Nicolas Martin

Luis Yannick

Jorge J'ai

Max

Ulysse, Elliot Alan Flo

Nicolas Bastien

Mes premières pensées vont vers Sepideh et Jean-Michel. Je tiens à vous faire part de ma profonde gratitude. La liberté que vous m'avez donné dans le choix des thématiques de recherche, vos précieux conseils, votre grande disponibilité, votre soutien indéfectible ainsi que votre confiance ont fait de cette thèse une expérience formatrice au cours de laquelle j'ai pris énormément de plaisir. Vous m'avez montré que le monde de la recherche n'est pas qu'une affaire académique mais bel et bien humaine. Pour toutes ces raisons, je vous suis profondément reconnaissant.

I would like to thanks Enrico Valdinoci and Hatem Zaag to accept to report this thesis. It is an honour for me and I deeply thank them for the precious time they have spent on it. De même, je suis honoré par la présence dans mon jury de Ariane Trescases, Benoit Perthame, Yannick Privat et Robin Aguilée. Je souhaite remercier Robin aussi pour le temps qu'il m'a accordé pendant ma thèse. Nos discussions ont été de véritables sources d'inspirations dans le travail que je vous propose ici.

Je tiens également à remercier l'IMT et toutes les personnes qui oeuvrent au bon accueil des doctorants. Je pense bien sûr aux différentes équipes, aux personnels du laboratoires et de l'école doctorale. J'ai su trouver dans l'enceinte du laboratoire un environnement propice au travail. De même, je souhaite remercier les deux établissements qui m'ont accueilli lors de longs séjours de recherche : le CIRM à Marseille et l'université du Texas à Austin. J'aimerais à ce propos remercier les organisateurs du CEMRACS. I would also like to thank Luis Caffareli for his hospitality.

Ces laboratoires ne seraient rien sans les géniales équipes de doctorants qui les composent. Ainsi, je veux remercier l'ensemble des doctorants toulousains et plus particulièrement les doctorants du 3ème étage. Vous avez toujours été là pour répondre à mes questions (même les plus stupides). Je retiens surtout les rires et le parkour sur les bureaux ! Merci à : Guillaume, Baptiste, Joachim, Mathias, Louis et Michelle. Au passage, merci beaucoup à toi Michelle, mon éternelle relectrice pour ton aide en anglais et surtout pour les Baklavas ! Enfin, je souhaite remercier ma grande soeur de thèse, Susely qui a toujours su me donner de supers conseils.

J'aimerais remercier les personnes avec qui j'ai travaillé durant le CEMRACS : Claire, Antoine,

Chapter 1 Introduction

A. Objet de la thèse

Ce manuscrit de thèse s'inscrit dans le domaine de l'étude mathématique de la dynamique des populations en écologie et biologie évolutive. Plus précisément, nous nous intéresserons à la dynamique d'une seule population modélisée par sa densité n. Nous porterons une attention particulière au rôle que peut jouer une diffusion non-locale dans un habitat fragmenté. Dans les deux prochains paragraphes, nous expliquerons brièvement quelles sont les principales équations et questions abordées dans cette thèse. Dans un second temps, pour chacune de ces deux équations, nous donnerons une description précise des modèles en jeu. Ensuite, nous rappelerons le contexte scientifique dans lequel ce manuscrit s'inscrit avant de présenter les principaux résultats obtenus.

Les premiers travaux de cette thèse décrivent la dynamique spatiale d'une population soumise à l'équation de réaction diffusion fractionnaire suivante

∂ t n + (-∆) α n = f (x, n) pour x ∈ Ω, t > 0, n = 0 pour x ∈ Ω c , t > 0. (1.1)
Pour fixer les idées, nous prendrons dans toutes cette introduction f (x, n) = n -n 2 ou f (x, n) = µ(x)n -n 2 : une non-linéarité de type Fisher-KPP. L'opérateur de diffusion (-∆) α est défini comme suit :

(-∆) α n(x) = C d,α R d n(x) -n(y) |x -y| d+2α dy avec C d,α = 4 α Γ( d 2 + α) π d 2 |Γ(-α)| . (1.2)
Ce type d'opérateur décrit un processus de diffusion longue portée avec une grande fréquence. Nous nous interesserons particulièrement au critère de survie de l'espèce ainsi qu'aux phénomènes de colonisation voir d'invasion du territoire Ω.

Dans cette thèse, nous étudions également un modèle venant de la biologie évolutive, où nous considérons une population structurée phénotypiquement à l'équilibre et sujette à deux types de déplacement spatiaux : des déplacements locaux et non locaux. De plus, nous prenons en compte deux autres phénomènes. Le premier est la prise en compte de mutations qui modifient légèrement le trait phénotypique θ de la progéniture, ce phénomène crée de la variabilité dans la population. Le second phénomène est la compétition entre les individus, qui mène à

Chapter 1 : Introduction une sélection naturelle. Nous cherchons à observer l'émergence de traits phénotypiques dominants lorsque les mutations deviennent petites. L'équation choisie pour modéliser les phénomènes biologiques impliqués est :

           -∂ xx n ε -ε 2 ∂ θθ n ε + Ln ε = n ε [R(x, θ) -ρ ε ] pour x ∈ Ω, -A < θ < A, ρ ε (x) = A -A n ε (x, θ)dθ pour x ∈ Ω, ∂ νx n ε = ∂ ν θ n ε = 0 pour (x, θ) ∈ ∂ (Ω c ×] -A, A[) .
(1.3)

La diffusion spatiale est ici modélisée par le Laplacien par rapport à la variable x pour la diffusion locale et un opérateur intégral L qui sera décrit par la suite. Les mutations sont modélisées par le Laplacien par rapport à la variable θ. Enfin, les termes R et ρ désignent respectivement le taux de croissance intrinsèque des individus, et la taille totale de la population. Nous prenons en compte en effet via ce dernier terme un taux de mortalité dû à la compétition entre les individus.

B. L'équation de Fisher-KPP fractionnaire dans un environnement fragmenté

Rappelons l'équation considérée dans toute cette partie :

∂ t n + (-∆) α n = f (x, n) pour x ∈ Ω, t > 0, n = 0 pour x ∈ Ω c , t > 0.
(1.1)

B.1 Description du modèle et motivations biologiques B.1.a Les motivations biologiques

Nous considérons ici une espèce biologique se développant dans un environnement fragmenté. Nous supposons que cette espèce biologique est sujette à une dispersion longue portée. La dispersion longue portée désigne un mode de dispersion où les individus ne se déplacent pas seulement de proche en proche mais aussi sur de grandes distances dans un laps de temps relativement court. De plus, la fréquence de ces sauts est supposée être grande. Ce type de phénomène a été observé par des biologistes comme par exemple dans l'article [START_REF] Hastings | The spatial spread of invasions : new developments in theory and evidence[END_REF] qui introduit de multiples exemples d'espèces biologiques ayant une telle dispersion tel que le scarabée des céréales. En effet, la dispersion de ce scarabé est accélérée par l'action de l'homme. Un autre exemple est celui d'arbres présenté dans l'article [START_REF] Kremer | Long-distance gene flow and adaptation of forest trees to rapid climate change[END_REF] dont le pollen peut être dispersé sur de grandes distances sous l'effet de grands vents.

Les environnements fragmentés sont considérés pour illustrer une réalité tangible : les espèces biologiques ne peuvent pas s'implémenter partout. De ce fait, leurs habitats peuvent être séparés par un (ou des) obstacle(s) et s'en retrouver fragmentés. Par exemple, une espèce d'arbres ne pourra s'implémenter dans un environnement urbain. Un autre exemple est celui d'un archipel pour une espèce terrestre. Remarquons que sous l'effet de la dispersion longue portée, l'ensemble de l'environnement peut être colonisé par l'espèce. Ceci est faux si l'on suppose que les individus se déplacent seulement de proche en proche. Pour de tels phénomènes de diffusion longue portée, l'article [START_REF] Hastings | The spatial spread of invasions : new developments in theory and evidence[END_REF] met en évidence des vitesses d'invasion exponentielles en temps. Au contraire, si la diffusion est une dispersion standard (i.e. déplacement des individus uniquement de proche en proche) alors les vitesses d'invasion sont constantes (c'est à dire que le front d'invasion avance de façon linéaire). Enfin, soulignons que pour parler d'invasion il faut deux phénomènes : une colonisation de l'ensemble du territoire et une autosuffisance (ou autonomie) des individus sur les territoires colonisés. Ainsi, les questions biologiques précises motivant cette étude et prenant en compte l'ensemble de ces phénomènes sont :

1. Peut on mettre en évidence un critère assurant la survie/l'extinction de l'espèce biologique ?

2. Lorsqu'il y a survie de l'espèce, peut on prédire la distribution asymptotique de la population ?

3. Si une petite quantité d'individus arrive dans un environnement vierge, est ce que l'espèce va coloniser cet environnement ?

4. Si il y a colonisation, dans quel cas peut on parler d'invasion ? Quelles sont les caractéristiques de cette invasion ?

Enfin, soulignons que nous tenterons d'apporter des réponses à ces questions à travers un modèle simplifié. Ce modèle ne prend pas en compte quantités de facteurs biologiques. Cependant, cette simplification permet de mettre en évidence des réponses qualitatives. Ces réponses qualitatives se retrouvent ensuite quantitavement dans certains exemples issus de la biologie comme des phénomènes d'accélération de la propagation. Evidemment, pour coller à un jeu spécifique de données, il faudrait pousser plus loin le travail de modélisation en prenant en compte dans (1.1) des phénomènes biologiques plus fins. Ce type de travail de modélisation n'est pas fait ici. Je tiens à souligner que j'ai eu au cours de ma thèse un bref aperçu de ce type de travail de modélisation. En effet, lors de mon stage au CEMRACS, nous avons modélisé la dynamique de moustiques tigres (infectés ou non par la bactérie Wolbachia) en prenant en compte certains phénomènes propre à l'insecte comme l'incompréssibilité cytoplasmique. Nous renvoyons à la partie D pour une première présentation du modèle.

B.1.b Description du modèle

Nous étudions ici l'équation (1.1). Elle modélise la dynamique au cours du temps d'une densité de population biologique n sujette à une dispersion longue portée dans un domaine fragmenté Ω. Nous parlerons de domaine fragmenté lorsque le domaine Ω modélisant l'environnement de vie de l'espèce biologique est composé de plusieurs composantes connexes (patchs). Commme la densité de population est supposée nulle en dehors du domain Ω (i.e. dans Ω c ), nous imposons une condition de Dirichlet extérieur. Ceci est à mettre en opposition avec une dispersion locale. Dans le cas d'une diffusion locale, l'espèce peut coloniser seulement les composantes connexes où elle est présente initialement. L'hypothèse commune pour modéliser une dispersion longue portée avec une grande fréquence

Chapter 1 : Introduction est de considérer une équation intégro-différentielles dont le noyau de l'opérateur de diffusion K est à queue lourde. Le Laplacien fractionnaire, défini comme :

(-∆) α n(x) = C d,α R d n(x) -n(y) |x -y| d+2α dy avec C d,α = 4 α Γ( d 2 + α) π d 2 |Γ(-α)|
, rentre dans cette famille d'opérateur à queue lourde (l'intégrale est à prendre ici au sens de la valeur principale). Notons que le noyau est singulier à l'origine : la fréquence de dispersion proche de x est très grande. Du point de vue de l'espèce, les individus vont se déplacer de proche en proche et avoir des sauts en espace avec une grande fréquence. D'un point de vu stochastique, la dispersion suit un processus de type Lévy. Le terme de réaction dans l'équation (1.1) est de type Fisher-KPP. Ce type de terme de réaction sert à modéliser une croissance logistique. Par exemple, si f (x, n) = µ(x)n -n 2 alors µ(x)n est un terme de naissance et -n 2 est un terme de mort modélisant la compétition pour les ressources. Les premières études d'une équation de type Fisher-KPP (∂ t n -∆n = n -n 2 ) sont dues à Fisher [START_REF] Fisher | The wave of advance of advantageous genes[END_REF] et Kolmogorov, Petrovsky et Piskunov [START_REF] Kolmogorov | A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem[END_REF] en 1937.

B.2 Etat de l'art

Nous décomposons l'état de l'art en deux sous-parties : les articles traitant de l'existence et de l'unicité d'une solution stationnaire bornée à une équation de Fisher-KPP et ceux discutant des phénomènes d'invasion. Il est évident qu'un même article peut rentrer dans les deux catégories, néanmoins, il est pratique de distinguer les deux littératures afin de présenter les outils en jeu.

B.2.a Existence et unicité de la solution stationnaire de Fisher-KPP

Lors de la preuve de la convergence vers un front d'invasion, un des ingrédients essentiels de la preuve est l'existence et l'unicité de la solution stationnaire positive bornée. L'existence s'appuie souvent sur un argument spectral. En effet, il suffit de regarder le signe de la valeur propre principale λ 0 (Ω, µ) de l'opérateur -∆-µ dans Ω ou dans un sous domaine avec des conditions de Dirichlet aux bords (respectivement extérieur). Si λ 0 (Ω, µ) < 0 alors il existe une unique solution stationnaire positive et bornée (construite par une méthode de sous et sur-solutions [START_REF] Smoller | Shock waves and reaction-diffusion equations[END_REF]). Sinon pour λ 0 (Ω, µ) ≥ 0, l'unique solution stationnaire positive bornée est la solution triviale. Cette valeur propre principale existe si l'on a des hypothèses de compacité sur Ω ou de périodicité sur µ. A l'initiative de Berestycki, des travaux [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF], [START_REF] Berestycki | Asymptotic spreading for general heterogeneous fisher-kpp type equations[END_REF], [START_REF] Berestycki | Persistence criteria for populations with non-local dispersion[END_REF], [START_REF] Berestycki | Generalized principal eigenvalues for heterogeneous road-field systems[END_REF] définissent une généralisation de cette valeur propre principale lorsque celle-ci n'est pas bien définie comme dans le cas d'un domaine non borné. Cette nouvelle notion de valeur propre principale généralisée n'affecte pas le critère d'existence d'une solution stationnaire bornée non-triviale. Dans le cas d'une diffusion fractionnaire et d'un terme de naissance µ périodique, Berestycki, Roquejoffre et Rossi établissent l'existence d'une valeur propre principale dans [START_REF] Berestycki | The periodic patch model for population dynamics with fractional diffusion[END_REF]. Pour un opérateur de diffusion (locale ou fractionnaire), l'unicité de la solution stationnaire de (1.1) pour Ω = R d découle de l'application du principe du maximum / principe de comparaison (voir [START_REF] Berestycki | Le nombre de solutions de certains problèmes semi-linéaires elliptiques[END_REF]). Remarquons que pour un opérateur intégral du type le Laplacien fractionnaire, le principe du maximum est trivial tandis que le lemme de Hopf découle de la comparaison avec des fonctions barrières bien choisies (voir les travaux de Ros-Oton et Serra [START_REF] Ros-Oton | The Dirichlet problem for the fractional Laplacian : regularity up to the boundary[END_REF] ou de Grecco et Servadei [START_REF] Greco | Hopf's lemma and constrained radial symmetry for the fractional Laplacian[END_REF]). Enfin, pour des opérateurs intégraux dont le noyau est moins singulier, Berestycki, Coville et Vo [START_REF] Berestycki | Persistence criteria for populations with non-local dispersion[END_REF] ou Coville, Davilla et Martinez [START_REF] Coville | Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity[END_REF] ont montré l'existence et l'unicité d'une solution stationnaire d'une équation de type Fisher-KPP non-locale. Le critère d'existence est basé sur le signe d'une valeur propre principale (généralisée ou non) tandis que l'unicité est montrée par la continuité et l'intégrabilité du noyau de l'opérateur de diffusion.

Dans la suite nous parlerons de domaine Ω et de taux de naissance µ viables lorsque λ 0 (Ω, µ) < 0. Si Ω = R d (respecitvement µ = 1), on écrira alors λ 0 (Ω, µ) = λ 0 (µ) (respectivement λ 0 (Ω, µ) = λ 0 (Ω)).

B.2.b Phénomène d'invasion

Comme mentionné précédemment, l'équation (1.1) a été introduite par Fisher [START_REF] Fisher | The wave of advance of advantageous genes[END_REF] et Kolmogorov, Petrovsky et Piskunov [START_REF] Kolmogorov | A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem[END_REF] en 1937 dans le cas d'une diffusion standard (∂ t n -∆n = n -n 2 ) avec pour domaine Ω = R d . Kolmogorov, Petrovsky et Piskunov prouvent l'existence d'ondes planes progressives. Elles sont un type de solution particulière connectant la solution stationnaire stable 1 à la solution stationnaire instable 0. Ces dernières sont de la forme n(x, t) = N (x -ct) avec c ≥ c * = 2 f (0). En 1978, Aronson et Weinberger [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF] prouvent que n'importe quelle condition initiale positive à support compact engendre un phénomène d'invasion à vitesse constante c * . Cela peut se traduire ainsi : si un observateur se déplace le long d'une direction donnée à une vitesse constante c < c * alors, au bout d'un certain temps, il verra l'environnement saturé partout autour de lui. En revanche, si l'observateur se déplace à une vitesse c > c * alors, au bout d'un certain temps, il ne verra personne autour de lui. Aronson et Weinbergen prouvent leurs résultats en construisant des sous et sur-solutions adaptées à leurs problèmes. Dans [START_REF] Evans | A PDE approach to geometric optics for certain semilinear parabolic equations[END_REF], Evans et Souganidis retrouvent ce résultat en utilisant une approche provenant de l'optique géométrique. Ils exhibent naturellement les deux états, à savoir, saturation de l'environnement (n = 1) ou absence de population (n = 0) en introduisant un changement de variable qui suit le front

n ε (x, t) = n x ε , t ε .
En effet, n ε est solution de ∂ t n ε -ε∆n ε = n ε -n 2 ε . Afin d'obtenir des estimations sur n ε et passer à la limite en ε, Evans et Souganidis utilisent une transformation de Hopf-Cole en introduisant u ε = ε log(n ε ).

Ce type d'approche sera utilisé dans les chapitres 2 et 3 de la thèse. Il est à noter que le résultat d'invasion précédent est isotropique : aucune direction n'est privilégiée. Ceci est dû à l'isotropie des termes de naissance et de diffusion. Si on considère un terme de naissance périodique viable de la forme µ(x)n (qui privilégie certaines directions plutôt que d'autres), alors Freidlin et Gartner ont montré dans [START_REF] Gärtner | The propagation of concentration waves in periodic and random media[END_REF] qu'une condition initiale positive à support compact entraîne un phénomène d'invasion avec une vitesse d'invasion c * (e) qui dépend de la direction choisie e ∈ S d-1 . La preuve de l'invasion de Freidlin et Gartner est basée sur une approche probabiliste. Rossi propose dans [START_REF] Rossi | The Freidlin-Gärtner formula for general reaction terms[END_REF] une preuve déterministe de ce dernier résultat (voir aussi [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF]). La Figure 1.1 est une illustration du phénomène d'invasion dans le cas d'un environement homogène (à gauche) ou périodique (à droite). Les phénomènes d'accélération du front de propagation dans des équations Fisher-KPP fractionnaires ont d'abord été observés numériquement (voir [START_REF] Del-Castillo-Negrete | Front propagation and segregation in a reactiondiffusion model with cross-diffusion[END_REF], [START_REF] Mancinelli | Front propagation in reactive systems with anomalous diffusion[END_REF] et [START_REF] Hastings | The spatial spread of invasions : new developments in theory and evidence[END_REF]). En effet, ces investigations numériques laissent penser que la vitesse d'invasion est exponentielle en temps. Ce résultat a été démontré par Cabré et Roquejoffre [START_REF] Cabré | The influence of fractional diffusion in Fisher-KPP equations[END_REF] Ce résultat est retrouvé par une approche d'optique géométrique par Mirrahimi et Méléard [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF]. L'idée est d'introduire un changement de variable qui suit le front n ε (x, t) = n(|x| 1 ε -1 x, t ε ) et ensuite de montrer la convergence de la suite à l'aide d'une transformation de Hopf-Cole. Ce type de phénomène d'accélération a aussi été retrouvé dans le cadre d'opérateurs à queues lourdes réguliers par Garnier [START_REF] Garnier | Accelerating solutions in integro-differential equations[END_REF]. Ce premier résultat d'invasion est établi dans un milieu homogène. Il a été étendu aux milieux hétérogènes périodiques (i.e. aux termes de réaction de la forme f (x, n) = µ(x)n -n 2 ) par Cabré, Coulon et Roquejoffre [START_REF] Cabré | Propagation in Fisher-KPP type equations with fractional diffusion in periodic media[END_REF]. Sous l'hypothèse que µ est un taux de naissance viable (i.e. λ 0 (µ) < 0), il existe une unique solution stationnaire bornée n + et il s'ensuit Remarquons que contrairement au cas d'une diffusion locale, la diffusion non-locale induit toujours une propagation isotropique même dans un environnement hétérogène. Soulignons enfin que [START_REF] Cabré | Propagation in Fisher-KPP type equations with fractional diffusion in periodic media[END_REF] donne une description précise des lignes de niveaux en temps longs.

Nous clôturons l'état de l'art sur les phénomènes d'invasion avec plusieurs remarques. Premièrement, pour une équation avec une diffusion locale (∂ t n-∆n = n-n 2 ), en partant d'une donnée initiale ayant des queues lourdes (i.e. n 0 (x) ∼ 1 |x| d+2α ), il a été démontré par Hamel et Roques [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF] que la vitesse d'invasion est exponentielle en temps. De façon heuristique, lorsque la donnée initiale est à support compact, nous verrons que le principal rôle de la diffusion fractionnaire est « au début ». En effet, contrairement à la diffusion locale qui donne une solution en temps court avec des queues exponentielles (i.e. n(x, t = 1) ∼ e -c|x| 2 ), le Laplacien fractionnaire entraine une solution en temps court à queue lourde (i.e. n(x, t = 1) ∼ 1 |x| d+2α ). Ces queues lourdes amènent ensuite à une propagation exponentielle en temps. Deuxièmement, d'autres questions plus complexes peuvent être considérées à partir de l'équation de Fisher-KPP comme -l'équation de Fisher-KPP avec une ligne de diffusion rapide ( [START_REF] Berestycki | The influence of a line with fast diffusion on Fisher-KPP propagation[END_REF], [START_REF] Berestycki | The effect of a line with nonlocal diffusion on Fisher-KPP propagation[END_REF], [START_REF] Pauthier | The influence of nonlocal exchange terms on Fisher-KPP propagation driven by a line of fast diffusion[END_REF]), -la charactérisation du second membre d'un développement asymptotique du front d'invasion (voir Barmson [START_REF] Bramson | Convergence of solutions of the Kolmogorov equation to travelling waves[END_REF] ou Hamel, Nolen, Roquejoffre, Ryznik [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF], [START_REF] Hamel | The logarithmic delay of KPP fronts in a periodic medium[END_REF] et [START_REF] Nolen | Convergence to a single wave in the Fisher-KPP equation[END_REF]),

-la stabilité du front d'invasion (voir Sattinger [START_REF] Sattinger | On the stability of waves of nonlinear parabolic systems[END_REF], Gallay [START_REF] Gallay | Local stability of critical fronts in nonlinear parabolic partial differential equations[END_REF] ou Faye et Holzer [START_REF] Faye | Asymptotic stability of the critical Fisher-KPP front using pointwise estimates[END_REF]).

Enfin nous avons dans cet état de l'art seulement mentionné les équations de réaction-diffusion (fractionnaire ou non) de type Fisher-KPP. Il existe bien entendu des équations de réaction diffusion moins simplistes comme les équations de type combustions, bistables... Nous renvoyons les lecteurs intéréssés aux livres de Smoller [START_REF] Smoller | Shock waves and reaction-diffusion equations[END_REF] et de Berestycki et Hamel (à paraître).

Les phénomènes d'invasion pour ce type d'équation existent mais sont plus complexes à étudier.

B.3 Présentations des résultats pour l'équation de Fisher-KPP fractionnaire

Les principaux résultats concernant l'équation de Fisher-KPP fractionnaire sont introduits ici. Ces résultats sont regroupés par chapitre de thèse.

B.3.a Chapitre 2-Equation de Fisher-KPP fractionnaire dans un environnement périodique

Le premier résultat de ce manuscrit introduit dans le chapitre 2 est un résultat d'invasion Ce résultat (tel qu'énoncé dans le manuscrit) est étendu à des opérateurs intégraux singuliers plus généraux que le Laplacien fractionnaire. De même, le terme de réaction est de type KPP périodique plus général que µ(x)n -n 2 . Nous pouvons décomposer la preuve de ce résultat en 4 grandes étapes :

A-S'assurer de l'existence et de l'unicité d'une solution stationnaire de (1.1).

Chapter 1 : Introduction B-Donner une estimation de la solution au temps t = 1.

C-Trouver une sous-solution qui va « pousser » la solution jusqu'à n + dans {|x| < e ct } ainsi qu'une sur-solution qui va « écraser » la solution proche de 0 dans |x| > e Ct . D-Conclure.

L'intérêt du Théorème 1.1 réside principalement dans la preuve des étapes C et D. En effet, le Théorème 1.1 est une extension du résultat de Cabré, Coulon et Roquejoffre [START_REF] Cabré | Propagation in Fisher-KPP type equations with fractional diffusion in periodic media[END_REF] à des opérateurs plus généraux via l'approche développée par Mirrahimi et Méléard [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF] pour le Laplacien fractionnaire. La preuve du Théorème 1.1 montre la robustesse des techniques de solutions de viscosités employées pour l'étude d'équations générales de type Fisher-KPP fractionnaires. L'étape A (existence et unicité d'une solution stationnaire) s'appuie sur le signe de la valeur propre principale et le principe du maximum. Tandis que l'étape B (estimation de la solution en temps cours) repose sur une estimation du noyau de la chaleur fractionnaire (comme énoncé dans Bogdan [START_REF] Bogdan | Heat kernel estimates for the fractional laplacian with Dirichlet conditions[END_REF] ou Chen et Kumagai [START_REF] Chen | Heat kernel estimates for stable-like processes on d-sets[END_REF]). L'étape C peut être abordée via une approche issue de l'optique géométrique (comme Evans et Souganidis [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF]). L'idée générale est d'effectuer un changement de variable qui suit le front d'invasion. Ainsi, nous considérerons l'équation satisfaite par n ε (x, t) = n(|x| 1 ε -1 x, t ε ) (à l'image du changement de variable effectué par Méléard et Mirrahimi dans [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF]). Après ce changement de variable, une sous (respectivement sur)-solution de ce problème peut être trouvée aisément en considérant φ 0 × G où G est une solution d'une équation différentielle ordinaire perturbée et φ 0 une fonction propre principale associée à l'opérateur (-∆) αµ. En effet, lorsque ε → 0, les termes de diffusions fractionnaires tendent vers 0 laissant formellement une simple équation différentielle ordinaire. Une autre méthode développée par Cabré, Coulon et Roquejoffre (voir [START_REF] Cabré | The influence of fractional diffusion in Fisher-KPP equations[END_REF] et [START_REF] Cabré | Propagation in Fisher-KPP type equations with fractional diffusion in periodic media[END_REF]) aurait été de considérer l'équation satisfaite par n φ 0 (r(t)•) où r(t) décroit exponentiellement vite. L'étape D peut aussi être abordée de deux façons différentes. Cabré Coulon Roquejoffre ( [START_REF] Cabré | Propagation in Fisher-KPP type equations with fractional diffusion in periodic media[END_REF]) propose une preuve directe. La preuve proposée dans ce manuscrit s'appuie sur des techniques provenant de la théorie des solutions de viscosité et des fonctions tests perturbées. Il est à noter que cette approche a été développée en même temps par Souganidis et Tarfulea [START_REF] Souganidis | Front propagation for non-local KPP reaction diffusion equations in periodic media[END_REF] et Bouin, Garnier, Henderson et Patout [START_REF] Bouin | Thin front limit of an integro-differential Fisher-KPP equation with fat-tailed kernels[END_REF]. Souganidis et Tarfulea prouvent un résultat de propagation pour une famille d'opérateurs intégraux singuliers semblables au Laplacien fractionnaire. Cette étude est sensiblement proche du travail présenté. Le travail de Bouin, Garnier, Henderson et Patout [START_REF] Bouin | Thin front limit of an integro-differential Fisher-KPP equation with fat-tailed kernels[END_REF] montrent la robustesse des techniques sur une famille d'opérateurs intégraux à queues lourdes non-singuliers dans un environnement homogène.

Soulignons que le résultat précédent est établi pour Ω = R d : l'environnement considéré n'est pas fragmenté. Toutefois, la périodicité de µ permet formellement de diviser l'environnement en deux : un domaine où l'environnement est favorable à l'espèce Ω -= x ∈ R d | µ(x) ≥ 0 et un domaine défavorable Ω + = x ∈ R d | µ(x) < 0 (les signes en indice de Ω + et Ω -renvoient aux signes de la valeur propre principale associée à chacun des domaines : λ(Ω + ) > 0 et λ(Ω -) < 0). Si on considère une suite de taux de naissance µ k telle que µ k (x) -→ k→+∞ -∞ lorsque x ∈ Ω + , alors l'environnement défavorable à l'espèce devient létal lorsque k → +∞. Ainsi, si l'ensemble Ω -est fragmenté (i.e. composé de composantes connexes bornées), il s'ensuit formellement qu'à la limite k → +∞, l'espèce vit dans l'environnement fragmenté Ω -avec des conditions de Dirichlet dans Ω + . L'étude de phénomènes d'invasions dans un environnement périodique fragmenté Ω est réalisée dans le Chapitre 3.

B.3.b Chapitre 3-Equation de Fisher-KPP fractionnaire dans un environnement périodique fragmenté

Nous dirons qu'un environnement est périodique et fragmenté s'il existe Ω 0 ⊂ R d borné vérifiant l'hypothèse de la boule intérieure et extérieure uniforme tel que

Ω = k∈Z d (Ω 0 + a k ), avec a k une suite périodique de R d telle que (Ω 0 + a i ) ∩ (Ω 0 + a j ) = ∅ ⇐⇒ i = j.
Nous rappelons que la condition de la boule intérieure (respectivement extérieure) uniforme réfère à l'existence d'un rayon r 0 > 0 tel que

∀x ∈ Ω, ∃z x ∈ Ω tel que B(z x , r 0 ) ⊂ Ω (respectivement ∀x ∈ Ω c , ∃z x ∈ Ω c tel que B(z x , r 0 ) ⊂ Ω c ).
Le résultat d'invasion est alors le suivant : Le schéma de preuve du Théorème 1.2 suit la progression A-B-C-D énoncée plus haut. Toutefois de nouvelles difficultés apparaissent du fait que la solution n et la solution stationnaire n + s'annulent dans Ω c . De ce fait, la preuve d'unicité de la solution stationnaire ne repose pas seulement sur le principe du maximum. Elle s'appuie également sur un Lemme de Hopf fractionnaire (voir [START_REF] Greco | Hopf's lemma and constrained radial symmetry for the fractional Laplacian[END_REF]) et une estimation précise par au-dessus et en-dessous de n + . Cette estimation est la suivante

cδ(x) α ≤ n + (x) ≤ Cδ(x) α avec c, C > 0 et δ(x) = dist(x, ∂Ω)1 Ω (x). (1.4)
L'inégalité (1.4) est obtenue via le principe de comparaison et fonctions barrières développées par [START_REF] Ros-Oton | The Dirichlet problem for the fractional Laplacian : regularity up to the boundary[END_REF].

Ceci permet de conclure à la preuve de l'étape A du plan de démonstration annoncée à la suite du Théorème 1.1. L'étape B pourrait être résolue en appliquant des résultats généraux sur les estimations du noyau de la chaleur du Laplacien fractionnaire développés par Bogdan, Grywny et Ryznar [START_REF] Bogdan | The boundary Harnack principle for the fractional Laplacian[END_REF] ou Chen, Kim et Song [START_REF] Chen | Heat kernel estimates for the Dirichlet fractional Laplacian[END_REF]. Malgré cela, nous développons notre propre estimation du noyau de la chaleur. Cette dernière est moins générale que celles énoncées dans [START_REF] Bogdan | The boundary Harnack principle for the fractional Laplacian[END_REF] et [START_REF] Chen | Heat kernel estimates for the Dirichlet fractional Laplacian[END_REF]. En effet, elle ne s'applique uniquement en temps courts pour des données initiales à support compact. Toutefois, la preuve est relativement courte et s'appuie uniquement sur un raisonnement déterministe (contre un raisonnement probabiliste dans [START_REF] Bogdan | The boundary Harnack principle for the fractional Laplacian[END_REF] et [START_REF] Chen | Heat kernel estimates for the Dirichlet fractional Laplacian[END_REF]).

Chapter 1 : Introduction

La preuve du point C peut encore une fois être développée de deux manières différentes : via une approche inspirée par [START_REF] Cabré | Propagation in Fisher-KPP type equations with fractional diffusion in periodic media[END_REF] ou par [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF]. Nous avons choisi d'utiliser la seconde approche pour les mêmes raisons évoquées précédemment. Cependant, nous avons amélioré la précision des sous et sur-solutions afin d'obtenir une estimation précise des lignes de niveaux. Nous proposons une preuve pour l'étape D qui est une adaptation de [START_REF] Cabré | Propagation in Fisher-KPP type equations with fractional diffusion in periodic media[END_REF] aux nouvelles difficultés qu'engendrent la présence d'un domaine hostile. Notons que dans cette preuve, le changement de variable (x, t) → |x| 1 ε -1 , t ε n'intervient pas. La deuxième preuve est une preuve où le changement de variable est préservé. Pour chacune d'entre elles, le fait que n et n + s'annulent sur Ω c entraîne de nouvelles difficultés par rapport à ce qui a été fait dans les travaux antérieurs ( [START_REF] Cabré | Propagation in Fisher-KPP type equations with fractional diffusion in periodic media[END_REF], [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF] ou [START_REF] Léculier | A singular limit in a fractional reaction-diffusion equation with periodic coefficients[END_REF]).

B.3.c Chapitre 4-Equation de Fisher-KPP fractionnaire dans un environnement fragmenté général

Le troisième résultat présenté dans cette introduction est un résultat d'existence et d'unicité de solution stationnaire bornée de l'équation (1.1) pour des domaines Ω généraux inclus dans

R d et f (x, n) = n -n 2 .
Ces problèmes sont abordés dans le chapitre 4 de ce manuscrit. Le résultat d'existence repose sur l'existence d'un sous-domaine Ω ⊂ Ω borné viable (i.e. de valeur propre principale λ 0 (Ω) négative). Si un tel domaine existe, il est facile de donner une sous-solution (une fonction propre positive associée à la valeur propre λ 0 (Ω)) et une sursolution (une constante suffisamment grande). A partir de ces sous-solutions et sur-solutions, la construction d'une solution stationnaire non-triviale s'ensuit (voir [START_REF] Smoller | Shock waves and reaction-diffusion equations[END_REF]). Il est donc important de savoir donner des critères donnant le signe de λ 0 (Ω). Il est bien connu que pour un domaine connexe, la valeur propre du domaine est décroissante par rapport à la taille du domaine (i.e. λ 0 (tΩ) décroît lorsque t grandit). Cependant, comme le Laplacien fractionnaire est un opérateur non-local, la valeur propre principale est aussi définie pour des domaines non-connexes. Le premier résultat du Chapitre 4 répond aux questions suivantes pour des intervalles bornés dans R : Si l'on considère a > 0 tel que le segment ]0, a[ est viable et le segment ]0, a 2 [ est nonviable, est ce qu'il existe une distance d 0 > 0 tel que pour toute distance

d < d 0 , l'ensemble ]0, a 2 [∪] a 2 + µ, a + µ[ soit viable ? En d'autres termes, est ce que l'application (µ ∈ R + → λ α (]0, a 2 [∪] a 2 + µ, a + µ[
) est continue en 0 ? Le Théorème suivant répond à ces deux questions : La preuve en dehors de 0 repose essentiellement sur le quotient de Rayleigh. La preuve en 0 pour α < 1 2 repose également sur un quotient de Rayleigh et sur la densité des fonctions de H α (]0, a[) qui vérifient φ( a 2 ) = 0. En revanche pour α ≥ 1 2 , la preuve revient à montrer que a 2 est une singularité effaçable de (-∆) α -Id dans ]0, a[\ a 2 . Pour cela, nous utilisons le relèvement du Laplacien fractionnaire au demi plan supérieur introduit par Caffareli et Silvestre [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF]. Remarquons que l'importance de la dimension 1, ne se retrouve que pour montrer la continuité en µ = 0. Ainsi, nous généralisons certains cas du théorème précédent :

Théorème 1.3 (L.-Roquejoffre (Chapitre 4) ). L'application (µ ∈ R + * → λ α (]0, a 2 [∪] a 2 + µ, a + µ[) est
Ω µ,e = Ω 1 ∪ (Ω 2 + t µ e) avec (t µ , e) ∈ R + × S d-1 tel que dist(Ω 1 , (Ω 2 + t µ e)) = µ.
Alors l'application (µ ∈ R + * → λ α (Ω µ,e )) est continue et croissante sur tout son ensemble de définition.

Une conséquence du Théorème 1.4 est que la distance entre les différentes composantes connexes de Ω joue un rôle primordial dans l'établissement d'un sous-domaine Ω viable.

Le résultat d'unicité (établi dans R d ) est plus complexe à démontrer que celui d'existence. Il s'appuie sur l'encadrement de n'importe quelle solution stationnaire bornée non-trivial par en-dessous et au-dessus par la même fonction G (à une multiplication par une constante près)

i.e. ∃ G : Ω → R + et c, C > 0, tel que c G(x) ≤ n + (x) ≤ C G(x).
(1.5)

Par exemple, si Ω = R d alors G = 1. Si Ω est périodique et fragmenté, l'inégalité (1.4) donne G(x) = δ(x) α .
Le but de ce chapitre est de démontrer l'assertion (1. Par domaine viable, on entend qu'en tout point x ∈ Ω -, il existe un sous domaine Ω -tel que

λ α (Ω -) < 0 et x ∈ Ω -⊂ Ω.
De plus, ces sous-ensembles viables Ω -sont de nombres finis (à translation près). Par domaine non-viable, on entend qu'il existe λ 0 > 0, tel que pour tout point

z ∈ R d et pour tout rayon R > 0, on a λ α (Ω + ∩ B(z, R)) > λ 0 .
Insistons sur le fait que le signe en indice de Ω -renvoie au signe de la valeur propre principale d'un sous ensemble viable inclu dans Ω -. De même, le signe de Ω + renvoie au signe de la valeur propre principale de

λ α (Ω + ∩ B(z, R)) pour tout centre z ∈ R d et R positif.
G(x) = min(δ(x) α , 1) × 1 Ω -(x) + Ω - 1 Ω + (x) |x -y| d+2α dy alors pour toute solution stationnaire bornée non-triviale n + il existe c, C > 0 tel que c G(x) ≤ n + (x) ≤ C G(x).
Conséquence : Il existe une unique solution stationnaire bornée non-triviale de l'équation (1.1).

Soulignons que le Théorème 1.5 implique que selon le choix de Ω, le comportement asymptotique de n + peut varier complètement. Nous présentons dans cette introduction trois exemples où les comportements asymptotiques diffèrent. Fixons la dimension

d = 1. Si Ω est pério- dique fragmenté alors G(x) = δ(x) α . Si Ω -est borné et Ω + non borné alors n + décroit comme δ(x) α 1+|x| 1+2α . Si Ω -= R -et Ω + est non borné alors n + décroît comme δ(x) α 1+max(0,x) 2α .
Remarquons également que Ω -et Ω + ne sont pas fixés. Sous les hypothèses plus générales énoncées dans le chapitre 4, nous pouvons toujours définir de nouveaux ensembles Ω + et Ω - différents de Ω + et Ω -mais vérifiant toujours les hypothèses requises par le Theorème 1.5. Par exemple, si Ω + est borné alors on peut définir Ω -= Ω et Ω + = ∅. Enfin, pour des domaines comportant des composantes non-viables (i.e. Ω + ne peut pas être réduit à l'ensemble vide), nous faisons la distinction entre invasion et colonisation. Si Ω + ne peut pas être réduit à l'ensemble vide, alors l'espèce colonise le territoire qui lui est disponible mais il n'y a pas de phénomènes d'invasion. En effet, on ne peut pas éviter un phénomène de type source-puit.

B.4 Retour sur les motivations biologiques

Au vue des résultats mathématiques présentés, nous répondons ici aux questions biologiques ennoncées dans la section B.1.a. Si l'espèce biologique considérée rentre dans les hypothèses de modélisation, alors un critère de survie de l'espèce est donné par le signe de la valeur propre principale λ α (O) où O est un sous-domaine bien choisie de Ω. De plus, sous certaines hypothèses sur l'environnement, nous pouvons prédire la distribution en temps grand de l'espèce en fonction de l'environnement considéré. Enfin, lorsque la structure de l'environnement est particulière (par exemple, un environnement périodique fragmenté), alors, nous pouvons parler d'invasion avec une vitesse d'invasion exponnentielle en temps.

C. Un problème d'évolution Darwinienne dans un environnement spatial fragmenté prenant en compte une dispersion non-locale.

Rappelons que l'équation considérée est :

           -∂ xx n ε -ε 2 ∂ θθ n ε + Ln ε = n ε [R(x, θ) -ρ ε ] pour x ∈ Ω, -A < θ < A, ρ ε (x) = A -A n ε (x, θ)dθ pour x ∈ Ω, ∂ νx n ε = ∂ ν θ n ε = 0 pour (x, θ) ∈ ∂ (Ω c ×] -A, A[) (1.3) avec Ω ⊂ R.

C.1. Description du modèle et motivations biologiques

L(n)(x) = Ω [n(x) -n(y)]K(x -y)dy avec K(x) = K(-x), 0 < c < K < C et régulier.
Un tel opérateur non-local, nous permet de considérer des environnements fragmentés. Toutefois, nous restreindrons l'étude à un domaine de R, borné et comprenant un nombre fini de composantes connexes :

i.e. Ω = m i=1 ]a i , b i [ avec a 1 < b 1 < a 2 < ... < a n < b n .
Le choix de décomposer la modélisation de la diffusion spatiale en deux opérateurs, à savoir un local et un non-local, plutôt qu'un opérateur intégral singulier prenant en compte les deux dynamiques (comme le Laplacien fractionnaire) est essentiellement technique. Nous reviendrons sur ce point dans les perspectives de la thèse.

R(x, θ) = r -g(bx -θ) 2 .
Le terme r est un taux de crossance maximale. Le trait optimal à la position x est donné par

C.2. Etat de l'art

De nombreux travaux étudient des modèles de sélection-mutation en utilisant des équations intégro-différentiels (voir par exemple [START_REF] Magal | Mutation, selection, and recombination in a model of phenotype evolution[END_REF], [START_REF] Calsina | Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics[END_REF] ou [START_REF] Desvillettes | On selection dynamics for continuous structured populations[END_REF]). Ici nous nous intéressons à une approche basée sur des équations de Hamilton-Jacobi initiée en 2005 par Dieckmann, Jabin, Mishler et Perthame [START_REF] Diekmann | The dynamics of adaptation : An illuminating example and a Hamilton-Jacobi approach[END_REF]. Les auteurs utilisent une approche via les équations de Hamilton-Jacobi pour étudier la dynamique des solutions d'équations de réaction-diffusion modélisant une population biologique dans un régime de petites mutations. Grâce à cette méthode, Barles et Perthame obtiennent des résultats de concentration pour une équation similaire à (1.3) ne faisant pas intervenir de structure spatiale. En effet, ils démontrent que sous certaines hypothèses sur les coefficients, à mesure que les mutations deviennent petites (i.e. ε → 0), la solution n ε (θ) converge vers une masse de Dirac δ θ . En d'autres termes, lorsque les mutations deviennent petites, la population tend vers une population monomorphe. Ce résultat a été étendu par Barles, Mirrahimi et Perthame [START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations : a general convergence result[END_REF] à des hypothèses plus générales. Cette approche via des équations de Hamilton-Jacobi pour démontrer l'emergence de traits domainants à mesure que les mutations deviennent petites est robuste. En effet, elle a été adaptée à de multiples situations comme par exemple : l'étude d'une population évoluant dans un environnement changeant de façon périodique au cours du temps (tel qu'un cycle jours/nuit) ou linéaire (changement climatique) par Figueroa Iglesias et Mirrahimi [START_REF] Figueroa Iglesias | Long time evolutionary dynamics of phenotypically structured populations in time-periodic environments[END_REF] et [START_REF] Figueroa Iglesias | Selection and mutation in a shifting and fluctuating environment[END_REF], l'étude d'une population évoluant dans un environnement organisé spatialement de façon discrète (présence de deux habitats avec des termes de migrations entre les habitats) par Mirrahimi [88] et Mirrahimi et Gandon [START_REF] Mirrahimi | Evolution of specialization in heterogeneous environments : Equilibrium between selection, mutation and migration[END_REF] ou l'étude d'une population évoluant dans un environnement spatiale continu et convexe par Perthame et Souganidis [START_REF] Perthame | Rare mutations limit of a steady state dispersal evolution model[END_REF]. Les travaux présentés dans cette partie sont dans la continuité de [88] et [START_REF] Perthame | Rare mutations limit of a steady state dispersal evolution model[END_REF]. Dans la suite, nous présentons les premiers travaux sur les modèles de sélection-mutation prenant en compte un environement spatial hétérogène avant de revenir plus précisément sur les résultats de [88] et [START_REF] Perthame | Rare mutations limit of a steady state dispersal evolution model[END_REF].

Un premier modèle prenant en compte une diffusion locale en espace et en trait a d'abord été développé par Champagnat et Méléard dans [START_REF] Champagnat | Invasion and adaptive evolution for individual-based spatially structured populations[END_REF]. Dans cet article, les auteurs dérivent d'un modèle stochastique une première équation de réaction-diffusion semblable à (1.3). Arnold, Desvilettes et Prévost [START_REF] Arnold | Existence of nontrivial steady states for populations structured with respect to space and a continuous trait[END_REF] démontrent l'existence d'une solution stationnaire non-triviale à un système de réaction-diffusion similaire à celui développé par Champagnat et Méléard [START_REF] Champagnat | Invasion and adaptive evolution for individual-based spatially structured populations[END_REF] via un théorème de point fixe de Schaudder. Dans [88], l'auteur considère une population pouvant vivre dans deux habitats hétérogènes avec un terme d'échange entre deux habitats. En fonction du terme de reproduction dans chacun des habitats, l'auteur met en évidence la possible apparition du polymorphisme dans la population (émergence de deux traits dominants) à mesure que les mutations deviennent petites. Souganidis et Perthame dans [START_REF] Perthame | Rare mutations limit of a steady state dispersal evolution model[END_REF] considèrent un environement spatial convexe continu. Ils mettent en évidence l'emergence d'un trait dominant lorsque les mutations deviennent petites. Le même type de résultat (i.e. apparititon d'un trait dominant pour une espèce organisée spatialement et phénotypiquement lorsque les mutations deviennent petites) a été établi en même temps que [START_REF] Perthame | Rare mutations limit of a steady state dispersal evolution model[END_REF] par Lam et Lou dans [START_REF] Lam | An integro-PDE model for evolution of random dispersal[END_REF]. Cependant, ce dernier article n'adopte pas une approche via des équations de Hamilton-Jacobi.

De multiples autres questions peuvent être abordées par l'étude d'équations de réactiondiffusion modélisant une espèce biologique dépendant d'une variable spatiale x et d'une autre variable phénotypique θ. Nous présentons ici brièvement le type de questions qui ont été traitées. L'existence de front d'invasion a été montrée par Alfaro, Coville et Raoul [START_REF] Alfaro | Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait[END_REF] et par Berestycki, Jin et Silvestre [START_REF] Berestycki | Propagation in a non local reaction diffusion equation with spatial and genetic trait structure[END_REF]. L'étude de persistance dans un environnement changeant a aussi été étudié par Alfaro, Berestycki et Raoul [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF] pour un environnement sujet à un changement climatique linéaire au cours du temps. Citons également les traveaux traitant de l'invasion du crapaud buffle en Australie ( [START_REF] Turanova | On a model of a population with variable motility[END_REF], [START_REF] Bouin | Travelling waves for the cane toads equation with bounded traits[END_REF], [START_REF] Berestycki | Analysis of the periodically fragmented environment model : I-Species persistence[END_REF] ou encore [START_REF] Bouin | The Bramson logarithmic delay in the cane toads equations[END_REF]). Des études biolo- Il est également possible de modéliser des mutations par un noyau intégral, pour prendre en compte des sauts non-locaux (voir par exemple [START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations : a general convergence result[END_REF] ou [START_REF] Desvillettes | On selection dynamics for continuous structured populations[END_REF] qui étudient de tels modèles de sélection-mutation dans des environnements homogènes). Champagnat et Méléard toujours dans [START_REF] Champagnat | Invasion and adaptive evolution for individual-based spatially structured populations[END_REF] dérivent également un modèle impliquant un terme intégral pour modéliser les mutations. Ce second modèle a aussi fait l'objet de plusieurs travaux : Arnold, Desvillettes et Prévost dans [START_REF] Arnold | Existence of nontrivial steady states for populations structured with respect to space and a continuous trait[END_REF] ont démontré l'existence d'une solution stationnaire pour un tel modèle. Lorsque le terme de mutation est modélisé par un Laplacien fractionnaire, l'étude de l'emergence de traits dominants par une approche de Hamilton-Jacobi avec contraintes est le sujet de [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF] par Méléard et Mirrahimi et de [START_REF] Mirrahimi | Singular limits for models of selection and mutations with heavy-tailed mutation distribution[END_REF] par Mirrahimi. Notons que Berestycki, Jin et Silveststre [START_REF] Berestycki | Propagation in a non local reaction diffusion equation with spatial and genetic trait structure[END_REF] considèrent aussi un modèle impliquant le Laplacien fractionnaire pour modéliser les mutations.

