in honor of jean-michel coron henri berestycki · the e ect of di usion on a line on fisher-kpp...
TRANSCRIPT
The effect of diffusion on a line on Fisher-KPPpropagation
In honor of Jean-Michel Coron
Henri Berestycki
EHESS, PSL Research University, Paris, ReaDi project - ERC
IHP, Paris, 22 June 2016
1/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 1 / 56
Fisher - KPP equation
Homogeneous reaction-diffusion equations
∂tv − d ∆v = f (v) x ∈ RN , t > 0
2/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 2 / 56
Fisher - KPP equation
Fisher – KPP case
∂tv = d∆v + f (v) t > 0, x ∈ RN
v |t=0 = v0 x ∈ RN
with v0 ≥ 0, 6≡ 0
,
.
fit "
>p
3/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 3 / 56
Fisher - KPP equation
Homogeneous equation – Spreading properties in KPP case
Invasion: v(x , t)→ 1 as t →∞, locally uniformly in x as soon asv0 6≡ 0.
Asymptotic speed of propagation: ∃w∗ such that for any v0 havingcompact support
∀c > w∗ sup|x |≥ct
v(x , t)→ 0 as t →∞
∀c < w∗ sup|x |≤ct
|v(x , t)− 1| → 0 as t →∞.
Fisher - KPP case: w∗ = cK = 2√
d f ′(0)
4/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 4 / 56
Fisher - KPP equation
Homogeneous equation – Spreading properties in KPP case
Invasion: v(x , t)→ 1 as t →∞, locally uniformly in x as soon asv0 6≡ 0.
Asymptotic speed of propagation: ∃w∗ such that for any v0 havingcompact support
∀c > w∗ sup|x |≥ct
v(x , t)→ 0 as t →∞
∀c < w∗ sup|x |≤ct
|v(x , t)− 1| → 0 as t →∞.
Fisher - KPP case: w∗ = cK = 2√
d f ′(0)
4/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 4 / 56
Fisher - KPP equation
Homogeneous equation – Spreading properties in KPP case
Invasion: v(x , t)→ 1 as t →∞, locally uniformly in x as soon asv0 6≡ 0.
Asymptotic speed of propagation: ∃w∗ such that for any v0 havingcompact support
∀c > w∗ sup|x |≥ct
v(x , t)→ 0 as t →∞
∀c < w∗ sup|x |≤ct
|v(x , t)− 1| → 0 as t →∞.
Fisher - KPP case: w∗ = cK = 2√
d f ′(0)
4/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 4 / 56
Fisher - KPP equation
Asymptotic position of level sets
@÷ w*sc ,
x : ult .x1 .
.a ) ,
ocaci .
Level sets for t >> 1
5/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 5 / 56
Fisher - KPP equation
The effect of a “road” with fast diffusion on Fisher-KPPpropagation
Joint work with Jean-Michel Roquejoffre and Luca Rossi
J. Math. Biology (2013)
Nonlinearity (2013)
Comm. Math. Phys. (2016)
Nonlinear Anal. (2016)
6/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 6 / 56
motivation
The system (B, Roquejoffre and Rossi)
Ω: upper half-plane R× R+.∂tu − D∂xxu = νv |y=0 − µu, x ∈ R, t ∈ R∂tv − d∆v = f (v), (x , y) ∈ Ω, t ∈ R−d∂y v |y=0 = µu − νv |y=0 , x ∈ R, t ∈ R.
Note: v |y=0 := limy0
v .
Birth/death rate: Logistic law (KPP type term) f :
f > 0 on (0, 1), f (0) = f (1) = 0, f (v) ≤ f ′(0)v .
7/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 7 / 56
motivation
Basic properties
Comparison principle
If (u1, v1) and (u2, v2) are solutions of the Cauchy problem with u1 ≤ u2
and v1 ≤ v2 at t = 0, then u1 ≤ u2 and v1 ≤ v2 for all t ≥ 0.
“Monotone system” (kind of)
8/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 8 / 56
motivation
Liouville-type result for stationary solutions
Steady states −D∆xU = νV |y=0 − µU, x ∈ RN
−d∆V = f (V ), (x , y) ∈ Ω,
−d∂y V |y=0 = µU − νV |y=0 , x ∈ RN .
where Ω = x = (x1, . . . , xN , y); y = xN+1 > 0.
Theorem
The only bounded steady states are (U ≡ 0,V ≡ 0) and(U = ν/µ,V ≡ 1).
Proof rests on a sliding method (HB - L. Nirenberg)
9/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 9 / 56
motivation
Liouville-type result for stationary solutions
Steady states −D∆xU = νV |y=0 − µU, x ∈ RN
−d∆V = f (V ), (x , y) ∈ Ω,
−d∂y V |y=0 = µU − νV |y=0 , x ∈ RN .
where Ω = x = (x1, . . . , xN , y); y = xN+1 > 0.
Theorem
The only bounded steady states are (U ≡ 0,V ≡ 0) and(U = ν/µ,V ≡ 1).
Proof rests on a sliding method (HB - L. Nirenberg)
9/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 9 / 56
motivation
Long time behaviour: invasion
∂tu − D∂xxu = νv |y=0 − µu, x ∈ R, t ∈ R∂tv − d∆v = f (v), (x , y) ∈ Ω, t ∈ R−d∂y v |y=0 = µu − νv |y=0 , x ∈ R, t ∈ R.
Theorem
Let (u, v) be a solution of the Cauchy problem with initial datum(u0, v0) 6≡ (0, 0) (nonnegative and bounded). Then,
(u(x , t), v(x , y , t))→ (ν/µ, 1), as t →∞,
locally uniformly in (x , y) ∈ Ω.
