two examples of reaction-diffusion front propagation in ...apauthie/contenu_du_site/defense.pdf ·...

Two examples of reaction-diffusion front propagation inheterogeneous media

Soutenance de thèse

Antoine Pauthier

Institut de Mathématique de Toulouse

Thèse dirigée par Henri Berestycki (EHESS) et Jean-Michel Roquejoffre (IMT)Soutenue par le projet ERC ReaDi

Antoine Pauthier (IMT) Soutenance de thèse 20 juin 2016 1 / 37

Outline

1 Fisher-KPP propagation driven by a line of fast diffusion: non-local exchangesPresentation of the modelResults

Robustness of the BRR-modelSpecific Properties of the modelUniform dynamics under a singular limit

Conclusion

2 Bistable entire solutionsIntroductionPrevious and current results

Entire solution in cylinder-like domainsA one dimensional case study

Conclusion


Nonlocal model

Model under study∂tu − D∂xx u = −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(1)

Hypotheses:The function f is of KPP-type: f (0) = f (1) = 0, f nonnegative, concave on ]0, 1[.Introduced by A. Kolmogorov, I. Petrovsky, and N. Piskounov (1937). In: Bull.Univ. Etat Moscou for the equation ∂tu − ∂xx u = u(1− u).

ν, µ ≥ 0, continuous, even and compactly supported. µ =∫µ, ν =

∫ν.

0 1 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 diffusionwith nonlocal exchanges.

Road of fast diffusion : ∂tu − D∂xx u = 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


Biological motivationInfluence of transportation network on biological invasion

Seismic lines in Alberta forest. Copyright (c) Province of British Columbia. All rights reserved.

Reproduced with permission of the Province of British Columbia.


GPS observation: wolves move and concentrate along these lines H.W. McKenzie et al.

(2012). In: Interface Focus. Original picture by Santiago Atienza, licence Cc-by-2.0


Initial model

Introduced in 2012 by H. Berestycki, J.-M. Roquejoffre, and L. Rossi.

The road∂tu − D∂xx u = νv − µu

∂tv − d∆v = f (v)KPP reaction-diffusion

The field

d∂y v = µu − νv

Our model deals with nonlocal exchange terms.


Initial model

Berestycki-Roquejoffre-Rossi∂tu − D∂xx u = νv(t , x , 0)− µu x ∈ R, t > 0∂tv − d∆v = f (v) (x , y) ∈ R× R∗, t > 0

(2)

v(t , x , 0+) = v(t , x , 0−), x ∈ R, t > 0−d∂y v(t , x , 0+)− ∂y v(t , x , 0−)

= µu(t , x)− νv(t , x , 0) x ∈ R, t > 0.

(3)

Faster diffusion on the road: D > d .

Exchange coefficients at the boundary: µ, ν.

Reaction term f of KPP type.

Initial questionDoes the road enhance the spreading ?


The homogeneous caseReaction-diffusion with KPP nonlinearity:

∂tu = d∆u + f (u). (4)

Theorem - definition D. G. Aronson and H. F. Weinberger (1978). In: Adv. Math.

Let u(t , x) be solution of (4) with 0 ≤ u0 ≤ 1, compactly supported. Then there existsc∗ such that

∀c > c∗, limt→∞ sup|x|>ct u(t , x) = 0

∀c < c∗, limt→∞ inf|x|<ct u(t , x) = 1

with c∗ = 2√

df ′(0).

In homogeneous media, propagation in every direction at speed cKPP := 2√

df ′(0).

Expansion of the muskrat in Europe.J. G. SKELLAM (1951). In: Biometrika


Results of Berestycki-Roquejoffre-Rossi

Theorem H. Berestycki, J.-M. Roquejoffre, and L. Rossi (2013). In: Journal of Mathematical Biology

There exists c∗ = c∗(µ, ν, d ,D) > 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

(u(t , x), v(t , x , y)) = (ν/µ, 1).

Moreover:

if D ≤ 2d , then c∗(µ, ν, d ,D) = cKPP := 2√

df ′(0) ;

if D > 2d , then c∗(µ, ν, d ,D) > cKPP and limD→∞ c∗(µ, ν, d ,D)/√

D exists and ispositive.

Enhancement of the spreading in the direction of the road.

Threshold D = 2d .


Questions

1 Can we retrieve the same kind of results for the nonlocal model ?2 How do nonlocal exchanges modify the spreading speed ?3 How can we retrieve the initial model from the nonlocal one ?




Conclusion



Conclusion


Stationary solutions

Proposition

(1) admits a unique nonnegative bounded stationary solution (Us,Vs(y)) 6≡ (0, 0). Thissolution is x−invariant.

Us = 1

µ

∫ν(y)Vs(y)dy

−dV ′′s (y) = f (Vs(y)) + Usµ(y)− Vs(y)ν(y)

Vs(±∞) = 1.

