arXiv subjects

Vincent Calvez

Publications and source records attributed to Vincent Calvez.

At least 37 records · Page 2Linked to original sources

arXiv2018

A well-balanced scheme for chemotactic travelling waves at the mesoscopic scale

We investigate numerically a model consisting in a kinetic equation for the biased motion of bacteria following a run-and-tumble process, coupled with two reaction-diffusion equations for chemical signals. This model exhibits asymptotic propagation at a constant speed. In particular, it admits travelling wave solutions. To capture this propagation, we propose a well-balanced numerical scheme based on Case's elementary solutions for the kinetic equation, and L-splines for the parabolic equations. We use this scheme to explore the Cauchy problem for various parameters. Some examples far from the diffusive regime lead to the coexistence of two waves travelling at different speeds. Numerical tests support the hypothesis that they are both locally asymptotically stable. Interestingly, the exploration of the bifurcation diagram raises counter-intuitive features.

math.AP↗

arXiv2018

Concentration waves of chemotactic bacteria: the discrete velocity case

The existence of travelling waves for a coupled system of hyperbolic/ parabolic equations is established in the case of a finite number of velocities in the kinetic equation. This finds application in collective motion of chemotactic bacteria. The analysis builds on the previous work by the first author (arXiv:1607.00429) in the case of a continuum of velocities. Here, the proof is specific to the discrete setting, based on the decomposition of the population density in special Case's modes. Some counter-intuitive results are discussed numerically, including the coexistence of several travelling waves for some sets of parameters, as well as the possible non-existence of travelling waves.

math.AP↗

arXiv2018

Uniqueness of the viscosity solution of a constrained Hamilton-Jacobi equation

In quantitative genetics, viscosity solutions of Hamilton-Jacobi equations appear naturally in the asymptotic limit of selection-mutation models when the population variance vanishes. They have to be solved together with an unknown function I(t) that arises as the counterpart of a non-negativity constraint on the solution at each time. Although the uniqueness of viscosity solutions is known for many variants of Hamilton-Jacobi equations, the uniqueness for this particular type of constrained problem was not resolved, except in a few particular cases. Here, we provide a general answer to the uniqueness problem, based on three main assumptions: convexity of the Hamiltonian function H(I, x, p) with respect to p, monotonicity of H with respect to I, and BV regularity of I(t).

math.AP↗

arXiv2017

Traveling Wave and Aggregation in a Flux-Limited Keller-Segel Model

Flux-limited Keller-Segel (FLKS) model has been recently derived from kinetic transport models for bacterial chemotaxis and shown to represent better the collective movement observed experimentally. Recently, associated to the kinetic model, a new instability formalism has been discovered related to stiff chemotactic response. This motivates our study of traveling wave and aggregation in population dynamics of chemotactic cells based on the FLKS model with a population growth term. Our study includes both numerical and theoretical contributions. In the numerical part, we uncover a variety of solution types in the one-dimensional FLKS model additionally to standard Fisher/KPP type traveling wave. The remarkable result is a counter-intuitive backward traveling wave, where the population density initially saturated in a stable state transits toward an unstable state in the local population dynamics. Unexpectedly, we also find that the backward traveling wave solution transits to a localized spiky solution as increasing the stiffness of chemotactic response. In the theoretical part, we obtain a novel analytic formula for the minimum traveling speed which includes the counterbalancing effect of chemotactic drift vs. reproduction/diffusion in the propagating front. The front propagation speeds of numerical results only slightly deviate from the minimum traveling speeds, except for the localized spiky solutions, even for the backward traveling waves. We also discover an analytic solution of unimodal traveling wave in the large-stiffness limit, which is certainly unstable but exists in a certain range of parameters.

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arXiv2017

Existence of recombination-selection equilibria for sexual populations

We study a birth and death model for the adapatation of a sexual population to an environment. The population is structured by a phenotypical trait, and, possibly, an age variable. Recombination is modeled by Fisher's infinitesimal operator. We prove the existence of principal eigenelements for the corresponding eigenproblem. As the infinitesimal operator is 1-homogeneous but nor linear nor monotone, the general Krein-Rutman theory cannot be applied to this problem.