10/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 10 / 56
motivation
Long time behaviour: invasion
∂tu − D∂xxu = νv |y=0 − µu, x ∈ R, t ∈ R∂tv − d∆v = f (v), (x , y) ∈ Ω, t ∈ R−d∂y v |y=0 = µu − νv |y=0 , x ∈ R, t ∈ R.
Theorem
Let (u, v) be a solution of the Cauchy problem with initial datum(u0, v0) 6≡ (0, 0) (nonnegative and bounded). Then,
(u(x , t), v(x , y , t))→ (ν/µ, 1), as t →∞,
locally uniformly in (x , y) ∈ Ω.
10/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 10 / 56
asymptotic speed of propagation in the direction of the line
The effect of roads on propagation
Theorem
There exists an asymptotic speed of propagation in the direction of thex-axis, w∗ = w∗(µ, d ,D) > 0. That is: let (u0, v0) be a compactlysupported initial datum (nonnegative, nontrivial). Then, locally in y :
∀c > w∗, sup|x |≥ct
|(u(x , t), v(x , y , t))| → 0 as t →∞
∀c < w∗, sup|x |≤ct, 0≤y≤a
|(u(x , t), v(x , y , t))− (ν/µ, 1)| → 0 as
t →∞.
Theorem
Let cK := 2√
d f ′(0) be the (homogenous) KPP speed of propagation.
If D ≤ 2d then w∗(µ, ν, d ,D, f ′(0))=cK .
If D > 2d then w∗(µ, d ,D)>cK .
The limit limD→+∞
w∗(D)/√
D exists and is positive.
11/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 11 / 56
asymptotic speed of propagation in the direction of the line
The effect of roads on propagation
Theorem
There exists an asymptotic speed of propagation in the direction of thex-axis, w∗ = w∗(µ, d ,D) > 0.
That is: let (u0, v0) be a compactlysupported initial datum (nonnegative, nontrivial). Then, locally in y :
∀c > w∗, sup|x |≥ct
|(u(x , t), v(x , y , t))| → 0 as t →∞
∀c < w∗, sup|x |≤ct, 0≤y≤a
|(u(x , t), v(x , y , t))− (ν/µ, 1)| → 0 as
t →∞.
Theorem
Let cK := 2√
d f ′(0) be the (homogenous) KPP speed of propagation.
If D ≤ 2d then w∗(µ, ν, d ,D, f ′(0))=cK .
If D > 2d then w∗(µ, d ,D)>cK .
The limit limD→+∞
w∗(D)/√
D exists and is positive.
11/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 11 / 56
asymptotic speed of propagation in the direction of the line
The effect of roads on propagation
Theorem
There exists an asymptotic speed of propagation in the direction of thex-axis, w∗ = w∗(µ, d ,D) > 0. That is: let (u0, v0) be a compactlysupported initial datum (nonnegative, nontrivial). Then, locally in y :
∀c > w∗, sup|x |≥ct
|(u(x , t), v(x , y , t))| → 0 as t →∞
∀c < w∗, sup|x |≤ct, 0≤y≤a
|(u(x , t), v(x , y , t))− (ν/µ, 1)| → 0 as
t →∞.
Theorem
Let cK := 2√
d f ′(0) be the (homogenous) KPP speed of propagation.
If D ≤ 2d then w∗(µ, ν, d ,D, f ′(0))=cK .
If D > 2d then w∗(µ, d ,D)>cK .
The limit limD→+∞
w∗(D)/√
D exists and is positive.
11/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 11 / 56
asymptotic speed of propagation in the direction of the line
The effect of roads on propagation
Theorem
There exists an asymptotic speed of propagation in the direction of thex-axis, w∗ = w∗(µ, d ,D) > 0. That is: let (u0, v0) be a compactlysupported initial datum (nonnegative, nontrivial). Then, locally in y :
∀c > w∗, sup|x |≥ct
|(u(x , t), v(x , y , t))| → 0 as t →∞
∀c < w∗, sup|x |≤ct, 0≤y≤a
|(u(x , t), v(x , y , t))− (ν/µ, 1)| → 0 as
t →∞.
Theorem
Let cK := 2√
d f ′(0) be the (homogenous) KPP speed of propagation.
If D ≤ 2d then w∗(µ, ν, d ,D, f ′(0))=cK .
If D > 2d then w∗(µ, d ,D)>cK .
The limit limD→+∞
w∗(D)/√
D exists and is positive.
11/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 11 / 56
asymptotic speed of propagation in the direction of the line
The effect of roads on propagation
Theorem
There exists an asymptotic speed of propagation in the direction of thex-axis, w∗ = w∗(µ, d ,D) > 0. That is: let (u0, v0) be a compactlysupported initial datum (nonnegative, nontrivial). Then, locally in y :
∀c > w∗, sup|x |≥ct
|(u(x , t), v(x , y , t))| → 0 as t →∞
∀c < w∗, sup|x |≤ct, 0≤y≤a
|(u(x , t), v(x , y , t))− (ν/µ, 1)| → 0 as
t →∞.
Theorem
Let cK := 2√
d f ′(0) be the (homogenous) KPP speed of propagation.
If D ≤ 2d then w∗(µ, ν, d ,D, f ′(0))=cK .
If D > 2d then w∗(µ, d ,D)>cK .
The limit limD→+∞
w∗(D)/√
D exists and is positive.
11/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 11 / 56
asymptotic speed of propagation in the direction of the line
The effect of roads on propagation
Theorem
There exists an asymptotic speed of propagation in the direction of thex-axis, w∗ = w∗(µ, d ,D) > 0. That is: let (u0, v0) be a compactlysupported initial datum (nonnegative, nontrivial). Then, locally in y :
∀c > w∗, sup|x |≥ct
|(u(x , t), v(x , y , t))| → 0 as t →∞
∀c < w∗, sup|x |≤ct, 0≤y≤a
|(u(x , t), v(x , y , t))− (ν/µ, 1)| → 0 as
t →∞.