C.3 Présentation des résultats

L'étude porte sur le comportement asymptotique de (1.3) lorsque ε → 0. Nous cherchons à démontrer l'existence de trait(s) dominant(s) : nous nous attendons au fait que n ε converge au sens de la mesure vers une somme masses de de Dirac en θ i.e.

n ε (x, θ) -→ ε→0 K i=1 N i (x)δ θ i (θ)
   -|∂ θ u(θ)| 2 = -λ(θ, ρ), max θ∈[-A,A] u(θ) = 0, (1.7) avec λ(θ, ρ) la valeur propre principale associée à l'opérateur -∂ xx + L -[R(•, θ) -ρ]
avec des conditions aux bords de Neumann.

Ω × {θ ∈] -A, A[ | u(θ) = 0}.
Pour montrer la convergence de u ε , nous utilisons le fait que u ε est solution de

                         - 1 ε ∂ xx u ε - |∂ x u ε | 2 ε 2 -ε∂ θθ u ε -|∂ θ u ε | 2 + Ω 1 -e uε(y)-uε(x) ε K(x -y)dy = (R(x, θ) -ρ ε (x)) dans Ω×] -A, A[, ρ ε (x) = ]-A,A[ n ε (x, θ)dθ dans ] -A, A[, ∂ νx u ε (x, θ) = 0 sur ∂Ω×] -A, A[, ∂ ν θ u ε (x, ±A) = 0.
(1.8) Premièrement, donnons un argument heuristique qui permet de retrouver (1.7). Faisons un développement asymptotique de u ε et ρ ε :

u ε (x, θ) = u 0 (x, θ) + εu 1 (x, θ) + o(ε) et ρ ε (x) = ρ 0 (x) + o ε (1).
Si nous remplaçons u ε et ρ ε par de tels développement asymptotique dans (1.8), il s'ensuit

1 ε -∂ xx u 0 - |∂ x u 0 | ε -2|∂ x u 0 ∂ x u 1 | + Ω [1 -e u 0 (y,θ)-u 0 (x,θ) ε +u 1 (y,θ)-u 1 (x,θ)+oε(1) ]K(x -y)dy -|∂ θ u 0 | 2 -∂ xx u 1 -|∂ x u 1 | 2 -[R -ρ 0 ] + o ε (1) = 0.
Cette équation suggère que u 0 (x, θ) = u 0 (θ). De plus, en passant formellement à la limite ε → 0, nous trouvons

-|∂ θ u 0 (θ)| 2 = [R(x, θ) -ρ 0 (x)] + ∂ xx u 1 (x, θ) + |∂ x u 1 (x, θ)| 2 - ω [1 -e u 1 (y,θ)-u 1 (x,θ) ]K(x -y)dy.
(1.9) Or comme la partie gauche de l'équation (1.9) ne dépend pas de x, cela laisse à penser que u 1 = log(ψ θ ) avec ψ θ la fonction propre principale (prise positive) de l'opérateur -∂ xx -L -[R(•, θ) -ρ] et des conditions aux bords de Neumann. Si cette affirmation est vraie, il s'ensuit

-|∂ θ u 0 (θ)| 2 = [R(x, θ) -ρ 0 (x)]ψ θ (x) + ∂ xx ψ θ (x) -L(ψ θ )(x) (ψ θ (x)) -1 .
Chapter 1 : Introduction Nous pouvons alors formellement conclure que la limite u 0 est solution de l'équation de Hamilton-Jacobi (1.7). La preuve détaillée du Théorème 1.6 s'appuie principalement sur une inégalité de type Harnack et des estimations Lipschitz de u ε . Cette inégalité et ces estimations sont introduites de façon précise dans le Chapitre 5 de la thèse.

D. Présentation du projet de CEMRACS

Durant l'été de ma première année de thèse, j'ai participé au CEMRACS dont le thème était « Modèles numériques et mathématiques pour des applications médicales : descriptions déterministes, probabilistes et statistiques ». Le but était de réaliser une première étude numérique sur le contrôle d'une population de Aedes aegypti (moustique tigre) en prenant en compte une dispersion spatiale. L'enjeu est d'ordre sanitaire : cette espèce de moustique est le principal vecteur de la Dengue. La motivation biologique est de relâcher une population de moustiques « transformée » par une bactérie tueuse de virus : la bactérie Wolbachia. En effet, les moustiques infectés par Wolbachia ne transmettent plus la Dengue. De plus, la reproduction entre des moustiques mâles (infectés ou non) avec des femelles infectées entraînent une descendance infectée, tandis que la proportion d'oeufs issus de la reproduction entre mâles infectés et femelles saines a une faible probabilité d'éclore (1 -s H avec s H ∈]0, 1]). En partant de ce constat, le but est de remplacer la population de moustiques résidente par une population de moustique infectée par Wolbachia. L'équation considérée est la suivante 2 dx, (i.e. p(x, T ) = 1 signifie que la population de moustique au temps T est composée seulement de moustiques infectés) avec u tel que

       ∂ t p -∆p = f (p) -ug(p) avec x ∈ Ω, t ∈]0, T [, ∂ ν p = 0 avec x ∈ ∂Ω, p(x, 0) = 0, (1.10) où Ω est un convexe borné de R 2 et f (p) = p(1 -p) d 1 b 2 -b 2 d 1 (1 -s H p) b 1 (1 -p)(1 -s H p) + b 2 p et g(p) = 1 κ b 1 (1 -p)(1 -s H p) b 1 (1 -p)(1 -s H p) +
J(u) = Ω (1 -p(x, T ))
u ∈ L ∞ (Ω × [0, T ]), 0 ≤ u ≤ M,

Introduction

The equation.

We are interested in the following equation :

   ∂ t n(x, t) + L α (n)(x, t) = µ(x)n(x, t) -n(x, t) 2 , (x, t) ∈ R d × [0, +∞) n(x, 0) = n 0 (x) ∈ C ∞ c (R d , R + ).
(2.1)

In the above setting, µ is a 1-periodic function, α ∈ (0, 1) is given and the term L α denotes a fractional elliptic operator which is defined as follows :

L α (n)(x, t) := -P V R d (n(x + h, t) -n(x, t))β(x, h |h| ) dh |h| d+2α , ( 2.2)
where

β : R d × S d-1 → R is a 1-periodic smooth function such that for all (x, θ) ∈ R d × S d-1 β(x, θ) = β(x, -θ) and 0 < b ≤ β(x, θ) ≤ B,
with b and B positive constants. When β is constant, we recover the classical fractional Laplacian (-∆) α .

The main aim of this paper is to describe the propagation front associated with (2.1). We show that the stable state invades the unstable state with an exponential speed.

The motivation

Equation (2.1) models the growth and the invasion of a species subject to non-local dispersion in a heterogeneous environment. Such models describe the situations where individuals can jump (move rapidly) from one point to the other, for instance because of the wind for Chapter 2 : A singular limit in a fractional reaction-diffusion equation with periodic coefficients seeds or human transportation for animals. The function n stands for the density of the population in position x at time t. The diffusion term represented by the operator L α describes the motions of individuals. The "logistic term" µ(x)n(x, t) -n(x, t) 2 represents the growth rate of the population. The heterogeneity of the environment is modeled by the periodic function µ. The regions where µ is positive represent areas where the species are favored whereas µ negative prevents the growth of the species. Conversely, the term -n 2 characterizes the death term because of some "logistic" considerations, as for example the quantity of food.

The operator L α generalizes the fractional Laplacian (-∆) α which models "homogeneous" jumps : the individuals jump in every direction with the same frequency. Whereas the operator L α models "heterogeneous" jumps : the individuals prefer to jump in the direction where β is high. Also, the frequency of jumps will depend on the position x of the individuals. Note that for the one dimensional case, for a regular bounded function n,

(1 -α)L α (n)(x) tends to -β(x)
4 n (x) as α tends to 1 -which corresponds to a heterogeneous local diffusion. Moreover, the function β will affect the principal eigenvalue λ 1 of L α -µ(x)Id (and consequently the negativity of λ 1 which is a criterion for the existence of a positive bounded stationary state). However, the hypothesis 0 < b ≤ β ≤ B implies that the techniques used for the fractional Laplacian are robust and can be extended to the case of the operator L α . Equation (2.1) was first introduced by Fisher in [START_REF] Fisher | The wave of advance of advantageous genes[END_REF] (1937) and Kolmogorov, Petrovskii and Piscunov in [START_REF] Kolmogorov | A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem[END_REF] (1938) in the particular case of a homogeneous environment (µ = 1) and a standard diffusion (L α = -∆) which corresponds to the case α = 1 and β = 1. In [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF], Aronson and Weinberger proved a first similar result to our result for the case introduced by Fisher and Kolmogorov, Petrovskii and Piscunov. In this case, the propagation is with a constant speed independent of the direction. In [START_REF] Gärtner | The propagation of concentration waves in periodic and random media[END_REF], Freidlin and Gärtner studied the question with a standard Laplacian in a heterogeneous environment (µ periodic). Using a probabilistic approach, they showed that the speed of the propagation is dependent on the chosen direction e ∈ S d-1 . But, the speed c(e) in the direction e is constant. Other proofs of this result, using PDE tools, can be found in [START_REF] Berestycki | The speed of propagation for KPP type problems. I. Periodic framework[END_REF] and [START_REF] Rossi | The Freidlin-Gärtner formula for general reaction terms[END_REF]. In the case of the fractional Laplacian and a constant environment, Cabré and Roquejoffre in [START_REF] Cabré | The influence of fractional diffusion in Fisher-KPP equations[END_REF] proved the front position is exponential in time (see also for instance [START_REF] Del-Castillo-Negrete | Front propagation and segregation in a reactiondiffusion model with cross-diffusion[END_REF] for some heuristic and numerical works predicting such behavior and [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF] for an alternative proof). Then in [START_REF] Cabré | Propagation in Fisher-KPP type equations with fractional diffusion in periodic media[END_REF], Cabré, Coulon and Roquejoffre investigate the speed of propagation in a periodic environment modeled by equation (2.1) but considering the fractional Laplacian instead of the operator L α . One should underline the fact that in the fractional case, the speed of propagation does not depend anymore on the direction. They proved that the speed of propagation is exponential in time with a precise exponent depending on a periodic principal eigenvalue.

The objective of this work is to provide an alternative proof of this property using an asymptotic approach known as "approximation of geometric optics". We will be interested in the long-time behavior of the solution n. We demonstrate that in the set (x, t) | |x| < e |λ 1 |t d+2α , as t tends to infinity, n converges to a stationary state n + and outside of this domain n tends to zero. The main idea in this approach is to perform a long time-long range rescaling to catch the effective behavior of the solution (see for instance [START_REF] Freidlin | Limit theorems for large deviations and reaction-diffusion equations[END_REF], [START_REF] Evans | A PDE approach to geometric optics for certain semilinear parabolic equations[END_REF] and [START_REF] Barles | Wavefront propagation for reaction-diffusion systems of PDE[END_REF] for the classical Laplacian case). This paper is closely related to [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF] where the authors Méléard and Mirrahimi have introduced such an "approximation" for a model with the fractional Laplacian and a simpler reaction term (n -n 2 ). A very recent work, [START_REF] Bouin | Thin front limit of an integro-differential Fisher-KPP equation with fat-tailed kernels[END_REF], uses also the techniques introduced in [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF] (known as the introduction of an adapting rescaling and the investigation of adapted 2.1. Introduction sub and super solution) to investigate an integro-differential homogeneous Fisher-KPP-type equation : the operator L α is replaced by J * n -n where the kernel J is fat tailed but does not have singularity at the origin.

This paper was initially written with a fractional Laplacian. At its completion, we became aware of a preprint by Souganidis and Tarfulea [START_REF] Souganidis | Front propagation for non-local KPP reaction diffusion equations in periodic media[END_REF] which proves a result quite close to ours in the case of spatially periodic stable operators. Our proof is quite different since their approach is based more on the theory of viscosity solutions. We have verified that our approach works for the model treated in [START_REF] Souganidis | Front propagation for non-local KPP reaction diffusion equations in periodic media[END_REF] with no additional idea. We present our result with the operator L α given by (2.2), where the proof for the fractional Laplacian applies almost word by word. In the course of the paper, we explain the points of our proof that allow to reach the generality of [START_REF] Souganidis | Front propagation for non-local KPP reaction diffusion equations in periodic media[END_REF].

The assumptions.

For the initial data we will assume

n 0 ∈ C ∞ c (R d , R + ), n 0 ≡ 0. (H1)
The function µ is a 1-periodic function, i.e.

∀k ∈ {1, ..., d} , µ(x 1 , ...,

x k + 1, ..., x d ) = µ(x 1 , ..., x d ). ( H2
)
Under the assumptions on β, the operator L α -µ(x)Id admits a principal eigenpair (φ 1 , λ 1 ) by the Krein-Rutman Theorem (see [START_REF] Berestycki | The periodic patch model for population dynamics with fractional diffusion[END_REF]) that is

L α φ 1 (x) -µ(x)φ 1 (x) = λ 1 φ 1 (x), x ∈ R d , φ 1 periodic, φ 1 > 0, φ 1 = 1. (2.3)
To assure the existence of a bounded, positive and periodic steady solution n + for (2.1), we will assume that the principal eigenvalue λ 1 is negative :

λ 1 < 0. ( H3
)
Note that such stationary solution is unique in the class of positive and periodic stationary solutions (see [START_REF] Berestycki | Le nombre de solutions de certains problèmes semi-linéaires elliptiques[END_REF]).

In section 4, we will study a more general equation :

   ∂ t n(x, t) + L α (n)(x, t) = F (x, n(x, t)), (x, t) ∈ R d × [0, +∞) n(x, 0) = n 0 (x) ∈ C ∞ c (R d , R + ).
(2.4)

We make the following assumptions for F :

                 (i) ∀s ∈ R, x → F (x, s) is periodic, (ii) F (x, 0) = 0, (iii) ∃c, C > 0 such that ∀(x, s) ∈ R d × R, -c ≤ ∂ s ( F (x, s) s ) ≤ -C, (iv) ∃M > 0, ∀(x, s) ∈ R d × [M, +∞[, F (x, s) < 0. ( H4
)
We will denote ∂ s (F )(x, 0) by µ(x) and we still denote by (λ 1 , φ 1 ) the principal eigenvalue and eigenfunction of L α -µ(x)Id. We also still suppose (H3) (i.e. λ 1 is strictly negative).

Chapter 2 : A singular limit in a fractional reaction-diffusion equation with periodic coefficients

2.1.4

The main result and the method.

We introduce the following rescaling (x, t) -→ |x| . The change of variable (2.5) will therefore respect the geometries of these sets. We then rescale the solution of (2.1) as follows

n ε (x, t) = n(|x| 1 ε x |x| , t ε )
and a new steady state n +,ε (x) = n + (|x|

1 ε x |x| ).
We prove :

Theorem 2.1. Assuming (H1), (H2) and (H3), let n be the solution of (2.1).

Then (i) n ε → 0, locally uniformly in A = (x, t) ∈ R d × (0, ∞)| |λ 1 | t < (d + 2α) log |x| , (ii) n ε n +,ε → 1, locally uniformly in B = (x, t) ∈ R d × (0, ∞)| |λ 1 | t > (d + 2α) log |x| .
To provide the main idea to prove Theorem 2.1, we first explain the main element of the proof in the case of constant environment which was introduced in [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF]. A central argument to prove such a result in the case of a constant environment, is that, using the rescaling (2.5), as ε → 0, the term ((-∆) α (n)n -1 ) (|x| 1 ε -1 x, t ε ) vanishes. More precisely, one can provide a sub and a super-solution to the rescaled equation which are indeed a sub and a super-solution to a perturbation of an ordinary differential equation derived from (2.1) by omitting the term (-∆) α . They also have the property that when one applies the operator f → (-∆) α (f )f -1 to such functions, the outcome is very small and of order O(ε 2 ) as ε tends to 0. In the case of periodic µ, we use the same idea. However, in this case, the sub and supersolutions are multiplied by the principal eigenfunction and, the term (L α (n)n -1 ) (|x| 1 ε -1 x, t ε ) will not just tend to 0 as in [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF] but also compensate the periodic media. To prove the convergence of n ε , dealing with this periodic term, we use the method of perturbed test functions from the theory of viscosity solutions and homogenization (introduced by Evans in [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF] and [START_REF] Evans | Periodic homogenisation of certain fully nonlinear partial differential equations[END_REF]). Note that we also generalize the arguments of [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF] to deal with a more general integral term L α while in [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF], only the case of the fractional Laplacian was considered. In the last part, we will also generalize Theorem 2.1 to the case of Fisher-KPP reaction term : Theorem 2.2. Assuming (H1), (H2), (H3) and (H4), let n be the solution of (2.4).

Then (i) n ε → 0, locally uniformly in A = (x, t) ∈ R d × (0, ∞)| |λ 1 | t < (d + 2α) log |x| , (ii) n ε n +,ε → 1, locally uniformly in B = (x, t) ∈ R d × (0, ∞)| |λ 1 | t > (d + 2α) log |x| .
The proof of this Theorem follows from an adaptation of the proof of Theorem 2.1.

In section 2, we introduce preliminary results and technical tools. In section 3, after the rescaling, we provide a sub and a super-solution and demonstrate Theorem 2.1. In section 4, we provide the points of the proof of Theorem 2.2 that differ from the proof of Theorem 2.1.

Preliminary results

Preliminary results

We first state a classical result on the fractional heat kernel. Proposition 2.1. [START_REF] Chen | Heat kernel estimates for stable-like processes on d-sets[END_REF] There exists a positive constant C larger than 1 such that the heat kernel p α (x, y, t) associated with the operator ∂ t + L α verifies the following inequalities for t > 0 :

C -1 × min(t -d 2α , t |x -y| d+2α ) ≤ p α (x, y, t) ≤ C × min(t -d 2α , t |x -y| d+2α ).
(2.6)

The proof of this proposition is given in [START_REF] Chen | Heat kernel estimates for stable-like processes on d-sets[END_REF]. Now we use this proposition to demonstrate that beginning with a positive compactly supported initial data leads to a solution with algebraic tails. Proposition 2.2. Assuming (H1), then there exist two constants c m and c M depending on n 0 , d, α and µ such that :

c m 1 + |x| d+2α ≤ n(x, 1) ≤ c M 1 + |x| d+2α .
Proof. First, we define M := max(max n 0 , max |µ|), we easily note that the constant functions 0 and M are respectively sub and super-solution to our problem. Then, thanks to the comparison principle (which is given in [START_REF] Cabré | The influence of fractional diffusion in Fisher-KPP equations[END_REF]), we have the following inequalities, for all

(x, t) ∈ R d × [0, +∞[ : 0 ≤ n(x, t) ≤ M.
Let n and n be the solutions of the two following systems :

∂ t n + L α n = -2M n, n(x, 0) = n 0 (x), (2.7)
and

∂ t n + L α n = max |µ| n, n(x, 0) = n 0 (x). ( 2
C -1 × n 0 (y) min(t -d 2α , t |x -y| d+2α )dy ≤ n(x, t) ⇒ e -2M supp(n 0 ) C -1 × n 0 (y) min(1, 1 |x -y| d+2α )dy ≤ n(x, 1).
Thanks to the dominated convergence theorem, we have :

(1 + |x| d+2α )e -2M supp(n 0 ) C -1 n 0 (y) min(1, 1 |x -y| d+2α )dy -→ |x|→∞ e -2M supp(n 0 ) C -1 n 0 (y)dy.
Therefore, we conclude by a compactness argument that for any x ∈ R d :

e -2M C -1 (1 + |x| d+2α ) ≤ n(x,
c m 1 + |x| d+2α ≤ n(x, 1) ≤ c M 1 + |x| d+2α .
(2.12)

We next provide a technical lemma which will be useful all along the article. The main ideas of the proof of the lemma come from [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF] by S. Méléard and S. Mirrahimi for Point (i) and [START_REF] Coulon Chalmin | Fast propagation in reaction-diffusion equations with fractional diffusion[END_REF] by A.C. Coulon Chalmin for Point (ii). To this end, we first introduce the computation of L α of a product of functions :

L α (f g)(x, t) = f (x, t)L α g(x, t) + g(x, t)L α f (x, t) -K(f, g)(x, t), with, K(f, g)(x, t) := C P V R d (f (x, t) -f (x + h, t))(g(x, t) -g(x + h, t)) |h| d+2α β(x, h |h| )dy. Lemma 2.1. Let γ be a positive constant such that γ ∈ [0, 2α[ if α < 1 2 ]2α -1, 1[ if 1 2 ≤ α < 1, χ : R → R d be a C 1 periodic
, strictly positive function and g(x) := 1 1+|x| d+2α . Then there exists a positive constant C, which does not depend on x, such that, for all x ∈ R d : (i) for all a > 0,

|L α g(ax)| ≤ a 2α Cg(ax), (ii) for all a ∈]0, 1], | K(g(a.), χ)(x)| ≤ Ca 2α-γ 1 + (a|x|) d+2α = Ca 2α-γ g(ax).
The proof is given in the Appendix. Note that we will not use the assumption b ≤ β in the proof of Lemma 2.1 (but only β ≤ B). The assumption b ≤ β is necessary for Proposition 2.1 and also to ensure the existence and the positiveness of φ 1 .

Remark. If we want to reach the same level of generality as in [START_REF] Souganidis | Front propagation for non-local KPP reaction diffusion equations in periodic media[END_REF], we just have to adapt the previous Lemma to an operator L α with a kernel β of the form β(x, y) where β is a 1-periodic with respect to x, smooth function from

R d × R d such that for all (x, y) ∈ R d × R d β(x, y) = β(x, -y) and 0 < b ≤ β(x, y) ≤ B,
with b and B positive constants. The interested reader can verify that the proof of Lemma 2.1 is robust enough and can easily be adapted to such kernels.

The proof of Theorem 2.1

In this section we will provide the proof of Theorem 2.1. Let us rewrite (2.1) with respect to the rescaling given by (2.5)

ε∂ t ( n(|x| 1 ε x |x| , t ε ) ) = -L α (n)(|x| 1 ε x |x| , t ε ) + n(|x| 1 ε x |x| , t ε )[µ(|x| 1 ε x |x| ) -n(|x| 1 ε x |x| , t ε )]. (2.13)
Notation. For any function v : R d × R + → R and w : R d → R we denote by v ε and w ε the rescaled functions given by :

v ε (x, t) := v(|x| 1 ε x |x| , t ε ) and w ε (x) = w(|x| 1 ε -1 x).
One can write the first term in the right-hand side of (2.13) in terms of n ε in the following way.

L α (n)(|x| 1 ε -1 x, t ε ) = -P V R d n(|x| 1 ε -1 x + h, t ε ) -n(|x| 1 ε -1 x, t ε ) |h| 2α+d × β(|x| 1 ε -1 x, h |h| )dh = -P V R d   n ε   |x| 1 ε -1 x + h ε (|x| 1 ε -1 x + h) ||x| 1 ε -1 x + h| , t   -n ε (x, t)   β ε (x, h |h| )dh |h| 2α+d .
We can hence define :

L α ε (n ε )(x, t) := L α (n)(|x| 1 ε x |x| , t ε ),
which allows us to write (2.13) as below :

ε∂ t n ε (x, t) = -L α ε n ε (x, t) + n ε (x, t)[µ ε (x) -n ε (x, t)].
(2.14)

In the same way we define

K ε (n ε , χ ε )(x, t) := K(n, χ)(|x| 1 ε -1 x, t ε ).
Moreover, according to the inequalities (2.12), we can consider n(x, 1) as our initial data instead of n(x, 0). So we can replace the assumption (H1) by :

c m 1 + |x| d+2α ≤ n 0 (x) ≤ c M 1 + |x| d+2α ⇒ c m 1 + |x| d+2α ε ≤ n 0,ε (x) ≤ c M 1 + |x| d+2α ε . (H1')
In the next subsection we are going to provide sub and super-solutions to (2.14) which will allow us to demonstrate Theorem 2.1 in a second subsection.

Sub and super-solution to (2.14).

Theorem 2.3. We assume (H2) and (H3) and we choose positive constants

C m < |λ 1 | max φ 1 and C M > |λ 1 |
min φ 1 and δ such that

0 < δ ≤ min( C M min φ 1 -|λ 1 |, |λ 1 | -C m max φ 1 ).
Then there exists a positive constant ε 0 < δ such that for all ε ∈]0, ε 0 [ we have :

(i) f M ε (t, x) = φ 1,ε (x) × C M 1+e -t ε (|λ 1 |+ε 2 )-δ ε |x| d+2α ε
is a super-solution of (2.14), Chapter 2 : A singular limit in a fractional reaction-diffusion equation with periodic coefficients

(ii) f m ε (x, t) = φ 1,ε (x) × Cme -δ ε 1+e -t ε (|λ 1 |-ε 2 )-δ ε |x| d+2α ε
is a sub-solution of (2.14).

(iii) Moreover, if we assume (H1') and

C m < c m max |φ 1 | and C M > c M min |φ 1 |
where c m and c M are given by (H1') then for all (x, t) ∈ R d × [0, +∞[,

φ 1,ε (x) × C m e -δ ε -εt 1 + e -|λ 1 |t+δ ε |x| d+2α ε ≤ n ε (x, t) ≤ φ 1,ε (x) × C M e εt 1 + e -|λ 1 |t+δ ε |x| d+2α ε . (2.15)
Proof. Since the proofs of (i) and (ii) follow from similar arguments, we will only provide the proof of (i) and (iii).

Proof of (i).

We define :

ψ(x, t) := C M 1 + e -t(|λ 1 |+ε 2 )-δ ε |x| d+2α . (2.16)
Then, noticing that φ 1 is independent of t, we first bound ∂ t ψ ε from below,

∂ t ψ ε (x, t) = C M (|λ 1 |+ε 2 ) ε e -t (|λ 1 |+ε 2 ) ε -δ ε |x| d+2α ε (1 + e -t (|λ 1 |+ε 2 ) ε -δ ε |x| d+2α ε ) 2 = ψ ε (x, t) ε [(|λ 1 | + ε 2 ) e -t (|λ 1 |+ε 2 ) ε -δ ε |x| d+2α ε 1 + e -t |λ 1 |+ε 2 ) ε -δ ε |x| d+2α ε ] ≥ ψ ε (x, t) ε [|λ 1 | + ε 2 -ψ ε (x, t)φ 1,ε (x)].
(2.17)

The last inequality is obtained from the definition of C M and ε. Actually, for such C M and ε, we have, for any positive non-null constant A, the following relation :

A(|λ 1 | + ε 2 ) 1 + A ≥ |λ 1 | + ε 2 - C M min φ 1 1 + A , because, |λ 1 | + ε 2 - C M min φ 1 1 + A = (1 + A)(|λ 1 | + ε 2 ) -C M min φ 1 1 + A = A(|λ 1 | + ε 2 ) -(C M min φ 1 -|λ 1 | -ε 2 ) 1 + A ≤ A(|λ 1 | + ε 2 ) 1 + A .
We also compute

L α ε (f M ε )(x, t) as a fractional Laplacian of a product of functions, L α ε (f M ε )(x, t) = φ 1,ε (x)L α ε ψ ε (x, t) + ψ ε (x, t)L α ε φ 1,ε (x) -K ε (ψ ε , φ 1,ε )(x, t) (2.18)
with K given in section 2. Replacing this in equation (2.14) and using the two previous results (2.17) and (2.18), we find :

ε∂ t f M ε (x, t) + L α ε f M ε (x, t) -f M ε (x, t)[µ ε (x) -f M ε (x, t)] ≥ f M ε (x, t)(|λ 1 | + ε 2 -f M ε (x, t)) + φ 1,ε (x)L α ε ψ ε (x, t) + ψ ε (x, t)L α ε φ 1,ε (x) -K ε (ψ ε , φ 1,ε )(x, t) -µ ε (x)f M ε (x, t) + f M ε (x, t) 2 = ε 2 f M ε (x, t) + φ 1,ε (x)L α ε ψ ε (x, t) -K ε (ψ ε , φ 1,ε )(x, t),
where we have used (2.3) and (H3) for the last equality.

In order to control L α ε ψ ε (x, t) and K ε (ψ ε , φ 1,ε )(x, t), we are going to use Lemma 2.1. For,

L α ε ψ ε (x, t), noticing that ψ ε (x, t) = C M g(e -t(|λ 1 |+ε 2 )-δ ε(1+2α) |x| 1 ε -1 x)
, and thanks to the point (i) of Lemma 2.1, we obtain :

-Ce -2α t(|λ 1 |+ε 2 )+δ ε ψ ε (t, x) ≤ L α ε ψ ε (t, x).
But, comparing the growths, there exists ε 1 > 0 such that for ε < ε 1 and for all t ≥ 0 :

C M × Ce -2α t(|λ 1 |+ε 2 )+δ ε(d+2α) - ε 2 3 ≤ 0, hence : - ε 2 3 ψ ε (x, t) ≤ L α ε ψ ε (x, t).
Now we deal with K ε (ψ ε , φ 1,ε )(x, t) in a similar fashion. Thanks to Lemma 2.1 (ii), we find :

K ε (ψ ε , φ 1,ε )(x, t) = K(ψ, φ 1 )(|x| 1 ε x |x| , t ε ) ≤ Ce -(2α-γ)[t(|λ 1 |+ε 2 )+δ] ε ψ(|x| 1 ε x |x| , t ε ) = Ce -(2α-γ)[t(|λ 1 |+ε 2 )+δ] ε ψ ε (x, t).
Then, noticing that for any choice of α, 2α -γ is strictly positive, we deduce that there exists ε 2 > 0 such that for all ε < ε 2 :

Ce -(2α-γ) t(|λ 1 |+ε 2 )+δ ε - ε 2 min φ 1 3 ≤ 0.
We deduce that

K ε (ψ ε , φ 1,ε )(x, t) ≤ ε 2 3 ψ ε (x, t) min φ 1 ≤ ε 2 3 ψ ε (x, t)φ 1,ε (x).
We set :

ε 0 = min(ε 1 , ε 2 ).
Then, we conclude that for ε ≤ ε 0 , we have :

ε∂ t f M ε (x, t) + L α ε f M ε (x, t) -f M ε (x, t)[µ ε (x) -f M ε (x, t)] ≥ ε 2 f M ε (x, t) + φ 1,ε (x)L α ε ψ ε (x, t) -K ε (ψ, φ 1 )(x, t) ≥ ε 2 f M ε (x, t) - ε 2 3 φ 1,ε (x)ψ ε (x, t) - ε 2 3 φ 1,ε (x)ψ ε (x, t) ≥ ε 2 3 f M ε (x, t) ≥ 0.
Therefore f M ε is a super-solution of (2.14) and this concludes the proof of the point (i).

Chapter 2 : A singular limit in a fractional reaction-diffusion equation with periodic coefficients

Proof of (iii).

From (H1'), since max |φ 1 |C m < c m and c M < C M min |φ 1 |, we have :

f m ε (x, 0) = φ 1,ε (x) × C m e -δ ε 1 + e -δ ε |x| d+2α ε = φ 1,ε (x) × C m e δ ε + |x| d+2α ε ≤ c m 1 + |x| d+2α ε ≤ n ε (x, 0) ≤ f M ε (x, t).
Then, according to the comparison principle, we obtain :

φ 1,ε (x) × C m e -δ ε 1 + e -t ε (|λ 1 |-ε 2 )-δ ε |x| d+2α ε ≤ n ε (x, t) ≤ φ 1,ε (x) × C M 1 + e -t ε (|λ 1 |+ε 2 )-δ ε |x| d+2α ε
, and hence

φ 1,ε (x) × C m e -δ ε -εt 1 + e -|λ 1 |t+δ ε |x| d+2α ε ≤ n ε (x, t) ≤ φ 1,ε (x) × C M e εt 1 + e -|λ 1 |t+δ ε |x| d+2α ε . (2.15)

Convergence to the stationary state.

Thanks to the inequalities (2.15), we can now prove Theorem 2.1. To prove this theorem, we are going to follow the ideas of Méléard and Mirrahimi in [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF].

Proof. (Proof of Theorem 2.1.) First, we perform a Hopf-Cole transformation

u ε (x, t) := ε log n ε (x, t) and u +,ε (x) := ε log n +,ε (x).
(

Taking the logarithm in (2.15) and multiplying by ε, we find :

-ε 2 t + ε log C m φ 1,ε -ε log(1 + e -|λ 1 |t+δ ε |x| d+2α ε ) -δ ≤ u ε (x, t) and u ε (x, t) ≤ ε 2 t + ε log C M φ 1,ε -ε log(1 + e -|λ 1 |t+δ ε |x| d+2α ε ). Define u(x, t) = lim inf ε→0 u ε (x, t), u(x, t) = lim sup ε→0 u ε (x, t), for all (x, t) ∈ R d × (0, +∞).
Letting ε → 0, we obtain min(0,

|λ 1 | t + δ -(d + 2α) log |x|) -δ ≤ u(x, t) ≤ u(x, t) ≤ min(0, |λ 1 | t + δ -(d + 2α) log |x|).
We then let δ → 0 and we obtain

u(x, t) := u(x, t) = u(x, t) = min(0, |λ 1 | t -(d + 2α) log |x|).
We deduce that u ε converges locally uniformly in R d × [0, +∞[ to u since the above limits are locally uniform in ε.

Proof of (i).

For any compact set K in A, there exists a positive constant a such that for all (x 0 , t 0 ) ∈ K, we have u(x 0 , t 0 ) < -a. It is thus immediate from (2.19) that n ε converges uniformly to 0 in K ⊂ A. This concludes the proof of (i).

The proof of Theorem 2.1

Proof of (ii). We divide (2.14) by n ε and we obtain

∂ t u ε + L α ε n ε n -1 ε = µ ε -n ε ,
that we rewrite as below,

n ε = n +,ε + (-∂ t u ε -L α ε n ε n -1 ε + µ ε -n +,ε ). (2.20)
Step 1 :

n ε (x 0 , t 0 ) n +,ε (x 0 ) ≥ 1 + o(1) in every compact set of B.
Let K be a compact set of B and (x 0 , t 0 ) ∈ K. We choose ν a positive constant small enough such that for all (y, s) ∈ K,

(d + 2α) log |y| < |λ 1 |s -2ν and 2ν < |λ 1 |s.
(2.21)

First, we define

ϕ(x, t) := min(0, -(d + 2α) log |x| + |λ 1 |t 0 -ν) -(t -t 0 ) 2 .
It is easy to verify that u -ϕ achieves a local strict in t and a global in x minimum at (x 0 , t 0 ). Then, we define

ϕ ε (x, t) := -ε log(1 + e -|λ 1 |t 0 -ν ε |x| d+2α ε ) -(t -t 0 ) 2 .
Thus, (ϕ ε ) ε converges locally uniformly to ϕ. Moreover, since n + is periodic and strictly positive, we have that u +,ε converges to 0, hence

u ε -(ϕ ε + u +,ε ) -→ ε→0 u -ϕ locally uniformly.
Thus, there exists (

x ε , t ε ) ∈ R d × [0, +∞[ such that (x ε , t ε ) is a minimum point (local in t and global in x) of (u ε -ϕ ε -u +,ε ) and (u ε -ϕ ε -u +,ε )(x ε , t ε ) → 0. Since (x 0 , t 0 ) is a strict in t local minimum of u -ϕ, one can choose t ε such that t ε → t 0 . We deduce that ∂ t u ε (x ε , t ε ) = ∂ t ϕ ε (x ε , t ε ) = -2(t ε -t 0 ) = o(1). (2.22)
One should ensure that (x ε ) ε→0 have all their accumulation points in B(0, e |λ 1 |t 0 -ν d+2α ) as ε tends to 0. This is the case because, at time t = t 0 , in B(0, e

|λ 1 |t 0 -ν d+2α ), u ε -ϕ ε -u +,ε tends to 0, whereas in B(0, e |λ 1 |t 0 -ν d+2α ) c , u ε -ϕ ε -u +,ε
tends to a strictly positive function.

We deduce that there exists ε 1 > 0 such that for all ε < ε 1 we have

x ε ∈ B(0, e |λ 1 |t 0 -ν 2 d+2α
). Then we continue by proving (-

L α ε (n ε )n -1 ε + µ ε -n +,ε )(x ε , t ε ) ≥ o(1), -L α ε (n ε )n -1 ε (x ε , t ε ) = R d (e uε |xε| 1 ε -1 xε+h ε-1 (|xε| 1 ε -1 xε+h),tε -uε(xε,tε) ε -1) β ε (x, h |h| )dh |h| d+2α .
From the definition of (x ε , t ε ), we have for all y ∈ R d :

(u ε -ϕ ε -u +,ε )(x ε , t ε ) ≤ (u ε -ϕ ε -u +,ε )(y, t ε ),
and thus

(ϕ ε + u +,ε )(y, t ε ) -(ϕ ε + u +,ε )(x ε , t ε ) ≤ u ε (y, t ε ) -u ε (x ε , t ε ).
Therefore, from (2.19) we have

-L α ε (e ϕ ε ε n +,ε )(e ϕ ε ε n +,ε ) -1 (x ε , t ε ) ≤ -L α ε (n ε )n -1 ε (x ε , t ε ).
Chapter 2 : A singular limit in a fractional reaction-diffusion equation with periodic coefficients

Finally, using that n +,ε is a solution of the stationary equation, we obtain

(-L α ε (n ε )n -1 ε + µ ε -n +,ε )(x ε , t ε ) ≥ (-L α ε (e ϕ ε ε n +,ε ))((e ϕ ε ε n +,ε ) -1 + µ ε -n +,ε )(x ε , t ε ) = (-L α ε (e ϕ ε ε )(e - ϕ ε ε ) -L α ε (n +,ε )(n +,ε ) -1 + K ε (e ϕ ε ε , n +,ε )((e ϕ ε ε n +,ε ) -1 + µ ε -n +,ε )(x ε , t ε ) = -L α ε (e ϕ ε ε )(e - ϕ ε ε )(x ε , t ε ) + K ε (e ϕ ε ε , n +,ε )(e ϕ ε ε n +,ε ) -1 (x ε , t ε ).
In order to control the last two terms of the above inequality, we are going to use Lemma 2.1. Note that, we have the following link between e ϕ ε ε and g

(x) = 1 1+|x| d+2α : e ϕ ε ε (x, t) = e -(t-t 0 ) 2 ε 1 + e -|λ 1 |t 0 -ν ε |x| d+2α ε = e -(t-t 0 ) 2 ε g(e -|λ 1 |t 0 -ν (d+2α)ε |x| 1 ε -1 x).
And so, we can deduce from Lemma 2.1 that :

o(1) = -Ce -2α(|λ 1 |t 0 -ν) (d+2α)ε ≤ -L α ε (e ϕ ε ε )(e - ϕ ε ε )(x ε , t ε ), and, o(1) = -C e -(2α-γ)(|λ 1 |t 0 -ν) ε n +,ε (x ε , t ε ) -1 ≤ K ε (e ϕ ε ε , n +,ε )(e ϕ ε ε n +,ε ) -1 (x ε , t ε ).
We deduce that :

o(1) ≤ (-L α ε (n ε )n -1 ε + µ ε -n +,ε )(x ε , t ε ). (2.23)
Finally, combining the above inequality with (2.20) and (2.22), we obtain that

1 + o(1) ≤ n ε (x ε , t ε ) n +,ε (x ε ) .
Now, we want to bring back this inequality at the point (x 0 , t 0 ). There are two cases :

Case 1 : |x ε | ≥ |x 0 | Because of the definition of ϕ ε , we have : ϕ ε (x ε , t ε ) ≤ ϕ ε (x 0 , t 0 ). Since (x ε , t ε ) is a minimum point of u ε -(ϕ ε + u ε,+ ), we deduce that u ε (x ε , t ε ) -u +,ε (x ε ) ≤ u ε (x 0 , t 0 ) -u +,ε (x 0 ). Thanks to (2.19), it follows that 1 + o(1) ≤ n ε (x ε , t ε ) n +,ε (x ε ) ≤ n ε (x 0 , t 0 ) n +,ε (x 0 ) . Case 2 : |x ε | < |x 0 |
In this case, since (x 0 , t 0 ) ∈ K and thanks to (2.21), we have that = o [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF]. We deduce that

-|λ 1 |t 0 + ν + (d + 2α) log(|x ε |) ≤ -|λ 1 |t 0 + ν + (d + 2α) log(|x 0 |) ≤ -ν < 0,
e - ϕ ε (xε,tε) ε = e (tε-t 0 ) 2 ε (1 + e -|λ 1 |t 0 +ν+(d+2α) log(|xε|) ε ) ≥ 1.
Moreover, following similar computations, we obtain that e ϕ ε (x 0 ,t 0 )

ε = 1 + o(1)
. Hence, from the definition of (x ε , t ε ), we get

u ε (x ε , t ε ) -u +,ε (x ε ) + ϕ ε (x 0 , t 0 ) -ϕ ε (x ε , t ε ) ≤ u ε (x 0 , t 0 ) -u +,ε (x 0 ).
Thanks to (2.19), we obtain that

1 + o(1) ≤ n ε (x ε , t ε ) n +,ε (x ε ) × e ϕ ε (x 0 ,t 0 ) ε e ϕ ε (xε,tε) ε ≤ n ε (x 0 , t 0 ) n +,ε (x 0 ) .
So we have proved, in all cases

1 + o(1) ≤ n ε (x 0 , t 0 ) n +,ε (x 0 ) .
Step 2 :

n ε (x 0 , t 0 ) n +,ε (x 0 ) ≤ 1 + o(1) in every compact set of B.
This step is very similar to the first one. We pick (x 0 , t 0 ) ∈ B and let ν be a positive constant. As before, we define

ϕ(x, t) := min(0, |λ 1 |t 0 + ν -(d + 2α) log |x|) + (t -t 0 ) 2 .
It is easy to verify that u -ϕ achieves a local and strict in t and a global in x maximum at (x 0 , t 0 ). Then, defining

ϕ ε (x, t) := -ε log(1 + e -|λ 1 |t 0 +ν ε |x| d+2α ε ) + (t -t 0 ) 2 ,
we have that (ϕ ε ) ε converges locally uniformly to ϕ. Moreover, we know that u +,ε tends to 0 and so u ε -

(ϕ ε + u +,ε ) -→ ε→0 0 uniformly in B. Thus, there exists (x ε , t ε ) ∈ R d × [0, +∞[ such that (x ε , t ε ) is a maximum point, global in x and local in t, of (u ε -ϕ ε -εu +,ε ) and (u ε -ϕ ε -u +,ε )(x ε , t ε ) → 0. (2.24) Since (x 0 , t 0 ) is a strict in t local maximum of u -ϕ, one can choose t ε such that t ε → t 0 . We deduce that ∂ t u ε (x ε , t ε ) = ∂ t ϕ ε (x ε , t ε ) = 2(t ε -t 0 ) = o(1). (2.25)
One should ensure that (x ε ) ε→0 have all their accumulation points in B(0, e

|λ 1 |t 0 + ν 4 d+2α
). This is the case because for ε small enough, in B(0, e

|λ 1 |t 0 d+2α ), u ε -ϕ ε -u +,ε tends to 0 whereas in B(0, e |λ 1 |t 0 + ν 4 d+2α ) c , u ε -ϕ ε -u +,ε
is lower than a strictly negative function. We deduce that there exists ε 2 > 0 such that for all ε < ε 2 we have

x ε ∈ B(0, e |λ 1 |t 0 + ν 2 d+2α ). (2.26)
Then we continue by showing (-

L α ε (n ε )n -1 ε +µ ε -n +,ε ) ≤ o(1)
. With similar computations as in step 1, we obtain :

(-L α ε (n ε )n -1 ε + µ ε -n +,ε )(x ε , t ε ) ≤ -L α ε (e ϕ ε ε )(e -ϕ ε ε )(x ε , t ε )
Chapter 2 : A singular limit in a fractional reaction-diffusion equation with periodic coefficients

+ K ε (e ϕ ε ε , n +,ε )(e ϕ ε ε n +,ε ) -1 (x ε , t ε ). Since e ϕ ε ε (x, t) = e (t-t 0 ) 2 ε 1 + e -|λ 1 |t 0 +ν ε |x| d+2α ε = e (t-t 0 ) 2 ε g(e -|λ 1 |t 0 +ν ε(d+2α) |x| 1 ε -1 x),
we can deduce thanks to Lemma 2.1 that :

(-L α ε (e ϕ ε ε )(e -ϕ ε ε )(x ε , t ε ) ≤ Ce -2α(|λ 1 |t 0 +ν) (d+2α)ε = o(1),
and,

K ε (e ϕ ε ε , n +,ε )(e ϕ ε ε n +,ε ) -1 (x ε , t ε ) ≤ C e -(2α-γ)(|λ 1 |t 0 +ν) ε(d+2α) n +,ε (x ε , t ε ) -1 = o(1).
Finally, combining the two previous inequalities and (2.25) in (2.20) we have obtained

n ε (x ε , t ε ) n +,ε (x ε ) ≤ 1 + o(1).
Then, there are two cases to bring it back to the point (x 0 , t 0 ) :

Case 1 : |x ε | ≤ |x 0 | By definition of ϕ ε , we have : ϕ ε (x 0 , t 0 ) ≤ ϕ ε (x ε , t ε ). Since (x ε , t ε ) is a maximum point of u ε -(ϕ ε + u ε,+ ), we deduce that u ε (x 0 , t 0 ) -u +,ε (x 0 ) ≤ u ε (x ε , t ε ) -u +,ε (x ε ).
Thanks to (2.19), it follows that,

n ε (x 0 , t 0 ) n +,ε (x 0 ) ≤ n ε (x ε , t ε ) n +,ε (x ε ) ≤ 1 + o(1). Case 2 : |x ε | > |x 0 | Thanks to (2.26
), there exists ε 2 such that for all positive ε < ε 2 there holds

|x ε | ≤ e |λ 1 |t 0 + ν 2 d+2α ⇒ -|λ 1 |t 0 -ν + (d + 2α) log |x ε | < - ν 2 .
And thus,

e -ϕ ε (xε,tε) ε = e -(tε-t 0 ) 2 ε (1 + e -|λ 1 |t 0 -ν+(d+2α) log(|xε|) ε ) ≤ 1 + e -ν 2ε ≤ 1 + o(1).
Moreover, we know by definition that e

ϕ ε (x 0 ,t 0 ) ε = 1 + o(1)
. Furthermore, by definition of (x ε , t ε ), we have

u ε (x 0 , t 0 ) -u +,ε (x 0 ) ≤ u ε (x ε , t ε ) -u +,ε (x ε ) + ϕ ε (x 0 , t 0 ) -ϕ ε (x ε , t ε ),
Combining the above inequalities and thanks to (2. [START_REF] Berestycki | The influence of a line with fast diffusion on Fisher-KPP propagation[END_REF]) we obtain that

n ε (x 0 , t 0 ) n +,ε (x 0 ) ≤ n ε (x ε , t ε ) n +,ε (x ε ) × e ϕ ε (x 0 ,t 0 ) ε e ϕ ε (xε,tε) ε ≤ 1 + o(1).

So we have proved, in all cases

n ε (x 0 , t 0 ) n +,ε (x 0 ) ≤ 1 + o(1).
Passing up to the limit, we finally obtain the result of (ii).

2.4. Generalization to KPP-type reaction terms

Generalization to KPP-type reaction terms

We can generalize our result to a model with a reaction term F (x, s) which verifies the Fisher KPP assumptions given by (H4).

Example 2.1. Obviously we can take as before

F (x, s) = µ(x)s -s 2 .
Example 2.2. We can generalize it to the classical example :

F (x, s) = µ(x)s -ω(x)s 2 .
Where µ is a continuous periodic function, and ω is a continuous periodic strictly positive function.

Of course, we keep the main idea of the previous proof : the rescaling (2.5). So the equation (2.4) becomes :

ε∂ t n ε (x, t) = -L α ε n ε (x, t) + F ε (x, n ε (x, t)). (2.27)
As before, according to the comparison principle, the point (ii) and (iii) of (H4) imply

µ(x)n -cn 2 ≤ F (x, n) ≤ µ(x)n -Cn 2 .
(2.28)

If we associate this result with (2.6), one can still obtain that the solution will have algebraic tails at time t = 1 and hence one can replace the assumption (H1) by (H1') :

c m 1 + |x| d+2α ε ≤ n 0,ε (x) ≤ c M 1 + |x| d+2α ε . (H1')
Therefore, we still have the same sub and super-solutions :

Theorem 2.4. We assume (H2), (H3) and (H4) and if we choose

C m < |λ 1 | c max φ 1 and C M > |λ 1 | C min φ 1
where c and C are given by the assumptions (iii) of (H4) and a positive constant δ such that

0 < δ ≤ min( CC M min φ 1 -|λ 1 |, |λ 1 | -cC m max φ 1 );
then there exists a positive constant ε 0 < δ such that for all ε ∈]0, ε 0 [ we have :

(i) f M ε (x, t) = φ 1,ε (x) × C M 1+e -t ε (|λ 1 |+ε 2 )-δ ε |x| d+2α ε is a super-solution of (2.27), (ii) f m ε (x, t) = φ 1,ε (x) × Cme -δ ε 1+e -t ε (|λ 1 |-ε 2 )-δ ε |x| d+2α ε is a sub-solution of (2.

