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30 results on '"Matthieu Alfaro"'

1. Populations facing a nonlinear environmental gradient: Steady states and pulsating fronts

Author

Matthieu Alfaro and Gwenaël Peltier

Subjects
010101 applied mathematics, Applied Mathematics, Modeling and Simulation, 010102 general mathematics, 0101 mathematics, 01 natural sciences
Abstract

We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth and nonlocal competition. This population is facing an environmental gradient: to survive at location [Formula: see text], an individual must have a trait close to some optimal trait [Formula: see text]. Our main focus is to understand the effect of a nonlinear environmental gradient. We thus consider a nonlocal parabolic equation for the distribution of the population, with [Formula: see text], [Formula: see text]. We construct steady states solutions and, when [Formula: see text] is periodic, pulsating fronts. This requires the combination of rigorous perturbation techniques based on a careful application of the implicit function theorem in rather intricate function spaces. To deal with the phenotypic trait variable [Formula: see text] we take advantage of a Hilbert basis of [Formula: see text] made of eigenfunctions of an underlying Schrödinger operator, whereas to deal with the space variable [Formula: see text] we use the Fourier series expansions. Our mathematical analysis reveals, in particular, how both the steady states solutions and the fronts (speed and profile) are distorted by the nonlinear environmental gradient, which are important biological insights.

Published
2021
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2. Interplay of nonlinear diffusion, initial tails and Allee effect on the speed of invasions

Author

Thomas Giletti, Matthieu Alfaro, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
AMS Subject Classifications: 35K65, Acceleration (differential geometry), 92D25, 01 natural sciences, Theoretical Computer Science, Allee effect, symbols.namesake, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), Position (vector), Reaction–diffusion system, 35K67, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Nonlinear diffusion, Statistical physics, 0101 mathematics, Diffusion (business), spreading properties, Mathematics, porous medium diffusion, 35B40, 010102 general mathematics, acceleration, heavy tails, 010101 applied mathematics, Nonlinear system, reaction-diffusion equations, fast diffusion, symbols, Porous medium, Analysis of PDEs (math.AP)
Abstract

We focus on the spreading properties of solutions of monostable equations with non-linear diffusion. We consider both the porous medium diffusion and the fast diffusion regimes. Initial data may have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity may involve a weak Allee effect, which tends to slow down the process. We study the balance between these three effects (nonlin-ear diffusion, initial tail, KPP nonlinearity/Allee effect), revealing the separation between "no acceleration" and "acceleration". In most of the cases where acceleration occurs, we also give an accurate estimate of the position of the level sets., arXiv admin note: text overlap with arXiv:1505.04626

Published
2020
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3. When the Allee threshold is an evolutionary trait: persistence vs. extinction

Author

Matthieu Alfaro, Lionel Roques, Léo Girardin, François Hamel, Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Biostatistique et Processus Spatiaux (BioSP), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), ANR-11-LABX-0056,LMH,LabEx Mathématique Hadamard(2011), ANR-14-CE25-0013,NONLOCAL,Phénomènes de propagation et équations non locales(2014), ANR-11-IDEX-0001,Amidex,INITIATIVE D'EXCELLENCE AIX MARSEILLE UNIVERSITE(2011), ANR-20-CE40-0011,DEEV,Modèles intégro-différentiels venant de la biologie évolutive(2020), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay (LMO), and Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)

Subjects
0106 biological sciences, Persistence (psychology), General Mathematics, Population, Mathematics::Analysis of PDEs, 010603 evolutionary biology, 01 natural sciences, Allee effect, Quantum nonlocality, symbols.namesake, Mathematics - Analysis of PDEs, Reaction–diffusion system, FOS: Mathematics, Initial value problem, Quantitative Biology::Populations and Evolution, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, 0101 mathematics, education, Mathematics, education.field_of_study, Extinction, Applied Mathematics, 010102 general mathematics, reaction-diffusion, structured population, evolutionary rescue, Trait, symbols, MSC 35K57, 35R09, 92D15, 92D25, Analysis of PDEs (math.AP)
Abstract

International audience; We consider a nonlocal parabolic equation describing the dynamics of a population structured by a spatial position and a phenotypic trait, submitted to dispersion , mutations and growth. The growth term may be of the Fisher-KPP type but may also be subject to an Allee effect which can be weak (non-KPP monostable nonlinearity, possibly degenerate) or strong (bistable nonlinearity). The type of growth depends on the value of a variable θ : the Allee threshold, which is considered here as an evolutionary trait. After proving the well-posedness of the Cauchy problem, we study the long time behavior of the solutions. Due to the richness of the model and the interplay between the various phenomena and the nonlocality of the growth term, the outcomes (extinction vs. persistence) are various and in sharp contrast with earlier results of the existing literature on local reaction-diffusion equations.

Published
2021
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4. Travelling waves for a non-monotone bistable equation with delay: existence and oscillations

Author

Thomas Giletti, Arnaud Ducrot, and Matthieu Alfaro

Subjects
Physics, education.field_of_study, Diffusion equation, Degree (graph theory), Bistability, General Mathematics, 010102 general mathematics, Population, 01 natural sciences, 010101 applied mathematics, Nonlinear system, Traveling wave, 0101 mathematics, Non monotone, education, Constant (mathematics), Mathematical physics
Abstract

We consider a bistable ($0\textless{}\theta\textless{}1$ being the three constant steady states) delayed reaction diffusion equation, which serves as a model in population dynamics. The problem does not admit any comparison principle. This prevents the use of classical technics and, as a consequence, it is far from obvious to understand the behaviour of a possible travelling wave in $+\infty$. Combining refined {\it a priori} estimates and a Leray Schauder topological degree argument, we construct a travelling wave connecting 0 in $-\infty$ to \lq\lq something" which is strictly above the unstable equilibrium $\theta$ in $+\infty$. Furthemore, we present situations (additional bound on the nonlinearity or small delay) where the wave converges to 1 in $+\infty$, whereas the wave is shown to oscillate around 1 in $+\infty$ when, typically, the delay is large.

