The effect of diffusion on a line on Fisher-KPP propagation In - - PowerPoint PPT Presentation

The effect of diffusion on a line on Fisher-KPP propagation

In honor of Jean-Michel Coron Henri Berestycki

EHESS, PSL Research University, Paris, ReaDi project - ERC

IHP, Paris, 22 June 2016

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, ≡ 0

fit

"

Homogeneous equation – Spreading properties in KPP case

Invasion: v(x, t) → 1 as t → ∞, locally uniformly in x as soon as v0 ≡ 0.

Homogeneous equation – Spreading properties in KPP case

Invasion: v(x, t) → 1 as t → ∞, locally uniformly in x as soon as v0 ≡ 0. Asymptotic speed of propagation: ∃w∗ such that for any v0 having compact support ∀c > w∗ sup

|x|≥ct

v(x, t) → 0 as t → ∞ ∀c < w∗ sup

|x|≤ct

|v(x, t) − 1| → 0 as t → ∞.

Homogeneous equation – Spreading properties in KPP case

Invasion: v(x, t) → 1 as t → ∞, locally uniformly in x as soon as v0 ≡ 0. Asymptotic speed of propagation: ∃w∗ such that for any v0 having compact 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

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• caci

The effect of a “road” with fast diffusion on Fisher-KPP propagation

Joint work with Jean-Michel Roquejoffre and Luca Rossi

• J. Math. Biology (2013)

Nonlinearity (2013)

• Comm. Math. Phys. (2016)

Nonlinear Anal. (2016)

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 := lim

yց0 v.

Birth/death rate: Logistic law (KPP type term) f : f > 0

• n

(0, 1), f (0) = f (1) = 0, f (v) ≤ f ′(0)v.

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)

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}.

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)

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.

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) ≡ (0, 0) (nonnegative and bounded). Then, (u(x, t), v(x, y, t)) → (ν/µ, 1), as t → ∞, locally uniformly in (x, y) ∈ Ω.

The effect of roads on propagation

The effect of roads on propagation

Theorem There exists an asymptotic speed of propagation in the direction of the x-axis, w∗ = w∗(µ, d, D) > 0.

The effect of roads on propagation

Theorem There exists an asymptotic speed of propagation in the direction of the x-axis, w∗ = w∗(µ, d, D) > 0. That is: let (u0, v0) be a compactly supported 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 → ∞.

The effect of roads on propagation

Theorem There exists an asymptotic speed of propagation in the direction of the x-axis, w∗ = w∗(µ, d, D) > 0. That is: let (u0, v0) be a compactly supported 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.

The effect of roads on propagation

Theorem There exists an asymptotic speed of propagation in the direction of the x-axis, w∗ = w∗(µ, d, D) > 0. That is: let (u0, v0) be a compactly supported 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.

The effect of roads on propagation

Theorem There exists an asymptotic speed of propagation in the direction of the x-axis, w∗ = w∗(µ, d, D) > 0. That is: let (u0, v0) be a compactly supported 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 effect of roads on propagation

Theorem There exists an asymptotic speed of propagation in the direction of the x-axis, w∗ = w∗(µ, d, D) > 0. That is: let (u0, v0) be a compactly supported 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 lim

D→+∞ w∗(D)/

√ D exists and is positive.

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∂yv|y=0 = µu − νv|y=0 x ∈ R, t ∈ R,

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∂yv|y=0 = µu − νv|y=0 x ∈ R, t ∈ R,

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∂yv|y=0 = µu − νv|y=0 x ∈ R, t ∈ R, The KPP hypothesis f (v) ≤ f ′(0)v ⇒ solutions of linearized system are supersolutions of nonlinear one.

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∂yv|y=0 = µu − νv|y=0 x ∈ R, t ∈ R, The KPP hypothesis f (v) ≤ f ′(0)v ⇒ solutions of linearized system are supersolutions 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.

