Hélène Leman

Convergence of an innite dimensional stochastic process to a spatially structured trait substitution sequence

Keywords: structured population, birth and death diusion process, large deviations studies, exit time, branching diusion processes, nonlinear reaction diusion equations, weak topology, Trait Substitution Sequence Mathematics Subject Classication (2000) 60J25, 60J60 (Primary), 35K57, 92D25 (Secondary)

We consider an individual-based spatially structured population for Darwinian evolution in an asexual population. The individuals move randomly on a bounded continuous space according to a reected brownian motion. The dynamics involves also a birth rate, a density-dependent logistic death rate and a probability of mutation at each birth event. We study the convergence of the microscopic process in a long time scale when the population size grows to +∞ and the mutation probability decreases to 0. We prove the convergence towards a jump process that jumps in the innite dimensional space containing the monomorphic stable spatial distributions. The proof requires specic studies of the microscopic model. First, we study the extinction time of the branching diusion processes that approximate small size populations. Then, we examine the upper bound of large deviation principle around the deterministic large population limit of the microscopic process. Finally, we nd a lower bound on the exit time of a neighborhood of a stationary spatial distribution.

Introduction

The spatial aspect is an important issue in ecology [START_REF] Tilman | Spatial ecology: the role of space in population dynamics and interspecic interactions[END_REF][START_REF] Durrett | Spatial aspects of interspecic competition[END_REF]. The inuence of the heterogeneity of the environment on the phenotypic evolution has been explored for a long time [START_REF] Endler | Geographic variation, speciation, and clines[END_REF][START_REF] Futuyma | The evolution of ecological specialization[END_REF][START_REF] Kassen | The experimental evolution of specialists, generalists, and the maintenance of diversity[END_REF]. For example, the emergence of phenotypic H. Leman CMAP, Ecole Polytechnique, UMR 7641, Université Paris-Saclay, route de Saclay, 91128 Palaiseau Cedex-France E-mail: helene.leman@polytechnique.edu clusters under a heterogeneous space has been extensively studied [START_REF] Doebeli | Speciation along environmental gradients[END_REF][START_REF] Polechová | Speciation through competition: a critical review[END_REF][START_REF] Leimar | Evolution of phenotypic clusters through competition and local adaptation along an environmental gradient[END_REF].

In [START_REF] Doebeli | Speciation along environmental gradients[END_REF][START_REF] Leimar | Evolution of phenotypic clusters through competition and local adaptation along an environmental gradient[END_REF], the authors suggest that clustering and aggregation of individuals can be a consequence of the spatial competition between individuals and generate structured populations based on isolated patches. In [START_REF] Polechová | Speciation through competition: a critical review[END_REF], the authors draw attention to the inuence of the boundary of the spatial environment.

The sensibility to heterogeneously distributed resources is also a key point to study the spatial dynamics of population [START_REF] Grant | Unpredictable evolution in a 30-year study of darwin's nches[END_REF]. In this context, the eect of a spatial structure on the evolution of a population is fundamental.

In this paper, we are concerned with the interplay between spatial structure and Darwinian evolution under three main biological assumptions : rare mutations, large population size and the impossibility of coexistence of two traits for a long time scale. We use an individual-based model rst introduced by [START_REF] Champagnat | Invasion and adaptative evolution for individual-based spatially structured populations[END_REF] which describes a spatially and phenotypically structured asexual population. Our main result describes the convergence of the microscopic model in the mutation scale to a jump process that jumps in an innite dimensional space containing the spatial proles of the population. This result is correlated with several works on adaptive dynamics and in particular with the model of Trait Substitution Sequence (TSS) introduced by Metz and al. [START_REF] Jaj | Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction[END_REF]. The TSS model describes the succession of invading advantageous phenotypic traits as a jump Markov process in the space of phenotypic traits. The derivation of the TSS model from the microscopic model has been rigorously proved by [START_REF] Champagnat | A microscopic interpretation for adaptive dynamics trait substitution sequence models[END_REF] and then generalized in dierent contexts, as aged-structured population [START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF], multi-resources chemostat population [START_REF] Champagnat | Adaptation in a stochastic multi-resources chemostat model[END_REF] or prey-predator population [START_REF] Costa | Stochastic eco-evolutionary model of a prey-predator community[END_REF]. Our paper generalized it in a spatial context. The spatial structured of the population adds new non trivial diculties. Our processes have values in innite dimensional spaces and the trajectories followed by the individuals in those spaces are random, contrary to [START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF] where the age structure is deterministic.

