Part I of the thesis has been devoted to a rigorous analysis of the asymptotic location of the level sets of the solution to two different problems.
**In** Chapter 1, we have applied our method on a Fisher-KPP model **in** periodic media **with** **fractional** **diffusion**. We have been able to construct precise explicit subsolutions and supersolutions. Thus, we have proved that the transition between the unstable state and the stable one occurs exponentially **fast** **in** time, and we have obtained the precise exponent that appears **in** this exponential speed of **propagation**. This has led to the proof of the convergence of the solution to its stationary state on a set that expands **with** an exponential **in** time speed. Numerical simulations have been carried out to understand the dependence of the speed of **propagation** on the initial condition at lower order **in** time. Although the different numerical results, done for the homogeneous model **in** dimension two, have given a precise idea of what is happening, a mathematical proof should be undertaken. Indeed, it seems that there is a symmetrisation of the solution, **in** the sense of Jones **in** [ 77 ]. Proving this observation requires an estimate of the gradient of the solution, which is not done **in** this thesis. This geometric result of symmetrisation could also be studied **in** periodic media. Moreover, as suggested by numerical investigations, it seems that the diffusive term of the **reaction**-**diffusion** equation only plays a role for small times. It would be interesting to show it rigorously. Finally, one could think of further perspectives. A first one consists **in** getting similar results for integro-differential **equations**, and thus obtaining more precise asymptotics as the ones proved **in** [ 65 ]. More general heterogeneous media might also be analysed, media for which the notion of generalised eigenvalues is needed.

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It now remains to obtain a similar bound for a given 4B0 1 < λ < 1. Such estimate can be obtained by redoing the argument **in** Section 3.
Acknowledgement. J.Coville carried out this work **in** the framework of Archim`ede Labex (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government program managed by the French National Research Agency (ANR). He has also received funding from the ANR DEFI project NONLOCAL (ANR-14-CE25-0013). C. Gui is supported by NSF grants DMS-1601885 and DMS-1901914 and Simons Foundation Award 617072. M. Zhao is supported by the National Natural Science Foundation of China No.11801404.

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Such a **reaction**-**diffusion** equation is often used **in** ecology and population dynamics to model spatial **propagation** phenomena [ 1 ]. **In** this context, the unknown function u denotes the density of some species, and the parameter α stands for its per capita growth rate. The function χ stands for the motility of individuals, and therefore the heterogeneity means that the motility depends on local environmental conditions. **In** the past decade, there has been several series of works dedicated to the study of a variation of ( 1.1 ) **with** constant motility (say χ ≡ 1 to fix the ideas) but **with** shifting growth rates, that typically reads

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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 longtime behavior of the solution n. We demonstrate that **in** the set n (x, t) | |x| < e d+2α |λ1|t

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the unique positive zero of f . **In** the moving frame **with** speed c to the right, a traveling front u(t, x) = φ(x − ct) is invariant **in** time. Such fronts are known to exist if and only if c ≥ 2pD 0 f 0 (0) and they are stable **with** respect to some natural classes of perturbations,
see e.g. [8, 28, 29, 42, 46]. For heterogeneous **equations** such as (1.1) **in** periodic media, standard traveling fronts do not exist **in** general and the notion of fronts is replaced by the more general one of pulsating fronts [45]. For (1.1), **in** the case of the existence (and uniqueness) of a periodic positive steady state p(x) of (2.2), a pulsating front connecting 0 and p is a solution of the type u(t, x) = φ(x · e − ct, x) **with** c 6= 0, e is a unit vector (e ∈ S N −1 ),

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Bibliographical remarks Existence of travelling waves for **reaction**-**diffusion** **equations** have been studied since the well known works of Kolmogorov, Petrovskii and Piskunov [70]. It has been generalised **in** the late seventies by Aronson and Weinberger [2] and Fife and McLeod [44] for the specific problem of a one dimensional bistable wave. Since these pioneering works, an important effort has been done to study front **propagation** and transition waves **in** inhomogeneous **reaction**-**diffusion** **equations** and it is impossible to cite them all. For our purpose let us mention the paper of Berestycki and Nirenberg [20] where they showed existence of travelling waves **in** cylinders **with** an inhomogeneous advection field. The notion of transition fronts, or invasion fronts, has been introduced **in** [76] and precisely defined and studied **in** [12]. The specific problem of transition waves when heterogeneities come from the boundary have received a much more recent attention. **In** [32], the authors showed that for a bistable nonlinearity, if the domain Ω is a succession of two half rectangular cylinders, one can find conditions on them width for the **propagation** to be blocked. The problem of transition fronts for exterior domains has been treated **in** [14] where the authors devised geometrical conditions for the invasion to be complete. Finally, as already said, cylinder **with** varying cross sections has just been studied **in** [5].