Asymptotic speed of propagation

The system on (α, β, γ) reads    −Dα2 + cα = γ − µ −d(α2 + β2) + cα = f ′(0) dβγ = µ − γ

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 + dβ d(α2 + β2) − cα + f ′(0) =

Exponential solutions - equations in (α, β)

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

system) ⇒ w∗ ≤ c∗

truncation of the support

e’s theorem)

→ truncation of the support

supported 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∂yV (x, 0) = µU(x) − V (x, 0) x ∈ R V (x, L) = 0 x ∈ R.

Exponential solutions of truncated system

Solutions of the form (1, γ(y))eαx. Existence iff following system has a solution:            −Dα2 + cα + (1 + e−2βL)dβµ 1 − e−2βL + (1 + e−2βL)dβ = −d(α2 + β2) + cα = f ′(0) dβ(γ1 − γ2) = µ − (γ1 + γ2) γ1e−βL + γ2eβL = 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 Rouch´ e’s theorem to derive the case of finite large L from the half plane case. Not a simple structure...

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.

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, stationary solution (U, V ). Moreover, U ≡ constant and V ≡ V (y). (Spreading). There are asymptotic speeds of propagation w∗

− towards

left and w∗

+ towards right.

(Spreading velocity). If D 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 ).

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

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

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 the stream: 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 the stream: 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

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 now reads: (Spreading velocity): If D d > 2 − g′(0) f ′(0) ∓ q

• df ′(0)

then w∗

± > ±cK, else w∗ + = cK (resp.

w∗

− = −cK).

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 now reads: (Spreading velocity): If D 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; requires both f and g concave.

On the mysterious 2 in the D > 2d condition

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,

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

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.

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,

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

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 supported initial datum (u0, v0) (nonnegative, nontrivial), and for all ε > 0: sup

dist( 1

v(x, y, t) → 0 as t → ∞ sup

dist( 1

|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 the asymptotic speed of propagation in direction ξ If W exists then W ∩ {y = 0} = [−w∗, w∗] × {0} Homogeneous 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 KPP propagation in field when optimal. W ⊃ WS := ([−w∗, w∗] × {0}) ∪ BcK

• .

Lower bound for W

The short-cut strategy: first go on the road, then standard KPP propagation in field when optimal. W ⊃ WS := ([−w∗, w∗] × {0}) ∪ BcK

• .

Lower bound for W

The short-cut strategy: first go on the road, then standard KPP propagation in field when optimal. W ⊃ WS := ([−w∗, w∗] × {0}) ∪ BcK

• .

Lower bound for W

The short-cut strategy: first go on the road, then standard KPP propagation in field when optimal. W ⊃ WS := ([−w∗, w∗] × {0}) ∪ BcK

• .

Lower bound for W

The short-cut strategy: first go on the road, then standard KPP propagation in field when optimal. W ⊃ WS := Conv

• ([−w∗, w∗] × {0}) ∪ BcK
• .

Main result

Theorem

W := {ρ(sin ϑ, cos ϑ) : −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].

Otherwise, ϑ0 < π/2 and ϑ0 is a strictly decreasing function of D.

Main result

Theorem

W := {ρ(sin ϑ, cos ϑ) : −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].

Otherwise, ϑ0 < π/2 and ϑ0 is a strictly decreasing function of D. ⇛ Critical angle phenomenon

Main result

Theorem

W := {ρ(sin ϑ, cos ϑ) : −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].

Otherwise, ϑ0 < π/2 and ϑ0 is a strictly decreasing function of D. ⇛ Critical angle phenomenon

Asymptotic expansion shape

Asymptotic expansion shape

A case where the Huyghens principle fails !

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, L´ eonard Monsaigeon, Maolin Zhou Same 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. Valdinoci Three populations u(x, t), ˜ u(x, t), v(x, y, t), with (x, y) ∈ R × (−R, R)

A variant: strip and 2 roads

Work of L. Rossi, A. Tellini and E. Valdinoci Three populations u(x, t), ˜ u(x, t), v(x, y, t), with (x, y) ∈ R × (−R, R) u u

• R

R v

A variant: strip and 2 roads

Work of L. Rossi, A. Tellini and E. Valdinoci Three 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) − µu d vy(x, R, t) = µu − νv(x, R, t) ˜ ut − D˜ uxx = νv(x, −R, t) − µ˜ u −d vy(x, −R, t) = µ˜ u − 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

Main result

Theorem This problem admits an A.S.P. c∗ = c∗(D, d, µ, ν, R, N) > 0.

