Publications
302 results found
Aceves-Sánchez P, Bailo R, Degond P, et al., 2024, Pedestrian models with congestion effects, Mathematical Models and Methods in Applied Sciences, ISSN: 0218-2025
We study the validity of the dissipative Aw–Rascle system as a macroscopic model for pedestrian dynamics. The model uses a congestion term (a singular diffusion term) to enforce capacity constraints in the crowd density while inducing a steering behavior. Furthermore, we introduce a semi-implicit, structure-preserving, and asymptotic-preserving numerical scheme which can handle the numerical solution of the model efficiently. We perform the first numerical simulations of the dissipative Aw–Rascle system in one and two dimensions. We demonstrate the efficiency of the scheme in solving an array of numerical experiments, and we validate the model, ultimately showing that it correctly captures the fundamental diagram of pedestrian flow.
Degond P, Diez A, 2023, Topological Travelling Waves of a Macroscopic Swarmalator Model in Confined Geometries, Acta Applicandae Mathematicae, Vol: 188, ISSN: 0167-8019
We investigate a new class of topological travelling-wave solutions for a macroscopic swarmalator model involving force non-reciprocity. Swarmalators are systems of self-propelled particles endowed with a phase variable. The particles are subject to coupled swarming and synchronization. In previous work, the swarmalator under study was introduced, the macroscopic model was derived and doubly periodic travelling-wave solutions were exhibited. Here, we focus on the macroscopic model and investigate new classes of two-dimensional travelling-wave solutions. These solutions are confined in a strip or in an annulus. In the case of the strip, they are periodic along the strip direction. Both of them have non-trivial topology as their phases increase by a multiple of 2 π from one period (in the case of the strip) or one revolution (in the case of the annulus) to the next. Existence and qualitative behavior of these solutions are investigated.
Barker M, Degond P, Martin R, et al., 2023, A mean field game model of firm-level innovation, Mathematical Models and Methods in Applied Sciences, Vol: 33, Pages: 929-970, ISSN: 0218-2025
Knowledge spillovers occur when a firm researches a new technology and that technology is adapted or adopted by another firm, resulting in a social value of the technology that is larger than the initially predicted private value. As a result, firms systematically under-invest in research compared with the socially optimal investment strategy. Understanding the level of under-investment, as well as policies to correct it, is an area of active economic research. In this paper, we develop a new model of spillovers, taking inspiration from the available microeconomic data. The model developed is a mean field game model, which allows for heterogeneity in the productivity of a firm and allows for a novel approach to describing sector-level spillovers. The model is constructed from a network of interacting firms, whose connections represent knowledge transfers. We prove existence and uniqueness of solutions to the model, and we conduct some initial simulations to understand how indirect spillovers contribute to the productivity of a sector.
Degond P, Frouvelle A, Merino-Aceituno S, et al., 2023, HYPERBOLICITY AND NONCONSERVATIVITY OF A HYDRODYNAMIC MODEL OF SWARMING RIGID BODIES, QUARTERLY OF APPLIED MATHEMATICS, ISSN: 0033-569X
Degond P, Manhart A, Merino-Aceituno S, et al., 2022, How environment affects active particle swarms: a case study, ROYAL SOCIETY OPEN SCIENCE, Vol: 9, ISSN: 2054-5703
Degond P, Diez A, Walczak A, 2022, Topological states and continuum model for swarmalators without force reciprocity, ANALYSIS AND APPLICATIONS, Vol: 20, Pages: 1215-1270, ISSN: 0219-5305
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- Citations: 2
Degond P, Hecht S, Romanos M, et al., 2022, Multi-species viscous models for tissue growth: incompressible limit and qualitative behaviour, JOURNAL OF MATHEMATICAL BIOLOGY, Vol: 85, ISSN: 0303-6812
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- Citations: 1
Degond P, Jin S, Zhu Y, 2022, An uncertainty quantification approach to the study of gene expression robustness, Methods and Applications of Analysis, Vol: 28, Pages: 195-220, ISSN: 1073-2772
We study a chemical kinetic system with uncertainty modeling a gene regulatory network in biology. Specifically, we consider a system of two equations for the messenger RNA and micro RNA content of a cell. Our target is to provide a simple framework for noise buffering in gene expression through micro RNA production. Here the uncertainty, modeled by random variables, enters the system through the initial data and the source term. We obtain a sharp decay rate of the solution to the steady state, which reveals that the biology system is not sensitive to the initial perturbation around the steady state. The sharp regularity estimate leads to the stability of the generalized Polynomial Chaos stochastic Galerkin (gPC‑SG) method. Based on the smoothness of the solution in the random space and the stability of the numerical method, we conclude the gPC‑SG method has spectral accuracy. Numerical experiments are conducted to verify the theoretical findings.
