"Spatial propagation for nonlocal non-autonomous Fisher-KPP equation"
Arnaud Ducrot (Université Le Havre Normandie, France)
In this lecture we discuss existence results of travelling wave solutions and spreading speed for a non-autonomousFisher-KPP equation with nonlocal diffusion. We prove that under suitable time averaging properties for the coefficients,
the equation exhibits a definite spreading speed. We also study non-autonomous Fisher-KPP equation on a lattice anddeduce from our analysis some spreading phenomenon for some predator-prey systems on lattice.

Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Spatial propagation for nonlocal
non-autonomous Fisher-KPP equation
Arnaud Ducrot
Université Le Havre Normandie
UR 3821 LMAH
Joint works with Zhucheng Jin
ICMAA 2023
International Conference on"Reaction-diffusion systems: from the past to the future"
in memory of Prof. Mimura
Meiji University, MIMS.

October 31, 2023
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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Introduction
Non-local diffusion
Fisher-KPP equation
Travelling waves
Spreading speed
Fisher-KPP equation on a lattice and applications
Fisher-KPP equation
Application to predator-prey systems on lattice

A. Ducrot

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Introduction
Non-local diffusion
Fisher-KPP equation
Travelling waves
Spreading speed
Fisher-KPP equation on a lattice and applications
Fisher-KPP equation
Application to predator-prey systems on lattice

A. Ducrot

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Non-local diffusion

Discrete diffusion

On the lattice Z, individuals can jump from j to i with probabilityp(j, i). Hence the density of population at i is given byXX∂u(t, i)=p(j, i)u(t, j) −p(i, j)u(t, i)
∂t
j∈Z
j∈Z
|
{z
}
|
{z
}
Coming at i from any j

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leaving i to somewhere

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Non-local diffusion

Discrete diffusion

On the lattice Z, individuals can jump from j to i with probabilityp(j, i). Hence the density of population at i is given byXX∂u(t, i)=p(j, i)u(t, j) −p(i, j)u(t, i)
∂t
j∈Z
j∈Z
|
{z
}
|
{z
}
Coming at i from any jleaving i to somewhereIf p(j, i) = J(i − j) depends on the distance between the point j
and i, we get:XX∂u(t, i)=J(i − j)u(t, j) − J(j − i) u(t, i)
∂t
j∈Z
j∈Z
|
{z
}
discrete convolution
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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Non-local diffusion

Continuous version

The discrete lattice is changed to the line and the probability
transitions are given with the kernel J(x − y) from y to x.
Non-local heat equation∂u(t, x)=∂tZZ
J(x − y)u(t, y)dy − u(t, x)RJ(y − x)dyRor equivalently,∂u(t, x)=∂tZJ(x − y) u(t, y) − u(t, x) dy.

R

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Non-local diffusion

Non-local heat kernel

If J has a unit mass, the fundamental solution K(t, x − x0 ) is
given by:• Using Fourier transform:b\·)(ξ) = et(J(ξ)−1)K(t,.

• Using exponential etJ∗· :−t

K(t, x) = e δ0 (dx) + Ψ(t, x) with Ψ(t, x) = e

−t

∞
X
t
k=1

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J ∗k (x)6/44

Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Non-local diffusion

Non-local heat kernel

If J has a unit mass, the fundamental solution K(t, x − x0 ) is
given by:• Using Fourier transform:b\·)(ξ) = et(J(ξ)−1)K(t,.

• Using exponential etJ∗· :−t

K(t, x) = e δ0 (dx) + Ψ(t, x) with Ψ(t, x) = e

−t

∞
X
t
k=1

k!

J ∗k (x)The solution with initial u0 reads asu(t, x) = e−t u0 (x) + rather smooth fonction.
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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

Fisher-KPP equation with nonlocal diffusionZ
J(x−y) [u(t, y) − u(t, x)] dy+f (u(t, x)),∂t u(t, x) =t > 0, x ∈ R,

R

where f (0) = f (1) = 0, f is KPP type, that is:
f is smooth, f (0) = f (1) = 0 and 0 < f (u) ≤ f 0 (0)u foru ∈ (0, 1).
Typical example: f (u) = u(1 − u).

KPP

Monostable
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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

Travelling wave solutions

Travelling waves: Special solution of the formu(t, x) = φ(x − ct)
which connects the stationary solutions 0 and 1
φ is the profile and c is the wave speed
Wave profile equation: φ = φ(z) satisfies
Z
 J(y) [φ(z − y) − φ(z)] dy + cφ0 (z) + f (φ(z)) = 0, z ∈ RRφ(−∞) = 1 and φ(+∞) = 0.

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

Travelling wave solutions for thin tailed kernel• Wave profile φ satisfies(R
0
R J(y) [φ(z − y) − φ(z)] dy + cφ (z) + f (φ(z)) = 0,φ(−∞) = 1 and φ(+∞) = 0.

• At the leading edge φ ≈ 0 andR00
R K(y) [φ(z − y) − φ(z)] dy + cφ (z) + f (0)φ(z) ≈ 0.

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

Travelling wave solutions for thin tailed kernel• Wave profile φ satisfies(R
0
R J(y) [φ(z − y) − φ(z)] dy + cφ (z) + f (φ(z)) = 0,φ(−∞) = 1 and φ(+∞) = 0.

• At the leading edge φ ≈ 0 andR00
R K(y) [φ(z − y) − φ(z)] dy + cφ (z) + f (0)φ(z) ≈ 0.
For φ(z) = e−λz , with λ > 0 we get:Z
J(y)[eλy − 1]dy − λc + f 0 (0) = 0.
R
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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

Known results: travelling waves

Theorem (Schumacher 1980; Carr, Chmaj 2004;...)
If f is ofRKPP type and J is a thin-tailed kernel, that is∃β > 0, R eβ|y| J(y)dy < ∞, there exists a travelling wave
solution with speed c if and only if c ≥ c∗ . The travelling wave
solution is unique up to translation. The minimal wave speed c∗
is characterized by
Z

∗
−1
λy
0
c := inf λJ(y)[e − 1]dy + f (0) .
λ>0R• Monostable type: Coville, Dupaigne 2007; Liang, Zhao
2007; Yagisita 2010; Fang, Zhao 2014;...
• Bistable or Ignition type: Bates, Fife, Ren, Wang 1997;
Chen 1997; Coville, Dupaigne 2005;...
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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

Known results: spreading speeds∂t u = J ∗ u − u + f (u),t > 0, x ∈ R, u(0, x) = u0 (x).

Theorem (Lutscher, Pachepsky, Lewis 2005; Liang, Zhao
2007;...)
Assume that J is thin-tailed and symmetric. If the initial data u0
is compactly supported, then there exists c∗∗ > 0 such that
lim sup u(t, x) = 0,∀c > c∗∗ ,t→∞|x|≥ct
lim sup |1 − u(t, x)| = 0, ∀c ∈ [0, c∗∗ ).
t→∞|x|≤ctc∗∗ is called the spreading speed and we have c∗∗ = c∗ .
Remark: Non-symmetric kernel J: Xu, Li, Ruan 2021.
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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

Our goal

Study the propagation for the non-autonomous problemZ∂t u(t, x) =K(t, x − y) [u(t, y) − u(t, x)] dy + f (t, u(t, x)).
R

• Generalized travelling waves• Spreading speed• discrete convolution: Fisher-KPP on lattice• Application for predator-prey system on lattice.

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

Our goal

Study the propagation for the non-autonomous problemZ∂t u(t, x) =K(t, x − y) [u(t, y) − u(t, x)] dy + f (t, u(t, x)).
R

• Generalized travelling waves• Spreading speed• discrete convolution: Fisher-KPP on lattice• Application for predator-prey system on lattice.
for simplicity we choose f (t, u) = µ(t)u(1 − u).

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

Time periodic coefficientsZ
K(x − y) [u(t, y) − u(t, x)] dy + f (t, u(t, x)).

∂t u(t, x) =R

∃T > 0 f (t + T, u) ≡ f (t, u) of KPP type and thin-tailed kernel.
• Pulsating wave [Shigesada, Kawasaki, Teramoto 1986;
Alikakos, Bate, Chen 1999]:
u(t, x) = φ(t, x − ct),φ(t, ·) = φ(t + T, ·),φ(t, −∞) = 1 and φ(t, +∞) = 0, unif. for t ∈ R.

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

More general time heterogeneities

Definition (Berestycki, Hamel 2007, 2012; Matano 2003;
Shen 2004)
A generalized transition front connecting 0 and 1 is an entire
solution u = u(t, x) and an interface function X : R → R s.t.
(
1, as x → −∞,u(t, x + X(t)) →unif. for t ∈ R.
0, as x → +∞;A. Ducrot

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

More general time heterogeneities• Existence of generalized transition front [Shen, Shen 2016]
Z∂t u(t, x) =K(x − y) [u(t, y) − u(t, x)] dy + f (t, u(t, x)),R

where f is of KPP type and K is a thin-tailed kernel.

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

More general time heterogeneities

Definition (Berestycki, Hamel 2007, 2012; Matano 2003;
Shen 2004)
A generalized transition front connecting 0 and 1 with interface
X is a solution u = u(t, x)
lim u(t, x+X(t)) = 1 andx→−∞

lim u(t, x+X(t)) = 0, unif. for t ∈ R.

x→+∞

Definition (Nadin, Rossi 2012; Shen 2011)
A generalized travelling wave solution u with speed functionc = c(t) ∈ L∞ (R) is nothing but a generalized transition front
with a globally Lipschitz continuous interface function
Z t
X(t) =
c(s) ds.
0
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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

Least mean value and mean value• Least mean value [Nadin, Rossi 2012]: for g ∈ L∞ (R),1
bgc := lim inf
T →+∞ t∈R T

A. Ducrot

Z Tg(t + s) ds.
0

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

Least mean value and mean value• Least mean value [Nadin, Rossi 2012]: for g ∈ L∞ (R),1
bgc := lim inf
T →+∞ t∈R T

Z T
g(t + s) ds.
0• Mean value: for g ∈ L∞ (R),1hgi := lim
T →+∞ TZ T
g(t + s)ds, unif. for t ∈ R.
0

Remark:1

2

Some typical classes of heterogeneities admit a meanvalue.
(e.g. periodic, almost periodic, so on...)If g ∈ L∞ (R) admits a mean value, then hgi = bgc.
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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Introduction
Non-local diffusion
Fisher-KPP equation
Travelling waves
Spreading speed
Fisher-KPP equation on a lattice and applications
Fisher-KPP equation
Application to predator-prey systems on lattice

A. Ducrot

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

The problemZ∂t u =

K(t, x−y) [u(t, y) − u(t, x)] dy+µ(t)u (1 − u) , (t, x) ∈ R2 .

R

• Aim: existence of generalized travelling wave solution withspeed c = c(t) ∈ L∞ (R).

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

The problemZ∂t u =

K(t, x−y) [u(t, y) − u(t, x)] dy+µ(t)u (1 − u) , (t, x) ∈ R2 .

R

• Aim: existence of generalized travelling wave solution withspeed c = c(t) ∈ L∞ (R).
R
• Wave profile φ(t, z) := u t, z + 0t c(s) ds (if smoothenough) satisfiesZ∂t φ−c(t)∂z φ =K(t, y)[φ(t, z−y)−φ(t, z)] dy+µ(t)φ(1−φ),Rwith
φ(t, −∞) = 1 and φ(t, +∞) = 0, unif. for t ∈ R.
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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Assumptions

Assumption:µ ∈ L∞ (R) and inf µ(t) > 0.
t∈RAssumption:(i) K is nonnegative and K ∈ L1y (R, L∞
t (R)).
(ii) (Thin-tailed) For some λ > 0,Z
kK(·, y)k∞ eλy dy < ∞.
R
R

(iii)

lim supλ→abs(K)−λy
R kK(·,y)k∞ e dy

λ= ∞, whereR
abs(K) := sup λ ≥ 0 : R kK(·, y)k∞ eλy dy < ∞ .
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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Linear determination of speed function• Linearized equationZ∂t ψ −c(t)∂z ψ − K(t, y)[ψ(t, z −y)−ψ(t, z)] dy −µ(t)ψ = 0.
R

• Ansatz
ψ(z) := e−λz , λ ∈ (0, abs(K)).
• DeriveRc(t) = c(λ)(t) :=λy − 1] dy + µ(t)R K(t, y)[eλA. Ducrot, λ ∈ (0, abs(K)).

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Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Linear determination of speed function• Linearized equationZ∂t ψ −c(t)∂z ψ − K(t, y)[ψ(t, z −y)−ψ(t, z)] dy −µ(t)ψ = 0.
R

• Ansatz
ψ(z) := e−λz , λ ∈ (0, abs(K)).
• DeriveRc(t) = c(λ)(t) :=λy − 1] dy + µ(t)R K(t, y)[eλ, λ ∈ (0, abs(K)).

Lemma (Decreasing property)
There exists λ∗ ∈ (0, abs(K)) s.t.

λ > 0 : ∃λ0 > λ, ∀k ∈ (λ, λ0 ], bc(λ) − c(k)c > 0 = (0, λ∗ ).
Moreover, λ 7→ bc(λ)c is decreasing in (0, λ∗ ).
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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Results

Theorem (Existence)
For each λ ∈ (0, λ∗ ), there exists a generalized travelling wave
solution with speed function c(λ) ∈ L∞ (R).
Remark:(bc(λ∗ )c, ∞) ⊂ {least mean value of admissible speed}

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Results

Theorem (Existence)
For each λ ∈ (0, λ∗ ), there exists a generalized travelling wave
solution with speed function c(λ) ∈ L∞ (R).
Remark:(bc(λ∗ )c, ∞) ⊂ {least mean value of admissible speed}

Theorem (Nonexistence)R
Set c(λ)(t) := R K(t, y)eλy y dy. There is no generalized
travelling wave solution with speed function c ∈ L∞ (R)satisfying
bcc < bc(λ∗ )c.

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Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Minimal wave speed

Corollary
If K(·, y) for each y ∈ R and µ(·) admit a mean value, then we
getc∗ := bc(λ∗ )c = bc(λ∗ )cand{least mean value of admissible speed} = either (c∗ , ∞) or [c∗ , ∞).

Open question for the mean value c∗ .

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Sketch of proof of existence theorem• For λ ∈ (0, λ∗ ),noSuper-solution: φ̄(t, z) := min 1, e−λz ,noSub-solution: φ(t, z) := max 0, e−λz − eb(t) e−(λ+h)z ,where h > 0 and b ∈ W 1,∞ (R).

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Sketch of proof of existence theorem• For λ ∈ (0, λ∗ ),noSuper-solution: φ̄(t, z) := min 1, e−λz ,noSub-solution: φ(t, z) := max 0, e−λz − eb(t) e−(λ+h)z ,where h > 0 and b ∈ W 1,∞ (R).
• The solution un (t, x) of Cauchy problem(R∂t u(t, x) = R K(t, y) [u(t, x − y) − u(t, x)] dy + µ(t)u(1 − u),R −nu(−n, x) = φ(−n, x − 0 c(λ)(s) ds).

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Sketch of proof of existence theorem• For λ ∈ (0, λ∗ ),noSuper-solution: φ̄(t, z) := min 1, e−λz ,noSub-solution: φ(t, z) := max 0, e−λz − eb(t) e−(λ+h)z ,where h > 0 and b ∈ W 1,∞ (R).
• The solution un (t, x) of Cauchy problem(R∂t u(t, x) = R K(t, y) [u(t, x − y) − u(t, x)] dy + µ(t)u(1 − u),R −nu(−n, x) = φ(−n, x − 0 c(λ)(s) ds).
• Regularity estimates for {un }n≥1 : Lipschitz estimatesobtained by the comparison principle.
• passing to the limit n → ∞A. Ducrot

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Introduction
Non-local diffusion
Fisher-KPP equation
Travelling waves
Spreading speed
Fisher-KPP equation on a lattice and applications
Fisher-KPP equation
Application to predator-prey systems on lattice

A. Ducrot

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

The problem

Aim: Study the spreading speed for the problem:(R∂t u = R K(y) [u(t, x − y) − u(t, x)] dy + µ(t)u (1 − u) , t > 0, x ∈ R,
u(0, x) = u0 (x),0 ≤ u0 ≤ 1, u0 6≡ 0, and compact support.

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

The problem

Aim: Study the spreading speed for the problem:(R∂t u = R K(y) [u(t, x − y) − u(t, x)] dy + µ(t)u (1 − u) , t > 0, x ∈ R,
u(0, x) = u0 (x),0 ≤ u0 ≤ 1, u0 6≡ 0, and compact support.
Assumption: The function µ is uniformly continuous, bounded,
and inf t≥0 µ(t) > 0.

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Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

The problem

Aim: Study the spreading speed for the problem:(R∂t u = R K(y) [u(t, x − y) − u(t, x)] dy + µ(t)u (1 − u) , t > 0, x ∈ R,
u(0, x) = u0 (x),0 ≤ u0 ≤ 1, u0 6≡ 0, and compact support.
Assumption: The function µ is uniformly continuous, bounded,
and inf t≥0 µ(t) > 0.
Assumption:
The kernel K : R → [0, ∞) is continuous, integrable andR(i) ∃α > 0 s.t. R K(y)eαy dy < ∞.
(ii) K(0) > 0. R(iii) bµc > K := R K(y)dy.
R(iv)

lim supλ→abs(K)−λy
R K(y)e dy

λ

= ∞.

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Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Linear determination of speed function• Similarly, introduce
R
K(y)eλy dy − K + µ(t), λ ∈ (0, abs(K)).
c(λ)(t) := Rλ• There exists λ∗ ∈ (0, abs(K)) s.t.
bc(λ∗ )c =inf

bc(λ)c.

λ∈(0,abs(K))• bµc > K =⇒ bc(λ∗ )c > 0.

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Main results

Theorem
Let u be a solution with a compactly supported initial data
0 ≤ u0 ≤ 1. Then we have(i) limt→∞

supu(t, x) = 0, ∀σ > 0,Rx≥ 0t c(λ∗ )(s)ds+σt(ii) lim sup |1 − u(t, x)| = 0, ∀c ∈ [0, bc(λ∗ )c).
t→∞ x∈[0,ct]

Remark:• If the coefficients admit a mean value, then one has
R
1 t
∗
∗
t 0 c(λ )(s)ds → bc(λ )c as t → ∞. We obtain the exact
spreading speed bc(λ∗ )c.
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Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Main results

Theorem (Slower exponential decay initial data)
Let u(t, x) be the solution with an initial data u0 , 0 ≤ u0 ≤ 1,
u0 6≡ 0 and such that u0 (x) ∼ Ce−λx as x → ∞ for λ ∈ (0, λ∗ ).
Then(i) limt→∞u(t, x) = 0, ∀σ > 0;supRx≥ 0t c(λ)(s)ds+σt(ii) lim sup |1 − u(t, x)| = 0, ∀0 < c < bc(λ)c.
t→∞ x∈[0,ct]

Remark: symmetrical argument yields the spreading properties
to the left.
Different decay rates can be considered in the two sides:→ different speeds of propagation.
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Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Lower wave speed estimate: a key persistence lemma

Lemma
Let u ∈ BU C([0, ∞) × R) be the solution. Define the limit set
ω(u) by ũ ∈ ω(u) if we have {(tn , xn )} with tn → ∞ andu(t + tn , x + xn ) → ũ(t, x), loc. unif. for (t, x) ∈ R2 , as n → ∞.
Let X = X(t) : [0, ∞) → R+ continuous and assume that
(H1) lim inf u(t, 0) > 0;t→∞(H2) ∃ε > 0 s.t. lim inf ũ(t, 0) ≥ ε, ∀ũ ∈ ω(u) \ {0};t→∞(H3) lim inf u(t, X(t)) > 0.
t→∞

Then for any k ∈ (0, 1), one has
lim inf

inf

t→∞ 0≤x≤kX(t)A. Ducrotu(t, x) > 0.
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Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Remarks on the persistence lemma• Examples of propagating path X(t): X(t) = ct,
Rt
X(t) = 0 c(s)ds,...
• Without the thin-tailed kernel assumption.
• The proof of the persistence lemma is inspired by Ducrot,
Giletti, Matano (CVPDE 2019); Ducrot, Giletti, Guo,
Shimojo (Nonlinearity 2021).

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Introduction
Non-local diffusion
Fisher-KPP equation
Travelling waves
Spreading speed
Fisher-KPP equation on a lattice and applications
Fisher-KPP equation
Application to predator-prey systems on lattice

A. Ducrot

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

The problem

We consider the problem for t ≥ 0, i ∈ Z:X
dw(t, i) =J(t, j)[w(t, i−j)−w(t, i)]+µ(t)w(t, i)(1−w(t, i)),dtj∈Zwith an initial data w(0, i) = w0 (i) compactly supported.
We assume that:• Assumptions for the non-negative kernel: J ∈ l1 (Z, L∞ (0, ∞))is symmetric, thin-tailed, inf t≥0 J(t, ±1) > 0. It has a meanvalue andX
λ−1hJ(·, i)ieλi → ∞ as λ → abs(J).
i∈Z• Assumption for t 7→ µ(t): uniformly continuous and has amean value.
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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

Linear speed function

Approximation w(t, i) ≈ e−λ(i−X(t) with λ > 0 yields
X
λX 0 (t) =J(t, j)[eλj − 1] + µ(t),j∈ZDefinecλ (t) = λ−1X

J(t, j)[eλj − 1] + λ−1 µ(t)j∈Zand its mean valuecλ = λ−1XhJ(., j)i[eλj − 1] + λ−1 hµi.

j∈Z

Then there exists λ∗ ∈ (0, abs(J)) such that(
decreasing on (0, λ∗ )λ 7→ cλ isincreasing on (λ∗ , abs(J))
A. Ducrot

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

Spreading result

TheoremIf the initial data w0 is compactly supported then the solution
satisfies
lim sup w(t, i) = 0 for all c > c∗ := cλ∗ ,t→∞ |i|≥ct

andlim sup |1 − w(t, i)| = 0 for 0 < c < c∗ .

t→∞ |i|≤ct

Extend previous results Weinberger 1982; Liang and Zhao
2007; Fang, Wei, Zhao 2010; Cao and Shen 2017; Liang andZhou 2020, ...
A. Ducrot

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Fisher-KPP equation

Idea for the proof of the lower bound• For the lower bound of the speed, we construct a suitablesub-solution with small speed.
• Then coupling with dynamical system arguments and limitingarguments. For limiting arguments, we have to take care of all
possible limit equations obtained by time shift t + tn withtn → ∞.

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Application to predator-prey systems on lattice

Predator-prey systems

Aim: Study the spreading speeds for the prey-predator system
for t > 0, i ∈ ZPdu(t, i) =J1 (t, j)[u(t, i − j) − u(t, i)]

dt


j∈Z






+r(t)u(t, i) 1 − u(t, i) −p(t)u(t, i)v(t, i),Pdv(t, i) =J2 (t, j)[v(t, i − j) − v(t, i)]
dt


j∈Z




+q(t)u(t, i)v(t, i) − ν(t)v(t, i),u(0, i) = u (i) and v(0, i) = v (i),i ∈ Z.
00• Prey: u = u(t, i) and Predator: v = v(t, i).

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Application to predator-prey systems on lattice

Predator-prey systems

Aim: Study the spreading speeds for the prey-predator system
for t > 0, i ∈ ZPdu(t, i) =J1 (t, j)[u(t, i − j) − u(t, i)]

dt


j∈Z






+r(t)u(t, i) 1 − u(t, i) −p(t)u(t, i)v(t, i),Pdv(t, i) =J2 (t, j)[v(t, i − j) − v(t, i)]
dt


j∈Z




+q(t)u(t, i)v(t, i) − ν(t)v(t, i),u(0, i) = u (i) and v(0, i) = v (i),i ∈ Z.
00• Prey: u = u(t, i) and Predator: v = v(t, i).
• Initial data u0 and v0 are both compactly supported.

A. Ducrot

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Application to predator-prey systems on lattice

Predator-prey systems

Aim: Study the spreading speeds for the prey-predator system
for t > 0, i ∈ ZPdu(t, i) =J1 (t, j)[u(t, i − j) − u(t, i)]

dt


j∈Z






+r(t)u(t, i) 1 − u(t, i) −p(t)u(t, i)v(t, i),Pdv(t, i) =J2 (t, j)[v(t, i − j) − v(t, i)]
dt


j∈Z




+q(t)u(t, i)v(t, i) − ν(t)v(t, i),u(0, i) = u (i) and v(0, i) = v (i),i ∈ Z.
00• Prey: u = u(t, i) and Predator: v = v(t, i).
• Initial data u0 and v0 are both compactly supported.
All coefficients are uniformly continuous and uniformly
positive;
all admit a mean value and inf t≥0 q(t) − ν(t) > 0.
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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Application to predator-prey systems on lattice

Assumptions

As for the Fisher-KPP equation, we assume that the kernels
are:
Jk are nonnegative, Jk ∈ l1 (Z, L∞
t (0, ∞)),
thin-tailed and symmetric, inf t≥0 Jk (t, ±1) > 0,mean value andX
λ−1
hJk (·, j)ieλj → ∞ as λ → abs(Jk ).
j∈Z

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Application to predator-prey systems on lattice

Two linear speeds• If v ≡ 0, then u satisfiesXdu(t, i) =J1 (t, j)[u(t, i − j) − u(t, i)] + r(t)u(t, i) 1 − u(t, i) .
dtj∈Z

Spreading speed: c∗u :=

inf
λ∈(0,abs(J1 ))

−1

λ


X
λj
hJ1 (·, j)i[e − 1] + hri .
j∈Z

A. Ducrot

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Application to predator-prey systems on lattice

Two linear speeds• If v ≡ 0, then u satisfiesXdu(t, i) =J1 (t, j)[u(t, i − j) − u(t, i)] + r(t)u(t, i) 1 − u(t, i) .
dtj∈Z

Spreading speed: c∗u :=

inf
λ∈(0,abs(J1 ))

−1

λ


X
λj
hJ1 (·, j)i[e − 1] + hri .
j∈Z• If u ≡ 1, then v satisfiesXdv(t, i) =J2 (t, j)[v(t, i − j) − v(t, i)] + (q(t) − ν(t)) v(t, i).
dtj∈Z

Spreading speed: 

X
∗
−1
γj
cv :=
inf
γ
hJ2 (·, j)i[e − 1] + hq − νi .
γ∈(0,abs(J2 ))j∈Z
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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Application to predator-prey systems on lattice

Main results: slow predator case i.e. c∗u > c∗v

Theorem (Slow predator case)
Assume c∗u > c∗v . If the initial data are compactly supported then(u, v) satisfies:(i) lim sup u(t, i) = 0, ∀c > c∗u ;t→∞ |i|≥ct(ii) for all c∗v < c1 < c2 < c∗u and for all c > c∗v one has:

lim

sup

t→∞ c t≤|i|≤c t
1|1 − u(t, i)| = 0 and lim sup v(t, i) = 0;t→∞ |i|≥ct2(iii) for all c ∈ [0, c∗v ) one has:
lim inf inf v(t, i) > 0.
t→∞ |i|≤ct

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Application to predator-prey systems on lattice

Main results: slow predator case i.e. c∗u > c∗v

Theorem (Slow predator case)
Assume c∗u > c∗v . If the initial data are compactly supported then(u, v) satisfies:(i) lim sup u(t, i) = 0, ∀c > c∗u ;t→∞ |i|≥ct(ii) for all c∗v < c1 < c2 < c∗u and for all c > c∗v one has:

lim

sup

t→∞ c t≤|i|≤c t
1|1 − u(t, i)| = 0 and lim sup v(t, i) = 0;t→∞ |i|≥ct2(iii) for all c ∈ [0, c∗v ) one has:
lim inf inf v(t, i) > 0.
t→∞ |i|≤ct

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Application to predator-prey systems on lattice

Main results: fast predator case i.e. c∗u ≤ c∗v

Theorem (Fast predator case)
Assume c∗u ≤ c∗v and (u0 , v0 ) are compactly supported. Then(u, v) satisfies:(i) lim sup [u(t, i) + v(t, i)] = 0, ∀c > c∗u ;t→∞ |i|≥ct(ii) for all c ∈ [0, c∗u ) one has:
lim inf inf v(t, i) > 0,t→∞ |i|≤ct

0 < lim inf inf u(t, i) ≤ lim sup sup u(t, i) < 1.
t→∞ |i|≤ctt→∞

A. Ducrot|i|≤ct41/44

Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Application to predator-prey systems on lattice

Main results: fast predator case i.e. c∗u ≤ c∗v

Theorem (Fast predator case)
Assume c∗u ≤ c∗v and (u0 , v0 ) are compactly supported. Then(u, v) satisfies:(i) lim sup [u(t, i) + v(t, i)] = 0, ∀c > c∗u ;t→∞ |i|≥ct(ii) for all c ∈ [0, c∗u ) one has:
lim inf inf v(t, i) > 0,t→∞ |i|≤ct

0 < lim inf inf u(t, i) ≤ lim sup sup u(t, i) < 1.
t→∞ |i|≤ctt→∞

A. Ducrot|i|≤ct41/44

Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Application to predator-prey systems on lattice

Sketch of proof: comparision with F-KPP equation

Since u ≈ 0 implies v ≈ 0 we obtain

Lemma ( Pointwise estimates–(i) )For each δ > 0, there exist Mδ > 0 and Tδ > 0 s.t.
v(t, i) ≤ δ + Mδ u(t, i), ∀t ≥ Tδ , ∀i ∈ Z.

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Application to predator-prey systems on lattice

Sketch of proof: comparision with F-KPP equation

Since u ≈ 0 implies v ≈ 0 we obtain

Lemma ( Pointwise estimates–(i) )For each δ > 0, there exist Mδ > 0 and Tδ > 0 s.t.
v(t, i) ≤ δ + Mδ u(t, i), ∀t ≥ Tδ , ∀i ∈ Z.

X
du(t, i) =J1 (t, j)[u(t, i − j) − u(t, i)] + r(t)u(1 − u) − p(t)uv,
dt
j∈Z
X
≥
J1 (t, j)[u(t, i − j) − u(t, i)] + r(t)u(1 − u) − p(t)u(δ + M u)j∈ZHence u spreads faster than speed c∗u .
A. Ducrot

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Application to predator-prey systems on lattice

Sketch of proof

If c < c∗u and v ≈ 0 then u ≈ 1.

Lemma ( Pointwise estimates–(ii) )Fix c ∈ [0, c∗u ), for each α > 0, there exist Mα > 0 and Tα > 0 s.t.
1 − u(t, i) ≤ α + Mα v(t, i), ∀t ≥ Tα , ∀|i| ≤ ct.

A. Ducrot

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Application to predator-prey systems on lattice

Sketch of proof

If c < c∗u and v ≈ 0 then u ≈ 1.

Lemma ( Pointwise estimates–(ii) )Fix c ∈ [0, c∗u ), for each α > 0, there exist Mα > 0 and Tα > 0 s.t.
1 − u(t, i) ≤ α + Mα v(t, i), ∀t ≥ Tα , ∀|i| ≤ ct.

X
dv(t, i) =J2 (t, j)[v(t, i − j) − v(t, i)] + q(t)uv − ν(t)v
dt
j∈Z
X
≥
J2 (t, j)[v(t, i − j) − v(t, i)] + v(q(t) − ν(t) − α − M v), |i| ≤ ct.
j∈Z

KPP type equation on a growing domain |i| ≤ ct.
A. Ducrot

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Introduction

Travelling waves

Spreading speed

Fisher-KPP equation on a lattice and applications

Application to predator-prey systems on latticeThank you for your attention.

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"Ecocultural range-expansion model of modern humans in Paleolithic"
Joe Yuichiro Wakano (Meiji University, Japan)
Modern human range expansion and the resulting extinction or assimilation of archaic humans such as Neanderthals
took place roughly 50,000 years ago. This phenomenon is recently very actively studied by using genetic methods suchas ancient DNA analysis. In this talk, a reaction diffusion model is proposed with emphasis on archaeological andecological aspects. Range expansion dynamics are studied as the traveling wave solutions in the system.

This PDF is modified from the original slides

Ecocultural range-expansion model ofmodern humans in Paleolithic
Joe Yuichiro Wakano
School of Interdisciplinary Mathematical Sciences, Meiji Univin collaboration with

Kenichi Aoki (Meiji Univ), Seiji Kadowaki (Nagoya Univ)
William Gilpin, Marcus Feldman (Stanford Univ)
2023/10/31 - 11/2 ICMMA2023 (10/31 10:40-11:20)"International Conference on "Reaction-diffusion systems: from the past to the future“— in memory of Prof. Masayasu Mimura —Self introduction of J. Y. Wakano and my memory of Mayan 1998-2001 Ph.D student (Centre for Ecological Research, Kyoto Univ)
 Evolutionary game theory, models in behavioral ecology, etc.
 Japan Association for Mathematical Biology (JAMB), found in 1989
1989-1992 Chief: N. Shigesada, 1992-1994 Chief: M. Mimura 2001-2007 post-Doc (Univ. of Tokyo)
 Pattern formation of bacterial colony 2001 JAMB-SMB Joint Conference on Mathematical Biology（Hawaii, Hiro）
 2003 JSMB found (Japanese Society for Mathematical Biology）
Frequent visit to Mimura-sensei's lab at Ikuta Campus, Meiji University 2004 Mayan became a professor at School of Science and Technology, Meiji Univ.
 His ambition: foundation of a new research area
 2007-2013 Specially appointed associate professor, Meiji Univ.
 Meiji Institute for Advanced Study of Mathematical Sciences (MIMS), found in 2007
 Daishin Ueyama (professor) and H. Izuhara (grad student)
 2008-2012
 Global COE Program "Formation and Development of Mathematical Sciences Based onModeling and Analysis" （現象数理学の形成と発展） Leader: M. Mimura
 Mayan, Daishin, and I were central members to prepare the application
 Mayan looked so nervous before and after the interview. He looked so happy when wegot a good news
 K. Ikeda (post-Doc)Self introduction of J. Y. Wakano and my memory of Mayan 2009-2012
 PRESTO (さきがけ) "Innovative model of biological processes and its
development"
Mayan was one of advisory professors, and pushed me hard to win this grant Face-to-face lessons on how to talk, a treasure in my life 2011 Graduate School of Advanced Mathematical Sciences, Meiji Univ Y. Tanaka (a grad student)
 2013 Interdisciplinary Mathematical Sciences, Meiji Univ
 Department of Mathematical Sciences Based on Modeling and Analysis （現象数理学科） 2013-2014 JSMB President: M. Mimura

Chief: J .Y. Wakano

 2021 Mayan passed away
Zu J, Mimura M & Wakano JY (2010) The evolution of phenotypic traits in a predator-preysystem subject to Allee effect. Journal of
Theoretical Biology 262:528-543

Wakano JY, Ikeda K, Miki T & Mimura M (2011) Effective dispersal rate is a function of habitat size and corridor shape: mechanistic
formulation of a two-patch compartment model for spatially continuous systems. Oikos 120: 1712-1720

Scotti T, Mimura M & Wakano JY (2015) Avoiding Toxic Prey May Promote Harmful Algal Blooms. Ecological Complexity 21:157165

joint project of archeology, environmental sciences, cultural anthropology, andmathematical modeling and analysis

Integrative research on the formative processes of modern human cultures in Asia

Range expansion of modern humans

Second wave of out-of-Africa

Neanderthals

Denisovan/
Neanderthals/
H. erectus

100～50 Kya.