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Generally speaking, for **diffusion** and **reaction** coefficients a(x) and f (x, u) satisfying (1.2), the questions of the existence of front-like solutions connecting 0 and 1 and of the existence of **propagation** speeds are still open, even under the additional assumption (1.11). Based on the work **in** this present paper and the companion paper [16], one can conclude that the spatial period L plays an important role **in** answering these questions. Namely, under various assumptions, there are transition fronts propagating **with** nonzero speeds (pulsating fronts **with** nonzero speeds) when L is small, while there are both stationary and non-stationary transition fronts propagating **with** zero speed when L is large. However, so far there is no explicit condition to guarantee the existence of transition fronts **in** general, and no example is known to show their non-existence.

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1 Introduction and main results
1.1 Motivation: models on climate change
**Reaction** **diffusion** problems are often used to model the evolution of biological species. **In** 1937, Kolmogorov, Petrovskii and Piskunov **in** [15], Fisher **in** [10] used **reaction** diffu- sion to investigate the **propagation** of a favourable gene **in** a population. One of the main notions introduced **in** [15, 10] is the notion of travelling waves, i.e solution of the form u(t, x) = U (x − ct) for x ∈ R, t > 0 and some constant c ∈ R. Since then a lot of papers have been dedicated to **reaction** **diffusion** **equations** and travelling waves **in** settings mod- elling all sorts of phenomena **in** biology.

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Existence results for transition fronts
Several articles study the existence of transition fronts **in** non homogeneous settings. Shen [106, 107], Nadin and Rossi [92] and Berestycki and Hamel [12] prove the existence of transition fronts (almost planar) when f is heterogeneous **with** respect to time **in** the bistable, monostable and KPP case. Vakulenko and Volpert [117] also proved the existence of another type of generalised transition fronts for perturbed bistable **reaction** **diffusion** systems. Mellet, Roquejoffre and Sire [86], Nolen and Ryzhik [97], Mellet et al [85] and Zlatoˇs [126] study the existence and the stability of transition fronts for combustion type heterogeneities. **In** all these papers it has been proved that for combustion like **equations** **in** dimension 1, transition fronts always exist, are unique and stable. Nolen et al [96] and Zlatos [125] build a multitude of transition fronts for non homogeneous KPP **equations** using the linearisation around 0. **In** a recent paper Tuo, Zhu and Zlatos [112] use this same approach to built transition fronts of monostable type **equations**. Thus there exist several heterogeneous frameworks where the existence of transitions fronts is proved, nevertheless this existence is questionned by Nolen et al [96], for some class of monostable type **reaction** **diffusion** **equations** **in** dimension 1. Indeed they show that when the nonlinearity f has a strongly localised heterogeneity then there exist some situations where transition fronts do not exist. This last result show that the existence of transition fronts **in** heterogeneous settings is not guarantee for monostable equation **in** dimension 1. Moreover Lewis and Keener [73] highlight the existence of blocking phenomena for **reaction** **diffusion** **equations** **in** dimension 1 **in** media **with** heterogeneous excitability. The same kind of phenomena is observed by Grindrod and Lewis [52] **in** dimension 2 **in** domains **with** varying diameters, studying an eikonal approximation as the limit of a **propagation** model. Lou, Matano and Nakamura [79, 74] study the existence of front-like solutions through the notion of recurrent waves, for curvature dependent motions of plane curves and prove that the speed of **propagation** is slowed down **in** the presence of undulations. All these papers thus indicates that **in** a heterogeneous framework, a multitude of phenomena can be observed (non existence of transition fronts, slowing down or failure of the **propagation**...). Chapters 1 and 3 of this thesis come up **in** this context. We study the existence of transition fronts for heterogeneous bistable **reaction** **diffusion** **equations** where the problem is set **in** a domain Ω Ĺ R n .