27).

(iii) Moreover, if we assume (H1') and

C m < c m max |φ 1 | and C M > c M min |φ 1 |
with c m and c M given by (H1') then for all

(x, t) ∈ R d × [0, +∞[, φ 1,ε (x) × C m e -δ ε -εt 1 + e -|λ 1 |t+δ ε |x| d+2α ε ≤ n ε (x, t) ≤ φ 1,ε (x) × C M e εt 1 + e -|λ 1 |t+δ ε |x| d+2α ε . (2.29)
Proof. Here is the main step of the proof of the point (i). As in the proof of Theorem 2.3, we put :

f M ε (t, x) = φ 1,ε (x) × ψ ε (x, t),
Chapter 2 : A singular limit in a fractional reaction-diffusion equation with periodic coefficients where ψ ε is given by (2.16), but with a constant C M given in Theorem 2.4. Then, with similar computations as before, we find :

∂ t f M ε ≥ f M ε ε (|λ 1 | + ε 2 -Cf M ε ).
Therefore, using (2.28) and Lemma 2.1, we get :

ε∂ t f M ε (x, t) + L α ε f M ε (x, t) -F ε (x, f ε (x, t)) ≥ f M ε (x, t)(|λ 1 | + ε 2 -Cf M ε (x, t)) + φ 1,ε (x)L α ε ψ ε (x, t) + ψ ε (x, t)L α ε φ 1,ε (x) -K ε (ψ, φ 1 )(x, t) -µ ε (x)f M ε (x, t) + Cf M ε (x, t) 2 ≥ ε 2 f M ε (x, t) - ε 2 3 f M ε (x, t) - ε 2 3 f M ε (x, t) + ψ ε (x, t) [L α ε φ 1,ε (x) -(λ 1 + µ ε (x))φ 1,ε (x)] ≥ 0.
Thus, we have demonstrated the point (i). The proof of the point (ii) follows similar arguments. We do not give the proof of the point (iii) because this is similar to the proof of (iii) of Theorem 2.3 : the main argument is the comparison principle. Thus, we can perform the Hopf-Cole transformation (2.19) and we obtain that u ε converges locally uniformly to :

u(x, t) = min(0, |λ 1 | t -(d + 2α) log |x|).
Therefore the part (i) of Theorem 2.2 can be proved following similar arguments as in the proof of (i) of Theorem 2.1. The proof of (ii) changes a little bit so we are going to provide the demonstration.

Proof. (Proof of (ii) of Theorem 2.2.) Dividing by n ε in (2.14), we obtain

∂ t u ε + L α ε n ε n -1 ε = F ε (x, n ε ) n ε .
(2.30)

Step 1 :

n ε (x 0 , t 0 ) n +,ε (x 0 ) ≥ 1 + o(1) in every compact set of B.
The main difference with the proof of Theorem 2.1 is that from (2.30), we do not obtain

directly n ε (x ε , t ε ) n +,ε (x ε ) ≥ 1 + o(1) but we deduce F ε (x ε , n +,ε ) n +,ε (x ε , t ε ) - F ε (x ε , n ε ) n ε (x ε , t ε ) ≥ o(1).
Let K be a compact set of B and (x 0 , t 0 ) ∈ K. We choose ν a positive constant small enough such that for all (y, s) ∈ K,

(d + 2α) log |y| < |λ 1 |s -2ν and 2ν < |λ 1 |s.
(2.31)

First, we define

ϕ(t, x) := min(0, -(d + 2α) log |x| + |λ 1 |t 0 -ν) -(t -t 0 ) 2 .
It is easy to verify that u -ϕ achieves a local strict in t and a global in x minimum at (x 0 , t 0 ). Then, we define

ϕ ε (x, t) := -ε log(1 + e -|λ 1 |t 0 -ν ε |x| d+2α ε ) -(t -t 0 ) 2 .
Thus, (ϕ ε ) ε converges locally uniformly to ϕ. We know that u +,ε tends to 0 and so u ε -

(ϕ ε + u +,ε ) -→ ε→0 u -ϕ locally uniformly. Thus, there exists (x ε , t ε ) ∈ B such that (x ε , t ε ) is a 2.4. Generalization to KPP-type reaction terms minimum point of (u ε -ϕ ε -u +,ε ) and (u ε -ϕ ε -u +,ε )(x ε , t ε ) → 0. Since (x 0 , t 0 ) is a strict local minimum of u -ϕ in t, we can choose t ε such that t ε → t 0 . Then ∂ t u ε (x ε , t ε ) = ∂ t ϕ ε (x ε , t ε ) = -2(t ε -t 0 ) = o(1). (2.32)
With the same reasoning as in the proof of Theorem 2.1, we get that there exists

ε 1 > 0 such that for ε < ε 1 , x ε ∈ B(0, e |λ 1 |t 0 -ν 2 d+2α
). Then we continue by proving

(-L α ε (n ε )n -1 ε + F ε (x, n ε ) n ε )(x ε , t ε ) ≥ ( F ε (x, n ε ) n ε - F (x ε , n +,ε ) n +,ε )(x ε , t ε ) + o(1).
We know that

-L α ε (n ε )n -1 ε (x ε , t ε ) = R d (e uε |xε| 1 ε -1 xε+h ε-1 (|xε| 1 ε -1 xε+h),tε -uε(xε,tε) ε -1) β ε (x ε , h |h| )dh |h| d+2α .
Note that, from the definition of (x ε , t ε ), we have for all y ∈ R d :

(u ε -ϕ ε -u +,ε )(x ε , t ε ) ≤ (u ε -ϕ ε -u +,ε )(y, t ε ),
and thus by (2.19)

-L α ε (e ϕ ε ε n +,ε )(e ϕ ε ε n +,ε ) -1 (x ε , t ε ) ≤ -L α ε (n ε )n -1 ε (x ε , t ε ).
Finally, we obtain

(-L α ε (n ε )n -1 ε + F ε (x, n ε ) n ε )(x ε , t ε ) ≥ (-L α ε (e ϕ ε ε n +,ε ))((e ϕ ε ε n +,ε ) -1 + F ε (x, n ε ) n ε )(x ε , t ε ) ≥ (-L α ε (e ϕ ε ε )(e - ϕ ε ε ) - F ε (x, n +,ε ) n +,ε + K ε (e ϕ ε ε , n +,ε )((e ϕ ε ε n +,ε ) -1 + F ε (x, n ε ) n ε )(x ε , t ε ) ≥ (o(1) + F ε (x, n ε ) n ε - F ε (x, n +,ε ) n +,ε )(x ε , t ε ).
(2.33)

We have to note that thanks to Lemma 2.1, in the last inequality, we have controlled the terms :

o(1) ≤ K ε (e ϕ ε ε , n +,ε )(e ϕ ε ε n +,ε ) -1 (x ε , t ε ) and o(1) ≤ -L α ε (e ϕ ε ε )(e - ϕ ε ε )(x ε , t ε ).
Finally combining (2.32) and (2.33), we obtain

o(1) ≤ F ε (x ε , n +,ε ) n +,ε (x ε , t ε ) - F ε (x ε , n ε ) n ε (x ε , t ε ). (2.34)
We are going to prove by contradiction that (2.34) implies o(1)

+ n +,ε (x ε ) ≤ n ε (x ε , t ε ).
Let's suppose that there exists a subsequence (ε k ) k∈N and a positive constant C such that

n ε k (x ε k , t ε k ) + C < n +,ε k (x ε k ).
Chapter 2 : A singular limit in a fractional reaction-diffusion equation with periodic coefficients Then, thanks to the strict monotony of the function s → F (x,s) s (assumption (iv) in (H4)) and the mean value Theorem, there exists a sequence y k such that

o ε k (1) ≤ F (x ε k , n +,ε k ) n +,ε k (x ε k , t ε k ) - F (x ε k , n ε k ) n ε k (x ε k , t ε k ) = ∂ s ( F (x ε k , s) s )(y ε k )(n +,ε k -n ε k )(x ε k , t ε k ) ≤ -C × C.
This is a contradiction. Therefore, for ε small enough,

n +,ε (x ε ) + o(1) ≤ n ε (x ε , t ε ) ⇒ 1 + o(1) ≤ n ε (x ε , t ε ) n +,ε (x ε ) .
To bring back this inequality at the point (x 0 , t 0 ), we use exactly the same arguments as for the proof of Theorem 2.1 by considering a disjunction of cases

|x ε | < |x 0 | and |x 0 | < |x ε |.
We do not provide the details of this disjunctions of cases since they are the same. So we have proved, in all cases

1 + o(1) ≤ n ε (x 0 , t 0 ) n +,ε (x 0 ) .
The second step can also be proved following similar arguments as in the previous step, thus we do not provide the demonstration.

Appendix -Proof of Lemma 2.1

All along the appendix, we will denote by C positive constants that can change from line to line.

Proof of Lemma 2.1. Proof of (i).

We are going to follow the appendix A of [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF].

Let δ < 1 2 be a positive constant. By a compactness argument, we only have to prove it for |x| > 1. We compute

L α (g)(x) g(x) = R d 1 + |x| d+2α 1 + |x + h| d+2α -1 β(x, h |h| )dh |h| d+2α ≤ R d \[B(-x,δ|x|)∪B(0,δ)] 1 + |x| d+2α 1 + |x + h| d+2α -1 β(x, h |h| )dh |h| d+2α + B(-x,δ|x|)\B(0,δ) 1 + |x| d+2α 1 + |x + h| d+2α -1 β(x, h |h| )dh |h| d+2α + B(0,δ) 1 + |x| d+2α 1 + |x + h| d+2α -1 β(x, h |h| )dh |h| d+2α = I 1 + I 2 + I 3 .
Let us begin by approximating I 1 .

I 1 = R d \[B(-x,δ|x|)∪B(0,δ)] 1 + |x| d+2α 1 + |x + h| d+2α -1 β(x, h |h| )dh |h| d+2α ≤ R d \[B(-x,δ|x|)∪B(0,δ)] C δ d+2α -1 β(x, h |h| )dh |h| d+2α ≤ (C + 1) δ d+2α B mes(S d-1 ) +∞ δ dh |h| 1+2α = C δ d+4α .
For I 2 , we write :

I 2 = B(-x,δ|x|)\B(0,δ) 1 + |x| d+2α 1 + |x + h| d+2α -1 β(x, h |h| )dh |h| d+2α ≤ B B(-x,δ|x|)\B(0,δ) 1 + |x| d+2α 1 + |x + h| d+2α -1 dh |h| d+2α ≤ B B(-x,δ|x|)\B(0,δ) |x| d+2α -|x + h| d+2α 1 + |x + h| d+2α dh |h| d+2α ≤ B B(-x,δ|x|)\B(0,δ) |x| d+2α + |x + h| d+2α 1 + |x + h| d+2α dh |h| d+2α ≤ B B(-x,δ|x|)\B(0,δ) |x| d+2α + |δx| d+2α 1 + |x + h| d+2α dh |h| d+2α ≤ C B(-x,δ|x|)\B(0,δ) 1 1 + |x + h| d+2α |x| d+2α |h| d+2α dh.
But we know that h ∈ B(-x, δ|x|)\B(0, δ), using that δ < 1 2 < |x|, we deduce that

|x|(1 -δ) ≤ |h| ≤ (1 + δ)|x| ⇒ x h ≤ 1 1 -δ .
Thus, we deduce

I 2 ≤ C (1 -δ) d+2α B(-x,δ|x|)\B(0,δ) 1 1 + |x + h| d+2α dh ≤ C (1 -δ) d+2α δ|x| 0 r d-1 1 + r d+2α dr ≤ C (1 -δ) d+2α ∞ 0 r d-1 1 + r d+2α dr.
To control I 3 , we write I 3 in the following form :

I 3 = C B(0,δ) 1 + |x| d+2α 1 + |x + h| d+2α + 1 + |x| d+2α 1 + |x -h| d+2α -2 β(x, h |h| )dh |h| d+2α . Next, we define f (x, h) := 1 + |x| d+2α 1 + |x + h| d+2α .
Since for all x ∈ R d , the map that (h → f (x, h)) is C 1+2α , we know that I 3 is well defined. Moreover for every h ∈ B(0, δ)\ {0}, when the parameter |x| tends to ∞, we have that

(f (x,h)+f (x,-h)-2)β(x, h |h| ) |h| d+2α
tends to 0. So we deduce thanks to the dominated convergence Theorem, that (x → I 3 (x)) tends to 0 when |x| tends to ∞. According to the continuity of the maps (x → f (x, h)) and (x → β(x, θ)), we deduce that the map (x → I 3 (x)) is continuous Chapter 2 : A singular limit in a fractional reaction-diffusion equation with periodic coefficients and so we conclude that I 3 is bounded independently of x. We refer to [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF] for more details (see the Annex A1). Combining the above inequalities, we obtain that there exists a constant C such that for all

x ∈ R d , |L α g(x)| ≤ Cg(x). (2.35)
Using the above inequality, we can conclude with a change of variable z = ay :

|L α g(ax)| = | R d g(ax) -g(ax + ay) |y| d+2α β(ax, y |y| )dy| = | R d g(ax) -g(ax + z) |a -1 z| d+2α a -d β(ax, a -1 z |a -1 z| )dz| = a 2α |L α (g)(ax)| ≤ Ca 2α g(ax).
Finally, we obtain

|L α g(ax)| ≤ Ca 2α g(ax).
Proof of (ii). Since all the functions involved in K are differentiable, and thanks to the dominated convergence theorem, we deduce that K is continuous. We can note the following fact :

|∇g(x)| = O(|x| -(d+2α+1) ) as |x| → +∞. (2.36)
With the change of variable x = ay, we find :

K(g(a.), χ)(x) = a 2α C P V R d (g(ax) -g( x))(χ(x) -χ(a -1 x)) |ax -x| d+2α β(x, ax -x |ax -x| )d x. (2.37) Since χ ∈ C 1 (R d ) ∩ L ∞ (R d ), β ∈ L ∞ (R d × S d-1 ) and g ∈ C 1 (R d ) ∩ L ∞ (R d ) this integral converges in R d . For x ∈ R d , we have to estimate J(x) = a 2α C P V R d (g(x) -g( x))(χ(a -1 x) -χ(a -1 x)) |x -x| d+2α β(a -1 x, x -x |x -x| )d x at point ax. We define for x ∈ R d J 1 (x) = a 2α C P V B(x,1) (g(x) -g( x))(χ(a -1 x) -χ(a -1 x)) |x -x| d+2α β(a -1 x, x -x |x -x| )d x,
and

J 2 (x) = a 2α C P V R d \B(x,1) (g(x) -g( x))(χ(a -1 x) -χ(a -1 x)) |x -x| d+2α β(a -1 x, x -x |x -x| )d x, so that J = J 1 + J 2 .
We split the proof in two parts : when |x| ≤ M and when |x| > M with M a positive constant arbitrarily large.

For |x| ≤ M , according to (2.37)

K(g(a•), χ)(x) = a 2α J(ax).
First we prove the existence of a constant C large enough such that

∀x ∈ B(0, M ), |J(x)| ≤ Cg(x). (2.38)
Since |J| is continuous, we deduce that in B(0, M ), |J| is bounded by a constant D. Thus, since g is decreasing, if we take C larger than D × (1 + M d+2α ), the assertions (2.38) holds true. Since a < 1, we conclude that for all x ∈ B(0, M ) :

| K(g(a•), χ)(x)| = a 2α |J(ax)| ≤ a 2α C 1 + |ax| d+2α ≤ a 2α-γ C 1 + |x| d+2α 1 + |ax| d+2α g(x) ≤ a 2α-γ Cg(x).
For |x| > M , we first study J 1 and then J 2 . Estimate of J 1 : From the formula (2.36), for |x| > M , since χ is C 1 (R d ) and periodic, and since γ < 1 and 2α -γ is strictly positive, we have :

|J 1 (x)| ≤ CB B(x,1) a 2α-γ |x -x| γ |x -x| d+2α sup z∈[x; x] |∇g(z)||x -x|d x ≤ C a 2α-γ |x| d+2α B(x,1) 1 |x -x| d+2α-γ-1 d x ≤ a 2α-γ D 1 1 + |x| d+2α .
Estimate of J 2 : Since χ is bounded and a < 1, we obtain :

|J 2 (x)| ≤ a 2α CB |y|≥1 g(x) |y| d+2α dy + a 2α CB |y|≥1 g(x + y) |y| d+2α dy ≤ a 2α CBg(x) + a 2α CB |y|≥ |x| 2 g(x + y) |y| d+2α dy + a 2α CB 1≤|y|≤ |x| 2 g(x + y) |y| d+2α dy ≤ a 2α CB |x| d+2α + a 2α 2 d+2α CB |x| d+2α R d g(y)dy + a 2α CB 1≤|y|≤ |x| 2 g( x 2 ) |y| d+2α dy ≤ a 2α CB |x| d+2α + a 2α 2 d+2α CB |x| d+2α R d g(y)dy + a 2α CB2 d+2α g(x) |y|≥1 1 |y| d+2α dy ≤ a 2α-γ D 2 1 + |x| d+2α .
The third line is obtained noting that for |y| ≤ |x| 2 , we have |x| 2 ≤ |x| -|y| ≤ |x + y|. Putting all together we find the existence of C such that (ii) holds.

Chapter 3 Propagation in a fractional reaction-diffusion equation in a periodically hostile environment

Ce chapitre va être publié dans Journal of Dynamics and Differential Equations dans un article co-signé avec Sepideh Mirrahimi et Jean-Michel Roquejoffre.

Introduction

Model and question

We focus on the following equation :

       ∂ t n(x, t) + (-∆) α n(x, t) = n(x, t)(1 -n(x, t)) for (x, t) ∈ Ω×]0, ∞[, n(x, t) = 0 for (x, t) ∈ Ω c × [0, ∞[, n(x, 0) = n 0 (x), (3.1)
where Ω is a periodic domain of R d that will be specified later on, n 0 a compactly supported initial data and (-∆) α the fractional Laplacian with α ∈]0, 1[ which is defined as follows :

∀(x, t) ∈ R d ×]0, +∞[, (-∆) α n(x, t) = C α P V R d n(x, t) -n(y, t) |x -y| d+2α dy where C α = 4 α Γ( d 2 + α) π d 2 |Γ(-α)|
. The main aim of this paper is to describe the propagation front associated to (3.1). We show that the stable state invades the unstable state with an exponential speed.

Equation (3.1) models the dynamic of a species subject to a non-local dispersion in a periodically hostile environment. The quantity n(x, t) stands for the density of the population at position x and time t. The fractional Laplacian describes the motion of individuals, it takes into account the possibility of "large jump" (move rapidly) of individuals from one point to another with a high rate, for instance because of human activities for animals or because of the wind for seeds. The term (1 -n(x, t)) represents the growth rate of the population at position x and time t. The originality of this model is the following,the reachable areas for the species are disconnected and periodic. Here, we assume that the regions where the species can develop itself are homogeneous.

Chapter 3 : Propagation in a fractional reaction-diffusion equation in a periodically hostile environment Many works deal with the case of a standard diffusion (α = 1, see [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF] for a proof of the passage from the non-local to the local character of (-∆) α ) with homogenous or heterogeneous environment (see [START_REF] Fisher | The wave of advance of advantageous genes[END_REF], [START_REF] Kolmogorov | A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem[END_REF], [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF] and [START_REF] Gärtner | The propagation of concentration waves in periodic and random media[END_REF]). Closer to this article, Guo and Hamel in [START_REF] Guo | Propagation and blocking in periodically hostile environments[END_REF] focus on a Fisher-KPP equation with periodically hostile regions and a standard diffusion. The authors prove that the stable state invades the unstable state in the connected component of the support of the initial data. In our work, thanks to the non-local character of the fractional Laplacian, contrary to what happens in [START_REF] Guo | Propagation and blocking in periodically hostile environments[END_REF], we show that there exists a unique non-trivial positive bounded stationary state, supported everywhere in the domain. Moreover, this steady state invades the unstable state 0 with an exponential speed.

Assumptions, notations and results

The domain Ω is a smooth non-connected periodic domain of R d i.e. Ω =

k∈Z d Ω 0 + a k , with Ω 0 a smooth bounded domain of R d and a k ∈ R d . (3.2)
We assume that

(Ω 0 + a i ) ∩ (Ω 0 + a j ) = ∅ if and only if i = j.
Moreover, if we denote e i the i th vector of the canonical basis of R d then we assume that for all k ∈ Z d there holds a k+e i -a k = a e i . Moreover, we assume that the principal eigenvalue

λ 1 of the Dirichlet operator (-∆) α -Id in Ω 0 is negative i.e. λ 1 < 0. (H1)
We also introduce the eigenvalue problem associated to the whole domain Ω. It is well known (thanks to the Krein Rutman theorem) that the principal eigenvalue λ 0 of the Dirichlet operator (-∆) α -Id in Ω is simple in the algebraic and geometric sense and moreover, the associated principal eigenfunction φ 0 , solves i.e.

       ((-∆) α -Id) φ 0 = λ 0 φ 0 in Ω, φ 0 = 0
in Ω c , φ 0 has a constant sign that can be chosen positive. (3.3) The first result of this paper ensures the existence and the uniqueness of a positive bounded stationary state n + of (3.1) :

i.e.

(-∆)

α n + = n + -n 2 + in Ω, n + = 0 in Ω c . (3.4)
Theorem 3.1. Under the assumption (H1), there exists a unique positive and bounded stationary state n + to (3.1). Moreover, we have 0 ≤ n + ≤ 1 and n + is periodic.

The existence is due to the negativity of the principal eigenvalue of the Dirichlet operator (-∆) α -Id in Ω 0 which allows to construct by an iterative method a stationary state (see [START_REF] Smoller | Shock waves and reaction-diffusion equations[END_REF] for more details). As for the uniqueness, the main step is to prove that thanks to the non-local character of the fractional Laplacian, any positive bounded stationary state behaves like δ(x

) α = dist(x, ∂Ω) α 1 Ω (x). (3.5)
Then, a classical argument (see [START_REF] Berestycki | Le nombre de solutions de certains problèmes semi-linéaires elliptiques[END_REF] and [START_REF] Berestycki | Analysis of the periodically fragmented environment model : I-Species persistence[END_REF]) relying on the maximum principle and the Hopf lemma provides the result. We should underline that the uniqueness is clearly due to the non-local character of the operator (-∆) α , and it does not hold in the case of a standard diffusion term (α = 1). A direct consequence of the existence of a stationary solution is Once we have established a unique candidate to be the limit of n(x, t) as t tends to +∞, we prove the invasion phenomena. First, we prove that starting from

n 0 ∈ C ∞ 0 (Ω) ∩ C c (R d ) and n 0 ≡ 0 (H2)
the solution has algebraic tails at time t = 1. To prove it, we provide an estimate of the heat kernel at time t = 1 for a general multi-dimensional domain which satisfies the uniform interior and exterior ball condition :

       ∂ t p(x, t) + (-∆) α p(x, t) = 0 for all (x, t) ∈ O×]0, +∞[, p(x, t) = 0 for all (x, t) ∈ O c × [0, +∞[, p(x, t = 0) = n 0 (x) ∈ C ∞ 0 (O, R + ) ∩ C 0 c (R d ), (3.6)
then there exists c > 0 and C > 0 such that for all x ∈ O,

c × min(δ(x) α , 1) 1 + |x| d+2α ≤ p(x, t = 1) ≤ C × min(δ(x) α , 1) 1 + |x| d+2α . (3.7)
Once Theorem 3.2 is established, we are able to state the main result of the paper.

Theorem 3.3. Assume (H1) and (H2). Then for all µ > 0 there exists a time t µ > 0 such that :

(i) for all c < |λ 0 | d+2α and all (x, t) ∈ {|x| < e ct } ×]t µ , +∞[ |n(x, t) -n + (x)| ≤ µ. (ii) for all C > |λ 0 | d+2α and all (x, t) ∈ |x| > e Ct ×]t µ , +∞[ |n(x, t)| ≤ µ.
We detail the general strategy to prove Theorem 3.3 in the next section. Our last result concerns the level sets of the solution n. Theorem 3.4. Let ν > 0 and Ω ν := {x ∈ Ω | δ(x) > ν}. There exists a constant c ν which depends only on ν, n 0 , Ω such that for all µ ∈]0, c ν [, there exist t µ , c µ , C µ > 0 such that

{(x, t) ∈ Ω ν ×]t µ , +∞[ | n(x, t) = µ} ⊂ (x, t) ∈ Ω ν ×]t µ , +∞[ | c µ e |λ 0 |t ≤ |x| d+2α ≤ C µ e |λ 0 |t .

Discussion on the main results

Theorem 3.2 is an application of general results about the fractional Dirichlet heat kernel estimates given for instance in [START_REF] Chen | Heat kernel estimates for the Dirichlet fractional Laplacian[END_REF] or in [START_REF] Bogdan | Heat kernel estimates for the fractional laplacian with Dirichlet conditions[END_REF]. Both of the two cited articles use a probabilistic approach. We propose in this work a deterministic proof of the lower bound of the fractional Dirichlet kernel estimates. Our proof is quite simple but the result is not as general as those presented in [START_REF] Chen | Heat kernel estimates for the Dirichlet fractional Laplacian[END_REF] and [START_REF] Bogdan | Heat kernel estimates for the fractional laplacian with Dirichlet conditions[END_REF]. In particular, it is only valid for finite time. It relies on a well adapted decomposition of the fractional Laplacian. We do not provide the proof of the upper bound of the fractional Dirichlet kernel estimates since there is no difficulties to obtain such bound.

Theorem 3.3 can be seen as a generalisation of the results of [START_REF] Coulon Chalmin | Fast propagation in reaction-diffusion equations with fractional diffusion[END_REF] or [START_REF] Léculier | A singular limit in a fractional reaction-diffusion equation with periodic coefficients[END_REF]. Indeed, if we study a non-local Fisher KPP equation in the whole domain R d with a reaction term depending on a parameter such that the reaction term becomes more and more unfavorable in Ω c then we recover Theorem 3.3. This is fully in the spirit of [START_REF] Guo | Propagation and blocking in periodically hostile environments[END_REF]. In fact, if we study the equation :

∂ t n + (-∆) α n = µ δ (x)n -n 2 in R d ×]0, +∞[, n(x, t = 0) = n 0 (x), with µ δ (x) =              1 if x ∈ Ω, 1 -(δ + 1)dist(x, Ω) if 0 < dist(x, Ω) ≤ 1 δ , - 1 δ if 1 δ < dist(x, Ω).
Then, denoting by λ δ the principal eigenvalue of the operator ((-∆) α -µ δ ) we claim that

λ δ -→ δ→0 λ 0 . (3.8)
It is then possible to obtain the result of Theorem 3.3 from such approximate problems in the spirit of [START_REF] Guo | Propagation and blocking in periodically hostile environments[END_REF]. Although we do not use such method, similar difficulties would arise to treat the problems with this approximation procedure. Our method can indeed be adapted to study those problems in a uniform way.

Strategy, comparison tools and outline of the paper

The general strategy

The general strategy to establish the results of Theorem 3.3 is the following :

A-Identify the unique candidate to be the limit. This is the content of Theorem 3.1. B-Starting from a compactly supported initial data, the solution n has algebraic tails immediatly after t = 0. This is the content of Theorem 3.2.

C-Establish a sub and a super-solution which bound the solution n from below and above. D-Use the sub-solution to "push" the solution n to the unique non-trivial stationary state

n + in |x| < e |λ 0 |t d+2α
and use the super-solution to "crush" the solution n to 0 in |x| > e |λ 0 |t d+2α .

Strategy, comparison tools and outline of the paper

The proof of C can be done with two different approaches. The first one is introduced in [START_REF] Cabré | Propagation in Fisher-KPP type equations with fractional diffusion in periodic media[END_REF] by Cabré, Coulon and Roquejoffre. The idea is to consider the quantity

v(x, t) = φ 0 (r(t)x) -1 n(r(t)x, t)
where the eigenfunction φ 0 is introduced in (3.3) and r(t) decreases exponentially fast. Next, the problem can be formally reduced to a transport equation leading to the fact that v is of the form φ 0 (x) 1+b(t)|x| d+2α . The idea is therefore to look for a sub-solution v and a supersolution v of the form

v(x, t) = aφ 0 (x) 1 + b(t)|x| d+2α and v(x, t) = aφ 0 (x) 1 + b(t)|x| d+2α
(where the positive constants a, a and the function b, b have to be adjusted).

The second approach is introduced in [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF] by Méléard and Mirrahimi (in order to extend the Geometric optics approach of [START_REF] Freidlin | Functional integration and partial differential equations[END_REF] and [START_REF] Freidlin | Limit theorems for large deviations and reaction-diffusion equations[END_REF], put to work in the PDE framework in [START_REF] Evans | A PDE approach to geometric optics for certain semilinear parabolic equations[END_REF]). The main idea is to perform the following scaling on equation (3.1)

(x, t) → |x| 1 ε x |x| , t ε , ε > 0 small. (3.9)
One of the interest of this scaling is to catch the effective behavior of the solution. Indeed, this scaling keeps invariant the set

B = (x, t) ∈ R × R + | (d + 2α) log |x| < |λ 0 |t
where λ 0 is defined by (3.3). In the region B, we expect that the solution n is close to the stationary state n + and in the region B c , we expect that the solution n is closed to 0. Then, we look for sub/super-solutions on the form

φ 0 (x) × G(x, t)
where G needs to be determined. Taking G with an algebraic tail gives that, once the scaling is performed, the fractional Laplacian of G vanishes as the parameter ε tends to 0. Indeed, the sub and super solutions are just perturbations of a simple ODE and are valid only for small ε. We choose the second method for the two reasons. First, the scaling reveals two invariant regions (B and B c ) which catch the effective behavior of the solutions. The alternative approach does not account for these features as clearly. Secondly, this method gives very clear indications to construct sub-and super-solutions. It highlights the important terms in the equation to establish sub-and super-solutions. Once the scaling is performed, the fractional Laplacian terms in the computations vanish as the parameter ε tends to 0. This means that the only role of the fractional Laplacian in determining the invasion speed is at initial time where it determines the algebraic tails of the solution. This is indeed very different from the classical Fisher-KPP equation where the diffusion not only determines the exponential tails of the solution but it also modifies the invasion speed in positive times (see [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF]). This is why in the asymptotic study of the classical Fisher KPP equation, one obtains a Hamilton-Jacobi equation [START_REF] Evans | A PDE approach to geometric optics for certain semilinear parabolic equations[END_REF] while in the fractional KPP equation the limit is a simple ordinary differential equation. Thus scaling the equation as in (3.9) is not a mere artefact, it really helps in the understanding of the phenomenon. Moreover, we improve the precision of the sub and super-solutions compared with the ones obtained in [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF]. Therefore, the exhibated sub and super-solutions and the ones that could be obtained with the alternative approach have the same level of precision. x |x| , t ε using the method of perturbed test functions from the theory of viscosity solutions and homogenization (introduced by Evans in [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF] and [START_REF] Evans | Periodic homogenisation of certain fully nonlinear partial differential equations[END_REF] and by Mirrahimi and Méléard in [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF] for the fractional Laplacian). Since the proof is technical, long and not easy to grasp (the domain moves also with the parameter ε), we prefer to drop the scaling and to perform the inverse scaling on our sub and super solutions. Therefore, we provide a direct proof of D by adapting the proof of Theorem 1.6 in [START_REF] Coulon Chalmin | Fast propagation in reaction-diffusion equations with fractional diffusion[END_REF]. In this proof, the author proves thanks to a subsolution that there exists σ > 0 and t σ > 0 such that

σ < inf (x,t)∈ |x|<e (|λ 0 |-δ)t d+2α ×]tσ,+∞[ n(x, t).
This last claim is obviously false in our case since the solution vanishes on the boundary. This is the main new difficulty that we will encounter. We overcome it by establishing the same kind of estimates away from the boundary. Theorem 3.4 is the consequence of the precision of the sub and super-solutions established in step C. Note that such level sets can not be established with the sub and super-solutions involved in [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF] or [START_REF] Léculier | A singular limit in a fractional reaction-diffusion equation with periodic coefficients[END_REF]. However, the new ingredients presented here to establish the sub and the super-solutions can be adapted to the framework of [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF] or [START_REF] Léculier | A singular limit in a fractional reaction-diffusion equation with periodic coefficients[END_REF]. In the framework of [START_REF] Léculier | A singular limit in a fractional reaction-diffusion equation with periodic coefficients[END_REF], we recover Theorem 1.1.5 of [START_REF] Coulon Chalmin | Fast propagation in reaction-diffusion equations with fractional diffusion[END_REF].

The comparison tools and some notations

∀(x, t) ∈ Ω×]a, b[, ∂ t n + (-∆) α n ≤ f (n),
and

∂ t n + (-∆) α n ≥ f (n), ∀(x, t) ∈ Ω c ×]a, b[, n(x, t) ≤ n(x, t), and ∀x ∈ R d , n(x, t = a) ≤ n(x, t = a) then ∀(x, t) ∈ Ω×]a, b[, n(x, t) ≤ n(x, t).
In the same spirit, we recall the fractional Hopf Lemma stated in [START_REF] Greco | Hopf's lemma and constrained radial symmetry for the fractional Laplacian[END_REF].

Lemma (The fractional Hopf Lemma [START_REF] Greco | Hopf's lemma and constrained radial symmetry for the fractional Laplacian[END_REF]).

lim inf x →z x∈O u(x) δ(x) α > 0.
All along the article, for any set U and any positive constant ν, we introduce the following new sets :

U ν = {x ∈ U| dist(x, ∂U) > ν} , U -ν = x ∈ R d | dist(x, U) < ν .
(3.10)

The constants denoted by c or C may change from one line to another when there is no confusion possible. Also, we drop the constant C α and the Cauchy principal value P.V. in front of the fractional Laplacian for better readability.

Outline of the paper

In section 3, we demonstrate Theorem 3.1. Next, section 4 is dedicated to the proof of Theorem 3.2. The first part of section 5 introduces the scaling and provides the sub and super-solutions. Finally, the second part of section 5 is devoted to the proof of Theorems 3.3 and 3.4.

Uniqueness of the stationary state n +

First, we state a proposition which gives the shape of any non-trivial bounded sub and super-solution to (3.4) near the boundary. Then, we use this result to prove the uniqueness result. Since the proof of the existence is classical we do not provide it.

Proposition 3.1. (i) If u is a smooth positive bounded function such that u(x) = 0 for all x ∈ Ω c and (-∆) α u(x) ≤ u(x) -u(x) 2 for all x ∈ Ω, then there exists C > 0 such that for all x ∈ R d u(x) ≤ Cδ(x) α . (ii) If v is a smooth positive bounded function such that v(x) = 0 for all x ∈ Ω c , (-∆) α v(x) ≥ v(x) -v(x) 2 for all x ∈ Ω and v ≡ 0 then there exists c > 0 such that for all x ∈ R d cδ(x) α ≤ v(x).
Proof of Proposition 3.1. Proof of (i). Let u be a continuous positive bounded function such that u = 0 in Ω c and (-∆) α u ≤ u -u 2 in Ω. Let x be a point of the boundary. Let z x ∈ R d and r 1 > 0 be the elements provided by the uniform exterior ball condition such that B(z x , r 1 ) ⊂ Ω c and x ∈ B(z x , r 1 ) ∩ ∂Ω.

We rescale and translate a barrier function (provided for instance in Annex B of [START_REF] Ros-Oton | The Dirichlet problem for the fractional Laplacian : regularity up to the boundary[END_REF]). This barrier function φ satisfies the following properties :

             (-∆) α φ ≥ 1 in B(z x , 4r 1 )\B(z x , r 1 ), φ ≡ 0 in B(z x , r 1 ), 0 ≤ φ ≤ C(|z x -x| -r) α in B(z x , 4r 1 )\B(z x , r 1 ), max u ≤ φ ≤ C in R d \B(z x , 4r 1 ). (3.11)
We prove that u ≤ φ in R d . By construction we have u ≤ φ in (B(z x , 4r 1 )\B(z x , r 1 )) c . Assume by contradiction that there exists

x 0 ∈ (B(z x , 4r 1 )\B(z x , r 1 )) ∩ Ω such that (φ -u)(x 0 ) < 0.
Then, there exists

x 1 ∈ (B(z x , 4r 1 )\B(z x , r 1 )) ∩ Ω such that (φ -u)(x 1 ) = min x∈R d (φ -u)(x) < 0. Thus, we obtain (-∆) α (φ -u)(x 1 ) < 0 and (-∆) α (φ -u)(x 1 ) ≥ 1 -u(x 1 ) + u(x 1 ) 2 ≥ 0, a contradiction.
Proof of (ii). Let v be a continuous positive bounded function such that v = 0 in Ω c and (-∆) α v ≥ v -v 2 in Ω. An easy but important remark is the following : thanks to the Chapter 3 : Propagation in a fractional reaction-diffusion equation in a periodically hostile environment non-local character of the fractional Laplacian, since v ≡ 0, we deduce that v > 0 in the whole domain Ω. Otherwise, the following contradiction holds true :

∃x ∈ Ω such that v(x) = 0 and (-∆) α v(x) -v(x) + v(x) 2 = - R d v(y) |x -y| d+2α dy < 0.
Next, let k be any element of Z d . We introduce

w k : (x, t) ∈ R d × [0, +∞[ → w k (x, t) ∈ R as the solution of        ∂ t w k + (-∆) α w k = w k -w 2 k in (Ω 0 + a k )×]0, +∞[, w k (x, t) = 0 in R d \(Ω 0 + a k ) × [0, +∞[ w k (x, 0) = v(x) in (Ω 0 + a k ), (3.12)
where Ω 0 and a k are introduced in (3.2). Thanks to the remark above, and recalling (H1), we deduce thanks to Theorem 5.1 in [START_REF] Berestycki | The periodic patch model for population dynamics with fractional diffusion[END_REF] that w k (., t) -→ t→+∞ w stat (.) with w stat the solution of

   (-∆) α w stat = w stat -w 2 stat , in (Ω 0 + a k ), w stat = 0 in R d \(Ω 0 + a k ). (3.13)
Note that the above w stat does not depend on the choice of k, i.e. w k (•, t) converges as t tends to +∞ to the same w stat (up to a translation). Then, we conclude thanks to the comparison principle that

w stat (x) ≤ v(x), ∀x ∈ R d .
Since, (Ω 0 + a k ) is bounded, we apply the results of [START_REF] Ros-Oton | The Dirichlet problem for the fractional Laplacian : regularity up to the boundary[END_REF] to find that there exists a constant c > 0 such that cδ(x

) α 1 (Ω 0 +a k ) (x) ≤ w stat (x) ≤ v(x).
The previous analysis holds for every k ∈ Z d . We conclude that

cδ(x) α ≤ v(x). (3.14)
Proof of Theorem 3.1. The argument relies on the fact that two steady solutions are comparable everywhere thanks to Proposition 3.1. This is in the spirit of [START_REF] Berestycki | Le nombre de solutions de certains problèmes semi-linéaires elliptiques[END_REF] and [START_REF] Berestycki | Analysis of the periodically fragmented environment model : I-Species persistence[END_REF] in the context of standard diffusion. Let u and v be two bounded steady solutions of (3.4). By the maximum principle, we easily have that for all x ∈ R d , u(x) ≤ 1 and v(x) ≤ 1.

We will assume that v(x 0 ) ≤ u(x 0 ) where x 0 ∈ Ω 0 .

(3.15)

Thanks to Proposition 3.1, we deduce the existence of two constants 0 < c ≤ C such that :

cδ(x) α ≤ u(x) ≤ Cδ(x) α and cδ(x) α ≤ v(x) ≤ Cδ(x) α .
Thus there exists a constant λ > 1 such that for all x ∈ R d ,

u(x) ≤ λv(x). ( 3
:= inf λ ≥ 1| ∀x ∈ R d , u(x) ≤ λv(x) .
The point is to prove by contradiction that l 0 = 1. It implies that x 0 is a contact point, and will allow us to conclude thanks to the fractional maximum principle that u = v. We assume by contradiction that l 0 > 1. Next, we define :

w = inf x∈Ω (l 0 v -u)(x) δ(x) α ≥ 0. (3.17)
There are two cases to be considered. Case 1 : w > 0. We show in this case that we can construct l 1 < 1 such that u(x) ≤ l 1 l 0 v(x) for all x ∈ R d : a contradiction. If w > 0, we claim that there exists µ ∈]0, 1[ and ν > 0 such that for all x ∈ Ω\Ω ν (we recall that Ω ν is defined by (3.10)),

w 2 ≤ (µl 0 v -u)(x) δ(x) α . (3.18)
Indeed, if there does not exist such couple (µ, ν), we deduce that for all n ∈ N, there exists

(x n ) n∈N ∈ Ω, such that δ(x n ) ≤ 1 n and ((1 -1 n )l 0 v -u)(x n ) δ(x n ) α < w 2 .
Passing to the liminf we get the following contradiction :

0 < w ≤ w 2 .
And so, the existence of the couple (µ, ν) implies that

(µl 0 v -u)(x) ≥ 0, ∀x ∈ Ω\Ω ν . (3.19)
Next, we claim that

∃ρ > 0 such that ∀x ∈ Ω ν , we have ρ ≤ (l 0 v -u)(x). (3.20)
Indeed, if such ρ does not exist then there exists a sequence (

x n ) n∈N ∈ Ω such that δ(x n ) ≥ ν and (l 0 v -u)(x n ) -→ n→+∞ 0. Then we obtain (l 0 v -u)(x n ) δ(x n ) α ≤ (l 0 v -u)(x n ) ν α -→ n→+∞ 0
which is in contradiction with the hypothesis w > 0. The existence of such ρ implies that for all x ∈ Ω ν

(1 - ρ max l 0 v )l 0 v -u (x) ≥ 0. (3.21)
Finally, if we define l 1 = max(µ, 1 -ρ max l 0 v+1 ) then we obtain the desired contradiction. Therefore this case cannot occur.

Chapter 3 : Propagation in a fractional reaction-diffusion equation in a periodically hostile environment Case 2 : w = 0. We consider (x n ) n∈N a minimizing sequence of w. There are 3 subcases : a subsequence of (x n ) n∈N converges to x 0 ∈ Ω, a subsequence of (x n ) n∈N converges to x b ∈ ∂Ω and any subsequence of (x n ) n∈N diverges.

Subcase a : There exists x 0 ∈ Ω, such that (l 0 v-u)(x 0 ) δ(x 0 ) α = 0. Since x 0 ∈ Ω we deduce that (l 0 v -u)(x 0 ) = 0. Hence, by the maximum principle, u = l 0 v. We deduce that l 0 v is a solution of (3.4) and we conclude that :

l 0 (v -v 2 ) = l 0 (-∆) α (v) = (-∆) α (l 0 v) = l 0 v -(l 0 v) 2 . (3.22)
This equation leads to l 0 = 1, a contradiction.

Subcase b :

There exists x b ∈ ∂Ω, such that lim inf

x→x b , x∈Ω (l 0 v-u)(x) δ(x) α = 0.
Here is a summary of what we know :

(i) l 0 v -u ≥ 0, (ii) (-∆) α (l 0 v -u) ≥ -l 0 (l 0 v -u), (iii) (l 0 v -u)(x b ) = 0.
According to the fractional Hopf Lemma, the previous assumptions leads to lim inf

x→x b , x∈Ω (l 0 v -u)(x) δ(x) α > 0.
However, we have assumed that lim inf

x→x b , x∈Ω (l 0 v-u)(x) δ(x) α = 0, a contradiction.
Subcase c : There exists a minimizing sequence (x n ) n∈N such that |x n | tends to the infinity. First, we set

x k = x k -a x k ,
where x ∈ Z d is such that x ∈ Ω 0 + a x . Since x k ∈ Ω 0 , we deduce that up to a subsequence x k converges to x ∞ ∈ Ω 0 . Then we define :

u k (x) = u(x + x k ) and v k (x) = v(x + x k ).
We also define the following set :

Ω ∞ = {x ∈ R | x + x ∞ ∈ Ω} .
By fractional elliptic regularity (see [START_REF] Ros-Oton | Regularity theory for general stable operators[END_REF]), we deduce that up to a subsequence (u n ) n∈N and

(v n ) n∈N converges to u ∞ and v ∞ solutions that verifies ∀x ∈ Ω ∞ , (-∆) α u ∞ (x) = u ∞ (x) -u ∞ (x) 2 , (-∆) α v ∞ (x) = v ∞ (x) -v ∞ (x) 2 and ∀x ∈ Ω c ∞ , u ∞ (x) = v ∞ (x) = 0. Remark that l 0 v ∞ -u ∞ ≥ 0 and lim inf x→0 x∈Ω∞ (l 0 v ∞ -u ∞ )(x) dist(x, ∂Ω ∞ ) α = 0. Hence, if x ∞ ∈ Ω 0 then 0 ∈ Ω ∞ and we fall in the subcase a). If x ∞ /
∈ Ω 0 then 0 ∈ ∂Ω ∞ and we fall in the subcase b). Both cases lead to a contradiction.

Thus, we conclude that l 0 = 1.

3.4. The fractional heat kernel and the preparation of the initial data

Remark.

Noticing that for all (x, k) ∈ Ω × Z d , we have

(-∆) α (n + (. + a k ))(x) = R n + (x + a k ) -n + (y + a k ) |x + a k -(y + a k )| d+2α dy = n + (x + a k ) -n + (x + a k ) 2 ,
we deduce by uniqueness of the solution of (3.4) that n + is periodic.

The fractional heat kernel and the preparation of the initial data

We first introduce some requirements in order to achieve the proof of the lower bound of Theorem 3.2. Once we have established Theorem 3.2, we apply it to the initial data. Let

u ∈ C ∞ (R d ×]0, +∞[), then we set for all (x, t) ∈ R d ×]0, +∞[ L α (u)(x, t) = B(0,ν) u(x, t) -u(y, t) |y| d+2α dy. (3.23)
We also introduce φ ν as the principal positive eigenfunction of the operator L α associated to the principal eigenvalue µ ν i.e.

         L α φ ν = µ ν φ ν in B(0, ν) φ ν = 0 in B(0, ν) c , φ ν ≥ 0, φ ν ∞ = 1.
Next, we state two intermediate technical results.

Lemma 3.1. Let w be the solution of the equation

       ∂ t w + L α w = 1 in B(0, ν)×]0, +∞[ w(x, t) = 0 in B(0, ν) c × [0, +∞[, w(x, t = 0) = 0 in B(0, ν). (3.24)
Then there exists a constant c ν > 0 such that

c ν × φ ν (x) ≤ w(x, t = 1). Proof. We define τ (t) = 1 µ ν (1 -e -µν t ) such that τ (t) + µ ν τ (t) = 1, τ (0) = 0.
Thanks to this choice of τ (t), the application w(x, t) := τ (t) × φ ν (x) is a sub-solution to (3.24). Actually, we have

(∂ t + L α )(w) -1 = τ φ ν + µ ν τ φ ν -1 ≤ τ φ ν + µ ν τ φ ν -φ ν = φ ν (τ + µ ν τ -1) = 0.
Since w(t = 0) = 0 ≤ w(t = 0), we can conclude thanks to the comparison principle that for all (x, t) ∈ R d × [0, +∞[, we have w(x, t) ≤ w(x, t). Setting the time t = 1 in the last inequality leads to

w(x, 1) = 1 µ ν (1 -e -µν )φ ν (x) = c ν φ ν (x) ≤ w(x, 1).
Chapter 3 : Propagation in a fractional reaction-diffusion equation in a periodically hostile environment

Next, we establish a barrier function for L α in the spirit of the one introduced in [START_REF] Ros-Oton | The Dirichlet problem for the fractional Laplacian : regularity up to the boundary[END_REF].

Lemma 3.2. There exists a function ψ such that

                           L α ψ ≤ 0 in B(0, ν)\B(0, ν 2 ), ψ = 0 in B(0, ν) c , ψ ≤ 1 in B(0, ν 2 ), c(ν -|x|) α ≤ ψ in B(0, ν), ψ is continuous in B(0, ν)\B(0, ν 2 ).
(3.25)

Proof. Choose C large enough such that the first point and the third point of (3.25) holds true with the following ψ :

ψ(x) := 1 C (ν 2 -|x| 2 ) α + 1 2 1 B(0, ν 4 ) (x) 1 B(0,ν) (x).
Indeed, defining f (x) := (ν 2 -|x| 2 ) α , we have for C large enough and x ∈ B(0, ν)\B(0, ν 2 )

L α ψ(x) ≤ L α f (x) C - 1 2 B(0, ν 4 ) 1 |x -y| d+2α dy ≤ sup B(0,ν)\B(0, ν 2 ) |L α f | C - m(B(0, ν 4 )) 2 × 4 ν d+2α < 0.
The other conditions follow.