Theorem
Let cK := 2√
d f ′(0) be the (homogenous) KPP speed of propagation.
If D ≤ 2d then w∗(µ, ν, d ,D, f ′(0))=cK .
If D > 2d then w∗(µ, d ,D)>cK .
The limit limD→+∞
w∗(D)/√
D exists and is positive.
11/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 11 / 56
asymptotic speed of propagation in the direction of the line
The effect of roads on propagation
Theorem
There exists an asymptotic speed of propagation in the direction of thex-axis, w∗ = w∗(µ, d ,D) > 0. That is: let (u0, v0) be a compactlysupported initial datum (nonnegative, nontrivial). Then, locally in y :
∀c > w∗, sup|x |≥ct
|(u(x , t), v(x , y , t))| → 0 as t →∞
∀c < w∗, sup|x |≤ct, 0≤y≤a
|(u(x , t), v(x , y , t))− (ν/µ, 1)| → 0 as
t →∞.
Theorem
Let cK := 2√
d f ′(0) be the (homogenous) KPP speed of propagation.
If D ≤ 2d then w∗(µ, ν, d ,D, f ′(0))=cK .
If D > 2d then w∗(µ, d ,D)>cK .
The limit limD→+∞
w∗(D)/√
D exists and is positive.
11/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 11 / 56
asymptotic speed of propagation in the direction of the line
Construction of super and subsolutions
Original system:∂tu − D∂xxu = νv |y=0 − µu x ∈ R, t ∈ R∂tv − d∆v = f (v) (x , y) ∈ Ω, t ∈ R−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t ∈ R,
The KPP hypothesis f (v) ≤ f ′(0)v ⇒ solutions of linearized system aresupersolutions of nonlinear one.Look for exponential travelling wave solutions:
(u(t, x), v(t, x , y)) = (e−α(x−ct) , γe−α(x−ct)−βy )
with c , α > 0, γ > 0 and β ∈ R.
12/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 12 / 56
asymptotic speed of propagation in the direction of the line
Construction of super and subsolutions
Linearized system about v ≡ 0:∂tu − D∂xxu = νv |y=0 − µu x ∈ R, t ∈ R∂tv − d∆v = f ′(0)v (x , y) ∈ Ω, t ∈ R−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t ∈ R,
The KPP hypothesis f (v) ≤ f ′(0)v ⇒ solutions of linearized system aresupersolutions of nonlinear one.Look for exponential travelling wave solutions:
(u(t, x), v(t, x , y)) = (e−α(x−ct) , γe−α(x−ct)−βy )
with c , α > 0, γ > 0 and β ∈ R.
12/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 12 / 56
asymptotic speed of propagation in the direction of the line
Construction of super and subsolutions
Linearized system about v ≡ 0:∂tu − D∂xxu = νv |y=0 − µu x ∈ R, t ∈ R∂tv − d∆v = f ′(0)v (x , y) ∈ Ω, t ∈ R−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t ∈ R,
The KPP hypothesis f (v) ≤ f ′(0)v ⇒ solutions of linearized system aresupersolutions of nonlinear one.
Look for exponential travelling wave solutions:
(u(t, x), v(t, x , y)) = (e−α(x−ct) , γe−α(x−ct)−βy )
with c , α > 0, γ > 0 and β ∈ R.
12/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 12 / 56
asymptotic speed of propagation in the direction of the line
Construction of super and subsolutions
Linearized system about v ≡ 0:∂tu − D∂xxu = νv |y=0 − µu x ∈ R, t ∈ R∂tv − d∆v = f ′(0)v (x , y) ∈ Ω, t ∈ R−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t ∈ R,
The KPP hypothesis f (v) ≤ f ′(0)v ⇒ solutions of linearized system aresupersolutions of nonlinear one.Look for exponential travelling wave solutions:
(u(t, x), v(t, x , y)) = (e−α(x−ct) , γe−α(x−ct)−βy )
with c , α > 0, γ > 0 and β ∈ R.
12/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 12 / 56
asymptotic speed of propagation in the direction of the line
Asymptotic speed of propagation
The system on (α, β, γ) reads−Dα2 + cα = γ − µ
−d(α2 + β2) + cα = f ′(0)dβγ = µ− γ
−Dα2 + cα =µdβ
1 + dβd(α2 + β2)− cα + f ′(0) = 0
13/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 13 / 56
asymptotic speed of propagation in the direction of the line
Asymptotic speed of propagation
The system on (α, β, γ) reads−Dα2 + cα = γ − µ
−d(α2 + β2) + cα = f ′(0)dβγ = µ− γ−Dα2 + cα =µdβ
1 + dβd(α2 + β2)− cα + f ′(0) = 0
13/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 13 / 56
asymptotic speed of propagation in the direction of the line
Exponential solutions - equations in (α, β)
Figure: Exponential solutions of linearized system (u, v) = (eα(x+ct), eα(x+ct)−βy ).