Empty

Sapiens

Empty

5

Range expansion of modern humans

Extinction of archaic humans～30 Kya.

sapiens

Empty

sapienssapiens

Empty
sapiens6

Range expansion of modern humans

Oppenheimer 2012

Pruffer et al. (2014)

Slon et al. (2018)

Svante Paebo (2022 Nobel Prize)

8Bae et al. (2017, Science)

9

Cultural Evolution in Paleolithic
10kya

Upper Homo
Paleolithic transition (上部旧石器革命）
sapienssapiens50kya
Homosapiens
neanderthalensis
250kyaHomoerectus10Spatial dynamics of genes (species)
vsSpatial dynamics of culture• Culture is not completely determined by genes (species)– cultural inheritance is not identical to genetic inheritance• "Modern humans"• "Modern humans"by their DNAsby their behavior and culture• Here we define modern humans by their DNAs, and aim to
study how spatial dynamics of modern human culture are11
related to those of modern humans

What makes cultural differences ?
learning hypothesis• Innate (genetic) difference in learning ability• Wakano & Miura (2014), Aoki & Feldman (2014), Lehmann et al. (2013), Aoki etal. (2012), … and much more

modernhumansNeanderthals

smart ??

stupid ??

12

What makes cultural differences ?
population size hypothesis• Innate (genetic) difference in learning ability• Wakano & Miura (2014), Aoki & Feldman (2014), Lehmann et al. (2013), Aoki etal. (2012), … and much more

• Small population results in low culture– Henrich (2004) claims that the loss of adaptive culture is triggered bydecreased population size by using empirical data in Tasmania andanalyzing mathematical models.
– population decrease ⇒ lower culture– population increase ⇒ higher culture– Many theoretical studies (Shennan 2001; Henrich 2004; Strimling et al.
2009; Mesoudi 2011; Lehmann et al. 2011; Aoki et al. 2011; Kobayashi &Aoki 2012; Forgaty et al 2015)13

Positive feedback loop
between culture and population size
• Low/high culture results in small/large population– Previous studies• Ghirlanda & Enquist 2007; Aoki 2015; Gilpin et al. 2016

– Positive spiral• population increase ⇒ higher culture ⇒ population even increases ⇒ ...

– Negative spiral• population decrease ⇒ lower culture ⇒ population even decreases ⇒ ...

• Different states can appear in genetically homogeneous
population.
• Previous models of positive feedback loop
did not explicitly model spatial structure
14

Single species model

15

Model Assumptions• Culture of an individual is either skilled or non-skilled– Newborns socially learn culture from a random individual of the same
species at the same location– Skill is lost at rate γ (bad memory, lack of practice)– Non-skilled changes to skilled at rate δ (individual learning)• Carrying capacity is increased when local population
carries more skilled individuals– Skill gives benefit to local population– Population size hypothesis• Individuals randomly migrate in 1-D continuous space– modeled by diffusion equation– migration of skilled individuals results in spatial spread of skills
16

Reaction diffusion system(single species case)skilledchanges tonon-skilled

population density(skilled + non-skilled)



2
N 
N ( x, t )  D 2 N  rN 1 

t
x
M
(
Z
)



density ofskilled individuals



2
N 
Z ( x, t )  D 2 Z  rZ 1  Z   ( N  Z )tx M (Z ) increase of skilled
due to newborns
learning skill

carrying capacity

ML
M (Z )  
M H( Z  Z *)( Z  Z *)non-skilledchanges to skilled

Low, when skilled < Z*
High, when skilled > Z*
17

Critical density of skilled, Z*N

MH
total density

two locallystable equilibria( N , Z )  ( M L ,M L )( N , Z )  ( M H ,M H )
ML




 

Z

Z*
density of skilled individuals

18

Traveling wave solution (TWS)
competition between populations at "high" and "low" equilibriums


2
N 
N ( x, t )  D 2 N  rN 1 tx
 M (Z ) 


2
N 
Z ( x, t )  D 2 Z  rZ 1 
  Z   ( N  Z )
t
x
M
(
Z
)



「High density +MHHigh skilled density」

density Nv ?
skilled Z

ML「Low density +Low skilled density」

x
19

Special solution (invariant manifold)


2
N 
N ( x, t )  D 2 N  rN 1 tx
 M (Z ) 


2
N 
Z ( x, t )  D 2 Z  rZ 1 
  Z   ( N  Z )
t
x
M
(
Z
)



Z ( x, t )  N ( x, t )N ( x, t )  u ( x, t )Z ( x, t )  u ( x, t )




 

is a special solution of the systeminnovation-forgetting balance


u
 2u
u 
 D 2  ru 1 

t
x
M
(

u
)


This invariant manifold is globally attracting in PDE sense (maximum principle).
20

Bistable TWS

u
 2u
u 
 D 2  ru 1 

t
x
M
(

u
)



e.g.) Allen-Cahn eq.

M
M (Z )   L
M H( Z  Z *)( Z  Z *)Reaction term dynamicsu0u  MHu  MLu  Z * /The direction of TWS is determined by the sign of3
 M H 2  M L2  1  1
MH 

u 
u 
1  Z * 
M L u1  M L du  z*/ u1  M H du   6   3  M L  M H   z */ 

21

Traveling wave can progress in either direction

MH

u 0

ML

Z * /

MH
ML

MH

u 0

ML

Z * / M H

ML

The high equilibrium does not always win.
22

M H  18

ML  6

9 skilled + 9 non-skilled

3 skilled + 3 non-skilled

AfricaAsia

Both high and low equilibria are locally stable.
23M H  18

ML  6

9 skilled + 9 non-skilled

3 skilled + 3 non-skilled

Z*  4AfricaAsia

When they start to randomly migrate,
high equilibrium (advanced culture) does or does not spread.
24

M H  18

ML  6

9 skilled + 9 non-skilled

3 skilled + 3 non-skilled

Z*  8AfricaAsia

When they start to randomly migrate,
high equilibrium (advanced culture) does or does not spread.
25

Condition for high equilibrium to spatially invade
low equilibrium
 M M (M H  M L ) 
Z*    L H

2





 

Z *    (1   )  K      2 

1
3

, M H  M L  K

1
3

• When lower number of skilled is required
to transition to high carrying capacity• or when skill is easier to obtain / harder to lose
• or when skill has larger impact on carrying capacity26

Two species model

27

(additional) Assumptions• Two species: modern humans and Neanderthals• No innate difference in abilities of the two species– the same learning ability, the same demographic ability• Within each species, carrying capacity is increased whenlocal population carries more skilled individuals• Ecological resource competition between the two species– weaker than intraspecific competition– different species use difference niches with some overlap

28

Lotka-Volterra type reaction diffusion system(Neanderthal vs. modern humans)

Neanderthals

modernhumans N  bN 2 2
N1 ( x, t )  D 2 N1  rN1 1  1

t
x
M
(
Z
)
1


 N1  bN 2 2
Z1 ( x, t )  D 2 Z1  rZ1 1 
  Z1   ( N1  Z1 )tx
M ( Z1 )  N  bN1 2
N 2 ( x, t )  D 2 N 2  rN 2 1  2

t
x
M
(
Z
)
2


 N 2  bN1 2Z 2 ( x, t )  D 2 Z 2  rZ 2 1 
  Z 2   ( N 2  Z 2 )tx
M (Z 2 ) (Intraspecific) Carrying capacity(Interspecific) Competition coefficientML
M (Z )  M H
0  b 1

( Z  Z *)( Z  Z *)29

niche overlap

 N  bN 2 2
N1 ( x, t )  D 2 N1  rN1 1  1tx
M ( Z1 )  N  bN 2 2
Z1 ( x, t )  D 2 Z1  rZ1 1  1
  Z1   ( N1  Z1 )
t
x
M
(
Z
)
1


 N  bN1 2
N 2 ( x, t )  D 2 N 2  rN 2 1  2tx
M (Z 2 )  N  bN1 2Z 2 ( x, t )  D 2 Z 2  rZ 2 1  2
  Z 2   ( N 2  Z 2 )tx
M (Z 2 ) ZN
N i i2  Z i 2i

1
x
x
Ai ( x, t )   (1  Ai )  D
2
t

Ni
2

Zi ( x, t )  Ni ( x, t )Ai ( x, t ) :Z i ( x, t )N i ( x, t )Global attractor exists: lim t  Ai ( x, t )  innovation-forgetting balance N1  bN 2 
N1
 2 N1DrN
1 1 

2
t
x
M
(

N
)
1 

 N 2  bN1 
N 2
2 N2
D
 rN 2 1 

2
t
x
M
(

N
)
2 


30

When interspecific competition (b) is very weak

ML 

z*



 MH 

ML
b

N2
ML /b



MH


 

z* / 

MLO

ML

z* / 

MH

ML /b

N1

31

When interspecific competition (b) is intermediate

ML 
N2

z*





ML
 MH
b


MH


 

ML /b
z* / 

MLO

ML

z* / 

ML /b

MH

N1

32

Spatial dynamics of population and culture• Depending on initial distribution, various dynamics occur.
• For explanation, we initially set
Modern humans only,
high equilibrium

AfricaNeanderthals only,
low equilibrium

space(For simplicity, we only consider cases when high equilibrium defeats low.)

Eurasia33

Skilled densities (Z1, Z2)
are not shown since we always
observed quick convergence toInitial condition (t=0)

Zi ( x, t )  Ni ( x, t )density

African side (left)
No Neanderthals. Modern humans at high-skill-high-density equilibrium
Eurasian side (right)
Neanderthals at low-skill-low-density equilibrium. No modern humans.

ModernAfrica𝑁2

Nean 𝑁1𝑥

Eurasia34

Modern Humandensity

Neanderthal densityinterspecific competition, bvery weakintermediate

Modern Humans

Neanderthals

36

Case A : very small b
Modern Humans

Neanderthals

Between the initial two states,
two additional states appear(N, MH) = (low, high) : Final state, coexistence of the two species(N, MH) = (low, low)

: Temporal stateThis state can last very long at locations
very far away from the initial contact

Temporal symmetric coexistence might accelerate genetic interbreeding.
Final asymmetric coexistence might result in complete assimilation of
Neanderthals by modern humans.

Very weak interspecific competition ⇒ assimilation scenario

Case B : intermediate b
Modern Humans

Between the initial two states,
one additional state appear

Neanderthals(N, MH) = (zero, high) : Final state(N, MH) = (low, low)

: Temporal stateThis state can last very long at locations
very far away from the initial contact

Finally, modern humans replace Neanderthals.
Local temporal coexistence persists long only at locations very far.
Limited genetic interbreeding is expected.

Intermediate interspecific competition ⇒ Replacement

The direction of the second wave:
bistable case
N2
ML /b

•

MH

Approximating a TWS trajectory by aline segment, we obtain N 2  b(M L  bN 2 ) 
N 2 2 N2
D
 rN 2 1 

t
x 2
M
(

N
)
2



z* / 

•MLOz* / 

ML

ML /b

MH

This yields the (approximate)condition for ‘high’ equilibrium
spatially spreads:
N1

P( z*)  ( M H bM L ) /(1b 2 )M L /(1b )High density region spreads in TWS if
32(1  b 2 ) Z * n  b( M L  bn2 ) 
rn2 1  2
dn2
M 2 ( 2 n2 )

is small s.t.



 (1  b 2 ) Z *  (1  b 2 ) Z *   3bM L   M L M H 2  (1  3b) M H M L  b3 M L 2  0
2







39

Numerics agree with these analytic predictions

linearconjecture
speed

analytically approximated Z* value
for which zero-speed wave exits

Comparison with archaeological data41

Early Epipaleolithic

Bladelets

Tor Hamar F (n = 4907)

EUP (Early Ahmarian)new

Tor Hamar G (n = 4291)

IUP-EUPTor Fawaz (n = 1494)

IUP (Emiran)
Wadi Aghar (Kadowaki 2017)(n = 317)

Late MPTor Faraj(n = 767)

oldTwo dispersals of modern humans
Archaeological records from Levant to Europe

Hubrin 2015kya

Second Dispersal(42ka）First dispersal(48ka）

Bohunician

Bachokirian

Proto-Aurignacian
Early Ahmarian

Emiran /44
Initial Upper Palaeolithic

First Dispersal (Initial Upper Paleolithic)
Emiran（Levant）

Belfer-Cohen and Goring-Morris
2007

Bohunician（Europe)

Skrdla 2003 (Stranska skala)45

Second Dispersal（42ka）

Mellars 2011 (Nature, 479)

Early Ahmarian（Levant）

BladeletBladeletBordes 2006
(Le Piage level K)

47
Goring-Morris and Belfer Cohen 2006

Summary• A model of spatial dynamics of human population and culture– Positive feedback loop between population density and culture– Cognitive and demographic equivalence of Neanderthals and modern
humans– They interact through ecological resource competition• High-density-high-culture equilibrium does not always spatiallyspread even if it is locally stable• Raplacement of Neanderthals by modern humans is possible when
ecological niche overlap is high (ML/MH<b<1)• Range-expansion of modern human might have occurred in two
major waves with different speeds– “First” wave : Ecological invasion due to different niche utilization– “Second” wave : Driven by high-density-high-culture equilibrium48

Thank you

Seiji
Kadowaki

Wakano JY, Gilpin W, Kadowaki S, Feldman MW, Aoki K (2018) Ecocultural range-expansion
scenarios for the replacement or assimilation of Neanderthals by modern humans
Theoretical Population Biology 119:3-14

49

"Traveling wave solutions for a three-species diffusive competition system"
Jong-Shenq Guo (Tamkang University, Taiwan)
We shall discuss the existence and stability of traveling waves for a three-species diffusive competition system.
This talk is based on some recent joint works with Karen Guo and Masahiko Shimojo.

Introduction
Existence
Stability

Traveling wave solutions for a three-species
diffusive competition systemJong-Shenq Guo (TKU)
Tamsui, New Taipei City, Taiwan
Joint work with K. Guo (PU) and M. Shimojo (TMU)

Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

1 / 29

Introduction
Existence
Stability§1. Introduction
Consider the following diffusive 3-species competition systemu = uxx + r1 u(1 − u − a2 v − a3 w), x ∈ R, t > 0,

 t
(1)
vt = vxx + r2 v(1 − b1 u − v − b3 w), x ∈ R, t > 0,
wt = wxx + r3 w(1 − c1 u − c2 v − w), x ∈ R, t > 0,
where u, v, w are the densities of three competing species.
r1 , r2 , r3 the intrinsic growth rates, and a2 , a3 , b1 , b3 , c1 , c2
the interspecific competition coefficients are positive
the carrying capacity of each species is normalized to be 1
we assume the diffusivities of all species are equal to 1
Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

2 / 29

Introduction
Existence
Stability

Due to the normalization of the carrying capacities, thata2 > 1(a2 < 1) means v is a strong (weak) competitor to u.
For example, {a2 > 1, b1 > 1} means u, v are strong
competing species, etc.
When b3 < 1 and c2 < 1, we have the semi-coexistencestate Ecu := (0, vc , wc ):vc =1 − b31 − c2∈ (0, 1), wc =∈ (0, 1).
1 − b3 c2
1 − b3 c2

It is also possible to have the co-existence stateE∗ := (u∗ , v∗ , w∗ ) with u∗ , v∗ , w∗ ∈ (0, 1).

Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

3 / 29

Introduction
Existence
Stability

Traveling wave solution
A traveling wave of (1) is a solution (u, v, w) of (1) in the form(u, v, w)(x, t) = (ϕ1 , ϕ2 , ϕ3 )(z), z := x − st,for some constant (the wave speed) s ∈ R and some functions(the wave profiles) {ϕ1 , ϕ2 , ϕ3 }.
Problem: find unknown {s, ϕ1 , ϕ2 , ϕ3 } such that
ϕ′′ + sϕ′1 + r1 ϕ1 (1 − ϕ1 − a2 ϕ2 − a3 ϕ3 ) = 0, z ∈ R,

 1
(2)
ϕ′′2 + sϕ′2 + r2 ϕ2 (1 − b1 ϕ1 − ϕ2 − b3 ϕ3 ) = 0, z ∈ R,
ϕ′′ + sϕ′ + r3 ϕ3 (1 − c1 ϕ1 − c2 ϕ2 − ϕ3 ) = 0, z ∈ R.
3
3

Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

4 / 29

Introduction
Existence
Stability

Two asymptotic states:(3)(ϕ1 , ϕ2 , ϕ3 )(±∞) = E± ,where E± are two different constant states of (1).
The wave is called a monostable wave if one of {E± } isunstable and the other is stable
It is bistable if both states E± are stable
When s > 0, we call E− the invading state and E+ theinvaded state. Roles are exchanged if s < 0.

Difficulty: comparison principle does not hold for 3-species
competition systems.
Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

5 / 29

Introduction
Existence
Stability

Mimura’s works on 3-species competition systems
Chen, Hung, Mimura, Tohma, Ueyama, Exact travelling wave solutions of three-species
competition-diffusion systems, DCDS-B (2012)
Chen, Hung, Mimura, Tohma, Ueyama, Semi-exact equilibrium solutions for three-species
competition-diffusion systems, Hiroshima Math. J. (2013)
Mimura, Tohma, Dynamic coexistence in a three-species competition-diffusion system, Ecol. Compl. (2015)
Contento, Mimura, Tohma, Two dimensional travelling waves arising from planar front interaction in a three
species competition diffusion system, JJIAM (2015)
Contento, Hilhorst, Mimura, Ecological invasion in competition-diffusion systems when the exotic species iseither very strong or very weak, J. Math. Biol. (2018)
Chang, Chen, Hung, Mimura, Ogawa, Existence and stability of non-monotone travelling wave solutions forthe diffusive Lotka-Volterra system of three competing species, Nonlinearity (2020)
Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

6 / 29

Introduction
Existence
Stability

In the work in 2020 (Nonlinearity) for general di , they studyu, w are strong competitors: a3 > 1, b1 > 1, b3 > 1, c1 > 1
v is a very weak competitor: 0 < a2 , c2 ≪ 1
TWS with E− = (1, 0, 0), E+ = (0, 0, 1): a bistable wave
by the bifurcation theory and the method of super-sub-solutions
See also Chang, Chen, JDDE (2023) for a related work.
The sign of wave speed is an open question?
Our aim: monostable waves?? easier!

Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

7 / 29

Introduction
Existence
Stability§2-1. Main results: existence
Assume there are two aboriginal weak competing species
v and w in the sense b3 < 1 and c2 < 1.
Let u be an alien species.
Assume that the semi-coexistence state Ecu = (0, vc , wc ) is
unstable for the ODE system of (1):β := 1 − a2 vc − a3 wc > 0.

(4)

Note that condition (4) can be achieved, e.g., whena2 + a3 < 1.

Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023(5)8 / 29

Introduction
Existence
Stability

Theorem 1
Let b3 < 1 and c2 < 1. Suppose, in addition to condition (4), thatr1 β ≥ max{r2 (b1 + b3 c2 vc ), r3 [c1 + c2 (1 − vc )]}.

(6)√
Set s∗ := 2 r1 β. Then there is a positive solution (ϕ1 , ϕ2 , ϕ3 ) of(2) satisfying(ϕ1 , ϕ2 , ϕ3 )(+∞) = Ecu = (0, vc , wc )for any s ≥ s∗ .

Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

9 / 29

Introduction
Existence
Stability

Assume w is an aboriginal weak competitor.
Let σ+ := max{0, σ} for σ ∈ R.
Theorem 2
Let a3 < 1 and b3 < 1. Assumer2 (1 − b3 ) = r1 (1 − a3 ) ≥ r3 (c1 + c2 − 1)+ .

(7)p
Set s∗ := 2 r1 (1 − a3 ). Then there exists a positive solution(ϕ1 , ϕ2 , ϕ3 ) of (2) satisfying(ϕ1 , ϕ2 , ϕ3 )(+∞) = Ew = (0, 0, 1)
for any s ≥ s∗ .
Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

10 / 29

Introduction
Existence
Stability

A strong alien competitor u is introduced to the habitat of two
aboriginal competing species v and w:
Theorem 3
Let {s, (ϕ1 , ϕ2 , ϕ3 )} be a traveling wave obtained in Theorem 1
with s ≥ s∗ or Theorem 2 with s ≥ s∗ . Then
(ϕ1 , ϕ2 , ϕ3 )(−∞) = Eu = (1, 0, 0), if we assume
a2 b1 ≥ 1, a3 c1 ≥ 1, a2 + a3 < 1.

(8)

Theorem 3 shows that system (1) has traveling waves of
mixed front-pulse type connecting Ew and Eu for anys ≥ s∗ , under conditions in Theorem 2 and (8).
Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

11 / 29

Introduction
Existence
Stability• Three species are all weak competitors:
Under the assumption
a2 + a3 < 1, b1 + b3 < 1, c1 + c2 < 1,(9)the co-existence state E∗ = (u∗ , v∗ , w∗ ) exists and E∗ is stable.
Theorem 4
Let {s, (ϕ1 , ϕ2 , ϕ3 )} be a traveling wave obtained in either
Theorem 1 with s ≥ s∗ or Theorem 2 with s ≥ s∗ . Then
(ϕ1 , ϕ2 , ϕ3 )(−∞) = (u∗ , v∗ , w∗ ), if condition (9) is enforced.

Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

12 / 29

Introduction
Existence
Stability§2-2. Proofs for existence
The main idea of the proofs of Theorems 1 and 2 is toconstruct suitable (generalized) upper-lower solutions
to capture the unstable tails Ecu and Ew at +∞, resp.
For the derivation of the stable tail limit at z = −∞, weapply the classical method of contracting rectangles(cf. e.g., Huang-Lin (JMAA14), Lin-Ruan (JDDE14),
Chen-G.-Yao (JMAA17), G.-Nakamura-Ogiwara-Wu(NA-RWA20), Chen-Giletti-G. (JDE21)).
Existence can allow non-equal diffusivities.

Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

13 / 29

Introduction
Existence
Stability

Definition 5
A pair of continuous functions (ϕ1 , ϕ2 , ϕ3 ) and (ϕ1 , ϕ2 , ϕ3 ) is
called a pair of upper-lower solutions of (2), if′′′
ϕi (z), ϕ′′i (z), ϕi (z), ϕ′i (z), i = 1, 2, 3, exist such that
′′

′

′′

′

′′

′

U1 (z) := ϕ1 (z) + sϕ1 (z) + ϕ1 (z)g1 (ϕ1 , ϕ2 , ϕ3 )(z) ≤ 0,
U2 (z) := ϕ2 (z) + sϕ2 (z) + ϕ2 (z)g2 (ϕ1 , ϕ2 , ϕ3 )(z) ≤ 0,
U3 (z) := ϕ3 (z) + sϕ3 (z) + ϕ3 (z)g3 (ϕ1 , ϕ2 , ϕ3 )(z) ≤ 0,
L1 (z) := ϕ′′1 (z) + sϕ′1 (z) + ϕ1 (z)g1 (ϕ1 , ϕ2 , ϕ3 )(z) ≥ 0,
L2 (z) := ϕ′′2 (z) + sϕ′2 (z) + ϕ2 (z)g2 (ϕ1 , ϕ2 , ϕ3 )(z) ≥ 0,
L3 (z) := ϕ′′3 (z) + sϕ′3 (z) + ϕ3 (z)g3 (ϕ1 , ϕ2 , ϕ3 )(z) ≥ 0
hold for z ∈ R except for a finite subset E of R.
Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

14 / 29

Introduction
Existence
Stability

Proposition 1
Given s > 0. Suppose that (2) has a pair of upper-lower
solutions (ϕ1 , ϕ2 , ϕ3 ) and (ϕ1 , ϕ2 , ϕ3 ) such that ϕi ≤ ϕi and′′lim ϕi (z) ≤ lim ϕi (z), lim ϕ′i (z) ≤ lim ϕ′i (z), ∀ z ∈ E,

z→z +

z→z −

z→z −

z→z +

for i = 1, 2, 3. Then (2) has a solution (ϕ1 , ϕ2 , ϕ3 ) such thatϕi ≤ ϕi ≤ ϕi , i = 1, 2, 3.
A pair of upper-lower solutions serves as the upper andlower bounds for the domain of the integral operator
corresponding to differential system (2) so that Schauder’s
fixed point theorem can be applied to obtain a solution of(2) in this domain (an idea of Ma (JDE01)).
Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

15 / 29

Introduction
Existence
Stability

The construction of upper-lower solutions relies on thelinearization of the equation of alien species in (2) at theunstable state, e.g., E+ = Ecu :λ2j + sλj + r1 β = 0, j = 1, 2, λ2 ≤ λ1 < 0 ↔ eλ1 z , zeλ1 z
so that for s > s∗ :ϕ1 (z) := min{1, eλ1 z }, ϕ1 (z) := max{0, eλ1 z − peµz },ϕ2 (z) := min{1, vc + (1 − vc )eλ1 z }, ϕ2 (z) := max{0, vc (1 − eλ1 z )},ϕ3 (z) := min{1, wc + c2 vc eλ1 z }, ϕ3 (z) := max{0, wc (1 − eλ1 z )},where p and µ are suitably chosen constants.
Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

16 / 29

Introduction
Existence
Stability

To derive the stable tail limit at z = −∞, we first let+
ϕ−i := lim inf ϕi (z), ϕi := lim sup ϕi (z), i = 1, 2, 3.
z→−∞

z→−∞

Since ϕi ≥ 0, i = 1, 2, 3, by the maximum principle we have0 ≤ ϕi ≤ 1, i = 1, 2, 3. Hence+0 ≤ ϕ−i ≤ ϕi ≤ 1, i = 1, 2, 3.

Then we haveϕ−
1 ≥ γ1 := 1 − a2 − a3 > 0,

(10)provided a2 + a3 < 1.
Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

17 / 29

Introduction
Existence
Stability

We only give a proof of Theorem 3.
For this, we define m1 (θ) := (1 − θ)γ1 /2 + θ, θ ∈ [0, 1].
Since γ1 < 1, m1 (θ) is increasing in θ ∈ [0, 1] such thatm1 (1) = 1. Let
A := {θ ∈ [0, 1) | ϕ−
1 > m1 (θ)}.
By (10), 0 ∈ A and so the quantity θ0 := sup A is well-definedsuch that θ0 ∈ (0, 1]. Then ϕ−
1 ≥ m1 (θ0 ) and+ϕ+
2 ≤ max{0, 1 − b1 m1 (θ0 )}, ϕ3 ≤ max{0, 1 − c1 m1 (θ0 )}.

Finally, a contradiction argument leads to θ0 = 1.
Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

18 / 29

Introduction
Existence
Stability§3-1. Main results: stability
Given positive constants {σi }, define a distance function (to Φ)
K[U ] :=3X

σi Ki [Ui ],Ki [Ui ] := Ui − ϕi − ϕi ln

i=1

Ui
,
ϕi

(11)

for any positive function U = (U1 , U2 , U3 ) defined on R.
Note that K[U ](z) ≥ 0 for all z ∈ R and K[U ](z) = 0 if and only if
U (z) = Φ(z) for some z ∈ R.
For a positive constant R, we let√√−s + s2 − 4R
λ = λ(s; R) :=, s ≥ 2 R.
2
Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023(12)
19 / 29

Introduction
Existence
Stability

Suppose that there exists a set of positive constants{σ1 , σ2 , σ3 } such that
− I :=3X

σi (ui − vi ) {gi (u) − gi (v)} ≤ 0

(13)

i=1

for all u = (u1 , u2 , u3 ), v = (v1 , v2 , v3 ) ∈ I := [0, 1]3 , whereg (u , u , u ) := r1 (1 − u1 − a2 u2 − a3 u3 ), 1 1 2 3
g2 (u1 , u2 , u3 ) := r2 (1 − b1 u1 − u2 − b3 u3 ),g3 (u1 , u2 , u3 ) := r3 (1 − c1 u1 − c2 u2 − u3 ).
Using the moving coordinate z = x − st, (1) is written as(ui )t = (ui )zz + s(ui )z + ui gi (u), z ∈ R, t > 0, i = 1, 2, 3. (14)Jong-Shenq Guo (TKU)

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Introduction
Existence
Stability

Theorem 6 (3 weak competitors)
Assume, in addition to (9), that either a2 b3 c1 = a3 b1 c2 or
a2 + b1 + a3 + c1 ≤ 2, a2 + b1 + b3 + c2 ≤ 2, a3 + c1 + b3 + c2 ≤ 2.
Let Φ be a positive traveling wave of (1) connecting
E+ ∈ {(0, 0, 1), (0, vc , wc )} and E− = E∗ with wave speed√s ≥ 2 R, where R := max{r1 , r2 , r3 }. Then {s, Φ} is stable inthe sense that u(z, t) → Φ(z) as t → ∞ locally uniformly forz ∈ R for any solution u of (14) with initial data u0 at t = 0
satisfying e−λz K[u0 ] ∈ L1 (R), where λ = λ(s; R) is defined in(12) and the positive constants {σ1 , σ2 , σ3 } in (11) are chosen
so that (13) holds.
Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

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Introduction
Existence
Stability

Theorem 7 (2 weak vs 1 strong competitors)
Assume, in addition to
a2 + a3 < 1, b3 , c2 < 1, b1 ≥ 1/a2 , c1 ≥ 1/a3 ,(15)

a2 b1 = 1 = a3 c1 , 2a2 a3 = a22 b3 + a23 c2 .

(16)

that

Let Φ be a positive traveling wave of (1) connecting
E+ ∈ {(0, 0, 1), (0, vc , wc )} and E− = Eu with wave speed√s ≥ 2 R, where R := max{r1 , r2 , r3 }. Then {s, Φ} is stable inthe sense described in Theorem 6.

Jong-Shenq Guo (TKU)

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Introduction
Existence
Stability

At the stable tail of traveling wave, i.e., z = −∞, theperturbation is allowed to be arbitrarily large due tocondition e−λz K[u0 ] ∈ L1 (R). Note that λ < 0.
However, at the unstable tail (z = ∞), the perturbation can
only be made with decay rate faster than eλz .
Note that the exponent λ is a function of the wave speed s.
This is a typical phenomenon in the stability of monostablewaves in many reaction-diffusion systems, including theirdiscrete analogues.
Note that R > r1 (1 − a3 ) and R > r1 β. The stability of√wave with speed s < 2 R is still left open.

Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

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Introduction
Existence
Stability§3-2. Proofs for stability
Let Ψ(z, t) := K[u(·, t)](z), z ∈ R, t > 0. Since
max sup{gi (Φ(z))} ≤ max{r1 , r2 , r3 } := R,1≤i≤3 z∈R

we can check that Ψ satisfies
Ψt ≤ Ψzz + sΨz + RΨ, z ∈ R, t > 0,(17)

if (13) holds for some {σ1 , σ2 , σ3 }.
From (17) and e−λz K[u0 ] ∈ L1 (R), by comparison, Theorems 6
and 7 are proved.

Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

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Introduction
Existence
Stability

Check the condition (13): I ≥ 0

Let u := (u1 , u2 , u3 ), v := (v1 , v2 , v3 ) andX := u1 − v1 , Y := u2 − v2 , Z := u3 − v3 .
Then I is computed as
I = σ1 r1 X 2 + σ2 r2 Y 2 + σ3 r3 Z 2 + (σ1 r1 a2 + σ2 r2 b1 )XY+(σ2 r2 b3 + σ3 r3 c2 )Y Z + (σ1 r1 a3 + σ3 r3 c1 )XZ.

Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

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Introduction
Existence
Stability

Three weak competitors
Case 1. a2 b3 c1 = a3 b1 c2 : choose
σ1 =

1
a2
a3
, σ2 =, σ3 =
,
r1
r2 b1
r3 c1

then we can writea2a3(1 − b1 − b3 )Y 2 + (1 − c1 − c2 )Z 2
b1
c1
a2 b3
+a2 (X + Y )2 + a3 (X + Z)2 +(Y + Z)2 .
b1

I = (1 − a2 − a3 )X 2 +Jong-Shenq Guo (TKU)

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Introduction
Existence
Stability

Case 2. condition
a2 +b1 +a3 +c1 ≤ 2, a2 +b1 +b3 +c2 ≤ 2, a3 +c1 +b3 +c2 ≤ 2. (18)
Choosing σi = 1/ri , i = 1, 2, 3, we obtain that
I = X 2 + Y 2 + Z 2 + (a2 + b1 )XY + (a3 + c1 )XZ + (b3 + c2 )Y Z, 1(a2 + b1 )/2 (a3 + c1 )/2 X = [X, Y, Z] (a2 + b1 )/2
1(b3 + c2 )/2   Y  .
(a3 + c1 )/2 (b3 + c2 )/2

1

Z

Then I ≥ 0 under condition (18), by Gerschgorin’s Theorem.

Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

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Introduction
Existence
Stability

Two weak and one strong competitors
Choosing
σ1 =

a2
a3
1
, σ2 =, σ3 =
,
r1
b1 r2
c1 r3

from (16) and (15), we can write
 

1
a2
a3
X

 
2
I = [X, Y, Z] a2 a2 a2 a3   Y  .
a3 a2 a3 a23
Z
Since B has the eigenvalues {0, 0, 1 + a22 + a23 } and so B issymmetric positive semi-definite, we obtain I ≥ 0.
In fact, we have I = (X + a2 Y + a3 Z)2 . Q: any other conditions?
Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

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Introduction
Existence
Stability

Thank you for your listening!

Jong-Shenq Guo (TKU)

ICMMA2023, MIMS, 10/31-11/2/2023

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"Stability of single transition layer solutions in mass-conserving
reaction-diffusion systems with bistable nonlinearity"
Hideo Ikeda（University of Toyama, Japan）
Mass-conserving reaction-diffusion systems with bistable nonlinearity are considered under general assumptions,which are useful models for studying cell polarity formation, whose process is key in cell division and differentiation.
The existence of stationary solutions with a single internal transition layer is shown by using the analytical singular
perturbation theory. Moreover, a stability criterion for the stationary solutions is provided by calculating the Evansfunction. This is a joint work with Masataka Kuwamura of Kobe University.

Stability of single transition layer solutions
in mass-conserving reaction-diffusion systems
with bistable nonlinearity
University of Toyama
Department of Mathematics
Hideo Ikeda
Jointly with
Masataka Kuwamura
Kobe University
ICMMA2023
Reaction-diffusion systems: from the past to the future
dedicated to the memory of Prof. Masayasu Mimura.
October 31 – November 2, 2023, on the Nakano
Campus of Meiji University, Tokyo.