Degond P, Frouvelle A, Liu J-G, 2022, FROM KINETIC TO FLUID MODELS OF LIQUID CRYSTALS BY THE MOMENT METHOD, KINETIC AND RELATED MODELS, Vol: 15, Pages: 417-465, ISSN: 1937-5093
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- Citations: 1
Hecht S, Perez-Mockus G, Schienstock D, et al., 2022, Mechanical constraints to cell-cycle progression in a pseudostratified epithelium, CURRENT BIOLOGY, Vol: 32, Pages: 2076-+, ISSN: 0960-9822
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- Citations: 1
Barker M, Degond P, Wolfram M-T, 2022, Comparing the best reply strategy and mean field games: the stationary case, European Journal of Applied Mathematics, Vol: 33, Pages: 79-110, ISSN: 0956-7925
Mean-field games (MFGs) and the best-reply strategy (BRS) are two methods of describing competitive optimisation of systems of interacting agents. The latter can be interpreted as an approximation of the respective MFG system. In this paper, we present an analysis and comparison of the two approaches in the stationary case. We provide novel existence and uniqueness results for the stationary boundary value problems related to the MFG and BRS formulations, and we present an analytical and numerical comparison of the two paradigms in some specific modelling situations.
Degond P, Diez A, Na M, 2022, Bulk Topological States in a New Collective Dynamics Model, SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS, Vol: 21, Pages: 1455-1494, ISSN: 1536-0040
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- Citations: 1
Danesin C, Ferreira MA, Degond P, et al., 2021, Anteroposterior elongation of the chicken anterior trunk neural tube is hindered by interaction with its surrounding tissues, CELLS & DEVELOPMENT, Vol: 168, ISSN: 2667-2901
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- Citations: 1
Degond P, Jin S, Zhu Y, 2021, AN UNCERTAINTY QUANTIFICATION APPROACH TO THE STUDY OF GENE EXPRESSION ROBUSTNESS, Publisher: INT PRESS BOSTON, INC
Aceves Sanchez P, Aymard B, Peurichard D, et al., 2021, A new model for the emergence of blood capillary networks, Networks and Heterogeneous Media, Vol: 16, Pages: 91-138, ISSN: 1556-1801
We propose a new model for the emergence of blood capillary networks. We assimilate the tissue and extra cellular matrix as a porous medium, using Darcy's law for describing both blood and interstitial fluid flows. Oxygen obeys a convection-diffusion-reaction equation describing advection by the blood, diffusion and consumption by the tissue. Discrete agents named capillary elements and modelling groups of endothelial cells are created or deleted according to different rules involving the oxygen concentration gradient, the blood velocity, the sheer stress or the capillary element density. Once created, a capillary element locally enhances the hydraulic conductivity matrix, contributing to a local increase of the blood velocity and oxygen flow. No connectivity between the capillary elements is imposed. The coupling between blood, oxygen flow and capillary elements provides a positive feedback mechanism which triggers the emergence of a network of channels of high hydraulic conductivity which we identify as new blood capillaries. We provide two different, biologically relevant geometrical settings and numerically analyze the influence of each of the capillary creation mechanism in detail. All mechanisms seem to concur towards a harmonious network but the most important ones are those involving oxygen gradient and sheer stress. A detailed discussion of this model with respect to the literature and its potential future developments concludes the paper.
