# Nicolas Vauchelet's research while affiliated with Sorbonne Université and other places

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## Publications (97)

In this paper, we investigate an initial-boundary value problem of a reaction–diffusion equation in a bounded domain with a Robin boundary condition and introduce some particular parameters to consider the non-zero flux on the boundary. This problem arises in the study of mosquito populations under the intervention of the population replacement met...

This work is devoted to the mathematical study of an optimization problem regarding control strategies of mosquito population in a heterogeneous environment. Mosquitoes are well known to be vectors of diseases, but, in some cases, they have a reduced vector capacity when carrying the endosymbiotic bacterium Wolbachia. We consider a mathematical mod...

In this paper, we investigate an initial-boundary-value problem of a reaction-diffusion equation in a bounded domain with a Robin boundary condition and introduce some particular parameters to consider the non-zero flux on the boundary. This problem arises in the study of mosquito populations under the intervention of the population replacement met...

The Sterile Insect Technique (SIT) is a classic vector control method that has been successfully applied to fight against diverse insect plagues since the 1950s . In recent years, this strategy has been used to control mosquito populations, in order to limit the spread of the diseases they transmit. In this paper, we consider a system of reaction-d...

The sterile insect technique consists in massive release of sterilized males in the aim to reduce the size of mosquitoes population or even eradicate it. In this work, we investigate the feasibility of using the sterile insect technique as a barrier against reinvasion. More precisely, we provide some numerical simulations and mathematical results s...

Mosquitoes are responsible for the transmission of many diseases such as dengue fever, zika or chigungunya. One way to control the spread of these diseases is to use the sterile insect technique (SIT), which consists in a massive release of sterilized male mosquitoes. This strategy aims at reducing the total population over time, and has the advant...

Two-dimensional dissipative and isotropic kinetic models, like the ones used in neutron transport theory, are considered. Especially, steady-states are expressed for constant opacity and damping, allowing to derive a scattering $S$-matrix and corresponding ``truly 2D well-balanced'' numerical schemes. A first scheme is obtained by directly implemen...

In this article, we are interested in the analysis and simulation of solutions to an optimal control problem motivated by population dynamics issues. In order to control the spread of mosquito-borne arboviruses, the population replacement technique consists in releasing into the environment mosquitoes infected with the Wolbachia bacterium, which gr...

In order to prevent the propagation of human diseases transmitted by mosquitoes (such as dengue or zika), one possible solution is to act directly on the mosquito population. In this work, we consider an invasive species (the mosquitoes) and we study two strategies to eradicate the population in the whole space by a local intervention. The dynamics...

This work was devoted to the study of a relaxation limit of the so-called aggregation equation with a pointy potential in one-dimensional space. The aggregation equation is today widely used to model the dynamics of a density of individuals attracting each other through a potential. When this potential is pointy, solutions are known to blow up in f...

This work is devoted to the study of a relaxation limit of the so-called aggregation equation with a pointy potential in one dimensional space. The aggregation equation is by now widely used to model the dynamics of a density of individuals attracting each other through a potential. When this potential is pointy, solutions are known to blow up in f...

The aim of this article is to understand how to apply partial or total containment to SIR epidemic model during a given finite time interval in order to minimize the epidemic final size, that is the cumulative number of cases infected during the complete course of an epidemic. The existence and uniqueness of an optimal strategy are proved for this...

We formulate a general SEIR epidemic model in a heterogeneous population characterized by some trait in a discrete or continuous subset of a space [Formula: see text]. The incubation and recovery rates governing the evolution of each homogeneous subpopulation depend upon this trait, and no restriction is assumed on the contact matrix that defines t...

This work deals with a mathematical analysis of sodium's transport in a tubular architecture of a kidney nephron. The nephron is modelled by two counter-current tubules. Ionic exchange occurs at the interface between the tubules and the epithelium and between the epithelium and the surrounding environment (interstitium). From a mathematical point o...

In this article, we are interested in the analysis and simulation of solutions to an optimal control problem motivated by population dynamics issues. In order to control the spread of mosquito-borne arboviruses, the population replacement technique consists in releasing into the environment mosquitoes infected with the Wolbachia bacterium, which gr...

Mosquitoes are responsible for the transmission of many diseases such as dengue fever, zika or chigungunya. One way to control the spread of these diseases is to use the sterile insect technique (SIT), which consists in a massive release of sterilized male mosquitoes. This strategy aims at reducing the total population over time, and has the advant...

