Two examples of reaction-diffusion front propagation in heterogeneous media

Transcription

1 Two examples of reaction-diffusion front propagation in heterogeneous 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 / 37

2 Outline 1 Fisher-KPP propagation driven by a line of fast diffusion: non-local exchanges Presentation of the model Results Robustness of the BRR-model Specific Properties of the model Uniform dynamics under a singular limit Conclusion 2 Bistable entire solutions Introduction Previous and current results Entire solution in cylinder-like domains A one dimensional case study Conclusion Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

3 Nonlocal model Model under study { 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) R 2, 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 xxu = u(1 u). ν, µ 0, continuous, even and compactly supported. µ = µ, ν = ν. f (u) 0 1 u The functions ν and µ model exchanges of densities between the road and the field exchange functions. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

4 Initial question Enhancement of biological invasion by heterogeneities: effect of a line of fast diffusion with nonlocal exchanges. y The Field x Road of fast diffusion : tu D xxu = exchange terms The Field Exchanges area (support of µ or ν) nonlocal equation KPP Reaction-Diffusion tv d v = f (v) Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

5 Biological motivation Influence 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. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

6 GPS observation: wolves move and concentrate along these lines (2012). In: Interface Focus. Original picture by Santiago Atienza, licence Cc-by-2.0 H.W. McKenzie et al. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

7 Initial model Introduced in 2012 by H. Berestycki, J.-M. Roquejoffre, and L. Rossi. tu D xxu = νv µu d y v = µu νv The road tv d v = f (v) KPP reaction-diffusion The field Our model deals with nonlocal exchange terms. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

8 Initial model Berestycki-Roquejoffre-Rossi { tu D xxu = νv(t, x, 0) µu x R, t > 0 tv d v = f (v) (x, y) R R, t > 0 { v(t, x, 0 + ) = v(t, x, 0 ), x R, t > 0 d { yv(t, x, 0 + ) yv(t, x, 0 ) } = µu(t, x) νv(t, x, 0) x R, t > 0. (2) (3) Faster diffusion on the road: D > d. Exchange coefficients at the boundary: µ, ν. Reaction term f of KPP type. Initial question Does the road enhance the spreading? Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

9 The homogeneous case Reaction-diffusion with KPP nonlinearity: Theorem - definition tu = d u + f (u). (4) D. G. Aronson and H. F. Weinberger (1978). In: Adv. Math. Let u(t, x) be solution of (4) with 0 u 0 1, compactly supported. Then there exists c such that c > c, lim t sup x >ct u(t, x) = 0 c < c, lim t inf x <ct u(t, x) = 1 with c = 2 df (0). In homogeneous media, propagation in every direction at speed c KPP := 2 df (0). Expansion of the muskrat in Europe. J. G. SKELLAM (1951). In: Biometrika Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

10 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, lim for all c < c, lim Moreover: sup t x ct inf t x ct (u(t, x), v(t, x, y)) = (0, 0); (u(t, x), v(t, x, y)) = (ν/µ, 1). if D 2d, then c (µ, ν, d, D) = c KPP := 2 df (0) ; if D > 2d, then c (µ, ν, d, D) > c KPP and lim D c (µ, ν, d, D)/ D exists and is positive. Enhancement of the spreading in the direction of the road. Threshold D = 2d. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

11 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? Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

12 1 Fisher-KPP propagation driven by a line of fast diffusion: non-local exchanges Presentation of the model Results Robustness of the BRR-model Specific Properties of the model Uniform dynamics under a singular limit Conclusion 2 Bistable entire solutions Introduction Previous and current results Entire solution in cylinder-like domains A one dimensional case study Conclusion Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

13 Stationary solutions Proposition (1) admits a unique nonnegative bounded stationary solution (U s, V s(y)) (0, 0). This solution is x invariant. V s(± ) = 1. { U s = 1 µ ν(y)vs(y)dy dv s (y) = f (V s(y)) + U sµ(y) V s(y)ν(y) Reminder: in the initial BRR-case, (U s, V s) = ( ) ν µ, 1. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

14 Robustness of the BRR-result Theorem There exists c = c (µ, ν, d, D, f (0)) > 0 such that: for all c > c, lim sup (u(t, x), v(t, x, y)) = (0, 0) ; t x ct for all c < c, lim Moreover, c satisfies: inf t x ct if D 2d, c = c KPP := 2 df (0) ; if D > 2d, c > c KPP. (u(t, x), v(t, x, y)) = (Us, Vs). Remark The threshold is still D = 2d. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

15 Main tool: construction of planar waves They serve as supersolutions (f (v) f (0)v). Linearised system { tu D xxu = 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) R 2, (5) Exponential solutions of the form ( ) ( ) u(t, x) = e λ(x ct) 1, (6) v(t, x, y) φ(y) With nonnegative λ, c, φ H 1 (R). In the BRR model, they were given by an algebraic computation. Here we are led to a nonlinear eigenvalue problem. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