Proof of Theorem 3.2. The aim is to prove that there exists a constant c > 0 such that ∀x ∈ O, we have c min(δ(x) α , 1) 1 + |x| d+2α ≤ p(x, 1). (3.26) To achieve the proof, there will be 4 steps. First, up to a translation and possibily a scaling of n, we prove (3.26) in {|x| < 1 + 2ν} where ν = min( 1 4 , r 1 ) (with r 1 the radius provided by the uniform interior ball). Next, we introduce a suitable decomposition of the fractional Laplacian (involving L α ) to prove the existence of

c 1 > 0 such that            c 1 1 + |x| d+2α ≤ ∂ t p(x, t) + L α p(x, t) + λp(x, t) for all (x, t) ∈ (O\ {|x| > 1 + ν}) ×]0, 1], p(x, t) ≥ 0 for all (x, t) ∈ (O\ {|x| > 1 + ν}) c × [0, 1], p(x, t = 0) = n 0 (x) ∈ C ∞ 0 (O, R + ) ∩ C c (R d ) (3.27)
where L α is defined by (3.23) and λ = R d \B(0,ν) 1 |y| d+2α dy. In a third step, we will show that

∃c 2 > 0 such that c 2 1 + |x| d+2α ≤ p(x, t = 1) for all x ∈ (O ν ∩ {|x| > 1 + 2ν}) .
(3.28)

Finally, we prove the same kind of result near the boundary :

∃c 3 > 0 such that c 3 δ(x) α 1 + |x| d+2α ≤ p(x, t = 1) for all x ∈ (O\O ν ∩ {|x| > 1 + 2ν}) . (3.29)
3.4. The fractional heat kernel and the preparation of the initial data

Step 1. First, note that thanks to a translation and possibly a scaling, we can suppose the following hypothesis :

∃σ > 0 such that σ < n 0 (x) for all x ∈ B(0, 2).

(3.30)

Next, we claim that inf t∈(0,1) z∈B(0,1+2ν) p(z, t) > 0.

(3.31) Indeed, let φ 2 be the first positive eigenfunction of the Dirichlet fractional Laplacian in B(0, 2) and λ 2 the associated eigenvalue i.e.

       (-∆) α φ 2 = λ 2 φ 2 for x ∈ B(0, 2), φ 2 = 0 for x ∈ B(0, 2) c , φ 2 ∞ = 1. Then the function p(x, t) := σ × φ 2 (x) × e -λ 2 t
is a sub-solution to (3.6) (where σ is defined by (3.30)). According to the comparison principle, we have for all (x, t) ∈ B(0,

1 + 2ν) × [0, 1] 0 < min s∈ [0,1] y∈B(0,1+2ν) p(y, s) = σ × min B(0,1+2ν) φ 2 × e -|λ 2 | ≤ p(x, t) ≤ p(x, t).
We deduce that if c is small enough, then (3.26) holds true for all x ∈ B(0, 1 + 2ν).

Step 2. In this step we prove (3.27) which is a key element to prove (3.26) for x ∈ {|x| > 1 + 2ν} ∩ O .

Then, we focus on {|x| > 1 + ν}. We split the fractional Laplacian into 2 parts :

(-∆) α p(x, t) = R d \B(0,ν) p(x, t) -p(x + y, t) |y| d+2α dy + L α p(x, t) = I 1 (x, t) + L α p(x, t). (3.32)
For I 1 , we obtain :

I 1 (x, t) = R d \B(0,ν) p(x, t) -p(x + y, t) |y| d+2α dy = λp(x, t) - R d \B(0,ν) p(x + y, t) |y| d+2α dy. Since |x| > 1 + ν, we have inf t∈(0,1) z∈B(0,1+ν) p(z, t) B(0,1) 1 |z -x| d+2α dz ≤ B(-x,1) p(x + y, t) |y| d+2α dy ≤ R d \B(0,ν) p(x + y, t) |y| d+2α dy.
(3.33) Equation (3.33) ensures the existence of a positive constant c 1 > 0 such that we have for all (x, t)

∈ (O ∩ {|x| > 1 + ν}) × [0, 1[ c 1 1 + |x| d+2α ≤ R d \B(0,ν) p(x + y, t)
|y| d+2α dy.

It follows that

I 1 (x, t) ≤ λp(x, t) - c 1 1 + |x| d+2α . ( 3
           c 1 1 + |x| d+2α ≤ ∂ t v(x, t) + L α v(x, t) for (x, t) ∈ (O ∩ {|x| > 1 + ν}) ×]0, 1], v(x, t) ≥ 0 for (x, t) ∈ (O ∩ {|x| > 1 + ν}) c × [0, 1], v(x, t = 0) = n 0 (x) ∈ C ∞ 0 (O, R + ).
(3.35)

Step 3. By uniform continuity of x → 1 1+|x| d+2α in R d , we deduce the existence of c 1 > 0 such that for all x 0 ∈ (O ν ∩ {|x| > 1 + ν}) and all (x, t)

∈ (O ν ∩ {|x| > 1 + 2ν}) ×]0, 1] we have c 1 1 + |x 0 | d+2α 1 B(0,ν) (x -x 0 ) ≤ c 1 1 + |x| d+2α ≤ ∂ t v(x, t) + L α v(x, t).
(3.36)

Inequality (3.36) gives that for all (x, t)

∈ (O ν ∩ {|x| > 1 + ν}) ×]0, 1] 1 B(0,ν) (x -x 0 ) ≤ ∂ t ( 1 + |x 0 | d+2α c 1 v(x, t)) + L α ( 1 + |x 0 | d+2α c 1 v(x, t)).
Then, according to the comparison principle and Lemma 3.1, we deduce that

∀x ∈ (O ν ∩ {|x| > 1 + ν}) , c ν φ ν (x -x 0 ) ≤ 1 + |x 0 | d+2α c 1 v(x, t = 1). (3.37)
If we evaluate (3.37) at x = x 0 , we obtain

c ν c 1 e -λ φ ν (0) 1 + |x 0 | d+2α ≤ p(x 0 , t = 1).
Defining c 2 = c ν c 1 e -λ φ ν (0) leads to (3.28).

Step 4. As in the proof of Proposition 3.1, we can show by contradiction that there exists a positive constant c 0 such that for all x ∈ R d ,

c 0 ψ(x) ≤ φ ν (x)
where ψ is defined in Lemma 3.2. Then we take x 1 ∈ (O\O ν ) ∩ {|x| > 1 + 2ν}. Since O satisfies the uniform interior ball condition, there exists

x 0 ∈ ∂O ν such that x 1 ∈ B(x 0 , ν), B(x 0 , ν) ⊂ O ∩ {|x| > 1 + ν} and ν -|x 1 -x 0 | = δ(x 1
). Thanks to (3.37) and the fourth point of Lemma 3.2, we deduce

c ν c 0 cνδ(x 1 ) α ≤ c ν c 0 ψ(x 1 -x 0 ) ≤ c ν φ ν (x 1 -x 0 ) ≤ (|x 0 | + 1) d+2α c 1 v(x 1 , t = 1).
We deduce that there exists c 3 > 0 such that (3.29) holds true.

Combining (3.28), (3.29) and (3.31) yields the conclusion of the Theorem.

We apply Theorem 3.2 to show that starting from n(x, 0)

∈ C ∞ 0 (Ω) ∩ C ∞ c (R)
, the solution of (3.1) n(•, t = 1) has algebraic tails.

Proposition 3.2.

There exists two constants c m and c M depending on n 0 such that for all x ∈ Ω, we have

c m δ(x) α 1 + |x| d+2α ≤ n(x, 1) ≤ c M δ(x) α 1 + |x| d+2α . (3.38)
Proof. Defining M := max(max n 0 , 1), the solution n belongs to the set [0, M ] (0 is a subsolution and M is a super-solution).

3.5. The proof of Theorem 3.3

We begin with the proof that cmδ(x) α 1+|x| d+2α ≤ n(x, 1). Let n be the solution of :

       ∂ t n(x, t) + (-∆) α n(x, t) = -M n(x, t) for all (x, t) ∈ Ω×]0, +∞[, n(x, t) = 0 for all (x, t) ∈ Ω c × [0, +∞[, n(x, 0) = n 0 (x) for all x ∈ R d , (3.39)
Thanks to the comparison principle, we deduce that for all (x, t) ∈ R × [0, +∞[, we have

n(x, t) ≤ n(x, t).
Moreover, if we define p(x, t) = e M t n(x, t), we find that p is solution of (3.6). Since Ω fulfils the uniform interior and exterior ball condition, we deduce thanks to Theorem 3.2 that there exists c m > 0 such that

c m δ(x) α 1 + |x| d+2α ≤ n(x, t = 1) ≤ n(x, t = 1).
(3.40)

The proof works the same for the other bound.

3.5

The proof of Theorem 3.3

Rescaling and preparation

The aim of this subsection is to establish the following Theorem.

Theorem 3.5. We assume (H1) and (H2) then for all ν > 0, the following holds true 1. For all c < |λ 0 | d+2α , there exists a constant σ > 0 and a time t σ > 0 such that

∀(x, t) ∈ Ω ν ∩ |x| < e ct ×]t σ , +∞[ we have σ < n(x, t). (3.41)
2. For all C > |λ 0 | d+2α , there exists three constants C 1 , C 2 , κ > 0 such that we have for all

(x, t) ∈ |x| > e Ct ×]1, +∞[ n(x, t) ≤ C 1 1 + C 2 e κt .
(3.42)

First we establish sub and super-solutions by performing the rescaling (3.9). Finally, we prove Theorem 3.5 by performing the inverse of this rescaling on the sub and super-solutions.

We rescale the solution of (3.1) as follows :

n ε (x, t) = n |x| 1 ε x |x| , t ε . (3.43)
Next, the equation becomes

       ε∂ t n ε + (-∆) α ε n ε = n ε (1 -n ε ) for (x, t) ∈ Ω ε ×]0, +∞[, n ε (x, t) = 0 for (x, t) ∈ Ω ε c ×]0, +∞[, n ε (x, 0) = n 0,ε (x) ∈ C ∞ 0 (Ω ε , R + ).
(1 ε )

Chapter 3 : Propagation in a fractional reaction-diffusion equation in a periodically hostile environment

where (-∆) α ε n ε (x, t) = (-∆) α n |x| 1 ε x |x| , t ε and Ω ε = x ∈ R d | |x| 1 ε -1 x ∈ Ω . Next, we set g(x) := 1 1 + |x| d+2α .
We state the behavior of g under the fractional Laplacian in the spirit of [START_REF] Cabré | Propagation in Fisher-KPP type equations with fractional diffusion in periodic media[END_REF].

Lemma 3.3. Let γ be a positive constant in ]0, α[ such that 2α -γ < 1. Let χ ∈ C α (R d
) be a periodic positive function. Then there exists a positive constant C, such that we have for all x ∈ R : (i) for all a > 0,

|(-∆) α g(ax)| ≤ a 2α Cg(ax), (ii) for all a ∈]0, 1], | K(g(a.), χ)(x)| ≤ Ca 2α-γ 1 + (a|x|) d+2α = Ca 2α-γ g(a|x|), where K(u, v)(x) = R (u(x)-u(y))(v(x)-v(y)) |x-y| d+2α
dy is such that

(-∆) α (u × v) = (-∆) α (u) × v + u × (-∆) α (v) -K(u, v).
Since, the same kind of result can be found in the appendix A of [START_REF] Léculier | A singular limit in a fractional reaction-diffusion equation with periodic coefficients[END_REF], we do not provide the proof of this lemma. Note that here, the lemma is stated with less regularity on χ such than in [START_REF] Léculier | A singular limit in a fractional reaction-diffusion equation with periodic coefficients[END_REF]. Nevertheless, there is no difficulty to adapt the proof.

Notation. As we have introduced

(-∆) α ε n ε (x, t) = (-∆) α n |x| 1 ε x |x| , t ε , we introduce K ε (u, v)(x, t) = K(u, v)(|x| 1 ε -1 , t ε ).
For any application h : R → R, we define

h ε : R d → R x → h(|x| 1 ε -1 x).
For any set U, we will denote

U ε = x ∈ R d | |x| 1 ε -1 x ∈ U . (3.44)
For reasons of brevity, we will always denote (U ν ) ε by U ε ν . In the following, we denote by c 0 and C 0 the positive constants provided by [START_REF] Ros-Oton | The Dirichlet problem for the fractional Laplacian : regularity up to the boundary[END_REF] such that ) then there exists ε 0 > 0 such that for all ε < ε 0 , the following holds true.

c 0 δ(x) α ≤ φ 0 (x) ≤ C 0 δ(x) α . ( 3
1

. For t ∈]0, 4 ε 2 [, if f m ε is defined as f m ε (x, t) = C m min(e -1 ε + εt 4 , 1) 1 + e -(|λ 0 |-ε 2 )t ε -1 ε |x| d+2α ε × (φ 0,ε (x) + ε), then it is a sub-solution of (1 ε ) in Ω ε ×]0, 4 ε 2 [. 3.5. The proof of Theorem 3.3 2. For t ∈] 4 ε 2 , +∞[, if f m ε is defined as f m ε (x, t) = C m 1 + e -|λ 0 |t ε +ε arctan(t-4 ε 2 )+ 3 ε |x| d+2α ε × (φ 0,ε (x) + ε), then it is a sub-solution of (1 ε ) in Ω ε ×] 4 ε 2 , +∞[. 3. If f M ε is defined as f M ε (x, t) = C M × φ 0,ε (x) 1 + e -|λ 0 |t ε -ε arctan(t)-1 ε |x| d+2α ε , then it is a super-solution of (1 ε ) in Ω ε ×]0, +∞[. 4. For all (x, t) ∈ R d × [0, +∞[, f m ε (x, t) ≤ n ε (x, t + ε) + ε and n ε (x, t + ε) ≤ f M ε (x, t).
Remark.

1. In the establishment of the sub and the super solutions, the choice of arctan is not primordial. We only need a positive increasing and smooth function h which satisfies e

-t ε ≤ εh (t)
for all t > t 0 and ε small enough. In [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF], h(t) = t but it does not allow to recover the level set of the solution as precisely as in [START_REF] Coulon Chalmin | Fast propagation in reaction-diffusion equations with fractional diffusion[END_REF]. The choice of a bounded function h (such as arctan) allows to recover the same level of precision in the establishment of the level sets.

2. The study (and the definition) of f m ε is split into two parts. For small time, the term e -1 ε in the denominator is necessary in order to control the term (-∆) α f m ε for small time. But, to use the comparison principle (and establish 4.), we have to check that the initial data are ordered in the right way. This is why, the term e -1 ε is needed in the numerator. However, this last term is an obstacle to establish the level sets result. Therefore, the trick is to "kill" this term for large time by replacing it by min(e εt 4 -1 ε , 1). This is why, we split the study of f m ε into two parts : when t is small (i.e. t < 4 ε 2 ) and when t is large (i.e. t > 4 ε 2 ).

Proof. We begin by proving (1). Let (x, t) be in Ω ε ×]0, 4 ε 2 [. We define :

ψ ε (x, t) = C m 1 + e -t ε (|λ 0 |-ε 2 )-1 ε |x| d+2α ε = C m g ε (e -t(|λ 0 |-ε 2 )-1 d+2α
x) and a(t

) = e -1 ε + εt 4 thus f m ε (x, t) = a(t) × ψ ε (x, t) × (φ 0,ε (x) + ε).
First, we bound ε∂ t ψ ε from above :

ε∂ t ψ ε (x, t) = ε C m (|λ 0 |-ε 2 ) ε e -t ε (|λ 0 |-ε 2 )-1 ε |x| d+2α ε (1 + e -t ε (|λ 0 |-ε 2 )-1 ε |x| d+2α ε ) 2 = ψ ε (x, t)   (|λ 0 | -ε 2 ) e -t ε (|λ 0 |-ε 2 )-1 ε |x| d+2α ε 1 + e -t ε (|λ 0 |-ε 2 )-1 ε |x| d+2α ε   ≤ ψ ε (x, t)[|λ 0 | -ε 2 -ψ ε (x, t)(φ 0,ε (x) + ε)] ≤ ψ ε (x, t)[|λ 0 | -ε 2 -f ε (x, t)].
(3.46)

Chapter 3 : Propagation in a fractional reaction-diffusion equation in a periodically hostile environment

The last inequalities hold because a(t) ≤ 1 and denoting by

D = e -t ε (|λ 0 |-ε 2 )-1 ε |x| d+2α ε
and using the definition of C m , we obtain for all ε < min( |λ 0 | 2 , 1)

|λ 0 | -ε 2 -ψ ε (φ 0,ε + ε) -|λ 0 | -ε 2 D 1 + D = |λ 0 | -ε 2 -C m (φ 0,ε + ε) 1 + D ≥ |λ 0 | -ε 2 -|λ 0 | 2 1 + D ≥ 0. Next, we compute (-∆) α ε f m ε (x, t) (-∆) α ε f m ε (x, t) =a(t)(φ 0,ε (x) + ε)(-∆) α ε ψ ε (x, t) + a(t)ψ ε (x, t)(-∆) α ε φ 0,ε (x) -a(t) K ε (ψ, (φ 0 + ε))(x, t).
Combining (3.46) and the above equality we find :

ε∂ t f m ε (x, t) + (-∆) α ε f m ε (x, t) -f m ε (x, t)(1 -f m ε (x, t)) ≤ f m ε (x, t)(|λ 0 | - 3ε 2 4 -f m ε (x, t)) + a(t)(φ 0,ε (x) + ε)(-∆) α ε ψ ε (x, t) + a(t)ψ ε (x, t)(-∆) α ε φ 0,ε (x) -a(t) K ε (ψ, φ 0 + ε)(x, t) -f m ε (x, t)(1 -f m ε (x, t)) = f m ε (x, t)(|λ 0 | - 3ε 2 4 ) + a(t)(φ 0,ε (x) + ε)(-∆) α ε ψ ε (x, t) + a(t)(λ 0 + 1)ψ ε (x, t)φ 0,ε (x) -f m ε (x, t) -a(t) K ε (ψ, φ 0 + ε)(x, t) = f m ε (x, t)(|λ 0 | - 3ε 2 4 ) + a(t)(λ 0 + 1)ψ ε (x, t)(φ 0,ε (x) + ε)(x, t) -f m ε (x, t) -a(t)ε(λ 0 + 1)ψ ε (x, t) + a(t)(φ 0,ε (x) + ε)(-∆) α ε ψ ε (x, t) -a(t) K ε (ψ, φ 0 + ε)(x, t) = - 3ε 2 4 f m ε (x, t) -a(t)ε(λ 0 + 1)ψ ε (x, t) + a(t)(φ 0,ε (x) + ε)(-∆) α ε ψ ε (x, t) -a(t) K ε (ψ, φ 0 + ε)(x, t).
(3.47) Thanks to Lemma 3.3, we obtain

|(-∆) α ε ψ ε (x, t)| = |C m (-∆) α ε (g ε (e -[t(|λ 0 |-ε 2 )+1] d+2α .))(x)| ≤ |C m e -2α[t(|λ 0 |-ε 2 )+1] ε(d+2α) (g ε )(e -[t(|λ 0 |-ε 2 )+1] d+2α
x)|.

We deduce that there exists ε 1 > 0 such that for all ε < ε 1 :

|(-∆) α ε ψ ε (x, t)| ≤ ε 2 4 ψ ε (x, t). (3.48)
Since (φ 0 + ε) is periodic, positive and C α according to [START_REF] Ros-Oton | The Dirichlet problem for the fractional Laplacian : regularity up to the boundary[END_REF] (Proposition 1.1), we conclude from Lemma 3.3 that there exists γ ∈]0, α[ and a constant C such that

| K ε (ψ, φ 0 + ε)(x, t)| ≤ Ce -[t(|λ 0 |-ε 2 )+1](2α-γ) ε(d+2α) ψ ε (x, t).
We deduce the existence of ε 2 > 0 such that for all ε < ε 2 , we have

| K ε (ψ, φ 0 + ε)(x, t)| ≤ ε 3 4 ψ ε (x, t) = ε 2 min(φ 0,ε + ε) 4 ψ ε (x, t). ( 3
:= min(ε 1 , ε 2 , |λ 0 | 2 , 1) and (x, t) ∈ Ω ε ×]0, 4 ε 2 [ we have : ε∂ t f m ε (x, t) + (-∆) α ε f m ε (x, t) -f m ε (x, t) + f m ε (x, t) 2 ≤ - 3ε 2 4 f m ε (x, t) -a(t)ε(λ 0 + 1)ψ ε (x, t) + a(t)(φ 0,ε + µ)(x)(-∆) α ψ ε (x, t) -a(t) K ε (ψ, φ 0 + ε)(x, t) ≤ - 3ε 2 4 f m ε (x, t) + ε 2 4 f ε (x, t) + ε 2 4 f ε (x, t) ≤ - ε 2 4 f m ε (x, t) ≤ 0. Therefore, f m ε is a sub-solution of (1 ε ) for (x, t) ∈ Ω ε ×]0, 4 ε 2 [
. It concludes the proof of (1). We continue by proving (2). Let (x, t) be in Ω ε ×] 4 ε 2 , +∞[. We define :

ψ ε (x, t) = C m 1 + e -|λ 0 |t ε +ε arctan(t-4 ε 2 )+ 3 ε |x| d+2α ε = C m g ε (e - |λ 0 |t+ε 2 arctan(t-4 ε 2 )+3 d+2α x) thus f m ε (x, t) = ψ ε (x, t) × (φ 0,ε (x) + ε).
First, we bound ε∂ t ψ ε from above :

ε∂ t ψ ε (x, t) = ε C m (|λ 0 |- ε 2 1+(t-4 ε 2 ) 2 ) ε e -|λ 0 |t ε +ε arctan(t-4 ε 2 )+ 3 ε |x| d+2α ε (1 + e --|λ 0 |t ε +ε arctan(t-4 ε 2 )+ 3 ε |x| d+2α ε ) 2 = ψ ε (x, t)   (|λ 0 | - ε 2 1 + (t -4 ε 2 ) 2 ) e -|λ 0 |t ε +ε arctan(t-4 ε 2 )+ 3 ε |x| d+2α ε 1 + e -|λ 0 |t ε +ε arctan(t-4 ε 2 )+ 3 ε |x| d+2α ε   ≤ ψ ε (x, t)[|λ 0 | - ε 2 1 + (t -4 ε 2 ) 2 -ψ ε (x, t)(φ 0,ε (x) + ε)] ≤ ψ ε (x, t)[|λ 0 | - ε 2 1 + (t -4 ε 2 ) 2 -f ε (x, t)].
(3.50)

Denoting by D = e

-|λ 0 |t ε +ε arctan(t-4 ε 2 )+ 3 ε |x| d+2α ε
and using the definition of C m , we obtain indeed for all ε < min( |λ 0 | 2 , 1)

|λ 0 | - ε 2 1 + (t -4 ε 2 ) 2 -ψ ε (φ 0,ε + ε) -|λ 0 | - ε 2 1 + (t -4 ε 2 ) 2 D 1 + D = |λ 0 | - ε 2 1+(t-4 ε 2 ) 2 -C m (φ 0,ε + ε) 1 + D ≥ |λ 0 | -ε 2 -|λ 0 | 2 1 + D ≥ 0. Next, we compute (-∆) α ε f m ε (x, t) (-∆) α ε f m ε (x, t) = (φ 0,ε + ε)(x)(-∆) α ε ψ ε (x, t) + ψ ε (x, t)(-∆) α ε φ 0,ε (x) -K ε (ψ, (φ 0 + ε))(x, t).
Chapter 3 : Propagation in a fractional reaction-diffusion equation in a periodically hostile environment

Combining (3.50) and the above equality we find, following similar computations as in [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF],

ε∂ t f m ε (x, t) + (-∆) α ε f m ε (x, t) -f m ε (x, t)(1 -f m ε (x, t)) ≤ - ε 2 1 + (t -4 ε 2 ) 2 f m ε (x, t) + (φ 0,ε (x) + ε)(-∆) α ε ψ ε (x, t) -K ε (ψ, φ 0 + ε)(x, t).
(3.51)

Thanks to Lemma 3.3, we obtain

|(-∆) α ε ψ ε (x, t)| = |C m (-∆) α ε (g ε (e [-t|λ 0 |t+ε 2 arctan(t-4 ε 2 )+3] d+2α .))(x)| ≤ |C m e - 2α[-|λ 0 |t+ε 2 arctan(t-4 ε 2 )+3] ε(d+2α) (g ε )(e [-|λ 0 |t+ε 2 arctan(t-4 ε 2 )+3] d+2α x)|.
Noticing that since t ≥ 4 ε 2 , there exists ε 3 > 0 such that for all ε < ε 3

-|λ 0 |t + ε 2 arctan(t - 4 ε 2 ) + 3 ≤ -|λ 0 |t 2 - 2|λ 0 | ε 2 + ε 2 π 2 + 3 ≤ -|λ 0 |t 2 .
We deduce the existence of ε 4 < ε 3 such that for all ε < ε 4 :

|(-∆) α ε ψ ε (x, t)| ≤ e -α|λ 0 |t ε(d+2α) ψ ε (x, t) ≤ ε 2 3(1 + (t -4 ε 2 ) 2 ) ψ ε (x, t). (3.52)
Following similar computations, we deduce the existence of ε 5 > 0 such that for all ε < ε 5 , we have

| K ε (ψ, φ 0 + ε)(x, t)| ≤ ε 3 3(1 + (t -4 ε 2 ) 2 ) ψ ε (x, t) = ε 2 min(φ 0,ε + ε) 3(1 + (t -4 ε 2 ) 2 ) ψ ε (x, t). ( 3
ε∂ t f m ε (x, t) + (-∆) α ε f m ε (x, t) -f m ε (x, t) + f m ε (x, t) 2 ≤ - ε 2 (1 + (t -4 ε 2 ) 2 ) f m ε (x, t) + (φ 0,ε + µ)(x)(-∆) α ψ ε (x, t) -K ε (ψ, φ 0 + ε)(x, t) ≤ - ε 2 3(1 + (t -4 ε 2 ) 2 ) f m ε (x, t) ≤ 0. Therefore, f m ε is a sub-solution of (1 ε ) for (x, t) ∈ Ω ε ×] 4 ε 2 , +∞[.
The proof of (3) follows the same arguments as the proof of (2).

For the proof of (4), we have to check that the initial data are ordered in the right way. According to (3.38), (3.45) and the definition of C m , we have that for all x ∈ Ω ε ,

f m ε (x, 0) = C m (φ 0,ε (x) + ε) e 1 ε + |x| d+2α ε ≤ c m δ ε (x) α 1 + |x| d+2α ε + ε ≤ n ε (x, ε) + ε. Furthermore, ∀(x, t) ∈ (Ω ε ) c × [0, +∞[, we know that f m ε (x, t) ≤ ε ≤ n ε (x, t + ε) + ε. (3.54)
3.5. The proof of Theorem 3.3

Thus we conclude from the comparison principle that for all (x, t

) ∈ R d × [0, 4 ε 2 [, we have f m ε (x, t) ≤ n ε (x, t + ε) + ε. (3.55)
Since, we have that for all x ∈ R d lim

t→ 4 ε 2 , t< 4 ε 2 f m ε (x, t) = lim t→ 4 ε 2 , t> 4 ε 2 f m ε (x, t)
and recalling that f m ε is also a subsolution in Ω ε ×] 4 ε 2 , +∞[ and the inequality (3.54), we deduce thanks to the comparison principle that for all (x, t

) ∈ R d × [0, +∞[ f m ε (x, t) ≤ n ε (x, t + ε) + ε. (3.56)
The other inequality can be obtained following similar arguments.

A direct consequence of (3.56) is that if ε fulfills the assumption of Proposition 3.3 then

∀(x, t) ∈ R d ×] 4 ε 2 , +∞[ C m × φ 0,ε (x) 1 + e -|λ 0 |t ε +ε arctan(t-4 ε 2 )+ 3 ε |x| d+2α ε ≤ n ε (x, t + ε) + ε. (3.57)
Next, we establish some consequences of Theorem 3.3 on the solution n without the scaling (3.9).

Proof of Theorem 3.5. First, we prove the first point by using the sub-solution f m ε . It is sufficient to prove it for ν < r 0 (where r 0 is the radius of the uniform interior ball condition satisfied by Ω 0 ). Proof of 1. Set ν > 0, c < |λ 0 | d+2α and ε 0 > 0 provided by Proposition 3.3. we deduce that for

ε = min ε 0 , 4(|λ 0 |-(d+2α)c) π 2 +3 , Cmmin Ων φ 0 4 2
and for all (x, t)

∈ (Ω ε ν ∩ {|x| < e ct }) ×] 4 ε 2 , +∞[ we have C m φ 0 (|x| 1 ε -1 x) 1 + e -|λ 0 |t ε +ε arctan(t-4 ε 2 )+ 3 ε |x| d+2α ε ≤ n ε (x, t + ε) + ε which implies C m min Ων φ 0 1 + e t ε ((d+2α)c-|λ 0 |)+ε arctan(t-4 ε 2 )+ 3 ε ≤ n ε (x, t + ε) + ε.
If we perform the inverse scaling to (3.9), since

ε < 4(|λ 0 |-(d+2α)c) π 2 +3
, it follows that for all

t > 4 ε 3 t(c(d + 2α) -|λ 0 |) + ε arctan(εt - 4 ε 2 ) + 3 ε < 0.
We deduce that for all (x, t)

∈ (Ω ν ∩ {|x| < e ct }) ×] 4 ε 3 + 1, +∞[ C m min Ων φ 0 2 ≤ C m min Ων φ 0 1 + e t(c(d+2α)-|λ 0 |)+ε arctan(εt-4 ε 2 )+ 3 ε ≤ n(x,
∀(x, t) ∈ R d ×]0, +∞[, n ε (x, t + ε) ≤ f M ε (x, t
). If we perform the inverse scaling to (3.9), it follows that

n(x, t + 1) ≤ C M × max φ 0 1 + e -|λ 0 |t-ε arctan(εt)-1 ε |x| d+2α
.

Then for all (x, t) ∈ {|x| > Ct} ×]0, +∞[ we have

n(x, t + 1) ≤ C M × max φ 0 1 + e -επ 2 -1 ε e t[(d+2α)C-|λ 0 |] . Defining C 1 := 2C M max φ 0 , C 2 := e -επ

The final argument

Proof of Theorem 3.3. We will prove (i) by splitting the proof into two parts : the upper bound and the lower bound. We will not provide the proof of (ii) since it is a direct application of 2. of Theorem 3.5.

Proof of (i). Let µ be a positive constant. We want to prove that there exists a time t µ > 0 such that for any c < |λ 0 | d+2α we have for all (x, t)

∈ {|x| < e ct } ×]t µ , +∞[ |n(x, t) -n + (x)| ≤ µ.
First we establish that there exists a time t 1 > 0 such that

∀(x, t) ∈ Ω×]t 1 , +∞[, n(x, t) -n + (x) ≤ µ (3.58)
Next, we prove the existence of a time t 2 > 0 such that

∀(x, t) ∈ Ω×]t 2 , +∞[, -µ ≤ n(x, t) -n + (x) (3.59)
The difficult part will be to establish (3.59). This is why, we do not provide all the details of the proof of (3.58).

Proof that (3.58) holds true. Thanks to (3.38) and Proposition 3.1, we deduce the existence of a constant C ≥ 1 such that n(x, t = 1) ≤ Cn + (x).

Moreover, the solution n of

       ∂ t n + (-∆) α n = n -n in Ω×]1, +∞[, n(x, t) = 0 in Ω c × [1, +∞[, n(x, t = 1) = Cn + (x)
in Ω is a super solution of (3.1). According to the comparison principle we deduce that

∀(x, t) ∈ R × [1,
Ω × [1, +∞[ |n(x, t) -n + (x)| ≤ n(x, t) + n + (x) ≤ n(x, t) + n + (x) ≤ (C + 1)n + (x).
According to Proposition 3.1, we deduce that for all (x, t)

∈ Ω × [1, +∞[ |n(x, t) -n + (x)| ≤ C(C + 1)δ(x) α .
We conclude that for all (x, t)

∈ Ω × [1, +∞[ such that δ(x) < µ C(C+1) 1 α := ν 1 we have |n(x, t) -n + (x)| ≤ µ.
The interior estimates. Thanks to Theorem 3.1, we know that n + ≤ 1 thus it is sufficient to prove the existence of t 2 > 0 such that

∀(x, t) ∈ |x| < e ct ∩ Ω ν 2 ×]t 2 , +∞[ 1 -µ ≤ n(x, t) n + (x)
where

ν 2 = min(ν 1 , r 1 )
where ν 1 is provided by the previous step and r 1 by the uniform interior ball condition.

The idea is to approximate n + by the solution of (3.4) on a ball of radius M . Noticing that thanks to (H1), there exists M 0 > 0 such that for M > M 0 , there exists a unique bounded positive solution n M,+ of

(-∆) α n M,+ = n M,+ -n 2 M,+ in Ω ∩ B(0, M ), n M,+ = 0 in (Ω ∩ B(0, M )) c (3.61)
We claim that

∃M 1 > M 0 , such that ∀M > M 1 , ∀x ∈ Ω 0,ν 2 , (1 -µ) 1 2 ≤ n M,+ (x) n + (x) . (3.62)
The proof of this claim is postponed to the end of this paragraph. Next, we approach n M,+ by the long time solution of the following equation :

       ∂ t n M,z + (-∆) α n M,z = n M,z -n 2 M,z in (Ω ∩ B(0, M )) ×]0, +∞[, n M,z (x, t) = 0 in (Ω ∩ B(0, M )) c ×]0, +∞[, n M,z (x, t = 0) = σ1 B(z, ν 2 4 ) (x). (3.63)
where σ is provided by Theorem 3.5 and z ∈ Ω 0, ν 2 2 will be fixed later on. We claim that

∃t µ > 0, such that ∀z ∈ Ω 0, ν 2 2 , ∀(x, t) ∈ Ω 0,ν 2 ×]t µ , +∞[, (1 -µ) 1 2 ≤ n M,z (x, t) n M,+ (x) . (3.64)
Chapter 3 : Propagation in a fractional reaction-diffusion equation in a periodically hostile environment

Again, the proof of this claim is postponed to the end of this section. Next, we define

t µ = t µ + t σ (3.65)
where t µ is defined by (3.64) and t σ by Theorem 3.5. Let (x, t) ∈ (Ω ν 2 ∩ {|x| < e ct }) ×]t µ , +∞[ and j ∈ Z d be such that x ∈ Ω 0 + a j . Since ν 2 < r 1 (the radius of the uniform interior ball condition), we deduce the existence of z x ∈ Ω 0, ν 2 2 such that

x ∈ B(z x + a j , ν 2 4
) and ∀y ∈ B(z x + a j , ν 2 4 ) there holds y ∈ (Ω 0 + a j ) ν 4 ∩ |y| < e ct . (3.66)

Remarking that n(x,t) n + (x) = n(x,t) n + (x-a j ) , we are going to control each terms of the following decomposition :

n(x, t) n + (x) = n(x, t) n M,zx (x -a j , t -t σ ) × n M,zx (x -a j , t -t σ ) n M,+ (x -a j ) × n M,+ (x -a j ) n + (x -a j ) = I × II ×
1 -µ) 1 2 ≤ n M,zx (x -a j , t -t σ ) n M,+ (x -a j ) . (3.68) Control of III. Since x -a j ∈ Ω 0 , we deduce thanks to (3.62) that (1 -µ) 1 2 ≤ n M,+ (x -a j ) n + (x -a j ) . ( 3
∈ Ω ν 2 ×]t µ , +∞[ ∩ (x, t) ∈ R d × R + | |x| < e ct , we obtain 1 -µ ≤ n(x, t) n M,zx (x -a j , t -t σ ) × n M,zx (x -a j , t -t σ ) n M,+ (x -a j ) × n M,+ (x -a j ) n + (x -a j ) = n(x, t) n + (x -a j ) = n(x, t) n + (x) .
This concludes the proof of Theorem 3.3. Proof of (3.62). The map (M ∈]M 0 , +∞[ → n M,+ ) is increasing as n M is a sub-solution to the equation for n M for M > M . It converges to a weak solution of (3.4). By fractional elliptic regularity, the limit is a strong solution of (3.4). We conclude thanks to the uniqueness of the solution of (3.4) stated in Theorem 3.1.

The proof of (3.64) relies on a compactness argument.

Proof of (3.64). For a fixed z ∈ Ω ν , the proof of convergence of n M,z to n M,+ is classical thanks to (H1). For each z ∈ Ω 0,ν , there exists t z > 0 such that

∀(x, t) ∈ R d ×]t z , +∞[, (1 -µ) 1 2 ≤ n M,z (x, t) n M,+ (x) .
We claim that sup z∈Ω 0,ν t z < +∞. This assertion is true by compactness of Ω 0,ν (otherwise there exists z ∈ Ω 0,ν such that t z = +∞ which is a contradiction).

The result on the level sets

In this section, we use the sub and super-solutions established in Section 3.5.1 to prove Theorem 3.4.

Proof. Let ν > 0 be such that Ω ν = ∅. We define , ε 0 , µ) where ε 0 is provided by Proposition 3.3. Next we define t µ = 4 ε 3 + 1. Let (x, t) ∈ Ω ν ×]t µ , +∞[ be such that n(x, t) = µ. First, we prove that there exits c > 0 (independant of the choice of (x, t)) such that ce |λ 0 |t ≤ |x| d+2α . Next, we prove the existence of C > 0 (independant of the choice of (x, t)) such that |x| d+2α ≤ Ce |λ 0 |t . Defining C µ = max(C, c -1 ), the conclusion follows. Existence of c. Thanks to Proposition 3.3, after the inverse scaling of (3.9), we obtain

C m (φ 0 (x) + ε) 1 + e -|λ 0 |(t-1)+ε arctan(ε(t-1)-4 ε 2 )+ 3 ε |x| d+2α ≤ n(x, t) + ε = µ + ε ⇒ C m min y∈Ων φ 0 (y) -µ + ε(C m -1) ≤ 2µe -|λ 0 |(t-1)+ε arctan(ε(t-1)-4 ε 2 )+ 3 ε |x| d+2α ⇒    Cm min y∈Ων φ 0 (y)-µ 2 e -επ 2 -3 ε -|λ 0 | 2µ    e |λ 0 |t ≤ |x| d+2α If we define c = (Cm min y∈Ων φ 0 (y)-µ)e -επ 2 -3 ε -|λ 0 | 4µ
then the conclusion follows. Existence of C. Thanks to Proposition 3.3, after the inverse scaling of (3.9), we obtain

µ = n(x, t) ≤ C M φ 0 (x) 1 + e -|λ 0 |(t-1)-ε arctan(ε(t-1))-1 ε |x| d+2α .
Chapter 4

Properties of steady states for a class of non-local Fisher-KPP equations in general domains

Ce chapitre a fait l'objet d'un dépot sur Arxiv et va faire l'objet d'une soumission dans un journal scientifique. Il a été co-écrit avec Jean-Michel Roquejoffre.

Introduction

Model, question, motivation

We investigate here existence and uniqueness of bounded positive solutions for the fractional Fisher-KPP equations of the form i.e.

(-∆)

α n + (x) = n + (x) -n 2 + (x) for x ∈ Ω, n + (x) = 0 for x ∈ Ω c . (4.1)
The domain Ω is an infinite union of patches, all of them but perhaps one being bounded. The operator (-∆) α is the fractional Laplacian :

(-∆) α φ(x) = C α P V R d φ(x) -φ(y) |x -y| d+2α dy with C α = 4 α Γ( d 2 + α) π d 2 |Γ(-α)| , ( 4.2)
we assume α < 1 throughout the work. Clearly, existence and uniqueness would be false if α = 1 (just think of a periodic union of large line segments), but nonlocality implies a solidarity between patches that may make existence and uniqueness become true. We will first see some effects of the nonlocality when dealing with principal eigenvalue problems, we will then try to understand how the solidarity forced by nonlocal diffusion eventually leads to existence and uniqueness. More important in our opinion, we will take advantage of the disconnectedness of the domain to derive precise estimates of possible solutions of (4.1) at infinity, that will eventually imply uniqueness. The evolution equation

       ∂ t n(x, t) + (-∆) α n(x, t) = n(x, t) -n(x, t) 2 for (x, t) ∈ Ω×]0, +∞[, n(x, t) = 0 for (x, t) ∈ Ω c × [0, +∞[, n(x, 0) = n 0 (x), (4.3)
Chapter 4 : Properties of steady states for a class of non-local Fisher-KPP equations in general domains models biological invasions. The variable n stands for a density of population. The fractional Laplacian models the fact that a species can jump from one point to another with a high rate. If a bounded solution of (4.3) converges as t tends to +∞, it is either to 0, either to a non-trivial stationary state of (4.3) : a solution of (4.1). Thus, if (4.1) does not admit a bounded positive solution, we deduce that the species modeled by n will extinct. On the other hand, if there exists a unique bounded positive solution n + to (4.1) to which the solution n converges to n + as t tends to +∞, the species will persist. An equation of type (4.3) was first introduced in 1937 by Fisher in [START_REF] Fisher | The wave of advance of advantageous genes[END_REF] and by Kolmogorov, Petrovskii and Piscunov in [START_REF] Kolmogorov | A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem[END_REF] in the whole domain R d and with a standard diffusion. Our model accounts for a situation where reproduction is allowed on some patches (perhaps an infinity of them) while the outside environment is lethal to the species. Moreover, the patches may be, individually, unfavourable to reproduction, and this may be true for all but one of them. The question is whether the species may survive in these conditions, and if such is the case, in which quantity. Of course this can only be possible because of nonlocal diffusion, survival being clearly impossible in the conditions just described, if the diffusion is the usual one. One of the main result of this work is that there is survival but if all the patches but one are individually unfavourable, the density of individuals will decay like a power of the distance to the favourable patch that we evaluate precisely. Before that, we account for specfic effects of the nonlocal diffusion for simple one-dimensional domains.

Some effects of nonlocality on the principal eigenvalue

We consider a domain made up of two patches of variable distance, and we wonder how the principal eigenvalue is affected by the distance between the patches. In particular, we ask whether it is continuous with respect to this distance. This question is of course irrelevant if the fractional Laplacian is replaced by the usual Laplacian, as both domains have a principal eigenvalue of their own. However, in the nonlocal case, there is a solidarity between the patches and the question becomes relevant. In the one dimensional case, we give a positive answer especially when the distance tends to 0. Our domain is of the form

Ω 1,2,µ = Ω 1 ∪ Ω 2 and dist(Ω 1 , Ω 2 ) = inf {|x 1 -x 2 | with x 1 ∈ Ω 1 , x 2 ∈ Ω 2 } = 2µ. (4.4)

Notation. For any smooth bounded set O, let ξ α (O) be the principal eigenvalue of the fractional Laplacian with an exponent α with Dirichlet conditions outside the domain

O i.e. (-∆) α φ α = ξ α (O)φ α in O, φ α = 0 in O c . (4.5)
The existence of such eigenvalue is ensured by the Krein-Rutman Theorem. In this part, we will adopt the following notation :

ξ i,α = ξ α (Ω i ).
We denote by φ 1,2,µ,α and φ i,α the eigenfunctions associated respectively to ξ α (Ω 1,2,µ ) and ξ i,α .

The main continuity result

As previously said, a two-piece domain has a first principal eigenvalue, and this eigenvalue is continuous under the mutual distance of the two pieces is intuitively obvious, as soon as 4.1. Introduction they remain far apart. When they are put together, continuity still holds : this result has of course no equivalent in the case of the standard Laplacian. Here is the precise statement. Theorem 4.1. Under the previous assumptions, the function (µ ∈]0, +∞[ → ξ α (Ω 1,2,µ )) is increasing and continuous. Moreover, it is continuous up to µ = 0 and

ξ α (Ω 1,2,0 ) = ξ α (]0, |Ω 1 | + |Ω 2 |[).
A first ingredient for the proof of Theorems 4.1 and the monotonicity of (µ

→ λ α (Ω 1,2,µ )) is the Rayleigh quotient R α (φ) = R (-∆) α φ(x)φ(x)dx R φ 2 (x)dx = 1 2 R R (φ(x)-φ(y)) 2 |x-y| 1+2α dydx R φ 2 (x)dx := [φ] α φ 2 2 . ( 4.6)
We will evaluate R α in the following spaces, for a general set O :

H α 0 (O) = φ ∈ L 2 (O) such that [φ] α < +∞ and φ |R\O = 0 . (4.7)
The link between the Rayleigh quotient and the principal eigenvalues is the following :

ξ α (Ω 1,2,µ ) = min φ∈H α 0 (Ω 1,2,µ )\{0} R α (φ) = R α (φ 1,2,µ,α ) and ξ i,α = min φ∈H α 0 (Ω i )\{0} R α (φ) = R α (φ i,α ).
(4.8) The proof of the continuity of ξ α (Ω 1,2,µ ) with respect to µ when µ > 0 is a consequence of standard uniqueness/compactness arguments. The continuity at µ = 0 is more involved, especially when α ≥ 1 2 . In this case, we have to prove that the contact point between the two domains Ω 1 and Ω 2 becomes a removable singularity. To achieve this result, we will use the extension on the upper half plane of the fractional Laplacian introduced by Caffarelli and Silvestre in [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF]. For the case α < 1 2 , the case µ = 0 can be treated as the case µ > 0 thanks to the density of the set of functions satisfying φ = 0 (see [START_REF] Tartar | An introduction to Sobolev spaces and interpolation spaces[END_REF]). We emphasise that the demonstration holds true up to µ = 0 because we work in a one dimensional space. Indeed, in one dimension, there is only one way to connect two intervals. For d > 2, the result holds true for distances µ > 0, with no real modification.

The limit α → 1 : Consequences of Theorem 4.1

It is well known (see [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF]) that for a smooth function n and for all x ∈ R, the function α → (-∆) α n(x) is continuous, and that

(-∆) α n(x) -→ α→1 -(-∆)n(x)
. The standard compactness/uniqueness argument yields the continuity of the function (α ∈ ]0, 1[→ ξ α (Ω 1,2,µ )). The next proposition describes, in our specific case, the dynamics of ξ α (Ω 1,2,µ ) when α tends to 1.

Proposition 4.1. The function α

∈]0, 1[ → ξ α (Ω 1,2,µ ) is continuous up to α = 1 with ξ 1 (Ω 1,2,µ ) = min (ξ 1 , ξ 2 ).
(4.9)

We now focus on the dynamics of ξ α (Ω 1,2,µ ) when (µ, α) converges both to (0, 1). There is a competition between the non-local character which becomes "less important" as α tends to 1 and the requirement of this non-local character which becomes also "less important" as µ tends to 0. Notations and assumptions on Ω Before giving the hypothesis and the results, we introduce some notations which will be used all along the article. Let O be a general smooth domain of R d , x be a point of R d and ν be a positive constant. Then we define the sets :

O + x = y ∈ R d such that y -x ∈ O and O ν = {y ∈ O such that dist(y, ∂O) > ν} (4.11)
where ∂O is the boundary of O. Since the distance to the boundary of the domain Ω under study will play an important role, we will denote it by δ, i.e. δ(x) = dist(x, ∂Ω)1 Ω (x).

(4.12)

When it is defined, the principal eigenvalue of the fractional Dirichlet operator (-∆) α -Id in O will play also an important role in the following. We will denote it by λ α (O) :

i.e.

   (-∆) α φ(x) -φ(x) = λ α (O)φ(x) for x ∈ O, φ(x) = 0 for x ∈ R d \O. (4.13)
We underline that λ α (O) = ξ α (O)-1. The principal eigenvalue λ α (O) will be a key ingredient of most of the upcoming results. Note that a such eigenvalue is well defined for instance when the domain O is smooth and bounded with a finite number of components. It is also well defined if O is smooth, periodic and such that the number of components in all compact sets is finite. For such domains, there is a dichotomy : either λ α (O) < 0 and (4.1) admits a positive bounded non-trivial solution, either λ α (O) ≥ 0 and the unique positive bounded solution of (4.1) is 0.

We assume that the domain Ω may be written as

Ω = k∈N Ω k (H1)
where the sets (Ω k ) k∈N are smooth, connected, bounded. Moreover, we assume that Ω satisfies the uniform interior and exterior ball condition. We assume that Ω can be decomposed in the following form :

Ω = Ω -∪ Ω + = k∈N (Ω -,k ∪ Ω +,k ). ( H3
)
In the following, when we pick x ∈ Ω ± , the integer k x will denote the only integer such that x ∈ Ω ±,kx . We assume that the domain Ω + is composed by some uniformly bounded clusters C +,k which are "far away" each other i.e. ∀k ∈ N,

∃z k ∈ R d such that C +,k ⊂ B(z k , r 0 ), |z i -z j | > R 0 + r 0 ( for i = j) and dist(Ω + , Ω -) > R 0 . (H4)
In what follows, the constant R 0 will be assume suitably large. Moreover, we assume that the eigenvalue of the Dirichlet operator (-∆) α -Id in x ∈ R d |dist(x, C +,k ) < ε 1 is uniformly bounded from below for some positive

ε 1 < ε 0 4 : i.e. 0 < λ 0 < λ α ( x ∈ R d |dist(x, C +,k ) < ε 1 ). ( H5
)
For Ω -, we assume that it is not empty and there exists a finite number of bounded sets (Ω -,n ) n∈{1,...,N } made up with a finite number of connected components such that ∀n ∈ {1, ..., N } , λ α (Ω -,n ) < 0,

and ∀x ∈ Ω -, ∃(y, n) ∈ R d × {1, ..., N } such that x ∈ y + Ω -,n and (y + Ω -,n ) ⊂ Ω.
(H6)

Remark. The assumptions (H1)-(H6) on Ω cover a large case of sets from the sets with only one bounded connected component to a general unbounded with an infinite number of components.

At the end of section 4.3, we present some examples of domains which satisfy such assumptions.

Remark. The decomposition Ω = Ω -∪ Ω + is not unique. For instance if Ω + is bounded, one can take Ω -= Ω and Ω + = ∅ as a new decomposition.