We study the incompressible limit for a two-species model with applications to tissue growth in the case of coupling through the so-called Brinkman's law in any space dimensions. The coupling through this elliptic equation accounts for viscosity effects among the individual species. In a recent paper Dębiec & Schmidtchen established said result in...

We formulate a general SEIR epidemic model in a heterogenous population characterized by some trait in a discrete or continuous subset of a space R d. The incubation and recovery rates governing the evolution of each homogenous subpopulation depend upon this trait, and no restriction is assumed on the contact matrix that defines the probability for...

We investigate the numerical discretization of a two-stream kinetic system with an internal state, such system has been introduced to model the motion of cells by chemotaxis. This internal state models the intracellular methylation level. It adds a variable in the mathematical model, which makes it more challenging to simulate numerically. Moreover...

Until a vaccine or therapy is found against the SARS-CoV-2 coronavirus, reaching herd immunity appears to be the only mid-term option. However, if the number of infected individuals decreases and eventually fades only beyond this threshold, a significant proportion of susceptible may still be infected until the epidemic is over. A containment strat...

The sterile insect technique consists in massive release of sterilized males in the aim to reduce the size of mosquitoes population or even eradicate it. In this work, we investigate the feasability of using the sterile insect technique as a barrier against reinvasion. More precisely, we provide some numerical simulations and mathematical results s...

This work deals with a mathematical analysis of sodium's transport in a tubular architecture of a kidney nephron. The nephron is modelled by two counter-current tubules. Ionic exchange occurs at the interface between the tubules and the epithelium and between the epithelium and the surrounding environment (interstitium). From a mathematical point o...

In order to prevent the propagation of human diseases transmitted by mosquitoes (as dengue or zika), a solution is to release mosquitoes infected by Wolbachia . In this study, we model the release and the propagation over time and space of such infected mosquitoes in a population of uninfected ones. The aim of this study is to investigate the best...

Two-dimensional dissipative and isotropic kinetic models, like the ones used in neutron transport theory, are considered. Especially, steady-states are expressed for constant opacity and damping , allowing to derive a scattering S-matrix and corresponding "truly 2D well-balanced" numerical schemes. A first scheme is obtained by directly implementin...

In this study we present a mathematical model describing the transport of sodium in a fluid circulating in a counter-current tubular architecture, which constitutes a simplified model of Henle's loop in a kidney nephron. The model explicitly takes into account the epithelial layer at the interface between the tubular lumen and the surrounding inter...

Dissipative kinetic models inspired by neutron transport are studied in a (1+1)-dimensional context: first, in the two-stream approximation, then in the general case of continuous velocities. Both are known to relax, in the diffusive scaling, toward a damped heat equation. Accordingly, it is shown that "uniformly accurate" L-splines discretizations...

In this study we present a mathematical model describing the transport of sodium in a fluid circulating in a counter-current tubular architecture, which constitutes a simplified model of Henle's loop in a kidney nephron. The model explicitly takes into account the epithelial layer at the interface between the tubular lumen and the surrounding inter...

In this article, we consider a simplified model of time dynamics for a mosquito population subject to the artificial introduction of {\itshape Wolbachia}-infected mosquitoes, in order to fight arboviruses transmission.Indeed, it has been observed that when some mosquito populations are infected by some {\itshape Wolbachia} bacteria, various reprodu...

We consider a system of two kinetic equations modelling a multicellular system : The first equation governs the dynamics of cells, whereas the second kinetic equation governs the dynamics of the chemoattractant. For this system, we first prove the existence of global-in-time solution. The proof of existence relies on a fixed point procedure after e...

In the fight against vector-borne arboviruses, an important strategy of control of epidemic consists in controlling the population of the vector, Aedes mosquitoes in this case. Among possible actions, two techniques consist either in releasing sterile mosquitoes to reduce the size of the population (Sterile Insect Technique) or in replacing the wil...

For a four-stream approximation of the kinetic model of radiative transfer with isotropic scattering, a numerical scheme endowed with both truly-2D well-balanced and diffusive asymptotic-preserving properties is derived, in the same spirit as what was done in [14] in the 1D case. Building on former results of Birkhoff and Abu-Shumays, [4], it is po...