Published
2017
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5. Propagation phenomena in monostable integro-differential equations: Acceleration or not?

Author

Jérôme Coville, Matthieu Alfaro, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Biostatistique et Processus Spatiaux (BioSP), Institut National de la Recherche Agronomique (INRA), 'ANR JCJC' project MODEVOL ANR-13-JS01-0009, ANR 'DEFI' project NONLOCAL ANR-14-CE25-0013, ANR-13-JS01-0009,MODEVOL,Modèles mathématiques pour la biologie évolutive(2013), ANR-14-CE25-0013,NONLOCAL,Phénomènes de propagation et équations non locales(2014), and Biostatistique et Processus Spatiaux (BIOSP)

Subjects
Kernel (set theory), Differential equation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Acceleration (differential geometry), 01 natural sciences, 010101 applied mathematics, Nonlinear system, Mathematics - Analysis of PDEs, Bounded function, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Initial value problem, 0101 mathematics, Dispersion (water waves), Degeneracy (mathematics), Analysis, Mathematics
Abstract

International audience; We consider the homogeneous integro-differential equation$\partial _t u=J*u-u+f(u)$ with a monostable nonlinearity $f$. Our interest is twofold: we investigate the existence/non existence of travelling waves, and the propagation properties of the Cauchy problem.When the dispersion kernel $J$ is exponentially bounded, travelling waves are known to exist and solutions of the Cauchy problem typically propagate at a constant speed \cite{Schumacher1980}, \cite{Weinberger1982}, \cite{Carr2004}, \cite{Coville2007a}, \cite{Coville2008a}, \cite{Yagisita2009}. %When the dispersion kernel $J$ is exponentially bounded, travelling waves are known to exist when $f$ belongs to one of the three main class of non-linearities (bistable, ignition or monostable), and solutions of the Cauchy problem typically propagate at a constant speed \cite{Schumacher1980}, \cite{Wei-82},\cite{Bates1997},\cite{Chen1997}, \cite{Carr2004}, \cite{Coville2007a}, \cite{Coville2008a}, \cite{Yagisita2009,Yagisita2009a}. On the other hand, when the dispersion kernel $J$ has heavy tails and the non-linearity $f$ is non degenerate, i.e $f'(0)>0$, travelling waves do not exist and solutions of the Cauchy problem propagate by accelerating \cite{Medlock2003}, \cite{Yagisita2009}, \cite{Garnier2011}. For a general monostable non-linearity, a dichotomy between these two types of propagation behaviour is still not known. The originality of our work is to provide such dichotomy by studying the interplay between the tails of the dispersion kernel and the Allee effect induced by the degeneracy of $f$, i.e. $f'(0)=0$. First, for algebraic decaying kernels, we prove the exact separation between existence and non existence of travelling waves. This in turn provides the exact separation between non acceleration and acceleration in the Cauchy problem. In the latter case, we provide a first estimate of the position of the level sets of the solution.

Published
2017
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6. Quantitative estimates of the threshold phenomena for propagation in reaction-diffusion equations

Author

Grégory Faye, Arnaud Ducrot, Matthieu Alfaro, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Biostatistique et Processus Spatiaux (BioSP), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Laboratoire de Mathématiques Appliquées du Havre (LMAH), Université Le Havre Normandie (ULH), Normandie Université (NU)-Normandie Université (NU), Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and ANR-16-IDEX-0006,MUSE,MUSE(2016)

Subjects
Physics, Work (thermodynamics), extinction, threshold phenomena, 010102 general mathematics, 01 natural sciences, 010101 applied mathematics, Mathematics - Analysis of PDEs, Extinction (optical mineralogy), Modeling and Simulation, Reaction–diffusion system, propagation, sharp threshold phenomena, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, 0101 mathematics, Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; We focus on the (sharp) threshold phenomena arising in some reaction-diffusion equations supplemented with some compactly supported initial data. In the so-called ignition and bistable cases, we prove the first sharp quantitative estimate on the (sharp) threshold values. Furthermore, numerical explorations allow to conjecture some refined estimates. Last we provide related results in the case of a degenerate monostable nonlinearity "not enjoying the hair trigger effect". AMS Subject Classifications: 35K57 (Reaction-diffusion equations), 35K15 (Initial value problems for second-order parabolic equations), 35B40 (Asymptotic behavior of solutions).

Published
2019
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7. Superexponential growth or decay in the heat equation with a logarithmic nonlinearity

Author

Matthieu Alfaro, Rémi Carles, Institut Montpelliérain Alexander Grothendieck (IMAG), and Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Logarithm, Applied Mathematics, Gaussian, 010102 general mathematics, Mathematical analysis, Double exponential function, Ode, 01 natural sciences, 010101 applied mathematics, Nonlinear system, symbols.namesake, Mathematics - Analysis of PDEs, Line (geometry), symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Heat equation, Uniqueness, 0101 mathematics, Analysis, Mathematics
Abstract

We consider the heat equation with a logarithmic nonlinearity, on thereal line. For a suitable sign in front of the nonlinearity, weestablish the existence and uniqueness of solutions of the Cauchyproblem, for a well-adapted class of initial data. Explicitcomputations in the case of Gaussian data lead to various scenariiwhich are richer than the mere comparison with the ODE mechanism,involving (like in the ODE case) double exponential growth or decayfor large time. Finally, we prove that such phenomena remain, in the case of compactlysupported initial data., Comment: 14 pages

Published
2017
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8. When fast diffusion and reactive growth both induce accelerating invasions