16 Equivalent system in λ, φ, c { Dλ 2 + λc + µ = ν(y)φ(y)dy dφ (y) + (λc dλ 2 f (0) + ν(y))φ(y) = µ(y). Goal First equation gives a map λ Ψ 1 (λ, c) := Dλ 2 + λc + µ. Second equation: at most one solution φ = φ(y; λ, c). Then set Ψ 2 (λ, c) := ν(y)φ(y)dy. Find λ, c such that the graphs of λ Ψ 1 (λ) and λ Ψ 2 (λ) intersect. Proposition If c > c KPP, then: 1 λ Ψ 2 (λ) defined on ]λ 2, λ+ 2 [ is positive, smooth, convex and symmetric with respect to the line {λ = c }. 2d 2 Ψ 2 (λ) λ λ 2 µ, dψ 2 (λ) dλ λ λ 2. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

17 The two curves Ψ 1 and Ψ 2 µ Ψ 1 Ψ 2 c 2D c D λ + λ 1 2 c 2d λ Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

18 Behaviour as c increases µ Ψ 1 Ψ 2 c 2D c D λ + λ 1 2 c 2d λ Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

19 Intersection for c = c µ Ψ 1 Ψ 2 λ(c ) λ Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

20 Slower spreading µ Ψ 1 Ψ 2 λ Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

21 Faster spreading µ Ψ 1 Ψ 2 λ Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

22 Natural question: influence of nonlocal exchanges on the spreading speed For fixed parameters d, D, f (0), µ, ν we consider the set of admissible exchanges Λ µ = {µ C 0 (R), µ 0, µ = µ, µ even}. Reminder: for µ Λ µ and ν Λ ν, there exists a spreading speed c (µ, ν). Let c 0 be the spreading speed for the initial BRR model (c 0 = c (µδ 0, νδ 0 )). Questions Can we compare c (µ, ν) with c 0? inf{c (µ, ν), µ Λ µ, ν Λ ν}? sup{c (µ, ν), µ Λ µ, ν Λ ν} = c 0? Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

23 Long range exchange terms: a new threshold For fixed parameters d, D, f (0), µ, ν we can get the infimum with ν R (y) = 1 R ν ( y R ), or µ R (y) = 1 ( y ) R µ, R + R Theorem Let us consider the nonlocal system (1) with fixed exchange masses µ and ν. Let c = c (µ, ν) be the spreading speed given by Theorem 1.1, depending on the repartition of µ or ν. 1 If D [ 2d, d ( 2 + µ f (0) ( 2 Fix D > d 2 + µ f (0) )], inf{c, µ, ν Λ µ,ν} = 2 df (0). ), then inf{c, µ, ν Λ µ,ν} > 2 df (0). Moreover, in both cases, minimizing sequences can be given by long range exchange terms of the form µ R (y) = 1 µ ( ) y R R or νr (y) = 1 ν ( ) y R R with R. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

24 Limit curve: infimum for the spreading speed Ψ 1 Ψ 2 λ + 1 = λ 2 = λ(c ) λ Ψ 1 is fixed. The extremal points of Ψ 2 do not depend on the repartition of µ and ν. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

25 First intermediate model tu D xxu = µu + νv(t, x, 0) x R, t > 0 tv d v = f (v) + µ(y)u(t, x) (x, y) R R, t > 0 v(t, x, 0 + ) = v(t, x, 0 ), x R, t > 0 d { yv(t, x, 0 + ) yv(t, x, 0 ) } = νv(t, x, 0) x R, t > 0. (7) Exchanges field road by boundary condition, id est ν = νδ y=0. Exchanges road field by a function µ with nontrivial support. We get the same general results (existence, spreading, minimal speed,...). Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

26 Maximum spreading speed Parameters d, D, f (0), ν, µ are fixed. We consider the set of admissible exchanges Λ µ = {µ C 0 (R), µ 0, µ = µ, µ even}. For µ Λ µ there exists c (µ) spreading speed. Let c0 be the BRR spreading speed. Proposition c 0 = sup{c (µ), µ Λ µ}. Fastest spreading for localised exchanges from the road to the field. The proof is an explicit computation. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

27 Second intermediate model tu D xxu = 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 > 0 v(t, x, 0 + ) = v(t, x, 0 ), x R, t > 0 d { yv(t, x, 0 + ) yv(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 = µδ 0 ). General theorems are preserved (existence, spreading,...). Do we get the same kind of results? Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

28 First case: selfsimilar exchanges For fixed parameters d, D, f (0), µ, ν we consider the set of admissible exchanges Λ ν = {ν C 0 (R), ν 0, ν = ν, ν even}. For a given function ν, we set ν ε(y) = 1 ε ν ( y ε ) = c (ε). Proposition Let us consider c as a function of the ε variable. Then there exists ε 0, ε < ε 0, 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 the spreading speed? Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

