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In this paper we investigate an optimal control problem involving a toy model for the protection on a crop field. Precisely, we consider a protection on a crop field and we want to place intervention zones represented by a control, in order to maximise the protection on the field during a given period. Using a relaxation method, we prove that there exists a control which maximises the protection and, moreover, it must be a bang-bang control. Furthermore, with additional assumptions on the crop field geometry, some results on the shape of the optimal intervention are proved using comparison results for elliptic equations via Schwarz and Steiner symmetrizations. Finally, some numerical simulations are performed in order to illustrate those results., Comment: 21 pages

In this article, we investigate a competitive reaction-diffusion system modelling the interaction between several species of mosquitoes. In particular, it has been observed that in tropical regions, Aedes aegypti mosquitoes are well established in urban area whereas Aedes albopictus mosquitoes spread widely in forest region. The aim of this paper is to propose a simple mathematical system modeling this segregation phenomenon. Moreprecisely, after modeling the dynamics by a competitive reaction-diffusion system with spatial heterogeneity, we prove that when there is a strong competition between these two species of mosquitoes, solutions to this system converge in long time to segregated stationary solutions. Then we study the influence of this strong competition on the success of a population replacement strategy using Wolbachia bacteria. Our theoretical results are also illustrated by some numerical simulations.

The Sterile Insect Technique (SIT) is one of the sustainable strategies for the control of disease vectors, which consists of releasing sterilized males that will mate with the wild females, resulting in a reduction and, eventually a local elimination, of the wild population. The implementation of the SIT in the field can become problematic when there are inaccessible areas where the release of sterile insects cannot be carried out directly, and the migration of wild insects from these areas to the treated zone may influence the efficacy of this technique. However, we can also take advantage of the movement of sterile individuals to control the wild population in these unreachable places. In this paper, we derive a two-patch model for Aedes mosquitoes where we consider the discrete diffusion between the treated area and the inaccessible zone. We investigate two different release strategies (constant and impulsive periodic releases), and by using the monotonicity of the model, we show that if the number of released sterile males exceeds some threshold, the technique succeeds in driving the whole population in both areas to extinction. This threshold depends on not only the biological parameters of the population but also the diffusion between the two patches.

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 model of a replacement strategy, consisting in rearing and releasing Wolbachia-infected mosquitoes to replace the wild population. We investigate the question of optimizing the release protocol to have the most effective replacement when the environment is heterogeneous. In other words we focus on the question: where to release, given an inhomogeneous environment, in order to maximize the replacement across the domain. To do so, we consider a simple scalar model in which we assume that the carrying capacity is space dependent. Then, we investigate the existence of an optimal release profile and prove some interesting properties. In particular, neglecting the mobility of mosquitoes and under some assumptions on the biological parameters, we characterize the optimal releasing strategy for a short time horizon, and provide a way to reduce to a one-dimensional optimization problem the case of a long time horizon. Our theoretical results are illustrated with several numerical simulations.

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 method, where the boundary condition takes into account the inflow and outflow of individuals through the boundary. Using phase-plane analysis, the present paper studies the existence and properties of non-constant steady-state solutions depending on several parameters. Then, we use the principle of linearized stability to prove some sufficient conditions for their stability. We show that the long-time efficiency of this control method depends strongly on the size of the treated zone and the migration rate. To illustrate these theoretical results, we provide some numerical simulations in the framework of mosquito population control.

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 of the population is modeled through a bistable reaction diffusion equation in an one dimensional setting and both strategies are based on the same idea : we act on a moving interval. The action of the first strategy is to kill as many individuals as we can in this moving interval. The action of the second strategy is to release sterile males in this moving interval. For both strategies, we manage to generate traveling waves that propagate in the opposite direction relative to the one of the natural invasive traveling wave. These cases correspond to succeeding in eradicating the invasive species. Furthermore, for the first strategy, we fully characterize the minimal size of the interval. All the results are illustrated by numerical simulations.

