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27 results on '"Matthieu Alfaro"'

1. The field-road diffusion model: Fundamental solution and asymptotic behavior

Author

Matthieu Alfaro, Romain Ducasse, and Samuel Tréton

Subjects
Mathematics - Analysis of PDEs, Applied Mathematics, FOS: Mathematics, Analysis, Analysis of PDEs (math.AP)
Abstract

We consider the linear field-road system, a model for fast diffusion channels in population dynamics and ecology. This system takes the form of a system of PDEs set on domains of different dimensions, with exchange boundary conditions. Despite the intricate geometry of the problem, we provide an explicit expression for its fundamental solution and for the solution to the associated Cauchy problem. The main tool is a Fourier (on the road variable)/Laplace (on time) transform. In addition, we derive estimates for the decay rate of the L $\infty$ norm of these solutions.

Published
2023
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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. On the modelling of spatially heterogeneous nonlocal diffusion: deciding factors and preferential position of individuals

Author

Matthieu Alfaro, Thomas Giletti, Yong-Jung Kim, Gwenaël Peltier, Hyowon Seo, Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematical Sciences, KAIST, KAIST, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Kyung Hee University (KHU), and ANR-20-CE40-0011,DEEV,Modèles intégro-différentiels venant de la biologie évolutive(2020)

Subjects
Applied Mathematics, nonlocal diffusion, deciding factors, focusing kernels, Agricultural and Biological Sciences (miscellaneous), shape of steady states, Diffusion, Mathematics - Analysis of PDEs, Modeling and Simulation, FOS: Mathematics, Humans, AMS Subject Classifications : 92B05 (General biology and biomathemat-ics), 45K05 (Integro partial differential equations), 35B36 (Pattern for-mations in context of PDEs), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], heterogeneity, Analysis of PDEs (math.AP)
Abstract

We develop general heterogeneous nonlocal diffusion models and investigate their connection to local diffusion models by taking a singular limit of focusing kernels. We reveal the link between the two groups of diffusion equations which include both spatial heterogeneity and anisotropy. In particular, we introduce the notion of deciding factors which single out a nonlocal diffusion model and typically consist of the total jump rate and the average jump length. In this framework, we also discuss the dependence of the profile of the steady state solutions on these deciding factors, thus shedding light on the preferential position of individuals.

Published
2022
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4. Confining integro-differential equations originating from evolutionary biology: ground states and long time dynamics

Author

Matthieu Alfaro, Pierre Gabriel, Otared Kavian, Université de Rouen Normandie (UNIROUEN), Normandie Université (NU), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ), and ANR-20-CE40-0011,DEEV,Modèles intégro-différentiels venant de la biologie évolutive(2020)

Subjects
Mathematics - Analysis of PDEs, Applied Mathematics, FOS: Mathematics, Discrete Mathematics and Combinatorics, Quantitative Biology::Populations and Evolution, eigenelements, long time behaviour, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], nonlocal diffusion, evolutionary genetics, Analysis of PDEs (math.AP)
Abstract

We consider nonlinear mutation selection models, known as replicator-mutator equations in evolutionary biology. They involve a nonlocal mutation kernel and a confining fitness potential. We prove that the long time behaviour of the Cauchy problem is determined by the principal eigenelement of the underlying linear operator. The novelties compared to the literature on these models are about the case of symmetric mutations: we propose a new milder sufficient condition for the existence of a principal eigenfunction, and we provide what is to our knowledge the first quantification of the spectral gap. We also recover existing results in the non-symmetric case, through a new approach.

Published
2021

5. Density dependent replicator-mutator models in directed evolution

Author

Matthieu Alfaro and Mario Veruete

Subjects
Applied Mathematics, Context (language use), Term (logic), Expression (computer science), Directed evolution, Mathematics - Analysis of PDEs, Mutation (genetic algorithm), FOS: Mathematics, Discrete Mathematics and Combinatorics, Initial value problem, Statistical physics, Diffusion (business), Selection (genetic algorithm), Analysis of PDEs (math.AP), Mathematics
Abstract

We analyze a replicator-mutator model arising in the context of directed evolution [23], where the selection term is modulated over time by the mean-fitness. We combine a Cumulant Generating Function approach [13] and a spatio-temporal rescaling related to the Avron-Herbst formula [1] to give of a complete picture of the Cauchy problem. Besides its well-posedness, we provide an implicit/explicit expression of the solution, and analyze its large time behaviour. As a by product, we also solve a replicator-mutator model where the mutation coefficient is socially determined, in the sense that it is modulated by the mean-fitness. The latter model reveals concentration or anti diffusion/diffusion phenomena., 19 pages, 7 figures