Aceves-Sanchez P, Aymard B, Peurichard D, et al., 2021, A NEW MODEL FOR THE EMERGENCE OF BLOOD CAPILLARY NETWORKS, Publisher: AMER INST MATHEMATICAL SCIENCES-AIMS
Barker M, Degond P, Martin R, et al., 2021, A mean field game model of firm-level innovation, Publisher: arXiv
Knowledge spillovers occur when a firm researches a new technology and thattechnology is adapted or adopted by another firm, resulting in a social valueof the technology that is larger than the initially predicted private value. Asa result, firms systematically under--invest in research compared with thesocially optimal investment strategy. Understanding the level ofunder--investment, as well as policies to correct it, is an area of activeeconomic research. In this paper, we develop a new model of spillovers, takinginspiration from the available microeconomic data. We prove existence anduniqueness of solutions to the model, and we conduct some initial simulationsto understand how indirect spillovers contribute to the productivity of asector.
Degond P, Diez A, Frouvelle A, et al., 2020, Phase Transitions and Macroscopic Limits in a BGK Model of Body-Attitude Coordination, Publisher: SPRINGER
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- Citations: 9
Degond P, Merino-Aceituno S, 2020, Nematic alignment of self-propelled particles: from particle to macroscopic dynamics, Mathematical Models and Methods in Applied Sciences (M3AS), Vol: 30, Pages: 1935-1986, ISSN: 0218-2025
Starting from a particle model describing self-propelled particles interacting through nematic alignment, we derive a macroscopic model for the particle density and mean direction of motion. We first propose a mean-field kinetic model of the particle dynamics. After diffusive rescaling of the kinetic equation, we formally show that the distribution function converges to an equilibrium distribution in particle direction, whose local density and mean direction satisfies a cross-diffusion system. We show that the system is consistent with symmetries typical of a nematic material. The derivation is carried over by means of a Hilbert expansion. It requires the inversion of the linearized collision operator for which we show that the generalized collision invariants, a concept introduced to overcome the lack of momentum conservation of the system, plays a central role. This cross-diffusion system poses many new challenging questions.
Aceves Sanchez P, Degond P, Keaveny E, et al., 2020, Large-scale dynamics of self-propelled particles moving through obstacles: model derivation and pattern formation, Bulletin of Mathematical Biology, Vol: 82, Pages: 1-39, ISSN: 0092-8240
We model and study the patterns created through the interactionof collectively moving self-propelled particles (SPPs) and elastically tetheredobstacles. Simulations of an individual-based model reveal at least three distinct large-scale patterns: travelling bands, trails and moving clusters. Thismotivates the derivation of a macroscopic partial differential equations modelfor the interactions between the self-propelled particles and the obstacles, forwhich we assume large tether stiffness. The result is a coupled system of non linear, non-local partial differential equations. Linear stability analysis showsthat patterning is expected if the interactions are strong enough and allowsfor the predictions of pattern size from model parameters. The macroscopicequations reveal that the obstacle interactions induce short-ranged SPP aggregation, irrespective of whether obstacles and SPPs are attractive or repulsive.