In the fight against vector-borne arboviruses, an important strategy of control of epidemic consists in controlling the population of vector, \textit{Aedes} mosquitoes in this case. Among possible actions, two techniques consist in releasing mosquitoes to reduce the size of the population (Sterile Insect Technique) or in replacing the wild populati...

The aggregation equation is a nonlocal and nonlinear conservation law commonly used to describe the collective motion of individuals interacting together. When interacting potentials are pointy, it is now well established that solutions may blow up in finite time but global in time weak measure valued solutions exist. In this paper we focus on the...

This paper investigates the incompressible limit of a system modelling the growth of two cells population. The model describes the dynamics of cell densities, driven by pressure exclusion and cell proliferation. It has been shown that solutions to this system of partial differential equations have the segregation property, meaning that two populati...

We study the biological situation when an invading population propagates and replaces an existing population with different characteristics. For instance, this may occur in the presence of a vertically transmitted infection causing a cytoplasmic effect similar to the Allee effect (e.g. Wolbachia in Aedes mosquitoes): the invading dynamics we model...

The flux limited Keller-Segel (FLKS) system is a macroscopic model describing bacteria motion by chemotaxis which takes into account saturation of the velocity. The hyper-bolic form and some special parabolic forms have been derived from kinetic equations describing the run and tumble process for bacterial motion. The FLKS model also has the advant...

Understanding mosquitoes life cycle is of great interest presently because of the increasing impact of vector borne diseases in several countries. There is evidence of oscillations in mosquito populations independent of seasonality, still unexplained, based on observations both in laboratories and in nature. We propose a simple mathematical model o...

We consider one dimensional coupled classical-quantum models for quantum semiconductor device simulations. The coupling occurs in the space variable: the domain of the device is divided into a region with strong quantum effects (quantum zone) and a region where quantum effects are negligible (classical zone). In the classical zone, transport in dif...

A numerical analysis of upwind type schemes for the nonlinear nonlocal aggregation equation is provided. In this approach, the aggregation equation is interpreted as a conservative transport equation driven by a nonlocal nonlinear velocity field with low regularity. In particular, we allow the interacting potential to be pointy, in which case the v...

This paper is concerned with diffusive approximations of peculiar numerical schemes for several linear (or weakly nonlinear) kinetic models which are motivated by wide-range applications, including radiative transfer or neutron transport, run-and-tumble models of chemotaxis dynamics, and Vlasov-Fokker-Planck plasma modeling. The well-balanced metho...

Starting from isentropic compressible Navier-Stokes equations with growth term in the continuity equation, we rigorously justify that performing an incompressible limit one arrives to the two-phase free boundary fluid system. The limiting system may be seen as a model of tumor, with the growth term describing the multiplication of tumoral cells.

This article is devoted to the analysis of some nonlinear conservative transport equations, including the so-called aggregation equation with pointy potential, and numerical method devoted to its numerical simulation. Such a model describes the collective motion of individuals submitted to an attractive potential and can be written as a continuity...

Several stains of the intracellular parasitic bacterium Wolbachia limit severely the competence of the mosquitoes Aedes aegypti as a vector of dengue fever and possibly other arboviroses. For this reason, the release of mosquitoes infected by this bacterium in natural populations is presently considered a promising tool in the control of these dise...

A mathematical model for tissue growth is considered. This model describes the dynamics of the density of cells due to pressure forces and proliferation. It is known that such cell population model converges at the incompressible limit towards a Hele-Shaw type free boundary problem. The novelty of this work is to impose a non-overlapping constraint...

Classical results from spectral theory of stationary linear kinetic equations are applied to efficiently approximate two physically relevant weakly nonlinear kinetic models: a model of chemotaxis involving a biased velocity-redistribution integral term, and a Vlasov-Fokker-Planck (VFP) system. Both are coupled to an attractive elliptic equation pro...

Kinetic-transport equations are, by now, standard models to describe the dynamics of populations of bacteria moving by run-and-tumble. Experimental observations show that bacteria increase their run duration when encountering an increasing gradient of chemotactic molecules. This led to a first class of models which heuristically include tumbling fr...

This paper deals with the analysis of the asymptotic limit toward the derivation of macroscopic equations for a class of equations modeling complex multicellular systems by methods of the kinetic theory. After having chosen an appropriate scaling of time and space, a Chapman-Enskog expansion is combined with a closed, by minimization, technique to...

