interesting in economic applications. Our model considers blind transfers of economic value (e.g. money) between individuals. The two models are the ``Robin Hood model'', where the richest individual unconditionally gives a fraction of their wealth to the poorest when a transfer occurs, and the other extreme, the ``Sheriff of Nottingham model'', where the richest unconditionally takes a fraction of the poorest's wealth. Between these two extreme cases is a continuum of intermediate models obtained by interpolating the kernels. We illustrate those models with numerical simulations and show that any small fraction of the ``Sheriff of Nottingham'' in the transfer rules leads to a segregated population with extremely poor and extremely rich individuals after some time. Although our study is motivated by economic applications, we believe that this study is a first step towards a better understanding of many transfer phenomena occurring in the life sciences.

math.AP↗

arXiv2023

Speed-up of traveling waves by negative chemotaxis

We consider the traveling wave speed for Fisher-KPP (FKPP) fronts under the influence of repulsive chemotaxis and provide an almost complete picture of its asymptotic dependence on parameters representing the strength and length-scale of chemotaxis. Our study is based on the convergence to the porous medium FKPP traveling wave and a hyperbolic FKPP-Keller-Segel traveling wave in certain asymptotic regimes. In this way, it clarifies the relationship between three equations that have each garnered intense interest on their own. Our proofs involve a variety of techniques ranging from entropy methods and decay of oscillations estimates to a general description of the qualitative behavior to the hyperbolic FKPP-Keller-Segel equation. For this latter equation, we, as a part of our limiting arguments, establish an explicit lower bound on the minimal traveling wave speed and provide a new construction of traveling waves that extends the known existence range to all parameter values.

math.AP↗

arXiv2023

Epidemic models in measure spaces: persistence, concentration and oscillations

We investigate the long-time dynamics of a SIR epidemic model in the case of a population of pathogens infecting a homogeneous host population. The pathogen population is structured by a genotypic variable. When the initial mass of the maximal fitness set is positive, we give a precise description of the convergence of the orbit, including a formula for the asymptotic distribution. When this initial mass is zero, we show the persistence of the population of infected and the concentration of the population of pathogens on the set of genotypic traits that maximize the fitness. We also investigate precisely the case of a finite number of regular global maxima and show that the initial distribution may have an influence on the support of the eventual distribution. In particular, the natural process of competition is not always selecting a unique species, but several species may coexist as long as they maximize the fitness function. In some configurations, species that maximize the fitness may still get extinct depending on the shape of the initial distribution and some other parameter of the model, and we provide a way to characterize when this unexpected extinction happens. Finally, we provide an example of a pathological situation in which the distribution never reaches a stationary distribution but oscillates forever around the set of fitness maxima.

math.DS↗

arXiv2023

Asymptotic behavior of an epidemic model with infinitely many variants

We investigate the long-time dynamics of a SIR epidemic model with infinitely many pathogen variants infecting a homogeneous host population. We show that the basic reproduction number $\mathcal{R}_0$ of the pathogen can be defined in that case and corresponds to a threshold between the persistence ($\mathcal{R}_0>1$) and the extinction ($\mathcal{R}_0\leq 1$) of the pathogen. When $\mathcal{R}_0>1$ and the maximal fitness is attained by at least one variant, we show that the systems reaches an equilibrium state that can be explicitly determined from the initial data. When $\mathcal{R}_0>1$ but none of the variants attain the maximal fitness, the situation is more intricate. We show that, in general, the pathogen is uniformly persistent and any family of variants that have a fitness which is uniformly lower than the optimal fitness, eventually gets extinct. We derive a condition under which the total pathogen population converges to a limit which can be computed explicitly. We also find counterexamples that show that, when our condition is not met, the total pathogen population may converge to an unexpected value, or the system can even reach an eternally transient behavior where the total pathogen population between several values. We illustrate our results with numerical simulations that emphasize the wide variety of possible dynamics.

q-bio.PE↗

arXiv2022

A Kermack-McKendrick model with age of infection starting from a single or multiple cohorts of infected patients