1

In Memoriam of Professor Mayan Mimura from my youth
・ First EncounterA camp-style C&A summer seminar in 1978. This seminar was a
study group where participants related to computation andanalysis from companies and universities gathered for threedays and two nights, eating and sleeping together.
(2泊3日の合宿セミナー)
My first impression of Mayan
He was serious and strict during the daytime seminar.
However, at night, he was very friendly and had a sense of humor.
I liked him at night.
This is a scene from that night.

Night scene in a Japanese-style room2

・ Opportunity to start the analytical singular perturbation method (1980)
He introduced the analytical singular perturbation method to Japan as a souvenir when hecame back from Oxford University. I was very interested in the method of constructing anapproximate solution for each domain and then successfully combining them at the end.
The content was an introduction of a paper by P.C. Fife

・Mayan‘s Intensive Lecture for Undergraduate Students at my university (1982)
The subject matter was something I had never heard before in a mathematics class.
The title was "A Model of Pulse Propagation on Neuronal Axons (Hodgkin-Huxley Equation).
------ Differential equations that won the Nobel Prize in Physiology and Medicine in 1963
I wondered for a moment if this was really mathematics. I thought for a moment,
but as I listened, differential equations naturally came to my face. At that time, I felt
freshness and interest in this kind of mathematics.

・Start of joint research with Mayan and T. Tsujikawa (1983)

3

Hodgkin-Huxley Equation

Calculated by NEC PC-9801 (MS-DOS) + BASIC

4

H. Ikeda, M. Mimura, T. Tsujikawa, Slow traveling wave solutions to the Hodgkin-Huxley equations.
Recent topics in nonlinear PDE, III (Tokyo, 1986), 1-73, North-Holland Math. Stud., 148, Lecture
Notes Numer. Appl. Anal., 9, North-Holland, Amsterdam, 1987.
H. Ikeda, M. Mimura, T. Tsujikawa, Singular perturbation approach to traveling wave solutions of
the Hodgkin-Huxley equations and its application to stability problems. Japan J. Appl. Math. 6(1989), no.1, 1-66.
T. Tsujikawa, T. Nagai, M. Mimura, R. Kobayashi, H. Ikeda, Stability properties of traveling pulse
solutions of the higher-dimensional FitzHugh-Nagumo equations. Japan J. Appl. Math. 6 (1989),
no.3, 341-366.

5

・One year research life at Hiroshima University as an in-country student in 1986---- with support from the Ministry of Education (MEXT)At that time, Mayan’s wife was returning to her parents' home for childbirth. Therefore, Mayan had nochoice but to go home early, so he stayed with me one-on-one until nighttime. However, we did nottalk about research all the time. He was absorbed in thinking of problems that he thought would beinteresting to solve. He would often say to me, “How about this problem? But you are the one whowill solve this problem”.
Thanks to him, I was able to write the following five papers.
H. Ikeda, On the asymptotic solutions for a weakly coupled elliptic boundary value problem with a small
parameter. Hiroshima Math. J. 16 (1986), no. 2, 227-250.
H. Ikeda, M. Mimura, Y. Nishiura, Global bifurcation phenomena of travelling wave solutions for somebistable reaction-diffusion systems. Nonlinear Anal. 13 (1989), no.5, 507-526.
H. Ikeda, M. Mimura, Wave-blocking phenomena in bistable reaction-diffusion systems.
SIAM J. Appl. Math. 49 (1989), no.2, 515-538.
Y. Nishiura, M. Mimura, H. Ikeda, H. Fujii, Singular limit analysis of stability of traveling wave solutions inbistable reaction-diffusion systems. SIAM J. Math. Anal. 21 (1990), no.1, 85-122.
H. Ikeda, M. Mimura, Stability analysis of stationary solutions of bistable reaction-variable diffusion
systems. SIAM J. Math. Anal. 22 (1991), no. 6, 1651-1678.

6

One-day trip to Miyajima with Mayan's family

BBQ + wine party on the roof of Mayan's apartment

7

Thanks to these, I received a Doctor of Science degree from Hiroshima University in 1989
・Deep acknowledgment to Mayan
I would like to take this opportunity to express my sincere gratitude to Mayan for giving
me a start in my research life and for his words of encouragement every time we met.

8

9

10

11

12

13

14

Existence of single transition layer solutions

By using the singular perturbation technique, we can obtainsatisfying
.

,

and
15

Stability analysis of the single transition layer solutions
When we apply the SLEP method 16

In this talk, we consider the full system (3).

This is a reason why it is convenient to show stability property------- to calculate the Evans function .

17

18

 Theorem 1 (Existence). There exists a solution

satisfying

19

Stability analysis of the single transition layer solutions

20

21

22

23

24

25

Case (I)26

27

Case (II) and Case (III)28

・Extension this method to reaction-diffusion system including a nonlocal terms.
The singular perturbation method is not good at dealing with nonlocal terms
because it is a method of constructing approximate solutions for each subdomain andthen successfully laminating them together at the end, but the method introduced here
can be extended to problems involving nonlocal terms.

29

Thanks for your kind attention

30

"Traveling wave solutions of combustion in a narrow channel"
Hirofumi Izuhara (Miyazaki University, Japan)
It is reported that combustion in a narrow channel shows a variety of char patterns depending on the airflow rate. In
this talk, we consider a mathematical model which describes the combustion experiment, and numerically study theexistence of traveling wave solution in one space dimension. In addition, we investigate the instability of its planar
combustion wave which is the onset of the variety of char patterns.

Hirofumi Izuhara University of Miyazaki
Joint work with
Kazunori Kuwana Tokyo University of Science
Tsuneyoshi Matsuoka Toyohashi University ofTechnology

ICMMA, Oct. 31st 2023, Meiji University

Paper density is uniform, ignition is uniform and supply of oxygen is
uniform.
Spreading of combustion is complicated even in a simple environment.
Can we understand the pattern forming mechanism?

Model aided understanding of combustion pattern formation

Fasano, Mimura and Primicerio proposed the following mathematicalmodel:
𝑢! = 𝐿𝑒∆𝑢 + 𝜙Λ𝑃𝑒𝑢" + 𝛽𝛾𝑘 𝑢 𝑣𝑤 − 𝑎(𝑢 − 𝑢)2
𝜙𝑣! = ∆𝑣 + 𝜙𝑃𝑒𝑣" − 𝛾𝑘 𝑢 𝑣𝑤
𝑤! = −𝐻# 𝛾𝑘 𝑢 𝑣𝑤

𝑡 > 0, (𝑥, 𝑦) ∈ Ω𝑢：temperature, 𝑣：oxygen concentration, 𝑤：paper density𝑘(𝑢)：Arrhenius law
exp(−𝜃/𝑢)𝑘 𝑢 =50𝑢 ≥ 𝑢∗𝑢 < 𝑢∗𝑢∗ ：ignition temperature𝑢∗𝑢Fasano, Mimura & Primicerio `09
Lu & Dong `11
Ijioma, Izuhara, Mimura & Ogawa`15

Parameter values:
𝐿𝑥 = 300
𝐿𝑦 = 100
𝐿𝑒 = 0.125
𝜙=1
Λ=0
𝛽 = 10𝛾 = 140𝑎 = 0.28
𝑢2 = 0.02
𝜃=3𝑢∗ = 0.08
𝐻# = 0.5𝑣0 = 0.1

Ω = (𝑥, 𝑦) 0 < 𝑥 < 𝐿" , 0 < 𝑦 < 𝐿&
Supply of oxygen𝐿𝑥
No paper region𝑥0Boundary conditions:𝑣 𝑡, 𝐿𝑥, 𝑦 = 𝑣0
The others are Neumann B.C.

Paper region
𝑤 0, 𝑥, 𝑦 = 𝑤%

No paper region0𝑦Initial conditions:
𝑢 0, 𝑥, 𝑦 = 𝑢2𝑣 0, 𝑥, 𝑦 = 𝑣%

𝐿𝑦

Ignition:
𝑢 0, 𝑥, 𝑦 = 𝑢∗ + 𝜀(𝑥, 𝑦)Ijioma, Izuhara & Mimura `17

Pe=3

Pe=1.5

Pe=0.3

Pe=0.17

Why do such complicated behaviors appear?
Smooth front

highConnected front

Fingering with
tip-splitting

Fingering without
tip-splitting

supply velocity

Question:
Can we capture the connected front as an instabilityof the planar wave?

lowTraveling wave solution
Introduce a moving coordinate 𝑧 = 𝑥 − 𝑐𝑡:𝑢∗ , 𝑣 ∗ , 𝑤 ∗ , 𝑐𝐿𝑒(𝑢'' + 𝑢&& ) + 𝑐𝑢' + 𝛽𝛾𝑘 𝑢 𝑣𝑤 − 𝑎 𝑢 − 𝑢2 = 0
𝑣'' + 𝑣&& + (𝑐𝜙 + 𝜙𝑃𝑒) 𝑣' − 𝛾𝑘 𝑢 𝑣𝑤 = 0
𝑐𝑤' − 𝐻# 𝛾𝑘 𝑢 𝑣𝑤 = 0𝑐Linearize the system around 𝑢∗ , 𝑣 ∗ , 𝑤 ∗ :%𝐿𝑒∆ + 𝑐 %&+ 𝛽𝛾𝑘′ 𝑢∗ 𝑣 ∗ 𝑤 ∗ − 𝑎ℒ: =𝛽𝛾𝑘 𝑢∗ 𝑤 ∗𝛽𝛾𝑘 𝑢∗ 𝑣 ∗−𝛾𝑘′ 𝑢∗ 𝑣 ∗ 𝑤 ∗%∆ + 𝑐𝜙 + 𝜙𝑃𝑒 %&− 𝛾𝑘 𝑢∗ 𝑤 ∗−𝛾𝑘 𝑢∗ 𝑣 ∗−𝐻( 𝛾𝑘′ 𝑢∗ 𝑣 ∗ 𝑤 ∗−𝐻( 𝛾𝑘 𝑢∗ 𝑤 ∗𝑈
𝑈Solve the eigenvalue problem ℒ 𝑉 = 𝜆 𝑉 .
𝑊
𝑊

%
𝑐 %&
− 𝐻( 𝛾𝑘 𝑢∗ 𝑣 ∗Periodic B.C.
𝑈
𝑈
Eigenfunction of ℒ 𝑉 = 𝜆 𝑉 is
𝑊
𝑊
T
𝑈 (𝑧)
𝑈(𝑧, 𝑦)!"#𝑉(𝑧, 𝑦) = 𝑒 ( $ & 𝑉U (𝑧) (𝑛 = 1,2,3 ⋯).
𝑊(𝑧, 𝑦)T (𝑧)𝑊(Taniguchi & Nishiura`94,
Ikeda, Nagayama & Ikeda. `04)𝑦

𝐿

Air supply

Propagation
direction𝑥

Therefore,

T
T
𝑈
𝑈
solve the eigenvalue problem ℒX 𝑉U = 𝜆 𝑉U ,
T
T
𝑊
𝑊
where
!

!
$%&
𝐿𝑒 !"
!# '

ℒ) =

!

!
+ 𝑐 !"+ 𝛽𝛾𝑘′ 𝑢∗ 𝑣 ∗ 𝑤 ∗ − 𝑎∗∗−𝛾𝑘′ 𝑢 𝑣 𝑤∗−𝐻) 𝛾𝑘′ 𝑢∗ 𝑣 ∗ 𝑤 ∗𝛽𝛾𝑘 𝑢∗ 𝑤 ∗
𝑑$
$
!
− $%&
+ 𝑐𝜙 + 𝜙𝑃𝑒 !"− 𝛾𝑘 𝑢∗ 𝑤 ∗$'𝑑𝑧−𝐻) 𝛾𝑘 𝑢∗ 𝑤 ∗𝛽𝛾𝑘 𝑢∗ 𝑣 ∗−𝛾𝑘 𝑢∗ 𝑣 ∗!
𝑐 !"− 𝐻) 𝛾𝑘 𝑢∗ 𝑣 ∗1D problem

𝑢! = 𝐿𝑒∆𝑢 + 𝛽𝛾𝑘(𝑢)𝑣𝑤 − 𝑎𝑢𝑣! = ∆𝑣 + 𝑃𝑒𝑣" − 𝛾𝑘(𝑢)𝑣𝑤
𝑤! = −𝐻# 𝛾𝑘(𝑢)𝑣𝑤
Parameter values :𝐿𝑒 = 0.3
𝛽 = 20.0
𝛾 = 5.0
𝐻# = 1.0
𝑎 = 0.28
𝑃𝑒 = 0.2

Traveling wave𝑢∗ , 𝑣 ∗ , 𝑤 ∗ , 𝑐

𝑐 = 1.25091

2D problem (𝑤)burnedWavy front

unburnedburnedunburnedRe𝜆10- and 11-modes aremost unstable.

𝑛Thank you for your kind attention!

"Two phase Stefan problems as the singular limit ofcompetition-diffusion systems arising in population dynamics"
Danielle Hilhorst (Université Paris-Saclay, France)
Competition-diffusion systems are coupled systems of nonlinear parabolic equations, where the unknown functions
represent the densities of interacting biological populations. We will first study the singular limit of of a two-component
competition-diffusion system in population dynamics when the interspecific competition rate tends to infinity [7], [8].
Using energy estimates, we will prove that the solution converges to the weak solution of a problem with a free boundary,which Mayan Mimura used to call a Stefan problem with zero latent heat [1], [2], [3]. In biological terms, this amounts
to proving that the habitats of two interacting populations become completely disjoint in the fast reaction limit. We willthen consider a three component competition-diffusion system and prove that its solution converges to a Stefan problem
with positive latent heat [4], [6].
Another question involves the limit of the Stefan problem as the latent heat coefficient tends to zero; we will show thatit converges to the Stefan problem with zero latent heat [5]. A question which we have started to study is then thefollowing : can we prove a similar result in the case that the partial differential equations in the Stefan problems are
perturbed by a white noise in time [9]References[1] Dancer, E. N.; Hilhorst, D.; Mimura, M.; Peletier, L. A. Spatial segregation limit of a competition-diffusion system. European J. Appl.
Math. 10 (1999), no. 2, 97-115.
[2] Crooks, E. C. M.; Dancer, E. N.; Hilhorst, D.; Mimura, M.; Ninomiya, H. Spatial segregation limit of a competition-diffusion systemwith Dirichlet boundary conditions. Nonlinear Anal. Real World Appl. 5 (2004), no. 4, 645-665.
[3] Hilhorst, Danielle; Martin, Sébastien; Mimura, Masayasu Singular limit of a competition-diffusion system with large interspecificinteraction. J. Math. Anal. Appl. 390 (2012), no. 2, 488-513.
[4] Hilhorst, Danielle; Iida, Masato; Mimura, Masayasu; Ninomiya, Hirokazu A competition-diffusion system approximation to the classicaltwo-phase Stefan problem. Recent topics in mathematics moving toward science and engineering. Japan J. Indust. Appl. Math. 18 (2001),
no. 2, 161-180.
[5] Hilhorst, Danielle; Mimura, Masayasu; Schätzle, Reiner, Vanishing latent heat limit in a Stefan-like problem arising in biology,
Nonlinear Anal. Real World Appl. 4 (2003), no. 2, 261–285.
[6] Hilhorst, Danielle; Iida, Masato; Mimura, Masayasu; Ninomiya, Hirokazu Relative compactness in Lp of solutions of some 2mcomponents competitiondiffusion systems. Discrete Contin. Dyn. Syst. 21 (2008), no. 1, 233-244.
[7] Hilhorst, Danielle; Mimura, Masayasu; Ninomiya, Hirokazu Fast reaction limit of competition-diffusion systems. Handbook ofdifferential equations: evolutionary equations. Vol. V, 105-168, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2009.
[8] Hilhorst, Danielle; Salin, Florian; Schneider, Victor; Gao, Yueyuan, Lecture notes on the singular limit of reaction-diffusion systems
dedicated to the Memory of Masayasu Mimura, Interdiscip. Inform. Sci. 29 (2023), no. 1, 1–53.
[9] Ciotir, Ioana; El Kettani, Perla; Goreac, Dan; Hilhorst, Danielle, The vanishing latent heat limit of stochastic Stefan problems inpopulation dynamics, in preparation.

Competition-diffusion sytems and two phase
Stefan problems in population dynamics
Danielle Hilhorst
CNRS and Université Paris-Saclay

October 31st 2023

Danielle Hilhorst

Competition diffusion systems

October 31st 2023

1 / 30Social life with MayanIn the occidental world, we always called Mimura sensei Mayan. I met
him very long ago, in the Netherlands. Most probably, we first met at a
Conference entitled : Conference on Analytical and Numericalapproaches to Asymptotic Problems, which took place at the Universityof Nijmegen, in the Netherlands, on June 9-13, 1980. Since Mayan
was an important Professor while I was just a fourth year PhD student,
I did not have much chance to discuss with him. But then, on
December 2nd 1981, when I defended my doctoral thesis at the
University of Leiden, also in the Netherlands, Mayan was there visiting
my PhD supervisor Bert Peletier. And Bert requested me to invite
Mayan both at my doctoral defense and at the dinner which I wasoffering at this occasion; it was then a tradition in the Netherlands toorganize a PhD dinner on the evening of the defense. This is how mycontacts with Mayan started.
Danielle Hilhorst

Competition diffusion systems

October 31st 2023

2 / 30

MayanDanielle Hilhorst

Competition diffusion systems

October 31st 2023

3 / 30Social life with Mayan
During many years, I often met Mayan, not only in France and in Japanbut also at various conferences all over the world. Mayan was always
full of life and happiness and had an incredible creativity. We
discussed about mathematics, and in particular about new problems
modeling biological phenomena posed by Mayan. I have worked a lot
about some of these problems together with PhD students, post-docs
as well as well-known mathematicians. I will show you some examples
in a little while.
Mayan loved to spend time in France, both for mathematics andeveryday life. He also showed an extraordinary hospitality to hiscolleagues from all over the world that he invited in Japan. We all keepa wonderful memory of these visits to Japan.

Danielle Hilhorst

Competition diffusion systems

October 31st 2023

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The model problem : Dancer, DH, Mimura, Peletier
We consider the competition-diffusion system∂t u = d1 ∆u + f (u) − kuv ,∂t v = d2 ∆v + g(v ) − αkuv ,Pk∂
u = 0, ∂ν v = 0, νu(·, 0) = u0k , v (·, 0) = v0k ,in Ω × R+in Ω × R+on ∂Ω × R+
on Ω,

where
f (s) = λs(1 − s), g(s) = µs(1 − s);k , α, d1 , d2 , λ, µ are positive constants;
u0k , v0k ∈ C(Ω), 0 ≤ u0k , v0k ≤ 1;
u0k ⇀ u0 , v0k ⇀ v0 in L2 (Ω) as k → ∞.
u, v are the densities of two biological populations;λ, µ are the intraspecific competition rates;k , αk are the interspecific competition rates.
Danielle Hilhorst

Competition diffusion systems

October 31st 2023

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The main question

Our main question:
Let (u k , v k ) denote the solution of Problem (P k ). What is the
singular limit of the solution pair (u k , v k ) as k → ∞?

Danielle Hilhorst

Competition diffusion systems

October 31st 2023

6 / 30References• [1] Dancer, E. N.; Hilhorst, D.; Mimura, M.; Peletier, L. A. Spatial
segregation limit of a competition-diffusion system. European J. Appl.
Math. 10 (1999), no. 2, 97-115.
• [2] Crooks, E. C. M.; Dancer, E. N.; Hilhorst, D.; Mimura, M.;
Ninomiya, H. Spatial segregation limit of a competition-diffusion
system with Dirichlet boundary conditions. Nonlinear Anal. Real World
Appl. 5 (2004), no. 4, 645-665.
• [3] Hilhorst, Danielle; Mimura, Masayasu; Ninomiya, Hirokazu Fast
reaction limit of competition-diffusion systems. Handbook of differential
equations: evolutionary equations. Vol. V, 105-168, Handb. Differ.
Equ., Elsevier/North-Holland, Amsterdam, 2009.
• [4] Hilhorst, Danielle; Salin, Florian; Schneider, Victor; Gao, Yueyuan,
Lecture notes on the singular limit of reaction-diffusion systems
dedicated to the memory of Masayasu Mimura, Interdiscip. Inform. Sci.
29 (2023), no. 1, 1-53.
Danielle Hilhorst

Competition diffusion systems

October 31st 2023

7 / 30A priori bounds

By a solution of Problem (P k ) we mean a pair of functions (u k , v k )such that u k , v k ∈ C(Q) ∩ C 2,1 (Ω × [δ, T ]) for all δ ∈ (0, T ). We begin
with a priori bounds for solutions of Problem (P k ).
Lemma. Let (u k , v k ) be a solution of Problem (P k ). Then0 ≤ u k ≤ 1 and 0 ≤ v k ≤ 1 in Q, where Q := Ω × R+ .
Proof. This assertion follows from the maximum principle.

Danielle Hilhorst

Competition diffusion systems

October 31st 2023

8 / 30Existence and uniqueness - a priori estimates
The existence and uniqueness of the solution (u k , v k ) of Problem (P k )follows from [Lunardi, Proposition 7.3.2].
Next we obtain a priori bounds for the solution (u k , v k ) of Problem (P k )which are uniform with respect to the parameter k in the equations.
This will enable us to study the properties of the family of solutions(u k , v k ) for large values of k .
Lemma. We have
Z TZ
0

uk v k ≤Ω|Ω|(ℓ0 T + 1).
k

Proof. Integration of the equation for u k over QT := Ω × (0, T ) yields
Z TZ
Z
Z
RT R
R T R ∂u k
k 0 Ω u k v k = d1 0 ∂Ω+f (u k )u k −u k (T ) +
u0k
∂ν
0
Ω
Ω
Ω
≤ (ℓ0 T + 1)|Ω|.
Danielle Hilhorst

Competition diffusion systems

October 31st 2023

9 / 30A priori estimates
Lemma. There exists a positive constant C, which does not depend on
k , such that
Z TZ
0|∇u k |2 ≤ C,

Ω

and
Z TZ
0

|∇v k |2 ≤ C.

Ω

Proof. We multiply the first equation in (P k ) by u and integrate on Ω.
This yields
Z
Z
Z
1d
2
2
u + d1|∇uk | + k
uk2 vk ≤ ℓ0 |Ω|,
2 dt Ω kΩΩwhere we have used the bounds on u k and v k . When we integrate on(0, T ) we obtain the first estimate. The second estimate can be provedin a similar way.
Danielle Hilhorst

Competition diffusion systems

October 31st 2023

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A way to eliminate the large parameter k
Next we consider the functionz k = uk −

1 k
v ,
α

which appears when we want to eliminate the terms involving k fromthe partial differential equation. It satisfies
ztk = d1 ∆u k −1d2∆v k + u k f (u k ) − v k g(v k ) in QTαα

together with the homogeneous Neumann boundary condition∂z k= 0 on ST := ∂Ω × (0, T ).
∂ν

Danielle Hilhorst

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October 31st 2023

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Where do we stand now?

We have L∞ (QT ) estimates for u k and v k , and L2 (QT ) estimates for|∇u k | and |∇v k |.
This is not sufficient to pass to the limit as k → ∞, even though it
would be if we would work on an elliptic problem. Here we need someextra knowledge about either the time derivative of the solution pair ordifferences of time translates of the solution pair.
We apply the Fréchet-Kolmogorov Theorem (see for instance the bookof Brezis on functional analysis).

Danielle Hilhorst

Competition diffusion systems

October 31st 2023

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Limit as k → ∞We deduce that the sequences {uk } and {vk } are relatively compact in
L2 (QT ). In particular, there exist subsequences of {u k } and {v k },which we denote again by {u k } and {v k }, and functions u and v in
L2 (0, T ; H 1 (Ω)) such that, as k → ∞, u k and v k converge to their limitsu and v strongly in L2 (QT ), a. e. in QT and weakly in L2 (0, T ; H 1 (Ω)).

Danielle Hilhorst

Competition diffusion systems

October 31st 2023

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Characterization of the limit functions
Lemma. Let T be an arbitrary positive number. The limit functions(u, v ) are such that
Z T Z n


 o
1 
d2 
1
u − v φt − ∇ d1 u − v ∇φ + uf (u) − v g(v ) φ
α
α
α
0
Ω
Z
v0
= − (u0 − )φ(0),αΩfor all functions φ ∈ C ∞ (QT ) such that φ(T ) = 0.
Proof. When we multiply the equation for z k by a test function
φ ∈ C0∞ (QT ) such that φ(T ) = 0, and integrate by parts, we obtain theidentityZ T Z n
1 
d2 
u k − v k φt − ∇ d1 u k − v k ∇φ
α
α
0
Ω
Z


o
vk
1
+ u k f (u k ) − v k g(v k ) φ = − (u0k − 0 )φ(0).
ααΩDanielle Hilhorst

Competition diffusion systems

October 31st 2023

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Characterization of the limit functions
We now let k → ∞ along a sequence for which u k and v k strongly
converge in L2 (QT ) and a. e. to their limits. Then, because
u k → u,and v k → v

as k → ∞ a.e. in QT ,and |u k |, |v k | ≤ 1 for all k ≥ 1, it follows by the dominated convergence
theorem that
Z TZ
Z TZ
k
k
uf (u) as k → ∞.
u f (u ) →
0

Ω

0

Ω

A similar result holds for the sequence {v k g(v k )}. Passing to the limit
in the equality above permits to complete the proof.

Danielle Hilhorst

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October 31st 2023

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Characterization of the limit functions
Next we show that the function
z := u −1v,
αis a weak solution of Problem (P) defined by+ ∂t z = ∇(d(z)∇z) + h(z), in Ω × R ,+∂ν z = 0,on ∂Ω × R ,(P) z(·, 0) = z0 := u0 − v0 ,
on Ω,
α
where
d(s) = {d1 if s > 0, d2 if s < 0},andh(s) = {f (s)sDanielle Hilhorst

if s > 0,g(−αs)s

Competition diffusion systems

if s < 0}.

October 31st 2023

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Characterization of the limit functions

Definition. A function z is called a weak solution of Problem (P) if
z ∈ L∞ (Ω × R+ )∩L2 (0, T , H 1 (Ω));
Z TZ
Z−{zφt − d(z)∇z∇φ + h(z)φ} =z0 φ(0),0Ω∞for all functions φ ∈ C (QT ) such that φ(T ) = 0 and for all T > 0.

Danielle HilhorstΩCompetition diffusion systems

October 31st 2023

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Characterization of the limit functions
Lemma. The function z is a weak solution of Problem (P).
Proof. We already know that z ∈ L∞ (Ω × R+ ) and thatz ∈ L2 (0, T ; H 1 (Ω)).
We observe thatd1 ∇u −

d2∇v = d(z) ∇zαand thatvg(v ) = h(z).
αTherefore z satisfies the integral equality
uf (u) −

Z TZ
0



Z
zφt − d(z)∇z∇φ + h(z)φ =

Ω

(u0 −
Ω

v0
)φ(0)
α

for all functions φ ∈ C ∞ (QT ) such that φ(T ) = 0 and for all T > 0.
Danielle Hilhorst

Competition diffusion systems

October 31st 2023

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Characterization of the limit functions

Thus z is a weak solution of Problem (P).
We also have the following result.
Lemma. Problem (P) has exactly one weak solution z, andz ∈ C β,β/2 (Ω × [0, ∞)) for all β ∈ (0, 1).

Danielle Hilhorst

Competition diffusion systems

October 31st 2023

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The limit problem as a free boundary problem

The limit problem is a free boundary problem. The free boundary
separates the regions where {u > 0, v = 0} and {v > 0, u = 0}.

Danielle Hilhorst

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October 31st 2023

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The limit free boundary problem

We can give both
• a weak form, where the free boundary does not explicitely appear;• a strong form, with explicit boundary conditions.

Danielle Hilhorst

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The convergence result
We have proved that as k → ∞,uk → z +,v k → αz − ,strongly in L2 (QT ), where the function z is the unique weak solution of
the problem
in Ω × R+ , ∂t z = ∆D(z) + h(z),∂ν z = 0,on ∂Ω × R+ ,(P)v

 w(·, 0) = z0 := u0 − 0 , on Ω,αwith D(s) := d1 s+ − d2 s− and h(s) := f (s+ ) − g(αs− ), wheres+ = max(s, 0) and s− = −min(s, 0).

Danielle Hilhorst

Competition diffusion systems

October 31st 2023

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The strong form of the limit free boundary problem
Assume that, at each time t ∈ [0, T ], there exists a close hypersurface
Γ(t) and two subdomains Ωu (t), Ωv (t) such thatΩ = Ωu (t) ∪ Ωv (t),z(·, t) > 0 on Ωu (t),Γ(t) = Ωu (t) ∩ Ωv (t),z(·, t) < 0 on Ωv (t).

Assume furthermore that t 7→ Γ(t) is smooth enough and that(u, v ) := (z + , αz − ) are smooth up to Γ(t). Then the functions u and v
satisfyS∂t u = d1 ∆u + f (u)in Qu := S {Ωu (t), t ∈ [0, T ]}
∂ v = d2 ∆v + g(v )in Qv :=S {Ωv (t), t ∈ [0, T ]} t
 u=v =0on Γ := {Γ(t), t ∈ t ∈ [0, T ]}
(P)
d2

on Γ
d1 ∂n u = − ∂n v


α



∂ν v = 0on ∂Ω × [0, T ] + initial conditions.
Danielle Hilhorst

Competition diffusion systems

October 31st 2023

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The two-component system

u1 ,u2

u1 ,u2

k!100

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

k!10000.2
x0.2

u1 ,u2

0.4

0.6

0.8

x

1

0.2
u1 ,u2

k!10000

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.4

0.6

0.8

1

0.8

1

k!100000

0.2
x
0.2

Danielle Hilhorst

0.4

0.6

0.8

1

x
0.2

Competition diffusion systems

0.4

0.6

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Comments on the limit free boundary problem
This is a Stefan problem with zero latent heat.
This led us to search for a reaction-diffusion system whose solution
converges to that of a "real" Stefan problem
D. Hilhorst, M. Iida, M. Mimura, H. Ninomiya, Japan J. Ind. Appl. Math.
(2001)To that purpose we need
• A three component system,• With two partial differential equations coupled to an ordinary
differential equation.

Danielle Hilhorst

Competition diffusion systems

October 31st 2023

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A three-component reaction-diffusion system
We consider the system∂t u = d1 ∆u + f (u) − ks1 uv − k λs1 (1 − w)u  ∂t v = d2 ∆v + g(v ) − ks2 uv − k λs2 wv∂t w = k (1 − w)u − kwvQk∂u = 0, ∂ν v = 0, νu(·, 0) = u0k , v (·, 0) = v0k , w(·, 0) = w0k

in Ω × R+in Ω × R+in Ω × R+on ∂Ω × R+
on Ω,

u0k , v0k ∈ C(Ω), w0k ∈ L∞ (Ω),
0 ≤ u0k , v0k , w0k ≤ 1;
u0k ⇀ u0 , v0k ⇀ v0 , w0k ⇀ w0 in L2 (Ω) as k → ∞.
The function w k approximates the characteristic function of the habitatof the population u k .

Danielle Hilhorst

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October 31st 2023

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The weak form of the limit free boundary problem
∂ z = ∆D(φλ (z)) + h(φλ (z)) in Ω × R+ t∂ν D(φλ (z)) = 0 on ∂Ω × R+Qλ z(·, 0) = u0 − v0 + λw0 on Ω,
s1
s2
with
D(r ) := d1 r + − d2 r − ,+h(r ) :=−f (s1 r + ) g(s2 r − )
−
,
s1
s2

φλ (r ) =(r − λ) − r .
We now have a Stefan problem with positive latent heat λ. We also
define the limit functionsu = s1 φλ (z + ),Danielle Hilhorstv = s2 φλ (z − ),Competition diffusion systemsw=

z − φλ (z).
λOctober 31st 2023

27 / 30

The strong form of the Stefan problem with positive
latent heat

The functions u and v satisfy∂t u = d1 ∆u + f (u)





 ∂t v = d2 ∆v + g(v ) u = v = 0
λ
d1
d2
Q
λVn = − ∂n u − ∂n v



s1
s2



∂
v
=
0

ν


+ initial conditions.

Danielle Hilhorst

S
in Qu := S {Ωu (t), t ∈ [0, T ]}in Qv :=S {Ωv (t), t ∈ [0, T ]}on Γ := {Γ(t), t ∈ [0, T ]}on Γ
on ∂Ω × [0, T ]Competition diffusion systems

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Present work with Ciotir, El Kettani and Goreac

With Mayan Mimura and Reiner Schätzle, we have proved that, as
λ → 0, the weak solution z λ of Problem (Qλ ) converges to the weaksolution z of Problem (P).
Hilhorst, Danielle; Mimura, Masayasu; Schätzle, Reiner, Vanishinglatent heat limit in a Stefan-like problem arising in biology, Nonlinear
Anal. Real World Appl. 4 (2003), no. 2, 261–285.
With Ciotir, El Kettani and Goreac, we are adding a multiplicative noise
involving a white noise in time on the right-hand-side of the partial
differential equation for z λ , and try to prove a similar result. This study
is rather technical but I believe that it will work out.

Danielle Hilhorst

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October 31st 2023

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Mayan, may you rest in peace.

Danielle Hilhorst

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"The floodgates to pattern formation problems"
Yasumasa Nishiura (Hokkaido University, Japan)
W. T. Gowers, a Fields medalist from Cambridge in 1998, eloquently expounded on the pivotal role of Paul Erdős
in his essay titled "The Two Cultures of Mathematics." He (Erdős) is famous not because it has large numbers ofapplications, nor because it is difficult, nor because it solved a long-standing open problem. Its fame rests on the factthat it opened the floodgates to probabilistic arguments in combinatorics.
In a similar vein, Mayan Mimura opened a parallel set of floodgates, ones that lead to the modeling and analysis ofreaction-diffusion equations. I am eager to trace the initial footsteps of Mayan and delve into the lasting impact of his
contributions on present-day research.

Mathematics can change our society!

The floodgates to pattern formation problems

Yasumasa Nishiura

ICMMA 2023 at Meiji University“In memory of Professor Mayan Mimura”

Encounter• Encounter with Mayan in early 70’s (Konan Univ)– Prof. Yamaguti‘s seminar– Debate on “what is a good math model?”• Scientific collaboration in 80’s and early 90’s (Hiroshima Univ)and discussion about the embryonic stage of the issues:– On the necessity to establish an applied mathematics major.
– Importance of computational approach– How should mathematics establish its relationship with various
sciences? （諸科学との関係）– Combining science and the humanities is quite challenging.
（文理融合は簡単ではない）

C.P. Snow “The Two Cultures and the Scientific Revolution”「２つの文化と科学革命」

人文的文化 vs

科学的文化

(The arts vs Sciences)“Third culture” by John Brockman

Thomas Kuhn “Paradigm Shift”• Those who aspire to become scientists acquire a specific paradigm
through long apprenticeship and, upon obtaining degrees, engage
in scientific research as full-fledged scientists. This is a form of
puzzle-solving in the sense of research guided by the paradigm.
Much of scientific research, in other words, normal science, isnothing but this kind of puzzle-solving.
• However, paradigms are not always secure indefinitely. Puzzles
gradually run out, and, conversely, anomalies that cannot be dealt
with by the paradigm accumulate. This eventually leads to a crisisof the paradigm, and from the midst of that confusion, a newparadigm emerges, leading to a "scientific revolution.”• It is not clear what is a paradigm in mathematics, namely once it
was proved rigorously, it becomes eternal. How about Kurt Gödel?