Artificial releases of Wolbachia-infected Aedes mosquitoes have been under study in the past years for fighting vector-borne diseases such as dengue, chikungunya and zika. Several strains of this bacterium cause cytoplasmic incompatibility (CI) and can also affect their host's fecun-dity or lifespan, while highly reducing vector competence for the...

Mathematical models have been widely used to describe the collective movement of bacteria by chemotaxis. In particular, bacterial concentration waves traveling in a narrow channel have been experimentally observed and can be precisely described thanks to a mathematical model at the macroscopic scale. Such model was derived in [1] using a kinetic mo...

An analysis of the error of the upwind scheme for transport equation with discontinuous coefficients is provided. We consider here a velocity field that is bounded and one-sided Lipschitz continuous. In this framework, solutions are defined in the sense of measures along the lines of Poupaud and Rascle's work. We study the convergence order of the...

Numerical resolution of two-stream kinetic models in strong aggregative setting is considered. To illustrate our approach, we consider an 1D kinetic model for chemotaxis in hyperbolic scaling and the high field limit of the Vlasov-Poisson-Fokker-Planck system. A difficulty is that, in this aggregative setting, weak solutions of the limiting model b...

We consider general models of coupled reaction-diffusion systems for interacting variants of the same species. When the total population becomes large with intensive competition, we prove that the frequencies (i.e. proportions) of the variants can be approached by the solution of a simpler reaction-diffusion system, through a singular limit method...

Various models of tumour growth are available in the literature. The first type describe the evolution of the cell number density when considered as a continuous visco-elastic material with growth. The second type describe the tumour as a set, and rules for the free boundary are given related to the classical Hele-Shaw model of fluid dynamics. Foll...

The nonlocal nonlinear aggregation equation in one space dimension is
investigated. In the so-called attractive case smooth solutions blow up in
finite time, so that weak measure solutions are introduced. The velocity
involved in the equation becomes discontinuous, and a particular care has to be
paid to its definition as well as the formulation of...

Various models of tumour growth are available in the literature. The first type describe the evolution of the cell number density when considered as a continuous visco-elastic material with growth. The second type describe the tumour as a set, and rules for the free boundary are given related to the classical Hele-Shaw model of fluid dynamics. Foll...

This paper is devoted to numerical simulations of electronic transport in nanoscale semiconductor devices for which charged carriers are extremely confined in one direction. In such devices, like DG-MOSFETs, the subband decomposition method is used to reduce the dimensionality of the problem. In the transversal direction electrons are confined and...

This paper deals with analysis and numerical simulations of a one-dimensional
two-species hyperbolic aggregation model. This model is formed by a system of
transport equations with nonlocal velocities, which describes the aggregate
dynamics of a two-species population in interaction appearing for instance in
bacterial chemotaxis. Blow-up of classic...

Les travaux menés portent sur la modélisation mathématique, l'analyse et la simulation numérique de systèmes d'équations aux dérivées partielles en application à la physique et à la biologie. Plus précisément, des modèles décrivant le mouvement d'électrons, l'agrégation de bactéries et la croissance cellulaire ont été considérés.
Tout d'abord, nou...

Existence and uniqueness of global in time measure solution for the
multidimensional aggregation equation is analyzed. Such a system can be written
as a continuity equation with a velocity field computed through a
self-consistent interaction potential. In Carrillo et al. (Duke Math J (2011)),
a well-posedness theory based on the geometric approach...

In this paper, we propose a kinetic model describing the collective motion by
chemotaxis of two species in interaction emitting the same chemoattractant.
Such model can be seen as a generalisation to several species of the
Othmer-Dunbar-Alt model which takes into account the run-and-tumble process of
bacteria. Existence of weak solutions for this t...

We focus in this work on the numerical discretization of the one dimensional
aggregation equation $\pa_t\rho + \pa_x (v\rho)=0$, $v=a(W'*\rho)$, in the
attractive case. Finite time blow up of smooth initial data occurs for
potential $W$ having a Lipschitz singularity at the origin. A numerical
discretization is proposed for which the convergence to...

The propagation of unstable interfaces is at the origin of remarkable patterns that are observed in various areas of science as chemical reactions, phase transitions, and growth of bacterial colonies. Since a scalar equation generates usually stable waves, the simplest mathematical description relies on two-by-two reaction-diffusion systems. The au...