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 final time. For this reason, measure-valued solutions have been defined. In this paper, we investigate an approximation of such measure-valued solutions thanks to a relaxation limit in the spirit of Jin and Xin. We study the convergence of this approximation and give a rigorous estimate of the speed of convergence in one dimension with the Newtonian potential. We also investigate the numerical discretization of this relaxation limit by uniformly accurate schemes.

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 greatly reduces the transmission of the virus to the humans. Spatial releases are then sought in such a way that the infected mosquito population invades the uninfected mosquito population. Assuming very high mosquito fecundity rates, we first introduce an asymptotic model on the proportion of infected mosquitoes and then an optimal control problem to determine the best spatial strategy to achieve these releases. We then analyze this problem, including the optimality of natural candidates and carry out first numerical simulations in one dimension of space to illustrate the relevance of our approach., Comment: ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, In press

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 advantage being specific to the targeted species, unlike the use of pesticides.In this article, we study the optimal release strategies in order to maximize the efficiency of this technique.We consider simplified models that describe the dynamics of eggs, males, females and sterile males in order to optimize the release protocol. We determine in a precise way optimal strategies, which allows us to tackle numerically the underlying optimization problem in a very simple way. We also present some numerical simulations to illustrate our results.

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 an individual of a given trait to be infected by an individual with another trait. Our goal is to derive and study the final size equation fulfilled by the limit distribution of the population. We show that this limit exists and satisfies the final size equation. The main contribution is to prove the uniqueness of this solution among the distributions smaller than the initial condition. We also establish that the dominant eigenvalue of the next-generation operator (whose initial value is equal to the basic reproduction number) decreases along every trajectory until a limit smaller than 1. The results are shown to remain valid in presence of diffusion term. They generalize previous works corresponding to finite number of traits (including metapopulation models) or to rank 1 contact matrix (modeling e.g. susceptibility or infectivity presenting heterogeneity independently of one another).

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 strategy is likely the best policy in the worst case where no vaccine or therapy is found. In order to keep the number of newly infected persons to a minimum, a possible strategy is to apply strict containment measures, so that the number of susceptible individuals remains close to herd immunity. Such an action is unrealistic since containment can only last for a finite amount of time and is never total. In this article, using a classical SIR model, we determine the (partial or total) containment strategy on a given finite time interval that maximizes the number of susceptible individuals over an infinite horizon, or equivalently that minimizes the total infection burden during the curse of the epidemic. The existence and uniqueness of the optimal strategy is proved and the latter is fully characterized. If applicable in practice, such a strategy would lead theoretically to an increase by 30% of the proportion of susceptible on an infinite horizon, for a containment level corresponding to the sanitary measures put in place in France from March to May 2020. We also analyze the minimum intervention time to reach a fixed distance from herd immunity, and show the relationship with the previous problem. Simulations are provided that illustrate and validate the theoretical results.

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, it has been shown that the macroscopic or mesoscopic quantities computed from this system converge to the Keller-Segel system at diffusive scaling or to the velocity-jump kinetic system for chemotaxis at hyperbolic scaling. Then we pay attention to propose numerical schemes uniformly accurate with respect to the scaling parameter. We show that these schemes converge to some limiting schemes which are consistent with the limiting macroscopic or kinetic system. This study is illustrated with some numerical simulations and comparisons with Monte Carlo simulations.

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 showing that performing the sterile insect technique on a band large enough may stop reinvasion.

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 of view, this model consists of a 5x5 semi-linear hyperbolic system. In the literature similar models neglect the epithelial layers. In this paper, we show rigorously that such models may be obtained by assuming that the permeabilities between lumen and epithelium are large. Indeed we show that when these grow, solutions of the 5x5 system converge in a certain way to solutions of a reduced 3x3 system where no epithelial layer is present. The problem is dened on a bounded spacial domain with initial and boundary data. Establishing BV compactness forces to introduce initial layers and to handle carefully the presence of lateral boundaries.