14/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 14 / 56
asymptotic speed of propagation in the direction of the line
Construction of super-solutions - Algebraic equations
15/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 15 / 56
asymptotic speed of propagation in the direction of the line
Case D > 2d
16/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 16 / 56
asymptotic speed of propagation in the direction of the line
Case D < 2d : super-solutions
17/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 17 / 56
asymptotic speed of propagation in the direction of the line
Case D = 2d : super-solutions
18/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 18 / 56
asymptotic speed of propagation in the direction of the line
Case D > 2d , c ∈ (cK , c∗)
19/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 19 / 56
asymptotic speed of propagation in the direction of the line
Case D > 2d , c = c∗
20/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 20 / 56
asymptotic speed of propagation in the direction of the line
Case D > 2d , c∗ − c > 0 small
21/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 21 / 56
asymptotic speed of propagation in the direction of the line
General strategy
1 Exponential solutions of the linearized system (about v ≡ 0)
2 Real exponential solutions exist for all speed c ≥ c∗ (Algebraicsystem) ⇒ w∗ ≤ c∗
3 Penalize the linearized system → subsolutions
4 Restrict to a strip R× (0, L) with Dirichlet condition at y = L →truncation of the support
5 For c∗− c > 0 small, complex solutions appear (by Rouche’s theorem)
6 Get solutions with support contained in infinite strips
7 Penalize the linearized system → subsolutions
8 Restrict to a strip R× (0, L) with Dirichlet condition at y = L→ truncation of the support
9 For c < c∗, use the real parts of complex solutions to get compactlysupported subsolutions ⇒ w∗ ≥ c∗
22/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 22 / 56
asymptotic speed of propagation in the direction of the line
Truncation in y
Horizontal strip ΩL := R× (0, L) with L > 0.−DU ′′ + cU ′ = V (x , 0)− µU x ∈ R−d∆V + c∂xV = f ′(0)V (x , y) ∈ ΩL
−d∂y V (x , 0) = µU(x)− V (x , 0) x ∈ RV (x , L) = 0 x ∈ R.
23/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 23 / 56
asymptotic speed of propagation in the direction of the line
Exponential solutions of truncated system
Solutions of the form (1, γ(y))eαx . Existence iff following system has asolution:
−Dα2 + cα +(1 + e−2βL)dβµ
1− e−2βL + (1 + e−2βL)dβ= 0
−d(α2 + β2) + cα = f ′(0)dβ(γ1 − γ2) = µ− (γ1 + γ2)
γ1e−βL + γ2eβL = 0
Unknowns α and β (look for γ under the form γ1e−βy + γ2eβy ).Sub-solution is
(u, v) := Re(1, γ(y))eαx
Delicate perturbative analysis adapting Rouche’s theorem to derivethe case of finite large L from the half plane case.
Not a simple structure...
24/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 24 / 56
effect of transport and reaction on the road
Adding transport q and mortality (rate ρ) on the road
∂tu − D∂xxu + q∂xu = −ρu + νv |y=0 − µu x ∈ R, t > 0
∂tv − d∆v = f (v) (x , y) ∈ Ω, t > 0
−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t > 0,
q ∈ R, ρ > 0.
Theorem
(Liouville-type result). There is a unique positive, bounded, stationarysolution (U,V ). Moreover, U ≡ constant and V ≡ V (y).
(Spreading). There are asymptotic speeds of propagation w∗− towardsleft and w∗+ towards right.
(Spreading velocity). IfD
d≤ 2+
ρ
f ′(0)∓ q√
df ′(0)then w∗± = ±cK ,
else w∗+ > cK (resp. w∗− < −cK ).
25/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 25 / 56
effect of transport and reaction on the road
Adding transport q and mortality (rate ρ) on the road
∂tu − D∂xxu + q∂xu = −ρu + νv |y=0 − µu x ∈ R, t > 0
∂tv − d∆v = f (v) (x , y) ∈ Ω, t > 0
−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t > 0,
q ∈ R, ρ > 0.
Theorem
(Liouville-type result). There is a unique positive, bounded, stationarysolution (U,V ). Moreover, U ≡ constant and V ≡ V (y).
(Spreading). There are asymptotic speeds of propagation w∗− towardsleft and w∗+ towards right.
(Spreading velocity). IfD
d≤ 2+
ρ
f ′(0)∓ q√
df ′(0)then w∗± = ±cK ,
else w∗+ > cK (resp. w∗− < −cK ).
25/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 25 / 56
effect of transport and reaction on the road
Effect of transport q and mortality ρ on the road
Case ρ = 0:The condition for enhancement of the invasion speed towards right, i.e.w∗+ > cK reads:
D
d> 2− q√
df ′(0)= 2(1− q
cK).
D > 2d enhancement occurs for all q ≥ 0
A drift q > cK always enhances the invasion speed
An invasion upstream, i.e. when q < 0, is never slowed down by thestream: w∗+ ≥ cK .But even in this case, a large D speeds up the invasion.
26/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 26 / 56
effect of transport and reaction on the road
Effect of transport q and mortality ρ on the road
Case ρ = 0:The condition for enhancement of the invasion speed towards right, i.e.w∗+ > cK reads:
D
d> 2− q√
df ′(0)= 2(1− q
cK).
D > 2d enhancement occurs for all q ≥ 0
A drift q > cK always enhances the invasion speed
An invasion upstream, i.e. when q < 0, is never slowed down by thestream: w∗+ ≥ cK .But even in this case, a large D speeds up the invasion.
26/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 26 / 56
effect of transport and reaction on the road
Effect of transport q and mortality ρ on the road
Case ρ = 0:The condition for enhancement of the invasion speed towards right, i.e.w∗+ > cK reads:
D
d> 2− q√
df ′(0)= 2(1− q
cK).
D > 2d enhancement occurs for all q ≥ 0
A drift q > cK always enhances the invasion speed
An invasion upstream, i.e. when q < 0, is never slowed down by thestream: w∗+ ≥ cK .But even in this case, a large D speeds up the invasion.
26/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 26 / 56
effect of transport and reaction on the road
Effect of transport q and mortality ρ on the road
Case ρ = 0:The condition for enhancement of the invasion speed towards right, i.e.w∗+ > cK reads:
D
d> 2− q√
df ′(0)= 2(1− q
cK).