Reminder: in the initial BRR-case, (Us,Vs) =

(ν

µ, 1).


Robustness of the BRR-result

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

(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 .

RemarkThe threshold is still D = 2d .


Main tool: construction of planar waves

They serve as supersolutions (f (v) ≤ f ′(0)v ).

Linearised system∂tu − D∂xx u = −µu +

∫ν(y)v(t , x , y)dy x ∈ R,

∂tv − d∆v = f ′(0)v + µ(y)u(t , x)− ν(y)v(t , x , y) (x , y) ∈ R2,(5)

Exponential solutions of the form(u(t , x)

v(t , x , y)

)= e−λ(x−ct)

(1

φ(y)

), (6)

With nonnegative λ, c, φ ∈ H1(R).

In the BRR model, they were given by an algebraic computation.

Here we are led to a nonlinear eigenvalue problem.


Equivalent system in λ, φ, c−Dλ2 + λc + µ =

∫ν(y)φ(y)dy

−dφ′′(y) + (λc − dλ2 − f ′(0) + ν(y))φ(y) = µ(y).

First equation gives a map λ 7→ Ψ1(λ, c) := −Dλ2 + λc + µ.

Second equation: at most one solution φ = φ(y ;λ, c). Then setΨ2(λ, c) :=

∫ν(y)φ(y)dy .

GoalFind λ, c such that the graphs of λ 7→ Ψ1(λ) and λ 7→ Ψ2(λ) intersect.

PropositionIf c > cKPP , then:

1 λ 7→ Ψ2(λ) defined on ]λ−2 , λ+2 [ is positive, smooth, convex and symmetric with

respect to the line λ = c2d .

2 Ψ2(λ) −→λ→λ∓2

µ, dΨ2dλ (λ) −→

λ→λ−2

−∞.


The two curves Ψ1 and Ψ2

µ

c2D

cD λ−2λ+

1c

2d λ

Ψ2Ψ1


Behaviour as c increases

µ

c2D

cD λ−2λ+

1c

2d λ

Ψ2Ψ1


Intersection for c = c∗

µ

λλ(c∗)

Ψ2Ψ1


Slower spreading

µ

λ

Ψ2Ψ1


Faster spreading

µ

λ

Ψ2Ψ1


Natural question: influence of nonlocal exchanges on the spreadingspeed

For fixed parameters d ,D, f ′(0), µ, ν we consider the set of admissible exchanges

Λµ = µ ∈ C0(R), µ ≥ 0,∫µ = µ, µ even.

Reminder: for µ ∈ Λµ and ν ∈ Λν , there exists a spreading speed c∗(µ, ν). Let c∗0 bethe spreading speed for the initial BRR model (c∗0 = c∗(µδ0, νδ0)).

QuestionsCan we compare c∗(µ, ν) with c∗0 ?

infc∗(µ, ν), µ ∈ Λµ, ν ∈ Λν ?

supc∗(µ, ν), µ ∈ Λµ, ν ∈ Λν = c∗0 ?


Long range exchange terms: a new threshold

For fixed parameters d ,D, f ′(0), µ, ν we can get the infimum with

νR(y) =1Rν( y

R

), or µR(y) =

1Rµ( y

R

), R → +∞

Theorem

Let us consider the nonlocal system (1) with fixed exchange masses µ and ν. Letc∗ = c∗(µ, ν) be the spreading speed given by Theorem 1.1, depending on therepartition of µ or ν.

1 If D ∈[2d , d

(2 +

µ

f ′(0)

)], infc∗, µ, ν ∈ Λµ,ν = 2

√df ′(0).

2 Fix D > d(

2 +µ

f ′(0)

), then infc∗, µ, ν ∈ Λµ,ν > 2

√df ′(0).

Moreover, in both cases, minimizing sequences can be given by long range exchangeterms of the form µR(y) = 1

Rµ( y

R

)or νR(y) = 1

R ν( y

R

)with R →∞.


Limit curve: infimum for the spreading speed

λ

Ψ2Ψ1

λ+1 = λ−2 = λ(c∗)

Ψ1 is fixed. The extremal points of Ψ2 do not depend on the repartition of µ and ν.


First intermediate model

∂tu − D∂xx u = −µu + νv(t , x , 0) x ∈ R, t > 0∂tv − d∆v = f (v) + µ(y)u(t , x) (x , y) ∈ R× R∗, t > 0v(t , x , 0+) = v(t , x , 0−), x ∈ R, t > 0−d∂y v(t , x , 0+)− ∂y v(t , x , 0−)

= −νv(t , x , 0) x ∈ R, t > 0.

(7)

Exchanges field→ road by boundary condition, id est ν = νδy=0.

Exchanges road→ field by a function µ with nontrivial support.

