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Transactions of the American Mathematical Society

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Singular measure traveling waves in an epidemiological model with continuous phenotypes


Author: Quentin Griette
Journal: Trans. Amer. Math. Soc. 371 (2019), 4411-4458
MSC (2010): Primary 35R09; Secondary 35C07, 35D30, 35Q92, 92D30, 92D15
DOI: https://doi.org/10.1090/tran/7700
Published electronically: November 13, 2018
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the reaction-diffusion equation

$\displaystyle u_t=u_{xx}+\mu \left (\int _\Omega M(y,z)u(t,x,z)dz-u\right ) + u\left (a(y)-\int _\Omega K(y,z) u(t,x,z)dz\right ) ,$    

where $ u=u(t,x,y) $ stands for the density of a theoretical population with a spatial ( $ x\in \mathbb{R}$) and phenotypic ( $ y\in \Omega \subset \mathbb{R}^n$) structure, $ M(y,z) $ is a mutation kernel acting on the phenotypic space, $ a(y) $ is a fitness function, and $ K(y,z) $ is a competition kernel. Using a vanishing viscosity method, we construct measure-valued traveling waves for this equation and present particular cases where singular traveling waves do exist. We determine that the speed of the constructed traveling waves is the expected spreading speed $ c^*:=2\sqrt {-\lambda _1} $, where $ \lambda _1 $ is the principal eigenvalue of the linearized equation. As far as we know, this is the first construction of a measure-valued traveling wave for a reaction-diffusion equation.

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Additional Information

Quentin Griette
Affiliation: IMAG, Université de Montpellier, 163 rue Auguste Broussonnet, 34090 Montpellier, France
Address at time of publication: IMB, Université de Bordeaux, 351 cours de la Libération, 33800 Talence, France
Email: quentin.griette@math.u-bordeaux.fr

DOI: https://doi.org/10.1090/tran/7700
Keywords: Traveling wave, singular measure, concentration, epidemiology
Received by editor(s): October 5, 2017
Received by editor(s) in revised form: June 1, 2018, and June 27, 2018
Published electronically: November 13, 2018
Article copyright: © Copyright 2018 American Mathematical Society