Problem solver vs Theory builder
Criticisms of Combinatorics（from core mathematics）
1. Lacks direction, or goals of a general kind
2. Not particularly deep
3. No interesting connections to other parts ofmathematics (core mathematics)
4. Many of them do not have applications

W. T. Gowers is a British
mathematician(combinatorialist) at Cambridge
awarded Fields medal 1998.
Since 2020, he is a professor at
the Collège de France, born in
1963.

Moreover, mathematicians in the theory-building areas often regard what they are doing as the central
core of mathematics, with subjects such as combinatorics thought of as peripheral and not particularly
relevant to the main aims of mathematics.
One can almost imagine a gathering of highly educated mathematicians expressing their incredulity
at the ignorance of combinatorialists, most of whom could say nothing intelligent about quantum
groups, mirror symmetry, Calabi-Yau manifolds, the Yang-Mills equation, solitons or even cohomology.

流れるままに
These criticisms can be answered in a similar way. Consider first the notionthat there are not general goals in combinatorics. I quote again from theinterview with Atiyah [A1]:“I was thinking more of the tendency today for people to develop whole areasof mathematics on their own, in a rather abstract fashion. They just go on
beavering away. （せっせと働く）
If you ask what is it all for, what is its significance, what does it connect withyou and that they don't know.
Atiyah was not particularly referring to combinatorics, but he makes apowerful point, and it is as important for combinatorialists as it is for anyone
else to show that they are doing more than merely beavering away.

Why should problem-solving subjects be less highly regarded than theoretical ones?
To answer this question we must consider a more fundamental one: what makes one pieceof mathematics more interesting than another? Once again, Atiyah writes very clearlyand sensibly on this matter (while acknowledging his debt to earlier great mathematicianssuch as Poincar e and Weyl). He makes the point (see for example [A2]) that so muchmathematics is produced that it is not possible for all of it to be remembered. The processesof abstraction and generalization are therefore very important as a means of making senseof the huge mass of raw data (that is, proofs of individual theorems) and enabling at least
some of it to be passed on. The results that will last are the ones that can be organizedcoherently and explained economically to future generations of mathematicians. Of course,
some results will be remembered because they solve very famous problems, but even these,
if they do not into an organizing framework, are unlikely to be studied in detail by morethan a handful of mathematicians.

抽象化と一般性がなければ，次へは進めない．
The important ideas of combinatorics do not usually appear in the form of precisely stated theorems,
but more often as general principles of wide applicability.

Combinatorics ではその現れ方は違う．例えばErdos の例ではThis result of Erdos [E] is famous not because it has large
numbers of applications, nor because it is difficult, nor because
it solved a long-standing open problem. Its fame rests on thefact that it opened the floodgates to probabilistic arguments incombinatorics.

Lower bound:probabilistic methodゴールが明確に示されている分野もある
Some branches of mathematics are dominated by a small number of problems of
universally acknowledged importance. One can justify many results by saying that,in however small a way, they shed light on the Riemann hypothesis, the BirchSwinnerton-Dyer conjecture, Thurston's geometrization conjecture, the Novikov
conjecture or something of the kind.

Combinatorics では少しそれとは異なる
It would be difficult to demonstrate that combinatorics had many general goals of
the sort just mentioned (with the one exception of the P=NP problem).
However, just as the true significance of a result in combinatorics is very often
not the result itself, but something less explicit that one learns from the proof, so
the general goals of combinatorics are not always explicitly stated.
Something similar?
The dichotomy of Applied Math vs Pure MathHow do you select a problem to study?

Floodgates• Finding of new things:– Finding of new phenomena• Quasi-crystal, chaos, …

– Finding new (simple) model• Landau model, Kuramoto model, …

– Finding of new methods• Connecting two different concepts– Important in Mathematics• Trigger to explore a new world– Introduce a new insight to a hard problem (like Paul Erdos)• Establish a common platform for applied mathematicians– Collaborations with interdisciplinary people– Incubator in all directions– Mayan was skilled at conversation and good at winning people over.

MIMS
•

Mayan’s dream of Applied Math Dept. with many twists and turns starting
around late 80’s, tough negotiations with deans and presidents– Hiroshima Univ.
• Dept. of Mathematical and Life Sciences (1999)
数理分子生命理学専攻• Graduate School of Integrated Sciences for Life (2019)
Program of Mathematical and Life Sciences– Meiji Univ.

• MIMS was established!

• Combining science and the humanities is quite challenging. （文理融合は簡単ではない）"Ecocultural range-expansion model of modern humans in Paleolithic" by Joe Yuichiro
Wakano (Archaeology based on ancient DNA analysis)

Soft science vs Hard scienceS. Huntington.

Soft science vs Hard science
Soft sciences are often harder than hard sciences
Discover (1987, August) by Jared Diamond
相互理解のギャップがもたらす不幸“pseudo Mathematics” by Huntington

S. Lang(NAS: National Academy of Sciences)Summary• The Two Cultures: C.P. Snow
– The arts vs Sciences:• The two cultures of Mathematics: W.T. Gowers– Michael Atiya’s Interview• Soft sciences vs Hard sciences: Jared Diamond– Huntington vs Serge Lang

• Border between pure and appl is disappearing.
– Appl. Math broadens the spectrum of pure math.

– Appl math demands all mathematics.

There still remains a gap, however,• Mayan Mimura opened the floodgates to patternformation problems!
– New modeling in mathematical biology– New collaborations among interdisciplinary fields– New institute where two cultures meet
Above all, it has bestowed upon me the joy and delight ofscholarly exploration
そして多くの研究の喜びと楽しさを伝えてくれた1988@Hiroshima

Ogawa-san’s cottage
2010, September

Mayan’s cottage

山口華楊

岡田節人

山口昌哉
寺本英

西田孝明
野木達夫
藤井宏
亀高惟倫
田端正久
俣野博
小林亮
岡本久
柳田英二

Mathematical biology
Numerical Analysis
Pattern dynamics
Mathematical modeling

Jim Murray
M. Bertch
V. Cappasso
Daniell Hilhorst

蔵本由紀

Singular perturbation
山口智彦 Interfacial Dynamics

三村昌泰
Man of action
太田隆夫甲斐昌一

自己組織化

重定奈々子
厳佐庸
望月敦史

学位論文

Bifurcation theory
Symmetry-breaking

Thank you for listening!

"Keller-Segel type approximation for nonlocal
Fokker-Planck equations in one-dimensional bounded domain"
Yoshitaro Tanaka (Future University Hakodate, Japan)
To describe biological phenomena such as cell migration and cell adhesion many evolutional equations are proposed inwhich an advective convolution term with a suitable integral kernel is imposed. It is well known that such nonlocal
equations can reproduce various behaviors depending on the shape of the integral kernel. These nonlocal evolutional
equations are often difficult to analyze, and the analytical method is developing. In the light of these background weapproximate the nonlocal Fokker-Planck equations by the combination of a Keller-Segel system which is a typical andlocally dynamical system. We will show that the solution of the nonlocal Fokker-Planck equation with any evencontinuous integral kernel can be approximated as a singular limit of the Keller-Segel system with specified parameters.

ICMMA 2023, "Reaction-diffusion systems: from the past to the future ” in memory of Prof. Masayasu
Mimura，Meiji University, 1 Nov. 2023. Chair: Toshiyuki OGAWA

Keller-Segel type approximation for nonlocal
Fokker-Planck equations in one-dimensionalbounded domain
Speaker：◯Yoshitaro Tanaka（Future University Hakodate）
Collaborator： Hideki Murakawa (Ryukoku University)

Dedicated to the memory of Professor Masayasu Mimura
Supported by JSPS, Grant number JP20K1436

Nonlocal interactions
Nonlocal interactions （spatially long range interactions） haveattracted attentions in various fields:◉ Neural firing phenomenon [S. Amari Bio. Cybernetics 1977], [C. Laing & W. Troy Phsica D 2003]，◉ Pigmentation pattern in animal skin [J.D. Murray, Springer 2003], [S. Kondo, J.T.B. 2017]
Typical modelings:
u = u(x, t) :some density, K = K(x): an integral kernel,<latexit sha1_base64="IHAXMzDThKGw+nHKd3epCuzuO9c=">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</latexit><latexit sha1_base64="YuFK6amdMtXWpz3Wg5+I0PCA9SM=">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</latexit><latexit sha1_base64="uAxYtv+Xy80zrTuRnhu6Yl5yZ7E=">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</latexit>

Spatial convolution with suitable integral kernel K ⇤ u =ut = K ⇤ ubu + · · · ,Dispersal, Growth rate・・・

K(xy)u(y, t)dy

Advective type

Normal type<latexit sha1_base64="+2rfkxM+yVen8CMvSIktef4XDvg=">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</latexit>

Z<latexit sha1_base64="Gi0xkv9Lk9+XMvuRYp2gjdeQtjM=">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</latexit>ut =r · (ur(K ⇤ u)) + · · ·Velocityb > 0 : a const.
<latexit sha1_base64="YXbn1kqgdQUSztO4gKNLIadeAUA=">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</latexit>[c.f. Ninomiya, T., Yamamoto, J.M.B 2017, J.J.I.A.M., 2018]9

Biological examples of nonlocal interactions◉ Cell size and cell projections
・Pigment cells in skin of zebra fish
[Yamanaka, Kondo, P.N.A.S., 2009][Yamanaka, Kondo, P.N.A.S. 2009][Kondo. J. Theor. Biol., 2017]

the pseudopodia. The xanthophores then extended
pseudopodia
projection
and chased the melanophores. 1.0
0.8
In our previous study, we observedthat a weak
repulsivel),re⇢" (|x|Cut out tail fin and anal fin from zebrafish
sponse of melanophores was triggeredby contact with xantho0.6⇢" 2 was
C01not,phores (15). In this system, we foundthat the interaction
0.4
Dissect finsasimplerepulsive
responsebutamore
activeandcontinuousB
run-and-chase movement. The 0.2run-and-chase movement was
Remove epidermal cells by trypsin,quite similar to the exclusive
movementof melanophores
in vivo-50
5digest collagen with collagenase(16). The similarity between these movements suggested that thecell movement observed in vitro was related to pattern formation
Filtrationin vivo.
Centrifugation with percoll
However, we observed the run-and-chase movement only between
xanthophores and melanophores that were paired by
Suspend pellet in serum free L-15 medium,[Togashi et al. J. Cell Biol. 2016][Murakawa,Togashi,J.T.B., 2015]
chancewhen pigment cells were spread as a mixture of xanthoculture harvested pigment cells in dishphores andfiguremelanophores.
Therefore, to analyze cell interactionofMovies of cell migration
and adhesions: 1 % FBS Shematic
serum free
in a controlled condition, we isolated single pigment cells andDEsensingfunction:transferredthem to another dish (Fig. 2B). Using this method,
we could arrange the cells as we liked and analyze the interactions in a controlled condition without interference from othercells (Fig. 2C). In this condition, the run-and-chase movement wasobserved similarly as in the mixed culture (Fig. 2D and Movie S4).
Hereafter, we analyzed cell movement using this system.

A

WT

1.2

C

<latexit sha1_base64="1Bl6a3r22ezVQZnVdDXU7xeVFnU=">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</latexit><latexit sha1_base64="m01ENzOkAPx+ARXp+YvZXnZyDoQ=">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</latexit>

・HEK 293 cells for experiments of cell adhesions

Size of cell body=Sensing function
Fig. 1. Harvesting pigment cells from zebrafish fins. (A) Adult WT zebrafish.
(B) Magnified image of the surface of a WT fish body. Black spots are
melanophores, and yellow spots are xanthophores. Clear gaps exist between

A

0:00
X

16:00

32:00

48:00

10

Mathematical models with advective non-local interactions① Aggregation diffusion model (Collective motion & cell migration)[Carrillo, Craig, Yao, Model. Simul. Sci. Eng. Technol., 2019 ]⇢m<latexit sha1_base64="Diy2dhaxBMNSH7+6TcjLE9RKkVY=">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</latexit>⇢t =r · (⇢r(K ⇤ ⇢)),

m1

advection

m = 1 → linear diffusion ⇢xx : Nonlocal Fokker-Planck equation<latexit sha1_base64="b9Cf0w7s/nczWetiLa/YGpx8sGE=">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</latexit><latexit sha1_base64="GU5Z4p0eFpeEIIEGoIFFnTDQ+og=">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</latexit>② Cell adhesion model [Togashi, Murakawa, J.T.B., 2015], [Carrillo, Murakawa,Sato, Togashi, Trush, J.T.B., 2019]<latexit sha1_base64="rGzslGna5Ed2H0FlmNKMl1ew8vE=">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</latexit>

@u
=
@t

u2

r · (u(1u)r(K ⇤ u)) + f (u),advection<latexit sha1_base64="2Xfw6t4yolMbx/vIYXm4WKTvMao=">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</latexit>

K(x)

1.2

1.0
<latexit sha1_base64="1LEY7dDRbM6621yNuSuJRMivsPk=">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</latexit>

K(x) = (R|x|) B(0,R) (x), R > 0 : Size of cell body,<latexit sha1_base64="9wXQftLJ9YQci8Biaq56xnDrnH8=">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</latexit><latexit sha1_base64="ssjKVNoXxmI5RHp0gCXYhlOrqoY=">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</latexit>

B(0, R) : Ball,<latexit sha1_base64="QsgqwZ/XKWRuVlEAcHa1pOPuOeQ=">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</latexit>

B(0,R) (x) =

(

1
0

0.8

0.6

if x 2 B(0, R),

otherwise

0.4

0.2

-4

-2

0

2

4

11

Motivation & Aim◉ Motivation: To analyze (multiple components) cell adhesion model<latexit sha1_base64="HP/ssvbOQDvfSNuk30lBlwoioio=">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</latexit>

@u
=
@t

u2

r · (u(1Porous medium typeu)r(W ⇤ u)) + f (u)advection・Difficulties: Nonlinear diffusivity & nonlocality
・Approximation for nonlinear diffusion by linear diffusion[Murakawa, J.J.I.A.M., 2018]◉ Aim: As a first step, we reveal whether the advective nonlocal
interaction in the nonlocal Fokker-Planck equation can beapproximated by a Keller-Segel system or not
Nonlocal Fokker-Planck equation<latexit sha1_base64="0sACRpLLOl2WApOZMqMieNgB3NQ=">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</latexit>⇢t = ⇢xx⌘@ ⇣ @⇢ (W ⇤ ⇢)
@x @x

12

Model: Nonlocal Fokker-Planck equation
We analyze the following nonlocal Fokker-Planck equation:

velocity
8
⇣
⌘
2
@
@
< @⇢ = @ ⇢⇢ (W ⇤ ⇢) ,t > 0, x 2 ⌦ := [ L, L],
@t
@x2
@x @x
:
⇢(x, 0) := ⇢0 (x) 2 C 2 (⌦),
<latexit sha1_base64="hkmnbblm2mVWkd3fhqNYVPzj1SY=">AAAGZXicnZTLbtNAFIaPCw3F0BsghMSCEVWrBKXRpOKmSkVVu4BFVdqmNzXTRr5M3FEd27Kd0GLlBXiBLliBxALxDKyQgBdg0SdAiGWR2LDgeGKSJm0TCVuxT37/35njM+PRPVsEIaVHSt+Fi/2pSwOX1StXB4eGR0avrQdu1Tf4muHarr+pawG3hcPXQhHafNPzuVbRbb6h783Hzzdq3A+E66yGBx7frmiWI8rC0EKUSqMKZTYvhyxSmc4t4USaLSyHm3V1gpV9zYgI8zQ/FJpNmL/rkvoJISR1QmZIh29nipy27qOK5slOM2l3oYfNCSudZOjtTpONe9KakWAmSxjDylFJ72cJzZDpGZmqRNP7GSYcMr8zlWbPK9zSMlmVccdsvTDzhbUb5lRGwicUE+EAMdFwT88UJxeyC9tZtTQyRnNUHuR0kE+CMUiOJXe0bwgYmOCCAVWoAAcHQoxt0CDAswh5oOChtg0Raj5GQj7nUAcV2Sq6ODo0VPfwauG/YqI6+D/OGUjawFFs/PlIEhin3+h7eky/0g/0B/1zbq5I5ohrOcC73mC5Vxp+davwuydVwXsIuy2qa80hlOGxrFVg7Z5U4rcwGnzt5eFxYXplPJqgb+lPrP8NPaKf8A2c2i/j3TJfeY3Zz69Ix2rq0hF7HNReyJ5UZJUOzkKEeswF6Ivjmux4QxGyd07S9W70/7IrCfmvZzqekVS7c1tncls9ucUzucWe3OqZ3GpPLu6GmbCttRg19e60jv6qXPWdfOtJ9wyWHInLeWjPUWx+KaZcfU9POLvnNJtrxcRZDnC24y/3AKMQr62v9Dy+0Fwv7d0sNEbG/STfuXucDtancvmHuQfL98dm55KdZQBuw11I4+7xCGbhGSzBGhjKofJR+ax86f+eGkzdSN1sWPuUhLkObUfqzl/nXn7F</latexit><latexit sha1_base64="lCY/0gm4rXLreOrcpgJBYzusprI=">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</latexit>(P)<latexit sha1_base64="gHrbVgNylNmRIhB+VT3j9YbpXvY=">AAAFUnicnZTNbtNAEMcnaUJLKW1KL0hIKCIqShGNNghEhYRUwQEOVdXmo6malsgf27CqvbbsdcBYuXHiBThwAokDcIQ34MIDwKGPgDgWCQlxYLw2CWkTR+patsf/nd/sePZDtQ3mCkIOU+mJTPbM5NTZ6XMz52fncvMXtlzLczRa1yzDcrZVxaUG47QumDDotu1QxVQN2lAP7of9jQ51XGbxmvBtumcqbc72maYIlFq5y41r3t1dxkUrWF7rPlrLN4pPl/0lr+hfF0u638oVSInIlj9plGOjAHHbsObTs7ALOliggQcmUOAg0DZAARevJpSBgI3aHgSoOWgx2U+hC9PIeuhF0UNB9QCfbfxqxirH7zCmK2kNRzHwdpDMwyL5Rt6RI/KFfCDfyZ+RsQIZI8zFx7casdRuzb24WP01ljLxLeBxn0rMWcA+rMhcGeZuSyX8Cy3iO89eHlXvVBaDq+QN+YH5vyaH5DP+Ae/81N5u0sorjD46IxWz6UqP0Iej9kTWxJRZcpyFAPWQc9EvtDuy4pHCZO14XPUk+rRsJSb/1UzFK5BqMrczlNsZy60P5dbHcrWhXG0sF1ZDj9n+Wgx6ejKtor8nV/1xvt+THKEtR6JyHgZjNHs7RZer78F/nskx9d5a0XGWXZztcOf6aAl89nfpKL7aWy+D1axGI+N5Uj5+epw0tm6UyrdKZPNmYfVefLJMwSW4AkU8PW7DKjyEDajj6M/hPXyET5mvmd/ZVHYick2nYmYBBlp25i8E/yU7</latexit>

where periodic B.C. is imposed and W ⇤ u =2L-periodic W 2 L1 (⌦).

Z Ly)u(y, t)dy for

W (x

L

<latexit sha1_base64="HX5tOclRTzaRj/rnKbNip5d2CjI=">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</latexit>◉ Typical examples: K(x) = (R<latexit sha1_base64="esvIbZK88eTsAj/85GbRGAO2Ntg=">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</latexit>

1
L |x|
p
cosh
,
dj sinh pL
dj|x|) B(0,R) (x), kj (x) = p<latexit sha1_base64="fgJ2vSRaY1+OfQrsJmub6HvnRq0=">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</latexit>

2<latexit sha1_base64="2rlkYi6hHkRLyIj9PZ9eQD3VWFc=">AAAFLXicnZQ7b9NQFMdPU0NLefTBgsRSERWVJbqpQCCmqixMVZtHW/pQZDu36VX9kn0TCFa/ACsDAzCAxIAY+BAsfIEOlRArQoxFYunQ/702CU0TR+Jato//9/zOPT73YQWOiCRjRyO5UePCxbHxSxOXr1y9Njk1PbMW+c3Q5lXbd/xwwzIj7giPV6WQDt8IQm66lsPXrf1Hqn+9xcNI+F5FtgO+45oNT+wK25SQnmzbe6K2NP/sTm0qzwpMt9nzRjE18pS2FX86N0bbVCefbGqSS5w8krAdMinCtUVFYhRA26EYWghL6H5OBzQBtgkvDg8T6j6eDXxtpaqHbxUz0rSNURzcIchZmmOH7CM7Zl/ZJ/aTnQyMFesYKpc23lbC8qA2+eJG+c9QysVb0l6XysxZ0i490LkK5B5oRf2FnfCt56+Oyw9Lc/Ft9p79Qv7v2BH7gj/wWr/tD6u89BrRB2dkIZsD7aF8PGhPdU1cnaWHWYihKy6Cn7JbuuKJInTtvLTqWfT/sqWU/FszC1es1Wxusy+3OZRb7sstD+UqfbnKUE5Vo56y3bUYd/Rs2oJ/U6/6Xr7bkx2hoUfieh56Y/zblx2l3lkddcxrhPlVe7UNS+LZ3ZeD+HJnhZytXzkZGSdIsfe8OG+sLRSK9wps9W5+cSk9S8bpJt2ieZwX92mRHtMKVTG6Sy/pDb01PhuHxjfje+KaG0mZ63SmGT9OAQJwHU8=</latexit>

dj

B (x) : characteristic func.

B(0, R) : a ball with radius R ,<latexit sha1_base64="cAjm+4XXT9quc3pzuBatYEllUfw=">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</latexit><latexit sha1_base64="2QsBxoar7fyD+q4R3wfBQhvDYfE=">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</latexit>x ⇤ ⇢velocity of advection term.
◉ (W ⇤ ⇢)x = W ⇤ ⇢x =isWthe<latexit sha1_base64="HAWyFfPr84AkovFVDrNdzZz/e48=">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</latexit>

Examples of W (x) :<latexit sha1_base64="6F2oEhxKTEEAcBQr5VWb6KzcXcE=">AAAFKHicnZS/b9NAFMdfU0NL+dEWFiSWiqioLNGlArViqtqFqWrzo6laqsp2rump/iX7Ekit/gMMrEViAokB+DNYGFkQysqGGIvEwsD3ziahaeJInGX7+Xvv8+753Q8rcEQkGeuM5caNS5cnJq9MXb12/cb0zOzNrchvhjav2r7jh9uWGXFHeLwqhXT4dhBy07UcXrOO1lR/rcXDSPheRbYDvueaDU8cCNuUSqotPLu/P5NnBabb3EWjmBp5StuGP5uboCdUJ59sapJLnDySsB0yKcK1S0ViFEDboxhaCEvofk4nNAW2CS8ODxPqEZ4NfO2mqodvFTPStI1RHNwhyDmaZ1/YO3bGPrEP7Dv7PTRWrGOoXNp4WwnLg/3p57fLv0ZSLt6SDntUZs6SDmhZ5yqQe6AV9Rd2wreOT8/Kj0rz8T32hv1A/q9Zh33EH3itn/bbTV56hejDM7KQzYn2UD4etKe6Jq7O0sMsxNAVF8FP2S1d8UQRunZeWvUs+n/ZUkr+rZmFK9ZqNrczkNsZya0P5NZHcpWBXGUkp6pRT9neWoy7ejZtwb+pV30/3+vJjtDQI3E9D/0x/u3LjlLvro465jXC/Kq92oYl8ezty2F8ubtCztevnIyME6TYf15cNLYWC8WHBbb5IL+ymp4lk3SH7tICzoslWqHHtEFVjH5IL+iUXhrvjc/GV6OTuObGUuYWnWvGtz9gWhtD</latexit><latexit sha1_base64="2Xfw6t4yolMbx/vIYXm4WKTvMao=">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</latexit>

K(x)

1.2

1.5

<latexit sha1_base64="Kt6rlIU9PzMC27fQc8oIBLy5ScU=">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</latexit>

kj (x)

1.0

0.8

1.0

0.6

0.4

0.5

0.2

-4

-2

0

2

4

-4

-2

2

4

15

Keller-Segel system for approximation
We introduce the auxiliary attractive and repulsive substancesvj" (x, t), (j = 1, · · · , M ) in the part of advective nonlocal interaction.
<latexit sha1_base64="erdAjIaRuuH/YayHI0m25RNq1MA=">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</latexit>[c.f. Ninomiya, T., Yamamoto, J.M.B 2017, J.J.I.A.M., 2018]
8
M
⇣ @ X
⌘
>
@
>
>
⇢"t = ⇢"xx⇢"
aj vj" ,>>>@x@x j=1<<latexit sha1_base64="epZwNWlltUQWrxINM2Ar8YPDjLg=">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</latexit><latexit sha1_base64="RSB0Lvt4t7xNEcrtXd+ZFG2qpX0=">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</latexit>(KS" )

1
>
"
"
"
"
>
(v
)
=
(d
(v
)
v
+
⇢
), (j = 1, . . . , M ),
t
j
xx
>
j
j
j
>
"
>
>
: "
⇢ (x, 0) = ⇢0 (x), vj" (x, 0) = (vj" )0 (x) 2 C 2 (⌦)

where 0 < " ⌧ 1, dj > 0 , and {aj }M
j=1 are nonzero constants.
<latexit sha1_base64="bx3GvPNCzj9VJnoWcjeFhmQFnRo=">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</latexit><latexit sha1_base64="S4Fgovv5GbKW/tfop0Zy1HmiF9w=">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</latexit>◉ If M = 1, (KS" ) becomes the Keller-Segel eq. with linear sensitive func.
<latexit sha1_base64="RSB0Lvt4t7xNEcrtXd+ZFG2qpX0=">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</latexit><latexit sha1_base64="JwiM70hOuL4tVzGTxUpenH4pFiw=">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</latexit>◉ Taking the limit of " ! 0 , we expect that 0 = dj (vj" )xx<latexit sha1_base64="Nt4tJXiWXliiu/SuriIvjMIMuek=">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</latexit><latexit sha1_base64="uSFB3bMz9Qtv9502VDU0JS8ZiwM=">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</latexit>

Base of non-local interaction

In fact, we have vj" (x, t) = (kj ⇤ ⇢" )(x, t),<latexit sha1_base64="+Zbz98G+K6uuW7sVS1DIV0tCkjY=">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</latexit><latexit sha1_base64="CDaAgBwzhTTCrgmsMKYZnVwas4U=">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</latexit>

1
L |x|
p
p
k
(x)
:=
cosh
.
where j2 dj sinh pLdj(P) with W (x) =<latexit sha1_base64="lCY/0gm4rXLreOrcpgJBYzusprI=">AAAFPHicnZQ7b9NQFMdPUwOlQF8sSCwVUauyRDcIBGKqYICpSvNoq6RV5cdtelX72rKdQLDyBZiRGBBIIDEgvgMLC1+AoTsLqpiKYGHo39cmIW3iSFzL9vH/nt+5x+c+DM8WQcjY4URuUjt3/sLUxelLl6/MzM7NL2wEbss3ec10bdffMvSA20LyWihCm295Ptcdw+abxsHDuH+zzf1AuLIadjy+4+hNKfaEqYeQ6tuO4T6NVko3u7tzeVZgqi2eNYqpkae0ldz53Axtk0UumdQihzhJCmHbpFOAq0FFYuRB26EImg9LqH5OXZoG24IXh4cO9QDPJr4aqSrxHccMFG1iFBu3D3KRlthX9oEdsy/sI/vO/oyMFakYcS4dvI2E5d7u7PNrld9jKQfvkPb7VGbOIe3RPZWrQO6eUuK/MBO+/ezlceV+eSlaZu/YEfJ/yw7ZZ/yBbP8036/z8itEH52RgWy6yiP2kdCeqJo4KkuJWYigx1wAv9huq4onilC1k2nVs+j/Zcsp+bdmBq5IqdlcfShXH8utDeXWxnLVoVx1LBdXw0rZ/lqMeno2bcC/pVb9ab7fkx2hqUbiah4GYzR6O8VSq+/RP57ZMa3eWrEwywFmO965HVghnv1dOoqv9NbLYDUrycg4T4qnT4+zxsatQvFOga3fzq8+SE+WKbpON2gFp8ddWqXHVKIaRpf0gl7TG+2T9k070n4krrmJlLlKA037dQJ+KiD0</latexit>vj" + ⇢" .

<latexit sha1_base64="2tNkuZgZIKJdm7KtVDdKrJ6Qaaw=">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</latexit>

M
X
j=1

dj

aj kj (x)16Main result 1: singular limit & order estimate
Theorem 1:
Let M be an arbitrary fixed natural number, and ⇢(x, t) be a solution<latexit sha1_base64="JVkocRivJqeRjvflg57eifBMMAc=">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</latexit><latexit sha1_base64="AY0IMfGb+y/BxtfaQgO7Cp6R8KE=">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</latexit><latexit sha1_base64="apaL5GPV0JUlMLCpDC392N/4vm8=">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</latexit>of (P) equipped with W =<latexit sha1_base64="lCY/0gm4rXLreOrcpgJBYzusprI=">AAAFPHicnZQ7b9NQFMdPUwOlQF8sSCwVUauyRDcIBGKqYICpSvNoq6RV5cdtelX72rKdQLDyBZiRGBBIIDEgvgMLC1+AoTsLqpiKYGHo39cmIW3iSFzL9vH/nt+5x+c+DM8WQcjY4URuUjt3/sLUxelLl6/MzM7NL2wEbss3ec10bdffMvSA20LyWihCm295Ptcdw+abxsHDuH+zzf1AuLIadjy+4+hNKfaEqYeQ6tuO4T6NVko3u7tzeVZgqi2eNYqpkae0ldz53Axtk0UumdQihzhJCmHbpFOAq0FFYuRB26EImg9LqH5OXZoG24IXh4cO9QDPJr4aqSrxHccMFG1iFBu3D3KRlthX9oEdsy/sI/vO/oyMFakYcS4dvI2E5d7u7PNrld9jKQfvkPb7VGbOIe3RPZWrQO6eUuK/MBO+/ezlceV+eSlaZu/YEfJ/yw7ZZ/yBbP8036/z8itEH52RgWy6yiP2kdCeqJo4KkuJWYigx1wAv9huq4onilC1k2nVs+j/Zcsp+bdmBq5IqdlcfShXH8utDeXWxnLVoVx1LBdXw0rZ/lqMeno2bcC/pVb9ab7fkx2hqUbiah4GYzR6O8VSq+/RP57ZMa3eWrEwywFmO965HVghnv1dOoqv9NbLYDUrycg4T4qnT4+zxsatQvFOga3fzq8+SE+WKbpON2gFp8ddWqXHVKIaRpf0gl7TG+2T9k070n4krrmJlLlKA037dQJ+KiD0</latexit>

M
X
j=1

aj kj (x) and the initial value ⇢0 (x) 2 C (⌦),
<latexit sha1_base64="H4JBGFNJdkhth0x9eiMLiqspHRM=">AAAFQHicnZTPT9RAFMcfSxHEHyx6MXpp3GCWy2aWKBhORC6eEPYHEFjYtN1hmdBOm3Z2YW1IPPsPePCkiSbGg/4NevEf8MB/oPFgDCYkxoNvpnUXlt1uwjRtX7/zPm9e3/wwPZsFgpCjodSwNnJpdOzy+JWr165PpCdvrAZuw7do2XJt1183jYDajNOyYMKm655PDce06Zq5tyj715rUD5jLS6Ll0S3HqHO2wyxDoFRN3674u26VZA+m9Qrj+uL2TLbyxKF1Y1qvpjMkR1TTzxv52MhA3JbdydQoVKAGLljQAAcocBBo22BAgNcm5IGAh9oWhKj5aDHVT+EQxpFtoBdFDwPVPXzW8WszVjl+y5iBoi0cxcbbR1KHKfKVvCPH5At5T76Tv31jhSqGzKWFbzNiqVedeH6reDKQcvAtYLdDJeYsYAceqlwZ5u4pRf6FFfHNpy+Oi/OFqfAeeU1+YP6vyBH5jH/Am7+tNyu08BKj98/IxGwOlYf04ajtq5o4KkuOsxCiLrkA/aTdVBWPFKZqx+OqJ9EXZQsx+b9mJl6hUpO5jZ7cxkBuqSe3NJAr9eRKAzlZjVrMdtZi2NaTaRP9G2rVd/OdnuQIdTUSVfPQHeN0X3KUWnt11HBeA5xfuVdbaAl8dvZlP77YXiFn61eMRsYTJN99Xpw3Vmdy+dncg5X7mYVH8VkyBnfgLmTxvJiDBXgMy1DG0Z/BW/gAH7VP2jftp/Yrck0NxcxNONO0P/8AqwgjLw==</latexit>2and ⇢" (x, t) be a solution of (KS" ) equipped with<latexit sha1_base64="io/jmGEoEbAPMrkv7OEL5zgJnbI=">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</latexit>⇣⌘<latexit sha1_base64="UL5NkR5Gi7UnKesBKp49NDtgGxM=">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</latexit>(1)<latexit sha1_base64="RSB0Lvt4t7xNEcrtXd+ZFG2qpX0=">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</latexit>"⇢" , v1" , . . . , vM
(x, 0) = (⇢0 , k1 ⇤ ⇢0 , . . . , kM ⇤ ⇢0 )(x).