The dynamics of the process is driven by a birth and death diusion process, in which the motion, birth, mutation and death of each individual i depend on its location X i t and its phenotypic trait U i t at time t ≥ 0. X i t lays in an open, bounded and convex subset X of R d with a C 2 -boundary. U i t belongs to a compact subset U of R q . The phenotype of an individual does not change during its life time contrary to its location.

The total population is represented at any time t by the nite measure

ν K t = 1 K Nt i=1 δ (X i t ,U i t ) , (1)
where δ y corresponds to the Dirac measure at y, N t is the number of individuals at time t. The parameter K scales the population size and the biological assumption of large population size is stated into mathematics by the convergence of K to +∞.

Any individual i with phenotypic trait u moves according to a diusion process driven by the following stochastic dierential equation normally reected at the boundary ∂X ,

dX i t = √ 2m u Id • dB t -n(X i t )dl t (2)
where B is a d-dimensional brownian motion, l t is the local time at the boundary ∂X , n is the outward normal to ∂X and m u is a function of the trait. An individual with location x ∈ X and trait u ∈ U gives birth at rate b(x, u), also denoted by b u (x) when u is xed. The ospring appears at the location of its parents. Furthermore, a mutation may occur with probability q K p, making the phenotypic trait of the ospring dierent. The law of the mutant trait is given by a kernel k(x, u, •). The parameter q K scales the mutation probability and the biological assumption of rare mutations is stated by q K → 0.

The natural death rate is d(x, u). The competition exerted by an individual (y, v) on an individual (x, u) depends on the location y and on the two traits through a competition kernel c : U × X × U → R + . For the population ν = 1

K n i=1 δ (xi,ui) ∈ M F (X × U), the competitive pressure exerted on individual (x, u) is c • ν(x, u) = 1 K n i=1 c(u, x i , u i ) = 1 K X ×U
c(u, y, v)ν(dy, dv).

Observe that the competition kernel does depend on y. This spatial dependence yields non-trivial mathematical diculties. Biologically, it is a way to describe the spatial heterogeneity of the competition induced, for example, by the defense of a specic location. Finally, the total death rate is d(x, u) + c • ν(x, u).

As above, we may use notation d u (x) and c uv (y) when u and v are xed.

Let us state the assumptions on the parameters.

Assumption 1 1. m, b, d, k and c are continuous and non-negative on their domains and b, d and c are Lipschitz functions with respect to x and y. 2. There exist m, b, b, d, c, c, k ∈ R such that for any (x, u, y, v) ∈ ( X × U) 2 , 0 < m u ≤ m, b < b(x, u) ≤ b, d(x, u) ≤ d, c ≤ c(u, y, v) ≤ c, k(x, u, v) ≤ k, and d is not the zero function. 3. The sequence of initial measures (ν K 0 ) K>0 , which belongs to M F (X × U), converges in law to some deterministic measure denoted by ξ 0 and it satises sup K E[ ν K 0 , 1 3 ] < +∞. [START_REF] Champagnat | Adaptation in a stochastic multi-resources chemostat model[END_REF]. q K tends to 0 when K tends to +∞.

Before going further, let us set and recall the notation, which we use in the entire paper.

Notation

• For all x ∈ ∂X , n(x) denotes the outward normal to the boundary of X at point x.