D > 2d enhancement occurs for all q ≥ 0
A drift q > cK always enhances the invasion speed
An invasion upstream, i.e. when q < 0, is never slowed down by thestream: w∗+ ≥ cK .
But even in this case, a large D speeds up the invasion.
26/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 26 / 56
effect of transport and reaction on the road
Effect of transport q and mortality ρ on the road
Case ρ = 0:The condition for enhancement of the invasion speed towards right, i.e.w∗+ > cK reads:
D
d> 2− q√
df ′(0)= 2(1− q
cK).
D > 2d enhancement occurs for all q ≥ 0
A drift q > cK always enhances the invasion speed
An invasion upstream, i.e. when q < 0, is never slowed down by thestream: w∗+ ≥ cK .But even in this case, a large D speeds up the invasion.
26/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 26 / 56
effect of transport and reaction on the road
More general reaction on the road
∂tu − D∂xxu + q∂xu = g(u) + νv |y=0 − µu x ∈ R, t > 0
∂tv − d∆v = f (v) (x , y) ∈ Ω, t > 0
−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t > 0
g(0) = 0 g(u) ≤ g ′(0)u
Theorem
Same results as before. The condition for enhancement of the speed nowreads: (Spreading velocity):
IfD
d> 2− g ′(0)
f ′(0)∓ q√
df ′(0)then w∗± > ±cK , else w∗+ = cK (resp.
w∗− = −cK ).
Actually, Liouville type theorem somewhat more complicated; requiresboth f and g concave.
27/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 27 / 56
effect of transport and reaction on the road
More general reaction on the road
∂tu − D∂xxu + q∂xu = g(u) + νv |y=0 − µu x ∈ R, t > 0
∂tv − d∆v = f (v) (x , y) ∈ Ω, t > 0
−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t > 0
g(0) = 0 g(u) ≤ g ′(0)u
Theorem
Same results as before. The condition for enhancement of the speed nowreads: (Spreading velocity):
IfD
d> 2− g ′(0)
f ′(0)∓ q√
df ′(0)then w∗± > ±cK , else w∗+ = cK (resp.
w∗− = −cK ).
Actually, Liouville type theorem somewhat more complicated; requiresboth f and g concave.
27/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 27 / 56
effect of transport and reaction on the road
More general reaction on the road
∂tu − D∂xxu + q∂xu = g(u) + νv |y=0 − µu x ∈ R, t > 0
∂tv − d∆v = f (v) (x , y) ∈ Ω, t > 0
−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t > 0
g(0) = 0 g(u) ≤ g ′(0)u
Theorem
Same results as before. The condition for enhancement of the speed nowreads: (Spreading velocity):
IfD
d> 2− g ′(0)
f ′(0)∓ q√
df ′(0)then w∗± > ±cK , else w∗+ = cK (resp.
w∗− = −cK ).
Actually, Liouville type theorem somewhat more complicated; requiresboth f and g concave.
27/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 27 / 56
effect of transport and reaction on the road
On the mysterious 2 in the D > 2d condition
In case q = 0 and g = f ,∂tu − D∂xxu = f (u) + νv |y=0 − µu x ∈ R, t > 0
∂tv − d∆v = f (v) (x , y) ∈ Ω, t > 0
−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t > 0,Then,
2− g ′(0)
f ′(0)= 1
threshold condition for enhancement:
D > d .
28/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 28 / 56
effect of transport and reaction on the road
On the mysterious 2 in the D > 2d condition
In case q = 0 and g = f ,∂tu − D∂xxu = f (u) + νv |y=0 − µu x ∈ R, t > 0
∂tv − d∆v = f (v) (x , y) ∈ Ω, t > 0
−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t > 0,
Then,
2− g ′(0)
f ′(0)= 1
threshold condition for enhancement:
D > d .
28/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 28 / 56
effect of transport and reaction on the road
On the mysterious 2 in the D > 2d condition
In case q = 0 and g = f ,∂tu − D∂xxu = f (u) + νv |y=0 − µu x ∈ R, t > 0
∂tv − d∆v = f (v) (x , y) ∈ Ω, t > 0
−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t > 0,Then,
2− g ′(0)
f ′(0)= 1
threshold condition for enhancement:
D > d .
28/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 28 / 56
effect of transport and reaction on the road
On the mysterious 2 in the D > 2d condition
In case q = 0 and g = f ,∂tu − D∂xxu = f (u) + νv |y=0 − µu x ∈ R, t > 0
∂tv − d∆v = f (v) (x , y) ∈ Ω, t > 0
−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t > 0,Then,
2− g ′(0)
f ′(0)= 1
threshold condition for enhancement:
D > d .
28/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 28 / 56
effect of transport and reaction on the road
Spreading enhancement along roads by pure growth effect
In case q = 0, D arbitrary, and reactions g , f ,∂tu − D∂xxu = g(u) + νv |y=0 − µu x ∈ R, t > 0
∂tv − d∆v = f (v) (x , y) ∈ Ω, t > 0
−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t > 0,
Threshold condition for speed-up:
D
d≥ 2− g ′(0)
f ′(0)
E.g. if D = d : g ′(0) > f ′(0) enhances speed of propagation
29/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 29 / 56
effect of transport and reaction on the road
Spreading enhancement along roads by pure growth effect
In case q = 0, D arbitrary, and reactions g , f ,∂tu − D∂xxu = g(u) + νv |y=0 − µu x ∈ R, t > 0
∂tv − d∆v = f (v) (x , y) ∈ Ω, t > 0
−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t > 0,
Threshold condition for speed-up:
D
d≥ 2− g ′(0)
f ′(0)
E.g. if D = d : g ′(0) > f ′(0) enhances speed of propagation
29/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 29 / 56
asymptotic shape of expansion
Propagation in other directions
30/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 30 / 56
asymptotic shape of expansion
Is this diffusion “trajectory” optimal?