Degond P, Herda M, Mirrahimi S, 2020, A Fokker-Planck approach to the study of robustness in gene expression, Mathematical Biosciences and Engineering, Vol: 17, Pages: 6459-6486, ISSN: 1547-1063
We study several Fokker-Planck equations arising from a stochastic chemical kinetic system modeling a gene regulatory network in biology. The densities solving the Fokker-Planck equations describe the joint distribution of the mRNA and μRNA content in a cell. We provide theoretical and numerical evidence that the robustness of the gene expression is increased in the presence of μRNA. At the mathematical level, increased robustness shows in a smaller coefficient of variation of the marginal density of the mRNA in the presence of μRNA. These results follow from explicit formulas for solutions. Moreover, thanks to dimensional analyses and numerical simulations we provide qualitative insight into the role of each parameter in the model. As the increase of gene expression level comes from the underlying stochasticity in the models, we eventually discuss the choice of noise in our models and its influence on our results.
Degond P, Merino-Aceituno S, Vergnet F, et al., 2020, Coupled Self-Organized Hydrodynamics and Stokes Models for Suspensions of Active Particles (vol 21, 6, 2019), JOURNAL OF MATHEMATICAL FLUID MECHANICS, Vol: 22, ISSN: 1422-6928
Barré J, Degond P, Peurichard D, et al., 2020, Modelling pattern formation through differential repulsion, Networks and Heterogeneous Media, Vol: 15, Pages: 307-352, ISSN: 1556-1801
Motivated by experiments on cell segregation, we present a twospeciesmodel of interacting particles, aiming at a quantitative description ofthis phenomenon. Under precise scaling hypothesis, we derive from the microscopicmodel a macroscopic one and we analyze it. In particular, we determinethe range of parameters for which segregation is expected. We compare ouranalytical results and numerical simulations of the macroscopic model to directsimulations of the particles, and comment on possible links with experiments.
Degond P, Diez A, Frouvelle A, et al., 2020, Phase transitions and macroscopic limits in a BGK model of body-attitude coordination, Journal of Nonlinear Science, ISSN: 0938-8974
In this article we investigate the phase transition phenomena that occur in a model of self-organisation through body-attitude coordination. Here, the body attitude of an agent is modelled by a rotation matrix in R3 as in Degond et al. (Math Models Methods Appl Sci 27(6):1005–1049, 2017). The starting point of this study is a BGK equation modelling the evolution of the distribution function of the system at a kinetic level. The main novelty of this work is to show that in the spatially homogeneous case, self-organisation may appear or not depending on the local density of agents involved. We first exhibit a connection between body-orientation models and models of nematic alignment of polymers in higher-dimensional space from which we deduce the complete description of the possible equilibria. Then, thanks to a gradient-flow structure specific to this BGK model, we are able to prove the stability and the convergence towards the equilibria in the different regimes. We then derive the macroscopic models associated with the stable equilibria in the spirit of Degond et al. (Arch Ration Mech Anal 216(1):63–115, 2015, Math Models Methods Appl Sci 27(6):1005–1049, 2017).
Degond P, Engel M, Liu J-G, et al., 2020, A Markov jump process modelling animal group size statistics, Communications in Mathematical Sciences, Vol: 18, Pages: 55-89, ISSN: 1539-6746
We translate a coagulation-framentation model, describing the dynamics of animal group size distributions, into a model for the population distribution and associate the nonlinear evolution equation witha Markov jump process of a type introduced in classic work of H. McKean. In particular this formalizesa model suggested by H.-S. Niwa [J. Theo. Biol. 224 (2003)] with simple coagulation and fragmentationrates. Based on the jump process, we develop a numerical scheme that allows us to approximate the equilibrium for the Niwa model, validated by comparison to analytical results by Degond et al. [J. NonlinearSci. 27 (2017)], and study the population and size distributions for more complicated rates. Furthermore,the simulations are used to describe statistical properties of the underlying jump process. We additionally discuss the relation of the jump process to models expressed in stochastic differential equations anddemonstrate that such a connection is justified in the case of nearest-neighbour interactions, as opposedto global interactions as in the Niwa model.