Author

Thomas Giletti, Matthieu Alfaro, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
92D25, 01 natural sciences, symbols.namesake, Acceleration, Mathematics - Analysis of PDEs, Reaction–diffusion system, 35K67, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Diffusion (business), spreading properties, Allee effect, Physics, Applied Mathematics, 35B40, 010102 general mathematics, self-similar so-lutions, General Medicine, Mechanics, Term (time), 010101 applied mathematics, Nonlinear system, reaction-diffusion equations, fast diffusion, acceleration AMS Subject Classifications: 35K65, symbols, Focus (optics), Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; We focus on the spreading properties of solutions of monostable equations with fast diffusion. The nonlinear reaction term involves a weak Allee effect, which tends to slow down the propagation. We complete the picture of [3] by studying the subtle case where acceleration does occur and is induced by a combination of fast diffusion and of reactive growth. This requires the construction of new elaborate sub and supersolutions thanks to some underlying self-similar solutions.

Published
2018
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9. EVOLUTIONARY BRANCHING VIA REPLICATOR-MUTATOR EQUATIONS

Author

Matthieu Alfaro, Mario Veruete, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques et de Modélisation de Montpellier (I3M), and Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)

Subjects
Class (set theory), 92B05, 92D15, 35K15, 45K05, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, [SPI.AUTO]Engineering Sciences [physics]/Automatic, symbols.namesake, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, Partial differential equation, Fitness function, Human evolutionary genetics, 010102 general mathematics, Eigenfunction, 010101 applied mathematics, Ordinary differential equation, Mutation (genetic algorithm), symbols, [SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing, Analysis, Schrödinger's cat, Analysis of PDEs (math.AP), [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

We consider a class of non-local reaction-diffusion problems, referred to as replicator-mutator equations in evolutionary genetics. For a confining fitness function, we prove well-posedness and write the solution explicitly, via some underlying Schr\"odinger spectral elements (for which we provide new and non-standard estimates). As a consequence, the long time behaviour is determined by the principal eigenfunction or ground state. Based on this, we discuss (rigorously and via numerical explorations) the conditions on the fitness function and the mutation rate for evolutionary branching to occur., Comment: 24 pages, 7 figures

Published
2018
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10. The effect of climate shift on a species submitted to dispersion, evolution, growth and nonlocal competition

Author

Matthieu Alfaro, Gaël Raoul, Henri Berestycki, Institut de Mathématiques et de Modélisation de Montpellier ( I3M ), Université Montpellier 2 - Sciences et Techniques ( UM2 ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Centre d'analyse et de mathématique sociale ( CAMS ), École des hautes études en sciences sociales ( EHESS ) -Centre National de la Recherche Scientifique ( CNRS ), Centre d’Ecologie Fonctionnelle et Evolutive ( CEFE ), Université Paul-Valéry - Montpellier 3 ( UM3 ) -Centre international d'études supérieures en sciences agronomiques ( Montpellier SupAgro ) -École pratique des hautes études ( EPHE ) -Institut national de la recherche agronomique [Montpellier] ( INRA Montpellier ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ) -Institut de Recherche pour le Développement ( IRD [France-Sud] ) -Institut national d’études supérieures agronomiques de Montpellier ( Montpellier SupAgro ), Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Centre d'Analyse et de Mathématique sociales (CAMS), École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS), Centre d’Ecologie Fonctionnelle et Evolutive (CEFE), Université Paul-Valéry - Montpellier 3 (UM3)-Institut National de la Recherche Agronomique (INRA)-Centre international d'études supérieures en sciences agronomiques (Montpellier SupAgro)-École pratique des hautes études (EPHE)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement (IRD [France-Sud])-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Université Paul-Valéry - Montpellier 3 (UPVM)-Institut National de la Recherche Agronomique (INRA)-Centre international d'études supérieures en sciences agronomiques (Montpellier SupAgro)-École Pratique des Hautes Études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement (IRD [France-Sud])-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut de Recherche pour le Développement (IRD [France-Sud])-Centre National de la Recherche Scientifique (CNRS)-École pratique des hautes études (EPHE)-Université de Montpellier (UM)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA)-Centre international d'études supérieures en sciences agronomiques (Montpellier SupAgro)-Université Paul-Valéry - Montpellier 3 (UM3), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre international d'études supérieures en sciences agronomiques (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA)-Université Paul-Valéry - Montpellier 3 (UPVM)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut de Recherche pour le Développement (IRD [France-Sud])

Subjects
Population, Space (mathematics), 01 natural sciences, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], FOS: Mathematics, Initial value problem, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Dispersion (water waves), education, Harnack's inequality, Mathematics, education.field_of_study, Extinction, Applied Mathematics, 010102 general mathematics, Global warming, Mathematical analysis, 010101 applied mathematics, Computational Mathematics, Variable (computer science), 13. Climate action, Analysis, Analysis of PDEs (math.AP)
Abstract

We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth and nonlocal competition. We introduce the climate shift due to {\it Global Warming} and discuss the dynamics of the population by studying the long time behavior of the solution of the Cauchy problem. We consider three sets of assumptions on the growth function. In the so-called {\it confined case} we determine a critical climate change speed for the extinction or survival of the population, the latter case taking place by \lq\lq strictly following the climate shift''. In the so-called {\it environmental gradient case}, or {\it unconfined case}, we additionally determine the propagation speed of the population when it survives: thanks to a combination of migration and evolution, it can here be different from the speed of the climate shift. Finally, we consider {\it mixed scenarios}, that are complex situations, where the growth function satisfies the conditions of the confined case on the right, and the conditions of the unconfined case on the left. The main difficulty comes from the nonlocal competition term that prevents the use of classical methods based on comparison arguments. This difficulty is overcome thanks to estimates on the tails of the solution, and a careful application of the parabolic Harnack inequality.