31/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 31 / 56
asymptotic shape of expansion
Short-cut! – Is the geometric optics trajectory optimal?
32/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 32 / 56
asymptotic shape of expansion
Asymptotic expansion set
Asymptotic expansion set (AES)
W such that for any solution (u, v) starting from a compactly supportedinitial datum (u0, v0) (nonnegative, nontrivial), and for all ε > 0:
supdist( 1
t(x ,y),W)>ε
v(x , y , t)→ 0 as t →∞
supdist( 1
t(x ,y),Ω\W)>ε
|v(x , y , t)− 1| → 0 as t →∞.
33/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 33 / 56
asymptotic shape of expansion
Upper level sets of v(·, t) ' tW for t large
The intersection of W with the line directed by ξ gives theasymptotic speed of propagation in direction ξ
If W exists then W ∩ y = 0 = [−w∗,w∗]× 0Homogeneous case (Fisher-KPP) : W = ball of radius 2
√df ′(0)
34/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 34 / 56
asymptotic shape of expansion
Lower bound for W
The short-cut strategy: first go on the road, then standard KPPpropagation in field when optimal.
W ⊃WS :=
Conv(
([−w∗,w∗]× 0) ∪ BcK
).
35/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 35 / 56
asymptotic shape of expansion
Lower bound for W
The short-cut strategy: first go on the road, then standard KPPpropagation in field when optimal.
W ⊃WS :=
Conv(
([−w∗,w∗]× 0) ∪ BcK
).
35/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 35 / 56
asymptotic shape of expansion
Lower bound for W
The short-cut strategy: first go on the road, then standard KPPpropagation in field when optimal.
W ⊃WS :=
Conv(
([−w∗,w∗]× 0) ∪ BcK
).
35/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 35 / 56
asymptotic shape of expansion
Lower bound for W
The short-cut strategy: first go on the road, then standard KPPpropagation in field when optimal.
W ⊃WS :=
Conv(
([−w∗,w∗]× 0) ∪ BcK
).
35/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 35 / 56
asymptotic shape of expansion
Lower bound for W
The short-cut strategy: first go on the road, then standard KPPpropagation in field when optimal.
W ⊃WS := Conv(([−w∗,w∗]× 0) ∪ BcK
).
35/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 35 / 56
asymptotic shape of expansion
Main result
Theorem
1 (Spreading) There exists an AES W2 (Expansion shape) The set W is convex and can be written as
W := ρ(sinϑ, cosϑ) : −π/2 ≤ ϑ ≤ π/2, 0 ≤ ρ ≤ w∗(ϑ),
with w∗ ∈ C 1([−π/2, π/2]) even and such that
∃ϑ0 ∈ (0, π/2], w∗ = cK in [0, ϑ0], (w∗)′ > 0 in (ϑ0, π/2].
3 (Directions with enhanced speed) If D ≤ 2d then ϑ0 = π/2.Otherwise, ϑ0 < π/2 and ϑ0 is a strictly decreasing function of D.
V Critical angle phenomenon
36/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 36 / 56
asymptotic shape of expansion
Main result
Theorem
1 (Spreading) There exists an AES W2 (Expansion shape) The set W is convex and can be written as
W := ρ(sinϑ, cosϑ) : −π/2 ≤ ϑ ≤ π/2, 0 ≤ ρ ≤ w∗(ϑ),
with w∗ ∈ C 1([−π/2, π/2]) even and such that
∃ϑ0 ∈ (0, π/2], w∗ = cK in [0, ϑ0], (w∗)′ > 0 in (ϑ0, π/2].
3 (Directions with enhanced speed) If D ≤ 2d then ϑ0 = π/2.Otherwise, ϑ0 < π/2 and ϑ0 is a strictly decreasing function of D.
V Critical angle phenomenon
36/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 36 / 56
asymptotic shape of expansion
Main result
Theorem
1 (Spreading) There exists an AES W2 (Expansion shape) The set W is convex and can be written as
W := ρ(sinϑ, cosϑ) : −π/2 ≤ ϑ ≤ π/2, 0 ≤ ρ ≤ w∗(ϑ),
with w∗ ∈ C 1([−π/2, π/2]) even and such that
∃ϑ0 ∈ (0, π/2], w∗ = cK in [0, ϑ0], (w∗)′ > 0 in (ϑ0, π/2].
3 (Directions with enhanced speed) If D ≤ 2d then ϑ0 = π/2.Otherwise, ϑ0 < π/2 and ϑ0 is a strictly decreasing function of D.
V Critical angle phenomenon
36/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 36 / 56
asymptotic shape of expansion
Asymptotic expansion shape
A case where the Huyghens principle fails !
37/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 37 / 56
asymptotic shape of expansion
Asymptotic expansion shape
A case where the Huyghens principle fails !
37/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 37 / 56
variants and further results
Variants
• Case when µ and ν are not constant. Consider the periodic case µ(x)and ν(x) as periodic functions of x .