We get the same general results (existence, spreading, minimal speed, ...).


Maximum spreading speed

Parameters d ,D, f ′(0), ν, µ are fixed. We consider the set of admissible exchanges

Λµ = µ ∈ C0(R), µ ≥ 0,∫µ = µ, µ even.

For µ ∈ Λµ there exists c∗(µ) spreading speed. Let c∗0 be the BRR spreading speed.

Proposition

c∗0 = supc∗(µ), µ ∈ Λµ.

Fastest spreading for localised exchanges from the road to the field.The proof is an explicit computation.


Second intermediate model

∂tu − D∂xx u = −µu +

∫ν(y)v(t , x , y)dy x ∈ R, t > 0

∂tv − d∆v = f (v)− ν(y)v(t , x , y) (x , y) ∈ R× R∗, t > 0v(t , x , 0+) = v(t , x , 0−), x ∈ R, t > 0−d∂y v(t , x , 0+)− ∂y v(t , x , 0−)

= µu(t , x) x ∈ R, t > 0

(8)

Exchanges field→ road by function ν with nontrivial support.

Exchanges road→ field by boundary condition (id est µ = µδ0).

General theorems are preserved (existence, spreading, ...).

Do we get the same kind of results ?


First case: selfsimilar exchanges

For fixed parameters d ,D, f ′(0), µ, ν we consider the set of admissible exchanges

Λν = ν ∈ C0(R), ν ≥ 0,∫ν = ν, ν even.

For a given function ν, we set

νε(y) =1εν(yε

)=⇒ c∗(ε).

PropositionLet us consider c∗ as a function of the ε variable. Then there exists ε0,

∀ε < ε0, c∗(ε) > c∗0 .

Localised exchanges seem to be a local minimizer for the spreading speed.

It does not depend on the function ν.

Is it a general result, that is, are localised exchange terms a local minimizer for thespreading speed ?


Second case: perturbation of a Dirac

Mixed exchanges field→ road: boundary condition + small nonlocal contribution.

ν(y) = (1− ε)δ0 + ευ(y), υ ∈ Λ1 −→ c∗(ε).

Theorem

There exist m1 > 2 depending on f ′(0), M1 depending on µ such that:1 If D < m1 there exists ε0 and υ ∈ Λ1 such that ∀ε < ε0, c∗0 < c∗(ε);

2 if µ > 4 and D, f ′(0) > M1 there exists ε0 such that ∀υ ∈ Λ1, ∀ε < ε0, c∗0 > c∗(ε).

No general result for this model.

Various behaviours may happen even in the neighbourhood of localisedexchanges.


The singular limit of concentrated exchanges

We consider two exchange functions ν, µ. For all ε > 0 we define

νε(y) =1εν(yε

), µε(y) =

1εµ(yε

).

The exchange functions tend to Dirac measures↔ boundary conditions.

Formal convergence of the system to the initial BRR system.

For fixed parameters d ,D, µ, f ′(0) and initial conditions (u0, v0) we have

a spreading speed c∗ε ;

a dynamical solution (uε(t , x), vε(t , x , y)) ;

a unique stationary solution (Uε,Vε(y)).

QuestionConvergence of the dynamics as ε→ 0 ?


Uniform dynamics

We write (u, v) the solution of the dynamical BRR system, c∗0 the spreading speed.

Propositionc∗ε → c∗0 as ε→ 0, locally uniformly in all the parameters.

(Uε,Vε) tends ( νµ, 1), uniformly in y.

TheoremLet c > c∗0 . ∀η > 0, ∃T0, ε0 such that ∀t > T0, ∀ε < ε0, sup

|x|>ct|uε(x , t)| < η.

Let c < c∗0 . ∀η > 0, ∃T0, ε0 such that ∀t > T0, ∀ε < ε0, sup|x|<ct

∣∣∣∣uε(x , t)− ν

µ

∣∣∣∣ < η.


The previous theorem gives the commutation of the limits

limt→∞

limε→0

uε(t , x + ct) = limε→0

limt→∞

uε(t , x + ct).

Main tool: convergence of the dynamical solutions.

Theorem

‖(u − uε)(t)‖L∞(R) + ‖(v − vε)(t)‖L∞(R2) −→ε→0

0 locally uniformly in t ∈ (0,+∞).

Convergence local in time, global in space.Idea of proof:

Convergence of the linear operator.

Duhamel’s formula, Gronwall type argument.


Conclusion

Persistence of the initial results of Berestycki, Roquejoffre and Rossi.

A new threshold for infinitely supported exchanges.

Differences between the two exchange functions and their influence on thedynamics.

Uniform dynamics for self-similar exchange terms.

Perspectives:

Including reaction on the road: persistence of the differences ?

Transition between classical and enhanced spreading for long range exchanges.

More general kernels.