Published
2020
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6. The spatio-temporal dynamics of interacting genetic incompatibilities. Part I: The case of stacked underdominant clines

Author

Matthieu Alfaro, Quentin Griette, Denis Roze, Benoît Sarels, Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), Normandie Université (NU)-Normandie Université (NU)-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), Station biologique de Roscoff (SBR), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), and ANR-20-CE40-0011,DEEV,Modèles intégro-différentiels venant de la biologie évolutive(2020)

Subjects
quasi linkage equilibrium, Models, Genetic, Applied Mathematics, genetic incompatibilities, AMS Subject Classifications:92D10 (Genetics and epigenetics), 35C07 (Traveling wavesolutions), 35B20 (Perturbations in context of PDEs), Agricultural and Biological Sciences (miscellaneous), Diploidy, heterozygote inferior case, Linkage Disequilibrium, Mathematics - Analysis of PDEs, Gene Frequency, Modeling and Simulation, standing wave, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], traveling wave, underdominance, perturbation analysis, Selection, Genetic, Analysis of PDEs (math.AP)
Abstract

We explore the interaction between two genetic incompatibilities (underdominant loci in diploid organisms) in a population occupying a one-dimensional space. We derive a system of partial differential equations describing the dynamics of allele frequencies and linkage disequilibrium between the two loci, and use a quasi-linkage equilibrium approximation in order to reduce the number of variables. We investigate the solutions of this system and demonstrate the existence of a solution in which the two clines in allele frequency remain stacked together. In the case of asymmetric incompatibilities (i.e. when one homozygote is favored over the other at each locus), these stacked clines propagate in the form of a traveling wave. We obtain an approximation for the speed of this wave which, in particular, is decreased by recombination between the two loci but is always larger than the speed of "one cline alone".

Published
2021

7. 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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8. Evolution equations involving nonlinear truncated Laplacian operators

Author

Matthieu Alfaro, Isabeau Birindelli, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Dipartimento di Matematica 'Guido Castelnuovo' [Roma I] (Sapienza University of Rome), and Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome]

Subjects
Hessian matrix, viscosity solutions, quenching phenomena, symbols.namesake, Mathematics - Analysis of PDEs, Dimension (vector space), FOS: Mathematics, Discrete Mathematics and Combinatorics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Eigenvalues and eigenvectors, Mathematics, Cauchy problem, heat equation, Applied Mathematics, Mathematical analysis, Fujita blow-up phenomena, Nonlinear system, Elliptic operator, Fully nonlinear elliptic operator, cauchy problem, symbols, Heat equation, Laplace operator, Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; We first study the so-called Heat equation with two families of elliptic operators whichare fully nonlinear, and depend on some eigenvalues of the Hessian matrix. The equationwith operators including the "large" eigenvalues has strong similarities with a Heatequation in lower dimension whereas, surprisingly, for operators including "small"eigenvalues it shares some properties with some transport equations. In particular, forthese operators, the Heat equation (which is nonlinear) not only does not have theproperty that "disturbances propagate with infinite speed" but may lead to quenchingin finite time. Last, based on our analysis of the Heat equations (for which we providea large variety of special solutions) for these operators, we inquire on the associated Fujita blow-up phenomena.

Published
2020

9. 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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10. 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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11. 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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12. 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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13. 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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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. 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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18. 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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19. 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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20. Convergence of a mass conserving Allen-Cahn equation whose Lagrange multiplier is nonlocal and local

Author

Pierre Alifrangis, 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 et de Modélisation de Montpellier ( I3M ), and Université Montpellier 2 - Sciences et Techniques ( UM2 ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
Mean curvature flow, Singular perturbation, Applied Mathematics, Zero (complex analysis), Internal layer, symbols.namesake, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Lagrange multiplier, Convergence (routing), FOS: Mathematics, symbols, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Allen–Cahn equation, Analysis of PDEs (math.AP), Mathematics
Abstract

International audience; We consider the mass conserving Allen-Cahn equation proposed in \cite{Bra-Bre}: the Lagrange multiplier which ensures the conservation of the mass contains not only nonlocal but also local effects (in contrast with \cite{Che-Hil-Log}). As a parameter related to the thickness of a diffuse internal layer tends to zero, we perform formal asymptotic expansions of the solutions. Then, equipped with these approximate solutions, we rigorously prove the convergence to the volume preserving mean curvature flow, under the assumption that classical solutions of the latter exist. This requires a precise analysis of the error between the actual and the approximate Lagrange multipliers.

Published
2014
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21. 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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22. 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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23. 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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24. 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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25. 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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26. 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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27. 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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