Singular limits for models of selection and mutations with heavy-tailed mutation distribution

Autor: Sepideh Mirrahimi
Přispěvatelé: 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), Centre National de la Recherche Scientifique (CNRS)-PRES Université de Toulouse-Université Toulouse III - Paul Sabatier (UPS), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS )
Rok vydání: 2020
Předmět:
Asymptotic analysis
viscosity solutions
nonlocal reaction term
General Mathematics
Dirac (software)
Population
01 natural sciences
Hamilton–Jacobi equation
WKB approximation
[ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP]
Mathematics - Analysis of PDEs
FOS: Mathematics
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
Applied mathematics
Quantitative Biology::Populations and Evolution
0101 mathematics
47G20
education
viscosity solutions AMS Class No: 35K57
Mathematics
education.field_of_study
49L25
Applied Mathematics
010102 general mathematics
Key-Words: Fractional reaction-diffusion equation
Hamilton-Jacobi equation
AMS Class No: 35K57
35B25
92D15
010101 applied mathematics
asymptotic analysis
Distribution (mathematics)
Mutation (genetic algorithm)
Viscosity solution
Analysis of PDEs (math.AP)
Zdroj: Journal de Mathématiques Pures et Appliquées
Journal de Mathématiques Pures et Appliquées, Elsevier, 2019, ⟨10.1016/j.matpur.2019.10.001⟩
Journal de Mathématiques Pures et Appliquées, 2019, ⟨10.1016/j.matpur.2019.10.001⟩
ISSN: 0021-7824
DOI: 10.1016/j.matpur.2019.10.001
Popis: In this article, we perform an asymptotic analysis of a nonlocal reaction-diffusion equation, with a fractional laplacian as the diffusion term and with a nonlocal reaction term. Such equation models the evolutionary dynamics of a phenotypically structured population. We perform a rescaling considering large time and small effect of mutations, but still with algebraic law. With such rescaling, we expect that the phenotypic density will concentrate as a Dirac mass which evolves in time. To study such concentration phenomenon, we extend an approach based on Hamilton-Jacobi equations with constraint, that has been developed to study models from evolutionary biology, to the case of fat-tailed mutation kernels. However, unlike previous works within this approach, the WKB transformation of the solution does not converge to a viscosity solution of a Hamilton-Jacobi equation but to a viscosity supersolution of such equation which is minimal in a certain class of supersolutions. Such property allows to derive the concentration of the population density as an evolving Dirac mass, under monotony conditions on the growth rate, similarly to the case with thin-tailed mutation kernels.
Databáze: OpenAIRE
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