Published
2017
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11. Slowing Allee effect versus accelerating heavy tails in monostable reaction diffusion equations

Author

Matthieu Alfaro, Institut de Mathématiques et de Modélisation de Montpellier (I3M), and Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)

Subjects
General Physics and Astronomy, Acceleration (differential geometry), 92D25, 01 natural sciences, Allee effect, symbols.namesake, Exponential growth, Position (vector), reaction diffusion equations, Reaction–diffusion system, acceleration AMS Subject Classifications: 35K57, Quantitative Biology::Populations and Evolution, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, 0101 mathematics, Algebraic number, spreading properties, Mathematical Physics, Mathematics, Applied Mathematics, 010102 general mathematics, 35B40, Statistical and Nonlinear Physics, heavy tails, 010101 applied mathematics, Multivibrator, Nonlinear system, symbols
Abstract

International audience; We focus on the spreading properties of solutions of monostable reaction-diffusion equations. Initial data are assumed to have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity involves a weak Allee effect, which tends to slow down the process. We study the balance between the two effects. For algebraic tails, we prove the exact separation between " no acceleration and acceleration ". This implies in particular that, for tails exponentially unbounded but lighter than algebraic , acceleration never occurs in presence of an Allee effect. This is in sharp contrast with the KPP situation [19]. When algebraic tails lead to acceleration despite the Allee effect, we also give an accurate estimate of the position of the level sets.

Published
2017
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12. Fujita blow up phenomena and hair trigger effect: the role of dispersal tails

Author

Matthieu Alfaro, Institut Montpelliérain Alexander Grothendieck ( IMAG ), Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Institut Montpelliérain Alexander Grothendieck (IMAG), and Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Physics, education.field_of_study, Diffusion equation, Fujita scale, Applied Mathematics, 010102 general mathematics, Population, Second moment of area, Type (model theory), 01 natural sciences, 010101 applied mathematics, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Bounded function, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Algebraic number, education, Mathematical Physics, Analysis, Kernel (category theory), Analysis of PDEs (math.AP), Mathematical physics
Abstract

We consider the nonlocal diffusion equation ∂ t u = J ⁎ u − u + u 1 + p in the whole of R N . We prove that the Fujita exponent dramatically depends on the behavior of the Fourier transform of the kernel J near the origin, which is linked to the tails of J. In particular, for compactly supported or exponentially bounded kernels, the Fujita exponent is the same as that of the nonlinear Heat equation ∂ t u = Δ u + u 1 + p . On the other hand, for kernels with algebraic tails, the Fujita exponent is either of the Heat type or of some related Fractional type, depending on the finiteness of the second moment of J. As an application of the result in population dynamics models, we discuss the hair trigger effect for ∂ t u = J ⁎ u − u + u 1 + p ( 1 − u ) .

Published
2017
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13. Replicator-mutator equations with quadratic fitness

Author

Matthieu Alfaro, Rémi Carles, Institut Montpelliérain Alexander Grothendieck (IMAG), and Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Work (thermodynamics), Applied Mathematics, General Mathematics, Gaussian, 010102 general mathematics, Harmonic potential, 01 natural sciences, Term (time), 010101 applied mathematics, symbols.namesake, Quadratic equation, Mathematics - Analysis of PDEs, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Heat equation, 0101 mathematics, Finite time, Mathematics
Abstract

This work completes our previous analysis on models arising in evolutionary genetics. We consider the so-called replicator-mutator equation, when the fitness is quadratic. This equation is a heat equation with a harmonic potential, plus a specific nonlocal term. We give an explicit formula for the solution, thanks to which we prove that when the fitness is non-positive (harmonic potential), solutions converge to a universal stationary Gaussian for large time, whereas when the fitness is non-negative (inverted harmonic potential), solutions always become extinct in finite time., Comment: 12 pages

Published
2016
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14. Pulsating fronts for Fisher-KPP systems with mutations as models in evolutionary epidemiology

Author

Quentin Griette, Matthieu Alfaro, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut Montpelliérain Alexander Grothendieck ( IMAG ), and Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
Applied Mathematics, 010102 general mathematics, General Engineering, General Medicine, 01 natural sciences, 010101 applied mathematics, Competition (economics), Computational Mathematics, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, Reaction–diffusion system, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Quantitative Biology::Populations and Evolution, Statistical physics, 0101 mathematics, Construct (philosophy), General Economics, Econometrics and Finance, Analysis, Bifurcation, Mathematics, Analysis of PDEs (math.AP)
Abstract

We consider a periodic reaction diffusion system which, because of competition between u and v , does not enjoy the comparison principle. It also takes into account mutations, allowing u to switch to v and vice versa. Such a system serves as a model in evolutionary epidemiology where two types of pathogens compete in a heterogeneous environment while mutations can occur, thus allowing coexistence. We first discuss the existence of nontrivial positive steady states, using some bifurcation technics. Then, to sustain the possibility of invasion when nontrivial steady states exist, we construct pulsating fronts. As far as we know, this is the first such construction in a KPP situation where comparison arguments are not available.

Published
2016
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15. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data

Author

Hiroshi Matano, Matthieu Alfaro, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Graduate School of Mathematical Sciences (GSMS), and The University of Tokyo (UTokyo)

Subjects
Large class, Singular perturbation, Applied Mathematics, 010102 general mathematics, Principal (computer security), Type (model theory), Fitzhugh nagumo, 01 natural sciences, Term (time), 010101 applied mathematics, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Discrete Mathematics and Combinatorics, Applied mathematics, 0101 mathematics, Asymptotic expansion, Allen–Cahn equation, Analysis of PDEs (math.AP), Mathematics
Abstract

International audience; Formal asymptotic expansions have long been used to study the singularly perturbed Allen-Cahn type equations and reaction-diffusion systems, including in particular the FitzHugh-Nagumo system. Despite their successful role, it has been largely unclear whether or not such expansions really represent the actual profile of solutions with rather general initial data. By combining our earlier result and known properties of eternal solutions of the Allen-Cahn equation, we prove validity of the principal term of the formal expansions for a large class of solutions.