Conclusion



Conclusion


Problem under studyBistable reaction-diffusion equation:

∂tu(t , x)−∆u(t , x) = f (u), t ∈ R, x ∈ Ω,

∂νu(t , x) = 0, t ∈ R, x ∈ ∂Ω.(9)

The domain Ω is assumed to be a smooth infinite domain in the x1−direction, i.e.

Ω =

(x1, x ′), x1 ∈ R, x ′ ∈ ω(x1) ⊂ RN−1.

We also make a cylinder-like assumption:

ω(x1) −→x1→−∞

ω∞

The domain is, in one direction at infinity, the straight cylinder R× ω∞.

x1

x ′

Figure : Example of asymptotically cylindrical domain


Assumptions:

The reaction term f is of bistablekind, with

∫ 10 f (s)ds > 0.

The domain Ω is diffeomorphic to thecylinder Ω∞ := R× ω∞. 0 1 u

f (u)

θ

QuestionExistence and uniqueness of an entire (i.e. eternal) solution in such a domainconnecting 0 to 1 ? Influence on the dynamics in such domains ?


Influence of the geometry: some biological motivations

Population dynamicsI Population going through an isthmusI or a straight (fishes)

Cortical Spreading DepressionI CSD: transient and large depolarisation of the membrane of neurons. Propagation in

the grey mater, absorption in the white matter.I Migraine with aura, stroke.I Blocking of CSD in rodent: inefficient in human.

These images are from the University of Wisconsin and Michigan State Comparative Mammalian Brain

Collections, and from the National Museum of Health and Medicine


Homogeneous case: the straight cylinder R× ω∞

Travelling waves: solutions connecting 0 and 1 of the form

u(t , x) = ϕ(x1 − ct).

System for (c, ϕ) −cϕ′ − ϕ′′ = f (ϕ)

ϕ(−∞) = 1, ϕ(+∞) = 0, ϕ(0) = θ.

c

u(t = 0) = ϕ(x) u(t = 1) = ϕ(x − c)

Bistable nonlinearity: there exists a unique (up to translation) couple (c, ϕ).

Theorem P. C. Fife and J. B. McLeod (1977). In: Arch. Ration. Mech. Anal.

The travelling wave attracts the dynamics: if u0 is front-like, there exists x0 such that

sup |u(t , x)− ϕ(x1 − ct + x0)| −→t→+∞

0.


Previous resultHenri Berestycki, Juliette Bouhours, and Guillemette Chapuisat (2016). In: Calc. Var. Partial Differential

EquationsUnder the assumption:

Ω ∩

x ∈ RN , x1 < 0

= R− × ω, ω ⊂ RN−1.

The domain is equal to a cylinder in the left half space.

ε

There exists a unique solution of (9) defined for all t ∈ R such that

u(t , x)− ϕ(x1 − ct) −→t→−∞

0, uniformly in Ω.

This solution is increasing in time and converges to a steady state u∞.Blocking phenomenon by a narrow passage.Partial invasion in sufficiently large domain.Complete invasion in star-shaped domains and domains with decreasingcross-section.


Result

Aim: get rid of the strong cylinder-like assumption. We suppose that ω(x1) convergesto ω∞ at some exponential rate for the C2,α topology.

TheoremThere exists a unique function u(t , x) defined for t ∈ R and x ∈ Ω such that

sup |u(t , x)− ϕ(x1 − ct)| , x ∈ Ω −→t→−∞

0.

The proof amounts to proving the stability of the bistable wave.

All the other results (propagation, complete or partial, blocking) are preserved.


A case study

One dimensional problem:

∂tu − ∂xx u = f (u) (1 + g(x)) , t ∈ R, x ∈ R (10)

where g is a bounded perturbation that satisfies the assumption

there exists κ > 0, |g(x)| ≤ eκx for all x ∈ R. (11)

Notation: ρ1 is the spectral gap of L := −∂xx − c∂x − f ′(ϕ).

TheoremSet $ = ρ1

‖f ′‖∞If g satisfies (11) and g > −$, then there exists a function

u∞ = u∞(t , x) defined for t ∈ R, x ∈ R solution of (10) which satisfies

‖u∞(t , .)− ϕ(.− ct)‖L∞(R) −→t→−∞0. (12)

Remark: other results for this type of equation concerning transition fronts A. Zlatoš

(2016). In: preprint.


Idea of proof

For the case study:1 long time stability of the wave under the perturbation ;2 compactness argument.

In a cylinder-like domain:1 ideas of the case study ;2 estimate of the solution ahead of the front.


Conclusion and perspectives

The spreading properties given by Berestycki-Bouhours-Chapuisat are preserved.

We ask for an exponential convergence, but at an arbitrary rate. What aboutweaker convergence ?

In the case study, link with the theory of transition fronts.

Time delay coming from the variation of the cross section.


Thank you for your attention.


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