• For suciently smooth f and for all (x, u)

∈ ∂X × U, ∂ n f (x, u) denotes the scalar product ∇ x f (x, u) • n(x). • C k,l n (X × U) represents the set of functions f ∈ C k,l (X × U) such that ∂ n f (x, u) = 0 for all (x, u) ∈ ∂ X ×U. We dene C k,l,j n (X ×U ×[0, T ]) similarly. • For any f ∈ C k,l,j n (X × U × [0, T ]), f s is the function on X × U such that f s (x, u) = f (x, u, s).
• For any compact set X, we denote the space of nite measures on X by M F (X).

• For all ν ∈ M F (X) and f ∈ C(X), we denote X f dν by ν, f . • C Lip (X) denotes the set of all positive Lipschitz-continuous functions f on X bounded by 1 and with a Lipschitz constant smaller than 1.

• We dene the Kantorovich-Rubinstein distance on M F (X) by : for any ν, µ ∈ M F (X),

W 1 (ν, µ) = sup f ∈C Lip (X) | ν, f -µ, f | .
As X is a compact set, this metric is a metrization of the topology of weak convergence. It is equivalent to the 1 st -Wasserstein distance.

• B(ν, γ) represents the ball of center ν and radius γ in M F (X) for the previous distance.

• D([0, T ], M F (X)) denotes the space of càdlàg functions from [0, T ] to M F (X), equipped with the Skorokhod topology.

• If ξ ∈ M F (X × {u, v}), we identify the two following ways of writing :

ξ(dx, dw) = ξ u (dx)δ u (dw) + ξ v (dx)δ v (dw) and ξ = (ξ u , ξ v ) ∈ (M F (X )) 2 .

Main theorem

A macroscopic approximation of the model described in Section 1 has been proved in [START_REF] Champagnat | Invasion and adaptative evolution for individual-based spatially structured populations[END_REF] as a large population limit.

Theorem 1 (Theorems 4.2 and 4.6 in [START_REF] Champagnat | Invasion and adaptative evolution for individual-based spatially structured populations[END_REF]) Suppose that Assumption 1 holds. For all T > 0, the sequence

(ν K ) K>0 of processes belonging to D([0, T ], M F (X × U)) converges in law to a deterministic and continuous function ξ, i.e. ξ ∈ C([0, T ], M F (X × U)) such that sup t∈[0,T ] ξ t , 1 < +∞ and ∀f ∈ C 2,0 n (X × U), ξ t , f = ξ 0 , f + t 0 X ×U b(x, u) -d(x, u) -c • ξ s (x, u) f (x, u) + m u ∆ x f (x, u) ξ s (dx, du)ds. (3)
Moreover, if U is nite, for any u ∈ U and t > 0, ξ t (., u) has a density with respect to Lebesgue measure which is a C 2 -function.

The limiting equation ( 3) is a nonlinear nonlocal reaction-diusion equation dened on the space of traits and locations. In [START_REF] Desvillettes | Innite dimensional reaction-diusion for population dynamics[END_REF][START_REF] Arnold | Existence of nontrivial steady states for populations structured with respect to space and a continuous trait[END_REF], the authors have studied the existence of the steady states of similar equations in the context of frequent mutations. Our study involves a rare mutations assumption and mutation terms disappear in the limit. The stability of the steady states and the long time behavior of the solutions to (3) have been characterized in [START_REF] Leman | Inuence of a spatial structure on the long time behavior of a competitive lotka-volterra type system[END_REF][START_REF] Coville | Convergence to equilibrium for positive solutions of some mutation-selection model[END_REF] in the particular cases of a monomorphic population (all individuals have a same phenotype) and a dimorphic population (two traits are involved). The stationary states and their stability are described using the following parameters.

Denition 1 For any u ∈ U, we dene H u by,

H u = - min φ∈H 1 (X ),φ ≡0 1 φ 2 L 2 (X ) m u X |∇φ| 2 - X (b u -d u )φ 2 , (4)
where H1 (X ) is the Sobolev space of order 1 on X . H u is the principal eigenvalue of the operator m u ∆ x • +(b u -d u )• with Neumann boundary condition on X . Let ḡu ∈ C 1 (X ) be the eigenfunction of the previous operator associated with the eigenvalue H u such that X c uu (y)ḡ u (y)dy = H u . If ḡu ≥ 0, we dene the associated measure in M F (X ) ξu (dx) := ḡu (x)dx.