Chapter 4 : Properties of steady states for a class of non-local Fisher-KPP equations in general domains

The steady solution n + : existence, accurate estimates, uniqueness

We begin by stating that there exists a bounded non-trivial solution n + Theorem 4.3. Under the assumption (H6), there exists a bounded non-trivial solution of (4.1).

Thanks to (H6), we can provide a subsolution : the solution of (4.1) where the domain Ω is switched with Ω -∩ B(0, R) (with R large enough such that λ α (Ω -∩ B(0, R)) < 0). Next, a supersolution is the constant function 1. Finally, we construct a solution by iteration from this sub and super solution (see [START_REF] Smoller | Shock waves and reaction-diffusion equations[END_REF]). A much less classical result is an estimate from above and below of any positive non-trivial bounded solutions of (4.1). Theorem 4.4. Assume that Ω fulfils (H1)-(H6) and that R 0 > C d,α,ε 0 ,ε 1 ,r 0 where C d,α,ε 0 ,ε 1 ,r 0 is a positive constant depending on the parameters d, α, ε 0 , ε 1 , r 0 then for any bounded positive solution n + of (4.1), there exists two positive constants

c 1 ∈]0, 1[, C 1 ∈ R + such that for all x ∈ R d we have c 1 G(x) ≤ n + (x) ≤ G(x) (4.15) with G(x) = min (min(C 1 δ(x) α , ε α ) × G(x), 1) . (4.16)
and

G(x) = 1 Ω -(x) + Ω - 1 |x -y| d+2α dy 1 Ω + (x).
(4.17)

We underline that the function G depends only on the decomposition (H3). We will devote a special section to its proof, its strategy being presented at the beginning. Thanks to Theorem 4.4, we prove uniqueness of the solution of (4.1).

Theorem 4.5. Assuming (H1)-(H6) and if R 0 is large enough then there exists a unique bounded positive non-trivial solution of (4.1).

The proof of uniqueness follows a general argument introduced by Berestycki in [START_REF] Berestycki | Le nombre de solutions de certains problèmes semi-linéaires elliptiques[END_REF]. Thanks to Theorem 4.4, we compare two solutions and we conclude thanks to the maximum principle or the fractional Hopf Lemma. A consequence is that the steady solution is a global attractor for the Cauchy Problem.

Corollary 4.1. Under the previous hypothesis, if n is solution of (4.3) and n 0 is positive, non-trivial and compactly supported initial data with the closure of the support included in the closure of Ω then there holds

∀x ∈ R d , lim t→+∞ n(x, t) = n + (x).
We underline that the proof will use the uniqueness result. From a biological point of view, we talk about a colonisation rather than an invasion phenomena. Indeed, invasion imply a colonisation phenomenon and an autonomy from the population in the neighborhood of the colonised area. This last fact does not hold in Ω + according to Theorem 4.4.

4.2. Dependence of the sign of ξ α (Ω 1,2,µ ) on the parameters

Discussion, comparison with existing results

Theorem 4.5 has a link with some recent results dealing with steady solutions of nonlocal Fisher-KPP equations in general environments. Closest to the result is the analysis of Berestycki, Coville and Vo [START_REF] Berestycki | Persistence criteria for populations with non-local dispersion[END_REF], where the dispersal is given by a smooth integral kernel, the domain is the whole space, but the reproduction term is inhomogeneous. Under essentially two assumptions, namely that a generalised principal eigenvalue (see [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] for the general definitions) is negative, and that the medium outside a ball is unfavourable, existence of uniqueness of a steady solution under a certain subsolution is established. We note a related work by Brasseur [32] that achieves a passage to the limit of a more and more concentrated dispersal. Our general setting is less general than the situation considered in [START_REF] Berestycki | Persistence criteria for populations with non-local dispersion[END_REF], in particular we have to make assumptions related to the fact that the environment is fragmented. In particular, whether Theorem 4.5 holds under the sole assumption on the sign of a generalised principal eigenvalue is an important question whose answer is unknown to us at the moment. Additional technical difficulties are present due to the fragmented environment (barriers at the boundary of the domain are sometimes tricky to devise) as well as the presence of the fractional Laplacian. On the other hand, we note that our uniqueness result holds in the whole class of bounded solutions due to the general estimate provided in Theorem 4.4, which we regard as one of the main results of this work. An aspect of the problem, that would call for further developments, is the detailed description of the invasion, in other words quantitative estimates on the convergence to the steady solution. This would be especially interesting if infinitely many patches are favourablea description of the steady states in this situation is, by the way, still to be developped, although not totally out of reach from the arguments of the present work. We mention the periodic setting treated in [START_REF] Léculier | Propagation in a fractional reaction-diffusion equation in a periodically hostile environment[END_REF], where exponential invasion is proved. To understand how things have to be modified outside this setting is still, to our knowledge, open.

Outline of the paper

We first provide in Section 4.2 a study on the one-dimensional case : the dependence of the sign of ξ α (Ω 1,2,µ ) on the parameters |Ω i,α |, µ and α. Next, in Section 4.3 the proof of Theorem 4.4. Section 4.4 is devoted to the proof of Theorem 4.5. In section 4.4.2, we establish Theorem 4.1. We also provide some numerics in order to illustrate Corollary 4.1 and Theorem 4.1. In order to be more readable, we do not write the principal eigenvalue P.V. and the constant C α in front of the fractional Laplacian. When there is no possible confusion, the constants denoted by c and C may change from one line to another. In all this work, when it is not precised, φ is usually used to denote a principal eigenfunction. Moreover, it is taken positive and with unit L 2 norm.

Dependence of the sign of ξ α (Ω 1,2,µ ) on the parameters

In subsection 4.2.1, we focus on the dependence of ξ α (Ω 1,2,µ ) on the size of Ω i and µ. We provide the proof of Theorem 4.1 and Proposition 4.2. Then, in subsection 4.2.2, we provide the proof of Theorem 4.1. Finally, subsection 4.2.3 is devoted to the investigation of the description of the dynamics of ξ α (Ω 1,2,µ ) when (µ, α) → (0, 1) at the same time.

Dependence on |Ω i | and µ

We start the proof of Theorem 4.1 with the following classical (and useful) bounds on ξ α (Ω 1,2,µ ).

Proposition 4.2. Under the previous hypothesis, there holds for all

µ ≥ 0 ξ α (Ω 1,2,µ ) ≤ min (ξ 1,α , ξ 2,α ).
Proof. For all µ > 0, it is straightforward that H α 0 (Ω i ) ⊂ H α 0 (Ω). Thus, we deduce that the function φ i,α ∈ H α 0 (Ω). We conclude thanks to the Rayleigh quotient that we have :

ξ α (Ω 1,2,µ ) = min φ∈H α 0 (Ω)\{0} R α (φ) ≤ R α (φ i,α ) = ξ i,α .
Proof of Theorem 4.1. We first show the monotonicity. Next, we prove the continuity for µ > 0 finally we demonstrate the continuity up to µ = 0. Proof of the monotonicity of (µ → ξ α (Ω 1,2,µ )). Let µ 1 and µ 2 be two positive constants such that µ 1 < µ 2 . We will consider Ω 1,2,µ 1 and Ω 1,2,µ 2 , that we write explicitely (possibly up to a translation) in order to fix ideas :

Ω 1,2,µ 1 =] -A 1 -µ 1 , -µ 1 [∪]µ 1 , A 2 + µ 1 [ and Ω 1,2,µ 2 =] -A 1 -µ 2 , -µ 2 [∪]µ 2 , A 2 + µ 2 [.
We define µ = µ 2 -µ 1 . We recall that for all j ∈ {1, 2}

ξ α (Ω 1,2,µ j ) = min φ∈H α 0 (Ω 1,2,µ j )\{0} R α (φ) = R α (φ 1,2,µ j ,α ).
The idea is to translate each component of the support of φ 1,2,µ 2 ,α in Ω 1,2,µ 1 . Thus we define

φ µ (x) := φ(x -µ 2 + µ 1 ) = φ 1,2,µ 2 ,α (x -µ) if x ∈] -∞, 0[, φ(x + µ 2 -µ 1 ) = φ(x + µ) if x ∈ [0, +∞[ (4.18)
for any φ ∈ H α 0 (Ω 1,2,µ 2 ). We easily observe that φ µ belongs to

H α 0 (Ω 1,2,µ 1 ). Next, we remark that φ µ 1,2,µ 2 ,α 2 = φ 1,2,µ 2 ,α 2 = 1. The aim is to show that [φ µ 1,2,µ 2 ,α ] α ≤ [φ 1,2,µ 2 ,α ] α . ( 4
R α (φ µ 1,2,µ 2 ,α ) = [φ µ 1,2,µ 2 ,α ] α φ µ 1,2,µ 2 ,α 2 2 ≤ [φ 1,2,µ 2 ,α ] α φ 1,2,µ 2 ,α 2 2 = R α (φ 1,2,µ 2 ,α ).
It allows us to conclude that

ξ α (Ω 1,2,µ 1 ) = min φ∈H α 0 (Ω 1,2,µ 1 )\{0} R α (φ) ≤ R α (φ µ 1,2,µ 2 ,α ) ≤ R(φ 1,2,µ 2 ,α ) = ξ α (Ω 1,2,µ 2 ).
Thus, we prove (4.19). We will denote

k ε (x) = max(ε, |x| 1+2α ) such that [φ] α = lim ε→0 1 2 R R (φ(x) -φ(y)) 2 k ε (x -y) dydx.
4.2. Dependence of the sign of ξ α (Ω 1,2,µ ) on the parameters For all ε > 0 and for all positive φ ∈ H α 0 (Ω 1,2,µ 2 ) thanks to the Fubini-Tonelli theorem, we have

1 2 R R (φ µ (x) -φ µ (y)) 2 k ε (x -y) dydx = 1 2 R R φ µ (x) 2 + φ µ (y) 2 -2φ µ (x)φ µ (y) k ε (x -y) dydx = R φ µ (x) 2 R 1 k ε (x -y) dydx - R R φ µ (x)φ µ (y) k ε (x -y) dydx = C ε R φ µ (x) 2 dx - Ω 1,2,µ 1 Ω 1,2,µ 1 φ µ (x)φ µ (y) k ε (x -y) dydx.
Here,

C ε = R 1 kε(z) dz is a constant independent of the choice of φ. Next we have : R R (φ µ (x) -φ µ (y)) 2 k ε (x -y) dydx = C ε R φ µ (x) 2 dx - Ω µ 1 1 Ω µ 1 1 φ µ (x)φ µ (y) k ε (x -y) dydx - Ω µ 1 2 Ω µ 1 2 φ µ (x)φ µ (y) k ε (x -y) dydx -2 Ω µ 1 2 Ω µ 1 1 φ µ (x)φ µ (y) k ε (x -y) dydx
(4.20) Thanks to a change of variable, we obtain that for all i ∈ {1, 2} :

Ω µ 1 i Ω µ 1 i φ µ (x)φ µ (y) k ε (x -y) dydx = Ω µ 2 i Ω µ 2 i φ(x)φ(y) k ε (x -y) dydx. (4.21) Since for (x, y) ∈ Ω µ 2 1 × Ω µ 2 2 we have k ε (x -y + 2µ) ≤ k ε (x -y), we obtain Ω µ 2 2 Ω µ 2 1 φ(x)φ(y) k ε (x -y) dydx ≤ Ω µ 2 2 Ω µ 2 1 φ(x)φ(y) k ε (x -y + 2µ) dydx = Ω µ 1 2 Ω µ 1 1 φ µ (x)φ µ (y) k ε (x -y) dydx.
α 0 (Ω 1,2,µ 2 ), we have R R (φ µ (x) -φ µ (y)) 2 k ε (x -y) dydx ≤ R R (φ(x) -φ(y)) 2 k ε (x -y) dydx.
Sending ε to 0, we conclude that (4.19) holds true and the conclusion follows.

Proof of the continuity of (µ → ξ α (Ω 1,2,µ )) for µ > 0. Let µ > 0 and (µ k ) k∈N with µ k -→ k→+∞ µ and µ k > 0. According to Proposition 4.2 and the Krein-Rutman Theorem, we

have 0 ≤ ξ α (Ω 1,2,µ ) < min(ξ 1 , ξ 2 ). Up to a subsequence, ξ α (Ω 1,2,µ k ) converges to ξ ∞ . We normalise φ 1,2,µ k ,α such that φ 1,2,µ k ,α L 2 = 1. Next, thanks to the Rayleigh quotient, we obtain that [φ 1,2,µ k ,α ] α ≤ ξ α (Ω 1,2,µ k ) ≤ min(ξ 1 , ξ 2 ).
Thus, we deduce that φ 1,2,µ k ,α H α is bounded. Up to a new subsequence, φ 1,2,µ k ,α converges strongly in L 2 (R) and weakly in H α (R) to φ ∞ . It is straightforward to obtain that φ ∞ ≥ 0 and φ ∞ = 0 in (Ω 1,2,µ ) c . Moreover, since Ω 1,2,µ k converges to Ω 1,2,µ we deduce that for all compact set K of Ω 1,2,µ , there exists k 0 ∈ N such that for k > k 0 , we have

K ⊂ Ω 1,2,µ k and 1 -ε ≤ φ 1,2,µ k ,α L 2 (K)
Chapter 4 : Properties of steady states for a class of non-local Fisher-KPP equations in general domains with ε as small as we want. We deduce that φ ∞ L 2 (Ω 1,2,µ ) = 1. With the same idea, we get that in all compact set K of Ω 1,2,µ , φ ∞ is a weak solution to

       (-∆) α φ ∞ = ξ ∞ φ ∞ for x ∈ K, φ ∞ = 0 for x ∈ Ω c 1,2,µ , φ ∞ ≥ 0, φ ∞ L 2 = 1.
Since it is true in all compact set of Ω 1,2,µ we conclude that

       (-∆) α φ ∞ = ξ ∞ φ ∞ for x ∈ Ω 1,2,µ , φ ∞ = 0 for x ∈ Ω c 1,2,µ , φ ∞ ≥ 0, φ ∞ L 2 = 1. (4.23)
Thanks to the fractional elliptic regularity (see [START_REF] Caffarelli | Regularity theory for fully nonlinear integro-differential equations[END_REF]), we obtain that (4.23) is true in the strong sense. By uniqueness of the eigenvalue associated to a strictly positive eigenfunction, we finally conclude that ξ ∞ = ξ α (Ω 1,2,µ ) and φ ∞ = φ 1,2,µ,α .

Proof of the continuity up to µ = 0. Let (µ k ) k∈N be a sequence such that µ k -→ k→+∞ 0 and µ k > 0. Following the same idea than in the previous part, we find that

ξ α (Ω 1,2,µ k ) converges to some ξ ∞ and φ 1,2,µ k ,α converges to φ ∞ with φ ∞ a bounded solution of        (-∆) α φ ∞ = ξ ∞ φ ∞ for x ∈] -A, A[\ {0} , φ ∞ = 0 for x ∈] -A, A[ c , φ ∞ ≥ 0, φ ∞ L 2 = 1. (4.24)
We consider two cases : α < 1 2 and α ≥ 1 2 . Case 1 α < 1 2 . It is sufficient to remark that φ 1,2,µ k ,α converges to φ ∞ in H α (] -A, A[) (see the comments after the proof of Lemma 16.1 p. 82 of [START_REF] Tartar | An introduction to Sobolev spaces and interpolation spaces[END_REF]). Indeed, since the set of functions

{ψ ∈ H α 0 (] -A, A[)|ψ(0) = 0} is dense in H α 0 (] -A, A[),
we conclude by compactness as in the case µ > 0. Case 2 α ≥ 1 2 . The idea is to show that 0 is a removable singularity in the extended problem, as introduced in [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF]. This will allow us to conclude that φ ∞ = φ 1,2,0,α and ξ ∞ = ξ α (Ω 1,2,0 ). The inspiration for the whole proof comes from Serrin [START_REF] Serrin | Removable singularities of solutions of elliptic equations[END_REF]. So, let v be the solution of

-div(y a ∇v) = 0 for (x, y) ∈ R × R + v(x, 0) = φ ∞ (x) for x ∈ R, with a = 1 -2α.
From [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF] we have that, for all x ∈] -A, A[\ {0} :

lim y→0 y a ∂ y v(x, y) = -(-∆) α φ ∞ (x) = -ξ ∞ φ ∞ (x) = -ξ ∞ v(x, 0).
We define w as the solution of the following equation (4.25) where L ∈]0, A[. That w exists and is unique is a consequence of the implicit function Theorem for L small enough. Define W = v -w such that

         -div(y a ∇w) = 0 for (x, y) ∈] -L, L[×]0, L[ w(x, y) = v(x, y) for (x, y) ∈ ∂(] -L, L[×]0, L[) ∩ {y > 0} , lim y→0 y a ∂ y w(x, y) = -lim y→0 ξ ∞ w(x, y) for x ∈] -L, L[,
         -div(y a ∇W ) = 0 for (x, y) ∈] -L, L[×]0, L[ W (x, y) = 0 for (x, y) ∈ ∂(] -L, L[×]0, L[) ∩ {y > 0} , lim y→0 y a ∂ y W (x, y) = -lim y→0 ξ ∞ W (x, y) for x ∈] -L, L[\ {0} . (4.26)
We are going to prove that W = 0. For this purpose, we split W into two parts :

W = W 1 +W 2 .
The function W 1 is solution of the following equation :

         -div(y a ∇W 1 ) = 0 for (x, y) ∈ B(0, ε) + , W 1 (x, y) = W (x, y) for (x, y) ∈ ∂B(0, ε) + , lim y→0 y a ∂ y W 1 (x, y) = 0 for x ∈] -ε, ε[ (4.27)
where ε ∈]0, L 2 [ will be chosen later on. Next, we focus the study of W 2 on the domain (]

-L, L[×]0, L[) \B(0, ε) + . The equation for W 2 is                -div(y a ∇W 2 ) = 0 for (x, y) ∈ (] -L, L[×]0, L[)\B(0, ε) + W 2 (x, y) = 0 for (x, y) ∈ ∂(] -L, L[×]0, L[) ∩ {y > 0} , W 2 (x, y) = W 1 (x, y) for (x, y) ∈ ∂B(0, ε) + lim y→0 y a ∂ y W 2 (x, y) = -lim y→0 ξ ∞ W (x, y) for x ∈] -L, L[\] -ε, ε[. (4.28)
We underline that W 2 -→ ε→0 W weakly. Therefore, since W 1 = W -W 2 , we deduce that W 1 -→ ε→0 0 weakly also. In the following, we denote by

D =] -L, L[×]0, L[\B(0, ε), Γ 1 = ∂ (] -L, L[×]0, L[) \ (] -L, L[× {0}) , Γ 2 =] -L, -ε[∪]ε, L[, and Γ 3 = ∂B(0, ε) + .
               -div(y a ∇W 2 ) = 0 for (x, y) ∈ D W 2 (x, y) = 0 for (x, y) ∈ Γ 1 , W 2 (x, y) = W 1 (x, y) for (x, y) ∈ Γ 3 lim y→0 y a ∂ y W 2 (x, y) = 0 for x ∈ Γ 2 (4.29)
and

               -div(y a ∇W 2 ) = 0 for (x, y) ∈ D W 2 (x, y) = 0 for (x, y) ∈ Γ 1 , W 2 (x, y) = 0 for (x, y) ∈ Γ 3 , lim y→0 y a ∂ y W 2 (x, y) = -lim y→0 ξ ∞ W (x, y) for x ∈ Γ 2 . (4.30)
Since, W 1 tends weakly to 0 as ε → 0, it follows that W 2 tends also weakly to 0. It remains to prove that W 2 vanishes. Let φ be a test function in H 1 0 (D, |y| a ) (see Chapter 1 of [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF] for the general framework of weighted Sobolev spaces). Noticing that for all τ > 0 and (x, y)

∈ D ∩ {y > τ }, we have div(φ|y| a ∇W 2 )(x, y) = ∇φ(x, y)|y| a ∇W 2 (x, y) + φ(x, y)div(|y| a ∇W 2 )(x, y) = ∇φ(x, y)|y| a ∇W 2 (x, y).
We deduce that

D ∇φ|y| a ∇W 2 dxdy = D∩{y>τ } ∇φ|y| a ∇W 2 dxdy + D∩{y≤τ } ∇φ|y| a ∇W 2 dxdy = D∩{y>τ } div(φ|y| a ∇W 2 )(x, y)dxdy + D∩{y≤τ } ∇φ|y| a ∇W 2 dxdy = ∂(D∩{y>τ }) φ|y| a ∇W 2 (x, y) • νdx + D∩{y≤τ } ∇φ|y| a ∇W 2 dxdy = D∩{y=τ } -φ(x, τ )|τ | a ∂ y W 2 (x, τ )dx + D∩{y≤τ } ∇φ|y| a ∇W 2 dxdy.
Next, if we let τ tends to 0, the second term of the right hand side tends to 0 thanks to the Cauchy-Schwarz inequality. Recalling that W = W 1 + W 2 + W 2 , we finally have obtained the following variational equation :

D ∇φ∇W 2 |y| a dxdy - Γ 2 φ(x, 0)ξ ∞ W 2 (x, 0)dx = Γ 2 ξ ∞ (W 1 (x, 0) + W 2 (x, 0))φ(x, 0)dx.
(4.31) In order to prove the existence and uniqueness of a solution of (4.31), we are going to apply the Lax-Milgram theorem. The linear map is the following :

Λ : φ ∈ H 1 0 (D, |y| a ) → ξ ∞ Γ 2 (W 1 (x, 0) + W 2 (x, 0))φ(x, 0)dx.
Since (W 1 + W 2 ) tends weakly to 0 as ε → 0, we deduce that

Λ H 1 0 (D) -→ ε→0 0. (4.32)
Next, we show that for L small enough, the bilinear form

b : (u, v) ∈ H 1 0 (D) × H 1 0 (D) → D ∇u∇v|y| a dxdy - Γ 2 u(x, 0)ξ ∞ v(x, 0)dx 4.2.
Dependence of the sign of ξ α (Ω 1,2,µ ) on the parameters is continuous and coercive. First, the Poincaré inequality implies that φ → ( D ∇φ 2 |y| a dxdy) 1 2 is a norm that is equivalent to the usual norm on H 1 0 (D, |y| a ) (see equation (1.5) p. 9 of [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF]). thus the continuity follows. Secondly, we prove that for L small enough, b is coercive. Indeed, since a = 1 -2α < 0, we have

Γ 2 u(x, 0) 2 dx = 2 Γ 2 L 0 u(x, y)∂ y u(x, y) |y| a |y| a dydx ≤ 2 L a Γ 2 L 0 u(x, y) 2 |y| a dy 1 2 L 0 ∂ y u(x, y) 2 |y| a dy 1 2 dx ≤ 2 L a Γ 2 L 0 u(x, y) 2 |y| a dy 1 2 L 0 ∂ y u(x, y) 2 |y| a dy dx ≤ 2 L a Γ 2 L 0 u(x, y) 2 |y| a dydx 1 2 Γ 2 L 0 ∂ y u(x, y) 2 |y| a dydx 1 2 ≤ 2Cdiam(D)L 2α-1 D |∇u| 2 |y| a dxdy.
Therefore, for L small enough, we deduce that there exists

c b > 0 such that b(u, u) > c b D |∇u| 2 |y| a dxdy.
We conclude thanks to the Lax-Milgram theorem that there exists a unique solution to (4.31). Moreover thanks to the estimates of the norm of the solution in the Lax-Milgram theorem and (4.32), we deduce that

W 2 H 1 0 (D) ≤ Λ H 1 0 (D) c b -→ ε→0 0.
The conclusion follows.

Dependence on α

In this subsection, we provide the proof of Proposition 4.1.

Proof of Proposition 4.1. The continuity for α < 1 is based on similar arguments as those presented in the proof of the continuity of ξ α (Ω 1,2,µ ) for µ > 0. Therefor, we focus on the continuity up to α = 1. First, we establish a monotonicity result. We claim that the function

(α ∈]0, 1] → diam(Ω 1,2,µ ) 2α ξ α (Ω 1,2,µ )) is increasing,
(where diam(Ω 1,2,µ ) designates the diameter of Ω 1,2,µ ). Indeed, since if α < β there holds

H β 0 (Ω 1,2,µ ) ⊂ H α 0 (Ω 1,2,µ ), we deduce that ξ α (Ω 1,2,µ ) ≤ min φ∈H α 0 (Ω 1,2,µ ) R α (φ) ≤ R α (φ 1,2,µ,β )
Chapter 4 : Properties of steady states for a class of non-local Fisher-KPP equations in general domains

= 1 2 R R (φ 1,2,µ,β (x) -φ 1,2,µ,β (y)) 2 |x -y| d+2α dydx = 1 2 R R,|x-y|<diam(Ω 1,2,µ ) (φ 1,2,µ,β (x) -φ 1,2,µ,β (y)) 2 |x -y| d+2α dydx + 1 2 R R,|x-y|≥diam(Ω 1,2,µ ) (φ 1,2,µ,β (x) -φ 1,2,µ,β (y)) 2 |x -y| d+2α dydx = diam(Ω 1,2,µ ) 2(β-α) 2 R R,|x-y|<diam(Ω 1,2,µ ) (φ 1,2,µ,β (x) -φ 1,2,µ,β (y)) 2 |x -y| d+2β dydx + 1 2 R φ 1,2,µ,β (x) R,|x-y|≥diam(Ω 1,2,µ ) 1 |x -y| d+2α dydx = diam(Ω 1,2,µ ) 2(β-α) 2 R R,|x-y|<diam(Ω 1,2,µ ) (φ 1,2,µ,β (x) -φ 1,2,µ,β (y)) 2 |x -y| d+2β dydx + diam(Ω 1,2,µ ) -2α 2 R φ 1,2,µ,β (x) 2 dx = diam(Ω 1,2,µ ) 2(β-α) 2 R R (φ 1,2,µ,β (x) -φ 1,2,µ,β (y)) 2 |x -y| d+2β dydx = diam(Ω 1,2,µ ) 2(β-α) ξ β (Ω 1,2,µ ).
(Note that this last result holds true for β = 1 thanks to the limit (2.8) in [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF]). Next, we prove the continuity result for α = 1. Let (α k ) k∈N ∈] 1 2 , 1[ be a sequence such that α k -→ k→+∞ 1. By replacing φ 1,2,µ,β by the φ 1 (with φ 1 the principal eigenfunction which corresponds to the principal eigenvalue min(ξ 1 (Ω 1 ), ξ 2 (Ω 2 )), we deduce that

0 < ξ α k (Ω 1,2,µ ) ≤ diam(Ω) 2-2α k min(ξ 1 (Ω 1 ), ξ 2 (Ω 2 )). (4.33)
We deduce that up to an extraction, ξ α k (Ω 1,2,µ ) converges to ξ ∞ . We recall that φ 1,2,µ,α k 2 = 1. Moreover, for all k ∈ N, we have

< (-∆) α k φ α k ,1,2,µ , φ α k ,1,2,µ > = ξ α k (Ω 1,2,µ ) φ α k ,1,2,µ 2 2 ≤ diam(Ω 1,2,µ ) 2(1-α k ) min(ξ 1 (Ω 1 ), ξ 1 (Ω 2 )).
Up to an extraction and thanks to the Sobolev embedding (see [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF]), φ α k ,1,2,µ converges to φ ∞ in L 2 (Ω 1,2,µ ). Moreover, the limit satisfies in a weak sense the following equation

(-∆)φ ∞ (x) = ξ ∞ φ ∞ (x) for x ∈ Ω 1,2,µ , φ ∞ (x) = 0 for x ∈ ∂Ω 1,2,µ . (4.34)
By the standard elliptic regularity, we find that (4.34) is true in a strong sense. Furthermore, thanks to (4.33), we deduce that ξ ∞ ≤ min(ξ 1 (Ω 1 ), ξ 2 (Ω 2 )). Since the L 2 norm of φ ∞ is not trivial and by positiveness of φ ∞ and uniqueness of φ 1 (up to a constant), we conclude that φ ∞ = φ 1 .

Dependence on µ and α

For this subsection, we denote by i ∈ {1, 2} the index of the larger domain (i.e. |Ω i | ≥ |Ω 3-i |). This subsection is devoted to the

Estimates for steady solutions

Proof of Proposition 4.2. We split the proof of (4.10) in three parts : first, we assume ξ * = ξ 1 (Ω 1,2,µ ), then we assume ξ * = ξ i,1 , and finally we do the general case. Proof when ξ * = ξ 1 (Ω 1,2,µ ). Let (α k ) k∈N be any increasing sequence such that α k → 1. According to Theorem 4.1, there exists µ k (α k ) such that

∀µ ≤ µ k (α k ), |ξ α k (Ω 1,2,µ ) -ξ α k (Ω 1,2,0 )| ≤ 1 k .
We conclude that (α k , µ k (α k )) gives the result.

Proof when ξ * = ξ i,1 . Let (ν k ) k∈N be any decreasing sequence such that ν k → 0. According to Theorem 4.1, there exists

β k (ν k ) < 1 such that ∀β ≥ β k (ν k ), |ξ β (Ω 1,2,ν k ) -ξ i,1 | ≤ 1 k .
We conclude that (ν k , β k (ν k )) gives the result. Proof when ξ * ∈]ξ 1 (Ω 1,2,0 ), ξ i,1 [. Let (µ k , α k ) be defined as follows :

                       (µ 0 , α 0 ) = (µ 1 , α 1 ) = (µ 2 , α 2 ) = ( 1 2 , 1 2 ), (µ 3k , α 3k ) = (µ 3k-1 , max(β k (µ 3k-1 ), α 3k-1 , 1 - 1 k )), (µ 3k+1 , α 3k+1 ) = (µ 3k+1 , α 3k ), (µ 3k+2 , α 3k+2 ) = (min(µ k (α 3k ), µ 3k , 1 k ), α 3k ),
where we will fix µ 3k+1 ∈ [µ 3k , µ 3k+2 ] later on. The constants µ k (α 3k ) and β k (µ 3k-1 ) are defined respectively in the two first parts of the proof. With a such choice of (µ k , α k ), we have

ξ α 3k (Ω 1,2,µ 3k ) -→ k→+∞ ξ i,1 and ξ α 3k+2 (Ω 1,2,µ 3k+2 ) -→ k→+∞ ξ 1 (Ω 1,2,0 ).
Let k 0 ∈ N be such that for all k > k 0 ,

ξ 1 (Ω 1,2,0 ) + 2 k < ξ * < ξ i,1 - 2 k . (4.35)
Inequalities (4.35) implies

ξ α 3k (Ω 1,2,µ 3k ) < ξ * < ξ α 3k+2 (Ω 1,2,µ 3k+2 ).
Next, we fix

µ 3k+1 = µ 3k for k ∈ {0, ..., k 0 }. Since µ → ξ α k (Ω 1,2,µ
) is continuous and increasing, and because α 3k = α 3k+1 , we deduce thanks to the intermediate value theorem that there exists µ 3k+1 ∈]µ 3k , µ 3k+2 [ such that ξ α 3k+1 (Ω 1,2,µ 3k+1 ) = ξ * . We conclude that the sequence (ξ k , γ k ) = (µ 3k+1 , α 3k+1 ) gives the result.

Estimates for steady solutions

This section is devoted to the proof of Theorem 4.4. First, we provide the general strategy of the proof, next we prove intermediate results and finally, we prove Theorem 4.4. We underline here that 0 ≤ n + ≤ 1

Chapter 4 : Properties of steady states for a class of non-local Fisher-KPP equations in general domains is easy to obtain thanks to the comparison principle. Therefore,we prove that there exists two constants c, C > 0 such that, if the function G(x) is given by (4.16), then we have

c min(δ(x) α , ε α 0 )G(x) ≤ n + (x) ≤ C min(δ(x) α , ε α 0 )G(x).
We highlight that the subdomains Ω -,k which satisfy (H6) may not be connected. It is one of the interest to consider non-local diffusion instead of local one. Moreover, it may happen that the principal eigenvalue of (-∆) α -Id defined in the connected components of Ω -,k are all positive however the principal eigenvalue of (-∆) α -Id defined in the whole domain Ω -,k is negative. A consequence of Theorem 4.1 is the following : if d = 1 and Ω -= Ω 1,2,µ (where Ω -refers to (H6) and Ω 1,2,µ to (4.4)), the following assertions hold :

1. we must have λ α (Ω 1,2,µ ) < 0, 2. if λ α (Ω 1 ) < 0 or λ α (Ω 2 ) < 0, then Ω -= Ω 1,2,µ
holds true for all µ > 0, 3. if λ α (Ω 1 ) > 0 and λ α (Ω 2 ) > 0, then there exists µ 0 > 0 such that µ < µ 0 .

The first point is a consequence of the continuity up to µ = 0 and the monotonicity of (µ → λ α (Ω

α (Ω 1 ), λ α (Ω 2 ) > 0, then there exists a distance µ 0 ≥ 0 such that λ α (Ω 1,2,µ 0 ) = 0.
The proof of this lemma involves lengthy but standard computations. Therefore, we postpone it at the end of the paper in the Appendix.

Strategy of the proof of Theorem 4.4

The lower part of (4.15) will be obtained from the fractional Poisson kernel in a ball (see for instance in [START_REF] Bogdan | The boundary Harnack principle for the fractional Laplacian[END_REF]) :

P r (x, y) = c d α r 2 -|x| 2 |y| 2 -r 2 α 1 |x -y| d 1 {|y|>r} (x). (4.36)
With this kernel, for any smooth function v and z ∈ R d we have that the solution of the equation

(-∆) α n(x) = 0 x ∈ B(z, r), n(x) = v(x) x ∈ B(z, r) c , is n(x) = R d P r (x -z, y -z)v(y)dy.
The difficult part of the proof will be to obtain the upper bound in Ω + of (4.15). First, we establish that : Lemma 4.2. Let n + be a positive bounded solution of (4.1). If R 0 > C d,α,ε 0 ,ε 1 ,r 0 then there exists a positive constant C 2 such that for all

x ∈ Ω + n + (x) ≤ C 2 Ω - 1 |x -y| d+2α dy.
(4.37)

Estimates for steady solutions

To prove Lemma 4.2, we localise the function n + in a cluster C k and we use hypothesis (H4) which essentially says that the cluster is at large distance from the others.

Next, the idea is to compare the solution n + with a translated and rescaled barrier function ψ. This particular barrier function satisfies

             (-∆) α ψ(x) > 1 for x ∈ T (0, 1, 4), ψ(x) = 0 for x ∈ B(0, 1), ψ(x) ≤ c(|x| -1) α for x ∈ T (0, 1, 4), 1 ≤ ψ ≤ C for x ∈ B(0, 4) c , (4.38)
where T (z, r, R) designates the torus of center z ∈ R d and of inner and outer radius 0

< r < R i.e. T (z, r, R) = y ∈ R d , r < |y -z| < R .
The construction of a such barrier function can be found in Appendix B of [START_REF] Ros-Oton | The Dirichlet problem for the fractional Laplacian : regularity up to the boundary[END_REF]. Then, we look for "a suitable" constant C k such that for all x ∈ C +,k we have

n + (x) ≤ C k ψ x -z x ε 0
where z x and ε 0 are introduced in (H2). We claim that there exists a positive constant C d,α,ε 0 ,ε 1 ,r 0 which depends only on the listed parameters such that

C k = sup x∈C +,k C d,α,ε 0 ,ε 1 ,r 0 Ω - 1 |x -y| d+2α dy . (4.39)
Thanks to (4.39), the third property of (4.38) and some Harnack type inequalities the conclusion follows.

Remark. In particular cases, some explicit and more tractable versions of G can be found. For instance, if Ω -is bounded, one can prove that G(x) = min(δ(x) α ,1) 1+|x| d+2α . We provide some examples at the end of this section.

Harnack type properties

First, we introduce the following function, defined for

x ∈ Ω + g(x) = Ω - 1 |x -y| d+2α dy, (4.40) such that G(x) = 1 Ω -(x) + g(x)1 Ω + (x).
We are going to prove some properties for g and G which are elementary in nature, but will be important for the sequel.

Proposition 4.3.

There exists a positive constant C > 0 such that for all k ∈ N,

sup x∈C +,k g(x) ≤ C inf x∈C +,k g(x). ( 4
+,k Ω - 1 |x -y| d+2α dy = Ω - 1 |x k -y| d+2α dy.
According to (H4), we have for all (x, y)

∈ C +,k × Ω -, |x -y| ≤ |x -x k | + |x k -y| ≤ 2r 0 + |x k -y|. It follows that 1 (2r 0 + |x k -y|) d+2α ≤ 1 |x -y| d+2α .
Since y ∈ Ω -, we deduce that R 0 ≤ |x k -y| and we conclude to the existence of a constant C (independent of x k and y) such that

1 |x k -y| ≤ C 2r 0 + |x k -y| .
The conclusion follows.

Corollary 4.2.

There exists ε 2 > 0 such that for all ε < ε 2 we have

inf x,y∈Ω |x-y|<ε 2 G(x) G(y) > 0. (4.42)
With similar arguments, we prove

Lemma 4.3. Let g : x ∈ Ω + → Ω∩B(zx,R 0 ) c 1 |x-y| d+2α dy. There exists C r 0 ,d,α depending on r 0 , d, α such that ∀x ∈ C +,k , max y∈C +,k g(y) ≤ C r 0 ,d,α g(x). Proposition 4.4. The map G is uniformly continuous in Ω + , that is ∀ε > 0, ∃µ ε such that ∀(x, y) ∈ Ω + , |x -y| < µ ε , then |G(x) -G(y)| < ε.
Proof. Let ε > 0 and R > 0 be such that

|y|>R 1 |y| d+2α dy = 1 R 2α ≤ ε 4 . ( 4.43)
Next, remarking that for all (x, y) ∈ Ω + ×Ω -we have R 0 ≤ |x-y|, by the uniform continuity of the application (t ∈]R 0 , +∞[ → t -(d+2α) ) we deduce the existence of a distance µ ε > 0 such that for all

x 1 , x 2 ∈ Ω + with |x 1 -x 2 | ≤ µ ε we have ∀y ∈ Ω -, | 1 |x 1 -y| d+2α - 1 |x 2 -y| d+2α | ≤ ε 2m(B(0, R + 2µ ε )) . ( 4.44)
Next, we conclude thanks to (4.43) and (4.44) that

| Ω - 1 |x 1 -y| d+2α dy - Ω - 1 |x 2 -y| d+2α dy| ≤ | Ω -∩B(x 1 ,R+2µε) 1 |x 1 -y| d+2α - 1 |x 2 -y| d+2α dy| + | Ω -∩B(x 1 ,R+2µε) c 1 |x 1 -y| d+2α dy| + | Ω -∩B(x 1 ,R+µε) c 1 |x 2 -y| d+2α dy| ≤ ε.

Estimates for steady solutions

Finally, we recall the following strong maximum principle for the fractional Laplacian :

Lemma 4.4. For any smooth bounded domain U ⊂ R d , and any smooth non-trivial function

v ∈ H α (U) such that 1. v ≥ 0, 2. ((-∆) α -Id)v > 0,
it follows v > 0 in the interior of the domain U.

The proof of Lemma 4.2

Step 1 : Localisation argument. Let χ k be a function such that

χ k ∈ C ∞ c (R d , [0, 1]
) and

χ k (x, z) = 1 if x ∈ B(z k , r 0 ), 0 if x ∈ B(z k , 2r 0 ) c (4.45)
where z k and r 0 are provided by (H4). We set n k = n + ×χ k . This function belongs to

H α 0 (C +,k ) ((-∆) α -Id)n k (x) = R d n k (x) -n k (y) |x -y| d+2α dy -n k (x) = R d n + (x) -n + (y) |x -y| d+2α dy + R d n + (y)(1 -χ k (y)) |x -y| d+2α dy -n + (x) = R d n + (y)(1 -χ k (y)) |x -y| d+2α dy -n + (x) 2 ≤ B(z k ,R 0 ) c ∩Ω n + (y) |x -y| d+2α dy.
In the following, we denote by

g k = max x∈C +,k B(z k ,R 0 ) c ∩Ω n + (y) |x -y| d+2α dy (4.46)
Let φ k be the principal eigenfunction of the Dirichlet operator (-∆) α -Id in the set

x ∈ R d |dist(x, C +,k ) < ε 1 .
Next, we define

ψ k (x) = g k cλ 0 min(ε 2α 1 , 1) φ k (x) (4.47)
where c is a positive constant such that min

y∈C +,k φ k (y) > c min(ε 2α 1 , 1)
. Indeed, since C +,k is uniformly bounded with respect to k, we deduce thanks to [START_REF] Ros-Oton | The Dirichlet problem for the fractional Laplacian : regularity up to the boundary[END_REF] (Proposition 3.5) and thanks to the interior ball condition that c exists and is positive. Moreover, we have obviously

g k ≤ ((-∆) α -Id)ψ k (x).
Since n + ≤ 1, we deduce the existence of C > 1 such that

∀x ∈ C +,k , n + (x) ≤ Cψ k (x).
Chapter 4 : Properties of steady states for a class of non-local Fisher-KPP equations in general domains

Let C * = inf {C > 1 | n + ≤ Cψ k }.
We claim that C * = 1. Indeed, if we assume by contradiction that C * = 1, it follows that there exists

x 0 ∈ C +,k such that C * ψ k (x 0 ) -n k (x 0 ) = 0. Recalling that n k|∂C +,k = 0 and ψ k |∂C +,k = 0, it follows that x 0 ∈ C +,k . Remarking that ((-∆) α -Id)(C * ψ k -n k )(x) ≥ C * g k -g(x) ≥ 0,
the existence of x 0 ∈ C +,k is in contradiction with Lemma 4.4. Therefore, we deduce that C * = 1. We conclude thanks to Lemma 4.3 that for all x ∈ C +,k

n + (x) ≤ n k (x) ≤ ψ k (x) = C r 0 ,d,α cλ 0 min(1, ε 2α 1 ) :=C 0 g(x). (4.48)
Step 2 : Concentration in

Ω -. Since n + ≤ 1, R 0 > C 1 2α
0 it follows by an immediate iteration of (4.48) that for all x ∈ Ω + :

n + (x) ≤ C 0 +∞ k=0 I k (x) with I k (x) = C 0 Ω + ∩B(x,R 0 ) c I k-1 (y) |x -y| d+2α dy, I 0 = Ω - 1 |x -y| d+2α dy.
Let us prove by iteration that

I k (x) ≤ C k 1 Ω - 1 |x -y| d+2α dy with C 1 < 1. (4.49)
The conclusion that we may only use the components Ω -will follows. It is clear that (4.49) holds true for k = 0. Next, we prove it for k = 1 :

I 1 (x) = C 0 Ω + ∩B(zx,R 0 ) c 1 |x -y 0 | d+2α Ω - 1 |y 0 -y 1 | d+2α dy 1 dy 0 = C 0 (x-y 0 )∈Ω + ,y 0 ∈B(0,R 0 ) c 1 |y 0 | d+2α Ω - 1 |x -y 0 -y 1 | d+2α dy 1 dy 0 .
Next, by the convexity of the function (t → |t| d+2α ) it follows

|x -y 1 | ≤ |x -y 0 -y 1 | + |y 0 | ⇒ 1 |x -y 0 -y 1 | d+2α ≤ 2 d-1+2α |x -y 1 | d+2α 1 + |y 0 | d+2α |x -y 0 -y 1 | d+2α .
Fubini's Theorem leads to

I 1 (x) ≤ C 0 Ω - 1 |x -y 1 | d+2α (x-y 0 )∈Ω + ,y 0 ∈B(0,R 0 ) c 2 d-1+2α |y 0 | d+2α 1 + |y 0 | d+2α |x -y 0 -y 1 | d+2α dy 0 dy 1 ≤ C 0 Ω - 2 d-1+2α |x -y 1 | d+2α y 0 ∈B(0,R 0 ) c 1 |y 0 | d+2α dy 0 + (x-y 0 )∈Ω + 1 |x -y 0 -y 1 | d+2α dy 0 dy 1 ≤ 2 d+2α C 0 R 2α 0 × Ω - 1 |x -y 1 | d+2α dy 1 . Assuming R 0 > (2 d+2α C 0 ) 1 2α
leads to the conclusion that (4.49) holds true for k = 1. If we assume that it is true for k ∈ N, we have by the recursive hypothesis on I k that

I k+1 (x) ≤ C 0 Ω + ∩B(x,R 0 ) c C k 1 |x -y 0 | d+2α Ω - 1 |y 0 -y 1 | d+2α dy 1 dy 0 .
The conclusions follows from computations that are similar to those of the case k = 1.

Estimates for steady solutions

Proof of Theorem 4.4

Before providing the proof of Theorem 4.4 we introduce an intermediate technical result : Lemma 4.5. There exist two constant C, σ > 0 such that for all x ∈ Ω + ,

Ω - 1 |x -y| d+2α dy ≤ C (Ω -)σ 1 |x -y| d+2α dy and C +,kx 1 |x -y| d+2α dy ≤ C (C +,kx )σ 1 |x -y| d+2α dy,
where O σ is defined in (4.11) for any set O.

Proof. We prove only the first inequality. The second one can be proved following similar computations. Thanks to the interior ball condition (H2), it is sufficient to prove that there exists σ, C > 0 such that there holds

B(z,ε 0 ) 1 |x -y| d+2α dy ≤ C B(z,ε 0 )σ 1 |x -y| d+2α dy (4.50)
for any x ∈ Ω + and B(z, ε) ⊂ Ω -. First, we remark that there exists two constants c 1 , c 2 > 0 such that for all t ∈]R 0 , +∞[,

c 1 t d+2α ≤ 1 (t + ε 0 ) d+2α and 1 (t -ε 0 ) d+2α ≤ c 2 t d+2α .
Next, denoting by m(E) the Lebesgue measure of the set E, we define

σ = ε 0 4 and C = 2c 2 m(B(0, ε 0 )) c 1 m(B(0, ε 0 2 ))
.

With a such choice of constants, it follows that

C B(z,ε 0 )σ 1 |x -y| d+2α dy - B(z,ε 0 ) 1 |x -y| d+2α dy ≥ Cm(B(0, ε 0 2 )) (|x -z| + ε 0 ) d+2α - m(B(0, ε 0 )) (|x -z| -ε 0 ) d+2α ≥ Cc 1 m(B(0, ε 0 2 )) |x -z| d+2α - c 2 m(B(0, ε 0 )) |x -z| d+2α ≥ c 2 m(B(0, ε 0 )) |x -z| d+2α ≥ 0.
Proof of Theorem 4.4. We split the study into two parts : the study in Ω + and the study in Ω -. Each part is split into two sub-parts :the lower and the upper bounds.

Part 1 : The study in Ω -. Subpart A : The lower bound. Let x ∈ Ω -and z x ∈ Ω -, k ∈ {1, ..., N } such that (H6) holds true. Since λ α (Ω k ) < 0, we deduce the existence of n k,z the solution of

(-∆) α n k,z = n k,zx -n 2 k,zx in Ω k + z x , n k,zx = 0 in (Ω k + z x ) c .
The maximum principle implies that

n k,zx (x) ≤ n + (x).
Moreover, since Ω k is bounded and regular, we deduce the existence of c k such that

c k min(δ(x) α , 1) ≤ n k,zx (x).
If we fix c = min k∈{1,...,N } c k then the conclusion follows.

Chapter 4 : Properties of steady states for a class of non-local Fisher-KPP equations in general domains Subpart B : The upper bound. From the maximum principle, it is clear that n + (x) ≤ 1. Therefore, we focus on what happens at the boundary : let x ∈ Ω -such that δ(x) < ε 0 and z x ∈ Ω c provided by the exterior ball condition such that

B( z x , ε 0 ) ∩ Ω = ∅, B( z x , ε 0 ) ∩ Ω = ∅ and x ∈ B( z x , 2ε 0 ).
Then, from the maximum principle applied to n + and ψ( •-zx ε 0 ) where ψ is defined by (4.38), it follows

n + (x) ≤ ψ( x -z x ε 0 ) ≤ Cδ(x) α .
The conclusion follows.

Part 2 : The study in Ω + . Subpart A : The lower bound. We prove that for all

x ∈ Ω + c min(δ(x) α , ε α 0 ) Ω - 1 |x -y| d+2α dy ≤ n + (x). (4.51)
Let x ∈ Ω + and z x ∈ Ω + provided by (H2) (remark that z x = x if δ(x) ≥ ε 0 ). We define n as the solution of

       (-∆) α n(y) = 0 for y ∈ B(z x , ε 0 ), n(y) = n + for y ∈ Ω -, n(y) = 0 for y ∈ R d \(B(z x , ε 0 ) ∪ Ω -).
The comparison principle gives that

n(x) ≤ n + (x).
We recall that thanks to (H6) for all positive σ small enough, there exists c σ > 0 such that c σ < n + (y) for all y ∈ (Ω -) σ . Formula (4.36) gives

n(x) = Ω - c d α ε 2 0 -|x -z x | 2 |y -z x | 2 -ε 2 0 α 1 |x -y| d n + (y)dy ≥ cc d α (ε 2 0 -|x -z x | 2 ) α Ω -,σ c σ |y -z x | d+2α dy.
If ε 0 ≤ δ(x), we deduce that z x = x and (4.51) holds true thanks to Lemma 4.5. Otherwise, we have

(ε 0 δ(x)) α ≤ (ε 0 + |x -z x |) α δ(x) α = (ε 2 0 -|x -z x | 2 ) α .
By uniform continuity and compactness of B(z x , ε 0 ), we deduce that for all y ∈ Ω -there holds

Subpart B :

The upper bound. Thanks to Lemma 4.2, we only have to consider x ∈ {y ∈ Ω + such that δ(y) < ε 0 }. Let k ∈ N and x ∈ C +,k be such that δ(x) < ε 0 . Assumption (H2) ensures the existence of z x ∈ R d such that

B( z x , ε 0 ) ∩ Ω = ∅, ∂B( z x , ε 0 ) ∩ ∂Ω = ∅ and x ∈ B( z x , 2ε 0 ) ∩ Ω. (4.52)
As mentioned in section 2.1, the aim of the proof is to prove that the constant C k defined by (4.39) verifies

n + (x) ≤ C k ψ x -z x ε 0 (4.53)

Estimates for steady solutions

where ψ k is defined by (4.38). Let χ k be a function of

C ∞ c (R d , [0, 1]) such that χ k (x) =        1 for x such that dist(x, C +,k ) < ε 0 4 0 for x such that dist(x, C +,k ) > ε 0 2 . (4.54)
In order to prove (4.53), we prove first that for all y ∈ T ( z x , ε 0 , 4ε 0 ) ∩ C +,k

(-∆) α n + χ k (y) -C k ψ( y -z x ε 0 ) < 0. (4.55)
Next, we prove that for all y ∈ (T ( z x , ε 0 , 4ε 0 ) ∩ C +,k ) c , we have

n + χ k (y) -C k ψ( y -z x ε 0 ) ≤ 0. ( 4.56)
The conclusion follows thanks to the maximum principle.