Then, for any " > 0 and T > 0 , there exist positive constants C1 and C2<latexit sha1_base64="iOnFcri+0Nf2upY/miB7G15ef4Y=">AAAFK3icnZS/b9NAFMdfg6Gl/OgPFiSWiKiIKXpBrUAMqIKlU9XmR1P1hyrbuYZT7bPlu6QKVv8Btk6V2gkkBoT4K1jYEUMRKwNiLBILA+/OJqFp4kicZfv5e+/z7vndDyf0uFSIp2O5S9blK+MTVyevXb9xc2p6ZnZNBq3IZTU38IJo3bEl87hgNcWVx9bDiNm+47G6s/dM99fbLJI8EFXVCdm2bzcF3+WurUiqb7FQPsH8znQBi2ha/qJRSo0CpG0lmMmNwxY0IAAXWuADAwGKbA9skHRtQgkQQtK2ISYtIoubfgYHMElsi7wYedik7tGzSV+bqSroW8eUhnZpFI/uiMg8zOFnfItn+BHf4Xf8PTRWbGLoXDr0dhKWhTtTL29Xfo2kfHoreN6jMnNWsAuPTK6ccg+Nov/CTfj2i6OzyuPyXHwPX+MPyv8VnuIH+gPR/um+WWXlE4o+PCOHsjkwHtpHkLZvauKbLAXNQky65iT5abttKp4o3NROpFXPov+XLafk35o5dMVGzeY2BnIbI7nlgdzySK46kKuO5HQ1GinbW4txV8+mHfJvmVXfz/d6siM0zUjMzEN/jH/7sqM0uqujQfMqaX71Xu2QpejZ25fD+Ep3hZyvXyUZmU6QUv95cdFYe1AsLRRxdb6w+DQ9SybgDtyF+3RePIRFWIIVqJnVfwjHcGK9tz5ZX6yviWtuLGVuwblmffsD9csccw==</latexit><latexit sha1_base64="jK8Wlu7Q1FSTlAfFWZQ4Ss5yHS4=">AAAFKHicnZS/b9NAFMdfg6Gl/GgLCxJLRFTEFF0qEIgBVbAwVW1+tFVLVdnONT3VPlv2JShY/QcYWIvEBBID8GewMLIglJUNMRaJhYHvnU1C08SROMv28/fe593zux9O6IlYMdabKpyxzp6bnjk/e+Hipctz8wtX1uOgHbm84QZeEG06dsw9IXlDCeXxzTDitu94fMM5eKT7Nzo8ikUg66ob8h3fbkmxJ1xbaan+gBV350uszEwrnjYqmVGirK0GC4VpekJNCsilNvnESZKC7ZFNMa5tqhCjENoOJdAiWML0czqkWbBteHF42FAP8GzhaztTJb51zNjQLkbxcEcgi7TIvrB37Jh9Yh/Yd/Z7bKzExNC5dPF2UpaHu3PPr9V+TaR8vBXtD6jcnBXt0T2Tq0DuoVH0X7gp33l2dFy7X11MbrI37Afyf8167CP+QHZ+um/XePUVoo/PyEE2h8ZD+0hoT01NfJOlxCwk0DUXw0/bHVPxVBGmdjKreh79v2w1I//WzMGVGDWf2xrJbU3kVkZyKxO5+kiuPpHT1Whm7GAtJn09n3bg3zarfpgf9ORHaJmRuJmH4Rj/9uVHafZXRxPzGmN+9V7twlJ4DvblOL7WXyEn61dLR8YJUhk+L04b60vlyp0yW7tdWn6YnSUzdJ1u0C2cF3dpmR7TKjUw+j69oCN6ab23PltfrV7qWpjKmKt0olnf/gA5aBsF</latexit><latexit sha1_base64="i4J2DQYwMf7kIjpkQe7orwEjylg=">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</latexit><latexit sha1_base64="g6awHi3kZHO01drnd3AHAh1U06U=">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</latexit>

which depend on aj and T , but are independent of " such that<latexit sha1_base64="fWZbARSEhuvh7SzRU4vTi441sP8=">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</latexit><latexit sha1_base64="d9+dtp2zDw90bjrfpiLH+8jYkzg=">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</latexit><latexit sha1_base64="BszaJalQAXJtVx2AFVAkCwBdWW4=">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</latexit>

sup k⇢" (·, t)<latexit sha1_base64="R8XCJFnapY04QyjD/lFazMTjxyE=">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</latexit>

t2[0,T ]<latexit sha1_base64="c3GYCsZ1QCD7SOBI1K0UzlXX3G0=">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</latexit>supt2[0,T ]vj" (·, t)⇢(·, t)kC(⌦)  C1 ",kj ⇤ ⇢(·, t) C(⌦)  C2 ".

17

A natural question

What is the relationship between any even potential andthe Keller-Segel system?
8
>
>
"
"
>
<⇢t = ⇢xx
<latexit sha1_base64="w/Skp5HC0qin/+HHYYhiPj6hnZI=">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</latexit><latexit sha1_base64="RSB0Lvt4t7xNEcrtXd+ZFG2qpX0=">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</latexit>(KS" )M⌘@ ⇣ " @ X⇢
aj vj" ,@x@x j=1>>>:(v " )t = 1 (dj (v " )xxjj"

vj" + ⇢" ), (j = 1, . . . , M ),18

Realization of any even kernel
1<latexit sha1_base64="aFN3QAxqCkZmNurBQ43D8JhFPq8=">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</latexit>

Setting dj =(jj<latexit sha1_base64="OdsvcyKOVMyx1YNqH/mymGYI2rk=">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</latexit>1), we have kj (x) = 2 sinh(j
2

1
1)L

cosh(j

|x|).

1)(L

We set d1 is sufficiently large.
<latexit sha1_base64="PcZjJatUOYGm65DqYFTLzpU85T4=">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</latexit>

Theorem 2 (cf. Existence of {cj }nj=0 : [Ninomiya, T., Yamamoto, J.M.B., 2017])<latexit sha1_base64="AEwmmuEtivtajiuoDsWNyHKjDs8=">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</latexit>

Assume that W (x)is even in ⌦ and in C ([0, L]) and, let f (x) bepf (x) := W (L log(x + x2 1)) = W (cosh 1 (L x)) . Then, for any n 2 N there
exist explicit {cjn}nj=0 such that⇣⌘<latexit sha1_base64="3IaPrxg4u1GeYKYlAHaFLMfccyE=">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</latexit><latexit sha1_base64="b1wzwEHbXFVGbMkQQwsNSLmLZ7s=">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</latexit><latexit sha1_base64="PZA64hqer3mdU2cbC7R11m1koO4=">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</latexit>n<latexit sha1_base64="tp/b6qTYCrWvcADDySGMqY/1p20=">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</latexit><latexit sha1_base64="ZNmxTC0UYy+Q2ZKb68rGBA3jnHY=">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</latexit><latexit 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sup

W (x)

x2[0,L]X

cj cosh j(Lj=01cosh Lx)  n2 (n + 1)!
2

1

n+1

max

y2[1,cosh L]|f (n) (y)|.

From this theorem, we can approximate the solution of (P) with any<latexit sha1_base64="lCY/0gm4rXLreOrcpgJBYzusprI=">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</latexit>

kernel by solution of (KS" ) with specified parameters.
<latexit sha1_base64="RSB0Lvt4t7xNEcrtXd+ZFG2qpX0=">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</latexit><latexit sha1_base64="HZrvlQc95DFo1LpM1GRCZaaeD0o=">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</latexit>

(

2Lc0

(j = 1),

We set aj = 2c sinh((jj 1kj (x) .
for1) (j = 2, . . . M )<latexit sha1_base64="c04+npCCafrlGrQoFI9x/xoXyRs=">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</latexit>

1)L)/(j19<latexit sha1_base64="Kt6rlIU9PzMC27fQc8oIBLy5ScU=">AAAFKnicnZS/b9NAFMdfg6Gl/GgLCxJLRVRUluiC+CWmChamqs2vVi1VZDvXcI19tuxLIFj9B5jYEDCBxIDgv2BhRmKIWFkQY5FYGPj6bBKaJo7Us2w/f+993j2/+2H5jggVY72p3Anj5KnpmdOzZ86eOz83v3ChFnrtwOZV23O8YNMyQ+4IyatKKIdv+gE3XcvhG1brfty/0eFBKDxZUV2f77hmU4pdYZsKUq1V31t+cq0+n2cFptviUaOYGnlK25q3kJumh9Qgj2xqk0ucJCnYDpkU4tqmIjHyoe1QBC2AJXQ/p32aBduGF4eHCbWFZxNf26kq8R3HDDVtYxQHdwBykZbYV/aeHbDP7AP7wf6MjRXpGHEuXbythOV+fe7ZpfLviZSLt6JHAyozZ0W7dEfnKpC7r5X4L+yE7zx9cVC+W1qKrrK37Cfyf8N67BP+QHZ+2e/Week1oo/PyEI2+9oj9pHQHuuauDpLiVmIoMdcCL/Y7uiKJ4rQtZNp1bPo47KllPxXMwtXpNVsbmsktzWRWx3JrU7kKiO5ykQurkYjZQdrMerr2bQF/7Ze9cP8oCc7QlOPxPU8DMf4vy87SqO/OhqY1xDzG+/VLiyF52BfjuPL/RVyuH7lZGScIMXh8+KoUbteKN4q3Fy/kV+5l54lM3SZrtAyzovbtEIPaI2qGH2PntNLemV8NL4YPeNb4pqbSpmLdKgZ3/8C0T0cOg==</latexit>

kj (x)

Realization of arbitrary kernel

1.5

1.0

0.5

Theorem 3:
For any even 2L-periodic C n ([0, L]) function W , any " > 0 and any
T > 0 , there exist a Keller-Segel system (KS" ) with M + 1 component,
and a positive constant CT independent of " such that
-4

-2

2

4

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sup k⇢" (·, t)t2[0,T ]⇢(·, t)kL2 (⌦)  CT "where ⇢ is the solution of (P) equipped with ⇢0 (x) 2 C 2 (⌦) and⇢" is the first component of (KS" ) equipped with (1).
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Numerical experiments:<latexit sha1_base64="lpsQkIv3BlVCkdXpTH6SLx4WICg=">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</latexit><latexit sha1_base64="Ujhk/gzdoLdGNPkKYOTfmlqRWrI=">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</latexit><latexit sha1_base64="lCY/0gm4rXLreOrcpgJBYzusprI=">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</latexit>(P)<latexit sha1_base64="9EmUq8EILzpBHNR9rCj3xxWoleM=">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</latexit>

W (x) = e<latexit sha1_base64="RSB0Lvt4t7xNEcrtXd+ZFG2qpX0=">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</latexit>(KS" )

5x2

Profiles of X
W and, ⇢0 (x) = 1 + ⇠(x), µ = 5.0, d1 = 1000000, M = 7

cj cosh(L

|x|)

20

Concluding remarks◉ Combining the solution of a Keller-Segel system with the linear
sensitive function (KS" ) can approximate the solution of thenonlocal Fokker-Planck eq. with any integral kernel in L2 .
<latexit sha1_base64="RSB0Lvt4t7xNEcrtXd+ZFG2qpX0=">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</latexit><latexit sha1_base64="4AuyHgxAYj20lueiIFV94w9rk0I=">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</latexit><latexit sha1_base64="lCY/0gm4rXLreOrcpgJBYzusprI=">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</latexit>(P)<latexit sha1_base64="RSB0Lvt4t7xNEcrtXd+ZFG2qpX0=">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</latexit>

8
>
>
"
"
>
< ⇢t = ⇢xx<latexit sha1_base64="ZMDVd0gneWrMin1JfdwroauYcGY=">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</latexit><latexit sha1_base64="0sACRpLLOl2WApOZMqMieNgB3NQ=">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</latexit>⇢t = ⇢xx⌘@ ⇣ @⇢ (W ⇤ ⇢)@x @x⇡<latexit sha1_base64="qD8KbQOWrvBNUCZndCQxZzGr1ZE=">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</latexit>>(KS" )M⌘@ ⇣ " @ X"⇢
aj vj ,
@x@x j=1>"!0+0 >: (v " ) = 1 (d (v " )<latexit sha1_base64="z70vQOi1ZFxI1aX2iKZdfPwmDog=">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</latexit>

j t

"

j

j xx

vj" + ⇢" ), (j = 1, . . . , M ),◉ The time of numerical simulation of (KS" ) is shorter than that of (P).
(Typical scheme of numerical integration was applied in (P).)<latexit sha1_base64="RSB0Lvt4t7xNEcrtXd+ZFG2qpX0=">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</latexit><latexit sha1_base64="lCY/0gm4rXLreOrcpgJBYzusprI=">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</latexit><latexit sha1_base64="lCY/0gm4rXLreOrcpgJBYzusprI=">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</latexit>◉ We will extend our theory to case of higher dimension and themodel of cell adhesion.

21

Numerical results in 2D<latexit sha1_base64="lCY/0gm4rXLreOrcpgJBYzusprI=">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</latexit>(P)<latexit sha1_base64="X1YNFC3dQ5wqTgsgds1IZ4iIYXo=">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</latexit>

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"Spreading fronts arising from the singular limit of reaction-diffusion systems"
Chang-Hong Wu (National Yang Ming Chiao Tung University, Taiwan)
To gain insight into the formation of spreading fronts of invasive species, in this presentation we will focus on the
singular limit of reaction-diffusion systems.
We investigate the dynamics of the limiting systems and give some interpretations for spreading fronts from themodeling viewpoint. The talk is based on joint works with Hirofumi Izuhara and Harunori Monobe.

Introduction

Competition-diffusion models

Numerical results

Spreading fronts arising from the singular limit ofreaction-diffusion systems

Chang-Hong Wu
National Yang Ming Chiao Tung University
Based on joint works with Hirofumi Izuhara and Harunori Monobe
Reaction-diffusion systems: from the past to the future
dedicated to the memory of Professor Masayasu Mimura
2023.10.31-11.2 ICMMA, Meiji University

SummaryIntroduction

Competition-diffusion models

OutlineIntroduction
Spatial segregation limit
Numerical results
Summary

Numerical results

SummaryIntroduction

Competition-diffusion models

Numerical results

Summary

The habitat segregation phenomenon
Mimura-Yamada-Yotsutani (1985,1986,1987) ut = d1 uxx + f (u), 0 < x < h(t), t > 0,
vt = d2 vxx + g(v ), h(t) < x < 1, t > 0,
 u(0, t) = M1 , v (1, t) = M2 , t > 0,u(h(t), t) = v (h(t), t) = 0, t > 0,
 h′ (t) = −µ1 ux (h(t), t) − µ2 vx (h(t), t), t > 0,h(0) = h0 ∈ (0, 1),u(x, 0) = u0 (x), 0 < x < h0 , v (x, 0) = v0 (x), h0 < x < 1.
The global existence, uniqueness, regularity, asymptotic behavior of solutions,and stability of stationary solutions

Classical Stefan problems arise in many physical problems such as the melting ofmaterials and freezing of liquid (Rubinstein 1971)

Introduction

Competition-diffusion models

Numerical results

The spreading of an invasive species
Du-Lin (2010):

ut = duxx + u(a − bu), 0 < x < h(t), t > 0,ux (0, t) = 0, u(h(t), t) = 0, t > 0,h′ (t) = −µux (h(t), t), t > 0,h(0) = h0 , u(x, 0) = u0 (x), 0 < x < h0 ,Spreading-Vanishing dichotomy
critical length: h∗ := π2

q

d
a

√
Semiwaves exist when c0 ∈ (0, 2 ad)
The global dynamics (Du-Matsuzawa-Zhou 2014, 2015)
If h∞ := limt→∞ h(t) = ∞, then u(x, t) → Uc0 (h(t) − x) uniformly in x, as t → ∞.

A biological explanation for FBC: population loss (Bunting-Du-Krakowski 2012)

SummaryIntroduction

Competition-diffusion models

Numerical results

Scalar equations: see also Kaneko-Yamada (2011), Bunting-Du-Krakowski(2012), Chang-Chen (2012), Du-Matano-Wang (2014), Du-Lou (2015),
Du-Matsuzawa-Zhou (2014,2015), Kaneko-Matsuzawa (2015,2018), Wang(2015,2016), Monobe-W. (2016), Zhao-Wang (2018),
Kaneko-Matsuzawa-Yamada (2020,2022,2023), El-Hachem-McCue-Simpson(2021) and more.
Guo-W. (2012): two species with the weak competition
ut = uxx + u(1 − u − kv ), 0 < x < s(t), t > 0,
vt = Dvxx + rv (1 − v − hu), 0 < x < s(t), t > 0,ux (0, t) = vx (0, t) = 0, u(s(t), t) = v (s(t), t) = 0, t > 0,
 ′s (t) = −µ1 ux (s(t), t) − µ2 vx (s(t), t), t > 0,

See also Guo-W. (2015), W. (2015, 2019), Du-W. (2018,2022), Wang-Zhang(2017), Liu-Huang-Wang (2019) for models with two different spreading fronts.

SummaryIntroduction

Competition-diffusion models

Numerical results

Summary

Question: can reaction-diffusion systems approximate these problems?
Fast-reaction limit, spatial segregation limit: Hilhorst-van der
Hout-Peletier (1996), Dancer-Hilhorst-Mimura-Peletier (1999),
Ei-Ikota-Mimura (1999), Hilhorst-Iida-Mimura-Ninomiya (2001),
Hilhorst-Mimura-Schatzle (2003),
Crooks-Dancer-Hilhorst-Mimura-Ninomiya (2004),
Crooks-Dancer-Hilhorst (2007), Alfaro-Hilhorst-Matano (2008),
Hilhorst-Mimura-Ninomiya (2009), Murakawa-Ninomiya (2011) and more

Introduction

Competition-diffusion models

Numerical results

Summary

Competition models with a small parameter
Izuhara-Monobe-W. (2023):

n
X
1


∂
u
=
d
∆u
+
f
(u
)
−
hij (ui , uj ) − Fi (ui , w),

t
i
i
i
i
i


ε

j=1,
j̸
=
i



n

X

βi
Fi (ui , w),(Pε ) ∂t w = dw ∆w + g(w) −
ε

i=1






∂ν ui = ∂ν w = 0,(ui (x, 0), w(x, 0)) = (ui,0 (x), w0 (x)),

in QT ,in QT ,on ∂QT ,in Ω,

i = 1, · · · , n, Ω is a bounded domain in RN with smooth boundary ∂Ω,QT := Ω × (0, T ], ∂QT := ∂Ω × (0, T ].
di > 0, βi > 0, ε > 0 and dw ≥ 0 (if dw = 0, ∂ν w = 0 is dropped)
fi and g stand for the intraspecific growth functions of ui and w, respectively;
hij measures the interspecific competition from the species uj to species ui ;Fi measures the interspecific competition between the species ui and w.
Izuhara-Monobe-W. (2021) considered n = 2, g ≡ 0 and dw = 0.

Introduction

Competition-diffusion models

Numerical resultsCompetition models with a small parameterIn this talk, we considerui 
fi (ui ) = ri ui 1 −, hij = αij ui uj ,Ki

Fi (ui , w) = γi ui w,w g(w) = rw w 1 −Kw

For initial data, 0 ≤ ui,0 ≤ Ki , 0 ≤ w0 ≤ Kw and ui,0 ∈ C(Ω),
w0 ∈ C(Ω) if dw > 0; w0 ∈ L∞ (Ω) if dw = 0.
Proposition
For any T > 0 and ε > 0, there exists a unique solution (uiε , w ε ) to (Pε ) with the following
regularity(i) if dw > 0,uiε , w ε ∈ C(QT ) ∩ C 2,1 (Ω × (0, T ])(ii) if dw = 0,uiε ∈ C(QT ) ∩ C 1 (0, T ]; C(Ω) ∩ C (0, T ]; W 2,p (Ω) , w ε ∈ C 1 ([0, T ]; L∞ (Ω))
for i = 1, 2, · · · , n and QT := Ω × (0, T ]. Moreover,0 ≤ uiε ≤ Ki , i = 1, ..., n,

and 0 ≤ w ε ≤ Kw .

SummaryIntroduction

Competition-diffusion models

Numerical results

The case dw = 0
We first consider dw = 0:

n
X
1

∂t ui = di ∆ui + fi (ui ) −hij (ui , uj ) − Fi (ui , w),



ε

j=1,
j̸
=
i



n

X

βi
Fi (ui , w),(Pε ) ∂t w = g(w) −
ε

i=1





∂ν ui = 0,





(ui (x, 0), w(x, 0)) = (ui,0 (x), w0 (x)),

in QT ,in QT ,on ∂QT ,in Ω,

Claim: there exist ûi , ŵ ∈ L2 (0, T ; H 1 (Ω)) such thatuiε → ûi ,w ε → ŵstrongly in L2 (QT ) and weakly in L2 (0, T ; H 1 (Ω))

as ε → 0 (up to a subsequence) andûi ŵ = 0

a.e. in QT ∀i(Segregation property)

Multiplying the equation of ui by βi and sum them up for i:nXβi ∂t ui =

i=1

n h
X

βi dui ∆ui + βi fi (ui ) −

i=1

n
X
j=1, j̸=ini Xβiβi hij (ui , uj ) −Fi (ui , w).
εi=1

We subtract the equation of w from the above equation,nX
i=1

βi ∂t ui − ∂t w =

n h
n
i
X
X
βi di ∆ui + βi fi (ui ) −βi hij (ui , uj ) − g(w).
i=1

j=1, j̸=i

SummaryIntroduction

Competition-diffusion models

Numerical results

Summary

We multiply it by a test function ζ. Then, by integrating it over QT , using integration byparts,
n
X

ZZ
QT

n h
n

i
X
X
βi ui − w ζt −βi di ∇ui · ∇ζ + βi fi (ui )ζ −βi hij (ui , uj )ζ

i=1

i=1

ZZ

Z X
n

−

g(w)ζ = −QTΣ

j=1, j̸=i

βi ui,0 − w0 ζ(x, 0),i=1

where (ui , w) = (uiε , w ε ).
Passing to the limit in ε along a subsequence,
n
X

ZZ
QT

n


X
βi dui ∇ûi · ∇ζ
βi ûi − ŵ ∂t ζ −

i=1

+

n
hX

i=1

βi fi (ûi ) +

i=1

=−

Z X
n
Ω

∞

n
n
X
X

i
βi hij (ûi , ûj ) − ŵg(ŵ) ζ dxdt

i=1 j=1,j̸=iβi ui,0 − w0 ζ(x, 0) dx

i=1

for any ζ ∈ C (QT ) with ζ(x, T ) = 0.

(2.1)

Introduction

Competition-diffusion models

Numerical results

Summary

Assume that ∃Ω1 (t), Ω2 (t) and Γ(t) are ’good enough’:Ω1 (t) := {x ∈ Ω| ûi (x, t) > 0, i = 1, 2, ..., n},Ω2 (t) := {x ∈ Ω| ŵ(x, t) > 0},Ω = Ω1 (t) ∪ Ω2 (t), Γ(t) := Ω1 (t) ∩ Ω2 (t),Next, we define[
QT1 :=Ω1 (t) × {t},0<t≤T

QT2 :=[Ω2 (t) × {t},

ΓT :=

0<t≤T

[

Γ(t) × {t}.

0<t≤T

By some assumptions on the smoothness of interfaces, we can separate QT into QT1and QT2 , respectively, in the integral (2.1), we have
ZZ
ζ

n
X

QT1

n
h
i
X
βi − ∂t ûi + dui ∆ûi + fi (ûi ) −hij (ûi , ûj ) dxdt

i=1

j=1,j̸=innoXζ ŵV +
βi dui ∂ν ûi dΩdt,

Z TZ

ZZ
ζ(∂t ŵ − g(ŵ)) dxdt −

+
QT2

0

Γ(t)

(2.2)

i=1

for any ζ ∈ C ∞ (QT ) with ζ(x, T ) = 0, where V denotes the normal velocity from Ω1 (t)to Ω2 (t) at Γ(t) and we further assume that initial conditions are segregated.

Introduction

Competition-diffusion models

Numerical results

Summary

Main result (dw = 0)
Theorem (Izuhara-Monobe-W. 2023)
Assume that dw = 0. Suppose that Γ(t) is a smooth closed oriented hypersurface satisfying∂Ω ∩ ∂Ω1 (t) = ∅ for all t ∈ [0, T ], and Γ(t) moves smoothly. Moreover, we suppose that ŵ
is also smooth in QT2 and ûi (i = 1, 2, ..., n) is smooth in QT1 . Then ŵ satisfiesdŵ(x, t) = g(ŵ),
dt(x, t) ∈ QT2 ,and (ûi , Ωi (t), Γ(t)) satisfies the following free boundary problem:

n
X



∂t ui = di ∆ui + fi (ui ) −hij (ui , uj )in QT1 , i = 1, 2, · · · , n,




j=1,
j̸
=
i




ui = 0on ΓT , i = 1, 2, · · · , n,



n

X
βi di ∂ν ui
on ΓT ,
ŵ(x, t)V = −



i=1



in QT2 ,w(x, t) = w0 (x)in Ω1 (0), i = 1, 2, · · · , n,u(x, 0) = ui,0 (x)w(x, 0) = w (x)in Ω2 (0),0where
QT1 :=[0<t≤TΩ1 (t) × {t},QT2 :=[0<t≤TΩ2 (t) × {t},

ΓT :=

[
0<t≤T

Γ(t) × {t}.

(2.3)

Introduction

Competition-diffusion models

Numerical results

Proposition (Izuhara-Monobe-W. 2023)
Assume that Ω = (−L, L) and
s± (0) = ±s0 , ui (x, 0) = ui,0 (x), x ∈ [−s0 , s0 ] for some s0 ∈ (0, L);ui,0 ∈ C 2 ([−s0 , s0 ]), ui,0 (±s0 ) = 0, w0 ∈ C α ([−L, L]), α ∈ (0, 1),
0 ≤ ui,0 (x) ≤ Ki , x ∈ [−s0 , s0 ]; 0 < w0 (x) ≤ Kw , x ∈ [−L, L],
Then the free boundary problem (2.3) admits a unique classical solutionαα(ui , s) ∈ [C 2+α,1+ 2 (DT )]n × C 1+ 2 ([0, T ]), i = 1, 2, · · · , n,for some small T > 0, whereDT := {(x, t)| s− (t) ≤ x ≤ s+ (t), t ∈ (0, T ]}.
Moreover, the unique solution can be extended up to a time T ∗ satisfying either
limt↗T ∗ s+ (t) = L or limt↗T ∗ s− (t) = −L.
Since w0 > 0 in [−L, L] and w satisfies the logistic equation, we have ki > 0 such thatk1 ≤ w ≤ k2 for all t ≥ 0.
Contraction mapping principle

SummaryIntroduction

Competition-diffusion models

Numerical results

Summary

Main result (dw > 0)
Using a similar argument as in dw = 0, we can obtain
Theorem (Izuhara-Monobe-W. 2023)
Assume that dw > 0. Suppose that Γ(t) is a smooth closed oriented hypersurface satisfying∂Ω ∩ ∂Ω1 (t) = ∅ for all t ∈ [0, T ], and Γ(t) moves smoothly. Moreover, we suppose that ŵ is alsosmooth in QT2 and ûi (i = 1, 2, ..., n) is smooth in QT1 . Then (ûi , ŵ, Ωi (t), Γ(t)) satisfies thefollowing free boundary problem:

n
X


∂t ui = dui ∆ui + fi (ui ) −hij (ui , uj )in QT1 ,
j=1, j̸=i∂t w = dw ∆w + g(w)in QT2 ,ui = w = 0
on ΓT ,

n
X
(2.4)

dw ∂ν w +βi dui ∂ν ui = 0
on ΓT ,




i=1




∂ν w = 0on ∂Ω × (0, T ),





in Ω1 (0),u(x, 0) = ui,0 (x)w(x, 0) = w0 (x)in Ω2 (0),When n = 1, the free boundary problem reduces to the one proposed byDancer-Hilhorst-Mimura-Peletier (1999).

Introduction

Competition-diffusion models

Numerical results

Summary

The revisit: the role of diffusion d1
We revisit the free boundary problem studied by Du and Lin (2010).
Considerb13ut = d1 uxx + u(a − bu) − ε uw, x ∈ Ω, t > 0,

b31
w
)−
uw, x ∈ Ω, t > 0,
wt = rw w(1 −


Kw
ε


∂ν u = 0, x ∈ ∂Ω, t > 0,
where u0 > 0 and w0 > 0 are spatial segregated.
Consider 1D case and spatial symmetry, we are led to study
ut = d1 uxx + u(a − bu), 0 < x < h(t), t > 0,ux (0, t) = 0, u(h(t), t) = 0, t > 0,
d1 b31h′ (t) = −ux (h(t), t), t > 0,w(h(t), t)b13
h(0) = h0 , u(x, 0) = u0 (x), 0 < x < h0 ,Q: the role of d1 in the spreading of u? We can revisit the free boundary problems
in the existing literature.

Introduction

Competition-diffusion models

Numerical results

Summary

The revisit: two weakly competitive species
A modeling perspective for the free boundary condition in Guo-W. (2012):
Considerb13
ut = d1 uxx + r1 u(1 − u − hv ) −uw, x ∈ Ω, t > 0,


ε



b23

vt = d2 vxx + r2 v (1 − v − ku) −vw, x ∈ Ω, t > 0,
ε

b31
b32
w


)−
uw −
vw, x ∈ Ω, t > 0,
wt = rw w(1 −


Kw
ε
ε


∂ν u = ∂ν v = 0, x ∈ ∂Ω, t > 0,
where u0 > 0 (resp., v0 > 0) and w0 > 0 are spatial segregated.
Then the limiting problem has the free boundary condition:h′ (t) = −d1 b31
d2 b32ux (h(t), t) −vx (h(t), t),w(h(t), t)b13
w(h(t), t)b23

where w obeys the logistic equation.

t > 0,

Introduction

Competition-diffusion models

Numerical results

Numerical results
Q: is there any difference between the cases dw > 0 and dw = 0?
We consider the interaction between three weakly competing species ui (i = 1, 2, 3)
and one strongly competing species w:

u1
1∂t u1 = d1 ∂xx u1 + r1 u1 1 −− α12 u1 u2 − α13 u1 u3 − γ1 u1 w,
K1
ε


u2
1
− α21 u2 u1 − α23 u2 u3 − γ2 u2 w,∂t u2 = d2 ∂xx u2 + r2 u2 1 −
K2
ε


u3
1
∂t u3 = d3 ∂xx u3 + r3 u3 1 −− α31 u3 u1 − α32 u3 u2 − γ3 u3 w,
K3
ε


w
1
∂t w = dw ∂xx w + rw w 1 −− (β1 γ1 u1 + β2 γ2 u2 + β3 γ3 u3 ) w,
Kw
εThe parameter values are all fixed except for dw and ε:
d1 = 0.75, d2 = 0.1, d3 = 1.0, r1 = 0.25, r2 = 0.35, r3 = 0.3, rw = 0.25,
K1 = 1.1, K2 = 0.9, K3 = 1.0, Kw = 1.0, α12 = 0.15, α13 = 0.25, α21 = 0.2,
α23 = 0.1, α31 = 0.1, α32 = 0.2, γ1 = γ2 = γ3 = 1.0, β1 = 0.8, β2 = 0.75, β3 = 0.7,
The coexistence state: (u1∗ , u2∗ , u3∗ ) ≈ (0.084760, 0.732021, 0.483733).

SummaryIntroduction

Numerical resultsCompetition-diffusion models

Numerical results

SummaryIntroduction

Numerical resultsCompetition-diffusion models

Numerical results

SummaryIntroduction

Numerical resultsCompetition-diffusion models

Numerical results

SummaryIntroduction

Numerical resultsCompetition-diffusion models

Numerical results

SummaryIntroduction

Numerical resultsCompetition-diffusion models

Numerical results

SummaryIntroduction

Competition-diffusion models

Numerical results

Summary

Summary

We consider the spatial segregation limit for two types ofcompetition-diffusion systems and derive two free boundary problems.
We revisit the free boundary problems in the literature and study the roleof diffusion.
We also present numerical results that demonstrate the complexity of
the role played by diffusion rates.
The interpretation of the parameter in the free boundary condition may
be helpful for its application to real-world data (Izuhara-Monobe-W.
2021).

Introduction

Competition-diffusion models

Numerical results

Thank you for your attention

Summary"Segregation pattern in a reaction-diffusion model of asymmetric cell division"
Yoshihisa Morita (Ryukoku University, Japan)
We deal with a mathematical model describing polarity in the asymmetric cell division of C. elegans embryo. In themaintenance phase of asymmet- ric cell division anterior PAR protein (aPAR) and posterior PAR protein (pPAR) areexclusively formed and a segregation pattern is created for the polarizations of aPAR and pPAR. Seirin-Lee and Shibata(2015) proposed a 4-component reaction-diffusion system with mass conservation as a model to describe the segregation
pattern. Later, some gradient-like dynamics and variational structure in a slightly modied model system were revealed
by Morita and Serin-Lee (2021). In this talk we review their work and report a recent progress.

Segregation pattern in a reaction-diffusion modelof asymmetric cell divisionInternational Conference on"Reaction-diffusion systems: from the past
to the future"— in memory of Prof. Masayasu Mimura —KAKENHI JP18H01139

Contents
1. Introduction
2. Basic results for the model equations
3. Stationary problem
4. Spectral comparison
5. Profile of stable solutions

This lecture is based on the works by:
・M–Seirin-Lee (2021)
S. Seirin-Lee (Kyoto University)

・M-Oshita (2023)

Y. Oshita (Okayama University)21. Introduction
In the asymmetric cell division of C. elegans embryo, anterior PAR
protein (aPAR) and posterior PAR protein (pPAR) are exclusively formed in
an asymmetrical manner, and in the maintenance phase a segregation
pattern is created for the polarizations of aPAR and pPAR.

PAR polarity of C. Elegans embryo cell

A Initial Phase

Maintenance Phase

Patterning Phase
sperm entry

A
aPAR

B

AP-axisSymmetrybreaking
point

pPAR

aPAR polarity

P

Asymmetriccell division
Concentrations
onpPAR polarity
6

7

C cell circumference

88
We remark that PAR proteins are mostly upstream6 regulators
that
7
!!
Model
control the downstream proteinsandaseriesoftheprocesses
of
6
6
4
"!
5
(1.1)
asymmetriccell division.
4

with

A

P

A

P

44
3

3

(by Seirin-Lee – Shibata 2015)

A model for asymmetric cell division

Membrane@ t Pm = D m r 2 Pm

Fof f Pm + Fon Pc<latexit sha1_base64="qwroBZOCG9o7GessIO8fyJh66vk=">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</latexit>AmPm<latexit sha1_base64="Lg8Cjp6+p9HOXZEl3QYH+AdkF5Q=">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</latexit><latexit sha1_base64="Z1nriykGd8WuXdsoHBJQILqo5L8=">AAACunichVFNSxtBGH6y/VBjW2N7KXgJDRZBDJMYsBQLKZXSoxqjgtFldp3EIbMf7E4CuuQP+Ac8eGqhiPQn9NiLf8CDxx41RwUvPfTdzYJU0b7LzjzzvO/zzjMzlq9kqBk7yxiPHj95OjQ8kh199vzFWG785WrodQJb1G1PecG6xUOhpCvqWmol1v1AcMdSYs1qf4rza10RhNJzV/SuLzYd3nJlU9pcE2Xm6g2fB1pyZer8R9P50PCoOm4WLfRMJ99wuaX4VjnOzdzkPvfMyGs2e8RO32JdIm0zV2BFlkT+LiiloIA0Fr3cERrYhgcbHTgQcKEJK3CE9G2gBAafuE1ExAWEZJIX6CFL2g5VCargxLZpbNFqI2VdWsc9w0Rt0y6K/oCUeUyyU3bMLtkJ+8HO2Z97e0VJj9jLLs3WQCt8c2z/de36vyqHZo2dG9WDnjWaeJd4leTdT5j4FPZA3907uKy9X56M3rJvrE/+v7Iz9otO4Hav7O9LYvkQWXqA0u3rvgtWy8XSbLG8VClUK+lTDGMCbzBF9z2HKr5gEXXa9yd+4wJ9Y96wDGm0B6VGJtW8wj9h6L93Rayf</latexit>

CytosolAcPc@t Pc = Dc r2 Pc + Fof f Pm

@ t Am = D m r 2 Am<latexit sha1_base64="Z3xwJOfjYGachz59HUtYiuJSmlo=">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</latexit>

Fon PcF of f Am + F on Ac@t Ac = Dc r2 Ac + F of f Am

F on Ac

Those o↵-rate functions depend on Am and Pm respectively.
<latexit sha1_base64="uvQzzlKq9pOJWeZ77ble1j/YQV4=">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</latexit>

K1 A2m
Fof f = ↵ +,K + A2m<latexit sha1_base64="pje+sZjBcy5jcuWWpRyeYIM5bAc=">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</latexit>

Fon = ,

<latexit sha1_base64="rMpZ1LwLqJ8Fir8BMmLgNh/IbbU=">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</latexit>

2
K 1 PmF of f = ↵ +,2
K + Pm<latexit sha1_base64="q89bFcNlteTA8bv1HUsVEDydHQ4=">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</latexit><latexit sha1_base64="GG0v50YUkq/yu8KKDZEQSX2ILZU=">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</latexit>

F on = ,4

Numerical simulations for the model demonstrating the segregation pattern

Neumannboundary conditions

pPAR (")
aPAR ($)

2.02
1.5

1.01
0.5
0
0.0
0.0
0

0.2

0.4

0.6

0.8 1.0
1

%

1.2

1.4

1.6

1.8

2
2.0

B
Concentrations

Concentrations

A

Periodic
boundary conditions

pPAR (")
aPAR ($)

2
2.0
1.5
1
1.0
0.5
0
0.0
0
0.0

0.2

0.4

0.6

0.8 1.0
1

1.2

1.4

1.6

1.8 2.02%

5

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In Seirin-Lee – Shibata 2015 they also consider a 2-component system which
is formally reduced from the 4-component system.
Although the two-component system is interesting by itself, the reduction
seems to be not so mathematically reasonable.