Published
2012
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16. Sharp interface limit of the Fisher-KPP equation

Author

Arnaud Ducrot, Matthieu Alfaro, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Institut de Mathématiques de Bordeaux (IMB), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Physics, Singular perturbation, Applied Mathematics, media_common.quotation_subject, 010102 general mathematics, Mathematical analysis, Motion (geometry), Fisher equation, General Medicine, Infinity, Space (mathematics), 01 natural sciences, 010101 applied mathematics, Mathematics - Analysis of PDEs, FOS: Mathematics, Traveling wave, Sharp interface, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, Analysis, Analysis of PDEs (math.AP), media_common
Abstract

We investigate the singular limit, as $\varepsilon\to 0$, of the Fisher equation $\partial_t u=\varepsilon\Delta u + \varepsilon^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus, possibly, perturbations very small as $||x|| \to \infty$. By proving both generation and motion of interface properties, we show that the sharp interface limit moves by a constant speed, which is the minimal speed of some related one-dimensional travelling waves. Moreover, we obtain a new estimate of the thickness of the transition layers. We also exhibit initial data "not so small" at infinity which do not allow the interface phenomena.

Published
2012
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17. Convergence rate of the Allen-Cahn equation to generalized motion by mean curvature

Author

Jérôme Droniou, Hiroshi Matano, Matthieu Alfaro, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Graduate School of Mathematical Sciences (GSMS), and The University of Tokyo (UTokyo)

Subjects
Off phenomenon, Mean curvature, 010102 general mathematics, Mathematical analysis, Motion (geometry), 01 natural sciences, 010101 applied mathematics, Mathematics (miscellaneous), Rate of convergence, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Computer Science::Data Structures and Algorithms, Allen–Cahn equation, Mathematical physics, Mathematics
Abstract

We investigate the singular limit, as \({\varepsilon \to 0}\), of the Allen-Cahn equation \({u^\varepsilon_t=\Delta u^\varepsilon+\varepsilon^{-2}f(u^\varepsilon)}\), with f a balanced bistable nonlinearity. We consider rather general initial data u0 that is independent of \({{\varepsilon}}\). It is known that this equation converges to the generalized motion by mean curvature — in the sense of viscosity solutions—defined by Evans, Spruck and Chen, Giga, Goto. However, the convergence rate has not been known. We prove that the transition layers of the solutions \({u^{\varepsilon}}\) are sandwiched between two sharp “interfaces” moving by mean curvature, provided that these “interfaces” sandwich at t = 0 an \({\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}\) neighborhood of the initial layer. In some special cases, which allow both extinction and pinches off phenomenon, this enables to obtain an \({\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}\) estimate of the location and the thickness measured in space-time of the transition layers. A result on the regularity of the generalized motion by mean curvature is also provided in the Appendix.

Published
2011
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18. Sharp interface limit of the Fisher-KPP equation when initial data have slow exponential decay

Author

Matthieu Alfaro, Arnaud Ducrot, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Tools of automatic control for scientific computing, Models and Methods in Biomathematics (ANUBIS), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Inria Bordeaux - Sud-Ouest, and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Applied Mathematics, Mathematical analysis, 010102 general mathematics, Constant speed, Fisher equation, Motion (geometry), Space (mathematics), 01 natural sciences, 010101 applied mathematics, Mathematics - Analysis of PDEs, FOS: Mathematics, Sharp interface, Traveling wave, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Discrete Mathematics and Combinatorics, Limit (mathematics), Exponential decay, 0101 mathematics, Analysis of PDEs (math.AP), Mathematics
Abstract

International audience; We investigate the singular limit, as $\ep \to 0$, of the Fisher equation $\partial _t u=\ep \Delta u + \ep ^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus perturbations with {\it slow exponential decay}. We prove that the sharp interface limit moves by a constant speed, which dramatically depends on the tails of the initial data. By performing a fine analysis of both the generation and motion of interface, we provide a new estimate of the thickness of the transition layers.

Published
2011
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19. Asymptotic analysis of a monostable equation in periodic media

Author

Thomas Giletti, Matthieu Alfaro, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Institut de Mathématiques et de Modélisation de Montpellier ( I3M ), Université Montpellier 2 - Sciences et Techniques ( UM2 ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Equations aux dérivées partielles ( EDP ), Institut Élie Cartan de Lorraine ( IECL ), and Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
Asymptotic analysis, General Mathematics, [ MATH.MATH-GM ] Mathematics [math]/General Mathematics [math.GM], Population, 35K57, 35R35, 35F21, 01 natural sciences, Hamilton–Jacobi equation, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], pulsating front, [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], Convergence (routing), propagating interface, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, education, Mathematics, periodic media, monostable nonlinearity, viscosity solution, education.field_of_study, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Hamilton-Jacobi equation, 010101 applied mathematics, Nonlinear system, Viscosity solution, Normal, Analysis of PDEs (math.AP)
Abstract

We consider a multidimensional monostable reaction-diffusion equation whose nonlinearity involves periodic heterogeneity. This serves as a model of invasion for a population facing spatial heterogeneities. As a rescaling parameter tends to zero, we prove the convergence to a limit interface, whose motion is governed by the minimal speed (in each direction) of the underlying pulsating fronts. This dependance of the speed on the (moving) normal direction is in contrast with the homogeneous case and makes the analysis quite involved. Key ingredients are the recent improvement \cite{A-Gil} %[4]of the well-known spreading properties \cite{Wein02}, %[32], \cite{Ber-Ham-02}, %[9],and the solution of a Hamilton-Jacobi equation.