Extension of the results about ASP in direction of the road:Thomas Gilletti, Leonard Monsaigeon, Maolin ZhouSame threshold D = 2d
• Existence of travelling fronts in the direction of the x-axis
38/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 38 / 56
variants and further results
A variant: strip and 2 roads
Work of L. Rossi, A. Tellini and E. ValdinociThree populations u(x , t), u(x , t), v(x , y , t), with (x , y) ∈ R× (−R,R)
u
u
R
-R
v
vt − d∆v = f (v)ut − Duxx = νv(x ,R, t)− µud vy (x ,R, t) = µu − νv(x ,R, t)ut − Duxx = νv(x ,−R, t)− µu−d vy (x ,−R, t) = µu − νv(x ,−R, t)
(PR)
39/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 39 / 56
variants and further results
A variant: strip and 2 roads
Work of L. Rossi, A. Tellini and E. ValdinociThree populations u(x , t), u(x , t), v(x , y , t), with (x , y) ∈ R× (−R,R)
u
u
R
-R
v
vt − d∆v = f (v)ut − Duxx = νv(x ,R, t)− µud vy (x ,R, t) = µu − νv(x ,R, t)ut − Duxx = νv(x ,−R, t)− µu−d vy (x ,−R, t) = µu − νv(x ,−R, t)
(PR)
39/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 39 / 56
variants and further results
A variant: strip and 2 roads
Work of L. Rossi, A. Tellini and E. ValdinociThree populations u(x , t), u(x , t), v(x , y , t), with (x , y) ∈ R× (−R,R)
u
u
R
-R
v
vt − d∆v = f (v)ut − Duxx = νv(x ,R, t)− µud vy (x ,R, t) = µu − νv(x ,R, t)ut − Duxx = νv(x ,−R, t)− µu−d vy (x ,−R, t) = µu − νv(x ,−R, t)
(PR)
39/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 39 / 56
variants and further results
The model
u
v
vt − d∆v = f (v) in Ω× (0,+∞)ut − D∆∂Ωu = νv − µu in ∂Ω× (0,+∞)
d ∂v∂n = µu − νv in ∂Ω× (0,+∞)
(PR)
with Ω = R× BR(0) ⊂ RN+1
40/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 40 / 56
Results and interpretation
Main result
Theorem
This problem admits an A.S.P. c∗ = c∗(D, d , µ, ν,R,N) > 0.
The function D 7→ c∗(D) is increasing and satisfies
limD↓0
c∗(D) = c0 > 0, limD→+∞
c∗(D)√D∈ (0,+∞).
The function R 7→ c∗(R) satisfies
limR↓0
c∗(R) = 0, limR→+∞
c∗(R) = c∗∞,
where c∗∞ is the A.S.P. of problem in half plane
c∗∞
= cKPP if D ≤ 2d ,
> cKPP if D > 2d .
41/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 41 / 56
Results and interpretation
Main result
Theorem
This problem admits an A.S.P. c∗ = c∗(D, d , µ, ν,R,N) > 0.
The function D 7→ c∗(D) is increasing and satisfies
limD↓0
c∗(D) = c0 > 0, limD→+∞
c∗(D)√D∈ (0,+∞).
The function R 7→ c∗(R) satisfies
limR↓0
c∗(R) = 0, limR→+∞
c∗(R) = c∗∞,
where c∗∞ is the A.S.P. of problem in half plane
c∗∞
= cKPP if D ≤ 2d ,
> cKPP if D > 2d .
41/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 41 / 56
Results and interpretation
Main result
Theorem
This problem admits an A.S.P. c∗ = c∗(D, d , µ, ν,R,N) > 0.
The function D 7→ c∗(D) is increasing and satisfies
limD↓0
c∗(D) = c0 > 0, limD→+∞
c∗(D)√D∈ (0,+∞).
The function R 7→ c∗(R) satisfies
limR↓0
c∗(R) = 0, limR→+∞
c∗(R) = c∗∞,
where c∗∞ is the A.S.P. of problem in half plane
c∗∞
= cKPP if D ≤ 2d ,
> cKPP if D > 2d .
41/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 41 / 56
Results and interpretation
Main result
Theorem
This problem admits an A.S.P. c∗ = c∗(D, d , µ, ν,R,N) > 0.
The function D 7→ c∗(D) is increasing and satisfies
limD↓0
c∗(D) = c0 > 0, limD→+∞
c∗(D)√D∈ (0,+∞).
The function R 7→ c∗(R) satisfies
limR↓0
c∗(R) = 0, limR→+∞
c∗(R) = c∗∞,
where c∗∞ is the A.S.P. of problem in half plane
c∗∞
= cKPP if D ≤ 2d ,
> cKPP if D > 2d .
41/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 41 / 56
Results and interpretation
Main result
If D ≤ 2d then R 7→ c∗(R) is increasing
If D > 2d there exists RM s.t. c∗(R) is increasing in (0,RM) anddecreasing in (RM ,+∞)
R
c¥*
c*HRL
42/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 42 / 56
Results and interpretation
Main result
If D ≤ 2d then R 7→ c∗(R) is increasing
If D > 2d there exists RM s.t. c∗(R) is increasing in (0,RM) anddecreasing in (RM ,+∞)
RMR
c¥*
cM*
c*HRL
42/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 42 / 56
Results and interpretation
Main result
If D ≤ 2d then R 7→ c∗(R) is increasing
If D > 2d there exists RM s.t. c∗(R) is increasing in (0,RM) anddecreasing in (RM ,+∞)
RMR
c¥*
cM*
c*HRL
42/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 42 / 56
Results and interpretation
Interpretation
The road acts as a barrier
If D is small (with respect to the diffusion in the field), it is moreconvenient to stay in the field (⇐ if the roads are separated, thespreading velocity increases)If D is large, there is a competitive effect between the reaction in thefield and the fast diffusion on the boundary
43/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 43 / 56
Results and interpretation
Nonlocal exchange terms – Antoine Pauthier
System with non-local exchanges
∂tu − D∂xxu = −µu +
∫ν(y)v(t, x , y)dy x ∈ R, t > 0
∂tv − d∆v = f (v) + µ(y)u(t, x)− ν(y)v(t, x , y) (x , y) ∈ R2, t > 0
Hypothesis :
f is of KPP-type.