We take a different approach to the 4-component system (M-Seirin-Lee 2021).
Assuming

K1 Am 2
K1 2
K1 2
4
A
Am ,
Fo↵ (Am ) = ↵ +
=
↵
+
+
O(A
)
⇡
↵
+
m
m
2
K
K
K + AmK 1 Pm 2
K1
K1
2
2
4
F o↵ (Pm ) = ↵ +
=
↵
+
+
O(P
)
⇡
↵
+
P
P
m
m
m
K
K
K + Pm 2

By putting

k := K1 /K

d2 := ⌧ Dm , D2 := ⌧ Dc ,,1 :=⌧ := (K/K 1 )k,and
u1 := Pm , v1 := Pc , u2 := Am , v2 := Ac , d1 := Dm , D1 := Dc ,,2 := ⌧, ↵1 = ↵, ↵2 := ⌧ ↵6

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1
and not identically zero.
ui,0 vi,0 (i = 1, 2) are L<latexit

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we consider the modified system:@ t u 1 = d1 u 1⌧ @ t u 2 = d2 u 2

We deal with this system in a bounded domain ⌦ ⇢ RN with the Neumannboundary condition and initial data:(u1 (x, 0) = u1,0 (x)v1 (x, 0) = v1,0 (x)(↵1 + ku22 )u1 +@t v1 = D1 v1 + (↵1 + ku22 )u1(↵2 + ku21 )u2 +⌧ @t v2 = D2 v2 + (↵2 + ku21 )u2

1 v1 ,

1 v1 ,(4system)
2 v2 ,

2 v2 ,

0, u2 (x, 0) = u2,0 (x) 0,0, v2 (x, 0) = v2,0 (x) 0(x 2 ⌦),

7

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Mass conservation:Z(u1 (x, t) + v1 (x, t))dx = constant,because ofZd(u1 (x, t) + v1 (x, t))dx = 0,dt ⌦<latexit

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2. Basic results for the model equations
Standing assumption:

d1 < D 1

⌦

and

Z

d2 < D 2

Lemma (M--Seirin-Lee)The system has a unique classical solution (u1 (x, t), u2 (x, t), v1 (x, t), v2 (x, t))satisfying

u1 (x, t), u2 (x, t), v1 (x, t), v2 (x, t) > 0 (x 2 ⌦)and it exists time global if 1  N  3 .

The system has the property:(u2 (x, t) + v2 (x, t))dx = constant.

⌦

d
dt
Z
(u2 (x, t) + v2 (x, t))dx = 0.

⌦8

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Put

m1 := hu1 i + hv1 i,Z
1h·i :=· dx|⌦| ⌦@t u1 = d1 u1(1⌧ @t u2 = d2 u2⌧ (1m2 := hu2 i + hv2 i,

By the change of variables

z1 = (d1 /D1 )u1 + v1 ,z2 = (d2 /D2 )u2 + v2

we convert the system to(↵1 + ku22 )u1(↵2 + ku21 )u2( 1 d1 /D1 )u1 +( 1 d2 /D2 )u2 +1 z1 ,

d1 /D1 )@t u1 + @t z1 = D1 z1 ,2 z2 ,

d2 /D2 )@t u2 + ⌧ @t z2 = D2 z2 .
9

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Lyapunov function

E(u, z) :=

where

=
Z
⌦

Z 
⌦

+

d2
↵1 + 1 d1 /D1 2
d1
2
2|ru1 | + |ru2 | +
u1
2
2
2

↵2 +
2 d2 /D22

u = (u1 , u2 ),u22 +k 2 2 ✓1 2 ✓2 2
u1 u2 + z1 + z2 dx,222z = (z1 , z2 ).

✓i :=
i

1
di /Di(i = 1, 2)

Indeed, we can checkdE(u(·, t), z(·, t))
dt[(@t u1 )2 + ⌧ (@t u2 )2 + ✓1 D1 |rz1 |2 + (✓2 /⌧ )D2 |rz2 |2 ]dx  0

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with<latexit

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3. Stationary problem

d1 u 1(↵1 + ku22 )u1( 1 d1 /D1 )u1 +1 z1 = 0,z1 = 0,d2 u 2(↵2 + ku21 )u2( 2 d2 /D2 )u2 +2 z2 = 0,z2 = 0,mi = (1wheredi /Di )hui i + hzi i

d1 u 1( 1 + ku22 )u1 +d2 u2( 2 + ku21 )u2 +i := ↵i +(u⇤1 , z1⇤ , u⇤2 , z2⇤ ) = (u⇤1 , m1
(1

1 {m1

2 {m2

i di /Di

gives an equilibrium solution to (4system).

(i = 1, 2).

These equations turn to be(1(1

d1 /D1 )hu1 i} = 0,

d2 /D2 )hu2 i} = 0.

(i = 1, 2)⇤⇤
We note that the corresponding to a solution (u1 , u2 ) to (SE)

d1 /D1 )hu⇤1 i, u⇤2 , m2
(1
d2 /D2 )hu⇤2 i)

11

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Variational characterization

DefineEs (u) :=
Z ⇢
⌦

d2
k
d1
1
2
|ru1 |2 + |ru2 |2 + u21 + u22 + u21 u22 dx
2
2
2
2
2
1 |⌦|
2
{m1 (1 d1 /D1 )hu1 i}+
2(1 d1 /D1 )2 |⌦|
2{m2 (1 d2 /D2 )hu2 i} .
+2(1 d2 /D2 )It is easy to see that (SE) with the Neumann boundary condition is theEuler-Lagrange equation of Es .

(Q) A local minimizer is stable in the 4-component system?

12

(1

Then given " > 0, there existsk(u(·, 0), z(·, 0))k(u(·, t), z(·, t))

In the view of Lemma (stability), in order to prove the existence ofstable nonconstant equilibrium solutions of (4system), it suffices toshow the existence of a nonconstant minimizer of Es .
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Lemma (stability) M--Serin-Lee

Let u⇤ be a local minimizer of Es and let z ⇤ = (z1⇤ , z2⇤ ) be defined as
di /Di )hu⇤i i(i = 1, 2).

> 0 such that(u⇤ , z ⇤ )kH 1 <implies(u⇤ , z ⇤ )kH 1 < C̃"(t0),

for a constant C̃ > 0.

13

Key lemma (Latos-Suzuki)

Let u⇤ = (u⇤1 , u⇤2 ) be a local minimizer of Es (u) (u 2 H 1 (⌦)2 ).
Then there exists an "1 > 0 such that for any " 2 (0, "1 /4] we can takeu⇤ kH 1 < " holds if
1 = 1 (") > 0 so that ku<latexit sha1_base64="MzEYpowwJq7x7gt8P4RxECK7Nug=">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</latexit>Es (u)Es (u⇤ ) <1with kuu⇤ kH 1 < "1 .

Es<latexit sha1_base64="zRnOypSGSeRQYhAuwX7Kb8+/wpU=">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</latexit><latexit sha1_base64="4Rh5pDSoNkJANrR5zptJ0wNf3to=">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</latexit><latexit sha1_base64="d5pwzjXnNRluZuSokpDPa/QJWcM=">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</latexit>

"

1

u⇤
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Lemma (stability) is proved by using this lemma.

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L1<latexit

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4. Spectral comparison
Around an equilibrium solution (u⇤1 , v1⇤ , u⇤2 , v2⇤ ) , we consider the
linearized operator

d1
B D1
B 1C
C
B
L @ A := B
@ d2
2
D2
2
0

1

✓
'1
'2

1

◆
:=

0

Dom(L) ={( 1 ,

✓
⇤ 2
⇤ ⇤
1 + (↵1 + k(u2 ) ) 1 + 2k(u1 u2 ) 2
1 1(↵1 + k(u⇤2 )2 ) 1 2k(u⇤1 u⇤2 ) 2 + 1 1 C
1
C
⇤ 2
⇤ ⇤
A
2 + (↵1 + k(u1 ) ) 2 + 2k(u1 u2 ) 1
2 2(↵1 + k(u⇤1 )2 ) 2 2k(u⇤1 u⇤2 ) 1 + 2 2
2

1

1,
2,
2)
T

@ i /@n = @2 H 2 (⌦; R4 ) :i /@n = 0

on @⌦,

h i i + h i i = 0 (i = 1, 2)}

d1 '1 + ( 1 + k(u⇤2 )2 )'1 + 2k(u⇤1 u⇤2 )'2 +d2 '2 + ( 1 + k(u⇤1 )2 )'2 + 2k(u⇤1 u⇤2 )'1 +
1 (1
2 (1

d1 /D1 )h'1 i
d2 /D2 )h'2 i◆

Dom(L1 ) ={( 1 , 2 )T 2 H 2 (⌦; R2 ) :@ i /@n = 0 on @⌦ (i = 1, 2)}15

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We let { j }j=1,2,...and {⌫j }j=1,2,... be sets of eigenvalues of
L and L1 respectively.

If
1 

Moreover, if
2  ··· j 6= 0j =0j 
j+1  · · ·⌫1  ⌫2  · · ·  ⌫j  ⌫j+1  · · ·We have the following spectral comparison result:
Theorem (M-Oshita)or ⌫j 6= 0, thenj ⌫j > 0,| j | < |⌫j | holds.

, then ⌫j = 0 holds, and vice versa.

This implies that the numbers of unstable eigenvalues to L and L1
coincide.

Proof can be done by modifying the arguments in M (2012).
(cf. Bates-Fife (1990), Ohinishi-Nishiura (1998))

16

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sha1_base64="m+xTPD0BlGcmSpnDBDmTUliBoK0=">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</latexit><latexit

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sha1_base64="V48Rh1zrcLWDlAE8ALnkn6Atn7U=">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</latexit><latexit

4. Profile of stable solutions
We first introduce previous results.

Lemma (minimizer of the reduced problem)

There is a minimizer u⇤ = (u⇤1 , u⇤2 ) of Es satisfyingu⇤i (x) > 0↵i  ↵,di  d,(x 2 ⌦),di /↵i  r

i = 1, 2.

Theorem (M-S.-Lee)Let ⌦ ⇢ RN (1  N  3) be a cylindrical domain as ⌦ = {x = (x1 , x0 ) 2(0, L) ⇥ D}, where D is a bounded domain of RN 1 with smooth boundary.
For the di↵usion coefficients assume di < Di (i = 1, 2). Then there are positive
numbers ↵, d and r such that for(i = 1, 2),

the system (4system) in ⌦ with N.B.C possesses a stable nonconstant equilibrium solution.
17

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di ⌧ D i ,U1 (x; `) := µ1 (`) 1

U2 (x; `) := µ2 (`) 1µ1 (`) := 1 d /D

1

µ2 (`) := 1 d /D

L
2
1

L

2

{`

{L
`

i ⌧1

( i = ↵i +✓cosh(x/!1 )cosh(`/!1 )✓cosh((Lcosh((L!1 tanh(`/!1 )} +m2

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Assumedi / ii di /Di ,

◆

m1

1/ 1

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Define<latexit

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Put<latexit

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Sketch of Proof:
For simplicity assume ⌦ = (0, L) .

(i = 1, 2)(i = 1, 2),x)/!2 )`)/!2 )`)/!2 )} +` > !1 ,so ↵i ⇡

2/ 2,

L
` > !2 .

i),U1 (x; 0.75)

U2 (x; 0.75)

◆

,

0.75
18

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sha1_base64="APl6pfShgItT/ABLhCkYj+A+9UA=">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</latexit><latexit
sha1_base64="aMfKF9X5ZVOLO7KFHnSzzrmuEVA=">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</latexit><latexit

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DefineThend1 (Ũ1 )xx

d2 (Ũ2 )xx

Es (Ũ1 , Ũ2 ) =

+
1L

2(1
d1 /D1 )sha1_base64="evvoa135/qY2euRuuhVeFFLTzMk=">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</latexit><latexitŨ1 (x) :=(U1 (x; `)0(0  x  `),(`  x  L),

Ũ2 (x) :=(0
U2 (x; `)(0  x  `),(`  x  L).

( 1 + k(Ũ2 )2 )Ũ1 +

Z `✓
0

m1
1
d1 /D1L
1 {m1( 2 + k(Ũ1 )2 )Ũ2 +
2 {m2

◆
◆
Z L✓
1
1
1
2
((U1 )x )2 + U12 dx +((U2 )x )2 + U22 dx
2
2
2
2
`
Z `
0

(1

(1

U1 (x)dx!2

d1 /D1 )hŨ1 i} =(d2 /D2 )hŨ2 i} =
(

+
2L

2(1
d2 /D2 )

0
1 1

2 2

0

m2
1

(0 < x < `),(` < x < L),(0 < x < `),(` < x < L).

andd2 /D2LZ L`

U2 (x)dx
19!2

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i = " i ("),˜
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di ⌧ 1,↵i = " ↵˜i,sha1_base64="MbGsg6I8q034LwT2dXf1AGQXFvY=">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</latexit>

sha1_base64="c2EyZ1ksfcyOdXzPoLFk7wJMbak=">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</latexit><latexit↵i ⌧ 1,<latexit

sha1_base64="sNXDK8WnePU4R+0J2YvlAoo2slI=">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</latexit><latexit

sha1_base64="9FvYdV/tRjUHy8mWJ3TPx3zpH+0=">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</latexit><latexit

If
di /↵i ⌧ 1 (i = 1, 2),i ⌧1(i = 1, 2) ),then for any constant solution (u1 , u2 )Es (Ũ1 , U˜2 ) < Es (u1 , u2 )holds.

This implies the existence of a stable nonconstant equilibrium solution.

In order to estimate the energy clearly, take the following scaling:di = "1+ d˜i ,0<1(i = 1, 2),Then

˜i (") := ↵˜ i + " d˜i i /Di (i = 1, 2)
20

sha1_base64="ZO8rDqLybwHKlUccFJMCN+vR++g=">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</latexit><latexit

sha1_base64="J6ZxPFrPi0pxOIX9O8UAg1tIXjo=">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</latexit><latexit

The equations are written as" d˜1 (u1 )xx( ˜1 (") + (k/")u22 )u1 + ( 1 /"){m1(1"1+ d˜1 /D1 )hu1 i} = 0,

" d˜2 (u2 )xx( ˜1 (") + (k/")u21 )u2 + ( 2 /"){m2(1"1+ d˜2 /D2 )hu2 i} = 0.

and1
Es (u) :="Z (⌦˜1 (")
˜2 (")
k
" d˜2
" d˜1
2
2
2
|(u1 )x | +|(u2 )x | +u1 +u22 + u21 u22
2
2
2
2
2"
n
o2
1L
1+ ˜
m1 (1 " d/D1 )hu1 i+1+
˜
2"(1 " d1 /D1 )
n
o2
2L
1+ ˜
m2 (1 " d2 /D2 )hu2 i .
+2"(1 "1+ d˜2 /D2 )
)
dx

21

sha1_base64="czqeaqQW/F7bFYLwUT0cKm5rbYg=">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</latexit><latexit

sha1_base64="w7i1t84CpWOrLrjgCdLSaDKbLKM=">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</latexit>µ1 (`) =<latexit

sha1_base64="QBTGZ9mH2VKy+zps6l1BvKJGt60=">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</latexit><latexit

sha1_base64="q4BcgOGD5S4UnBK4IOe5A6/fTM8=">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</latexit><latexit

In view of

Lm1+ O(" /2 ),`µ2 (`) =for ` = `⇤ :=Lm2+ O(" /2 )L `we have

1 (Lm2 )2 ↵1 (Lm1 )2 ↵1
˜1
˜2
Es (Ũ1 , Ũ2 ) =++ O(" /2 )
"
2
`
2 L `
p
p 2
L
(m1 ↵
˜ 1 + m2 ↵˜ 2 ) + O(" /2 )
2
Lm ↵
˜1
p 1
p
m1 ↵
˜ 1 + m2 ↵
˜2

p

22

sha1_base64="YJEo0ViRFtjl3zEHcLmaD1Krnbw=">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</latexit><latexit

where ⇢1 (") = O("1/2 ) and ⇢2 (") = O(" /2 ).
p
L p
( ↵
˜ 1 m1 + ↵˜ 2 m2 )22
sha1_base64="4wmkxuiwZ3hK/QfqJETWL50dQLo=">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</latexit>

p
p 2
L
1
"
"
˜ 1 + m2 ↵˜ 2 ) + O(" /2 )Es (u1 , u2 )  (m1 ↵2"<latexit

sha1_base64="NTWBeKQBZ7OuRcabtyLA7hIcnW0=">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</latexit><latexit

Since
sha1_base64="vMVV1RXMDmUQW7Tk//HSqBQafGk=">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</latexit><latexit

11""
Es (u"1 , u"2 )  Es (Ũ1 , Ũ2 ) for the minimizer (u1 , u2 )""we have an upper estimate(M– Seirin-Lee)

Moreover, we obtain

Lemma (M-Oshita)

Let u⇤" = (u⇤1 " , u⇤2 " ) be the minimizer of Es with (S). Then
p
1
L p
⇤
⇢1 (")  Es (u" )  ( ↵˜ 1 m1 + ↵˜ 2 m2 )2 + ⇢2 ("),"
223

sha1_base64="TfX9pMK+d8PpvKTzWlPKuEZuw7U=">AAACpnichVHLahRBFD1pNYmT16gbwU3jkDCBMFSH+MBVRISsJK/OBNKhU12pSYpUP+iuGYhN/4A/4MKVQgjiZ2STjUuVfIK4jODGhXd6GoIGk9t01bmn7rl1qipItMoMY2dD1o2bt4ZHRm/XxsYnJqfqd+5uZHE3FdIVsY7TzYBnUqtIukYZLTeTVPIw0LIdHLzor7d7Ms1UHK2bw0Ruh3wvUh0luCHKrz/3tLS9TspF7hS51+OpTDKl46jwQm72Bdf5y8LPmp5RelfmbuE7c/ZFMj9r+/UGa7Ey7MvAqUADVSzH9WN42EUMgS5CSEQwhDU4Mvq24IAhIW4bOXEpIVWuSxSokbZLVZIqOLEHNO5RtlWxEeX9nlmpFrSLpj8lpY1p9oV9ZOfslH1i39nv//bKyx59L4c0BwOtTPypN/fXfl2rCmk22L9QXenZoIOnpVdF3pOS6Z9CDPS912/P156tTucz7AP7Qf7fszN2QieIej/F0YpcfYcaPYDz73VfBhvzLedx69HKQmNxoXqKUTzAQzTpvp9gEUtYhkv7HuMzvuKb1bReWa7VHpRaQ5XmHv4Ka+cPQZ6jzA==</latexit><latexit

sha1_base64="NTWBeKQBZ7OuRcabtyLA7hIcnW0=">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</latexit><latexit

sha1_base64="d8L8TFIwgSHrKCFVZdqHFSwk8Gs=">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</latexit><latexit

sha1_base64="ylqzOBz5KLn2jD8mh01WulFiQAQ=">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</latexit>

and
p
L p
( ↵
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There exists an equilibrium solution u = (u1 , u2 ) satisfying<latexit

Theorem (M-Oshita)(0 < x < L)

p
1
L p
⇤
⇢1 (")  Es (u )  ( ↵˜ 1 m1 + ↵˜ 2 m2 )2 + ⇢2 (")"
2Ũ2 (x)24

sha1_base64="BVBc8yv0f0/OXJigJ8EiLgzdC3o=">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</latexit>(v) Free boundary problem in the singular limit " ! 0 .
<latexit(Future problems)(i) A rigorous proof to show that the equilibrium solutions with monotone
profile is a minimizer.
(ii) The existence of a stable/unstable solution with multi layers;
Although we can prove that solutions obtained by reflection of themonotone one is unstable, it remains to verify if there exists a stable
solution with multi-layers or not.

(iii) The existence of a stable solution with a transition layer in a higherdimensional domain;(iv) Different parameter regime allowing the segregation pattern should be
examined.

25

Thank you for sharing your time!

In memory of Professor Masayasu Mimura for his great
achievements in the theory of pattern formation arising inreaction-diffusion systems

26

"Speed-up of traveling waves by negative chemotaxis"
Quentin Griette (Université Le Havre Normandie, France)
We consider the traveling wave speed for Fisher-KPP (FKPP) fronts under the influence of chemotaxis and provide
an almost complete picture of its asymptotic dependence on parameters representing the strength and length-scale ofchemotaxis.
Our study is based on the convergence to the porous medium FKPP traveling wave and a hyperbolic FKPP-KellerSegel traveling wave in certain asymptotic regimes. In this way, it clarifies the relationship between three equations
that have each garnered intense interest on their own. Our proofs involve a variety of techniques ranging from entropy
methods and decay of oscillations estimates to a general description of the qualitative behavior to the hyperbolic FKPPKeller-Segel equation. For this latter equation, we, as a part of our limiting arguments, establish an explicit lower
bound on the minimal traveling wave speed and provide a new construction of traveling waves that extends the known
existence range to all parameter values.
This is a joint work with Chris Henderson and Olga Turanova.

Traveling waves in repulsive Keller-Segel models
Quentin Griette (Université Le Havre Normandie)
quentin.griette@univ-lehavre.fr

ICMMA2023: “Reaction-diffusion systems: from the past to the future”
in memory of Prof. Masayasu Mimura

November 2nd, 2023

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

PART I:The hyperbolic-elliptic model
joint works with Xiaoming Fu, Pierre Magal, Min Zhao.

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

1
Jennifer Pasquier et al. “Different Modalities of Intercellular Membrane Exchanges Mediate Cell-to-cell
P-glycoprotein Transfers in MCF-7 Breast Cancer Cells”. J. Biol. Chem. 287.10 (2012), pp. 7374–7387.
doi: 10.1074/jbc.M111.312157.
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

The cell-cell repulsion model∂t u − χ∇ · u∇P = u 1 − u ,P − σ 2 ∆P = u,ν · ∇P = 0,

∇P (t, x)ν(x)Π(t, 0; x0)x0

2
Xiaoming Fu, Quentin Griette, and Pierre Magal. “A cell-cell repulsion model on a hyperbolic
Keller-Segel equation”. J. Math. Biol. 80.7 (2020), pp. 2257–2300. doi: 10.1007/s00285-020-01495-w.
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Related models
The Patlak-Keller-Segel equation modeling chemotaxis (Patlak 1953, Keller andSegel 1970)ut = κ∆u − χ∇ · (u∇c)εct = η∆c + βn − αc,see also Calvez and Corrias 2008, Desvillettes et al 2019.
The porous medium equation with KPP source
ut = ∇ · (u∇u) + u(1 − u)see de Pablo and Vazquez 1991, Vazquez 2007.
in 2006, Armstrong, Painter and Sherratt proposed a model for cell-cell adhesion
modeled by a nonlocal gradient.
ut = uxx − (uK (u))x ,R1

where K (u) = α −1 g(u(x + x0 ))ω(x0 )dx0 .
A full model was proposed by Ducrot et al (2011) with a porous medium
equation with contact inhibition.
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Two-species model∂t u1 − χ1 ∇ · u1 ∇P = u1 r1 − a11 u1 − a12 u2 ,∂ u − χ ∇ · u ∇P  = u r − a u − a u ,
t 2
2
2
2 2
21 1
22 2
2

P
−
σ
∆P
=
u
+
u
,

1
2



ν · ∇P = 0,

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Preservation of segregation when χ1 = χ2t
Π(t, 0; x0)

t1

t0
−L

Π(t0, 0; x0)

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

LMonolayer cell experiment in the Petri dishsparsely seeded initial condition

densely seeded initial condition

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

The one-dimensional model: motivation∂t u(t, x ) − χ∂x (u(t, x )∂x p(t, x )) = u(1 − u), x ∈ R, t > 0p(t, x ) − σ 2 ∂xx p(t, x ) = u(t, x ),x ∈ R, t > 0u(t = 0, x ) = u0 (x ),x ∈ R.

Wound healing experiments

3James E. N. Jonkman et al. “An introduction to the wound healing assay using live-cell microscopy”.
Cell Adhesion & Migration 8.5 (2014). PMID: 25482647, pp. 440–451. doi: 10.4161/cam.36224.
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

The one-dimensional one-species model

We focus on the one-dimensional model(∂t u(t, x ) − χ∂x (u(t, x )∂x p(t, x )) = u(1 − u), x ∈ R, t > 0
x ∈ R.

u(t = 0, x ) = u0 (x ),As before p is determined by the following equationp(t, x ) − σ 2 ∂xx p(t, x ) = u(t, x ).
Equivalently,Zρ(x )u(t, x − y )dy ,p(t, x ) = (ρ ⋆ u)(t, x ) =ρ(x ) :=

R

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

1 − |x |e σ
2σSolution integrated along the characteristics
We transform the original model by using the characteristic curves:(∂t u(t, x ) − χ∂x u(t, x )∂x p(t, x ) = u 1 + χ

v −u
−u ,
σ2



x ∈ R, t > 0p(t, x ) = (ρ ⋆ u)(t, x )
|x |

1 − σ
where ρ(x ) := 2σe, is split in two equations: the equation of the characteristic
curves(dΠ(t, x ) = −χ(ρx ⋆ u)(t, Π(t, x )),dt
Π(t = 0, x ) = x ,(1)

and the local dynamics on a characteristicdû(t, x ) = û(t, x ) 1 + χ̂(ρ ⋆ u)(t, Π(t, x )) − (1 + χ̂) û(t, x ) ,
dt


(2)

where û(t, x ) = h t, Π(t, x ) and χ̂ = σχ2 . (1)–(2) define a solution integrated
along the characteristics.
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

The Cauchy problem
Here L1η is the L1 space for the measure η2 e −η|x | dx .
Theorem (Solution integrated along the characteristics)
Let u0 ∈ L∞ (R), u0 (x ) ≥ 0. There exists τ ∗ (u0 ) ∈ (0, +∞] such that1
For each τ ∈ [0, τ ), there is a unique u(t, x ) ∈ C 0 [0, τ ], L1η (R) which is asolution integrated along the characteristics and satisfiesu(t = 0, x ) = u0 (x ).
234For each t ≥ 0 we have u(t, ·) ∈ L∞ (R),The map t 7→ Tt u0 := u(t, ·) is a semigroup which is continuous for the
L1η (R) topology,
For each t ≥ 0, the map u0 ∈ L∞ (R) 7→ Tt u0 = u(t, ·) is continuous for
the L1η topology

In addition: preservation of monotony, continuity and differentiability (as well as
superior smoothness) of the initial data.

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Discontinuous traveling waves

Theorem (Existence of a sharp traveling wave)
Assume 0 < χ̂< χ̄. There exists a traveling wave traveling at speed
σ χ̂ σ χ̂2−c ∈ 2+
χ̂ , 2 . U satisfies U(x ) = 0 for all x ≥ 0, U(0 ) ≥ 2+χ̂ . Moreover U
is strictly decreasing and differentiable on (−∞, 0] and a classical solution to−cU ′ − χ(UP ′ )′ = U(1 − U),where P = (U ⋆ ρ).
In addition: for solutions of the Cauchy problem with compactly supported
initial data, convergence speed of the level sets to the separatric, estimate on
the jump size. Non-existence of sharp continuous traveling waves.

4
Xiaoming Fu, Quentin Griette, and Pierre Magal. “Sharp discontinuous traveling waves in a hyperbolic
Keller–Segel equation”. Mathematical Models and Methods in Applied Sciences (2021). to appear. doi:
10.1142/S0218202521500214.
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Propagation starting from initially square-like boundary
Initial condition u0 (x ) =(x − x0 )21[−L,x0 ] (x ), L = 20, x0 = −15.
(L + x0 )2

1.2
1
0.8
0.6
0.4
0.2
0
-20

-15

-10

Quentin Griette (LMAH)

-5

0

5

Traveling waves in Keller-Segel models

10Dependency on the initial steepness
1.2
1
0.8
0.6
0.4
0.2
0
-20

-15

-10

Quentin Griette (LMAH)-50

Traveling waves in Keller-Segel models

5Position of the level sets and empirical speed
25

distance

20
15
10
5
0
0

10

Quentin Griette (LMAH)

20

30

Traveling waves in Keller-Segel models

40Formation of a discontinuity
Theorem (Exponential convergence of the level sets)
Let u0 ∈ C 0 (R), u0 (x ) ≥ 0 be supported in (−∞, 0), and assume that the
behavior of u0 (x ) near x = 0 is polynomial:u0 (x ) ≥ γ|x |α ,for some γ > 0 and α ≥ 1. Let Π∗ (t) := Π(t, 0) be the characteristic startingfrom x = 0 and ξ(t, β) := sup{x ∈ R|u(t, x ) = β} be the level set at level β.
1Then for each β ∈ (0, 1+χ̂+αχ̂ ) we haveΠ∗ (t) −  α1ηβe − 2α t ≤ ξ(t, β) ≤ Π∗ (t),γχ̂where η := 1 − 1+χ̂+α∈ (0, 1).
βQuentin Griette (LMAH)

Traveling waves in Keller-Segel models

A cartoon for the formation of the discontinuityt = t2t = t1βξ(t1 , β)h∗ (t1 )Quentin Griette (LMAH)

ξ(t2 , β)h∗ (t2 )Traveling waves in Keller-Segel models

Divergence of characteristics
We quantify the divergence speed of characteristics near the separatrix. We
have, for x < 0,ddtΠ(t, 0; 0) − Π(t, 0; x ) ≤ −χ(ρx ⋆ u)(t, Π(t, 0; 0)) + χ(ρx ⋆ u)(t, Π(t, 0; x ))Z

ρx (Π(t, 0; x ) − z) − ρx (Π(t, 0; 0) − z) u(t, z)dz=χRZ Π(t,0;x )ρx (Π(t, 0; x ) − z) − ρx (Π(t, 0; 0) − z) u(t, z)dz=χ−∞Z Π(t,0;0)ρx (Π(t, 0; x ) − z) − ρx (Π(t, 0; 0) − z) u(t, z)dz+χ
Π(t,0;x )Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Divergence of characteristics
We quantify the divergence speed of characteristics near the separatrix. We
have, for x < 0,ddtΠ(t, 0; 0) − Π(t, 0; x ) ≤ −χ(ρx ⋆ u)(t, Π(t, 0; 0)) + χ(ρx ⋆ u)(t, Π(t, 0; x ))Z

ρx (Π(t, 0; x ) − z) − ρx (Π(t, 0; 0) − z) u(t, z)dz=χRZ Π(t,0;x )ρx (Π(t, 0; x ) − z) − ρx (Π(t, 0; 0) − z) u(t, z)dz

=χ
−∞

{z

|

}

≤0

Z Π(t,0;0)}ρx (Π(t, 0; x ) − z) − ρx (Π(t, 0; 0) − z) u(t, z)dz+χ
Π(t,0;x ){z|≤1/σ 2

|z|

|z|

1 −σ
−σ
where ρ(z) = 2σe, ρx (z) = − sign(z). Therefore
2σ 2 eΠ(t, 0; 0) > Π(t, 0; x ) ≥ Π(t, 0; 0) + xe χ̂εtwhere ε := supx ≤z≤0,s∈[0,t] u(s, Π(s, 0; z)).
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Dynamics on the characteristics

On each characteristic, u(t, Π(t, 0; x )) satisfiesdu(t, Π(t, 0; x )) = u(t, Π(t, 0; x )) 1 + χ̂(ρ ⋆ u) t, Π(t, 0; x ) − (1 + χ̂)u t, Π(t, 0; x )dt≥ u(t, Π(t, 0; x ))if u(t, Π(t, 0; x )) is sufficiently small. Thereforeu(t, Π(t, 0; x )) ≥ u0 (x )e t .

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Heuristic computation

Assume x is close to Π(t, 0; 0) so that u(t, x ) ≤ ε .
In the worst case scenario Π(t, 0; x ) ≈ Π(t, 0; 0) + xe χ̂εt so that
Π(0, t; x ) ≈ −(Π(t, 0; 0) − x )e −χ̂εt .
Next by using the dynamics on the characteristicsu(t, x ) = u(t, Π(t, 0; Π(0, t; x ))) ≥ u0 (Π(0, t; x ))e t≈ u0 − (Π(t, 0; 0) − x )e −χ̂εt e t ,therefore if u0 (z) is polynomial near 0: u0 (z) ≥ γ|z|α ,u(t, x ) ⪆ γ(Π(t, 0; 0) − x )α e (1−αχ̂ε)tQuentin Griette (LMAH)

Traveling waves in Keller-Segel models

Preservation of regularity
Proposition
Let u0 ∈ L∞ (R) and u(t, x ) be an integrated solution.
1

if 0 ≤ u0 (x ) ≤ 1, then 0 ≤ u(t, x ) ≤ 1. In particular τ ∗ (u0 ) = +∞.

2

if u0 (x ) is continuous, then u ∈ C 0 ([0, τ ] × R) for all τ < τ ∗ (u0 ).

