Singular measure traveling waves in an epidemiological model with continuous phenotypes

TL;DR

This paper constructs measure-valued traveling waves in a reaction-diffusion model with continuous phenotypes, revealing the existence of singular waves and determining their spreading speed, which is a novel achievement in the field.

Contribution

It introduces a vanishing viscosity method to construct measure-valued traveling waves, including singular cases, for the first time in reaction-diffusion equations with phenotypic structure.

Findings

Existence of measure-valued traveling waves including singular waves.

Traveling wave speed matches the expected spreading speed 2√(-λ₁).

First construction of measure-valued traveling waves for such equations.

Abstract

We consider the reaction-diffusion equation \begin{equation*} 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) , \end{equation*} 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.