q-bio.PE↗

arXiv2016

The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime

We consider an aggregation-diffusion equation modelling particle interaction with non-linear diffusion and non-local attractive interaction using a homogeneous kernel (singular and non-singular) leading to variants of the Keller-Segel model of chemotaxis. We analyse the fair-competition regime in which both homogeneities scale the same with respect to dilations. Our analysis here deals with the one-dimensional case and provides an almost complete classification. In the singular kernel case and for critical interaction strength, we prove uniqueness of stationary states via a variant of the Hardy-Littlewood-Sobolev inequality. Using the same methods, we show uniqueness of self-similar profiles in the sub-critical case by proving a new type of functional inequality. Surprisingly, the same results hold true for any interaction strength in the non-singular kernel case. Further, we investigate the asymptotic behaviour of solutions, proving convergence to equilibrium in Wasserstein distance in the critical singular kernel case, and convergence to self-similarity for sub-critical interaction strength, both under a uniform stability condition. Moreover, solutions converge to a unique self-similar profile in the non-singular kernel case. Finally, we provide a numerical overview for the asymptotic behaviour of solutions in the full parameter space demonstrating the above results. We also discuss a number of phenomena appearing in the numerical explorations for the diffusion-dominated and attraction-dominated regimes.

math.AP↗

arXiv2016

Hyperbolic traveling waves driven by growth

We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed $ε^{-1}$ ($ε\textgreater{}0$), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter $ε$: for small $ε$ the behaviour is essentially the same as for the diffusive Fisher-KPP equation. However, for large $ε$ the traveling front with minimal speed is discontinuous and travels at the maximal speed $ε^{-1}$. The traveling fronts with minimal speed are linearly stable in weighted $L^2$ spaces. We also prove local nonlinear stability of the traveling front with minimal speed when $ε$ is smaller than the transition parameter.

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arXiv2016

Equilibria of homogeneous functionals in the fair-competition regime

We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular/smooth kernel leading to variants of the Keller-Segel model of chemotaxis. We analyse the regime in which both homogeneities scale the same with respect to dilations, that we coin as fair-competition. In the singular kernel case, we show that existence of global equilibria can only happen at a certain critical value and they are characterised as optimisers of a variant of HLS inequalities. We also study the existence of self-similar solutions for the sub-critical case, or equivalently of optimisers of rescaled free energies. These optimisers are shown to be compactly supported radially symmetric and non-increasing stationary solutions of the non-linear Keller-Segel equation. On the other hand, we show that no radially symmetric non-increasing stationary solutions exist in the smooth kernel case, implying that there is no criticality. However, we show the existence of positive self-similar solutions for all values of the parameter under the condition that diffusion is not too fast. We finally illustrate some of the open problems in the smooth kernel case by numerical experiments.

math.AP↗

arXiv2016

Chemotactic waves of bacteria at the mesoscale

The existence of travelling waves for a model of concentration waves of bacteria is investigated. The model consists in a kinetic equation for the biased motion of cells following a run-and-tumble process, coupled with two reaction-diffusion equations for the chemical signals. Strong mathematical difficulties arise in comparison with the diffusive regime which was studied in a previous work. The cornerstone of the proof consists in establishing monotonicity properties of the spatial density of cells. Travelling waves exist under certain conditions on the parameters. Counter-examples to both existence and uniqueness are found numerically after careful analysis of the discrete velocity problem.

math.AP↗

arXiv2016

Blow-up phenomena for gradient flows of discrete homogeneous functionals

We investigate gradient flows of some homogeneous functionals in R^N , arising in the Lagrangian approximation of systems of self-interacting and diffusing particles. We focus on the case of negative homogeneity. In the case of strong self-interaction, the functional possesses a cone of negative energy. It is immediate to see that solutions with negative energy at some time become singular in finite time, meaning that a subset of particles concentrate at a single point. Here, we establish that all solutions become singular in finite time for the class of functionals under consideration. The paper is completed with numerical simulations illustrating the striking non linear dynamics when initial data have positive energy.

math.AP↗

arXiv2015

Pattern selection in a biomechanical model for the growth of walled cells

In this paper, we analyse a model for the growth of three-dimensional walled cells. In this model the biomechanical expansion of the cell is coupled with the geometry of its wall. We consider that the density of building material depends on the curvature of the cell wall, thus yield-ing possible anisotropic growth. The dynamics of the axisymmetric cell wall is described by a system of nonlinear PDE including a nonlin-ear convection-diffusion equation coupled with a Poisson equation. We develop the linear stability analysis of the spherical symmetric config-uration in expansion. We identify three critical parameters that play a role in the possible instability of the radially symmetric shape, namely the degree of nonlinearity of the coupling, the effective diffusion of the building material, and the Poisson's ratio of the cell wall. We also investigate numerically pattern selection in the nonlinear regime. All the results are also obtained for a simpler, but similar, two-dimensional model.

math.AP↗

arXiv2014

Propagation in a kinetic reaction-transport equation: travelling waves and accelerating fronts