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The setup we consider is **in** striking contrast **with** previous available results **in** homogenization of integro-differential problems. **In** [2, 3], Arisawa analyzed periodic homogenization for equa- tions **with** purely Lévy operators, and rather light interaction between the slow and **fast** variable. Homogenization results for nonlocal **equations** **with** variational structure have been recently stud- ied **in** [26, 30]. This paper is closely related to [32], where periodic homogenization for uniformly elliptic Bellman-Isaacs **equations** was obtained by Schwab. Later on these results were extended to stochastic homogenization **in** [31]. The arguments **in** both papers are completely different than ours, and are based on the obstacle problem method, previously introduced **in** [18, 19] **in** order to establish stochastic homogenization and rates of convergence for fully nonlinear, uniformly ellip- tic partial differential **equations**. Periodic homogenization for nonlocal Hamilton-Jacobi **equations** **with** coercive gradient terms has been addressed **in** [6], where techniques similar to ours appear, except that here we cannot rely on the gradient coercivity.

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s (x, 0) = µ 0 (φ(x)), **with** µ 0 a periodic function and φ a smooth increasing function such
that φ ′ (x) → 0 as x → +∞. If φ increases sufficiently **fast**, then these authors proved that
w ∗ = w ∗∗ and it is possible to compute this speed. We refer to [17] for a precise definition
of “sufficiently **fast**”. An example of such a φ is φ(x) = ln(1 + |x|) α , α > 1. This result is proved by constructing appropriate test-functions **in** the definition of the generalized principal eigenvalues (10). As **in** the random stationary ergodic setting, it is necessary to consider test- functions which are not necessarily bounded but satisfy 1 x ln φ(x) → 0 as |x| → +∞. When φ increases slowly for example, when φ(x) = ln(1 + |x|) α , α ∈ (0, 1) , then it was proved **in** [17] that w ∗ = 2 √ min R µ 0 and w ∗ = 2√max R µ 0 , which provides an example of coefficients

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Mots-clés : réaction-**diffusion**, ondes progressives, fronts, **propagation**, vitesse.
Abstract : the aim of the thesis is the study of enhancement of **propagation** **in** **reaction**- **diffusion** **equations**, through a new mechanism involving a line **with** **fast** **diffusion**. We answer the question of the influence of such a coupling **with** strong **diffusion** on **propagation** by gen- eralizing a result of Berestycki, Roquejoffre and Rossi (2013). The model under study was proposed to give a mathematical understanding of the influence of transportation networks on biological invasions. The first chapter shows existence and uniqueness of travelling waves solu- tions **with** a continuation method. The transition occurs through a singular perturbation – new **in** this context – connecting the system **with** a Wentzell boundary value problem. The second chapter is concerned **with** the speed of the waves : we show that it grows as the square root of the diffusivity on the line, generalizing and showing the robustness of the result by Berestycki, Roquejoffre and Rossi. Moreover, the growth ratio is characterized as the unique admissible velocity for the waves of an hypoelliptic a priori degenerate system. The last part is about the dynamics : we show that the waves attract a large class of initial data. **In** particular, we shed light on a new mechanism of attraction which enables the waves to attract initial data **with** size independent of the diffusivity on the line : this is a new result, **in** the sense than usually, enhancement of **propagation** has to be paid by strengthening the assumptions on the initial data for invasion to happen.

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the condition was algebraic and not asymptotic), there exists indeed such a pulsating front. While the forthcoming main ideas might be generalizable to systems **with** periodic **diffusion** and interspecific competition rates, an existence result is lacking. Therefore we naturally stick **with** the aforementioned system. Let us recall moreover that the fully heterogeneous problem (non-periodic non-constant coefficients) is, as far as we know, still completely open at this time. Let us recall as well that several important results about scalar **reaction**–**diffusion** **equations** **in** periodic media have been established recently (about “KPP”-type, see [16, 17, 117, 118, 121]; about “ignition”-type and monostable non-linearities, see [14]; about bistable non-linearities, see [57, 56, 144]). The first author used extensively the results about “KPP”-type **equations** **in** [Gir17]. **In** the forthcoming work, we will use the whole collection of results. Especially, we will use several times, **in** slightly different contexts, the sliding method of Berestycki–Hamel [14].