3

4

if u0 (x ) ∈ C 1 (R), then u ∈ C 1 ([0, τ ] × R) for all τ < τ ∗ (u0 ). In this caseu(t, x ) is a classical solution of the hyperbolic problem.
if u0 (x ) is monotone, then u(t, x ) has the same monotony for all
0 ≤ t < τ ∗ (u0 ).

The first property comes from an ad hoc argument. Properties 2 - 4 are shown
thanks to the following formulaRtu0 (Π(0, t; x )) exp 0 1 + χ̂p(l, Π(l, t; x ))dlRu(t, x ) =Rtl
1 + (1 + χ̂)u0 (Π(0, t; x )) 0 exp 0 1 + χ̂p(σ, Π(σ, t; x ))dσ dlwhich requires the a priori definition of Π and p.
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Explicit estimate of the jump size
Theorem (Estimate of the jump)
Let u0 ∈ L∞ (R) be a non-increasing profile supported in (−∞, 0] satisfyingu(−∞) ≤ 1,u(0− ) > 0.

Then:
2
,
2 + χ̂dσ χ̂
lim inf Π∗ (t) ≥,t→+∞ dt
2 + χ̂

lim inf u(t, Π∗ (t)) ≥t→+∞where Π∗ (t) = Π(t, 0) and χ̂ = σχ2 .
Note that this estimate is better than the one provided by the “convergence ofthe level sets” theorem. Indeed
2
1
1
1
=
>
≥
.
χ̂
2 + χ̂
1
+
χ̂
1
+
χ̂
+ αχ̂
1+ 2
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Non-existence of smooth sharp waves

Our last result is the non-existence of smooth sharp traveling waves. It confirmsthat discontinuous traveling waves are the “natural” asymptotic shape ofcompactly supported initial conditions.
Theorem (Non-existence of sharp smooth waves)
Let U ≥ 0 be a traveling wave and assume that U is continuous. ThenU ∈ C 1 (R), U is strictly positive and−χ(ρx ⋆ U)(x ) < c

Quentin Griette (LMAH)

for all x ∈ R.

Traveling waves in Keller-Segel models

Existence of non-sharp smooth traveling waves waves
This last result is a recent work with Pierre Magal and Min Zhao5 .
Theorem (Existence of a continuous traveling wave)
We assume thatc≥2p
σ 2 χ̂ (1 + χ̂).

There exists a traveling wave u(t, x ) = U(x − ct) with a continuous profilex → U(x ) is continuously differentiable and strictly decreasing, andlim U(x ) = 1, andlim U(x ) = 0,(3)

− c U ′ − χ(UP ′ )′ = U(1 − U), on R,(4)

P − σ 2 P ′′ = U, on R.

(5)

x →−∞

x →+∞

and satisfies traveling wave problem

where

5
Quentin Griette, Pierre Magal, and Min Zhao. Traveling waves with continuous profile for hyperbolic
Keller-Segel equation. 2022.
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Summary: Traveling waves, what is known?
χ.
σ2Suppose 0 < χ̂ ≤χ̄ (with χ̄ ≈ 1.045). Forσ χ̂ σ χ̂−+c∈,=: (csharp, csharp): existence of discontinuous waves
2 + χ̂ 2p+
For any c ≥ ccont := 2 σ 2 χ̂ (1 + χ̂) > csharp
there exists a continuous
traveling wave (no restriction on χ̂).

Recall χ̂ =0−csharp+csharpSharp TW

ccont
Smooth TW

So there is a gap between the two “zones of existence”.
Uniqueness of traveling waves and non-existence of waves: open questions.

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

PART II:
Speed-up of traveling waves bynegative chemotaxis
joint work with Chris Henderson and Olga Turanova6 .

6Quentin Griette, Christopher Henderson, and Olga Turanova. “Speed-up of traveling waves by negative
chemotaxis”. J. Funct. Anal. 285.10 (2023), Paper No. 110115, 67. doi: 10.1016/j.jfa.2023.110115.
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

The repulsive chemotaxis model
We aim at studying the model:(
Ut + χ(Vx U)x = Uxx + U(1 − U),− dVxx = U − V ,and more specifically the traveling waves(− c̄Ux + χ(Vx U)x = Uxx + U(1 − U),− dVxx = U − V ,with the conditions :U(−∞) = 1,

U(+∞) = 0,V ∈ L∞ (R).

χ will be negative: −χ > 0.

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Known results(− c̄Ux + χ(Vx U)x = Uxx + U(1 − U),− dVxx = U − V ,The speed in the absence of advection is cKPP = 2.
In is known7 that when −χ > 0, the minimal speed c̄χ,d satisfies c̄χ,d ≥ 2 (noslow down by repulsive chemotaxis). More precisely, it has been proved that :1

when d, −χd ≪ 1 then c̄χ,d = 22√ (so there is a speed-up)
when 1 ≪ −χ ≪ d then c̄χ,d ≈ 2|χ|d

We prove that845√when d ≪ −χ then c̄χ,d ⪆ cpm,ε −χ (−χ may remain finite)√when d ≪ −χ then c̄χ,d ⪆ cpm,ε −χ7Christopher Henderson. “Slow and fast minimal speed traveling waves of the FKPP equation withchemotaxis”. J. Math. Pures Appl. (9) 167 (2022), pp. 175–203. doi: 10.1016/j.matpur.2022.09.004.
8
Griette, Henderson, and Turanova, “Speed-up of traveling waves by negative chemotaxis”.
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

The rescaling(− c̄Ux + χ(Vx U)x = Uxx + U(1 − U),− dVxx = U − V ,We introduce a rescaling of the unknown functions√√v (x ) = V (x −χ),u(x ) = U(x −χ),so the new functions solve the equation

1
c̄

− √
ux − (vx u)x =uxx + u(1 − u),

−χ
−χ

d

−
vxx = u − v ,−χWe look at −χ and d as varying parameters which may go to +∞ or 0.

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Limit equations: Porous medium − cux − (vx u)x = 1 uxx + u(1 − u),−χ− νvxx = u − v ,c̄dwith c = √−χand ν = −χ.

⇝ First case: porous medium medium limit. Up to extraction,
ν→
− 0,

−χ →
−

1
, with ε ∈ [0, +∞).
εThe limit equation is local (porous medium-type) and the minimal speed is
known:(√√1 +2ε, if ε < 12 ,∗2
cpm,ε =√2 ε,
if ε ≥ 12 .

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Limit equations: Hyperbolic repulsive Keller-Segel − cux − (vx u)x = 1 uxx + u(1 − u),−χ− νvxx = u − v ,c̄dwith c = √−χand ν = −χ.

⇝ Second case: hyperbolic limit. Up to extraction,ν→− positive constant, ν,

−χ →
− 0.

The limit equation is non-local (hyperbolic type), we some properties but theminimal speed is unknown.
We derive a universal positive lower bound for the speed of the hyperbolic
problem, among other things.

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Results. I: the porous medium limit∗∗cχ,ν
is the minimal speed for the rescaled equation; cpm,εis the minimal speedof the porous medium limit.

TheoremFix any ε ≥ 0.
(i) The minimal speeds have the asymptotics:
lim inf−χ→ ε1 ,ν→0

∗
∗
cχ,ν
≥ cpm,ε,(ii) Consider any sequence (χn → χ, νn → ν) and traveling wave solutions(cn , un , vn ) to the rescaled equation. If lim sup cn < +∞, then up to
shifting (un , vn ) so thatmin un (x ) = un (0) = δ ∈ (0, 1),x ≤0there exists a (c, u) solution to the porour medium KPP equation and asubsequence nk withcnk → c,unk → u in L∞
loc ,Quentin Griette (LMAH)

1
and vnk ⇀ v in Hloc.

Traveling waves in Keller-Segel models

Results. IIa: hyperbolic waves, weak notion
We developed a new weak notion of hyperbolic traveling waves to deal withthe limit of the rescaled equation in a natural way.

 − (c + v )u = u 1 + v − u − u ,
x x
ν
(6)

 − νv = u − v ,xxGiven c > 0, u ∈ L∞ (R) and v ∈ W 2,∞ (R). We denote
Z = {x : c + vx (x ) = 0}.
We say that (c, u, v ) is a traveling wave if v solves the second equation in (6)1
on R, u ∈ Cloc
(Z c ) and solves (6) on Z c , and we have


v −u
u 1+
−u =0
ν
on Int(Z).
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Results. IIb: hyperbolic waves, regularity of weak solutions

The equation allows us to gain some regularity on the weak solutions.
Proposition
Let (c, u, v ) be a weak hyperbolic traveling wave. Suppose that u isnonconstant, that is, both {u > 0} and {u < 1} have positive measures. Thenc > 0 and there are only two possibilities:(i) Z = ∅. In that case, (u, v ) is a classical solution of the system.
(ii) Z = {x0 } consists of a single point. In that case, u has a single jump∞
discontinuity at x0 , with {u > 0} = (−∞, x0 ). Moreover u ∈ Cloc
(R\{x0 })and u satisfies, at the jump,u(x0− ) =Quentin Griette (LMAH)

ν + v (x0 ).
ν+1Traveling waves in Keller-Segel models

Results. IIc: hyperbolic waves, estimates on the speed.
It is known that, for any traveling wave (either sharp discontinuous or smooth)sup −vx (x ) ≤ c.
x ∈RPropositionFix −χ ∈ (−χ0 , +∞], 0 < νm < νM and CM > 0. Let (c, u, v ) be either asolution of the rescaled or hyperbolic equation, with c ∈ [0, CM ]. If
u(0) ≤

ν,ν+1then there exists θ > 0 and C > 0 depending only on −χ0 , νm , νM and CM withu(x ) ≤ Cu(0)e −θx for all x ≥ 0.
CorollaryFix 0 < νm < νM . There exists c, depending only on νm and νM , such that forany (c, u, v ) solution to the hyperbolic model with ν ∈ [νm , νM ], we havec ≥ c.
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Results. II: hyperbolic waves, limit speed.
TheoremLet νhyp > 0.
(i) The minimal speeds have the asymptotics:
lim inf−χ→+∞,ν→νhypc = c(νhyp ) > 0.

(ii) Consider any sequence (−χn → +∞, νn → νhyp ) and traveling wave
solutions (cn , un , vn ) to the rescaled equation. If lim sup cn < +∞, then up
to shifting (un , vn ) so thatmin un (x ) = un (0) = δ, with 0 < δ <
x ≤0

νhyp
,
νhyp + 1

there exists a solution (c, u, v ) to the hyperbolic model and a subsequence
nk withcnk → c,unk → u in C 1 (Z c ),and vnk → v in C 2 (Z c )∩Ww2,∞−∗ (R),

where either Z = ∅ or Z = {x0 }.
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Results IIIa: upper bound for the hyperbolic limit
We obtain upper bounds for the minimal speed by constructing the solutions of
the rescaled equation that converge to a specific solution of the hyperbolic andporous medium models.
TheoremFix νhyp > 0 and any sequence −χn → +∞ and νn → νhyp . There exists asolution (cn , un , vn ) of the rescaled equation and a solution (c, u, v ) to thehyperbolic model such that(i) Both (un , vn ) and (u, v ) are decreasing in x ,(ii) u is sharp and has a jump discontinuity at x = 0,(iii) Up to the extraction of a subsequence, we havelim(cn , un , vn ) = (c, u, v )in the topology of R, C 1 (R\{0}), and W 2,∞ (R) weak-∗Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Results IIIb: upper bound for the porous medium limit
We show that the hyperbolic discontinuous wave converge as ν → 0 to a sharp(and consequently minimal speed) wave for the porous medium model and use adouble limit to show that the asymptotic minimal speed in the porous medium
case is the expected one.
Theorem
Consider the decreasing family of sharp discontinuous traveling waves to thehyperbolic model with parmeter ν constructed previously, (c, u, v ). Then1lim (c, u) = √ , upm ,ν→02
where upm is the unique minimal speed traveling wave to the porous
medium-KPP equation with {upm > 0} = (−∞, 0).
Then thanks to a careful double limit, we obtain the corollary
Corollary
We havelim−χ→∞,ν→0

1
∗
∗
cχ,ν
= cpm,0=√ .
2Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Thank you !

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Fu, Xiaoming, Quentin Griette, and Pierre Magal. “A cell-cell repulsion model on a hyperbolic Keller-Segel equation”. J. Math. Biol. 80.7 (2020),
pp. 2257–2300. doi: 10.1007/s00285- 020- 01495- w.
—.“Sharp discontinuous traveling waves in a hyperbolic Keller–Segel equation”. Mathematical Models and Methods in Applied Sciences (2021). toappear. doi: 10.1142/S0218202521500214.
Griette, Quentin, Christopher Henderson, and Olga Turanova. “Speed-up of traveling waves by negative chemotaxis”. J. Funct. Anal. 285.10 (2023),
Paper No. 110115, 67. doi: 10.1016/j.jfa.2023.110115.
Griette, Quentin, Pierre Magal, and Min Zhao. Traveling waves with continuous profile for hyperbolic Keller-Segel equation. 2022.
Henderson, Christopher. “Slow and fast minimal speed traveling waves of the FKPP equation with chemotaxis”. J. Math. Pures Appl. (9) 167 (2022),
pp. 175–203. doi: 10.1016/j.matpur.2022.09.004.
Jonkman, James E. N. et al. “An introduction to the wound healing assay using live-cell microscopy”. Cell Adhesion & Migration 8.5 (2014). PMID:
25482647, pp. 440–451. doi: 10.4161/cam.36224.
Pasquier, Jennifer et al. “Different Modalities of Intercellular Membrane Exchanges Mediate Cell-to-cell P-glycoprotein Transfers in MCF-7 Breast
Cancer Cells”. J. Biol. Chem. 287.10 (2012), pp. 7374–7387. doi: 10.1074/jbc.M111.312157.

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

The Cauchy problem: Proof of the well-posedness
The proof of the well-posedness is technical. The main steps of the proof are as
follows:
First step: We fix U ⊂ R a conull set, u0 ∈ L∞ (U) and focus on the system
obtained by the change of variables w (t, x ) := u(t, Π(t, 0; x )) andp̂(t, x ) := p(t, Π(t, 0; x )). Here Π(t, s; x ) is the flow of the characteristic
equation: ∂ Π(t, s; x ) = −χp t, Π(t, s; x ),x∂t
Π(s, s; x ) = x .
Then we can show that (w , p̂) is a fixed-point of the operator

R
 T
t
1
+
χ̂p̂(l,
Π(l,
0;
x
))
−
(1
+
χ̂)w
(l,
x
)dl
u
(x
)
exp
0
0

Rt
TUτ [u0 ](w , p̂)(t, x ) = 
R 1 −|x −Π(t,0;z)|/σ
1−w (l,z)dl
0
e
u
(z)e
dz
0
2σ
For τ U (u0 ) sufficiently small this operator acting on the adequate set offunctions over [0, τ U (u0 )] is a contraction. τ U (u0 ) depends only on ∥u0 ∥L∞ (U ) .
However, we don’t have a semigroup property with this formulation.
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

The Cauchy problem: Proof of the well-posedness
Second step: In order to get a semi-group property, we need to go back to
the original formuation. However, the characteristics may not havepreserved the conull set U on which u0 is defined. Therefore we "go back"to u0 ∈ L∞ (R) and apply the construction of Step 1 to a particular choice ofU ⊂ R and û0 ∈ L∞ (U) such that ∥û0 ∥L∞ (U ) = ∥u0 ∥L∞ (R) . It can be shownthat the L∞ -class of u(t, x ) is independent of the choice of U and û0 .
Now u(t, x ) satisfies a semigroup property.
Third step: By the semigroup property and classical arguments, there exists a
maximal time of existence τ ∗ (u0 ) ∈ (0, +∞], such thateither τ ∗ (u0 ) = +∞ or

lim inf ∥u(t, ·)∥L∞ (R) = +∞.

t→τ ∗ (u0 )−We actually have a stronger result: that there exists a conull set U and a realfunction u0 ∈ L∞ (U) such that the w (t, ·) := u(t, Π(t, 0; ·)) is the L∞ class ofŵ (t, ·), where (ŵ , p̂) is the unique fixed point of the original map TUτ [u0 ] for
all 0 < τ ≤ τ ∗ (u0 ).
The L1η continuity of t 7→ u(t, ·) can be obtained from this property.
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

The Cauchy problem: Proof of the well-posedness
Fourth step: The remaining two properties, namely, the continuity of themap u0 7→ u(t, ·) for the L∞ (R) − L1η (R) topologies and the equivalence
between the solutions to the fixed-point problem and the integrated solutions
are done independently by ad-hoc methods. We refer to the paper for details.
This finished the proof of the Theorem.
∞Now we∗havein our hands a solution u(t, ·) 1∈ L (R) which is uniquely definedon 0, τ (u0 ) × R and continuous for the Lη (R)-topology. Moreover, the fieldp(t, x ) is well-defined and continuous for the W 1,∞ (R) topology and hasbounded second derivative:∥pxx (t, ·)∥L∞ (R) < +∞.
The flow of the characteristic curves Π(t, s; x ) is well-defined and Lipschitz
continuous with respect to x .

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Sketch of the proof of existence of a sharp traveling wave
The traveling wave equation is:(−c − χP ′ (z))U ′ (z) = U(z) 1 + χ̂P(z) − (1 + χ̂)U(z) ,where P(z) = (U ⋆ ρ)(z). We look for a profile
U(z) = 0 if z > 0, U(z) > 0 if z < 0.
To have a discontinuity on the profile the equation must be degenerate thereforec = −χP ′ (0).
The equation under consideration isχ(P ′ (0) − P ′ (z))U ′ (z) = U(z) 1 + χ̂P(z) − (1 + χ̂)U(z) .

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Sketch of the proof of existence of a sharp traveling waveχ(P ′ (0) − P ′ (z))U ′ (z) = U(z) 1 + χ̂P(z) − (1 + χ̂)U(z) .
We introduce a change of variable:(τ ′ (t) = χ P ′ (0) − P ′ (τ (t))
τ (0) = −1.
Then U(t) = U(τ (t)) satisfies:U ′ (t) = U(t) 1 + χ̂P(τ (t)) − (1 + χ̂)U(t) ,thereforeU(t) =1 Rt(1 + χ̂) −∞ exp − l 1 + χ̂P(τ (s))ds dl
Rt

We look for U as a fixed-point ofT (U)(z) := U(τ −1 (z))
Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

Sketch of the proof of existence of a sharp traveling waveχ(P ′ (0) − P ′ (z))U ′ (z) = U(z) 1 + χ̂P(z) − (1 + χ̂)U(z) .
Admissible profiles U are:1

continuous,

2

valued in [0, 1], and limz→0− U(z) ≥ 2+2 χ̂ ,3

sharp, i.e. U(z) = 0 for all z ≥ 0,

4

non-increasing.

It can be shown that this set A of admissible profiles is invariant by T .
Moreover T is continuous and compact on A for the topology induced by√∥U∥η := sup e ηz −zU(z).
z<0Since A is convex, the Schauder fixed-point theorem concludes that T has afixed-point on A.

Quentin Griette (LMAH)

Traveling waves in Keller-Segel models

"A Billiard Problem in Nonlinear Dissipative Systems"
Shin-Ichiro Ei (Hokkaido University, Japan)
The motion of camphor discs in a square domain is considered. Different motions from a usual Billiard problem areobserved such as the existence of a stable limit cycle. This talk is mainly done according to the content of the monographby Miyaji, E. and Mimura. The interaction of elliptic camphor discs is also mentioned.

A Billiard Problem in Nonlinear Dissipative Systems-for the memory of Prof. Masayasu Mimura-Shin-Ichiro Ei
Hokkaido University
Sapporo, Japan

Short summary of careers of Mimura sensei and I
Mimura sensei

I
Staffs:Tomoeda, Ito, Matano, Kobayashi,
Nishiura, Ogawa, …

-1973, Kyoto Univ.
1970-1980, Konan Univ.
1980-1993, Hiroshima Univ.

1982-1992, Hiroshima Univ.

1993-1998, Tokyo Univ.

1992-2004, Yokohama City Univ.

1998-2004, Hiroshima Univ.
2004-2017, Meiji Univ.
2017- , Musashino Univ., Meiji Univ.

2004-2014, Kyushu Univ.
2014 - , Hokkaido Univ.

Tsujikawa, Nakaki, Kan-on, I,
Tohma, Kuwamura, …
Nagayama, Izuhara, …
:members related to ReaDiNet

Sorry to other members !

Sensei looked very busy,
but enjoyed research,softball and tennis.
Many students in Mimura Lab.

Mimura’s Lab. in Hiroshima

In seminars
Simply show simple things.
(Simple things should be shown in a simple way)
Explain intuitively (and understandably).

sometimes
frequently

What is interesting ? What is the motivation ? absolutely(What is the salse point ?)overwhelmed by his active power and outstanding insightSpiritually mooring

The first paper for me

1984

by Nanako Shigesada

The first paper which Mimura sensei gave me !
Related to transient and asymptotic motions of solutions for a RD
with heterogeneous media.

The motivation fixed my research direction
PhD in 1987

1. S.-I. Ei and M. Mimura, Transient and large time behavors of solutions to heterogeneous reaction-diffusion equations, Hiroshima Math. J. vol.14,
649-678, 1984.
2. S.-I. Ei, M. Mimura and S. Takigawa, Transient and large time behaviors of solutions of a size-space distribution model including chemotactic
aggregation, Japan J. Appl. Math. vol. 6,223-244, 1989.
3. M. Mimura, S.-I. Ei and M. Kuwamura, Phase separation in an activator-inhibitor medium, Forma 4, 65-73, 1989.
4. S.-I. Ei and M. Mimura, Pattern formation in heterogeneous reaction-diffusion- advection systems with an application to population dynamics, SIAM
J. Math. Anal. vol. 21, 346-361, 1990.
5. M. Mimura, S.-I. Ei and Q. Fang, Effect of domain-shape on coexistence problems in a competition-diffusion system, J. Math. Biol. 29, 219-237, 1991.
6. S.-I. Ei and M. Mimura, Relaxation oscillations in combustion models of thermal self-ignition, J.D.D.E. vol.4, 191-229, 1992.
7. M. Kuwamura, S.-I. Ei and M. Mimura, Very slow dynamics for some reaction- diffusion systems of the activator-inhibitor type, J.JIAM vol.9, 35-77,
1992.
8. M. Mimura, K. Sakamoto and S.-I. Ei, Singular perturbation problems to a combustion equation in very long cylindrical domains, AMS/IP Studies inAdvanced Math. vol. 3(1997), 75-84.
9. S.-I. Ei, R. Ikota and M. Mimura, Segregating pattern problem in competition-diffusion systems, J. Interfaces and Free Boundaries vol. 1(1999), 5780.
10. S.-I. Ei, M. Mimura and M. Nagayama, Pulse-pulse interaction in reaction-diffusion systems, Physica D 165 (2002), 176-198.
11. S.-I. Ei, M. Mimura and M. Nagayama, Interacting Spots in reaction diffusion systems, DCDS 14 (2006), 31-62.
12. Xinfu Chen, S.-I. Ei and M. Mimura, SELF--MOTION OF CAMPHOR DISCS -MODEL AND ANALYSIS-, NETWORKS AND HETEROGENEOUS
MEDIA Volume 4, Number 1 (2009), 1-18.
13. S.-I. Ei, H. Izuhara and M. Mimura, Infinite dimensional relaxation oscillation in aggregation-growth systems, Discrete and Continuous Dynamical
Systems, Series B, 17 (2012) 1859 - 1887.
14. S.-I. Ei, H. Izuhara and M. Mimura, Infinite dimensional relaxation oscillation in reaction-diffusion systems, RIMS Kokyuroku Bessatsu B35 (2012),
31-40.
15. Shin-Ichiro Ei, Masayasu Mimura and Tomoyuki Miyaji, Reflection of a self-propelling rigid disk from a boundary, DCDS-A 2021, 14(3): 803-817
Special issue on recent topics in material, computer and life sciences. doi: 10.3934/dcdss.2020229

Short history of an ongoing joint work

springerprefaceMotion of pulses
Joint works with Mimura sensei for pulse/front dynamics. M. Mimura, K. Sakamoto and S.-I. Ei, Singular perturbation
problems to a combustion equation in very long cylindrical
domains, AMS/IP Studies in Advanced Math. vol. 3(1997), 75-84.

1D reflec.

. S.-I. Ei, R. Ikota and M. Mimura, Segregating pattern problem
in competition-diffusion systems, J. Interfaces and Free Boundaries
vol. 1(1999), 57-80.
. S.-I. Ei, M. Mimura and M. Nagayama,
Pulse-pulse interaction in reaction-diffusion systems,
Physica D 165 (2002), 176-198.

2D reflec.

. S.-I. Ei, M. Mimura and M. Nagayama,Interacting Spots in reaction diffusion systems, DCDS 14 (2006), 31-62.
. Xinfu Chen, S.-I. Ei and M. Mimura,
SELF--MOTION OF CAMPHOR DISCS -MODEL AND ANALYSIS-,
NETWORKS AND HETEROGENEOUS MEDIA Volume 4, Number 1 (2009), 1-18.

Drift bifurcation of
traveling pulse, spotkckPitch-fork type bifurcation
diagram of pulses, spots

Camphor Layer Model(樟脳)P・• Nagayama et.al.00
ut  Du  au  F (| x  P |),0 (u)  
1  cu P  x (u) |xP  P,
F
0

r
kc

• Mimura et.al.01
f 0u t   2 u  f (u, v)   f ,
10
vt  Dv  au   v,
1
1 tanh v
 f 
f dx, f (u, v)  u(1 u)(u  a(v)), a(v) 

| |
2

(Koya 00)kExperiment for camphor disk
By Kanda (02, Hiroshima Univ.)

Recorded by the video camera
at A and monitored by thedisplay at B to produce amovie.

Camphor disk problem isa nice example both frommodels and phenomena !

A trajectory of a camphor
disk of an experimentkc

k = 1/(content rate of camphor)Moving boundary (MB) model for camphor disk
Chen, E, Mimura 2009

air
water

・P

kc

k

WS in Lorentz center 2003?
Drift bifurcation was shown by Chen

First rigorous resultfor 2D reflec.

MB model which can be rigorously treated.

Interaction of camphor disksk  kc  
MB model

P2

k

P
1

kc

M 0 , M '0  0  Repulsive interaction
M 0  h  P1  ς 1  h e e ,
M 0  h  P2  ς 2  h e e , ς    W ( ς )  M '0 e   h e ,
1
 1
h

 ς 2    W ( ς 2 )  M '0 e   h e ,h| P2  P1 |h | P2  P1 |, e 
h

ς1
P
1

e

ς2
P2

Special Cases• On a line

Pj  ( p j ,0), ς j  ( j ,0)(E. Mimura, Nagayama 02)• Neuman boundaryP1  ( p, q), ς1  ( , ), P2  ( p, q), ς2  ( ,  ),(Matsumoto 02)

Dynamics of ODE
P1  ( p, q), ς1  ( , ), P2  ( p, q), ς2  ( ,  ),  T ( )   (?)
(   ) Mimura et.al p   , q    M 0 e  2 q ,

2q

2
Meaning?




(
M
|
ς
|
 M 2 ) ,1
difference from usual billiard problem ?
M '0  2  q
2




(
M
|
ς
|

M

)


e
,

1
2
2q


Dynamics in a square region

Mimura sensei discovered
the stable limit cycle
c.f. Billiard problem(Generically the region isdensely filled by orbits)

Experiment in Hiroshima
by Mimura and Lab. students 2002

Complicated motions of camphor diskba

4. Y. Kanda. Experiments and numerical analyses for motions of a camphor disk(in Japanese). Bachelor thesis, Hiroshima University, 2002.
5. M. Mimura, T. Miyaji, and I. Ohnishi, A billiard problem in nonlinear and nonequilibrium systems, Hiroshima Math. J. 37(2007) 343–384.
6. S. Nakata, Y. Iguchi, S. Ose, M. Kuboyama, T. Ishii, and K. Yoshikawa. Self-rotation of a camphor scraping on water: new insight into the old problem. Langmuir 13 (1997)
4454– 4458.
7. S. Nakata et al.(eds.) Self-organized Motion: Physicochemical Design based on Nonlinear Dynamics, Royal Society of Chemistry, 2019.
8. U. A. Rozikov, An Introduction to Mathematical Billiards, World Scientific, New Jersey, 2019
9. N. J Suematsu and S. Nakata, Evolution of Self-Propelled Objects: From the Viewpoint of Nonlinear Science, Chemistry–A European Journal 24 (2018) 6308–6324.
10. T. Vicsek et al., Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75 (1995) 1226. 11. T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep.
517 (2012) 71–140

Limiting problem
by considering sufficiently large regionAssume   

 
2

 

  T ( )2  T ( )Discrete time model

 
2

 

2  T (1)  T ( / 2 1)  T ( )






2



2

 2  T (1 )

2

*


2

1

2



1

2  T (1)  T ( / 2 1)

 '3

2

1


 '4

n1  T 4 (n ) n 1xn 1 xnn

 n1  T 4 (n )xn1  G(n , xn )

2

*

 '2


2

Numerically true (Mimura et.al.) *; stablen  *(n )*  n 1  T 4 ( n ) 2
 xn 1  G ( n , xn )Ifis stable

Prop.

*



*

2

*
*

x*

If     T (), then for  near *,| G( , x)  G( , y) | q | x  y |4holds for 0  q  1.
 *  T ( *), n   *, xn   x * (n  ) x*  G( *, x*)
Unique existence of stable limit cycle
In Discrete time model

Discrete time modelRemained problems

n1  T (n )

 '3

4

2

 '4

 n1
xn 1 xn

*



 '2



 

n

unsatisfactory

2



2

  T ( )?

Numerically true (Mimura et.al.)

MB model

 





?

2





 p   ,ODEparticle model  q    M 0 e  2  q ,


2q

2
    ( M 1 | ς |  M 2 ) ,

M '0  2  q
2e,    ( M 1 | ς |  M 2 ) 
2q
15. Ei, Masayasu Mimura and T. Miyaji, Reflection of a self-propelling rigid disk from a boundary, DCDS-A 2021, 14(3): 803-817
Special issue on recent topics in material, computer and life sciences. doi: 10.3934/dcdss.2020229

Aim to complete this book !

springerprefaceRelation between real phenomena and models
Real phenomena

more realistic but
complicated models(can not be analyzed,
many black boxes)+

Discrete time model

 '3

 '4

 n1
xn 1 xn

 '2
ncheck experimentally?
MB modelODEkkc

drift bifurcation

particle model p   , q    M 0 e  2  q ,

2q

2
    ( M 1 | ς |  M 2 ) ,

M '0  2  q
2e,    ( M 1 | ς |  M 2 ) 
2q


Synchronization of theory and experiment
In order to reproduce real phenomena truly

sophistication of model equations:many variables, complicated nonlinearity, etc

What are checked and measured in experiments ?
Physical phenomena: considerably accurate
Chemical phenomena: considerably accurate for low molecular compounds
Biological phenomena: qualitative properties e.g. monotonicity, on-off effect

Biological systems or systems close to biology
Many unknown factors: many black boxes, no explicit nonlinearities,
unknown number of necessary variables, etc

Necessity of modeling and analysis without
any specialization for biological partsmodels with black boxesu ; density of camphor expanding to water
generalization with black boxes

Rely on parts according to physical law
Corresponding to high molecular parts:
Generalize to vector values and treat as black box+
kkc

drift bifurcation

 p   ,particle model q    M 0 e  2  q ,

2q

2
    ( M 1 | ς |  M 2 ) ,

M '0  2  q
2e,    ( M 1 | ς |  M 2 ) 
2qInteraction of Non-radial Camphor tips

Shin-Ichiro Ei
Hokkaido University
Sapporo, Japan
Partially Joint work with
Nagayama, Kitahata, Koyano
Supported by JST, CREST JPMJCR14D3, Japan

Camphor tip on water surfaceair樟脳 (防虫剤)
Iida, Kitahata, Nagayama 13

wateru ; density of camphor expanding to water P ・

Fix centers

MotionE. Nagayama,of Interaction
Kitahata, Koyano 2018
Ex. m = 2
(ellipse)

Note:
appears in・・

Apsis or minor axis

Prop.
then
m次の第1種変形ベッセル関数

Motion of interaction : Mode 2m=2(ellipse)minor axis

Note:appears in・・Motion of interaction : Mode 3

m=3

Camphor tip on water surfaceair樟脳 (防虫剤)
Iida, Kitahata, Nagayama 13

wateru ; density of camphor expanding to water P ・
Rely on parts according to physical law
Corresponding to high molecular parts:
Generalize to vector values and treat as black box

Only assume the existence and stability of stationary solution whenF1 is given.

Motion of Interaction
Ex. m = 2
(ellipse)

Apsis or minor axis

Universal equation

Note:appears in・・

Nm is determined by experiments

Experimentexperiment

Fix centers

ExperimentInteraction of two elliptic camphors

angle

time

summary
・ Model for two camphor particles with deformation is considered.
・The equation describing the motion of angles are derived
for general reaction terms.
・In the case of small deformation from radial symmetry,the equation is explicitly derived, even with black boxes.