Main result

Theorem This problem admits an A.S.P. c∗ = c∗(D, d, µ, ν, R, N) > 0. The function D → c∗(D) is increasing and satisfies lim

D↓0 c∗(D) = c0 > 0,

lim

D→+∞

c∗(D) √ D ∈ (0, +∞).

Main result

Theorem This problem admits an A.S.P. c∗ = c∗(D, d, µ, ν, R, N) > 0. The function D → c∗(D) is increasing and satisfies lim

D↓0 c∗(D) = c0 > 0,

lim

D→+∞

c∗(D) √ D ∈ (0, +∞). The function R → c∗(R) satisfies lim

R↓0 c∗(R) = 0,

lim

R→+∞ c∗(R) = c∗ ∞,

where c∗

∞ is the A.S.P. of problem in half plane

Main result

Theorem This problem admits an A.S.P. c∗ = c∗(D, d, µ, ν, R, N) > 0. The function D → c∗(D) is increasing and satisfies lim

D↓0 c∗(D) = c0 > 0,

lim

D→+∞

c∗(D) √ D ∈ (0, +∞). The function R → c∗(R) satisfies lim

R↓0 c∗(R) = 0,

lim

R→+∞ 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 → c∗(R) is increasing

R

• cR

Main result

If D ≤ 2d then R → c∗(R) is increasing If D > 2d there exists RM s.t. c∗(R) is increasing in (0, RM) and decreasing in (RM, +∞)

R

• cM
• cR

Main result

If D ≤ 2d then R → c∗(R) is increasing If D > 2d there exists RM s.t. c∗(R) is increasing in (0, RM) and decreasing in (RM, +∞)

R

• cM
• cR

Interpretation

The road acts as a barrier

If D is small (with respect to the diffusion in the field), it is more convenient to stay in the field (⇐ if the roads are separated, the spreading velocity increases) If D is large, there is a competitive effect between the reaction in the field 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 and the field → exchange functions.

Initial question

Enhancement of biological invasion by heterogeneities: effect of a line of fast 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)) ≡ (0, 0). This solution is x−invariant, and satisfies Vs(±∞) = 1. Theorem there exists c∗ = c∗(µ, ν, d, D, f ′(0)) > 0 such that: for all c > c∗, lim

t→∞ sup |x|≥ct

(u(t, x), v(t, x, y)) = (0, 0) ; for all c < c∗, lim

t→∞

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.

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 be

the spreading speed for the local exchange model (i.e. c∗

0 = c∗(µδ0, νδ0)).

Questions inf{c∗(µ, ν), µ ∈ Λµ, ν ∈ Λν} ? Can we compare c∗(µ, ν) with c∗

0 ?

sup{c∗(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 North America

• T. de-Camino-Beck and M. Lewis - with data from Alberta province

Effects 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 condition µ√ρd ν + √ρd ≥ f ′(0), any solution starting from a bounded initial datum, tends to 0 as t → +∞, 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 condition µ√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,

Asymptotic speed of propagation

Theorem Under the same condition, there is a positive spreading speed w∗. In other words, let (u, v) be a solution with a nonnegative, compactly supported initial datum (u0, v0) ≡ (0, 0). Then, For all c > w∗, we have lim

t→+∞ sup |x|≥ct

(u(x, t), v(x, y, t)) = (0, 0), for all c ∈ [0, w∗), for all a > 0, we have lim

t→+∞

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 always persistence. Case µ > f ′(0) : explicit threshold value ν0 for ν so that there is persistence if and only if ν > ν0. Case µ > f ′(0) and ν is fixed. Critical threshold γ so that there is persistence of the species if and only if ρd < γ. Condition only involves ρd

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

HAPPY BIRTHDAY JEAN-MICHEL !