Published
2016
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20. Varying the direction of propagation in reaction-diffusion equations in periodic media

Author

Thomas Giletti, Matthieu Alfaro, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques et de Modélisation de Montpellier ( I3M ), Université Montpellier 2 - Sciences et Techniques ( UM2 ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Equations aux dérivées partielles ( EDP ), Institut Élie Cartan de Lorraine ( IECL ), and Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
Statistics and Probability, [ MATH.MATH-GM ] Mathematics [math]/General Mathematics [math.GM], Type (model theory), 01 natural sciences, law.invention, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], pulsating traveling front, [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], law, 0103 physical sciences, Reaction–diffusion system, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, spreading properties, 35K57, 35B10, monostable nonlinearity, periodic media, Physics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, General Engineering, ignition nonlin earity, Computer Science Applications, Ignition system, Multivibrator, 010307 mathematical physics, Analysis of PDEs (math.AP)
Abstract

International audience; We consider a multidimensional reaction-diffusion equation of either ignition or monostable type, involving periodic heterogeneity, and analyze the dependence of thepropagation phenomena on the direction. We prove that the(minimal) speed of the underlying pulsating fronts dependscontinuously on the direction of propagation, and so does itsassociated profile provided it is unique up to time shifts. Wealso prove that the spreading properties \cite{Wein02} areactually uniform with respect to thedirection.

Published
2015
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21. Explicit solutions for replicator-mutator equations: extinction vs. acceleration

Author

Matthieu Alfaro, Rémi Carles, Institut de Mathématiques et de Modélisation de Montpellier (I3M), and Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)

Subjects
0303 health sciences, Class (set theory), Extinction, [SDV.GEN.GPO]Life Sciences [q-bio]/Genetics/Populations and Evolution [q-bio.PE], Applied Mathematics, Gaussian, Acceleration (differential geometry), Function (mathematics), 01 natural sciences, 010101 applied mathematics, 03 medical and health sciences, symbols.namesake, Mathematics - Analysis of PDEs, symbols, Applied mathematics, Quantitative Biology::Populations and Evolution, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Heat equation, 0101 mathematics, Finite time, Quantitative Biology - Populations and Evolution, 030304 developmental biology, Mathematics
Abstract

We consider a class of nonlocal reaction-diffusion problems, referred to as replicator-mutator equations in evolutionary genetics. By using explicit changes of unknown function, we show that they are equivalent to the heat equation and, therefore, compute their solution explicitly. Based on this, we then prove that, in the case of beneficial mutations in asexual populations, solutions dramatically depend on the tails of the initial data: they can be global, become extinct in finite time or, even, be defined for no positive time. In the former case, we prove that solutions are accelerating, and in many cases converge for large time to some universal Gaussian profile. This sheds light on the biological relevance of such models., Comment: 15 pages

Published
2014
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22. Bistable travelling waves for nonlocal reaction diffusion equations

Author

Gaël Raoul, Jérôme Coville, Matthieu Alfaro, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Biostatistique et Processus Spatiaux (BIOSP), Institut National de la Recherche Agronomique (INRA), Centre d’Ecologie Fonctionnelle et Evolutive (CEFE), Université Paul-Valéry - Montpellier 3 (UM3)-Institut National de la Recherche Agronomique (INRA)-Centre international d'études supérieures en sciences agronomiques (Montpellier SupAgro)-École pratique des hautes études (EPHE)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement (IRD [France-Sud])-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut de Mathématiques et de Modélisation de Montpellier ( I3M ), Université Montpellier 2 - Sciences et Techniques ( UM2 ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Biostatistique et Processus Spatiaux ( BIOSP ), Institut National de la Recherche Agronomique ( INRA ), Centre d’Ecologie Fonctionnelle et Evolutive ( CEFE ), Université Paul-Valéry - Montpellier 3 ( UM3 ) -Centre international d'études supérieures en sciences agronomiques ( Montpellier SupAgro ) -École pratique des hautes études ( EPHE ) -Institut national de la recherche agronomique [Montpellier] ( INRA Montpellier ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ) -Institut de Recherche pour le Développement ( IRD [France-Sud] ) -Institut national d’études supérieures agronomiques de Montpellier ( Montpellier SupAgro ), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Biostatistique et Processus Spatiaux (BioSP), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre international d'études supérieures en sciences agronomiques (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA)-Université Paul-Valéry - Montpellier 3 (UPVM)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut de Recherche pour le Développement (IRD [France-Sud]), Université Paul-Valéry - Montpellier 3 (UPVM)-Institut National de la Recherche Agronomique (INRA)-Centre international d'études supérieures en sciences agronomiques (Montpellier SupAgro)-École Pratique des Hautes Études (EPHE), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement (IRD [France-Sud])-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)

Subjects
Bistability, 01 natural sciences, Stability (probability), Leray-Schauder topological degree, law.invention, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, law, Reaction–diffusion system, Traveling wave, FOS: Mathematics, Discrete Mathematics and Combinatorics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, Steady state, Applied Mathematics, 010102 general mathematics, ignition case, 010101 applied mathematics, Ignition system, Nonlinear system, bistable case, Classical mechanics, travelling waves, nonlocal reaction-diffusion equation, Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; We are concerned with travelling wave solutions arising in a reaction diffusion equation with bistable and nonlocal nonlinearity, for which the comparison principle does not hold. Stability of the equilibrium $u\equiv 1$ is not assumed. We construct a travelling wave solution connecting 0 to an unknown steady state, which is \lq\lq above and away\rq\rq\, from the intermediate equilibrium. For focusing kernels we prove that, as expected, the wave connects 0 to 1. Our results also apply readily to the nonlocal ignition case.