Several mathematical models of tumor growth are now commonly used to explain
medical observations and predict cancer evolution based on images. These models
incorporate mechanical laws for tissue compression combined with rules for
nutrients availability which can differ depending on the situation under
consideration, in vivo or in vitro. Numerical...

We formulate a Hele-Shaw type free boundary problem for a tumor growing under
the combined effects of pressure forces, cell multiplication and active motion,
the latter being the novelty of the present paper. This new ingredient is
considered here as a standard diffusion process. The free boundary model is
derived from a description at the cell lev...

We consider one dimensional coupled classical-quantum models for quantum semiconductor device simulations. The coupling occurs in the space variable : the domain of the device is divided into a region with strong quantum effects (quantum zone) and a region where quantum effects are negligible (classical zone). In the classical zone, transport in di...

The starting point of the study is the so-called Othmer-Dunbar-Alt kinetic model describing chemotaxis of cells. This mesoscopic approach allows to take into account the run-and-tumble process observed during chemotactic motion. A diffusive limit of such system leads to macroscopic equation of Keller-Segel type. A hyperbolic approximation can also...

Existence and uniqueness of global in time measure solution for a one
dimensional nonlinear aggregation equation is considered. Such a system can be
written as a conservation law with a velocity field computed through a
selfconsistant interaction potential. Blow up of regular solutions is now well
established for such system. In Carrillo et al. (Du...

In the recent biomechanical theory of cancer growth, solid tumors are considered as liquid-like materials comprising elastic components. In this fluid mechanical view, the expansion ability of a solid tumor into a host tissue is mainly driven by either the cell diffusion constant or the cell division rate, with the latter depending on the local cel...

Existence and uniqueness of global in time measure solution for a one
dimensional nonlinear aggregation equation is considered. Such a system
can be written as a conservation law with a velocity field computed
through a selfconsistant interaction potential. Blow up of regular
solutions is now well established for such system. In Carrillo et al.
(Du...

A coupled quantum-classical model describing the transport of electrons
confined in nanoscale semiconductor devices is considered. Using the subband
decomposition approach allows to separate the transport directions from the
confinement direction. The motion of the gas in the transport direction is
assumed to be classical. Then a hierarchy of adiab...

We propose a hybrid classical-quantum model to study the motion of electrons in ultra-scaled confined nanostructures. The transport of charged particles, considered as one dimensional, is described by a quantum effective mass model in the active zone coupled directly to a drift-diffusion problem in the rest of the device. We explain how this hybrid...

The hydrodynamic limit for a kinetic model of chemotaxis is investigated. The limit equation is a non local conservation law, for which finite time blow-up occurs, giving rise to measure-valued solutions and discontinuous velocities. An adaptation of the notion of duality solutions, introduced for linear equations with discontinuous coefficients, l...

In this work, we consider the computation of the boundary conditions for the linearized
Euler–Poisson derived from the BGK kinetic model in the small mean free path regime.
Boundary layers are generated from the fact that the incoming kinetic flux might be far
from the thermodynamical equilibrium. In [2], the authors propose a method to compute
num...

We propose in this paper to derive and analyze a self-consistent model
describing the diffusive transport in a nanowire. From a physical point of
view, it describes the electron transport in an ultra-scaled confined
structure, taking in account the interactions of charged particles with
phonons. The transport direction is assumed to be large compar...

We investigate existence and uniqueness of duality solutions for a scalar
conservation law with a nonlocal interaction kernel. Following the work of
Bouchut and James (Comm. Partial Diff. Eq., 24, 1999), a notion of duality
solution for such a nonlinear system is proposed, for which we do not have
uniqueness. Then we prove that a natural definition...

This work is dedicated to Naoufel Ben Abdallah, who was a talented researcher, an enthusiastic supervisor and a generous person. Abstract We propose in this paper to derive and analyze a self-consistent model describing the diffusive transport in a nanowire. From a physical point of view, it describes the elec-tron transport in an ultra-scaled conf...

Recent experiments for swarming of the bacteria Bacillus subtilis on nutrient-rich media show that these cells are able to proliferate and spread out in colonies exhibiting complex patterns as dendritic ramifications. Is it possible to explain this process with a model that does not use local nutrient depletion?
We present a new class of models whi...

The hydrodynamic limit of a one dimensional kinetic model describing chemotaxis is investigated. The limit system is a conservation law coupled to an elliptic problem for which the macroscopic velocity is possibly discontinuous. Therefore, we need to work with measure-valued densities. After recalling a blow-up result in finite time of regular solu...