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The proof of Theorem 1.1 consists **in** showing that the solution u of (1.1) and (1.5) **with** this function f satisfies the desired conclusions listed **in** Theorem 1.1. For the proof of (1.7), the strategy is to estimate u from above and below at some larger and larger times and at some further and further points. More precisely, on the one hand, the fact that µ takes its maximal value µ + on the large intervals [x n + 1, y n − 1] lead to the existence of time-growing bumps on these spatial intervals, whereas, on the other hand, the fact that µ takes its minimal value µ − on the large intervals [y n , x n+1 ] slows down the **propagation** **in** some time intervals when the growing bumps are still negligible (see the following lemmas for further details and Figure 1 below).

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L , + ∞) have been given
**in** [25, 28, 29].
Uniqueness of the speed and further qualitative properties of pulsating fronts
**In** this paper, we first discuss the question of the uniqueness of the speed of pulsating fronts for the bistable equation (1.1), under assumptions (1.2) and (1.3), as well as the monotonicity and the uniqueness of pulsating fronts **with** nonzero speed. **In** [9], qualitative properties of tran- sition fronts, which are more general than pulsating fronts, were investigated **in** unstructured heterogeneous media. Some results of [9] can be applied to the pulsating fronts of the periodic equation (1.1), provided that the **propagation** speeds are not zero. More precisely, [9, Theo- rems 1.11 and 1.14] (see also [47] for x-independent reactions f = f (u)) lead to the uniqueness of the speed and of the fronts (up to shifts **in** time) **in** the class of pulsating fronts **with** nonzero speed, as well as the negativity of ∂ ξ φ for a pulsating front φ(x −ct, x/L) **with** c 6= 0 and ξ = x−ct.

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∂ t u − ∆u = f (t, u), x ∈ R N , t ∈ R,
where f = f (t, u) is a KPP monostable nonlinearity which depends **in** a general way on t ∈ R. A typical f which satisfies our hypotheses is f (t, u) = µ(t)u(1 − u), **with** µ ∈ L ∞ (R) such that ess inf t∈R µ(t) > 0. We first prove the existence of generalized transition waves (recently defined **in** [4, 22]) for a given class of speeds. As an appli- cation of this result, we obtain the existence of random transition waves when f is a random stationary ergodic function **with** respect to t ∈ R. Lastly, we prove some spreading properties for the solution of the Cauchy problem.

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c ∗∗ = f ′ (0)/2α. All these results will be formally established **in** Chapter 1. Additionally we
recall the earlier work **in** the case α ∈ (0, 1) by Berestycki, Roquejoffre and Rossi [10], where it is proved that there is invasion of the unstable state by the stable one, also **in** [10], the authors derive a class of integro-differential **reaction**-**diffusion** **equations** from simple principles. They then prove an approximation result for the first eigenvalue of linear integro-differential opera- tors of the **fractional** **diffusion** type, they also prove the convergence of solutions of **fractional** evolution problem to the steady state solution when the time tends to infinity. For a large class of nonlinearities, Engler [25] has proved that the invasion has unbounded speed. For another type of integro-differential **equations** Garnier [27] also establishes that the position of the level sets moves exponentially **in** time for algebraically decaying dispersal kernels. And **in** a recent paper Stan and V´azquez [42] study the **propagation** properties of nonnegative and bounded solutions of the class of **reaction**-**diffusion** **equations** **with** nonlinear **fractional** **diffusion**.

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(Dated: September 4, 2016)
For **reaction**-**diffusion** processes **with** at most bimolecular reactants, we derive well-behaved, numerically-tractable, exact Langevin **equations** that govern a stochastic variable related to the response field **in** field theory. Using duality relations, we show how the particle number and other quantities of interest can be computed. Our work clarifies long-standing conceptual issues encoun- tered **in** field theoretical approaches and paves the way to systematic numerical and theoretical analyses of **reaction**-**diffusion** problems.

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These models are also amenable to modeling invasion phenomena, because (when posed on the whole space) they typically have solutions called travelling waves **in** the form y(x − ct) for certain speeds c, linking the states 0 and 1, see the pioneering work [ 10 ].
For such problems, it is thus a requirement for the solution to satisfy y > 0, a condition which is fulfilled **with** non-negative Dirichlet boundary conditions. We might consider using controls that are above 1, taking into account the possibility for releases at 0 or L to be above the capacity of the system.