ν, µ ≥ 0, continuous, even, compact support; µ =∫µ, ν =
∫ν.
u
f (u)
The functions ν and µ model exchanges of densities between the road andthe field → exchange functions.
44/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 44 / 56
Results and interpretation
Initial question
Enhancement of biological invasion by heterogeneities: effect of a line offast diffusion.
Road of fast diffusion : ∂tu − D∂xxu = exchange terms
The Field
The Field Exchanges area (support of µ or ν)nonlocal equation
KPP Reaction-Diffusion∂tv − d∆v = f (v)
x
y
45/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 45 / 56
Results and interpretation
Robustness of the BRR-result
Proposition
The system admits a unique nonnegative bounded stationary solution(Us ,Vs(y)) 6≡ (0, 0). This solution is x−invariant, and satisfiesVs(±∞) = 1.
Theorem
there exists c∗ = c∗(µ, ν, d ,D, f ′(0)) > 0 such that:
for all c > c∗, limt→∞
sup|x |≥ct
(u(t, x), v(t, x , y)) = (0, 0) ;
for all c < c∗, limt→∞
inf|x |≤ct;|y |<a
(u(t, x), v(t, x , y)) = (Us ,Vs).
Moreover, c∗ satisfies:
if D ≤ 2d, c∗ = cKPP := 2√
df ′(0) ;
if D > 2d, c∗ > cKPP .
The threshold is still D = 2d .46/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 46 / 56
Results and interpretation
Influence of nonlocal exchanges on the spreading speed
For fixed parameters d ,D, f ′(0), µ, ν, set of admissible exchanges
Λµ = µ ∈ C0(R), µ ≥ 0,
∫µ = µ, µ even.
For µ ∈ Λµ and ν ∈ Λν , there exists a spreading speed c∗(µ, ν). Let c∗0 bethe spreading speed for the local exchange model (i.e. c∗0 = c∗(µδ0, νδ0)).
Questions
infc∗(µ, ν), µ ∈ Λµ, ν ∈ Λν ?
Can we compare c∗(µ, ν) with c∗0 ?
supc∗(µ, ν), µ ∈ Λµ, ν ∈ Λν = c∗0 ?
For two last questions, split the system in two intermediate models. =⇒Dissymmetric results
47/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 47 / 56
Results and interpretation
Spreading of weeds
Example: scentless chamomile (matricaria perforata) weed in NorthAmericaT. de-Camino-Beck and M. Lewis - with data from Alberta provinceEffects of disturbed habitat: roadsides, farmland. . .
Picture credit: T. de-Camino-Beck
48/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 48 / 56
Results and interpretation
Propagation due to the road only
∂tu − D∂xxu = f (u) + νv |y=0 − µu x ∈ R, t > 0
∂tv − d∆v = −ρv (x , y) ∈ Ω, t > 0
−d∂y v |y=0 = µu − νv |y=0 x ∈ R, t > 0,
49/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 49 / 56
Results and interpretation
Propagation due to the road only
Theorem
Under the conditionµ√ρd
ν +√ρd≥ f ′(0),
any solution starting from a bounded initial datum, tends to 0 ast → +∞, uniformly in x ∈ R, y ≥ 0.
=⇒ no nonzero steady state exists in this range of parameters.
50/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 50 / 56
Results and interpretation
Existence of a non-zero stationary solution
Theorem
Under conditionµ√ρd
ν +√ρd
< f ′(0),
the system has a unique positive, bounded steady state (us , vs).Moreover, us is constant, equal to the only positive root of
µ√ρd
ν +√ρd
us = f (us),
and vs = vs(y) =µus
ν +√ρd
e−√ρ/d y .
−D∂xxU = f (U) + νV |y=0 − µU x ∈ R, t > 0
−d∆V = −ρV (x , y) ∈ Ω, t > 0
−d∂y V |y=0 = µU − νV |y=0 x ∈ R, t > 0,51/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 51 / 56
Results and interpretation
Asymptotic speed of propagation
Theorem
Under the same condition, there is a positive spreading speed w∗. In otherwords, let (u, v) be a solution with a nonnegative, compactly supportedinitial datum (u0, v0) 6≡ (0, 0). Then,
For all c > w∗, we have
limt→+∞
sup|x |≥ct
(u(x , t), v(x , y , t)) = (0, 0),
for all c ∈ [0,w∗), for all a > 0, we have
limt→+∞
sup|x |≤ct, 0≤y≤a
|u(x , t), v(x , y , t))− (us , vs(y))| = 0.
52/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 52 / 56
Results and interpretation
Biological interpretation
All else equal, Critical rate of loss from the road for extinction :
µ0 := f ′(0)[ν√ρd
+ 1]
such that there is extinction ⇐⇒ µ ≥ µ0.
If µ < f ′(0) (loss rate is below intrinsic growth rate), there is alwayspersistence.
Case µ > f ′(0) : explicit threshold value ν0 for ν so that there ispersistence if and only if ν > ν0.
Case µ > f ′(0) and ν is fixed. Critical threshold γ so that there ispersistence of the species if and only if ρd < γ.
Condition only involves ρd
53/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 53 / 56
conclusion
Conclusion
A model for the interaction of propagation on lines and in the plane(or surfaces and in the space)
Precise formula for the asyptotic speed of progagation along the road
Precise thresholds for propagation enhancement
Asymptotic shape of expansion (and ASP in every direction)
Critical angle phenomenon
Role of other factors
Existence of travelling fronts (B, Roquejoffre and Rossi)
Propagation along a favourable road in an unfavourable environment,discussion of parameters
Many variants and many open problems - and conjectures
54/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 54 / 56
conclusion
HAPPY BIRTHDAY JEAN-MICHEL !
55/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 55 / 56