(5)

Finally, for any (u, v) ∈ U, we set

κ vu := X c vu (y)ḡ u (y)dy X ḡu (y)dy -1 .
As proved in [START_REF] Leman | Inuence of a spatial structure on the long time behavior of a competitive lotka-volterra type system[END_REF], H u > 0 is the condition ensuring that a monomorphic population with trait u is able to survive. In that case, the stationary stable state is described by the positive spatial prole ḡu .

The dimorphic case implies four distinct stationary states : the trivial state (0, 0), two monomorphic states and a co-existence state. To ensure the impossibility of co-existence of two traits for a long time which is a principle also known as "Invasion-Implies-Fixation", we set the following assumption Assumption 2 Let u be in U, for almost all v ∈ U,

1. either, H v κ uu -H u κ vu < 0,

or,

H v κ uu -H u κ vu > 0 H u κ vv -H v κ uv < 0.
According to [START_REF] Leman | Inuence of a spatial structure on the long time behavior of a competitive lotka-volterra type system[END_REF], any solution to (3) converges either to (ḡ u , 0), or to (0, ḡv ) under Assumption 2. Furthermore, the authors of [START_REF] Leman | Inuence of a spatial structure on the long time behavior of a competitive lotka-volterra type system[END_REF] prove that condition 1 ensures the L 2 -stability of the equilibrium (ḡ u , 0). Thus, if a v-population with a small density function g v is emerging in a u-population with density function ḡu , it will not be able to survive. On the contrary, under condition 2, it will invade and replace the u-population since condition 2 ensures that any deterministic solution to (3) converges to the stable equilibrium (0, ḡv ) whatever the initial condition is. In the light of the previous considerations, we refer to H v κ uu -H u κ vu as the invasion tness of the individuals with type v in a resident population with type u.

Moreover, we will show that the probability of success of such an invasion can be described by means of the geographical birth position x 0 of the rst individual with trait v and the function φ vu dened for all u, v ∈ U by 2. if H v κ uu -H u κ vu > 0, φ vu is the unique positive solution to the elliptic equation on X

     m v ∆ x φ + b v -d v - X c vu (y)ḡ u (y)dy φ -b v φ 2 = 0, ∂ n φ(x) = 0, ∀x ∈ ∂X . (6)
We are now ready to state the main result of this paper. It describes the convergence to a spatial structured TSS under the separation of time scales introduced in [START_REF] Champagnat | A microscopic interpretation for adaptive dynamics trait substitution sequence models[END_REF].

Theorem 2 We suppose that Assumptions 1 and 2 hold. We also assume that the scaling parameters satisfy

Kq K log(K) -→ K→+∞ 0 and Kq K e KV -→ K→+∞ +∞, for any V > 0. ( 7
)
Then for any T > 0, ν K

(t/Kq K ) t∈[0,T ]
converges towards a jump Markov process (Λ t ) t≥[0,T ] as K → +∞. At any time t, Λ t belongs to the subspace

{ ξu δ u , u ∈ U} of M F (X × U)
, where for any u ∈ U, ξu ∈ M F (X ) is the spatial pattern dened in [START_REF] Champagnat | Invasion and adaptative evolution for individual-based spatially structured populations[END_REF]. The process (Λ t ) t≥0 jumps from the state characterized by the trait u ∈ U to the state characterized by v ∈ U at the innitesimal rate

X pb u (x)φ vu (x)ḡ u (x)k(x, u, v)dxdv.
This convergence holds in the sense of convergence of the nite dimensional distributions.

The limiting jump process describes an evolutionary phenomenon using a sequence of monomorphic equilibria characterized by their spatial patterns and their phenotypic trait.

