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research
Singular measure traveling waves in an epidemiological model with continuous phenotypes
Authors
Quentin Griette
Publication date
13 September 2018
Publisher
Doi
Cite
View
on
arXiv
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
)
u=u(t,x,y)
u
=
u
(
t
,
x
,
y
)
stands for the density of a theoretical population with a spatial (
x
∈
R
x\in\mathbb R
x
∈
R
) and phenotypic (
y
∈
Ω
⊂
R
n
y\in\Omega\subset \mathbb R^n
y
∈
Ω
⊂
R
n
) structure,
M
(
y
,
z
)
M(y,z)
M
(
y
,
z
)
is a mutation kernel acting on the phenotypic space,
a
(
y
)
a(y)
a
(
y
)
is a fitness function and
K
(
y
,
z
)
K(y,z)
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
−
λ
1
c^*:=2\sqrt{-\lambda_1}
c
∗
:=
2
−
λ
1
​
​
, where
λ
1
\lambda_1
λ
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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Last time updated on 23/03/2021