29 Second case: perturbation of a Dirac Mixed exchanges field road: boundary condition + small nonlocal contribution. ν(y) = (1 ε)δ 0 + ευ(y), υ Λ 1 c (ε). Theorem There exist m 1 > 2 depending on f (0), M 1 depending on µ such that: 1 If D < m 1 there exists ε 0 and υ Λ 1 such that ε < ε 0, c0 < c (ε); 2 if µ > 4 and D, f (0) > M 1 there exists ε 0 such that υ Λ 1, ε < ε 0, c0 > c (ε). No general result for this model. Various behaviours may happen even in the neighbourhood of localised exchanges. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

30 The singular limit of concentrated exchanges We consider two exchange functions ν, µ. For all ε > 0 we define ν ε(y) = 1 ε ν ( y ε ), µ ε(y) = 1 ( y ) ε µ. ε 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 (u 0, v 0 ) we have a spreading speed cε ; a dynamical solution (u ε(t, x), v ε(t, x, y)) ; a unique stationary solution (U ε, V ε(y)). Question Convergence of the dynamics as ε 0? Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

31 Uniform dynamics We write (u, v) the solution of the dynamical BRR system, c 0 the spreading speed. Proposition c ε c 0 as ε 0, locally uniformly in all the parameters. (U ε, V ε) tends ( ν, 1), uniformly in y. µ Theorem Let c > c0. η > 0, T 0, ε 0 such that t > T 0, ε < ε 0, sup u ε(x, t) < η. x >ct Let c < c0. η > 0, T 0, ε 0 such that t > T 0, ε < ε 0, sup uε(x, t) ν µ < η. x <ct Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

32 The previous theorem gives the commutation of the limits lim lim uε(t, x + ct) = lim lim uε(t, x + ct). t ε 0 ε 0 t Main tool: convergence of the dynamical solutions. Theorem (u u ε)(t) L (R) + (v v ε)(t) L (R 2 ) 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. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

33 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 the dynamics. 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. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

34 1 Fisher-KPP propagation driven by a line of fast diffusion: non-local exchanges Presentation of the model Results Robustness of the BRR-model Specific Properties of the model Uniform dynamics under a singular limit Conclusion 2 Bistable entire solutions Introduction Previous and current results Entire solution in cylinder-like domains A one dimensional case study Conclusion Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

35 Problem under study Bistable reaction-diffusion equation: { tu(t, x) u(t, x) = f (u), t R, x Ω, νu(t, x) = 0, t R, x Ω. The domain Ω is assumed to be a smooth infinite domain in the x 1 direction, i.e. { Ω = (x 1, x ), x 1 R, x ω(x 1 ) R N 1}. We also make a cylinder-like assumption: ω(x 1 ) x 1 ω The domain is, in one direction at infinity, the straight cylinder R ω. (9) x x 1 Figure : Example of asymptotically cylindrical domain Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

36 Assumptions: f (u) The reaction term f is of bistable kind, with 1 f (s)ds > 0. 0 The domain Ω is diffeomorphic to the cylinder Ω := R ω. 0 θ 1 u Question Existence and uniqueness of an entire (i.e. eternal) solution in such a domain connecting 0 to 1? Influence on the dynamics in such domains? Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

37 Influence of the geometry: some biological motivations Population dynamics Population going through an isthmus or a straight (fishes) Cortical Spreading Depression CSD: transient and large depolarisation of the membrane of neurons. Propagation in the grey mater, absorption in the white matter. Migraine with aura, stroke. 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 Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

38 Homogeneous case: the straight cylinder R ω Travelling waves: solutions connecting 0 and 1 of the form u(t, x) = ϕ(x 1 ct). System for (c, ϕ) { cϕ ϕ = f (ϕ) ϕ( ) = 1, ϕ(+ ) = 0, ϕ(0) = θ. u(t = 0) = ϕ(x) u(t = 1) = ϕ(x c) 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 u 0 is front-like, there exists x 0 such that sup u(t, x) ϕ(x 1 ct + x 0 ) t + 0. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

39 Previous result Henri Berestycki, Juliette Bouhours, and Guillemette Chapuisat (2016). In: Calc. Var. Partial Differential Equations Under the assumption: { } Ω x R N, x 1 < 0 = R ω, ω R N 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(x 1 ct) 0, uniformly in Ω. t 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 decreasing cross-section. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

40 Result Aim: get rid of the strong cylinder-like assumption. We suppose that ω(x 1 ) converges to ω at some exponential rate for the C 2,α topology. Theorem There exists a unique function u(t, x) defined for t R and x Ω such that sup { u(t, x) ϕ(x 1 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. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

41 A case study One dimensional problem: tu xxu = 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) e κx for all x R. (11) Notation: ρ 1 is the spectral gap of L := xx c x f (ϕ). Theorem Set ϖ = ρ 1 If g satisfies (11) and g > ϖ, then there exists a function f u = u (t, x) defined for t R, x R solution of (10) which satisfies u (t,.) ϕ(. ct) L (R) 0. (12) t Remark: other results for this type of equation concerning transition fronts (2016). In: preprint. A. Zlatoš Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

42 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. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

43 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 about weaker convergence? In the case study, link with the theory of transition fronts. Time delay coming from the variation of the cross section. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37

44 Thank you for your attention. Antoine Pauthier (IMT) Soutenance de thèse 20 juin / 37