Thank you for your attention
with a lot of memories of Mimura sensei

"Convergence, concentration and critical mass phenomena fora model of cell migration with signal production on the boundary"Philippe Souplet (Université Sorbonne Paris Nord, France)(joint work with Nicolas MEUNIER (Université d'Evry-Val d'Essonne))

CONVERGENCE, CONCENTRATION AND
CRITICAL MASS PHENOMENA FOR A MODEL OF
CELL MIGRATION WITH SIGNAL PRODUCTION AT THE BOUNDARYPhilippe Souplet
LAGA, Université Sorbonne Paris Nord
Joint work with Nicolas Meunier, Université Évry Val d’Essonne, France

ICMMA – Meiji University, November 2023
In memory of Professor Masayasu Mimura

BIOLOGICAL BACKGROUND: CELL MIGRATION• Cell migration: fundamental process in physiological and pathological functions(immune response, morphogenesis, cancer metastasis, etc)

BIOLOGICAL BACKGROUND: CELL MIGRATION• Cell migration: fundamental process in physiological and pathological functions(immune response, morphogenesis, cancer metastasis, etc)
• Cell migration obeys general principle:
Strong correlation between direction of trajectories and velocity of cells(fastest cells have more directional migration)

CELL MIGRATION MECHANISMS• Actin filaments: essential components of cytoskeleton of eukaryotic cells• Polymerize and grow at one end, depolymerize and retract at other end(near cell membrane)• Retrograde actin flow
Large, fast actin flows enhance cell polarity, hence cell persistence time
• Actin flows as a result of advection of polarity signals(= molecules involved in regulation of cytoskeleton activity)

MODELu = u(t, x) concentration of solute located in cell
(cytoplasmic protein controlling active force-generation/adhesion machinery of cell)

(P )








ut = ∇ · ∇u − A(t)u , t > 0, x ∈ Ω,

0 = ∇u − A(t)u · ν, t > 0, x ∈ ∂Ω,
u(x, 0) = u0 (x) ≥ 0,x ∈ Ω,

with convective vector fieldZ
A(t) = A(u(t)) :=f (u)νdσ∂ΩMODELu = u(t, x) concentration of solute located in cell
(cytoplasmic protein controlling active force-generation/adhesion machinery of cell)

(P )








ut = ∇ · ∇u − A(t)u , t > 0, x ∈ Ω,

0 = ∇u − A(t)u · ν, t > 0, x ∈ ∂Ω,
u(x, 0) = u0 (x) ≥ 0,x ∈ Ω,

with convective vector fieldZ
A(t) = A(u(t)) :=f (u)νdσ∂Ω• (P) describes feedback loop between actin fluxes A(t) and a molecular species- molecules advected by actin fluxes and activated at cell membrane- activated molecules affect speed of actin flow
- higher concentration gradient across cell =⇒ faster actin flow

MODELu = u(t, x) concentration of solute located in cell
(cytoplasmic protein controlling active force-generation/adhesion machinery of cell)

(P )








ut = ∇ · ∇u − A(t)u , t > 0, x ∈ Ω,

0 = ∇u − A(t)u · ν, t > 0, x ∈ ∂Ω,
u(x, 0) = u0 (x) ≥ 0,x ∈ Ω,

with convective vector fieldZ
A(t) = A(u(t)) :=f (u)νdσ∂Ω• (P) describes feedback loop between actin fluxes A(t) and a molecular species- molecules advected by actin fluxes and activated at cell membrane- activated molecules affect speed of actin flow
- higher concentration gradient across cell =⇒ faster actin flow
• References[Calvez, Meunier, Voituriez, CRAS 2010][Calvez, Hawkins, Meunier, Voituriez, SIAP 2012][P. Maiuri, R. Voituriez et al., Cell 2015][Lavi-Meunier-Voituriez-Casademunt, Phys. Rev. 2020]

MOTIVATIONS AND AIMS
• Investigate influence of nonlinearity f on global or nonglobal solvability• Can concentration occur (on ∂Ω) ? finite time blow-up ?
• Understand global and/or blow-up behavior• Local well-posedness for rough (Lq ) initial data (key tool !)Typical nonlinearities
f (u) = up ,f (u) ∼ up at ∞,p≥1

0 < p < 1 (f ∈ C 1 , f ≥ 0)
• Traveling waves for nonlinearities with saturation f (u) = u/(C + u) – ongoing project

BASIC FEATURES
1. Conservation of mass (total amount of solute).
No-flux condition =⇒ZZ
c0 (t, x) dx = M :=Ω

→ special role of space L1

c0 (x) dxΩ

BASIC FEATURES
1. Conservation of mass (total amount of solute).
No-flux condition =⇒ZZ
c0 (t, x) dx = M :=c0 (x) dxΩΩ→ special role of space L1
2. Order of nonlinearity
Divergence formula =⇒Z
A(t) :=
Eqn. becomes:Zf (u)νdσ =∂Ω

f 0 (u)∇u dxΩ

Z
ut − ∆u = p∇u ·up−1 ∇u dxΩ

Quadratic in ∇u (cf. so-called “natural growth”), nonlinearity of order p + 1

BASIC FEATURES
1. Conservation of mass (total amount of solute).
No-flux condition =⇒ZZ
c0 (t, x) dx = M :=c0 (x) dxΩΩ→ special role of space L1
2. Order of nonlinearity
Divergence formula =⇒Z
A(t) :=

Zf (u)νdσ =∂Ω

Eqn. becomes:

f 0 (u)∇u dxΩ

Z
ut − ∆u = p∇u ·up−1 ∇u dxΩ

Quadratic in ∇u (cf. so-called “natural growth”), nonlinearity of order p + 1
3. Scale invariance (e.g. half-space case Ω = {x ∈ Rn ; xn > 0})u solution=⇒ uλ (x, t) = λn/p u(λx, λ2 t) also solutionq

Invariant L -norm for q = p :kuλ (·, 0)kLp ≡ ku(·, 0)kLp ,λ>0(λ > 0)RELATED NONLOCAL PROBLEMS• Nonlocal Neumann problems with zero order nonlinearities(R
ut − ∆u = f1 (u) Ω f2 (u) dx , t > 0, x ∈ ΩRuν = g1 (u) Ω g2 (u) dx , t > 0, x ∈ ∂Ω
[Pao 98, Liu-Wu-Sun-Li 17, Gladkov 17]

RELATED NONLOCAL PROBLEMS• Nonlocal Neumann problems with zero order nonlinearities(R
ut − ∆u = f1 (u) Ω f2 (u) dx , t > 0, x ∈ ΩRuν = g1 (u) Ω g2 (u) dx , t > 0, x ∈ ∂Ω
[Pao 98, Liu-Wu-Sun-Li 17, Gladkov 17]• Equations with nonlocal gradient terms (and Zhomogenous boundary conditions)
ut − um ∆u = up|∇u|2 dxΩm = p = 1: model of replicator dynamics[Dlotko 91, S02, Kavallaris-Suzuki 18, Kavallaris-Lankeit-Winkler 17, Lankeit-Winkler 18]

RELATED NONLOCAL PROBLEMS• Nonlocal Neumann problems with zero order nonlinearities(R
ut − ∆u = f1 (u) Ω f2 (u) dx , t > 0, x ∈ ΩRuν = g1 (u) Ω g2 (u) dx , t > 0, x ∈ ∂Ω
[Pao 98, Liu-Wu-Sun-Li 17, Gladkov 17]• Equations with nonlocal gradient terms (and Zhomogenous boundary conditions)
ut − um ∆u = up|∇u|2 dxΩm = p = 1: model of replicator dynamics[Dlotko 91, S02, Kavallaris-Suzuki 18, Kavallaris-Lankeit-Winkler 17, Lankeit-Winkler 18]• Nonlocal problems with mass conservation (with Zhomogeneous Neumann conditions):
ut − ∆u = up − |Ω|−1

up dx

Ω

[Budd-Dold-Stuart 93, Hu-Yin 95, Jazar-Kiwan 08, Wang-Tian-Li 15]

RELATED NONLOCAL PROBLEMS• Nonlocal Neumann problems with zero order nonlinearities(R
ut − ∆u = f1 (u) Ω f2 (u) dx , t > 0, x ∈ ΩRuν = g1 (u) Ω g2 (u) dx , t > 0, x ∈ ∂Ω
[Pao 98, Liu-Wu-Sun-Li 17, Gladkov 17]• Equations with nonlocal gradient terms (and Zhomogenous boundary conditions)
ut − um ∆u = up|∇u|2 dxΩm = p = 1: model of replicator dynamics[Dlotko 91, S02, Kavallaris-Suzuki 18, Kavallaris-Lankeit-Winkler 17, Lankeit-Winkler 18]• Nonlocal problems with mass conservation (with Zhomogeneous Neumann conditions):
ut − ∆u = up − |Ω|−1

up dx

Ω

[Budd-Dold-Stuart 93, Hu-Yin 95, Jazar-Kiwan 08, Wang-Tian-Li 15]• Nonlocal problem with mass control:
ut − ∆u = R(related with Keller-Segel system)[Wolansky 97, Kavallaris-Suzuki 07, 18]

λeu
eu dx
Ω

OUR PROBLEM
Conservative formut = ∇ · ∇u − A(t)u , t > 0, x ∈ Ω,

0 = ∇u − A(t)u · ν, t > 0, x ∈ ∂Ω,







u(x, 0) = u0 (x) ≥ 0,x ∈ Ω,

Convective-diffusive form(P )

u − ∆u = −A(t) · ∇u, t > 0, x ∈ Ω, tuν = (A(t) · ν)u, t > 0, x ∈ ∂Ω,u(x, 0) = u0 (x) ≥ 0, x ∈ Ω,Z
Z
A(t) =
f (u)νdσ =f 0 (u)∇u dx∂ΩΩpf (u) = u(p > 0)LOCAL WELL-POSEDNESS FOR ROUGH DATAStandard theory =⇒ local well-posedness for smooth initial data (e.g. C 1 (Ω)).
Theorem 1.
Let p > 0, q ≥ max(p, 1), u0 ∈ Lq (Ω).
(i) Problem (P) admits a unique maximal classical positive solution
u ∈ C 2,1 (Ω × (0, T ∗ )) ∩ C([0, T ∗ ); Lq (Ω)).
(ii) If q > p and T ∗ < ∞, then limt→T ∗ ku(t)kq = ∞.

LOCAL WELL-POSEDNESS FOR ROUGH DATAStandard theory =⇒ local well-posedness for smooth initial data (e.g. C 1 (Ω)).
Theorem 1.
Let p > 0, q ≥ max(p, 1), u0 ∈ Lq (Ω).
(i) Problem (P) admits a unique maximal classical positive solution
u ∈ C 2,1 (Ω × (0, T ∗ )) ∩ C([0, T ∗ ); Lq (Ω)).
(ii) If q > p and T ∗ < ∞, then limt→T ∗ ku(t)kq = ∞.
Remarks• Critical role played by Lp norm:- continuation property (ii) fails for 1 ≤ q < p or q = p = 1,
- for p = 1, entropy blows up wheneverT ∗ < ∞, namelyZ(u log u)(t) dx = ∞

lim

t→T ∗

Ω

• Condition q ≥ p in Theorem 1 also natural in view of scaling properties• Open problem whether local existence may fail when 1 ≤ q < p• L1 scaling critical case belongs to local existence range(6= Fujita equation ! ut − ∆u = up , p = 1 + n2 , q = 1)

CRITICAL MASS PHENOMENON FOR p = 1

Theorem 2.
Let f (u) = u, u0 ∈ L1 (Ω) and set M = ku0 k1 .
(i) If M ≤ 1, then T ∗ = ∞ and supt≥1 ku(t)k∞ < ∞(ii) If M > 1 and Ω is a cylinder, then there exist u0 such that T ∗ < ∞.

CRITICAL MASS PHENOMENON FOR p = 1

Theorem 2.
Let f (u) = u, u0 ∈ L1 (Ω) and set M = ku0 k1 .
(i) If M ≤ 1, then T ∗ = ∞ and supt≥1 ku(t)k∞ < ∞(ii) If M > 1 and Ω is a cylinder, then there exist u0 such that T ∗ < ∞.

Remarks• Critical mass phenomenon with sharp threshold M = 1.
• Reminiscent of well-known situation for 2d Keller-Segel system.
Main difference:- critical mass phenomenon is dimension-independent- solutions with critical mass remain bounded, unlike critical mass case of 2d KS

CRITICAL ROLE OF EXPONENT p = 1
1ū0 :=
|Ω|

Z
u0 dx

(average of u0 )Ω

Theorem 3. Let u0 ∈ L1 .
(i) If 0 < p < 1, then T ∗ = ∞ and supt≥1 ku(t)k∞ < ∞(ii) If p > 1 and M > 0, then there exists u0 such that T ∗ = ∞ and ku0 k1 = M

CRITICAL ROLE OF EXPONENT p = 1
1ū0 :=
|Ω|

Z
u0 dx

(average of u0 )Ω

Theorem 3. Let u0 ∈ L1 .
(i) If 0 < p < 1, then T ∗ = ∞ and supt≥1 ku(t)k∞ < ∞(ii) If p > 1 and M > 0, then there exists u0 such that T ∗ = ∞ and ku0 k1 = M

Sign of nonlinearity is important ! :Theorem 4. Let f (u) = −um with m ≥ 1 and u0 ∈ Lm (Ω).
Then T ∗ = ∞ and supt≥1 ku(t)k∞ < ∞.

ASYMPTOTIC BEHAVIOR OF GLOBAL SOLUTIONS
Theorem 5.
(i) Let p = 1 and u0 ∈ L1 .
• If M < 1, thenlim ku(t) − ū0 k∞ = 0,

t→∞

1
ū0 =
|Ω|

• If M = 1, there exist nonconstant steady states.
(ii) Let p > 0 and u0 ∈ Lq , q = max(p, 1).
• If ku0 kq  1 then T ∗ = ∞ andlim ku(t) − ū0 k∞ ≤ Ce−λt .

t→∞• There also exist nonconstant steady states.

Z
u0 dx.
Ω

ASYMPTOTIC BEHAVIOR OF GLOBAL SOLUTIONS
Theorem 5.
(i) Let p = 1 and u0 ∈ L1 .
• If M < 1, thenlim ku(t) − ū0 k∞ = 0,

t→∞

1
ū0 =
|Ω|

Z
u0 dx.
Ω• If M = 1, there exist nonconstant steady states.
(ii) Let p > 0 and u0 ∈ Lq , q = max(p, 1).
• If ku0 kq  1 then T ∗ = ∞ andlim ku(t) − ū0 k∞ ≤ Ce−λt .

t→∞• There also exist nonconstant steady states.
Remark. Previous result [Calvez, Hawkins, Meunier, Voituriez, SIAP 2012]ut = (ux − u(0, t)u)x on half-line I = (0, ∞), with zero flux condition at x = 0
Existence of a global, suitable weak solution
for u0 ∈ L1 (I, (1 + x)dx), u0 log u0 ∈ L1 (I), M ≤ 1

ASYMPTOTIC BEHAVIOR OF BLOWUP SOLUTIONS
0⊂ Rn if n ≥ 2, or Ω = (0, L) if n = 1• Cylinder Ω = (0, L) × BR

• Initial data u0 ∈ C 1 (Ω), axisymmetric with respect to e1 (if n ≥ 2), ∂x1 u0 ≤6≡ 0
Theorem 6.
(i) Assume n ≥ 2. Thenu(x, t) ≤ Cx−1
1in Ω × (0, T ∗ ).

In particular, if T ∗ < ∞ then blow-up set ⊂ ∂Ω ∩ {x1 = 0}.
(ii) Assume n = 1. Thenu(x, t) ≤ (px)−1/p + C

in (0, L] × (0, T ∗ ).

In particular, if T ∗ < ∞ then blow-up set = {0}(iii) Lower blow-up estimate:ku(t)k∞ ≥ C(T ∗ − t)−1/2p , 0 < t < T ∗ASYMPTOTIC BEHAVIOR OF BLOWUP SOLUTIONS
0⊂ Rn if n ≥ 2, or Ω = (0, L) if n = 1• Cylinder Ω = (0, L) × BR

• Initial data u0 ∈ C 1 (Ω), axisymmetric with respect to e1 (if n ≥ 2), ∂x1 u0 ≤6≡ 0
Theorem 6.
(i) Assume n ≥ 2. Thenu(x, t) ≤ Cx−1
1in Ω × (0, T ∗ ).

In particular, if T ∗ < ∞ then blow-up set ⊂ ∂Ω ∩ {x1 = 0}.
(ii) Assume n = 1. Thenu(x, t) ≤ (px)−1/p + C

in (0, L] × (0, T ∗ ).

In particular, if T ∗ < ∞ then blow-up set = {0}(iii) Lower blow-up estimate:ku(t)k∞ ≥ C(T ∗ − t)−1/2p , 0 < t < T ∗Remark: power 1/p is optimal (open question for n ≥ 2)
Open problems (possibly difficult)- upper time rate estimate (type I/II ?)- single point BU in cylinder- BU for other domains (difficulty: monotonicity)- attractivity of nonconstant steady-states

IDEAS OF PROOFS: Theorem 1 (local existence for Lq data)• Approximation by smooth initial data
• Semigroup techniques / smoothing effects via representation formula
Z tu(t) = S(t)u0 +K∇ (t − s)[A(s)u(·, s)] ds, 0 < t < T,0with differentiated semigroupZ∇y G(x, y, t) · ψ(y) dy,[K∇ (t)ψ](x) =ψ ∈ (L1 (Ω))nΩ• Fixed point in fractional Sobolev spaces with time-weighted intermediate norm
sup t1/2q kui (t)k1/q,qt∈(0,T )• Use interpolation + trace inequality W 1/q,q (Ω) ⊂ Lq (∂Ω)
Remark. Critical case p = q (especially p = q = 1) more delicate

IDEAS OF PROOFS: Theorem 2 (blowup in a cylinder – case p = 1, M > 1)
• Assume u0,x1 ≤ 0. Maximum principle =⇒ ux1 ≤ 0• Auxiliary functional: first momentZ
x1 u dx ≥ 0φ(t) =Ω

IDEAS OF PROOFS: Theorem 2 (blowup in a cylinder – case p = 1, M > 1)
• Assume u0,x1 ≤ 0. Maximum principle =⇒ ux1 ≤ 0• Auxiliary functional: first momentZ
x1 u dx ≥ 0φ(t) =
Ω
0

Z

φ =

Z
x1 ∇ · (∇u − A(t)u) dx

x1 ut =ΩΩ

R
R
• IBP + B.C., mass conservation Ω u = M and A(t) = Ω ∇u dx =⇒
Z
Z
0
φ =−
e1 · (∇u − A(t)u) dx = (M − 1)ux1 dx ≤ 0ΩΩIDEAS OF PROOFS: Theorem 2 (blowup in a cylinder – case p = 1, M > 1)
• Assume u0,x1 ≤ 0. Maximum principle =⇒ ux1 ≤ 0• Auxiliary functional: first momentZ
x1 u dx ≥ 0φ(t) =
Ω
0

Z

Z

φ =

x1 ∇ · (∇u − A(t)u) dx

x1 ut =ΩΩ

R
R
• IBP + B.C., mass conservation Ω u = M and A(t) = Ω ∇u dx =⇒
Z
Z
0
φ =−
e1 · (∇u − A(t)u) dx = (M − 1)ux1 dx ≤ 0
Ω

Ω

Z 1
• For Ω = (0, 1): u(t, 0) ≥ M , u(t, 1) = u(t, 1)Z 12xdx ≤

0

2xu(t, x)dx = 2φ(t)0=⇒ φ0 ≤ −(M − 1)(M − 2φ(t))
Z 1
But 2φ(0) =Z 12xu0 (x)dx <

0

2

Z 1

2xu0 (x )dx =0u0 (z)dz = M0=⇒ φ0 ≤ −η < 0 : contradictionIDEAS OF PROOFS: Theorem 2 (global existence – case p = 1, M ≤ 1)
• Entropy function φ(t) :=RΩ(u log u + 1) dx > 0.

IDEAS OF PROOFS: Theorem 2 (global existence – case p = 1, M ≤ 1)
• Entropy function φ(t) :=RΩ(u log u + 1) dx > 0.

• Test with log u:
Z0

φ (t) = −

u

−1

2

|∇u| dx +

Ω

2

Z
∇u dx
Ω

Cauchy-Schwarz and M ≤ 1 =⇒
Z
Z
Z
Z
0
−1
2
−1
2
φ (t) ≤ −u |∇u| dx +u dxu |∇u| dx = (M − 1)
u−1 |∇u|2 dx ≤ 0
Ω

Ω

Ω

Ω

IDEAS OF PROOFS: Theorem 2 (global existence – case p = 1, M ≤ 1)
• Entropy function φ(t) :=RΩ(u log u + 1) dx > 0.

• Test with log u:
Z0

φ (t) = −

u

−1

2

|∇u| dx +

Ω

2

Z
∇u dx
Ω

Cauchy-Schwarz and M ≤ 1 =⇒
Z
Z
Z
Z
0
−1
2
−1
2
φ (t) ≤ −u |∇u| dx +u dxu |∇u| dx = (M − 1)
u−1 |∇u|2 dx ≤ 0ΩΩ• An ε-regularity property:(
u0 = u10 + u20
ku10 k1  1,ku20 k∞ ≤ KΩΩ=⇒ smoothing effect in L∞ with uniform time τ (K) > 0• Entropy bound + ε-regularity =⇒ uniform L∞ estimate

IDEAS OF PROOFS (continued)
Theorem 3 (global existence – case p < 1)
• L1 norm is scaling supercritical=⇒ L∞ smoothing effect from L1 bound (by local theory)• Mass control =⇒ uniform L∞ estimate

IDEAS OF PROOFS (continued)
Theorem 3 (global existence – case p < 1)
• L1 norm is scaling supercritical=⇒ L∞ smoothing effect from L1 bound (by local theory)• Mass control =⇒ uniform L∞ estimate
Theorem 5 (asymptotic behavior)• case p = 1, M < 1. Use entropy as Liapunov functional:
Z ∞Z
u−1 |∇u|2 dxdt < ∞1Ω=⇒ ω limits are space-independent hence ≡ ū0 by mass conservationR• case p > 1, ku0 kp  1. Use Ω up as Liapunov functional

IDEAS OF PROOFS (continued)
Theorem 3 (global existence – case p < 1)
• L1 norm is scaling supercritical=⇒ L∞ smoothing effect from L1 bound (by local theory)• Mass control =⇒ uniform L∞ estimate
Theorem 5 (asymptotic behavior)• case p = 1, M < 1. Use entropy as Liapunov functional:
Z ∞Z
u−1 |∇u|2 dxdt < ∞1Ω=⇒ ω limits are space-independent hence ≡ ū0 by mass conservationR• case p > 1, ku0 kp  1. Use Ω up as Liapunov functional
Theorem 6 (blow-up space asymptotics – case n = 1)
• Auxiliary functional
φ := ux + up+1 − K1 u − K2
Maximum principle =⇒ φ ≤ 0• Integrate in x =⇒ u(x, t) ≤ (px)−1/p + C

CONCLUSIONS• Optimal results on local well-posedness in Lq spaces (critical exponent q = p)
• Optimal results on global solvability (critical exponent p = 1)
• Sharp mass threshold phenomenon for p = 1 (critical mass M = 1)
• Asymptotic convergence to constants for subcritical mass or small data
• Nonconstant steady states• Partial information on blow-up set, rate and profiles• Open problems- upper time rate estimate (type I/II ?)- single point BU in cylinder- BU for other domains- attractivity of nonconstant steady-states- ...

CONCLUSIONS• Optimal results on local well-posedness in Lq spaces (critical exponent q = p)
• Optimal results on global solvability (critical exponent p = 1)
• Sharp mass threshold phenomenon for p = 1 (critical mass M = 1)
• Asymptotic convergence to constants for subcritical mass or small data
• Nonconstant steady states• Partial information on blow-up set, rate and profiles• Open problems- upper time rate estimate (type I/II ?)- single point BU in cylinder- BU for other domains- attractivity of nonconstant steady-states- ...
THANK YOU FOR YOUR ATTENTION !

"Front propagation in the presence of obstacles"
Hiroshi Matano (Meiji University, Japan)
In this talk I will discuss the effect of geometric obstacles on the propagation of fronts. Two types of fronts are
considered. One is a transition layer in a bistable reaction diffusion equation. The other is a curvature-dependent motionof plane curves. Both types of fronts are closely related. In the first part, I will present my joint work with Henri
Berestycki and François Hamel. I will then discuss the curvature-dependent motion of plane curves through an infinite
channel with saw-toothed boundaries. For this second topic, I will first recall my joint work with Ken-Ichi Nakamura
and Bendong Lou (2006, 2013), which deal with domains with mildly undulating boundaries. I will then discuss myongoing joint work with Ryunosuke Mori, which deals with domains whose boundaries have steeper bumps. In suchdomains, a new type of phenomenon, which we call “obstacle-induced propagation”, can be observed.

Front Propagationin the presence of obstacles
Hiroshi Matano（Meiji University)
ICMMA 2023 “Reaction-diffusion systems:
from the past to the future”
MIMS, Meiji University, November 2, 2023
Tokyo 2023

Chronology (Prof. Mimura) 1941: born in Takamatsu City.
 1965: graduated Kyoto University.
 1970-79: Konan University
 1976-77: visiting Oxford University
 1980: Hiroshima University

Masayasu Mimura(Mayan)(1993—1999: Univ. Tokyo)

Oct 11, 1941 – Apr 8, 2021

 2004: Meiji University
advisor: Prof. Masaya Yamaguti

Matano  1975: entered Master’s Course at Kyoto University

Tokyo 2023

Chronology (Prof. Mimura) 1941: born in Takamatsu City.
 1965: graduated Kyoto University.
 1970-79: Konan University
 1976-77: visiting Oxford University
 1980: Hiroshima University

Masayasu Mimura(Mayan)(1993—1999: Univ. Tokyo)

Oct 11, 1941 – Apr 8, 2021

 2004: Meiji University
advisor: Prof. Masaya Yamaguti

Matano  1975: entered Master’s Course at Kyoto University
In the summer of 1976, Mayan cameback from Oxford for a few days andvisited Kyoto University, where we
had intensive discussions.
That was a turning point in my life!

Tokyo 2023

Chronology (Prof. Mimura) 1941: born in Takamatsu City.
 1965: graduated Kyoto University.
 1970-79: Konan University
 1976-77: visiting Oxford University
 1980: Hiroshima University

Masayasu Mimura(Mayan)(1993—1999: Univ. Tokyo)

Oct 11, 1941 – Apr 8, 2021

 2004: Meiji University
advisor: Prof. Masaya Yamaguti

Matano  1975: entered Master’s Course at Kyoto University
In the summer of 1976, Mayan cameback from Oxford for a few days andvisited Kyoto University, where we
had intensive discussions.
That was a turning point in my life!

My work on dumbbell-shaped domain
was inspired by his question.Tokyo 2023

Chronology (Prof. Mimura) 1941: born in Takamatsu City.
 1965: graduated Kyoto University.
 1970-79: Konan University
 1976-77: visiting Oxford University
 1980: Hiroshima University

Masayasu Mimura(Mayan)(1993—1999: Univ. Tokyo)

Oct 11, 1941 – Apr 8, 2021

 2004: Meiji University
advisor: Prof. Masaya Yamaguti

Matano  1975: entered Master’s Course at Kyoto University
 1979 - 82: Univ. Tokyo

 1982 - 88: Hiroshima University (Mimura, Kobayashi, Ei, …)
 1988 - 2018: Univ. Tokyo
 2018 - 2023: Meiji University

Tokyo 2023

01 May 2005: at Taro Okamoto Museum, Kawasaki, JapanTokyo 2023

Outline of the talk1.

Introduction

2.

Front propagation through a perforated wall

3.

Propagation in a saw-toothed cylinder
Curvature-dependent motion of interfaces

4.

Previous results
Motion without singularity

5.

Main results
Traveling waves with singularities &Obstacle-induced propagation

6. Numerical simulation
Joint work with Ryunosuke Mori (Meiji University)

Tokyo 2023

1. Introduction
Propagation in the presence of obstacles

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Front propagation in the presence of obstacles
Theme 1

Effect of localized obstacles

K : obstacle
Incidental
distubanceplanar front

Q

What is the long-time effect of this incidental disturbance?

(Case1) K : compact obstacle

Berestycki-Hamel-M. (CPAM 2009)(Case2) K : perforated wall

subject of the present talkTokyo 2023

Theme 2

Q

Effect of boundary geometry on front propagation

Propagation: Which direction is faster?

Tokyo 2023

Theme 2

Effect of boundary geometry on front propagation

The bigger the
opening angles, theslower the speed

Q

Propagation: Which direction is faster?

Related problem:
Existence of a non-constantstable stationary solution in a
dumbbell-shaped domain

M (1979)
M – Mimura (1983)

competition sys
Tokyo 2023

RD equationf (u)f (u) : bistablebalanced:
Interface motion (N > 1)

0

a

1

u

unbalanced:
Approximate law of motion(A) f : balanced

V = (N - 1) H

mean curvature flow(B) f : unbalanced

V = (N - 1) H + A

V = normal velocity
H = mean curvature
N = space dimension
A : driving force
outward normal
vector

S. Allen & J. Cahn (1979)
K. Kawasaki & T. Ohta (1982)
Tokyo 2023

Effect of boundary geometry on front propagation

The bigger the
opening angles, theslower the speed

Q

Propagation: Which direction is faster?
Too rapid widening
blocks propagation.

The same fact was used, e.g., in the
study of spreading depression (SD)by Dronne et al (2004).

Tokyo 2023

Related result. If the holes are too small, then blocking occurs.

Idea of proof. Either by a variational
method or by a super-solution method.
cf. [M. 1979], [M.-Mimura 1983] (A)[Beres.-Hamel-M. 2009] (B)[Berestycki-Bouhours-Chapuisat 2016] (C)

(C)

(A)

(B)

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2. Propagation through aperforated wall

Joint work with Henri Berestycki (EHESS)
François Hamel (Aix-Marseille)

Tokyo 2023

Formulation of the problem

f (u)
0

a

1

u

f (u): bistable

Planar traveling front

1

0

Q

x

Under what conditions can the front
penetrate through the wall?

K
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Invasion / blocking dichotomy

In particular, there is no blocking profile that converges to 0 too slowly.

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Invasion / blocking dichotomy

Dichotomy theorem

De Giorgi type lemmaTokyo 2023

Theorem 3. If the holes are too small, then blocking occurs.

Idea of proof. Either by a variational
method or by a super-solution method.
cf. [M. 1979], [M.-Mimura 1983] (A)[Beres.-Hamel-M. 2009] (B)[Berestycki-Bouhours-Chapuisat 2016] (C)

(C)

(A)

(B)

Tokyo 2023

Three types of walls
1. Wall with large holes
A ball of radius R0 can pass through one of theholes in the wall, where R0 is a certain positive
number to be specified later.
2. Small capacity wall
K is close to a set of capacity 0 in a certain sense.
(Kε is in the ε neighborhood of a set K0 of capacity 0.)
3. Skeleton wall
K consists of thin panels parallel to the x1 axis.
More precisely, K0 is a locally finite union of
hypersurfaces parallel to the x1 axis and let Kεconverge to K0 in a certain sense.
Tokyo 2023

3. Propagation in a saw-toothedcylinderCurvature-dependent motion of interfaces
Joint work with Ryunosuke Mori (Meiji University)

Tokyo 2023

Curvature-dependent motion of a plane curveV

normal velocity
curvature
constant > 0• Evolution of a phase boundary.
• Singular limit of the Allen-Cahn equation

Tokyo 2023

Motion in an infinite strip with undulating boundaries

normal velocity
curvature
constant > 0

perpendicular

Motivation
1. To study the effect of geometry on the
speed of propagation.
2. Find conditions on the boundary shapefor propagation and blocking.
New discovery: obstacle-induced propagation
Tokyo 2023

Singular limit
sharp-interface limit

Transition layer(moving front)mean curvature flow

sharp-interface

front

Typical expample

Tokyo 2023

2. Previous results
Motion without singularity(the case of mildly undulating boundaries)[1] HM, K.-I. Nakamura, B. Lou(Networks & Heterogeneous Media 2006)[2] B. Lou, HM, K.-I. Nakamura(JDE 2013)

Tokyo 2023

normal velocity
curvature
constant

perpendicular

recurrent functions
If the curve is a graph: function
then the equation reduces to:Tokyo 2023

Notation
maximal opening angles
maximal closing angles

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Stationary solution

𝛾𝛾 : stationary
𝛾𝛾 is a circular arc of curvature −𝐴𝐴
whose ends meet the boundarywith the angle 𝜋𝜋/2 .
Definition
Blocking: The curve remains bounded as 𝑡𝑡 → ∞ .

Propagation: The curve (or at least its portion goes to infinity in
𝑦𝑦 direction as 𝑡𝑡 → ∞ .

Tokyo 2023

Slope condition in the previous works[1] M, Nakamura & Lou (2006)[2] Lou, M & Nakamura (JDE 2013)

Boundary slope condition
Assumption on
the initial curve
A unique classical solution of the following problem exists
globally in time and satisfies+B.C.

Motion without
singularity
Tokyo 2023

The results of MNL (2006)
𝑔𝑔 𝑦𝑦 periodic

1. Blocking occurs if and only if there isa stationary solution, or, equivalently,
if and only if
In this case, any solution converges
to a stationary solution as 𝑡𝑡 → ∞ .

2. If no stationary solution exists, thenthere exists a traveling wave solution,
which is unique up to time shift. In
this case, any solution converges to a
traveling wave as 𝑡𝑡 → ∞ .

3. Results on the homogenization limit.

Idea of proof: Strong maximum principle and the analysis of
the omega limit set.

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What are traveling waves?
The periodic case:Definition (periodic TW):

average speed

Remark: In the non-periodic case, traveling waves can be definedby using the notion of hull of a function.

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Homogenization

boundary shape

Homogenization limit

c0: limit speed
φ: limit profile
large
What determines the limit speed ?

small
The limit contact
angle plays the
key role.
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Difficulty: The two ends of the curve flips back and forth very
rapidly, which makes it difficult to derive a precise asymptotic
expansion even formally.

Strategy
Estimate the gradient ataway from boundary.

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Homogenization limit
Theorem [1] (homogenization).
Assumenormalized to satisfy(i)and letbe the periodic TW that is. Thenconverges to a function of the form
whose contact angle is.

(ii) The limit speedasis determined by

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Corollary. The limit speed

satisfies

The larger the
opening angle ,the slower
the speed.

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3. Main results• Traveling waves with singularities• Obstacle-induced propagation

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Removal of the slope condition

Boundary slope condition

The curve may touch the boundary
besides the endpoints, which forces
the curve to split, thus creating
singularities.

The solution can no longer be treated in the classical framework,so we consider the problem in the framework of viscosity solutions.
Tokyo 2023

Viscosity solution
Classical
setting

Level set
approach

Regard the curve 𝛾𝛾𝑡𝑡 as a level set of an auxiliary
function 𝑈𝑈 𝑥𝑥, 𝑡𝑡 .

Tokyo 2023

The function 𝑈𝑈 satisfies the following equation formally.

We consider viscosity solutions of the above equation and regard it
as a solution of the original problem in a generalized sense.
Note: we focus on the central component of the viscosity solution.
𝒰𝒰 𝛾𝛾0 : viscosity solution

𝒞𝒞𝒞𝒞 𝛾𝛾0 : central component

Tokyo 2023

Blocking and propagation
1. Blocking occurs if and only if there isa stationary solution.
2. If no stationary solution exists, thenthere exists a traveling wave solution,
which is unique up to time shift. In
this case, any solution without
fattening converges to a travelingwave as 𝑡𝑡 → ∞ .

3. Results on the homogenization limit.

Corollary. If the bumps are sufficiently dense,
then propagation occurs.
Obstacle-induced propagation(This idea was proposed by H. Ninomiya for RD.)No stationary
solution

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4. Numerical simulation
Propagation with singularities

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Which direction is faster?

Simulation by Steffen Plunder(ASHBi, Kyoto University)

Tokyo 2023

Which direction is faster?

Simulation by Steffen Plunder(ASHBi, Kyoto University)

Tokyo 2023

Which direction is faster?

Simulation by Steffen Plunder(ASHBi, Kyoto University)

Tokyo 2023

Obstacle-aided propagation

Simulation by Steffen Plunder(ASHBi, Kyoto University)

Tokyo 2023

Obstacle-aided propagation

Simulation by Steffen Plunder(ASHBi, Kyoto University)

Tokyo 2023

Obstacle-aided propagation

Simulation by Steffen Plunder(ASHBi, Kyoto University)

Obstacle-induced
propagation

Tokyo 2023

Thank you for yourattention!

Tokyo 2023