Proof that (4.55) holds true. Let y ∈ T ( z x , ε 0 , 4ε 0 )∩C +,k , then thanks to the properties of ψ (see (4.38)), and Lemma 4.2, we obtain

(-∆) α n + χ k (y) -C k ψ( y -z x ε 0 ) ≤(-∆) α n + (y) + C c +,k n + (y 0 )(1 -χ k (y 0 )) |y -y 0 | d+2α dy 0 - C k ε 2α 0 ≤n + (y) + Ω - 1 |y 0 -y| d+2α dy 0 + C c +,k ∩Ω + n + (y 0 ) |y -y 0 | d+2α dy 0 -C k ≤(C 2 + 1) Ω - 1 |y -y 0 | d+2α dy 0 + C 2 C c +,k ∩Ω + 1 |y -y 0 | d+2α Ω - 1 |y 0 -y 1 | d+2α dy 1 dy 0 -C k ≤ C 2 (1 + C d,α,r 0 R 2α 0 ) + 1 Ω - 1 |y -y 0 | d+2α dy 0 -C k .
We conclude thanks to (4.39) that (4.55) holds true.

Proof that (4.56) holds true. According to (4.52), it is straightforward that (n

+ χ k (y)- C k ψ( y-zx ε )) ≤ 0 for all y ∈ B( z x , ε 0 ) ∪ B( z x , 4ε 0 ) c ∩ C c +,k
. Therefore, we focus on y ∈ B( z x , 4ε 0 ) c ∩ C +,k . Thanks to Lemma 4.2 and the properties of ψ (introduced in (4.38)), we obtain

n + (y)χ k (y) -C k ψ( y -z x ε 0 ) ≤ n + (y) -C k ≤ C 2 Ω - 1 |y -y 0 | d+2α dy 0 -C k < 0.
We deduce that (4.56) holds true.

The conclusion follows thanks to Proposition 4.3.

Some examples where G is explicit

In this subsection, we detail some examples where the function G is more explicit. It highlights how the shape of the steady solution is strongly connected to the domain Ω.

1. Ω = R d . It is well known that in this case G(x) = 1.
Chapter 4 : Properties of steady states for a class of non-local Fisher-KPP equations in general domains 2. Ω is bounded. In this case, we have

G(x) = δ(x) α .
We recover here a consequence of [START_REF] Ros-Oton | The Dirichlet problem for the fractional Laplacian : regularity up to the boundary[END_REF].

3. Ω is unbounded and Ω + = ∅. Following Theorem 4.4, there holds

G(x) = min(1, δ(x) α ).
Remark that this case contains the periodic patchy domain (see [START_REF] Léculier | Propagation in a fractional reaction-diffusion equation in a periodically hostile environment[END_REF] for more details) :

Ω = k∈Z d (Ω 0 + a k ) where Ω 0 + a k = x ∈ R d | x -a k ∈ Ω 0 , Ω 0 bounded and such that λ α (Ω 0 ) < 0 and a k ∈ R d such that a k+e i -a k = a e i
where e i is the i th element of the canonical basis of Z d . [START_REF] Arnold | Existence of nontrivial steady states for populations structured with respect to space and a continuous trait[END_REF]. Ω is unbounded with Ω -bounded. In this case, up to a translation, we can assume that B(0, ε 0 ) ⊂ Ω -⊂ B(0, R 1 ) (with R 1 > 0 large enough). Then it is easy to observe that there exists two constants c, C > 0 such that

c min(δ(x) α , 1) 1 + |x| d+2α ≤ G(x) ≤ C min(δ(x) α , 1) 1 + |x| d+2α .

The dimension d = 1

and Ω -= R -. In this case, a straightforward computation leads to

c min(δ(x) α , 1) 1 Ω -(x) + 1 Ω + (x) 1 + |x| 2α ≤ G(x) ≤ C min(δ(x) α , 1) 1 Ω -(x) + 1 Ω + (x) 1 + |x| 2α .
Of course, the hypothesis on Ω allows more general domains. However, with these five examples, we can already observe how the behavior of the solution n + may change from one domain to another.

Uniqueness and attractivity of the steady state 4.4.1 Uniqueness

As mentioned before, the idea to prove uniqueness is to compare two solutions u + and v + thanks to Theorem 4.4. Before providing the details of the proof, we make an easy but important remark :

∀x ∈ R d , G(x) ≤ 1. (4.57)
In the case where Ω + = ∅, the proof would be very easy. Because of the possibly degeneracy of the solutions in Ω + , the proof is not just an adaptation of the case Ω + = ∅. It is inspired by the strategy of the proof of the fractional Hopf Lemma provided by Grecco and Servadei in [START_REF] Greco | Hopf's lemma and constrained radial symmetry for the fractional Laplacian[END_REF].

Uniqueness and attractivity of the steady state

Proof of Theorem 4.5. Let u + and v + be two bounded non trivial solutions of (4.1). Thanks to Theorem 4.4, there exists a constant L > 0 such that

u + ≤ Lv + .
We define l = inf {L > 1 such that u + ≤ Lv + }. We are going to prove by contradiction that l = 1 because if l = 1 then u + ≤ v + and with the same argument, v + ≤ u + and then the conclusion follows. Thus, we assume l > 1 by contradiction. Next we define

w = inf x∈Ω (lv + -u + )(x) G(x) .
Thanks to the definition of l, we deduce that w = 0 (otherwise we can construct l * < 1 such that u + < l * v + holds true, see the proof of Theorem 1 in [START_REF] Léculier | Propagation in a fractional reaction-diffusion equation in a periodically hostile environment[END_REF] for more details). Let (x n ) n∈N ∈ Ω be a minimizing sequence i.e. (lv

+ -u + )(x n ) G(x n ) -→ n→+∞ 0.
In order to localise where we use the maximum principle or the fractional Hopf Lemma, we will use the help from the bubble function β z,r :

β z,r (x) = C α (r 2 -|x -z| 2 ) α 1 B(z,r) (x) with C α = Γ( d 2 ) 4 α Γ(1 + α)Γ( d 2 + α) . ( 4.58)
Indeed, β z,r is solution of the equation

(-∆) α β z,r (x) = 1 for x ∈ B(z, r), β z,r (x) = 0 for x ∈ B(z, r) c . (4.59)
We distinguish 2 cases : (up to a subsequence) inf δ(x n ) > δ 0 > 0 and δ(x n ) -→ n→+∞ 0. In the first case, we define z n = x n and r = δ 0 , and in the second one, we define r = ε 0 and we use the interior ball condition (hypothesis (H2)) to deduce the existence of We prove by contradiction that there exists ρ > 0 such that

z n ∈ Ω such that δ(x n ) = ε 0 -|z n -x n |.
∀x ∈ R d , ρβ zn,r (x) ≤ (lv + -u + )(x n ) G(x n ) . ( 4.60)
Indeed, if (4.60) holds true for all n ∈ N, then we deduce the following contradiction

ρC α r α = inf x∈B(0,ε 0 ) ρβ 0,r (x) (r -|x|) α = inf x∈B(zn,r) ρβ zn,r (x) (r -|x -z n |) α ≤ (lv + -u + )(x n ) min(r 0 , δ(x n )) α G(x n ) = (lv + -u + )(x n ) G(x n ) -→ n→+∞ 0.
Assume by contradiction that (4.60) is false. Then since supp(β zn,r ) = B(z n , r), we deduce that

∀k ∈ N, ∃n k > n k-1 and y k ∈ B(z n k , r) such that (lv + -u + )(y k ) G(x n k ) < β zn k ,r (y k ) k .
Since for all x ∈ R d \B(z n k , r) we have

β zn k ,r (x) k - (lv + -u + )(x) G(x n k ) = - (lv + -u + )(x) G(x n k ) < 0 we deduce that βz n k ,r k -(lv + -u + ) G(xn k ) takes its maximum at some ξ k ∈ B(z n k , r). Remark that since (β zn k ,r ) k∈N is uniformly bounded, we have that 0 < (lv + -u + )(ξ k ) G(x n k ) < sup β zn k ,r k -→ k→+∞ 0. (4.61)
Next, if we compute (-∆) α (lv + -u + ), we find that

(-∆) α (lv + -u + ) = lv + -lv 2 + -u + + u 2 + = (lv + -u + ) -(lv + -u + )(v + + u + ) + u + v + (l -1) ≥ (lv + -u + )(1 -(u + + v + )).
Recalling that u + ≤ 1 and v + ≤ 1, we obtain

- (lv + -u + ) G(x n k ) ≤ (-∆) α (lv + -u + ) G(x n k ) . ( 4.62)
On one hand, if we evaluate (4.62) at ξ k we find thanks to (4.61) and Proposition (4.3) that for k large enough

-ε ≤ - C 1 G(ξ k )δ(ξ k ) α G(x n k ) ≤ - (lv + -u + )(ξ k ) G(x n k ) ≤ (-∆) α (lv + -u + )(ξ k ) G(x n k ) (4.63)
with ε as small as we want.

On the other hand, we claim that there exists c * > 0 such that (-∆) α (lv

+ -u + )(ξ k ) G(xn k )
≤ -c * + ε with ε as small as we want. Indeed, we first introduce

w k ∈ B(z n k , r) such that B(w k , r 4 ) ⊂ B(z n k , r), δ(w k ) > r 2 and ∀y ∈ B(w k , r 4 ), |y -ξ k | > r 4 (4.64)
4.4. Uniqueness and attractivity of the steady state (remark that one can take

w k = z n k if δ(x n k ) -→ k→+∞ 0). Next, we split (-∆) α (lv + -u + )(ξ k ) G(xn k )
in the following way

(-∆) α (lv + -u + )(ξ k ) G(x n k ) = 1 G(x n k ) B(w k , r 4 ) (lv + -u + )(ξ k ) -(lv + -u + )(y) |ξ k -y| d+2α dy + 1 G(x n k ) B(w k , r 4 ) c (lv + -u + )(ξ k ) -(lv + -u + )(y) |ξ k -y| d+2α dy =I 1 + I 2 .
For I 1 , according to (4.61), (4.64), Theorem 4.4 and Corollary 4.2 we deduce that for k large enough we have

I 1 = m(B(0, r 4 ))(lv + -u + )(ξ k ) G(x n k ) - B(w k , r 4 ) - (lv + -u + )(y) G(x n k )|ξ k -y| d+2α dy ≤ m(B(0, r 4 ))(lv + -u + )(ξ k ) G(x n k ) - B(w k , r 4 ) (c 1 + C 1 )G(y) min(δ(y) α , 1) G(x n k )|ξ k -y| d+2α dy ≤ ε 2 -r α (c 1 + C 1 ) inf y∈B(w k , r 4 ) G(y) G(x n k ) inf ξ∈T (0, r 4 , 3r 4 ) B(0, r 4 ) 1 |ξ -y| d+2α dy = ε 2 -c * . ( 4.65)
Note that c * depends only on r and not on k. For I 2 , we recall that max

x∈R d β zn k ,r (x) k - (lv + -u + )(x) G(x n k ) = β zn k ,r (ξ k ) k - (lv + -u + )(ξ k ) G(x n k ) .
Then we deduce that for k large enough we have

I 2 ≤ 1 k B(w k , r 4 ) c β zn k ,ε 0 (ξ k ) -β zn k ,ε 0 (y) |ξ k -y| d+2α dy = 1 k (-∆) α β zn k ,r (ξ k ) - B(w k , r 4 ) β zn k ,r (ξ k ) -β zn k ,r (y) |ξ k -y| d+2α dy ≤ 1 k 1 + max ξ∈T (0, r 4 , 3r 4 ) B(0, ε 0 2 ) β 0,r (y) |ξ -y| d+2α dy ≤ ε 2 . ( 4.66)
Finally, combining (4.63), (4.65) and (4.66), we deduce that for k large enough we have

-ε ≤ (-∆) α (lv + -u + )(ξ k ) G(x n k ) ≤ -c * + ε which is a contradiction for ε = c * 4 .

Convergence to the steady state

The idea is quite classical : it consists in enclosing the solution of (4.3) between an increasing (with respect to the time) sub-solution and a decreasing (with respect to the time) supersolution. The specific ingredient is, once again, that the behaviour of the solution of the Cauchy Problem at infinity has to match that of the steady solution. This is achieved through estimates of the heat kernel developed in [START_REF] Léculier | Propagation in a fractional reaction-diffusion equation in a periodically hostile environment[END_REF] (Theorem 2 p.3) for general domains which satisfy the uniform interior and exterior ball condition (here assumed in the assumption (H2)). The details being otherwise standard, we just present an overview of the proof.

The super-solution. Up to a translation, there is no loss of generality to assume that

B(0, ε 0 ) ⊂ Ω -.
On one hand thanks to Lemma 4.1, there exists some constants c > 0 (that may change from line to line) such that

n + (x) ≥ c min(δ(x) α , 1) × 1 Ω -(x) + Ω - 1 Ω + (x) |x -y| d+2α dy ≥ c min(δ(x) α , 1) × 1 Ω -(x) + B(0,ε 0 ) 1 Ω + (x) |x -y| d+2α dy ≥ c min(δ(x) α , 1) 1 + |x| d+2α . (4.67)
On the other hand, thanks to Theorem 2 in [START_REF] Léculier | Propagation in a fractional reaction-diffusion equation in a periodically hostile environment[END_REF], there exists a constant C > 0 such that

n(x, t = 1) ≤ C min(δ(x) α , 1) 1 + |x| d+2α . ( 4.68)
From (4.67) and (4.68), we deduce that there exists a constant C > 1 such that n(x, t = 1) ≤ Cn + (x).

Next, we define

       ∂ t n + (-∆) α n = n -n 2 for (x, t) ∈ Ω×]0, +∞[, n(x, t) = 0 for (x, t) ∈ Ω c × [0, +∞[, n(x, t = 0) = Cn + (x) for x ∈ Ω.
Since, in the distributional sense, we have

lim t→0 + ∂ t n(x, t) = lim t→0 + (Cn(x, t) -Cn(x, t) 2 -(-∆) α Cn(x, t)) = Cn + (x)(1 -C) ≤ 0.
We deduce that n is decreasing. Since it is bounded from below by n + , it converges to a nontrivial stationary state of (4.1) and by uniqueness of n + we deduce that n(x, t) -→

t→+∞ n + (x).
The sub-solution. As previously, we assume that B(0, ε 0 ) ⊂ Ω 1 ⊂ Ω -where Ω 1 is defined by (H6). Thanks to Theorem 2 in [START_REF] Léculier | Propagation in a fractional reaction-diffusion equation in a periodically hostile environment[END_REF], there exists a constant c > 0 such that c min(δ(x) α , 1) 1 + |x| d+2α ≤ n(x, t = 1).

It follows that there exists σ > 0 such that

σ < inf x∈Ω 1
n(x, t = 1).

Numerical illustrations, perspectives

Next, if we denote by φ 1 the principal eigenfunction of (-∆) α -Id in Ω 1 , it follows that the solution n of

           ∂ t n + (-∆) α n = n -n 2 for (x, t) ∈ Ω×]0, +∞[, n(x, t) = 0 for (x, t) ∈ Ω c × [0, +∞[, n(x, t = 0) = min(σ, |λ 1 |)φ 1 (x) φ 1 ∞ for x ∈ Ω.
is increasing with respect to the time. Indeed, it is sufficient to verify it at time t = 0. For x ∈ Ω 1 , there holds (in a distributional sense)

lim t→0 + ∂ t n(x, t) = lim t→0 + (n(x, t) -n(x, t) 2 -(-∆) α n(x, t)) = min(σ, |λ 1 |)φ 1 (x) φ 1 ∞ (|λ 1 | - min(σ, |λ 1 |)φ 1 (x) φ 1 ∞ ) ≥ 0.
For x ∈ Ω\Ω 1 , there holds (still in a distributional sense)

lim t→0 + ∂ t n(x, t) = lim t→0 + (n(x, t) -n(x, t) 2 -(-∆) α n(x, t)) = R d min(σ,|λ 1 |)φ 1 (y) φ 1 ∞ |x -y| d+2α dy > 0.
Finally, n is increasing and bounded therefore point-wise converging. By fractional elliptic regularity, the limit is a solution of (4.1) and by uniqueness of the non trivial stationary state n + we conclude that n(x, t) -→ t→+∞ n + (x).

Conclusion.

Since the initial datum are right ordered i.e. n(x, t = 0) ≤ n(x, t = 1) ≤ n(x, t = 0), we conclude thanks to the comparison principle and the conclusions of the two lasts parts that lim

t→+∞ n(x, t) = n + (x) ≤ lim t→+∞ n(x, t) ≤ lim t→+∞ n(x, t) = n + (x).
This ends the proof of Corollary 4.1.

Numerical illustrations, perspectives

We provide some numerical illustrations of the results developed in this section. More precisely, we investigate the large time of simulations of equation (4.3) with two 1-dimensional disconnected components modeled by a finite difference method. We vary the distance µ between the two components and the exponent of the diffusivity α. We recover that if the distance µ is to high or the constant α is to closed to 1 then the solution n of (4.3) tends to 0 which means that λ α (Ω 1,2,µ ) ≥ 0. Whereas if the distance µ is not to high and the constant α is not closed to 1 then the solution n tends to a non-trivial positive stationary state which means that λ α (Ω 1,2,µ ) < 0.

The first simulation (Figure 4.4) shows the numerical solution n at time T = 2000 and T = 4000. We can not distinguish the difference between the two drawings, we deduce that we have reached the stationary state.

Next, in Figure 4.5 we increase α. We put α = 0.8, and we find that the solution is almost 0.

The conclusion follows by the intermediate value Theorem. We start from the Rayleigh quotient defining λ i,α :

λ i,α = R R (φ i,α (x)-φ i,α (y)) 2 |x-y| 1+2α dy -(φ i,α (x)) 2 dx Ω i (φ i,α (x)) 2 dx = Ω i R (φ i,α (x)-φ i,α (y)) 2 |x-y| 1+2α dy -(φ i,α (x)) 2 dx Ω i (φ i,α (x)) 2 dx + R\Ω i Ω i (φ i,α (y)) 2 |x-y| 1+2α dydx Ω i (φ i,α (x)) 2 dx = Ω i Ω i (φ i,α (x)-φ i,α (y)) 2 |x-y| 1+2α dy -(φ i,α (x)) 2 dx Ω i (φ i,α (x)) 2 dx + 2 R\Ω i Ω i (φ i,α (y)) 2 |x-y| 1+2α dydx Ω i (φ i,α (x)) 2 dx .
Chapter 4 : Properties of steady states for a class of non-local Fisher-KPP equations in general domains

We continue in the same way by rewriting λ 1,2,µ,α :

λ α (Ω 1,2,µ ) = R R (φ 1,2,µ,α (x)-φ 1,2,µ,α (y)) 2 |x-y| 1+2α dy -(φ 1,2,µ,α (x)) 2 dx Ω 1,2,µ (φ 1,2,µ,α (x)) 2 dx = Ω 1,2,µ R (φ 1,2,µ,α (x)-φ 1,2,µ,α (y)) 2 |x-y| 1+2α dy -(φ 1,2,µ,α (x)) 2 dx Ω 1,2,µ (φ 1,2,µ,α (x)) 2 dx + R\Ω 1,2,µ R (φ 1,2,µ,α (x)-φ 1,2,µ,α (y)) 2 |x-y| 1+2α dydx Ω 1,2,µ (φ 1,2,µ,α (x)) 2 dx
.

Thus, we have found :

λ α (Ω 1,2,µ ) = 1 Ω 1,2,µ (φ 1,2,µ,α (x)) 2 dx (I 1 + I 2 ) . (4.69)
We rewrite I 1 and I 2 in order to involving the expression of λ 1,α and λ 2,α . We begin by rewriting I 1 :

I 1 = Ω 1,2,µ R (φ 1,2,µ,α (x) -φ 1,2,µ,α (y)) 2 |x -y| 1+2α dy -(φ 1,2,µ,α (x)) 2 dx = Ω 1 R (φ 1,2,µ,α (x) -φ 1,2,µ,α (y)) 2 |x -y| 1+2α dy -(φ 1,2,µ,α (x)) 2 dx + Ω 2 R (φ 1,2,µ,α (x) -φ 1,2,µ,α (y)) 2 |x -y| 1+2α dy -(φ 1,2,µ,α (x)) 2 dx = Ω 1 Ω 1 (φ 1,2,µ,α (x) -φ 1,2,µ,α (y)) 2 |x -y| 1+2α dy -(φ 1,2,µ,α (x)) 2 dx + Ω 1 R\Ω 1,2,µ (φ 1,2,µ,α (x)) 2 |x -y| 1+2α dydx + Ω 2 Ω 2 (φ 1,2,µ,α (x) -φ 1,2,µ,α (y)) 2 |x -y| 1+2α dy -(φ 1,2,µ,α (x)) 2 dx + Ω 2 R\Ω 1,2,µ (φ 1,2,µ,α (x)) 2 |x -y| 1+2α dydx + 2 Ω 1 Ω 2 (φ 1,2,µ,α (x) -φ 1,2,µ,α (y)) 2 |x -y| 1+2α dydx = Ω 1 Ω 1 (φ 1 1,2,µ,α (x) -φ 1 1,2,µ,α (y)) 2 |x -y| 1+2α dy -(φ 1 1,2,µ,α (x)) 2 dx + Ω 1 R\Ω 1 (φ 1 1,2,µ,α (x)) 2 |x -y| 1+2α dydx + Ω 2 Ω 2 (φ 2 1,2,µ,α (x) -φ 2 1,2,µ,α (y)) 2 |x -y| 1+2α dy -(φ 2 1,2,µ,α (x)) 2 dx + Ω 2 R\Ω 2 (φ 2 1,2,µ,α (x)) 2 |x -y| 1+2α dydx + 2 Ω 1 Ω 2 (φ 1,2,µ,α (x) -φ 1,2,µ,α (y)) 2 |x -y| 1+2α dydx - Ω 1 Ω 2 (φ 1 1,2,µ,α (x)) 2 |x -y| 1+2α dydx - Ω 2 Ω 1 (φ 2 1,2,µ,α (x)) 2 |x -y| 1+2α dydx.
Finally, we find that

I 1 = Ω 1 Ω 1 (φ 1 1,2,µ,α (x) -φ 1 1,2,µ,α (y)) 2 |x -y| 1+2α dy -(φ 1 1,2,µ,α (x)) 2 dx + Ω 1 R\Ω 1 (φ 1 1,2,µ,α (x)) 2 |x -y| 1+2α dydx + Ω 2 Ω 2 (φ 2 1,2,µ,α (x) -φ 2 1,2,µ,α (y)) 2 |x -y| 1+2α dy -(φ 2 1,2,µ,α (x)) 2 dx + Ω 2 R\Ω 2 (φ 2 1,2,µ,α (x)) 2 |x -y| 1+2α dydx + 2 Ω 1 Ω 2 (φ 1,2,µ,α (x) -φ 1,2,µ,α (y)) 2 |x -y| 1+2α dydx - Ω 2 Ω 1 (φ 2 1,2,µ,α (x)) 2 |x -y| 1+2α dydx - Ω 1 Ω 2 (φ 1 1,2,µ,α (x)) 2 |x -y| 1+2α dydx.
(4.70) With similar computations, we find that for I 2 :

I 2 = Ω 1 R\Ω 1 (φ 1 1,2,µ,α (y)) 2 |x -y| 1+2α dxdy + Ω 2 R\Ω 2 (φ 2 1,2,µ,α (y)) 2 |x -y| 1+2α dxdy - Ω 1 Ω 2 (φ 1 1,2,µ,α (y)) 2 |x -y| 1+2α dxdy - Ω 2 Ω 1 (φ 2 1,2,µ,α (y)) 2 |x -y| 1+2α dxdy.
λ α (Ω 1,2,µ ) = 2 Ω 1 Ω 2 (φ 1,2,µ,α (x)-φ 1,2,µ,α (y)) 2 |x-y| 1+2α dydx Ω 1,2,µ (φ 1,2,µ,α (x)) 2 dx + Ω 1 (φ 1 1,2,µ,α (x)) 2 dx   Ω 1 Ω 1 (φ 1 1,2,µ,α (x)-φ 1 1,2,µ,α (y)) 2 |x-y| 1+2α dy-(φ 1 1,2,µ,α (x)) 2 dx+2 Ω 1 R\Ω 1 (φ 1 1,2,µ,α (x)) 2 |x-y| 1+2α dydx Ω 1 (φ 1 1,2,µ,α (x)) 2 dx   Ω 1,2,µ (φ 1,2,µ,α (x)) 2 dx + Ω 2 (φ 2 1,2,µ,α (x)) 2 dx   Ω 2 Ω 2 (φ 2 1,2,µ,α (x)-φ 2 1,2,µ (y)) 2 |x-y| 1+2α dy-(φ 2 1,2,µ,α (x)) 2 dx+2 Ω 2 R\Ω 2 (φ 2 1,2,µ,α (x)) 2 |x-y| 1+2α dydx Ω 2 (φ 2 1,2,µ,α (x)) 2 dx   Ω (φ 1,2,µ,α (x)) 2 dx - 2 Ω 1 Ω 2 (φ 1 1,2,µ,α (x)) 2 |x-y| 1+2α dydx + 2 Ω 2 Ω 1 (φ 2 1,2,µ,α (x)) 2 |x-y| 1+2α dydx Ω 1,2,µ (φ 1,2,µ,α (x)) 2 dx ≥ Ω 1 (φ 1 1,2,µ,α (x)) 2 dx Ω 1,2,µ (φ 1,2,µ,α (x)) 2 dx λ 1,α + Ω 2 (φ 2 1,2,µ,α (x)) 2 dx Ω 1,2,µ (φ 1,2,µ,α (x)) 2 dx λ 2,α - 2 Ω 1 Ω 2 (φ 1 1,2,µ,α (x)) 2 |x-y| 1+2α dydx + 2 Ω 2 Ω 1 (φ 2 1,2,µ,α (x)) 2 |x-y| 1+2α dydx Ω 1,2,µ (φ 1,2,µ,α (x)) 2 dx ≥ min(λ 1,α , λ 2,α ) - 2 Ω 1 Ω 2 (φ 1 1,2,µ,α (x)) 2 |x-y| 1+2α dydx + 2 Ω 2 Ω 1 (φ 2 1,2,µ,α (x)) 2 |x-y| 1+2α dydx Ω 1,2,µ (φ 1,2,µ,α (x)) 2 dx .
But, for i ∈ {1, 2}, we have for all x ∈ Ω i and y ∈ Ω 3-i

2µ < |x -y| ⇒ - 1 (2µ) 1+2α ≤ - 1 |x -y| 1+2α .
We deduce that

λ α (Ω 1,2,µ ) ≥ min(λ 1,α , λ 2,α ) - 2 Ω 1 Ω 2 (φ 1 1,2,µ,α (x)) 2 |x-y| 1+2α dydx + 2 Ω 2 Ω 1 (φ 2 1,2,µ,α (x)) 2 |x-y| 1+2α dydx Ω 1,2,µ (φ 1,2,µ,α (x)) 2 dx
Chapter 4 : Properties of steady states for a class of non-local Fisher-KPP equations in general domains

≥ min(λ 1,α , λ 2,α ) - 2 Ω 1 Ω 2 (φ 1 1,2,µ,α (x)) 2 µ 1+2α dydx + 2 Ω 2 Ω 1 (φ 2 1,2,µ,α (x)) 2 µ 1+2α dydx Ω 1,2,µ (φ 1,2,µ,α (x)) 2 dx ≥ min(λ 1,α , λ 2,α ) - 4A (2µ) 1+2α -→ µ→+∞ min(λ 1,α , λ 2,α ).
Since min(λ 1,α , λ 2,α ) > 0, we deduce the existence of µ 1 > 0 such that for all µ > µ 1 ,

λ α (Ω 1,2,µ ) > 0.
Chapter 5

Adaptation to a heterogeneous environment with nonlocal dispersion

Ce chapitre est le résumé d'un travail en cours avec Sepideh Mirrahimi.

Introduction

In this work, we provide an asymptotic analysis of the equilibria of a non-local parabolic Lotka-Volterra type equation. Such equation arises in the study of adaptive evolution of phenotypically structured populations. The equation under study is the following

                   -∂ xx n ε -ε 2 ∂ θθ n ε + Ln ε = n ε (R(x, θ) -ρ ε (x)) in Ω×] -A, A[, ρ ε (x) = ]-A,A[ n ε (x, θ)dθ in ] -A, A[, Ln ε (x, θ) = Ω [n ε (x, θ) -n ε (y, θ)]K(x -y)dy in Ω×] -A, A[, ∂ νx n ε (x, θ) = 0 on ∂Ω×] -A, A[, ∂ ν θ n ε (x, ±A) = 0 (E)
with Ω a bounded subset of R. We have denoted by ∂ νx , ∂ ν θ the exterior derivatives with respect to the variables x and θ. Here, n ε (x, θ) stands for the density of a population at equilibrium at position x with a phenotypical trait θ. The term R(x, θ) corresponds to the intrinsic growth rate of individuals of phenotype θ at position x. The term ρ(x) corresponds to the total size of the population at position x. Via this term in the right hand side of (E), we take into account a mortality rate due to the competition between the individuals at the same position, independently of their traits. The trait of the parent is transmitted to the offspring. However the trait can be modified due to the mutations that we model by a Laplace term. We also consider that the species is subject to a local and a non-local dispersion in the space variable x. Indeed, in addition to a classical local dispersion term modeled by a Laplace term, we assume that the individuals can jump from position x to position y with a rate K(x -y). From a biological point of view, non-local dispersion may have an important role for instance for the tree species and because of the effect of the wind on the seeds or the pollens [START_REF] Kremer | Long-distance gene flow and adaptation of forest trees to rapid climate change[END_REF]. Finally, the Neumann boundary condition with respect to x models the fact that the species cannot leave the domain. The Neumann boundary condition with respect to θ means that the mutants cannot be born with a trait in ] -A, A[ c . Several questions motivate our analysis. Is it possible to determine extinction and survival criteria for such model ? Is it possible to characterize the population distribution at equilibrium ? What will be the impact of the non-local dispersion and a fragmented habitat on the phenotypical distribution of the population ? Several works have studied the properties of phenotypically structured populations subject to selection, migration and mutation. We will describe these related works in the following paragraphs. The originality of our work lies in the non-local spatial dispersion operator L. It allows to consider non-connected domains. The non-local dispersion may have antagonistic effects on the population dynamics. On the one hand, it may allow the population to reach new favorable geographic regions which are not accessible by a local diffusion. On the other hand, it may also impede local adaptation by bringing individuals with locally maladapted traits from other regions. While the role of the nonlocal dispersion is significant in many situations, as in the adaptation of forest trees to the climate change, very few theoretical works take it into account [START_REF] Kremer | Long-distance gene flow and adaptation of forest trees to rapid climate change[END_REF].

Integro-differential equations modeling the Darwinian evolution of phenotypically structured equations have been widely studied (see for instance [START_REF] Calsina | Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics[END_REF][START_REF] Desvillettes | On selection dynamics for continuous structured populations[END_REF][START_REF] Magal | Mutation, selection, and recombination in a model of phenotype evolution[END_REF]). In the context of space heterogeneous environments, a model closely related to (E) but with only a classical diffusion term for dispersion was derived from a stochastic individual based model by Champagnat and Méléard in [START_REF] Champagnat | Invasion and adaptive evolution for individual-based spatially structured populations[END_REF]. A first analysis of such deterministic model was performed by Arnold, Desvilettes and Prévost in [START_REF] Arnold | Existence of nontrivial steady states for populations structured with respect to space and a continuous trait[END_REF], where they investigated the existence of non-trivial steady states of the model. Propagation phenomena and existence of traveling front solutions for parabolic equations close to (E) were studied by Alfaro, Coville and Raoul in [START_REF] Alfaro | Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait[END_REF], by Bouin and Mirrahimi in [START_REF] Bouin | A Hamilton-Jacobi approach for a model of population structured by space and trait[END_REF] and by Berestycki, Jin and Silvestre in [START_REF] Berestycki | Propagation in a non local reaction diffusion equation with spatial and genetic trait structure[END_REF]. Alfaro, Berestycki and Raoul also studied a related model taking into account the effect of the climate change. A wide number of articles focus on a closely related equation known as the "cane-toads" model where the growth rate is independent of the trait, but the trait influences the ability of dispersal leading to a θ coefficient in front of the diffusion term in space (see for instance [START_REF] Turanova | On a model of a population with variable motility[END_REF][START_REF] Bouin | Travelling waves for the cane toads equation with bounded traits[END_REF][START_REF] Bouin | Super-linear spreading in local and non-local cane toads equations[END_REF][START_REF] Bouin | The Bramson logarithmic delay in the cane toads equations[END_REF]). This equation is motivated by the propagation of the cane toads in Australia by taking into account the role of a phenotypical trait : the size of the legs of the toads. More closely to our work, the steady states of a "cane-toads" type model, in the regime of small mutations, were studied by Perthame and Souganidis in [START_REF] Perthame | Rare mutations limit of a steady state dispersal evolution model[END_REF] and by Lam and Lou in [START_REF] Lam | An integro-PDE model for evolution of random dispersal[END_REF]. In another related project, a model where similarly to (E) the growth rate, and not the dispersion rate, depends on the phenotype, but considering a discrete spatial structure, was studied by Mirrahimi [88,[START_REF] Mirrahimi | A Hamilton-Jacobi approach to characterize the evolutionary equilibria in heterogeneous environments[END_REF]. In these works an asymptotic analysis of the steady states in the regime of small mutations was provided. In particular, it was shown that the presence of spatial heterogeneity can lead to polymorphic situations, that is the emergence of several dominant traits in the population.

In this work, we will use an approach based on Hamilton-Jacobi equations, which is adapted to study the small mutation regime (ε small). A closely related approach was first introduced in [START_REF] Freidlin | Limit theorems for large deviations and reaction-diffusion equations[END_REF][START_REF] Evans | A PDE approach to geometric optics for certain semilinear parabolic equations[END_REF], by Friedlin using probabilistic techniques and by Evans and Souganidis using deterministic tools, to study the propagation phenomena in reaction-diffusion equations. In the context of models from evolutionary biology and in the regime of small mutations, this method was suggested by Dieckmann, Jabin, Mishler and Perthame [START_REF] Diekmann | The dynamics of adaptation : An illuminating example and a Hamilton-Jacobi approach[END_REF]. In [START_REF] Barles | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF], Barles and Perthame provide the first rigorous results within this approach and obtain a concentration phenomena : as the mutations become small, the solution converges to a Dirac mass. In this case, the population at equilibrium is monomorphic (there is a single dominant trait in the population). We quote [START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations : a general convergence result[END_REF] which extends the main results of [START_REF] Diekmann | The dynamics of adaptation : An illuminating example and a Hamilton-Jacobi approach[END_REF]. This approach was then widely extended to study more general models with heterogeneity. In particular, in the context of the space heterogeneous environments, the works [START_REF] Bouin | A Hamilton-Jacobi approach for a model of population structured by space and trait[END_REF][START_REF] Turanova | On a model of a population with variable motility[END_REF][START_REF] Perthame | Rare mutations limit of a steady state dispersal evolution model[END_REF]88,[START_REF] Mirrahimi | A Hamilton-Jacobi approach to characterize the evolutionary equilibria in heterogeneous environments[END_REF] are within this framework.

In an ecological context, fragmented environments and non-local spatial dispersion phe-

Introduction

nomena were studied by Léculier, Mirrahimi and Roquejoffre [START_REF] Léculier | Propagation in a fractional reaction-diffusion equation in a periodically hostile environment[END_REF] and by Léculier and Roquejoffre [START_REF] Léculier | Properties of steady states for a class of non-local Fisher-KPP equations in general domains[END_REF]. Both mentioned works do not take into account any phenotypical structure. In [START_REF] Léculier | Propagation in a fractional reaction-diffusion equation in a periodically hostile environment[END_REF], the authors study invasion phenomena in a Fisher-KPP equation involving a fractional Laplacian arising in a fragmented periodic environment with Dirichlet exterior conditions. In [START_REF] Léculier | Properties of steady states for a class of non-local Fisher-KPP equations in general domains[END_REF], the authors study the existence and uniqueness of bounded positive steady-states in a Fisher-KPP equation involving a fractional Laplacian in general fragmented environment with Dirichlet exterior conditions. One of the perspectives of the present work is to study models with other operators of dispersion, as the fractional Laplacian (-∂ xx ) α , instead of -∂ xx + L, and considering Dirichlet exterior conditions. The results of [START_REF] Léculier | Propagation in a fractional reaction-diffusion equation in a periodically hostile environment[END_REF][START_REF] Léculier | Properties of steady states for a class of non-local Fisher-KPP equations in general domains[END_REF] would help us to attein this goal. Considering a non-local operator with a regular kernel as L makes a significant difference with the fractional Laplacian. However, we see this work as a first step before considering the problem with a fractional Laplacian and Dirichlet exterior conditions, in term of the technical difficulties.

The assumptions and the notations

The domain Ω ⊂ R is assumed to be bounded and composed of one or several connected components :

Ω = m i=1 ]a i , b i [ with a 1 < b 1 < a 2 < ... < a m < b m . (H1)
We assume that the growth rate verifies

R ∈ C 1 (Ω × [-A, A]) and R W 1,∞ (Ω×]-A,A[) < C R . ( H2
)
Example 5.1. A typical example of growth rate is written

R(x, θ) = r -g(bx -θ) 2 .
In this example, r is the maximal growth rate. The above quadratic term indicates that the optimal trait at position x is given by θ o = bx. The term b is the gradient of the environment : it indicates how fast the optimal trait varies as a function of a position in space. Moreover, g corresponds to the selection pressure. If g increases, the habitats becomes more hostile for unsuitable individuals. Remark, that depending on the choice of parameters r, b, g, the growth rate R can be negative.

We make the following assumptions on

K K ∈ C 1 (R), K > 0, K(x) = K(-x), 0 < c K < K < C K and |∂ x K| < C K . (H3)
We introduce here two eigenvalues problems associated to the equation (E) : let λ(θ, ρ) be the principal eigenvalue of the operator -∂ xx -L -[R(•, θ) -ρ]Id and µ ε be the principal eigenvalue of the operator -∂ xx -ε 2 ∂ θθ -L -R with Neumann boundary conditions :

i.e.    -∂ xx ψ θ + L(ψ θ ) -[R(•, θ) -ρ]ψ θ = λ(θ, ρ)ψ θ in Ω, ∂ νx ψ θ = 0 in ∂Ω (5.1) and -∂ xx ξ ε -ε 2 ∂ θθ ξ ε + Lξ ε -Rξ ε = µ ε ξ ε in Ω×] -A, A[, ∂ νx ξ ε = ∂ ν θ ξ ε = 0 on ∂(Ω×] -A, A[).
(5.2)

All along the article, we consider that the principal eigenfunctions (such as ψ or ξ ε ) are taken positive with a L 2 norm equals to 1. We make the following assumption : It follows obviously that

∃θ 0 ∈] -A, A[, such that min θ∈]-A,A[ λ(θ, 0) = λ(θ 0 , 0) < 0. ( H4
∃ε 0 > 0, ∀ε ∈]0, ε 0 [, µ ε < λ(θ 0 , 0) 2 < 0. (5.3)
We postpone the proof of Lemma 5.1 to the Appendix and we make the hypothesis that (5.3) holds true.

The results and the strategy

First, we prove the existence of a solution of (E) for all small value ε.

Theorem 5.1. Under the assumptions (H1)-(H4), for all ε ∈]0, ε 0 [, there exists a non-trivial positive bounded solution n ε of (E).

The proof of Theorem 5.1 follows the one of Theorem 2.1 by Lam and Lou in [START_REF] Lam | An integro-PDE model for evolution of random dispersal[END_REF] which treats the case of local diffusion. This proof relies on a topological degree argument. In section 5.2, we provide the additional arguments. which allows to adapt the proof of [START_REF] Lam | An integro-PDE model for evolution of random dispersal[END_REF] to the non-local operator L.

Next, we perform the Hopf-Cole transformation

n ε (x, θ) = e uε(x,θ) ε . (5.4)
This is the usual first step in the Hamilton-Jacobi approach (see [START_REF] Diekmann | The dynamics of adaptation : An illuminating example and a Hamilton-Jacobi approach[END_REF][START_REF] Barles | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF][START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations : a general convergence result[END_REF]). The main idea in this approach is to first study the limit of u ε as ε → 0, and next obtain from this limit, information on the limit of the phenotypic density n ε . The advantage of this transformation is that the limit of u ε is usually a continuous function which solves a Hamilton-Jacobi equation, while the limit of n ε is a measure. Performing such change of variable, we find that u ε is solution to where λ(θ, ρ) is the principal eigenvalue introduced in (5.1). Moreover, the limit u depends only on θ.

                         - 1 ε ∂ xx u ε - |∂ x u ε | 2 ε 2 -ε∂ θθ u ε -|∂ θ u ε | 2 + Ω 1 -e uε(y)-uε(x) ε K(x -y)dy = (R(x, θ) -ρ ε (x)) in Ω×] -A, A[, ρ ε (x) = ]-A,A[ n ε (x, θ)dθ in Ω, ∂ νx u ε (x, θ) = 0 on ∂Ω×] -A, A[, ∂ ν θ u ε (x, ±A) = 0. ( E
3. n ε converges to n in the sense of measures. Moreover, supp n ⊂ {u(θ) = 0}.

We present briefly heuristic arguments to understand how the second item of Theorem 5.2 can be recovered. First, we perform asymptotic developments of u ε and ρ ε with respect to the powers of ε i.e. u ε (x, θ) = u 0 (x, θ) + εu 1 (x, θ) + o(ε) and ρ ε (x) = ρ 0 (x) + o ε [START_REF] Alfaro | The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition[END_REF].

Next, we implement such asymptotic developments into (E HC ), it follows that

1 ε -∂ xx u 0 -2|∂ x u 0 ∂ x u 1 | - |∂ x u 0 | 2 ε + Ω [1 -e u 0 (y,θ)-u 0 (x,θ) ε +u 1 (y,θ)-u 1 (x,θ)+oε(1) ]K(x -y)dy -∂ xx u 1 -|∂ x u 1 | 2 -|∂ θ u 0 | 2 -[R -ρ 0 ] + o ε (1) = 0.
This equation suggests that u 0 (x, θ) = u 0 (θ) and passing to the limit ε → 0, we deduce that

-|∂ θ u 0 (θ)| 2 = [R(x, θ) -ρ 0 (x)] + ∂ xx u 1 (x, θ) -|∂ x u 1 (x, θ)| 2 - Ω [1 -e u 1 (
y,θ)-u 1 (x,θ) ]K(x -y)dy.

(5.6) Since the left hand side of (5.6) does not depend on x, it suggests that u 1 is such that u 1 = log(ψ θ ) with ψ θ the principal eigenfunction introduced in (5.1), that is

-|∂ θ u 0 (θ)| 2 = [R(x, θ) -ρ 0 (x)]ψ θ (x) + ∂ xx ψ θ (x) -L(ψ θ ) (ψ θ ) -1 .
(5.7)

We deduce that u 0 is a formal solution of (5.5).

From a technical point of view, the convergence of (u ε ) ε>0 is proved using the Arzela-Ascoli Theorem and a perturbed test function argument (see [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF]). To prove the uniform continuity, using the Bernstein's method, we prove that the first derivatives are bounded. These bounds rely on the establishment of Harnack type inequalities. Indeed, we prove the following result on the regularity of u ε Theorem 5.3. Under assumptions (H1)-(H4), the following results hold true.

[Harnack inequality]

There exists a constant C > 0 (independent of the choice of ε)

such that for all intervals I ⊂] -A, A[ with |I| = ε, there holds

sup (x,θ)∈Ω×I n ε (x, θ) ≤ C inf (x,θ)∈Ω×I n ε (x, θ).
(5.8) The associated eigenfunction ψ has a constant sign and is unique up to multiplication by a constant. Moreover, the function λ(θ, ρ) is continuous with respect to θ and the function ρ.

Existence of a principal eigenpair

In the following, we will consider that ψ is positive and of L 2 norm equal to 1.

Proof. First, we prove the existence of the principal eigenpair by verifying that we can apply the Krein Rutman Theorem (see [START_REF] Smoller | Shock waves and reaction-diffusion equations[END_REF] p 122). Since it is classical, we do not provide all the details. The cone of functions where we apply the Krein-Rutmann Theorem is

K = {u ∈ C 1+α (Ω) | u > 0 and ∂ νx u = 0}.
We define L(v) as the unique solution of

     -∂ xx L(v) + Ω [L(v)(x) -L(v)(y)]K(x -y)dy -(R(•, θ) -ρ -C)L(v) = v in Ω, ∂ νx L(v) = 0 on ∂Ω where C > sup x∈Ω (R(x, θ) -ρ(x)) and v ∈ K.
The operator L is linear, compact thanks to the elliptic estimates. We have to prove that ∀v ∈ K\ {0} , L(v) ∈ int(K).

Let v be in K with v not trivial. By elliptic regularity, it follows L(v) ∈ C 1+α and ∂ νx L(v) = 0. It remains to prove that L(v) > 0.

First we prove that if L(v) is constant then it is necessarily a positive constant. Next we prove that if L(v) varies then L(v) > 0.

Assume that L(v) = c. Let x ∈ Ω be such that v(x) > 0. Moreover, the choice of C gives

-(R(x, θ) -ρ(x) -C) > 0 and since -∂ xx L(v) = L(L(v)) = 0, we deduce that L(v)(x) = c = v(x) -(R(x, θ) -ρ(x) -C) > 0.
Next, we suppose that L(v) is not constant. Assume by contradiction that there exists x such that L(v)(x) ≤ 0. Let x ∈ Ω be such that

inf x∈Ω L(v)(x) = L(v)(x ).
Then either x ∈ Ω or x ∈ ∂Ω. In the first case, we deduce that

-∂ xx L(v)(x ) ≤ 0 and L(L(v))(x ) < 0,
which leads to the following contradiction

0 ≤ v(x ) = -∂ xx L(v)(x ) + L(L(v))(x ) -(R(x , θ) -ρ(x ) -C)L(v)(x ) < 0. If x ∈ ∂Ω, since L(v)
is not constant, we deduce from Lemma 5.2 that ∂ νx L(v)(x ) < 0. It is in contradiction with the Neumann boundary condition. We conclude that we can apply the Krein Rutman theorem and the conclusion follows.

Preliminary results

Next, we prove the continuity of λ(θ, ρ) with respect to θ. Let ρ be a smooth bounded function. By the uniform continuity of R, for any ε > 0, there exists δ > 0 such that for all

θ 1 , θ 2 ∈] -A, A[ with |θ 1 -θ 2 | < δ there holds sup x∈Ω |R(x, θ 1 ) -R(x, θ 2 )| < ε.
We prove that for a such couple of θ 1 , θ 2 , we have |λ(θ 1 , ρ) -λ(θ 2 , ρ)| < ε. We denote ψ θ i the principal eigenfunction associated to the eigenvalue θ i . Thanks to the Rayleigh quotient, we know that

λ(θ i , ρ) = Ω |∂ x ψ θ i (x)| 2 -[R(x, θ i ) -ρ(x)]ψ θ i (x) 2 dx + Ω×Ω [ψ θ i (x) -ψ θ i (y)] 2 2 K(x -y)dydx.
Moreover, if we define

R(θ, g) = Ω |∂ x g(x)| 2 dx + Ω×Ω [g(x) -g(y)] 2 2 K(x -y)dydx - Ω (R(x, θ) -ρ(x))g(x) 2 dx,
we have that λ(θ i , ρ) = min

g∈H 1 (Ω) R(θ i , g) = R(θ i , ψ θ i ).
We deduce that

λ(θ 2 , ρ) ≤ R(θ 2 , ψ θ 1 ) = λ(θ 1 , ρ) + Ω [R(x, θ 1 ) -R(x, θ 2 )]ψ θ 1 (x) 2 dx ≤ λ(θ 1 , ρ) + ε.
With similar computations, we obtain that λ(θ 1 , ρ) ≤ λ(θ 2 , ρ) + ε. We conclude that

|λ(θ 1 , ρ) -λ(θ 2 , ρ)| ≤ ε.
With similar computations, we conclude to the continuity with respect to ρ.