In this paper, we study the existence and stability of travelling wave solutions of a kinetic reaction-transport equation. The model describes particles moving according to a velocity-jump process, and proliferating thanks to a reaction term of monostable type. The boundedness of the velocity set appears to be a necessary and sufficient condition for the existence of positive travelling waves. The minimal speed of propagation of waves is obtained from an explicit dispersion relation. We construct the waves using a technique of sub- and supersolutions and prove their \eb{weak} stability in a weighted $L^2$ space. In case of an unbounded velocity set, we prove a superlinear spreading. It appears that the rate of spreading depends on the decay at infinity of the velocity distribution. In the case of a Gaussian distribution, we prove that the front spreads as $t^{3/2}$.

math.AP↗

arXiv2014

Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems

We study a growth maximization problem for a continuous time positive linear system with switches. This is motivated by a problem of mathematical biology (modeling growth-fragmentation processes and the PMCA protocol). We show that the growth rate is determined by the non-linear eigenvalue of a max-plus analogue of the Ruelle-Perron-Frobenius operator, or equivalently, by the ergodic constant of a Hamilton-Jacobi (HJ) partial differential equation, the solutions or subsolutions of which yield Barabanov and extremal norms, respectively. We exploit contraction properties of order preserving flows, with respect to Hilbert's projective metric, to show that the non-linear eigenvector of the operator, or the "weak KAM" solution of the HJ equation, does exist. Low dimensional examples are presented, showing that the optimal control can lead to a limit cycle.

math.AP↗

arXiv2014

Confinement by biased velocity jumps: aggregation of Escherichia coli

We investigate a linear kinetic equation derived from a velocity jump process modelling bacterial chemotaxis in the presence of an external chemical signal centered at the origin. We prove the existence of a positive equilibrium distribution with an exponential decay at infinity. We deduce a hypocoercivity result, namely: the solution of the Cauchy problem converges exponentially fast towards the stationary state. The strategy follows [J. Dolbeault, C. Mouhot, and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. AMS 2014]. The novelty here is that the equilibrium does not belong to the null spaces of the collision operator and of the transport operator. From a modelling viewpoint it is related to the observation that exponential confinement is generated by a spatially inhomogeneous bias in the velocity jump process.

math.AP↗

arXiv2014

Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up

We investigate a particle system which is a discrete and deterministic approximation of the one-dimensional Keller-Segel equation with a logarithmic potential. The particle system is derived from the gradient flow of the homogeneous free energy written in Lagrangian coordinates. We focus on the description of the blow-up of the particle system, namely: the number of particles involved in the first aggregate, and the limiting profile of the rescaled system. We exhibit basins of stability for which the number of particles is critical, and we prove a weak rigidity result concerning the rescaled dynamics. This work is complemented with a detailed analysis of the case where only three particles interact.

math.FA↗

arXiv2013

Travelling waves for the cane toads equation with bounded traits

In this paper, we study propagation in a nonlocal reaction-diffusion-mutation model describing the invasion of cane toads in Australia. The population of toads is structured by a space variable and a phenotypical trait and the space-diffusivity depends on the trait. We use a Schauder topological degree argument for the construction of some travelling wave solutions of the model. The speed $c^*$ of the wave is obtained after solving a suitable spectral problem in the trait variable. An eigenvector arising from this eigenvalue problem gives the flavor of the profile at the edge of the front. The major difficulty is to obtain uniform $L^\infty$ bounds despite the combination of non local terms and an heterogeneous diffusivity.

math.AP↗

arXiv2013

Self-similarity in a General Aggregation-Fragmentation Problem ; Application to Fitness Analysis

We consider the linear growth and fragmentation equation with general coefficients. Under suitable conditions, the first eigenvalue represents the asymptotic growth rate of solutions, also called \emph{fitness} or \emph{Malthus coefficient} in population dynamics ; it is of crucial importance to understand the long-time behaviour of the population. We investigate the dependency of the dominant eigenvalue and the corresponding eigenvector on the transport and fragmentation coefficients. We show how it behaves asymptotically as transport dominates fragmentation or \emph{vice versa}. For this purpose we perform suitable blow-up analysis of the eigenvalue problem in the limit of small/large growth coefficient (resp. fragmentation coefficient). We exhibit possible non-monotonic dependency on the parameters, conversely to what would have been conjectured on the basis of some simple cases.

math.AP↗

arXiv2013

Numerical simulation of the dynamics of molecular markers involved in cell polarisation

A cell is polarised when it has developed a main axis of organisation through the reorganisation of its cytosqueleton and its intracellular organelles. Polarisation can occur spontaneously or be triggered by external signals, like gradients of signaling molecules ... In this work, we study mathematical models for cell polarisation. These models are based on nonlinear convection-diffusion equations. The nonlinearity in the transport term expresses the positive loop between the level of protein concentration localised in a small area of the cell membrane and the number of new proteins that will be convected to the same area. We perform numerical simulations and we illustrate that these models are rich enough to describe the apparition of a polarisome.

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