During an epidemic, the infectiousness of infected individuals is known to depend on the time since the individual was infected, that is called the age of infection. Here we study the parameter identifiability of the Kermack-McKendrick model with age of infection which takes into account this dependency. By considering a single cohort of individuals, we show that the daily reproduction number can be obtained by solving a Volterra integral equation that depends on the flow of new infected individuals. We test the consistency our the method by generating data from deterministic and stochastic numerical simulations. Finally we apply our method to a dataset from SARS-CoV-1 with detailed information on a single cluster of patients. We stress the necessity of taking into account the initial data in the analysis to ensure the identifiability of the problem.

math.AP↗

arXiv2021

Propagation dynamics of solutions to spatially periodic reaction-diffusion systems with hybrid nonlinearity

In this paper we investigate the dynamical properties of a spatially periodic reaction-diffusion system {whose reaction terms are of hybrid nature in the sense that they are partly competitive and partly cooperative depending on the value of the solution. This class of problems includes various biologically relevant models and in particular many models focusing on the Darwinian evolution of species. We start by studying the principal eigenvalue of the associated differential operator and establishing a minimal speed formula for linear monotone systems. In particular, we show that the generalized Dirichlet principal eigenvalue and the periodic principal eigenvalue may not coincide when the reaction matrix is not symmetric, in sharp contrast with the case of scalar equations. We establish a sufficient condition under which equality holds for the two notions. We also show that the propagation speed may be different depending on the direction of propagation, even in the absence of a first-order advection term, again in a sharp contrast with scalar equations. Next we reveal the relation between the hair-trigger property of front propagation and the sign of the periodic principal eigenvalue. Finally, we discuss the linear determinacy of the propagation speed and also establish the existence of travelling waves travelling whose speeds greater than the minimal speed is also proved. We apply our results to an important class of epidemiological models with genetic mutations.

math.AP↗

arXiv2021

The spatio-temporal dynamics of interacting genetic incompatibilities. Part I: The case of stacked underdominant clines

We explore the interaction between two genetic incompatibilities (underdominant loci in diploid organisms) in a population occupying a one-dimensional space. We derive a system of partial differential equations describing the dynamics of allele frequencies and linkage disequilibrium between the two loci, and use a quasi-linkage equilibrium approximation in order to reduce the number of variables. We investigate the solutions of this system and demonstrate the existence of a solution in which the two clines in allele frequency remain stacked together. In the case of asymmetric incompatibilities (i.e. when one homozygote is favored over the other at each locus), these stacked clines propagate in the form of a traveling wave. We obtain an approximation for the speed of this wave which, in particular, is decreased by recombination between the two loci but is always larger than the speed of "one cline alone".

math.AP↗

arXiv2020

Sharp discontinuous traveling waves in a hyperbolic Keller--Segel equation

In this work we describe a hyperbolic model with cell-cell repulsion with a dynamics in the population of cells. More precisely, we consider a population of cells producing a field (which we call "pressure") which induces a motion of the cells following the opposite of the gradient. The field indicates the local density of population and we assume that cells try to avoid crowded areas and prefer locally empty spaces which are far away from the carrying capacity. We analyze the well-posedness property of the associated Cauchy problem on the real line. We start from bounded initial conditions and we consider some invariant properties of the initial conditions such as the continuity, smoothness and monotonicity. We also describe in detail the behavior of the level sets near the propagating boundary of the solution and we find that an asymptotic jump is formed on the solution for a natural class of initial conditions. Finally, we prove the existence of sharp traveling waves for this model, which are particular solutions traveling at a constant speed, and argue that sharp traveling waves are necessarily discontinuous. This analysis is confirmed by numerical simulations of the PDE problem.

math.AP↗

arXiv2020

Concentration estimates in a multi-host epidemiological model structured by phenotypic traits

In this work we consider an epidemic system modelling the evolution of a spore-producing pathogen within a multi-host population of plants. Here we focus our analysis on the study of the stationary states. We first discuss the existence of such nontrivial states by using the theory of global attractors. Then we introduce a small parameter epsilon that characterises the width of the mutation kernel, and we describe the asymptotic shape of steady states with respect to epsilon. In particular, we show that the distribution of spores converges to the singular measure concentrated on the maxima of fitness of the pathogen in each plant population. This asymptotic description allows us to show the local stability of each of the positive steady states in the regime of narrow mutations, from which we deduce a uniqueness result for the nontrivial stationary states by means of a topological degree argument. These analyses rely on a careful investigation of the spectral properties of some non-local operators.

math.AP↗