The existence of the solution of (5.2) is also due to the Krein-Rutman Theorem, therefore we do not provide the proof of existence.

Existence of a non-trivial solution of (E)

As mentioned in the introduction, we recall that the proof is an adaptation of the proof of Theorem 2.1 by Lam and Lou in [START_REF] Lam | An integro-PDE model for evolution of random dispersal[END_REF]. The major difference is the presence of the integral operator L. Therefore, we only provide the main elements dealing with the integral operator L.

Proof of Theorem 5.1. We fix ε ∈]0, ε 0 [ (where ε 0 is given by (5.3)). Let τ ∈ [0, 1] and n τ be a solution of

           -∂ xx n τ -ε 2 ∂ θθ n τ + Ln τ = n τ (R -τ ρ τ -(1 -τ )n τ ) in Ω×] -A, A[, ρ τ (x) = ]-A,A[ n τ (x, θ)dθ in Ω, ∂ νx n τ (x, θ) = 0 on ∂Ω×] -A, A[, ∂ ν θ n τ (x, ±A) = 0 in Ω. (E τ )
It is well known that for τ = 0, according to (5.3), there exists a non-trivial steady solution n 0 . As in [START_REF] Lam | An integro-PDE model for evolution of random dispersal[END_REF], we prove that there exists a constant C ε > 1 (which may depend on ε) such that we have for any τ ∈ [0, 1]

C -1 ε ≤ A -A Ω n τ dxdθ ≤ C ε .
Then one can conclude using a topological degree arguments.

The lower bound. Let v τ be such that n τ = ξ ε v τ (where ξ ε is provided by (5.2)). First, we remark that

L(v τ ξ ε ) = v τ L(ξ ε ) + ξ ε L(v τ ) + Λ(v τ , ξ ε ) and Λ(v τ , ξ ε )(x) = Ω [(v τ (x) -v τ (y))(ξ ε (y) -ξ ε (x))]K(x -y)dy. Then v τ is solution of -ξ ε ∂ xx v τ -2∂ x ξ ε ∂ x v τ -ε 2 ξ ε ∂ θθ v τ -2ε 2 ∂ θ ξ ε ∂ θ v τ + ξ ε L(v τ )+Λ(v τ , ξ ε ) + µ ε ξ ε v τ = -v τ ξ ε [τ ρ τ + (1 -τ )n τ ].
If we multiply it by ξε vτ , we obtain

-∂ x (ξ 2 ε ∂ x v τ ) -ε 2 ∂ θ (ξ 2 ε ∂ θ v τ ) + ξ 2 ε L(v τ ) + ξ ε Λ(v τ , ξ ε ) v τ = ξ 2 ε (-µ ε -τ ρ τ -(1 -τ )n τ ).
Next, we integrate over all the domain

A -A Ω -∂ x (ξ 2 ε ∂ x v τ ) -ε 2 ∂ θ (ξ 2 ε ∂ θ v τ ) + ξ 2 ε L(v τ ) + ξ ε Λ(v τ , ξ ε ) v τ dxdθ = A -A Ω -∂ x (ξ 2 ε ∂ x v τ ) -ε 2 ∂ θ (ξ 2 ε ∂ θ v τ ) v τ dxdθ + A -A Ω ξ 2 ε L(v τ ) + ξ ε Λ(v τ , ξ ε ) v τ dxdθ = I 1 + I 2 .
We next prove that I 1 and I 2 are negative. For I 1 , by an integration by part, we have

I 1 = A -A Ω -∂ x (ξ 2 ε ∂ x v τ ) -ε 2 ∂ θ (ξ 2 ε ∂ θ v τ ) v τ dxdθ = - A -A Ω ξ 2 ε v 2 τ |∂ x v τ | 2 + ε 2 |∂ θ v τ | 2 dxdθ ≤ 0.
For I 2 , using that K is even and the Fubini Theorem, we obtain

I 2 = A -A Ω ξ 2 ε (x)L(v τ )(x) + ξ ε (x)Λ(v τ , ξ ε )(x) v τ (x) dxdθ = A -A Ω ξ ε (x) v τ (x) Ω [(v τ (x) -v τ (y))ξ ε (y)]K(x -y)dydxdθ = - A -A Ω Ω ξ ε (x)ξ ε (y) v τ (x)v τ (y) (v τ (y) -v τ (x)) 2 K(x -y)dydxdθ -I 2
We deduce that

I 2 = - 1 2 A -A Ω Ω ξ ε (x)ξ ε (y) v τ (x)v τ (y) (v τ (y) -v τ (x)) 2 K(x -y)dydxdθ ≤ 0.
Therefore, we have that

A -A Ω ξ 2 ε [-µ ε -τ ρ τ -(1 -τ )n τ ]dxdθ ≤ 0.
Thanks to (5.3), we conclude that for ε small enough

|λ(θ 0 , 0)| 2 ≤ -µ ε = -µ ε A -A Ω ξ 2 ε dxdθ ≤ sup(ξ 2 ε )[τ + (1 -τ )] A -A Ω n τ dxdθ.

Regularity results

The upper bound. First, we remark that thanks to the Neumann boundary conditions and the parity of K, we have that

A -A Ω -∂ xx n τ -∂ θθ n τ + L(n τ )dxdθ = 0.
Therefore, if we integrate (E τ ) with respect to x and θ, we obtain

(1 -τ ) 2A|Ω| + τ |Ω| n τ 2 L 1 = τ |Ω| Ω ρ τ dx 2 + (1 -τ ) 2A|Ω| A -A Ω n τ dxdθ 2 ≤ τ Ω ρ 2 τ dx + (1 -τ ) A -A Ω n 2 τ dxdθ = A -A Ω Rn τ dxdθ ≤ C R A -A Ω n τ dxdθ = C R n τ L 1 .

Conclusion.

It follows that there exists a bounded non trivial solution n ε of (E). However, we have indeed proved that there exist constants c, C > 0 such that

c sup ξ 2 ε ≤ A -A Ω n ε dxdθ ≤ C.

Regularity results

In this section we prove Theorem 5.3. The sub-sections correspond respectively to the proof of the item 1.

0 ≤ ρ ε ≤ C R ,
(where C R is introduced in (H2)). Moreover, there exists C > 0 such that for all ε small enough ρ ε W 2,p (Ω) ≤ C.

Proof. The L ∞ -bounds. It is obvious that ρ ε ≥ 0. If we integrate (E) with respect to θ, we obtain

     -∂ xx ρ ε + Lρ ε = A -A R(•, θ)n ε (•, θ)dθ -ρ 2 ε in Ω, ∂ νx ρ ε = 0 in ∂Ω. (E ρ )
Recalling the L ∞ bounds on R (H2), it follows :

-∂ xx ρ ε + Lρ ε ≤ C R ρ ε -ρ 2 ε . We conclude thanks to the maximum principle that ρ ε ≤ C R .
The W 2,p (Ω) bounds. Thanks to the L ∞ bounds on R, K, ρ (assumptions (H2), (H3) and the previous inequality), we may write (E ρ ) on the following form (5.16)

-∂ xx ρ ε = f ε with f ε ∈ L ∞ (Ω)

A Harnack inequality

The first step to prove the first item of Theorem 5.3 is to prove the result in the interior of Ω×] -A, A[. Theorem 5.4. For all (x 0 , θ 0 ) ∈ Ω×] -A, A[, and R 0 > 0 such that

B 3R 0 (x 0 ) × B 3εR 0 (θ 0 ) ⊂ Ω×] -A, A[ there exists C(R 0 ) > 0 such that sup (x,θ)∈B R 0 (x 0 )×B εR 0 (θ 0 ) n ε (x, θ) ≤ C(R 0 ) inf (x,θ)∈B R 0 (x 0 )×B εR 0 (θ 0 ) n ε (x, θ).
(5.17)

Next, we extend the above result thanks to a reflective argument(see Remark 9 p.275 in [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]). We perform the following change of variable : n(x, θ) = n ε (x, εθ). Therefore, we consider the following scaled equation

-∂ xx n -∂ θθ n + L n = n[ R -ρ] in Ω×] -ε -1 A, εA[, ∂ νx n = ∂ ν θ n = 0 in ∂(Ω×] -ε -1 A, εA[. (E')
We have denoted by R the function R(x, θ) = R(x, εθ). Remark that R still verifies (H2).

Proof of Theorem 5.4.

Let (x 0 , θ 0 ) ∈ Ω×] -ε -1 A, ε -1 A[ and a radius R 0 > 0 be such that B 3R 0 (x 0 , θ 0 ) ⊂ Ω×] -ε -1 A, ε -1 A[. If we denote by f (x, θ) = Ω n(y, θ)K(y -x)dy, according to (H3) it follows that f ∈ L ∞ (B 2R 0 (x 0 , θ 0 )).
From the classical Harnack inequality, the Theorem 9.20 and 9.22 pp. 244-246 in [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF], and using (H2) we deduce the existence of C 1 > 0 (depending on R 0 ) such that sup

(x,θ)∈B R 0 (x 0 ,θ 0 ) n(x, θ) ≤ C 1 inf (x,θ)∈B R 0 (x 0 ,θ 0 ) n(x, θ) + C 1 sup (x,θ)∈B 2R 0 (x 0 ,θ 0 ) |f (x, θ)| ≤ C 1 inf (x,θ)∈B R 0 (x 0 ,θ 0 ) n(x, θ) + C 1 C K sup θ∈B 2R 0 (θ 0 ) Ω n(x, θ)dx.
(5.18)

The main element of the proof is to prove the following claim :

∃C > 0 such that sup

θ∈B 2R 0 (θ 0 ) Ω n(x, θ)dx ≤ C inf (x,θ)∈B R 0 (x 0 ,θ 0 ) n(x, θ). (5.19)
It is clear that if (5.19) holds true, the conclusion follows.

First, we integrate (E') with respect to x. It follows thanks to the Neumann boundary conditions that for all θ ∈ B 3R 0 (θ 0 ) we have

-∂ θθ Ω n(x, θ)dx = Ω n(x, θ) R(x, θ) -ρ(x) -Ω K(x -y)dy dx Ω n(x, θ)dx Ω n(x, θ)dx + Ω Ω n(y, θ)K(x -y)dydx Ω n(x, θ)dx Ω n(x, θ)dx.
Thanks to the L ∞ -bounds on K, R, ρ (assumptions (H2), (H3) and Lemma 5.3) and the Fubini theorem, we have

-C ≤ Ω n(x, θ) R(x, θ) -ρ(x) -Ω K(x -y)dy dx Ω n(x, θ)dx < C 5.3. Regularity results and Ω Ω n(y, θ)K(x -y)dydx Ω n(x, θ)dx ≤ C K |Ω|. It follows -C Ω n(x, θ)dx ≤ -∂ θθ Ω n(x, θ)dx ≤ C Ω n(x, θ)dx.
Hence, we apply the Harnack inequality to θ ∈ B 3R 0 (θ 0 ) → Ω n(x, θ)dx into the ball B 2R 0 (θ 0 ) and we deduce the existence of a constant C 2 > 0 such that sup

θ∈B 2R 0 (θ 0 ) Ω n(x, θ)dx ≤ C 2 inf θ∈B 2R 0 (θ 0 ) Ω n(x, θ)dx.
(5.20)

Next, thanks to the L ∞ -bounds on K, R, ρ (assumptions (H2), (H3) and Lemma 5.3), it follows that in Ω × B 2R 0 (θ 0 )

c K inf θ∈B 2R 0 (θ 0 ) Ω n(y, θ)dy ≤ c K Ω n(y, θ)dy ≤ Ω n(y, θ)K(x -y)dy ≤ (-∂ xx -∂ θθ ) n + C n.
From an inequality developed by Krylov (we refer to Theorem 7.1 p33 in [START_REF] Busca | Harnack type estimates for nonlinear elliptic systems and applications[END_REF] and the reference therein), we deduce the existence of a constant C 3 > 0 such that inf

B 2R 0 (θ 0 ) Ω n(x, θ)dx ≤ C 3 inf B R 0 (x 0 ,θ 0 ) n(x, θ). ( 5.21)
Combining the previous inequality with (5.20) and (5.21) yields to sup

θ∈B 2R 0 (θ 0 ) Ω n(x, θ)dx ≤ C 2 inf θ∈B 2R 0 (θ 0 ) Ω n(x, θ)dx ≤ C 2 C 3 inf (x,θ)∈B R 0 (x 0 ,θ 0 ) n(x, θ).
This concludes the proof.

Lipschitz estimates

We prove 2. of Theorem 5.3 by the Bernstein method.

Proof of 2. of Theorem 5.3. We recall the main equation satisfied by u ε :

-∂ xx u ε ε - |∂ x u ε | 2 ε 2 -ε∂ θθ u ε -|∂ θ u ε | 2 + Ω [1 -e uε(y)-uε(x) ε ]K(x -y)dy = R(x, θ) -ρ ε (5.22)
with Neumann boundary conditions. The first step is to differentiate (5.22) with respect to x and multiply it by ∂xuε ε 2 :

- ∂ xxx u ε ∂ x u ε ε 3 - ∂ x |∂xuε| 2 ε 2 ∂ x u ε ε 2 + Ω e uε(y,θ)-uε(x,θ) ε K(x -y)dy ∂ x u 2 ε ε 3 - ∂ x |∂ θ u ε | 2 ∂ x u ε ε 2 - ∂ xθθ u ε ∂ x u ε ε = Ω [e uε(y,θ)-uε(x,θ) ε -1]∂ x K(x -y)dy + ∂ x R -∂ x ρ ε ∂ x u ε ε 2 .
Remarking that

∂ xxx u ε ∂ x u ε = ∂ xx (|∂ x u ε | 2 ) 2 -(∂ xx u ε ) 2 and ∂ xθθ u ε ∂ x u ε = ∂ θθ (|∂ x u ε | 2 ) 2 -(∂ θx u ε ) 2 123
Chapter 5 : Adaptation to a heterogeneous environment with nonlocal dispersion yields to

- ∂ xx ( |∂xuε| 2 ε 2 ) 2ε + (∂ xx u ε ) 2 ε 3 - ∂ x |∂xuε| 2 ε 2 ∂ x u ε ε 2 + Ω e uε(y,θ)-uε(x,θ) ε K(x -y)dy (∂ x u ε ) 2 ε 3 - ∂ x |∂ θ u ε | 2 ∂ x u ε ε 2 + (∂ θx u ε ) 2 ε - ε∂ θθ ( |∂xuε| 2 ε 2 ) 2 = Ω [e uε(y,θ)-uε(x,θ) ε -1]∂ x K(x -y)dy + ∂ x R -∂ x ρ ε ∂ x u ε ε 2 .
(5.23) In the second step, we differentiate (5.22) with respect to θ and multiply by ∂ θ u ε . With computations similar to the ones presented above, we find

- ∂ xx (|∂ θ u ε | 2 ) 2ε + (∂ θx u ε ) 2 ε -∂ θ |∂ x u ε | 2 ε 2 ∂ θ u ε - ε 2 ∂ θθ (|∂ θ u ε | 2 ) + ε(∂ θθ u ε ) 2 -∂ θ |∂ θ u ε | 2 ∂ θ u ε + Ω (∂ θ u ε (x, θ) 2 -∂ θ u ε (x, θ)∂ θ u ε (y, θ)) ε e uε(y,θ)-uε(x,θ) ε K(x -y)dy = ∂ θ R∂ θ u ε .
(5.24) Next, we introduce

p ε (x, θ) = |∂ x u ε (x, θ)| 2 ε 2 + |∂ θ u ε (x, θ)| 2 .
(5.25)

If we combine (5.23) and (5.24) together and we rewrite it involving p ε , it follows

- ∂ xx p ε 2ε - ε∂ θθ p ε 2 + 1 ε Ω [p ε (x, θ) -∂ θ u ε (x, θ)∂ θ u ε (y, θ)]e uε(y,θ)-uε(x,θ) ε K(x -y)dy - ∂ x p ε ∂ x u ε ε 2 -∂ θ p ε ∂ θ u ε + 2(∂ xθ u ε ) 2 ε + (∂ xx u ε ) 2 ε 3 + ε(∂ θθ u ε ) 2 = Ω [e uε(y,θ)-uε(x,θ) ε -1]∂ x K(x -y)dy + ∂ x R -∂ x ρ ε ∂ x u ε ε 2 + ∂ θ R∂ θ u ε .
(5.26)

Let (x ε , θ ε ) be such that sup

(x,θ)∈Ω×]-A,A[ p ε (x, θ) = p ε (x ε , θ ε ).
Thanks to the Neumann boundaries conditions, we deduce that (x ε , θ ε ) / ∈ ∂Ω × ∂ (] -A, A[). Therefore, we distinguish three cases : either

(x ε , θ ε ) ∈ Ω×] -A, A[ or (x ε , θ ε ) ∈ ∂Ω×] -A, A[ or (x ε , θ ε ) ∈ Ω × {±A}. Case 1 : (x ε , θ ε ) ∈ Ω×] -A, A[. First,
we bound the right-hand-side of (5.26). Indeed, thanks to the Harnack inequality (first item of Theorem 5.3) and the L ∞ -bounds on the derivative of K, R and ρ ε (assumptions (H2), (H3) and Corollary 5.1), it follows that

Ω [e uε(y,θ)-uε(x,θ) ε -1]∂ x K(x -y)dy + ∂ x R -∂ x ρ ε ∂ x u ε ε 2 + ∂ θ R∂ θ u ε ≤ C √ p ε ε . (5.27)
Next, we evaluate (5.26) at (x ε , θ ε ). We claim that

-∂ xx p ε (x ε , θ ε ) ≥ 0, -∂ θθ p ε (x ε , θ ε ) ≥ 0, ∂ x p ε (x ε , θ ε ) = ∂ θ p ε (x ε , θ ε ) = 0 and 1 ε Ω [p ε (x ε , θ ε ) -∂ θ u ε (x ε , θ ε )∂ θ u ε (y, θ ε )]e uε(y,θε)-uε(xε,θε) ε K(x ε -y)dy ≥ 0. (5.28)
Indeed, the first inequalities follow easily since p(x ε , θ ε ) = max p ε and the last inequality holds true thanks to the following computations

1 ε Ω [p ε (x ε , θ ε ) -∂ θ u ε (x ε , θ ε )∂ θ u ε (y, θ ε )]e uε(y,θε)-uε(xε,θε) ε K(x ε -y)dy 5.3. Regularity results ≥ 1 ε Ω p ε (x ε , θ ε )e uε(y,θε)-uε(xε,θε) ε K(x ε -y)dy - 1 2 Ω ∂ θ u 2 ε (x ε , θ ε )e uε(y,θε)-uε(xε,θε) ε K(x ε -y)dy - 1 2 Ω ∂ θ u 2 ε (y, θ ε )e uε(y,θε)-uε(xε,θε) ε K(x ε -y)dy ≥ 1 2ε Ω p ε (x ε , θ ε )e uε(y,θε)-uε(xε,θε) ε K(x ε -y)dy - Ω p ε (y, θ ε )e uε(y,θε)-uε(xε,θε) ε K(x ε -y)dy ≥ 0.
We deduce thanks to (5.27) and (5.28) that 1 2ε

∂ xx u ε (x ε , θ ε ) ε + ε∂ θθ u ε (x ε , θ ε ) 2 ≤ 1 ε   ∂ xx u ε (x ε , θ ε ) ε 2 + (ε∂ θθ u ε (x ε , θ ε )) 2   ≤ C p ε (x ε , θ ε ) ε .
Hence, using the original equation (5.22), we deduce that

-p ε (x ε , θ ε ) + Ω (1 -e uε(y,θε)-uε(xε,θε) ε )K(x ε -y)dy -R(x ε , θ ε ) + ρ ε (x ε ) 2 ≤ C p ε (x ε , θ ε ).
(5.29) Thanks to the L ∞ -bounds on K, R and ρ ε (assumption (H2) (H3) and Lemma 5.3), it follows that p ε (x ε , θ ε ) is uniformly bounded with respect to ε. The conclusion follows. Case 2 :

(x ε , θ ε ) ∈ ∂Ω×] -A, A[. First remark that in this case, p ε (x ε , θ ε ) = |∂ θ u ε (x ε , θ ε )| 2 .
We claim that p ε verifies also the Neumann boundary conditions at (x ε , θ ε ). Indeed, according to the Neumann boundary conditions satisfied by u ε , we can use a reflective argument and differentiate p ε on the boundary. We obtain

∂ x p ε (x ε , θ ε ) = 2∂ x u ε (x ε , θ ε )∂ xx u ε (x ε , θ ε ) ε 2 + 2∂ θ u ε (x ε , θ ε )∂ xθ u ε (x ε , θ ε ) = 0 because ∂ x u ε (x ε , θ ε ) = 0 and ∂ xθ u ε (x ε , θ ε ) = 0. Since p(x ε , θ ε ) = max p ε , we deduce that -∂ xx p ε (x ε , θ ε ) ≥ 0.
We conclude that (5.28) and (5.27) hold also true in this case and the conclusion follows from the same computations as in the previous case. Case 3 : (x ε , θ ε ) ∈ Ω × {±A}. This case is treated in the same manner as the previous case.

The bounds on ρ ε

We recall the equation (E ρ ) satisfied by ρ ε : Next, if we multiply (E) by ξ ε k (introduced in (5.2)) and we integrate by part, we obtain

     -∂ xx ρ ε + Lρ ε = A -A R(x, θ)n ε (x, θ)dθ -ρ 2 ε in Ω, ∂ νx ρ ε = 0 on ∂Ω. (E ρ )
µ ε A -A Ω n ε k ξ ε k dxdθ = - A -A Ω ρ ε k n ε k ξ ε k dxdθ.
We deduce thanks to (5.3) that for k large enough, it holds

|λ(θ 0 , 0)| 2 ≤ -µ ε k ≤ sup ρ ε k A -A Ω n ε k ξ ε k dxdθ A -A Ω n ε k ξ ε k dxdθ .
It is in contradiction with the hypothesis sup ρ ε k -→ k→+∞ 0. Therefore, there exists a constant

c > 0 such that ∀ε ∈]0, ε 0 [, c ≤ sup ρ ε .
(5.30)

The bounds on ρ ε in the whole domain Ω. The upper bound is already proved in Lemma 5.3. Therefore, we concentrate on the lower bound. Let ε < ε 0 and x 0 ∈ Ω be such that ρ ε (x 0 ) = sup ρ ε .

We conclude thanks to (5.30) and the Lipschitz estimates obtained in the second item of Theorem 5.3 that for all x ∈ Ω

ρ ε (x) = A -A e uε(x,θ) ε dθ = A -A e uε(x,θ)-uε(x 0 ,θ)+uε(x 0 ,θ) ε dθ ≥ ρ ε (x 0 )e -C ≥ ce -C .

The bounds on u ε

Proof of 3. of Theorem 5.3. First, we prove that there exists a > 0 such that -a < u ε . Thanks to the third item of Theorem 5.3, we know that there exists c > 0 such that for all ε small enough we have

c < A -A n ε (x, θ)dθ.
We deduce the existence of (x 0 , θ 0

) ∈ Ω×] -A, A[ such that c 2A ≤ n ε (x 0 , θ 0 ).
Hence, it follows

ε log c 2A ≤ u ε (x 0 , θ 0 ).
We conclude thanks to the Lipschitz estimates established in the second item of Theorem 5.3 that We prove it by contradiction. Assume that there exists a > 0 and sequences ε

∀(x, θ) ∈ Ω×] -A, A[, -a ≤ -2CA + ε[log c 2A -C|Ω|] ≤ u ε (x, θ). ( 5
k , (x k , θ k ) such that ε k -→ k→+∞ 0 and a < u ε k (x k , θ k ).
Using the Lipschitz estimates provided by the second item of Theorem 5.3, it follows for all

θ ∈ B a 4C (θ k )∩] -A, A[ u ε k (x k , θ) = u ε k (x k , θ) -u ε k (x k , θ k ) + u ε k (x k , θ k ) ≥ -C|θ -θ k | + a ≥ a 2
where C corresponds to the Lipschitz estimate given by (5.9). We deduce that

ρ ε k (x k ) ≥ min(2A, a 4C )e a 2ε k .
We conclude that lim inf k→+∞ ρ ε k (x k ) = +∞. This is in contradiction with the L ∞ bounds on ρ ε established in the third item of Theorem 5.3.

Convergence to the Hamilton Jacobi equation

Proof of Theorem 5.2. We prove here the 3 items of Theorem 5.2.

Proof of 1. Thanks to the third item of Theorem 5.3, it follows that for ε small enough 0 < c ≤ ρ ε ≤ C and ρ ε W 2,p (Ω) ≤ C. We deduce from the classical Sobolev injection (see [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]) that ρ ε converges, along subsequences, strongly in W 1,p (Ω) and in particular uniformly to ρ and ρ verifies 0 < c ≤ ρ ≤ C.

Proof of 2.

From the Lipschitz estimates and the bounds established in the second and the fourth items of Theorem 5.3, we deduce thanks to the Arzela-Ascoli Theorem that up to a subsequence, (u ε ) ε>0 converges locally uniformly to some continuous function u. Moreover, the limit function u does not depend on x. We prove that u is a viscosity solution of

-|∂ θ u| 2 = -λ(θ, ρ)
with λ(θ, ρ) the principal eigenvalue of (5.1). We recall that for a fixed value θ, Proposition 5.1 provides the existence of a sequence of principal eigenvalues λ(θ, ρ ε ) associated with a sequence of positive eigenfunctions (ψ θ ε ) ε>0 of the operator

-∂ xx + L -(R(x, θ) -ρ ε ) with Neumann boundary conditions : i.e.    -∂ xx ψ θ ε + L(ψ θ ε ) -(R(x, θ) -ρ ε )ψ θ ε = λ(θ, ρ ε )ψ θ ε in Ω, ∂ νx ψ θ ε = 0 on ∂Ω. (5.32) Since ψ θ ε > 0, we introduce Ψ θ ε = ln(ψ θ ε ).
Chapter 5 : Adaptation to a heterogeneous environment with nonlocal dispersion Therefore, for any test function φ, let θ ∈] -A, A[ be such that u -φ has a strict maximum at θ. Then, there exists (

x ε , θ ε ) ∈ Ω×] -A, A[ such that θ ε -→ ε→0 θ and max (x,θ)∈Ω×]-A,A[ u ε (x, θ) -φ(θ) -εΨ θε ε (x) = u ε (x ε , θ ε ) -φ(θ ε ) -εΨ θε ε (x ε ).
We distinguish two cases : either x ε ∈ Ω or x ε ∈ ∂Ω. Case 1 : x ε ∈ Ω. Since u ε is a classical solution of (E HC ), we deduce that it is also a viscosity solution, therefore

- ∂ xx (φ(θ ε ) + εΨ θε ε (x ε )) ε - [∂ x (φ(θ ε ) + εΨ θε ε (x ε ))] 2 ε 2 + Ω [1 -e Ψ θε ε (y)-Ψ θε ε (xε) ]K(x ε -y)dy -ε∂ θθ (φ(θ ε ) + εΨ θε ε (x ε )) -[∂ θ (φ(θ ε ) + εΨ θε ε (x ε ))] 2 -R(x ε , θ ε ) + ρ ε (x ε ) ≤ 0.
Remarking that φ does not depend on x and the θ value is fixed in Ψ θε ε , we deduce that

-∂ xx Ψ θε ε (x ε ) -[∂ x Ψ θε ε (x ε )] 2 + Ω [1 -e Ψ θε ε (y)-Ψ θε ε (xε) ]K(x ε -y)dy -R(x ε , θ ε ) + ρ ε (x ε ) -ε∂ θθ φ(θ ε ) -[∂ θ φ(θ ε )] 2 ≤ 0.
(5.33) Next, we observe that (5.32) implies

-∂ xx Ψ θε ε (x ε )-[∂ x Ψ θε ε (x ε )] 2 + Ω [1-e Ψ θε ε (y)-Ψ θε ε (xε) ]K(x ε -y)dy -R(x ε , θ ε )+ρ ε (x ε ) = λ(θ ε , ρ ε ).
Therefore, passing to the limit ε → 0, thanks to the continuity of λ(θ, ρ) with respect to θ and ρ (Proposition 5.1), it follows that

-[∂ θ φ(θ)] 2 ≤ -λ(θ, ρ).
Case 2 : x ε ∈ ∂Ω. First, we remark that in this case,

-∂ x u ε (x ε , θ ε ) = -∂ x Ψ θε ε (x ε ) = 0.
Therefore, we deduce that

-∂ x [u ε (x ε , θ ε ) -φ(θ ε ) -εΨ θε ε (x ε )] = 0. Moreover, since (u ε -φ -εΨ θε ε )(x ε , θ ε ) = max(u ε -φ -εΨ θε ε ), we have firstly by a reflective argument that -∂ xx (u ε -φ -εΨ θε ε )(x ε , θ ε ) ≥ 0, (5.34)
and secondly, we have

u ε (y, θ ε ) -u ε (x ε , θ ε ) ≤ ε[Ψ θε ε (y) -Ψ θε ε (x ε )]. (5.35)
The inequalities (5.34) and (5.35) lead to

-∂ xx εΨ θε ε (x ε ) ≤ -∂ xx u ε (x ε , θ ε ) and Ω [1 -e Ψ θε ε (y)-Ψ θε ε (xε) ]K(x ε -y)dy ≤ Ω [1 -e uε(y,θε)-uε(xε,θε) ε ]K(x ε -y)dy.
Therefore, the conclusion follows from similar computation as above.

Discussions and numerics

Finally, u is a sub-solution of (5.5) in a viscosity sense. With similar arguments, u is also a super-solution. We conclude that u is a viscosity solution of (5.5).

Next, we prove that sup θ∈[-A,A]

u(θ) = 0.

If it does not holds true, it follows that sup θ∈[-A,A]

u(θ) < -a < 0. Hence for ε small enough, we deduce that max

(x,θ)∈Ω×[-A,A] u ε (x, θ) < -a 2
, which implies that ρ ε < c for ε sufficiently small. This is in contradiction with the third item of Theorem 5.3. We conclude that max

θ∈[-A,A] u(θ) = 0.
Proof of 3. Thanks to the L ∞ bounds on ρ ε (third point of Theorem 5.3), we deduce that

c ≤ n ε L 1 (Ω×]-A,A[) ≤ C.
It follows that n ε converges in the sense of the measure to n and the measure n is non-negative and not trivial. Finally, we prove that .36) We prove that Ω A -A φ(x, θ)n(x, θ)dxdθ = 0. To this end, we introduce -a = sup supp φ u. According to (5.36), it follows that a > 0. We deduce that for all ε small enough and all (x, θ) ∈ Ω × supp φ, we have

supp n ε ⊂ Ω × {θ ∈] -A, A[ | u(θ) = 0} . Indeed, let φ ∈ C ∞ c (Ω×] -A, A[) be any positive test function such that supp φ ⊂ Ω × {θ ∈] -A, A[ | u(θ) = 0} c . ( 5
u ε (x, θ) ≤ - a 2 . ( 5.37)
We conclude that

Ω A -A φ(x, θ)n(x, θ)dθdx = Ω supp φ(x,•) φ(x, θ)n(x, θ)dθdx = lim ε→0 Ω supp φ(x,•) φ(x, θ)n ε (x, θ)dθdx = lim ε→0 Ω supp φ(x,•) φ(x, θ)e uε(x,θ) ε dθdx ≤ lim ε→0 Ω supp φ(x,•) φ(x, θ)e -a 2ε dθdx = 0.
This concludes the proof of 3.

Discussions and numerics

Numerical resolutions

To find a numerical solution of (E), we solve numerically the following parabolic equation :

                 ∂ t n ε -∂ xx n ε -ε 2 ∂ θθ n ε + L(n ε ) = [R -ρ ε ]n ε in R + × Ω×] -A, A[, ρ ε (t, x) = A -A n ε (t, x, θ)dθ in R + × Ω, ∂ νx n ε = ∂ ν θ n ε = 0, n(t = 0, x, θ) = n 0 (x, θ). (E t )
Chapter 5 : Adaptation to a heterogeneous environment with nonlocal dispersion

For this numerical study, we take

R(x, θ) = r -g(bx -θ) 2 .
We recall from Example 5.1 that in this case, r is a growth rate and -g(bx -θ) 2 models the selection. The parameter g is the selection pressure whereas b is the gradient of the environment. We provide numerical examples where we vary this set of parameters. We implement equation (E t ) by a semi-implicit finite difference method. We stop the algorithm when we find a numerical steady state of (E t ) : a numerical solution of (E). First, we underline that in all the numerical resolutions, the density of the population concentrates around one or several distinct trait(s). Moreover, these optimal traits are present everywhere in space thanks to the local and the non-local migration. However, the density of the population at the position x with an optimal trait θ m depends on whether this trait θ m is adapted or not to the position x. 5.3). We observe that with a same set of parameters, considering a patchy domain leads to the apparition of more dominants traits compared with the population living in a single connected environment. Moreover, the part of the population living in a patch Ω 1 , with a dominant traits θ 2 adapted to the other patch Ω 2 , becomes weaker as the distance between both patches increases. Finally, we want to emphasize that the emergence of new dominant traits as a function of the distance between habitats depends also on the value of g. Indeed, for the same parameters value as in Figure 5.3 except for g that we take g = 0.1, we observe in Figure 5.4 that the stationary state is still monomorphic even though Ω =] -3, -1[∪]1, 3[.

g = 0.1 g = 5 g = 10

Introduction

Aedes aegypti is the main vector transmitting dengue viruses. This mosquito can also transmit chikungunya, yellow fever and Zika infection. According to the World Health Organization, 390 million people are infected by dengue every year and 3.9 billion people, in 128 countries, are at risk of infection by dengue viruses. As there is no treatment for dengue fever, the current method of preventing dengue virus transmission and epidemics is to target the vector, i.e. the mosquito. Beyond preventing mosquitoes from accessing egg-laying habitats by environmental management and modification, one of the most promising control techniques is to transform mosquito population with a virus-suppressing Wolbachia bacteria. The idea of using Wolbachia for disease control was first proposed in the 1960s [START_REF] Laven | Eradication of culex pipiens fatigans through cytoplasmic incompatibility[END_REF] but applying it to Aedes aegypti population is very recent. Wolbachia bacterium strains were isolated from Drosophila melanogaster in laboratory just before 2000 [START_REF] Min | Wolbachia, normally a symbiont of drosophila, can be virulent, causing degeneration and early death[END_REF][START_REF] Mcmeniman | Host adaptation of a wolbachia strain after long-term serial passage in mosquito cell lines[END_REF] but were introduced into Aedes aegypti embryos only on 2009 [START_REF] Mcmeniman | Stable introduction of a life-shortening wolbachia infection into the mosquito aedes aegypti[END_REF]. The capability of this bacteria to suppress dengue virus and other pathogens transmission by Aedes aegypti was shown in laboratory around 2010 [START_REF] Moreira | A wolbachia symbiont in aedes aegypti limits infection with dengue, chikungunya, and plasmodium[END_REF][START_REF] Bian | The endosymbiotic bacterium wolbachia induces resistance to dengue virus in aedes aegypti[END_REF][START_REF] Walker | The w mel wolbachia strain blocks dengue and invades caged aedes aegypti populations[END_REF]. It was also shown that this bacteria shortens life span [START_REF] Yeap | Dynamics of the "popcorn" wolbachia infection in outbred aedes aegypti informs prospects for mosquito vector control[END_REF] and most of the infected adults do not reach the infectious stage. But the most important modification induced by the bacteria is cytoplasmic incompatibility (CI) [START_REF] Mcmeniman | Stable introduction of a life-shortening wolbachia infection into the mosquito aedes aegypti[END_REF]. Cytoplasmic incompatibility is used by the bacteria to spread rapidly into natural population [START_REF] Turelli | Cytoplasmic incompatibility in populations with overlapping generations[END_REF] by producing non-viable eggs when uninfected females mate with infected males. Reproduction between infected males and females lead to infected eggs. As this bacteria is vertically transmitted (from mother to off-springs), uninfected males mating with infected females give rise only to infected eggs.

We are interested on optimizing the release of Wolbachia-infected mosquitoes into a wild host population of mosquitoes. Thus, the aim of the study is to model the propagation across time and space of the density of infected mosquitos, denoted n 2 , starting from a controlled release u into an existing population of uninfected. In what follows, we will denote by n 1 the density of uninfected mosquitos.

Formally, a proportion 1 -s h of uninfected female's eggs fertilized by infected males actually hatch. Cytoplasmic incompatibility is perfect when s h = 1. We denote by b 1 , respectively b 2 , the net fecundity rate of uninfected females, respectively infected females. Death rate for uninfected mosquitoes is denoted d 1 . As Wolbachia decreases lifespan, death rate of infected mosquitoes d 2 verifies d 2 > d 1 . Is is also observed that Wolbachia infected mosquitoes tend to have reduced fertility, then b 2 ≤ b 1 . Finally, we denote κ the carrying capacity. Cytoplasmic incompatibility and vertical transmission drive the spatial spread of the infected population producing a bistable dynamic of Wolbachia [START_REF] Turelli | Deploying dengue-suppressing wolbachia : Robust models predict slow but effective spatial spread in aedes aegypti[END_REF]. If the infected population is installed above a sufficient threshold frequency Θ compared to the uninfected population, it will spread and tend to increase to 1, otherwise it will tend to decline to zero. For fixed maximal time T > 0 and domain Ω, the system of equation that we consider is the following :

                   ∂ t n 1 -D∆n 1 = b 1 n 1 (1 -s h n 2 n 1 + n 2 )(1 - n 1 + n 2 κ ) -d 1 n 1 in Ω, ∂ t n 2 -D∆n 2 = b 2 n 2 (1 - n 1 + n 2 κ ) -d 2 n 2 + u in Ω, ∂ ν n 1 = ∂ ν n 2 = 0 on ∂Ω, n 1 (0, x) = n 0 1 (x), n 2 (0, x) = n 0 2 (x)
in Ω.

(6.1)

The equations driving the dynamics of n 1 and n 2 are bistable and monostable reactiondiffusion equations, respectively. Note that in the reaction term of the first equation the term n 2 n 1 +n 2 stands for the vertical transition of the disease whereas the coefficient s h models that this vertical transmission may or not be perfect because of the cytoplasmic incompatibility. More precisely, assuming homogeneous repartition of individuals, the probability to mate with an infected mosquito is n 2 n 1 +n 2 . Then, uninfected mosquitoes are generated from mating of uninfected mosquitoes with uninfected mosquito (probability n 1 n 1 n 1 +n 2 ) or uninfected mosquitoes with infected mosquitoes but with a probability (1 -s h )n 1 n 2 n 1 +n 2 . The first term in the right hand side is the sum of this latter quantities. The diffusion coefficient is denoted D ; it is assumed to be the same for both populations since both populations belongs to the same genus of mosquitoes. The last term of the second equation +u stands here to model the releases of infected mosquitoes developed in laboratory : it is on this control that we will act upon. More precisely, a question we want to address in this work is to know what should be the shape of the release function u to be as close as possible to the total invasion of the infected population into the domain.

The outline of this paper is the following. In the next section, we introduce the optimal control problem and prove the existence of an optimum for this problem. In Section 6.2, we consider a toy problem, which is a very simplified version of the full problem, for which we can solve explicitly the optimal problem and find the optimum. In Section 6.3, we investigate numerically the optimization of the spatial releases of mosquitoes. Finally, we end this paper with a conclusion and perspective for future works. An appendix is devoted to recalling the reduction of system (6.1).

Optimal Control Problem

We are going to simplify the problem. Instead of studying the coupled equations (6.1), we are going to follow the proportion of mosquitoes p(t, x) = n 2 (t,x) n 1 (t,x)+n 2 (t,x) as in [START_REF] Strugarek | Reduction to a single closed equation for 2-by-2 reaction-diffusion systems of lotka-volterra type[END_REF]. This reduction is clearly justified in the limit of large population in [START_REF] Strugarek | Reduction to a single closed equation for 2-by-2 reaction-diffusion systems of lotka-volterra type[END_REF] (see also [START_REF] Almeida | Optimal releases for population replacement strategies : application to wolbachia[END_REF]Section 2.3]). Details on the formal computation are provided in Appendix. In order to simplify the reading, The general optimal control problem we want to investigate involves the least-squares functional J defined by Ĵ(u) = 1 2 Ω (1 -p(T, x)) 2 dx, (6.4) which models that one aims at steering the system as close as possible to the target state. In some sense, it stands for the research of a control strategy ensuring the persistence of infected mosquitoes at the time horizon T . Of course, it is relevant from the biological point of view to impose several constraints on the control function u. Indeed, the production of Wolbachia-infected mosquitoes is limited, which imposes that the total number of mosquitoes released is bounded. Hence, the control function u is assumed to belong to the set Since this problem involves the minimization over function depending on time and space variables, it is difficult to study. Then, we will reduce it to a simpler one by assuming that the time distribution of the control function is given.

Modeling of the optimal control problem

In order to weaken the difficulty of Problem (P full ), we introduce a simpler, although still relevant, problem by assuming that :

-releases are done periodically in time (for instance every week) and are impulses in time1 ;

Chapter 6 : Optimal release of mosquitoes to control dengue transmission -at each release, the largest allowed amount of mosquitoes is released, corresponding to the maximal production capacity per week (which is relevant, according to the comparison principle). As a consequence, we will be interested in determining the optimal way of releasing spatially the infected mosquitoes. Let us denote by t 0 = 0 < t 1 < . . . < t N = T , t i = i∆T , the release times. Rewriting the L 1 constraint on the control as u, It is possible to recast System (6.2) without source measure terms, coming from the specific form of the control functions. For the sake of simplicity, we provide here a naive formal analysis, but claim that this can be proven rigorously by using a standard variational analysis. Let us approximate the Dirac measure at t = t i by the function 1 ε 1 [t i ,t i +ε] . Making the change of variable t = t i + τ ε, and introducing p given by p(τ, x) = p(t, x), one gets from system (6.2) that p solves Then, by a direct integration of (6.6) on [0, 1], we obtain

G(p(1, x)) = G(p(0, x)) + u i (x), x ∈ Ω.
Coming back on the function p yields p(t + i , x) = G -1 (G(p(t - i , x)) + u i (x)), x ∈ Ω.

Hence we arrive at the system where p is the solution of (6.7). In the next Section, we investigate the existence of solutions for this problem. Proof. For the sake of readability, we only provide the proof in the case N = 2. Indeed, there is no additional difficulty to deal with the general case whose proof follows exactly the same lines. The proof is divided into several steps. Let u n = {u n i } i∈{1,...,N } ∈ (V T,C,M ) N be a minimizing sequence for Problem (P reduced ). Notice that, since u belongs to V T,C,M and G -1 takes its value in [0, 1[, we infer from the maximum principle that 0 ≤ p(t, •) < 1 for a.e. t ∈ [0, T ] so that one has for all u ∈ V T,C,M 0 ≤ J(u) ≤ |Ω| 2 .

                
It follows that inf u∈V T,C,M J(u) belongs to (0, |Ω| 2 ) and, in particular, is finite.

Step 1 : Convergence of the minimizing sequence. Let p n be the solution to (6.7) associated to the control function u n and let us introduce v n 0 (.) = u n 0 (.) v n 1 (.) = G -1 (G(p n (t - 1 , .)) + u n 1 (.)).

By induction, one easily shows that v n is uniformly bounded in L ∞ . Since the class V T,C,M is closed for the L ∞ weak-star topology, there exists v ∞ ∈ V T,C,M such that, up to a subsequence, v n converges weakly-star to v ∞ in L ∞ . Here and in the sequel, we will denote similarly with a slight abuse of notation a given sequence and any subsequence.

Multiplying the main equation of (6.7) by p n and integrating by parts, we infer from the above estimates the existence of a positive constant C such that The sequence of functions p n is uniformly bounded in L 2 ([0, T ], H 1 (Ω)) thanks to the pointwise bounds on p n . Furthermore, by using (6.7), one gets that the sequence ∂ t p n is uniformly bounded in L 2 ([0, T ], W -1,1 (Ω)). According to the Aubin-Lions theorem (see [START_REF] Simon | Compact sets in the spacel p (o, t ; b)[END_REF]) we infer that p n converges (up to a subsequence) to p ∞ ∈ L 2 ([0, T ], H 1 (Ω)), strongly in

Optimal Control Problem

Proof. It is a direct application of the comparison principle. Let u * be a solution of Problem (P reduced ). By contradiction, let us assume that u * is not identically equal to M a.e. in Ω. Then, let t i be a release time for which the associated control function u * i is not identically equal to M in Ω. Recall that u * i ≤ M . Let us denote by p * the solution of the problem (6.2) associated to the control function u * . Let u M be the control function defined by u M i = M and u M j = u * j for all j ∈ {0, . . . , N -1}\{i}.

Let p M be the solution of (6.2) associated to u M identically. Since G -1 is an increasing function by the comparison principle we have for all time t ∈ [0, T ] and a.e. x ∈ Ω, 0 < p * (t, x) ≤ p M (t, x) < 1.

Evaluating this expression at time t = T , the expected conclusion follows by noting that the constant function equal to M on (0, T ) × Ω belongs to U T,C,M .

Computation of derivatives

As a preliminary remark, we claim that for any element u of the set V T,C,M and any admissible perturbation h, the mapping V T,C,M u → p ∈ L 2 (0, T, H 1 (Ω)), where p denotes the unique weak solution of (6.7), is differentiable in the sense of Gâteaux at u in the direction h. Indeed, proving such a property is standard in calculus of variations and rests upon an application of the implicit function theorem. In the sequel, and with no confusion possible, we will denote by ṗ the Gâteaux-differential of p at u in direction h and by dJ(u), h the Gâteaux-differential of J at u in direction h, namely dJ(u), h = lim ε 0 J(u + εh) -J(u) ε .

Let us make the cone of admissible perturbations precise. We call "admissible perturbation" any element of the tangent cone T u,V T,C,M to the set V T,C,M at u. Definition 6.1. The cone T u,V T,C,M is the set of N -tuples h = (h 0 , . . . , h N -1 ) ∈ (L ∞ (Ω)) N such that, for any i ∈ {0, . . . N -1} and for any sequence of positive real numbers ε n decreasing to 0, there exists a sequence of functions h n i ∈ L ∞ (0, T ) converging to h i as n → +∞, and u i + ε n h n i ∈ V T,C,M for every n ∈ N (see e.g. [START_REF] Cominetti | Tangent sets to unilateral convex sets[END_REF]). Proposition 6.2. Assume that N = 1. Let u = (u 0 ) ∈ V T,C,M and h = (h 0 ) ∈ T u,V T,C,M . One has dJ(u), h = Ω h(x)(G -1 ) (u 0 (x))q(0, x) dx, where q is the unique solution of the backward problem

      
-∂ t q(t, x) -∆q(t, x) -f (p(t, x))q(t, x) = 0, (t, x) ∈ (0, T ) × Ω, ∂ n q(t, x) = 0, (t, x) ∈ (0, T ) × ∂Ω, q(T, x) = p(T, x) -1, x ∈ Ω.

Proof. By using the preliminary discussion, one has dJ(u), h = Ω ṗ(T, x)(p(T, x) -1) dx, (6.9) where ṗ denotes the unique solution of the system

           ∂ ṗ ∂t -∆ ṗ = f (p) ṗ, t ∈ (0, T ), x ∈ Ω,
∂ ν ṗ(t, x) = 0, x ∈ ∂Ω, ṗ(0 + , •) = (G -1 ) (u 0 (•))h. (6.10) Let us multiply the main equation of this system by q and then integrate by parts with respect to the variables t and x. By using in particular the Green formula, we get successively that Remark. For practical purposes, it may be useful to notice that q(t, x) = q(T -t, x), t ∈ [0, T ], x ∈ Ω, where q denotes the solution of the initial boundary value problem

      
∂ t q(t, x) -∆ q(t, x) -f (p(T -t, x)) q(t, x) = 0, (t, x) ∈ (0, T ) × Ω, ∂ n q(t, x) = 0, (t, x) ∈ (0, T ) × ∂Ω, q(0, x) = p(T, x) -1, x ∈ Ω.

A toy Problem

This section is devoted to investigating a simpler version of (P full ) corresponding to the case N = 1 with f = 0. More precisely, let p be the solution of where p ∈ C 0 ([0, T ], L 2 (Ω)) is the unique solution of Equation (6.11). Note that the equation (6.11) has to be understood in a weak sense, since u 0 ∈ L ∞ (Ω) ⊂ L 2 (Ω) (see for example [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]Section 10.7]).

For this simple problem, we are able to solve explicitely the optimization problem : Proof. First, note that Problem (P toy ) has a solution. Indeed, it is standard that the mapping L 2 (Ω) u 0 → p ∈ C 0 ([0, T ], L 2 (Ω)) is continuous. Therefore, so is Ĵ by composition of continuous mappings. The conclusion follows by observing that V T,C,M is a compact subset of L 2 (Ω).