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 interstitium. In a specific range of parameters, we show that explicitly accounting for transport across the apical and basolateral membranes of epithelial cells, instead of assuming a single barrier, affects the axial concentration gradient , an essential determinant of the urinary concentrating capacity. We present the solution related to the stationary system, and we perform numerical simulations to understand the physiological behaviour of the system. We prove that when time grows large, our dynamic model converges towards the stationary system at an exponential rate. In order to prove rigorously this global asymptotic stability result, we study eigen-problems of an auxiliary linear operator and its dual.

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 reproductive alterations are induced in mosquitoes, including cytoplasmic incompatibility. Some of these {\itshape Wolbachia} bacteria greatly reduce the ability of insects to become infected with viruses such as the dengue ones, cutting down their vector competence and thus effectively stopping local dengue transmission.The behavior of infected and uninfected mosquitoes is assumed to be driven by a compartmental system enriched with the presence of an internal control source term standing for releases of infected mosquitoes, distributed in time. We model and design an optimal releasing control strategy with the help of a least square problem. In a nutshell, one wants to minimize the number of uninfected mosquitoes at a given horizon of time, under some relevant biological constraints. We derive properties of optimal controls, highlight a limit problem providing useful asymptotic properties of optimal controls. We numerically illustrate the relevance of our approach.

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 establishing some a priori estimates. Then, we investigate the hyperbolic limit after rescaling of the kinetic system. It leads to a macroscopic system of Cattaneo type. The rigorous derivation is established thanks to a compactness method.

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 population by a population carrying a bacteria, called \textit{Wolbachia}, blocking the transmission of viruses from mosquitoes to human. This paper is concerned with the question of optimizing the release protocol for these two strategies with the aim of getting as close as possible to the objectives. Starting from a mathematical model describing the dynamics of the population, we include the control function and introduce the cost functional for both \textit{population replacement} and \textit{Sterile Insect Technique} problems. Next, we establish some properties of the optimal control and illustrate them with some numerical simulations.

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 advantage that traveling pulse solutions exist as observed experimentally. It has attracted the attention of many authors recently. We design and prove a general derivation of the FLKS departing from a kinetic model under stiffness assumption of the chemotactic response and rescaling the kinetic equation according to this stiffness parameter. Unlike the classical Keller-Segel system, solutions of the FLKS system do not blow-up in finite or infinite time. Then we investigate the existence of radially symmetric steady state and long time behaviour of this flux limited Keller-Segel system.

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 of egg hatching enhancement by larvae which produces such oscillations that conveys a possible explanation. We propose both a theoretical analysis, based on slow-fast dynamics and Hopf bifurcation, and numerical investigations in order to shed some light on the mechanisms at work in this model.

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 velocity field may have discontinuities. Based on recent results of existence and uniqueness of a Filippov flow for this type of equations, we study an upwind finite volume numerical scheme and we prove that it is convergent at order 1/2 in Wasserstein distance. This result is illustrated by numerical simulations, which show the optimality of the order of convergence.

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 method applied to such kinetic equations leads to time-marching schemes involving a '' scattering S-matrix '' , itself derived from a normal modes decomposition of the stationary solution. One common feature these models share is the type of diffusive approximation: their macroscopic densities solve drift-diffusion systems, for which a distinguished numerical scheme is Il'in/Scharfetter-Gummel's '' exponential fitting '' discretization. We prove that the well-balanced schemes relax, within a parabolic rescaling, towards the Il'in exponential-fitting discretization by means of an appropriate decomposition of the S-matrix. This is the so-called asymptotic preserving (or uniformly accurate) property.

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.

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 diseases. Following works by M. Turelli [4] and subsequently M. Strugarek et al. [21, 22], we consider a simple scalar reaction-diffusion model describing the evolution of the proportion of infected mosquitoes, sufficient to reveal the bistable nature of the Wolbachia dynamics. A simple distributed feedback law is proposed, whose application on a compact domain during finite time is shown to be sufficient to invade the whole space. The corresponding stabilization result is established for any space dimension.

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. This constraint is important to be satisfied in many applications. One way to guarantee this non-overlapping constraint is to choose a singular pressure law. The aim of this paper is to prove that, although the pressure law has a singularity, the incompressible limit leads to the same Hele-Shaw free boundary problem.