Published
2013
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23. Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait

Author

Matthieu Alfaro, Gaël Raoul, Jérôme Coville, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Biostatistique et Processus Spatiaux (BioSP), Institut National de la Recherche Agronomique (INRA), Centre d’Ecologie Fonctionnelle et Evolutive (CEFE), Université Paul-Valéry - Montpellier 3 (UPVM)-Institut National de la Recherche Agronomique (INRA)-Centre international d'études supérieures en sciences agronomiques (Montpellier SupAgro)-École Pratique des Hautes Études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement (IRD [France-Sud])-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), ANR-2010-0112-01, ANR-08-BLAN-0333-01, Biostatistique et Processus Spatiaux (BIOSP), Institut de Recherche pour le Développement (IRD [France-Sud])-Centre National de la Recherche Scientifique (CNRS)-École pratique des hautes études (EPHE)-Université de Montpellier (UM)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA)-Centre international d'études supérieures en sciences agronomiques (Montpellier SupAgro)-Université Paul-Valéry - Montpellier 3 (UM3), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre international d'études supérieures en sciences agronomiques (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA)-Université Paul-Valéry - Montpellier 3 (UPVM)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut de Recherche pour le Développement (IRD [France-Sud]), Institut de Mathématiques et de Modélisation de Montpellier ( I3M ), Université Montpellier 2 - Sciences et Techniques ( UM2 ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Biostatistique et Processus Spatiaux ( BIOSP ), Institut National de la Recherche Agronomique ( INRA ), Centre d’Ecologie Fonctionnelle et Evolutive ( CEFE ), and Université Paul-Valéry - Montpellier 3 ( UM3 ) -Centre international d'études supérieures en sciences agronomiques ( Montpellier SupAgro ) -École pratique des hautes études ( EPHE ) -Institut national de la recherche agronomique [Montpellier] ( INRA Montpellier ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ) -Institut de Recherche pour le Développement ( IRD [France-Sud] ) -Institut national d’études supérieures agronomiques de Montpellier ( Montpellier SupAgro )

Subjects
Méthodologie, équation mathématique, analyse mathématique, Population, Space (mathematics), 01 natural sciences, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, Reaction–diffusion system, Traveling wave, Quantitative Biology::Populations and Evolution, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, education, Eigenvalues and eigenvectors, Mathematics, Variable (mathematics), Calcul, education.field_of_study, structured population, travelling waves, nonlocal reaction-diffusion equation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Methodology, Phenotypic trait, 010101 applied mathematics, mathématiques appliquées, réaction diffusion, Computation, densité de population, invasion biologique, équation Fisher-KPP, modèle dynamique, Analysis
Abstract

International audience; We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed $c^*>0$, and prove the existence of waves when $c\geq c^*$ and the non existence when $0\leq c < c^*$

Published
2013
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24. Interface dynamics of the porous medium equation with a bistable reaction term

Author

Matthieu Alfaro, Danielle Hilhorst, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Laboratoire de Mathématiques d'Orsay (LM-Orsay), and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)

Subjects
education.field_of_study, Partial differential equation, Bistability, Differential equation, General Mathematics, 010102 general mathematics, Mathematical analysis, Population, Degenerate energy levels, Zero (complex analysis), 01 natural sciences, 010101 applied mathematics, Classical mechanics, Mathematics - Analysis of PDEs, Convergence (routing), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, education, Porous medium, Mathematics, Analysis of PDEs (math.AP)
Abstract

International audience; We consider a degenerate partial differential equation arising in population dynamics, namely the porous medium equation with a bistable reaction term. We study its asymptotic behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We prove the rapid formation of transition layers which then propagate. We prove the convergence to a sharp interface limit whose normal velocity, at each point, is that of the underlying degenerate travelling wave.

Published
2012
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25. General fractal conservation laws arising from a model of detonations in gases

Author

Jérôme Droniou, Matthieu Alfaro, Institut de Mathématiques et de Modélisation de Montpellier (I3M), and Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)

Subjects
Physics, Conservation law, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mathematics::Analysis of PDEs, 01 natural sciences, Fractional power, 010101 applied mathematics, Computational Mathematics, Theoretical physics, Fractal, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Laplace operator, Entropy (arrow of time), Analysis
Abstract

International audience; We consider a model of cellular detonations in gases. They consist in conservation laws with a non-local pseudo-differential operator whose symbol is asymptotically $|\xi|^\lam$, where $0

Published
2012
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26. Rapid travelling waves in the nonlocal Fisher equation connect two unstable states

Author

Jérôme Coville, Matthieu Alfaro, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Biostatistique et Processus Spatiaux (BioSP), Institut National de la Recherche Agronomique (INRA), Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), and Biostatistique et Processus Spatiaux (BIOSP)

Subjects
Traveling waves, integro-differential equation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Fisher equation, turing instability, 01 natural sciences, 010101 applied mathematics, Mathematics - Analysis of PDEs, Maximum principle, Turing instability, Homogeneous, Integro-differential equation, Traveling wave, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], travelling waves, 0101 mathematics, Turing, computer, Kernel (category theory), Mathematics, computer.programming_language, Analysis of PDEs (math.AP)
Abstract

International audience; In this note, we give a positive answer to a question addressed in \cite{Nad-Per-Tan}. Precisely we prove that, for any kernel and any slope at the origin, there do exist travelling wave solutions (actually those which are \lq\lq rapid\rq\rq) of the nonlocal Fisher equation that connect the two homogeneous steady states 0 (dynamically unstable) and 1. In particular this allows situations where 1 is unstable in the sense of Turing. Our proof does not involve any maximum principle argument and applies to kernels with {\it fat tails}.

