in honor of jean-michel coron henri berestycki · the e ect of di usion on a line on fisher-kpp...

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



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



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)



Asymptotic position of level sets

@÷ w*sc ,

x : ult .x1 .

.a ) ,

ocaci .

Level sets for t >> 1



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)


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 .


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)


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)


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.


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.




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 :


|(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√





w∗(D)/√





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 :


|(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√





w∗(D)/√




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.




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,



with c , α > 0, γ > 0 and β ∈ R.





The KPP hypothesis f (v) ≤ f ′(0)v ⇒ solutions of linearized system aresupersolutions of nonlinear one.

Look for exponential travelling wave solutions:


with c , α > 0, γ > 0 and β ∈ R.







with c , α > 0, γ > 0 and β ∈ R.



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




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



Exponential solutions - equations in (α, β)

Figure: Exponential solutions of linearized system (u, v) = (eα(x+ct), eα(x+ct)−βy ).



Construction of super-solutions - Algebraic equations



Case D > 2d



Case D < 2d : super-solutions



Case D = 2d : super-solutions



Case D > 2d , c ∈ (cK , c∗)



Case D > 2d , c = c∗



Case D > 2d , c∗ − c > 0 small



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∗



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.



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...


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 ).



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.





D

d> 2− q√

df ′(0)= 2(1− q

cK).



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.





D

d> 2− q√

df ′(0)= 2(1− q

cK).



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.



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.



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 .





∂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


D > d .





∂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


D > d .



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


asymptotic shape of expansion

Propagation in other directions



Is this diffusion “trajectory” optimal?



Short-cut! – Is the geometric optics trajectory optimal?



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 →∞.



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)



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

).



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

).



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



Asymptotic expansion shape

A case where the Huyghens principle fails !


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



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)



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


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 .



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



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



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



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.



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



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


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



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



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,



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.



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



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.



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


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


conclusion

HAPPY BIRTHDAY JEAN-MICHEL !


in honor of jean-michel coron henri berestycki · the e ect of di usion on a line on fisher-kpp...

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