How can repulsive and attractive forces, acting on a conservative system, create stable travelling patterns or branching instabilities? We have proposed to study this question in the framework of the hyperbolic Keller–Segel system with logistic sensitivity. This is a model system motivated by experiments on cell communities auto-organization, a fie...

This paper is devoted to numerical simulations of a kinetic model describing chemotaxis. This kinetic framework has been investigated since the 80’s when experimental observations have shown that the motion of bacteria is due to the alternance of ‘runs and tumbles’. Since parabolic and hyperbolic models do not take into account the microscopic move...

Aquantum-classical coupled system which models the diffusive transport of electrons partially confined in semiconductors
nanostructures was presented in Ben Abdallah and Méhats (Proc. Edinb. Math. Soc. 49:513–549, 2006). In this model, electrons are assumed to behave like wave in the confinement direction and to have a classical behaviour
in a diff...

The modelling and the numerical resolution of the electrical charging of a spacecraft in interaction with the Earth magnetosphere is considered. It involves the Vlasov-Poisson system, endowed with non standard boundary conditions. We discuss the pros and cons of several numerical methods for solving this system, using as benchmark a simple 1D model...

In this work we present the mathematical modeling and the simulation of the diffusive transport of an electron gas confined
in a nanostructure. A coupled quantum-classical system is considered, where the coupling occurs in the momentum variable:
the electrons are like point particles in the direction parallel to the gas, while they behave like wave...

The paper is devoted to the analysis of a drift-diffusion-Schrödinger–Poisson (DDSP) system. From the physical point of view, it describes the transport of a quasi-bidimensional electron gas confined in a nanostructure. Existence, uniqueness and long-time behavior of a weak solution were already obtained in Ref. 8 for constant scalar diffusion matr...

A self-consistent model for charged particles, accounting for quantum con-finement, diffusive transport and electrostatic interaction is considered. The electrostatic potential is a solution of a three dimensional Poisson equation with the particle density as the source term. This density is the product of a two dimensional surface density and that...

A self-consistent model for charged particles, accounting for quantum confinement, diffusive transport and electrostatic interaction is considered. In this coupled quantum-classical system, the coupling occurs in the momentum variable: the electrons are like point particles in the direction parallel to the gas (classical transport) while they behav...

In this note we analyze the long time behavior of a drift-diffusion-Poisson system with a symmetric definite positive diffusion matrix, subject to Dirichlet boundary conditions. This system models the transport of electrons in semiconductor or plasma devices. By using a quadratic relative entropy obtained by keeping the lowest order term of the log...

A Drift-Diusion-Sc hrödinger-Poisson system is presented, which models the transport of a quasi bidimensional electron gas confined in a nanostructure. We prove the existence of a unique solution to this nonlinear system. The proof makes use of some a priori estimates due to the physical structure of the problem, and also involves the resolution of...

## Citations

... Following ideas in e.g. [4], [27], we model the mosquito population by a partially degenerate reaction-diffusion system for time t > 0, position x ∈ R: ...

... Truly two-dimensional well-balanced schemes are not much developed yet. We mention the recent paper [1], and [9] in the particular case of radiative transfer equation. However, the case at hand can be seen as a one dimension problem by introducing a new variable. ...

... The main result in [4] shows that if the initial wild mosquitoes distribution behaves as 1 R− and we release enough sterile males in some compact set (ct, L + ct) with a speed c < 0, then the wild population remains close to 0 in the set {x > L + ct} thanks to the assumed natural dynamics of the mosquitoes. We also quote [1,2], which was done before [4] where the authors studied the analogous system of reaction-diffusion equations to (1.1) in a bistable context taking into account the strong Allee effects. They proved that for large enough constant releases in a bounded interval, there exists a barrier that blocks the invasion of mosquitoes. ...

... The main result in [4] shows that if the initial wild mosquitoes distribution behaves as 1 R− and we release enough sterile males in some compact set (ct, L + ct) with a speed c < 0, then the wild population remains close to 0 in the set {x > L + ct} thanks to the assumed natural dynamics of the mosquitoes. We also quote [1,2], which was done before [4] where the authors studied the analogous system of reaction-diffusion equations to (1.1) in a bistable context taking into account the strong Allee effects. They proved that for large enough constant releases in a bounded interval, there exists a barrier that blocks the invasion of mosquitoes. ...