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 is bistable. After quantification of the propagules, a second question of major interest is the invasive power. How far can such an invading front go, and what can stop it? We rigorously show that a heterogeneous environment inducing a strong enough population gradient can stop an invading front, which will converge in this case to a stable front. We characterize the critical population jump, and also prove the existence of unstable fronts above the stable (blocking) fronts. Being above the maximal unstable front enables an invading front to clear the obstacle and propagate further. We are particularly interested in the case of artificial Wolbachia infection, used as a tool to fight arboviruses., Comment: 33 pages, 9 figures

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 derive hyperbolic models at the macroscopic level. The resulting macroscopic equations show how the macroscopic tissue behavior can be described by hyperbolic systems which seem the most natural in this context. We propose also an asymptotic-preserving well-balanced scheme for the one-dimensional hyperbolic model, in the two dimensional case, we consider a time splitting method between the conservative part and the source term where the conservative equation is approximated by the Lax-Friedrichs scheme.

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 fecundity or lifespan, while highly reducing vector competence for the main arboviruses .We consider and answer the following questions: 1) what should be the initial condition (i.e. size of the initial mosquito population) to have invasion with one mosquito release source? We note that it is hard to have an invasion in such case. 2) How many release points does one need to have sufficiently high probability of invasion? 3) What happens if one accounts for uncertainty in the release protocol (e.g. unequal spacing among release points)? We build a framework based on existing reaction-diffusion models for the uncertainty quantification in this context, obtain both theoretical and numerical lower bounds for the probability of release success and give new quantitative results on the one dimensional case.

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 model based on an accurate description of the mesoscopic run-and-tumble process. We extend this approach to study the behavior of the interaction between two populations of E. Coli. Separately, each population travels with its own speed in the channel. When put together, a synchronization of the speed of the traveling pulses can be observed. We show that this synchronization depends on the fraction of the fast population. Our approach is based on mathematical analysis of a macroscopic model of partial differential equations. Numerical simulations in comparison with experimental observations show qualitative agreement., Comment: in PLoS Computational Biology, Public Library of Science, 2016

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 upwind scheme in the Wasserstein distances. More precisely, we prove that in this setting the convergence order is 1/2. We also show the optimality of this result. In the appendix, we show that this result also applies to other "diffusive" "first order" schemes and to a forward semi-Lagrangian scheme.

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 and a relative compactness argument. As an example of application, we retrieve the classical bistable equation for Wolbachia's spread into an arthropod population from a system modeling interaction between infected and uninfected individuals.

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 the corresponding flux. When this is done, the notion of duality solutions allows to obtain global in time existence and uniqueness for measure solutions. An upwind finite volume scheme is also analyzed, and the convergence towards the unique solution is proved. Numerical examples show the dynamics of the solutions after the blow up time., Comment: in CoToCoLa, Jan 2015, Besan{\c c}on, France. arXiv admin note: text overlap with arXiv:1303.5836

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 classical solutions occurs in finite time. This raises the question to define measure-valued solutions for this system. To this aim, we use the duality method developed for transport equations with discontinuous velocity to prove the existence and uniqueness of measure-valued solutions. The proof relies on a stability result. In addition, this approach allows to study the hyperbolic limit of a kinetic chemotaxis model. Moreover, we propose a finite volume numerical scheme whose convergence towards measure-valued solutions is proved. It allows for numerical simulations capturing the behaviour after blow up. Finally, numerical simulations illustrate the complex dynamics of aggregates until the formation of a single aggregate: after blow-up of classical solutions, aggregates of different species are synchronising or nonsynchronising when collide, that is move together or separately, depending on the parameters of the model and masses of species involved.