Published
2012
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27. Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen-Cahn equation

Author

Matthieu Alfaro, Hiroshi Matano, Reiner Schätzle, Harald Garcke, Danielle Hilhorst, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Naturwissenschaftliche Fakultat I - Mathematik, Universität Regensburg (UR), Laboratoire d'Analyse Numérique, Université Paris-Sud - Paris 11 (UP11), Graduate School of Mathematical Sciences (GSMS), The University of Tokyo (UTokyo), Mathematisches Institut, Arbeitsbereich Analysis, and Eberhard Karls Universität Tübingen = Eberhard Karls University of Tuebingen

Subjects
Anisotropic diffusion, General Mathematics, Motion (geometry), 01 natural sciences, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Almost everywhere, 0101 mathematics, Anisotropy, Mathematics, ddc:510, Mean curvature, Weak solution, 58B20, 010102 general mathematics, Mathematical analysis, 510 Mathematik, 35B25, 010101 applied mathematics, Nonlinear system, 35K57, 35K55, Allen–Cahn equation, 35R35, Analysis of PDEs (math.AP)
Abstract

We consider the spatially inhomogeneous and anisotropic reaction–diffusion equation ut = m(x)−1 div[m(x)ap(x,∇u)] + ε−2f(u), involving a small parameter ε > 0 and a bistable nonlinear term whose stable equilibria are 0 and 1. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. More precisely, we show that, within the time-scale of order ε2|ln ε|, the unique weak solution uε develops a steep transition layer that separates the regions {uε ≈ 0} and {uε | 1}. Then, on a much slower time-scale, the layer starts to propagate. Consequently, as ε → 0, the solution uε converges almost everywhere (a.e.) to 0 in Ω−t and 1 in Ω+t , where Ω−t and Ω+t are sub-domains of Ω separated by an interface Гt, whose motion is driven by its anisotropic mean curvature. We also prove that the thickness of the transition layer is of order ε.

Published
2010
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28. Generation of interface for an Allen-Cahn equation with nonlinear diffusion

Author

Danielle Hilhorst, Matthieu Alfaro, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Laboratoire de Mathématiques d'Orsay (LM-Orsay), and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)

Subjects
Singular perturbation, Diffusion equation, Bistability, Interface (Java), media_common.quotation_subject, Population, 01 natural sciences, 03 medical and health sciences, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, education, 030304 developmental biology, Mathematics, media_common, 0303 health sciences, education.field_of_study, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Infinity, Term (time), Modeling and Simulation, Allen–Cahn equation, Analysis of PDEs (math.AP)
Abstract

International audience; In this note, we consider a nonlinear diffusion equation with a bistable reaction term arising in population dynamics. Given a rather general initial data, we investigate its behavior for small times as the reaction coefficient tends to infinity: we prove a generation of interface property.

Published
2010
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29. The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system

Author

Matthieu Alfaro, Hiroshi Matano, Danielle Hilhorst, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Laboratoire d'Analyse Numérique, Université Paris-Sud - Paris 11 (UP11), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Graduate School of Mathematical Sciences (GSMS), and The University of Tokyo (UTokyo)

Subjects
FitzHugh–Nagumo, Bistability, Motion (geometry), Scale (descriptive set theory), 01 natural sciences, Mathematics - Analysis of PDEs, Allen–Cahn, FOS: Mathematics, Order (group theory), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, Singular perturbation, Mathematics, Mathematical physics, Component (thermodynamics), Applied Mathematics, 010102 general mathematics, Mathematical analysis, Nonlinear PDE, 010101 applied mathematics, Nonlinear system, Reaction–diffusion system, Interface motion, Analysis, Allen–Cahn equation, Analysis of PDEs (math.AP)
Abstract

We consider an Allen–Cahn type equation of the form u t = Δ u + e −2 f e ( x , t , u ) , where e is a small parameter and f e ( x , t , u ) = f ( u ) − e g e ( x , t , u ) a bistable nonlinearity associated with a double-well potential whose well-depths can be slightly unbalanced. Given a rather general initial data u 0 that is independent of e , we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within the time scale of order e 2 | ln e | , and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order e . This is an optimal estimate that has not been known before for solutions with general initial data, even in the case where g e ≡ 0 . Next we consider systems of reaction–diffusion equations of the form { u t = Δ u + e −2 f e ( u , v ) , v t = D Δ v + h ( u , v ) , which include the FitzHugh–Nagumo system as a special case. Given a rather general initial data ( u 0 , v 0 ) , we show that the component u develops a steep transition layer and that all the above-mentioned results remain true for the u -component of these systems.

Published
2008
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30. Convergence to a propagating front in a degenerate Fisher-KPP equation with advection

Author

Elisabeth Logak, Matthieu Alfaro, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Analyse, Géométrie et Modélisation (AGM - UMR 8088), and CY Cergy Paris Université (CY)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Singular perturbation, Advection, Applied Mathematics, Chemotaxis, Drift effect, 010102 general mathematics, Mathematical analysis, Degenerate energy levels, Front (oceanography), Boundary (topology), Fisher-KPP equation, 01 natural sciences, 010101 applied mathematics, Mathematics - Analysis of PDEs, Convergence (routing), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, Diffusion (business), Density-dependent diffusion, Analysis, Mathematics, Analysis of PDEs (math.AP)
Abstract

International audience; We consider a Fisher-KPP equation with density-dependent diffusion and advection, arising from a chemotaxis-growth model. We study its behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We analyze, for small times, the emergence of transition layers induced by a balance between reaction and drift effects. Then we investigate the propagation of the layers. Convergence to a free-boundary limit problem is proved and a sharp estimate of the thickness of the layers is provided.

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