Degond P, Hecht S, Vauchelet N, 2020, Incompressible limit of a continuum model of tissue growth for two cell populations, Networks and Heterogeneous Media, Vol: 15, Pages: 57-85, ISSN: 1556-1801
This paper investigates the incompressible limit of a system modelling the growth oftwo cells population. The model describes the dynamics of cell densities, driven by pressureexclusion and cell proliferation. It has been shown that solutions to this system of partialdifferential equations have the segregation property, meaning that two population initiallysegregated remain segregated. This work is devoted to the incompressible limit of suchsystem towards a free boundary Hele Shaw type model for two cell populations.
Degond P, Ferreira M, Merino-Aceituno S, et al., 2020, A new continuum theory for incompressible swelling materials, SIAM: Multiscale Modeling and Simulation, Vol: 18, Pages: 163-197, ISSN: 1540-3459
Swelling media (e.g. gels, tumors) are usually described by mechanical constitutive laws (e.g. Hooke or Darcy laws). However, constitutive relations of real swelling media are not well-known. Here, we take an opposite route and consider a simple packing heuristics, i.e. the particles can’t overlap. We deduce a formula for the equilibrium density under a confining potential. We then consider its evolution when the average particle volume and confining potential depend on time under two additional heuristics: (i) any two particles can’t swap their position; (ii) motion should obey some energy minimization principle. These heuristics determine the medium velocity consistently with the continuity equation. In the direction normal to the potential level sets the velocity is related with that of the level sets while in the parallel direction, it is determined by a Laplace-Beltrami operator on these sets. This complex geometrical feature cannot be recovered using a simple Darcy law.
Aceves Sanchez P, Bostan M, Carrillo de la Plata JA, et al., 2019, Hydrodynamic limits for kinetic flocking models of Cucker-Smale type, Mathematical Biosciences and Engineering, Vol: 16, Pages: 7883-7910, ISSN: 1547-1063
We analyse the asymptotic behavior for kinetic models describing the collective behavior of animal populations. We focus on models for self-propelled individuals, whose velocity relaxes toward the mean orientation of the neighbors. The self-propelling and friction forces together with the alignment and the noise are interpreted as a collision/interaction mechanism acting with equal strength. We show that the set of generalized collision invariants, introduced in [39], is equivalent in our setting to the more classical notion of collision invariants, i.e., the kernel of a suitably linearized collision operator. After identifying these collision invariants, we derive the fluid model, by appealing to the balances for the particle concentration and orientation. We investigate the main properties of the macroscopic model for a general potential with radial symmetry.
Degond P, Pulvirenti M, 2019, Propagation of chaos for topological interactions, Annals of Applied Probability, Vol: 29, Pages: 2594-2612, ISSN: 1050-5164
We consider aN-particle model describing an alignment mechanism due to a topolog-ical interaction among the agents. We show that the kinetic equation, expected to holdin the mean-field limitN→∞, as following from the previous analysis in Ref. [3] can berigorously derived. This means that the statistical independence (propagation of chaos)is indeed recovered in the limit, provided it is assumed at time zero.
Chertock A, Degond P, Hecht S, et al., 2019, Incompressible limit of a continuum model of tissue growth with segregation for two cell populations, Mathematical Biosciences and Engineering, Vol: 16, Pages: 5804-5835, ISSN: 1547-1063
This paper proposes a model for the growth of two interacting populations of cells that do not mix. The dynamics is driven by pressure and cohesion forces on the one hand and proliferation on the other hand. Contrasting with earlier works which assume that the two populations are initially segregated, our model can deal with initially mixed populations as it explicitly incorporates a repul-sion force that enforces segregation. To balance segregation instabilities potentially triggered by the repulsion force, our model also incorporates a fourth order diffusion. In this paper, we study the influ-ence of the model parameters thanks to one-dimensional simulations using a finite-volume method for a relaxation approximation of the fourth order diffusion. Then, following earlier works on the single population case, we provide formal arguments that the model approximates a free boundary Hele Shaw type model that we characterise using both analytical and numerical arguments.
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