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 frequencies depending on the path-wise gradient of chemotactic signal. More recently, the biochemical pathways regulating the flagellar motors were uncovered. This knowledge gave rise to a second class of kinetic-transport equations, that takes into account an intra-cellular molecular content and which relates the tumbling frequency to this information. It turns out that the tumbling frequency depends on the chemotactic signal, and not on its gradient. For these two classes of models, macroscopic equations of Keller-Segel type, have been derived using diffusion or hyperbolic rescaling. We complete this program by showing how the first class of equations can be derived from the second class with molecular content after appropriate rescaling. The main difficulty is to explain why the path-wise gradient of chemotactic signal can arise in this asymptotic process. Randomness of receptor methylation events can be included, and our approach can be used to compute the tumbling frequency in presence of such a noise.

Various models of tumor growth are available in the litterature. A first class describes the evolution of the cell number density when considered as a continuous visco-elastic material with growth. A second class, describes the tumor as a set and rules for the free boundary are given related to the classical Hele-Shaw model of fluid dynamics. Following the lines of previous papers where the material is described by a purely elastic material, or when active cell motion is included, we make the link between the two levels of description considering the 'stiff pressure law' limit. Even though viscosity is a regularizing effect, new mathematical difficulties arise in the visco-elastic case because estimates on the pressure field are weaker and do not imply immediately compactness. For instance, traveling wave solutions and numerical simulations show that the pressure may be discontinous in space which is not the case for the elastic case., Comment: 17 pages

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 of gradient flows in measure metric spaces has been developed for mildly singular potentials at the origin under the basic assumption of being lambda-convex. We propose here an alternative method using classical tools from PDEs. We show the existence of a characteristic flow based on Filippov's theory of discontinuous dynamical systems such that the weak measure solution is the pushforward measure with this flow. Uniqueness is obtained thanks to a contraction argument in transport distances using the lambda-convexity of the potential. Moreover, we show the equivalence of this solution with the gradient flow solution. Finally, we show the convergence of a numerical scheme for general measure solutions in this framework allowing for the simulation of solutions for initial smooth densities after their first blow-up time in Lp-norms., Comment: 33 pages

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 two-species kinetic model is studied and the convergence of its diffusive limit towards a macroscopic model of Keller-Segel type is analysed.

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 towards duality solutions of the aggregation equation is proved. It relies on a careful choice of the discretized macroscopic velocity $v$ in order to give a sense to the product $v \rho$. Moreover, using the same idea, we propose an asymptotic preserving scheme for a kinetic system in hyperbolic scaling converging towards the aggregation equation in hydrodynamical limit. Finally numerical simulations are provided to illustrate the results.

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, growth of bacterial colonies. Since a scalar equation generates usually stable waves, the simplest mathematical description relies on two by two reaction-diffusion systems. Our interest is the extension of the Fisher/KPP equation to a two species reaction which represents reactant concentration and temperature when used for flame propagation, bacterial population and nutrient concentration when used in biology. We study in which circumstances instabilities can occur and in particular the effect of dimension. It is observed numerically that spherical waves can be unstable depending on the coefficients. A simpler mathematical framework is to study transversal instability, that means a one dimensional wave propagating in two space dimensions. Then, explicit analytical formulas give explicitely the range of paramaters for instability., Comment: 13 pages

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 solutions exhibit, as expected from medical observations, a proliferative rim and a necrotic core. However, their precise profiles are rather complex, both in one and two dimensions., Comment: 25 pages

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 level using the asymptotic of a stiff pressure limit. Compared to the case when active motion is neglected, the pressure satisfies the same complementarity Hele-Shaw type formula. However, the cell density is smoother (Lipschitz continuous), while there is a deep change in the free boundary velocity, which is no longer given by the gradient of the pressure, because some kind of 'mushy region' prepares the tumor invasion.

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. (Duke Math J (2011)), a theory of existence and uniqueness based on the geometric approach of gradient flows on Wasserstein space has been developped. We propose in this work to establish the link between this approach and duality solutions. This latter concept of solutions allows in particular to define a flow associated to the velocity field. Then an existence and uniqueness theory for duality solutions is developped in the spirit of James and Vauchelet (NoDEA (2013)). However, since duality solutions are only known in one dimension, we restrict our study to the one dimensional case.