The proof relies on a well-adapted rewriting of the criterion Ĵ. According to the spectral theorem, there exists an orthonormal family (φ j ) j≥1 consisting of (real-valued) eigenfunctions of -∆ N , associated with the non-decreasing sequence positive eigenvalues (λ j ) j≥1 . Moreover, by setting λ 0 = 0 and φ 0 = 1 √ |Ω| , the sequence (φ j ) j≥0 is a Hilbert basis of L 2 (Ω) and any solution p of (6.11) can be expanded in a unique way in L 2 (Ω) as p(t, x) = +∞ j=0 p(0, •), φ j L 2 (Ω) e -λ j t φ j (x) = +∞ j=0 u 0j e -λ j t φ j (x), (6.12)

with u 0j = u 0 , φ j L 2 (Ω) . By expanding the square in the definition of Ĵ, we then infer that where ψ(x) = -1 + +∞ j=0 e -2λ j T u 0j φ j (x). The first order optimality conditions reads d Ĵ(u 0 ), h ≥ 0, ∀h ∈ T u 0 ,V T,C,M . (6.13)

The analysis of such optimality condition is standard in optimal control theory (see for example [START_REF] Li | Necessary conditions for optimal control of distributed parameter systems[END_REF]) and yields the existence of a Lagrange multiplier ξ ≤ 0 such that -on {u 0 = M }, ψ(x) ≤ ξ, -on {u 0 = 0}, ψ(x) ≥ ξ, -on {0 < u 0 < M }, ψ(x) = ξ, ξ ( Ω u 0 (x) dx -C) = 0 (complementarity condition).

Gaussian Releases

The goal is then to find the best position of the releases and the optimization problem becomes inf (x 1 ,...,x K )∈Ω K J K (x 1 , ..., x K ) with J K (x 1 , ..., x K ) = 1 2 Ω (1 -p(T, x)) 2 dx , (P K )

where p ∈ C 0 ([0, T ], L 2 (Ω)) is the unique solution of (6.7) with control u K (•, x 1 , ..., x K ).

Remark.

Since Ω is a bounded domain in R 2 , the question of the existence of a minimizer is trivial. But, the uniqueness is still a challenging problem. Proposition 6.3. Let (x 1 , ..., x K ) ∈ Ω K . For k ∈ {1, ..., K}, one has

∂J K ∂x k (x 1 , ..., x K ) = Ω (G -1 ) (u K (x))q(0, x) ∂u K ∂x k (x, x 1 , ..., x K ) dx,
where q is the unique solution of the backward problem

    
-∂ t q(t, x) -∆q(t, x) -f (p(t, x))q(t, x) = 0, (t, x) ∈ (0, T ) × Ω, ∂ n q(t, x) = 0, (t, x) ∈ (0, T ) × ∂Ω, q(T, x) = p(T, x) -1,

x ∈ Ω.

Proof. It is an easy application of the chain rule. First, we notice that J K (x 1 , ..., x K ) = J(u K (x, x 1 , ..., x K )).

Next, using Proposition 6.2, we find thanks to the chain rule that for all k ∈ {1, ..., K} ∇J K (x 1 , ..., x K ) =< dJ(u(x, x 1 , ..., x K )), ∇u(x, x 1 , ...x K ) > = Ω (G -1 ) (u K (x, x 1 , ..., x K )q(0, x)∇u K (x, x, x 1 , ..., x K )dx.

We deduce the result from the last equality.

Numerical Resolution

We now present the computation of the numerical solution of (P K ). For this we use a direct method which consists in carrying out a discretization of Equation (6.7) and of the control in order to obtain a finite dimensional optimization problem with constraints. We can then compute an approximation of a local minimizer of (P K ) with a numerical optimization solver. Our results were obtained with the finite element toolbox FreeFem++ [START_REF] Hecht | New development in freefem++[END_REF] which contains an implementation of the optimization routine Ipopt [START_REF] Wächter | On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming[END_REF].

We therefore consider a finite element basis of functions (ϕ i ) i that allows us to discretize the control as u h (x, x 1 , ..., x K ) = i u i ϕ i (x) and the proportion of infected mosquitoes as p h (t, x) = i p i (t)ϕ i (x), the finite element approximation of the solution of the PDE (6.7) with initial condition G -1 (u h (x, x 1 , ..., x K )). The cost function can be computed with numerical integration as J h (x 1 , ..., x K ) = Ω (1 -p h (T, x)) 2 dx. In addition, Ipopt requires the gradient of the cost function and thanks to Proposition 6.3 we have

∂J h ∂x k = Ω (G -1
) (u h (x))q h (0, x) ∂u h ∂x k (x, x 1 , ..., x K ) dx where q h (0, x) is the finite element approximation of the solution of the backwards PDE. Le but de l'étude serait de donner des propriétés qualitatives des fonctions α et α.

Etudier les équilibres et les phénomènes d'invasions dans un modèle fractionnaire impliquant une non-linéarité bistable à la place d'une non linéarité du type Fisher-KPP (Chapitres 3 et 4).

Modèles de diffusion non-conventionnelle en écologie et biologie évolutive impliquant des environnements fragmentés

Abstract

In this thesis, we are interested in a qualitative mathematical study of problems from ecology and evolutionary biology. We study the influence of a non-local dispersion for a biological species living in a patchy environment. More precisely, we first establish a criterion whose ensures the survival of a biological species which dynamics are driven by a fractional Fisher-KPP equation in a fragmented domain with Dirichlet exterior conditions. This criterion relies on the sign of the principal eigenvalue of subsets included in the fragmented domain. Moreover, we demonstrate an existence and uniqueness result of the stationary state of a Fisher-KPP equation in general patchy domains belonging to the class of non-negative, bounded and non-trivial solutions. In the particular case of a periodic and patchy domain, we establish the existence of invasion phenomena with exponential speed. Finally, we consider a model dealing with a phenotypically structured biological species living in a patchy environment. This species is subject to small mutations of the phenotype and to local and non-local spatial dispersion. We demonstrate the emergence of phenotypical dominant traits as the mutations become small. Key words : Evolutionary biology, Unconventional diffusion kernel, Acceleration phenomena, Asymptotic analysis, Parabolic integro-differential equations

Chapter 1 :Figure 1 . 1 -

 111 Figure 1.1 -Illustration d'un phénomène d'invasion pour f (x, n) = n -n 2 (à gauche) et f (x, n) = µ(x)n -n 2 (à droite).

  pour l'équation ∂ t n + (-∆) α n = n -n 2 dans R d × R + . Pour une telle équation, les auteurs démontrent en construisant des sous et sur-solutions adaptées que ∀c > 1, lim t→+∞ sup |x|<e c -1 t d+2α |n(x, t) -1| = 0 et lim t→+∞ sup |x|>e ct d+2α |n(x, t)| = 0.

∀c > 1

 1 , lim t→+∞ sup |x|<e c -1 |λ 0 (µ)|t d+2α |n(x, t) -n + (x)| = 0 et lim t→+∞ sup |x|>e c|λ 0 (µ)|t d+2α |n(x, t)| = 0.

Théorème 1 . 1 (

 11 L. (Chapitre 2) ). Soit n la solution de (1.1) avec Ω = R d , f (x, n) de la forme µ(x)n -n 2 où µ est périodique. Si λ 0 la valeur propre principale de (-∆) α -µ est négative, alors, on a 1. ∀c < |λ 0 | d+2α , lim t→+∞ sup |x|<e ct |n(x, t) -n + (x)| = 0, 2. ∀C > |λ 0 | d+2α , limt→+∞ sup |x|>e Ct |n(x, t)| = 0.

Théorème 1 . 2

 12 (L.-Mirrahimi-Roquejoffre (Chapitre 3) ). Soit n la solution de (1.1) avec Ω un domaine périodique fragmenté et f (x, n) = n -n 2 . Si la valeur propre principale λ 0 de (-∆) α -Id dans Ω 0 avec des conditions de Dirichlet extérieures est négative, alors il existe une unique solution stationnaire bornée positive non-triviale n + . De plus, pour toute solution n avec n(t = 0) positive, non-triviale à support compacte dans Ω, nous avons que 1. ∀c < |λ 0 | d+2α , lim t→+∞ sup |x|<e ct |n(x, t) -n + (x)| = 0, 2. ∀C > |λ 0 | d+2α , lim t→+∞ sup |x|>e Ct |n(x, t)| = 0.

  continue croissante sur tout son ensemble de définition. De plus, elle peut être prolongée par continuité en 0.

Chapter 1 :Figure 1 . 2 -

 112 Figure 1.2 -Illustration des hypothèses précédentes.

  θ o (x) = bx. Lorsqu'un individu a un trait θ différent de ce trait optimal, il y a un coût pour son taux de croissance qui est donné par (θ -θ o (x)) 2 fois la constante g qu'on appelera la pression de la sélection. Les individus fortement inadaptés à la position x auront un taux de naissance faible (voir négatif). La constante b indique à quelle vitesse le trait optimal varie en tant que fonction de l'espace. Enfin le terme ρ désigne la taille totale de la population. Via le terme -ρ nous prenons en compte un taux de mortalité dû à la compétition entre les individus pour les ressources disponibles. Les conditions aux bords de Neumann par rapport à x modélisent le fait qu'aucun individu ne peut ni entrer ni sortir du territoire. De même, les conditions aux bords de Neumann par rapport à θ se traduisent par le fait qu'aucun mutant ne puisse apparaître en dehors de l'intervalle ] -A, A[. Le but du chapitre 5 est de présenter nos premiers résultats obtenus sur l'apparition de trait(s) dominant(s) (voir Figure 1.3 en exemple) lorsque le terme de mutation devient petit (i.e. ε → 0). .

Figure 1 . 3 -

 13 Figure 1.3 -Simulation numérique de n ε pour Ω =] -2, 2[, A = 3 et r = 5, b = 1, g = 5, ε = 0.005.

2. 3 .

 3 The proof of Theorem 2.1 and thus e -|λ 1 |t 0 +ν+(d+2α) log(|xε|) ε

3. 1 . Introduction Corollary 3 . 1 .

 131 The principal eigenvalue λ 0 of the Dirichlet operator (-∆) α -Id in Ω is negative.

Definition 3 . 1 (

 31 The uniform interior and exterior ball condition). A set O ⊂ R d with d ≥ 1 satisfies the uniform interior and exterior ball condition if there exists r 1 > 0 such that ∀x ∈ ∂O, ∃y x ∈ O such that x ∈ ∂B(y x , r 1 ) and B(y x , r 1 ) ⊂ O, and ∀z ∈ O c , ∃y z ∈ O c such that z ∈ B(y z , r 1 ) and B(y z , r 1 ) ⊂ O c . Theorem 3.2. Let O be a smooth domain of R d with d ≥ 1 satisfying the uniform interior and excterior ball condition. If we define p as the solution of the following equation

Chapter 3 :

 3 Propagation in a fractional reaction-diffusion equation in a periodically hostile environment The proof of D can be achieved with the rescaled solution n |x| 1 ε

  All along the article, we will use many times the comparison principle. We recall here what we mean by comparison principle. Theorem (The comparison principle). Let f be a smooth function, a ∈ [0, +∞[ and b ∈ ]0, +∞]. If n and n are such that

  Let O ⊂ R d be an open set satisfying the uniform interior and exterior ball condition at z ∈ ∂O and let c ∈ L ∞ (O). Consider a positive lower semi-continuous function u : R d → R satisfying (-∆) α u ≥ c(x)u point-wise in O. Then, either u vanishes identically in O, or there holds

. 45 ) 3 . 3 .

 4533 Proposition We assume (H1) and (H2). If we set C m = min( |λ 0 | 2(max φ 0 +1) , cm C 0 , 1) and C M = 2|λ 0 |+c M min(1,c 0 ) where c 0 , c m , C 0 and C M are introduced in (3.45) and (3.38

. 53 )

 53 Inserting(3.52) and (3.53) into (3.51) and defining ε 0 := min(ε 3 , ε 4 , ε 5 , |λ 0 | 2 , 1), we conclude that for all ε < ε 0 and all (x, t) ∈ Ω ε ×] 4 ε 2 , +∞[ we have :

2 - 1 ε

 21 and κ := (d + 2α)C -|λ 0 | then the conclusions follows.

3. 5 . 3

 53 The proof of Theorem 3.It remains to prove the claims (3.62) and(3.64). The proof of (3.62) relies on the uniqueness result stated in Theorem (3.1).

  c ν = C m min y∈Ων φ 0 (y) where the function C m is defined in Proposition 3.3. Set µ ∈]0, c ν [ and we define ε = min(

Definition 4 . 1 (Figure 4 . 1 -

 4141 Figure 4.1 -Illustration of a possible domain Ω

Chapter 4 :

 4 Properties of steady states for a class of non-local Fisher-KPP equations in general domains

(4. 22 )

 22 Inserting (4.21) and(4.22) in (4.20), we conclude that for all ε > 0 and all positive φ ∈ H

Figure 4 . 2 - 4 :

 424 Figure 4.2 -The set under study

c |x-y| d+2α ≤ 1 |zx

 1 -y| d+2α , thus (4.51) follows from Lemma 4.5.

Figure 4 . 3 -Chapter 4 :

 434 Figure 4.3 -The left picture illustrates the case where inf δ(ξ k ) > δ 0 (in this case, z k = x k ). The right picture illustrates the case inf δ(ξ k ) = 0. In this case, up to a subsequence, we can assume that δ(ξ k ) < ε 0 4 and w k = z k .

Chapter 4 :

 4 Properties of steady states for a class of non-local Fisher-KPP equations in general domains

( 4 . 71 )

 471 Combining (4.70) and (4.71) in (4.69), we deduce

) Lemma 5 . 1 .

 51 Under the assumptions (H1)-(H4)µ ε -→ ε→0 λ(θ 0 , 0).

  HC ) We prove the following Theorem 5.2. Under the assumptions (H1)-(H4), as ε → 0 along subsequences, there holds 5.1. Introduction 1. ρ ε converges uniformly to ρ with 0 < c ≤ ρ ≤ C, 2. u ε converges uniformly to u with u a viscosity solution of -|∂ θ u(θ)| 2 = -λ(θ, ρ), max u(θ) = 0, (5.5)

Proposition 5 . 1 .

 51 For a fixed bounded smooth function ρ and a fixed value θ ∈] -A, A[ there exists a principal eigenvalue λ(θ, ρ) of the operator -∂xx ψ + L(ψ) -(R(•, θ) -ρ)ψ with Neumann boundary conditions i.e. -∂ xx ψ + L(ψ) -(R(•, θ) -ρ)ψ = λ(θ, ρ)ψ in Ω,∂ νx ψ = 0 on ∂Ω.(5.15)

2 . 3 .Lemma 5 . 3 .

 2353 and 4. of Theorem 5.3. But, we need an intermediate result : ρ ε is uniformly bounded. Under the assumptions (H1) -(H4), we have that for all ε < ε 0

Corollary 5 . 1 .

 51 uniformly bounded. The result follows from the standard elliptic estimates. There exists a constant C > 0 such that for all ε small enough|∂ x ρ ε | ≤ C.

Chapter 5 :

 5 Adaptation to a heterogeneous environment with nonlocal dispersion Proof of 4. of Theorem 5.3. We start by proving that 0 < c ≤ sup ρ ε ≤ C. Next, we prove that c < ρ ε < C holds true in the whole domain Ω. The bounds on sup ρ ε . The upper bound is already obtained in Lemma 5.3. It remains to prove the lower bound. Assume by contradiction that there exists a sequence ε k such that ε k -→ k→+∞ 0 and sup ρ ε k -→ k→+∞ 0.

.31) 5 . 4 .

 54 Convergence to the Hamilton Jacobi equation Next, we prove that lim ε→0 sup (x,θ)∈Ω×]-A,A[ u ε (x, θ) ≤ 0.

Figure 5 . 1 -

 51 Figure 5.1 -Numerical solutions of (E) with 3 different parameters g (g = 0.1, 5, 10) and the other parameters are fixed as follows Ω =] -2, 2[, A = 3, r = 10, b = 1 and ε = 0.005.

Figure 5 . 2 -

 52 Figure 5.2 -Numerical resolution of (E) with the following set of parameters r = 10, b = 1, g = 10, A = 3, ε = 0.005 and Ω =] -2.2, -0.2[∪]0.2, 2.2[.

Figure 5 .

 5 1 illustrates the variation of the density of the population when Ω is connected and g varies. When the selection pressure g increases, we observe that several dominant traits appear. Figures 5.2 and 5.3 illustrate the influence of the distance when we consider a patchy space

Figure 5 . 3 -

 53 Figure 5.3 -Numerical resolution of (E) with the following set of parameters r = 10, b = 1, g = 10, A = 3, ε = 0.005 and Ω =] -3, -1[∪]1, 3[.

Figure 5 . 4 -

 54 Figure 5.4 -Numerical resolution of (E) with the following set of parameters r = 10, b = 1, g = 0.1, A = 3, ε = 0.005 and Ω =] -3, -1[∪]1, 3[.

6. 1 .• b 1 ( 1 -

 111 Optimal Control Problem we perform the scaling x = x √ D not to keep the diffusion coefficient along the computations. Obviously, for the numerical simulations performed in Section 6.3, we have to keep in mind this scaling.Denoting by p the proportion of infected mosquitoes, and u the release function, the dynamics is governed by the reaction-diffusion equation f (p) + ug(p), t ∈ (0, T ), x ∈ Ω,∂ ν p(t, x) = 0, x ∈ ∂Ω, p(0, x) = 0, x ∈ Ω,(6.2)wheref (p) = p(1 -p) d 1 b 2 -d 2 b 1 (1 -s h p) b 1 (1 -p)(1 -s h p) + b 2 p and g(p) = 1 κ p)(1 -s h p) b 1 (1 -p)(1 -s h p) + b 2 p . (6.3)

U

  T,C,M = u ∈ L ∞ ([0, T ] × Ω), 0 ≤ u ≤ M a.e. , T 0 Ω u(t, x) dxdt ≤ C . (6.5)modeling an upper limit on the instantaneous number of Wolbachia-infected individuals released at time t, as well as on the total number of released mosquitoes.We then deal with the following optimal control problem : inf u∈U T,C,M Ĵ(u). (P full )

  1 D ,D((0,T )×Ω) ≤ C, the control function readsu(t, x) = N -1 i=0 u i (x)δ {t=t i } , with N -1 i=0 Ω u i (x) dx ≤ C,where the pointwise constraint is modified into 0≤ u i (•) ≤ M .The new optimal design problem readsinf u∈V T,C,M J(u), where u = (u i ) 0≤i≤N -1 , J(u) = J N -1 i=0 u i (•)δ {t=t i } (P full )andV T,C,M = u = (u i ) 0≤i≤N -1 , with 0 ≤ u i ≤ M a.e.in Ω andN -1 i=0 Ω u i (x) dx ≤ C .

  ∂ p ∂τ -ε∆p = εf (p) + u i g(p). τ ∈ [0, 1], x ∈ Ω.Letting formally ε go to 0 and denoting, with a slight abuse of notation, still by p the formal limit of the system above yields∂ p ∂τ (τ, x) = u i (x)g(p(τ, x)), τ ∈ [0, 1], x ∈ Ω. (6.6)Let us denote G the anti-derivative of 1 g vanishing at 0, namely

2 Ω ( 1 -

 21 ∂p ∂t -∆p = f (p), t ∈ (0, T ) \ {t i } i∈{1,...,N -1} , x ∈ Ω, ∂ ν p(t, x) = 0, x ∈ ∂Ω, p(0 + , •) = G -1 (u 0 (•)), p(t + i , •) = G -1 G(p(t - i , •)) + u i (•) , i ∈ {1, . . . ,N -1} (6.7) 6.1. Optimal Control Problem and the optimization problem reads inf u∈V T,C,M J(u) with J(u) = 1 p(T, x)) 2 dx , (P reduced )

6. 1 . 2 Theorem 6 . 1 .

 1261 Existence of minimizers Problem (P reduced ) has a solution.

∂ 2 Ω

 2 t (p n (t, x) 2 )dxdt + T 0 Ω |∇p n (t, x)| 2 dxdt ≤ Cfor every n ∈ N, which also reads1 (p n (t, x)) 2 ) n (t, x)| 2 dxdt ≤ Cfor every n ∈ N.

  x) ṗ(T, x) dx -Ω q(0, x) ṗ(0 + , x) dx, dJ(u), h = Ω q(T, x) ṗ(T, x) dx = Ω q(0, x) ṗ(0 + , x) dx,yielding the desired conclusion.

  0, t ∈ (0, T ), x ∈ Ω, ∂ ν p(t, x) = 0, x ∈ ∂Ω, p(0 + , •) = u 0 (•).

(6. 11 ) 2 Ω ( 1 -

 1121 Then, the optimization toy problem reads infu 0 ∈V T,C,M Ĵ(u 0 ) with Ĵ(u 0 ) = 1 p(T, x)) 2 dx , (P toy )

6. 2 . A toy Problem Theorem 6 . 2 .

 262 Problem (6.11) has a unique solution u 0 , which is constant and equal to min 1, M, C |Ω| .

  For that purpose, let us introduce the Neumann operator -∆ N on Ω defined on D(-∆ N ) = {y ∈ H 2 (Ω) | ∂y ∂n |∂Ω = 0 and Ω y(x) dx = 0}.

Ĵ 2 .

 2 x)φ j (x) dx Let u be a solution of Problem (6.11) and h ∈ T u 0 ,V T,C,M . Then, one hasd Ĵ(u 0 ), h = -Ω h(x) dx + +∞ j=0 e -2λ j T Ω u 0 (x)φ j (x) dx Ω h(x)φ j (x) dx = Ω h(x)ψ(x) dx,

Chapter 7 :

 7 Conclusions and perspectivesl'hypothèse de boule intérieure et exterieure a été utilisée à maintes reprises afin d'obtenir des estimations uniformes au bord du domaine. Peut-on se passer d'une telle hypothèse ? De même, peut on se passer de l'hypothèse qui parait assez technique traitant des distances entre les clusters de Ω + et entre Ω + et Ω -? Dans le cas où les réponses sont négatives, est ce possible d'exhiber un contre-exemple de domaine qui admet deux solutions stationnaires distinctes ?Etendre le résultat de continuité de la valeur propre du Laplacien fractionnaire d'ordre α par rapport à la distance entre deux sous-domaines en 0 pour α ≥1 2 et la dimension d ≥ 2 (Chapitre 4).Toujours dans le chapitre 3, nous avons établi la continuité de l'application(µ ∈ R + → λ α (]a, -µ[∪]µ, b[)) jusqu'à µ = 0 (avec a < 0 < b). Pour α <1 2 la preuve s'appuie sur un argument de compacité et de densité des fonctions à trace nulle en x = 0. Un tel argument se généralise aisément à la dimension supérieure. En revanche pour α ≥ 1 2 , il a fallu montrer que le point à l'interface x = 0, est une singularité effaçable. L'avantage de la dimension 1 est qu'il n'y a qu'une façon d'obtenir une telle interface. A partir de la dimension 2, l'interface obtenue est moins claire. Lorsque celle-ci est un hyper-plan (ou l'image par un difféomorphisme d'un hyperplan), je m'attends à ce que l'argument de prolongement par continuité soit robuste. Il y a cependant des détails techniques qui apparaissent. En effet, considérer une interface ayant une forme d'hyper-plan en dimension d ≥ 2 fait intervenir des « coins » dans les sous-domaines (i.e. les sous domaines sont Lipschitziens et non C 1 ). A priori, ces points nécessitent un traitement particulier. En revanche, je ne suis pas sûr que le résultat reste vrai lorsque le recollement est singulier (par exemple en dimension deux, deux disques qui se rapprochent pour obtenir Ω = B(-1, 1) ∪ B(1, 1)). Il ne m'apparait pas alors évident que la méthode soit robuste.Pousser plus loin l'étude de la dynamique de λα (]a, -µ[∪]µ, b[) lorsque µ → 0 et α → 1 avec a < 0 < b (Chapitre 4). L'idée serait de trouver deux fonctions (µ ∈ R + → α(µ)) et (µ ∈ R + → α(µ)) telles que pour toutes suites (µ n , α n ) → (0, 1) avec α n < α(µ n ),on ait λ αn (]a, -µ n [∪]µ n , b[) -→ n→+∞ λ 1 (]a, b[), tandis que pour toutes suites (µ n , α n ) → (0, 1) avec α n > α(µ n ), on ait λ α n (]a, -µ n [∪]µ n , b[) -→ n→+∞ min(λ 1 (]a, 0[), λ 1 (]0, b[).

Auteur :

  Alexis Léculier Directeurs de thèse : Sepideh Mirrahimi et Jean-Michel Roquejoffre Date de la soutenance : le 07 Juillet 2020 Discipline : Mathématiques appliquées Résumé Dans cette thèse nous nous intéressons à une étude mathématique qualitative de problèmes issus d'écologie et de biologie évolutive. Nous étudions l'influence d'une dispersion non-locale pour une espèce biologique vivant dans un environnement fragmenté. Plus précisément, dans une première partie, nous établissons un critère de survie pour une espèce biologique dont la dynamique est régie par une équation de Fisher-KPP fractionnaire dans un domaine fragmenté avec des conditions extérieures de Dirichlet. Ce critère repose sur le signe de la valeur propre principale de sous ensembles inclus dans le domaine. De plus, nous démontrons un résultat d'existence et d'unicité de la solution stationnaire d'une équation de Fisher-KPP dans des domaines fragmentés généraux appartenant à la classe des solutions positives, bornées et non-triviales. Dans le cas particulier d'un domaine périodique et fragmenté, nous établissons l'existence d'un phénomène d'invasion à vitesse exponentielle. Enfin, dans une seconde partie, nous considérons un modèle traitant d'une espèce biologique organisée phénotypiquement vivant dans un environnement fragmenté. Cette espèce est sujette à des mutations à petits effets phénotypiques ainsi qu'à une dispersion spatiale à la fois locale et non-locale. Nous démontrons l'émergence de traits phénotypiques dominants lorsque les mutations ont de petits effets. Mots clefs : Biologie évolutive, Noyau de diffusion non-conventionnelle, Phénomène d'accélération, Analyse asymptotique, Equations integro-différentielles paraboliques

  Théorème 1.4 (L.-Roquejoffre (Chapitre 4) ). Soient Ω 1 et Ω 2 deux sous-ensembles réguliers et bornés de R d . Soit Ω µ,e l'ensemble défini par

Chapter 1 :

 1 IntroductionNous supposerons que le trait phénotypique du parent est transmis à la descendance. Toutefois, il est possible que ce trait subisse de petites mutations que nous modélisons par un Laplacien -ε 2 ∂ θθ . Le terme R(x, θ) est un taux de croissance prenant en compte la natalité et la mortalité intrinsèque. A titre d'exemple, nous pouvons prendre comme taux de croissance

Chapter 1 :

 1 Introductiongiques ont relevé que la vitesse d'invasion accélérait au cours du temps. Cette accélération serait due à un changement phénotypique : l'allongement de la taille des jambes accélérant la diffusivité des individus.

  b 2 p .

	La variable p représente la proportion de moustiques infectés, u est le contrôle (i.e. la pro-
	portion de moustiques infectés relâchés dans Ω), b 1 et d 1 (respectivement b 2 et d 2 ) sont des
	taux de morts et de naissances d'insectes sains (respectivement infectés) et 1 κ est la capacité
	de charge du territoire. L'objectif est de minimiser la fonctionnelle

Plan du manuscrit

  a.e. et

	Privat, Strugarek et Vauchelet. Dans cet article, les auteurs considèrent une équation dif-
	férentielle ordinaire semblable à (1.10) (abscence de structure spatiale et avec un terme de
	réaction identique à (1.10)). Les auteurs démontrent que la méthode optimale de relâché
	est « bang-bang » (saturation des contraintes). Nous citons également [105] par Strugarek,
	Vauchelet et Zubelli qui traitent d'un problème organisé en temps et en espace. Cependant,
	ce travail s'intéresse à une stratégie de remplacement en temps long contrairement à ce qui
	a été considéré au CEMRACS.
	Pendant le CEMRACS, nous avons supposé que les temps de libération de u étaient dis-
	crets et donnés. Nous avons prouvé l'existence théorique d'un minimiseur à notre problème.
	Nous n'avons cependant pas réussi à le retrouver numériquement. De ce fait, nous avons
	réduit la complexité du problème en supposant que les lâchés de moustiques sont modélisés
	par un nombre fini de distributions Gaussiennes. Les jeux de données considérées semblent
	tendre vers un remplacement de la population non-infectée par une population infectée par
	Wolbachia.
	Le chapitre 2 est consacré à la preuve du Théorème 1.1 : l'étude de l'invasion d'un territoire
	périodique pour une dynamique dictée par une équation de type Fisher-KPP fractionnaire.
	Le chapitre 3 est dédié à la démonstration du Théorème 1.2 : l'étude de l'invasion d'un
	territoire périodique et fragmenté lorsque la dynamique est dictée par une équation de Fisher-
	KPP fractionnaire.
	Le chapitre 4 se concentre sur les résultats d'existence et d'unicité de solutions stationnaires
	aux équations de Fisher-KPP fractionnaires dans des domaines fragmentés (Théorèmes 1.3
	et 1.5).
	Le chapitre 5 se concentre sur l'étude de problèmes d'évolutions dans des domaines fragmentés
	(Théorème 1.6).
	Enfin le chapitre 6 présente les résultats sur le remplacement de moustiques Aedes aegypti
	sains par des moustiques infectés par la bactérie Wolbachia.
	La présentation des chapitres de cette thèse adopte la présentation d'articles scientifiques.
	En effet, chaque chapitre fait l'objet d'une publication (chapitres 2, 3 et 6) ou d'un dépot
	de soumission (chapitre 4). L'étude faite dans le chapitre 5 n'est pas encore aboutie (nous
	détaillons dans les perspectives de la thèse les points que nous aimerions aborder avant de
	soumettre nos résultats à une revue scientifique). Enfin, nous cloturons ce manuscrit par la
	présentation des conclusions et des perspectives. Les perspectives se composent de nouvelles
	questions ouvertes apparues dans le cadre de ce travail ou de prolongements naturels que
	j'aimerais aborder.
	T
		u(x, t)dxdt < C,
	0	Ω
	pour des données positives M et C. La dérivation du problème est détaillée dans le chapitre
	6 (et dans les références données à l'intérieur). Citons tout de même l'article [3] par Almeida,

E.

Chapter 2 A singular limit in a fractional reaction-diffusion equation with periodic coefficients

	Ce chapitre a fait l'objet d'une publication scientifique dans Communication in Mathematical
	Sciences (Volume 17).

  .34) Chapter 3 : Propagation in a fractional reaction-diffusion equation in a periodically hostile environment Equations (3.32) and (3.34) lead to(3.27). Moreover, if we define v(x, t) = e λt × p(x, t), we find the following system :

  .49) 3.5. The proof of Theorem 3.3Noticing that (λ 0 + 1) > 0, inserting (3.48) and (3.49) into (3.47), we conclude that for all ε < ε 0

  We prove the second point by using the super-solutionf M ε . Proof of 2. Let C > |λ 0 |d+2α and we fix ε = ε 0 such that Proposition 3.3 holds true. It implies that

	Chapter 3 : Propagation in a fractional reaction-diffusion equation in a periodically
	hostile environment
	Cm min Ων	φ 0

t + 1) + ε. If we define σ = 4 and t σ = 4 ε 3 + 1, we conclude that (3.41) holds true.

  One can easily observe that n is periodic, decreasing in time and converges uniformly to n + in the whole domain Ω as t → +∞. Thus there exists a times t 1 > 1 such that We split this part of the proof into two subparts, what happens on the boundary and what happens in the interior. The boundary estimates. Since n is decreasing in time and thanks to (3.60), we deduce that for all (x, t) ∈

	3.5. The proof of Theorem 3.3
	The conclusion follows.
	Proof that (3.59) holds true.
	(3.60)

+∞[, n(x, t) ≤ n(x, t). ∀(x, t) ∈ Ω×]t 1 , +∞[, n(x, t) -n + (x) ≤ µ.

  Under the previous hypothesis, for all ξ* ∈ [ξ 1 (Ω 1,2,0 ), min(ξ 1,1 , ξ 2,1 )], there exists a sequence (µ k , α k ) k∈N such that (µ k , α k ) -→

	(4.10)
	4.1.3 Existence and uniqueness of a steady solution to (4.1)

Theorem 4.1 and Proposition 4.1 imply the following statement, which has once again no equivalent in the case of local diffusion : Chapter 4 : Properties of steady states for a class of non-local Fisher-KPP equations in general domains Theorem 4.2. k→+∞ (0, 1) and ξ α k (Ω 1,2,µ k ) -→ k→+∞ ξ * .

  .41) Chapter 4 : Properties of steady states for a class of non-local Fisher-KPP equations in general domains Proof. Let k ∈ N and x k ∈ C +,k be such that sup x∈C

We consider Dirac measures since at the time-level of the study (namely, some generations), the release can be considered as instantaneous.

Remerciements

As a conclusion to this work, let us recall that, after establishing conditions on a domain Ω which ensure the existence and the uniqueness of the stationary state of the fractional Fisher-KPP equation, we focus on the principal eigenvalue of (-∆) α -Id in one dimension of domain composed by two bounded connected components. This study is strongly related to the issue of existence and uniqueness of the stationary state of the fractional Fisher-KPP equation.

The perspective are the followings. We would like to relax the hypothesis on the domain Ω. Indeed, rather than a minimal distance between two patches, we would like to assume that ∃λ 0 > 0 such that ∀R > 0, λ α (Ω + ∩ B(0, R)) > λ 0 .

We also expect to prove the continuity result on the principal eigenvalue in the multidimensional case. Finally, we would like to have a better understanding of the dynamic of λ α (] -a -µ, -µ[∪]µ, a + µ[) when (α, µ) → (1, 0). We provide here the proof of Lemma 4.1. Before giving the proof, we introduce a new notation :

Notation. For all φ ∈ H α 0 (Ω 1,2,µ ) and i ∈ {1, 2}, we will denote by φ i the function φ restricted to the set Ω i and extended by 0 outside Ω i i.e. φ i (x) = φ(x)1 Ω i (x).

We also denote by λ i the principal eigenvalue of (-∆) α -Id in Ω i with 0 exterior Dirichlet conditions.

Remark. For all φ ∈ H α 0 (Ω 1,2,µ ), we have

Proof. The aim of the proof is to prove that for µ large enough, there exists C > 0 such that

Chapter 5 : Adaptation to a heterogeneous environment with nonlocal dispersion

[Lipschitz bounds]

There exists C > 0 such that for all ε small enough,

(5.9)

3. [Bounds on ρ ε ] For all ε small enough, ρ ε is uniformly bounded in W (5.11) with a > 0.

Note that the combination of the local and the non-local diffusion terms makes the establishment of such regularity estimates non-standard (see for instance [START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations : a general convergence result[END_REF] and [88] where such types of estimates where obtained for related models with a local diffusion term).

Remark. In 1. of Theorem 5.3, the interval I can be at the boundary of

Outline of the paper

In section 5.2, we provide some preliminary results and the existence of n ε by proving Theorem 5.1. Next, in section 5.3, we prove the regularity results given by Theorem 5.3. Section 5.4 is devoted to the proof of Theorem 5.2. We provide some numerical simulations and discuss the types of outcomes that may be obtained for the asymptotic populations density. The constants c and C are positive constants independent of the choice of ε and may change from line to line when there is no confusion possible.

Preliminary results

First, we establish a Hopf Lemma. It is obtained by a classical argument but for the sake of completeness and because of the presence of the less classical non-local operator L, we provide the proof. Next, we verify the existence of λ(θ, ρ) and then the existence of a non-trivial bounded solution n ε . We close this section with the proof of the convergence of ρ ε .

A Hopf Lemma

In this section we prove the following Hopf Lemma Lemma 5.2 (Hopf Lemma). Let u be a smooth function defined on Ω such that

(5.13)

Preliminary results

The proof is in the spirit of the classical proof of the Hopf Lemma (see [START_REF] Evans | Partial differential equations[END_REF] p.250).

Proof. Up to a scaling, there is no loss of generality if we assume that B(0, 1) ⊂ Ω and x 0 ∈ ∂B(0, 1). Next, we define

for λ a positive constant. We underline that v(x) = e -3 4 λ -1 in B(0, 1 2 ). Next, we claim that by taking λ large enough, for all x ∈ B(0, 1)\B(0, 3 4 ) there holds

(5.14) The first inequality of (5.14) follows a straightforward computation. For the second inequality, according to the assumption (H3), we have lim inf λ→+∞ B(0,1)

Therefore, if λ is large enough, (5.14) holds true.

Next, we claim that if n is not constant, the minimum can not be reached in the interior of Ω. Otherwise, we deduce the existence of x 1 ∈ Ω such that n(x 1 ) = min x∈Ω n < 0. Since c is positive, we have -∂ xx n(x 1 ) ≤ 0, Ln(x 1 ) < 0 and c(x 1 )n(x 1 ) ≤ 0.

Therefore, we deduce that -∂ xx n(x 1 ) + Ln(x 1 ) + c(x 1 )n(x 1 ) < 0. This is in contradiction with the assumption (5.12). We deduce that min x∈∂B(0, 3 4 )

n(x 0 ) -n(x) < 0. Next, taking ε small enough, there holds that

Since v = 0 on ∂B(0, 1) and by definition of x 0 , it follows

Moreover, according to (5.12) and (5.14) we have for all x ∈ B(0, 1)\B(0, 3 4 ) that

We deduce thanks to the maximum principle that n(x 0 ) -n(x) -εv(x) ≤ 0. We conclude that

Appendix -Proof of Lemma 5.1

The proof of Lemma 5.1 follows essentially the steps of the proof of the convergence of u ε (i.e. second item of Theorem 5.2). Therefore, we will only emphasize the differences between the two proofs. We made the choice to provide the proof of the convergence of u ε rather than Chapter 5 : Adaptation to a heterogeneous environment with nonlocal dispersion the convergence of µ ε because it is the result that motivated the current study.

Proof of Lemma 5.1. We recall the equation satisfied by µ ε and ξ

(5.

2)

The existence of ξ ε is ensured by the Krein-Rutman Theorem. Moreover, according to the Krein-Rutman Theorem, the sign of ξ ε is constant. Therefore, we consider that ξ ε > 0, ξ ε L 2 = 1 and we define

Next, we prove that µ ε is bounded from below and above respectively bysup R andinf R.

First, we focus on the upper bound. Let (x, θ) ∈ Ω×[-A, A] be such that sup

From (5.2), we deduce that

we conclude with a reflective argument and the same computations as in the previous case. In any case, for all ε > 0 we have

Next, we focus on the lower bound. Let (x, θ) ∈ Ω × [-A, A] be such that inf

With similar arguments as for the upper bound, we deduce that

Therefore, µ ε is uniformly bounded from below and above thus µ ε converges along subsequences to µ.

Next, as we have established Lipschitz and uniform bounds on u ε , we can prove that there exists a constant C > 0 such that

Therefore, we deduce that v ε converges along subsequences to v. Moreover, with similar computations as in the proof of the second item of Theorem 5.2, we deduce that v is a viscosity solution of

We postpone the proof of this claim to the end of this paragraph. Thanks to (5.38) and (5.39) we deduce that

It follows that ∂ θ v(θ m ) = 0 and -λ(θ m , 0) + µ = 0 = max (-λ(θ, 0) + µ). We deduce thanks to (H4) that

We conclude that λ(θ 0 , 0) = µ.

We finish the proof by remarking that the previous convergence result holds for any subsequence of µ ε . Therefore, we conclude that

It remains to prove (5.39). Let ψ θ µ be the principal eigenfunction associated to the principal eigenvalue of λ(θ, -µ) with µ a constant

µ > 0 and by the uniqueness of the positive eigenfunction of -∂ xx + L -R(•, θ) (up to a multiplication by a scalar), we deduce that λ(θ, -µ) + µ = λ(θ, 0).

Chapter 6

Optimal release of mosquitoes to control dengue transmission

Ce chapitre a été accepté pour être publié dans Esaim Proceding. Il a été co-écrit avec Luis Almeida, Antoine Haddon, Claire Kermorvant, Yannick Privat, Martin Strugarek, Nicolas Vauchelet et Jorge Zubelli.

Chapter 6 : Optimal release of mosquitoes to control dengue transmission L 2 ([0, T ], L 2 (Ω)) and weakly in L 2 ([0, T ], H 1 (Ω)). Passing to the limit in (6.7) yields that p ∞ is a weak solution to

It is standard that any solution to this bistable reaction-diffusion equation is continuous in time.

Introducing

A such equality holds true by passing to the limit as n → +∞ in the variational formulation on p n and by using adapted test-functions belonging to

. Indeed, this is a consequence of the continuity and convexity since one has

Step 2 : Conclusion. Let us first show that u ∞ belongs to V T,C,M . Since the derivative of G is 1/g which is positive, G is increasing and therefore, one has 0 ≤ u ∞ ≤ m a.e. in Ω. For the integral condition (namely, Ω u ≤ C), let us distinguish between two cases : Case 1 : if m|Ω| ≤ C, the conclusion follows immediately.

Case 2 : if m|Ω| > C, let us use that G is, as aforementioned, lower semi-continuous for the weak-star topology of L ∞ . Thus, we deduce that

It follows that u ∞ belongs to V T,C,M and one concludes by using the Fatou Lemma :

We finally infer that u ∞ solves Problem (P reduced ).

Remark. The uniqueness issue remains open, even for simple domain. It is likely that symmetries of the release domain play an important role.

It is interesting to notice that, in a very particular case, we have an explicit expression of the minimizer for this problem. Chapter 6 : Optimal release of mosquitoes to control dengue transmission Let us investigate the optimality of constant functions. To this aim, notice that the functional Ĵ is strictly convex 2 . It follows that the optimality conditions (6.13) are at the same time necessary and sufficient and that Problem (6.11) has a unique solution.

Let u 0 be an admissible constant function for Problem (P toy ). Then, u 0 ∈ [0, M ] whenever M ≤ C/|Ω| and u 0 ∈ [0, C/|Ω|] elsewhere.

Furthermore, if u 0 is constant, then,

Let us now investigate each case separately. If u 0 = 0, then, from the complementarity condition, ξ = 0 and ψ(x) = -1 which is in contradiction with the optimality conditions above. Let us assume that u 0 = 0.

- The conclusion follows.

Gaussian Releases

From a practical point of view, not all controls u ∈ V T,C,M correspond to a release that could actually be conducted, as for example the constant solution of the toy problem of the previous section. To guarantee a solution that could be implemented, we restrict here the admissible controls to more accurately model the way mosquitoes are released in practice.

We thus consider that there are K ∈ N simultaneous releases and that each one results in a Gaussian distribution of mosquitoes centered around the position of the release x k ∈ Ω for k = 1, ..., K. Then, the feasible controls are of the form

where the constants m and σ are chosen such that u K (•, x 1 , ..., x K ) ∈ V T,C,M . In particular, we choose to saturate the constraint on the total number of mosquitoes released, i.e. we take

2. The convexity results from the convexity of the square function combined with the linearity of u 0 → p(T, •). Furthermore,

and vanishes if, and only if, Ω h(x)φ j (x) dx = 0 for all j, meaning that h = 0 since (φ j ) j≥1 is a Hilbert basis of L 2 (Ω). The strict convexity of Ĵ follows.

Chapter 6 : Optimal release of mosquitoes to control dengue transmission

Remark. Because of Proposition 6.1, we were interested in the case M > C

|Ω| . In addition, we have fixed C such that the constant solution u = C

|Ω| leads to extinction (as T tends to +∞) but there exists R ∈]0, C πM [ such that the function u(x) = M × 1 B(0,R) (x) belongs to V T,C,M and leads to invasion (as T tends to +∞).

We now present numerical simulations for the parameters given in Table 6.1. The birth and death rates are given per day, whereas the unit of the carrying capacity is per m 2 and the diffusion coefficient is given per m 2 per day. The numerical values are taken from [START_REF] Almeida | Optimal releases for population replacement strategies : application to wolbachia[END_REF] and references therein. We consider a square domain of 1 hectare, a final time of 200 days and we set the total amount of mosquitoes released such that C < G(θ)|Ω|. In Figure 6.1 we show the control u K (•, x 1 , ..., x K ) for K = 3, 4, 5, 6 releases and for each case the same total amount of mosquitoes is released. For the case of 6 releases we display in Figure 6.2 the time dynamics of the proportion of infected mosquitoes p(t, •). As expected, it leads to the total invasion of the domain.

1.12 1.12 0.27 0.36 6 • 10 -2 2.5

Table 6.

-Model parameters

Our simulations seem to be very sensitive on the initial data given to Ipopt. Indeed, for most choices of initial datum in the optimization algorithm, the best solution provided by Ipopt has the "same shape" as the initial datum (more precisely, by assimilating the Gaussian releases to domains, the optimal solution seems to have the same number of connected components as the initialization). Heuristically, this suggest that the function J K we aim at minimizing mainly penalizes a lot the final time (here T = 200) and does not take into account what occurs at intermediate times. Since most of the initial data lead to invasion with the set of parameter we considered (in other words, the global minimum of J K is almost reached), the considered interior points algorithm (via the software Ipopt) tries some new configurations relatively close to the initial data to find out that it was already an "almost" global minimum of J K . We have tried unsuccessfully to make tests with a lower final time, and the results are similar. In a future work, to avoid such bad boundary effects, we foresee to consider another functional J K taking into account not only the final time but also several intermediate times.

Conclusion

We investigate in this work the optimization of the release of Wolbachia-infected mosquitoes into a host population in the aim to replace the wild population by a Wolbachia-infected population unable to transmit several diseases to human. To conduct this study, we first reduce the optimal problem under investigation by assuming that the time distribution is given. Then we obtain existence of a minimum for this latter problem. Finally, reducing again the control problem by considering that the releases are modeled by Gaussian distributions, some numerical computations are performed.

Optimization strategies for release protocols of mosquitoes have been investigated by several authors [START_REF] Thomé | Optimal control of aedes aegypti mosquitoes by the sterile insect technique and insecticide[END_REF][START_REF] Campo-Duarte | Optimal control approach for establishing wmelpop wolbachia infection among wild aedes aegypti populations[END_REF][START_REF] Bliman | Feedback control principles for biological control of dengue vectors[END_REF]. However, in these papers, only the time optimization of the releases is investigated. Up to our knowledge, this work is the first attempt in optimizing spatially the releases, which is of great interest for experiments in the field. The preliminary results obtained in this paper should be continued. In particular, the optimality conditions for the system (P K ) should be studied in a future work in the aim to find properties of the optimal solution. The numerical simulations should also be continued to have a better representation of what is observed in the field.

Appendix -Reduction of system (6.1) For the sake of completeness and for reader facility, we explain briefly in this appendix how to reduce system (6.1) to system (6.2). We will not provide all the details of this reduction but only the main steps. We refer to [START_REF] Strugarek | Reduction to a single closed equation for 2-by-2 reaction-diffusion systems of lotka-volterra type[END_REF] and [START_REF] Almeida | Optimal releases for population replacement strategies : application to wolbachia[END_REF]Section 2.3] for the interested reader. The starting point is to introduce a small parameter 0 < ε 1 modeling the ratio of the fertility on the death rate. Indeed for mosquitoes population, the fertility is large compared to death rates. System (6.1) reads then

As ε → 0, we expect from this system that n ε 1 + n ε 2 → κ. Hence we introduce the quantity

the proportion of infected mosquitoes. From

Chapter 6 : Optimal release of mosquitoes to control dengue transmission straightforward computations, we deduce the system satisfied by (n ε , p ε ) :

where we use the notations

> 0. Assuming that the sequences (n ε ) ε and (p ε ) ε admit limits denoted n and p respectively, we deduce from the first equation that, formally,

Passing into the limit into the equation satisfied by p, we get

Injecting the expression of n (6.15) into this latter equation, we recover the equation

with f and g defined in (6.3)

Chapter 7

Conclusions and perspectives

A. Conclusion

Dans cette thèse nous nous sommes intéressés à une étude mathématique de problèmes issus d'écologie et biologie évolutive. En effet, nous avons étudié l'influence d'une dispersion non-locale pour une espèce biologique vivant dans un domaine fragmenté. Plus précisément, nous avons établi un critère de survie pour une espèce biologique dont la dynamique est régie par une équation de Fisher-KPP fractionnaire dans un domaine fragmenté avec des conditions extérieures de Dirichlet. Ce critère repose sur le signe de la valeur propre principale de sous-ensembles inclus dans le domaine. De plus, nous avons démontré l'existence et l'unicité de la solution positive et bornée de l'équation de Fisher-KPP fractionnaire à l'équilibre . Dans le cas d'un domaine périodique et fragmenté, nous avons démontré qu'un phénomène d'invasion à vitesse exponentielle a lieu. Enfin, dans une seconde partie, nous avons considéré une espèce biologique organisée phénotypiquement et sujette à de petites mutations du phénotype et à une dispersion spaciale à la fois locale et non-locale. Nous avons montré l'apparition de traits dominants lorsque les mutations deviennent petites.

B. Perspectives

Je présente ici les prolongements naturels et les questions qui sont apparus à l'issu de ce travail. Dans le titre de chacune des perspectives, je mentionnerai le chapitre impliqué.

Etendre les résultats d'existence et d'unicité de la solution stationnaire de l'équation de Fisher-KPP fractionnaire au cadre de la valeur propre principale généralisée (Chapitre 4).

Une quantité qui apparaît naturelle à considérer est la valeur propre principale généralisée introduite par Berestycki et Rossi (voir [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] et [START_REF] Berestycki | Persistence criteria for populations with non-local dispersion[END_REF]). Sous le spectre de cette valeur propre généralisée, il est natuel de se demander quelles sont les hypothèses minimales pour établir un résultat d'existence et d'unicité via des estimations précises comme établi dans le Chapitre 4. Une condition nécéssaire est naturellement de considérer une valeur propre principale généralisée λ α (Ω) strictement négative. La question est : est-ce suffisant ? Dans le chapitre 4,