Although the structure of Theorem 2's proof is similar to the one of Theorem 1 in [START_REF] Champagnat | A microscopic interpretation for adaptive dynamics trait substitution sequence models[END_REF], the spatial structure of the process leads us to deal with innite dimensional processes. Two key points of the proof have to be approached differently. The rst point concerns the study of the dynamics of a population descended from a mutant which has appeared in a well-established monomorphic population. As long as the mutant population size is small, the competitive terms between mutants can be neglected and the dynamics of the mutant population will be compared with the dynamics of a branching diffusion process. Thus, the rst ingredient of the proof is to understand nely the extinction and survival probabilities of a branching diusion process. We will describe it by means of the eigenparameters dened in Denition 1 and Assumption 2. The second point which is approached dierently concerns the study of the process when it is close to a monomorphic deterministic equilibrium to (3) taking into account a small mutant population and the possibility of new mutations. In this way, we avoid the comparisons used in [START_REF] Champagnat | A microscopic interpretation for adaptive dynamics trait substitution sequence models[END_REF], where the behavior of the resident population process is compared with the behavior of a theoretical monomorphic population evolving alone. Those comparisons are much more involved when the population is spatially structured. We estimate the exit time of a neighborhood of a monomorphic equilibrium by the resident population by studying a large deviation upperbound of the stochastic process (ν K t ) t≥0 around its deterministic limit (3) when K is large. The large deviations studies for processes combining diusion process and jumps in a non-constant size population are still unresolved, to our knowledge. Those studies have thus their own interest.

The next section presents some numerical examples that illustrate Theorem 2. In Section 4, we evaluate the survival probability of a branching diusion process and we characterize the scale time under which its size is of order K. In Section 5, we describe an explicit large deviation upper bound in our innite dimensional framework by using ideas in [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF][START_REF] Léonard | On large deviations for particle systems associated with spatially homogeneous boltzmann type equations[END_REF][START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF]. Then we study the rate function associated with the large deviation principle. Section 6 deals with the exit time of a neighborhood of a stationary state to (3). Finally, Section 7 is devoted to the proof of Theorem 2. We detail two key propositions. The rst one deals with the dynamics of the individual-based process in the case where only one trait is involved. The second one gives the dynamics of the process after the time of the rst mutation but as long as only at most two traits are involved.

Numerics

In this section, we illustrate Theorem 2 with two numerical examples. For both examples, the location space is X = (0, 1) and the trait space is U = [0, 1].

The simulations are computed using an iterative construction which gives an eective algorithm of the process.

The evolution of ecological niches

We consider here a set of parameters similar to the one in [START_REF] Doebeli | Speciation along environmental gradients[END_REF] and [START_REF] Leman | Inuence of a spatial structure on the long time behavior of a competitive lotka-volterra type system[END_REF] in which, for any trait u, the growth rate is maximal for the location x = u. For instance, the location space state can represent a variation of resources, as seed size for some birds, and two populations with two dierent traits are not best-adapted to the same resources. For some bird species, a gradual variation of seed size can determine a gradual variation of beak size [START_REF] Grant | Unpredictable evolution in a 30-year study of darwin's nches[END_REF]. Moreover, the maximum value of the growth rate function is the same for all traits but when u decreases, the birth rate function goes faster to 0, as follows :

b(x, u) = max{4 -160(1 -u)(x -u) 2 , 0}, d(x, u) = 1.
That is, the birds with a large trait value are more generalists than the ones with a small trait value : they are able to feed themselves on a larger set of seed size [START_REF] Futuyma | The evolution of ecological specialization[END_REF]. All individuals move with the same diusion coecient m u = 0.003.

The competition kernel is a constant c = 10 and the mutation kernel k(x, u, w) is the probability density of a Gaussian random variable N (u, 0.05) conditioned on staying in U. Fig. 1 Simulations with K = 100000, q K = 10 -5 . The initial population is composed of K individuals with trait 0.4 at location 0.5. We observe the evolution of the spatial niche occupied by the individuals over time.

In Figure 1

The evolution of the diusion coecient

The evolution of dispersal is a classical question in evolutionary biology since it has profound eects on geographical distributions and eective population sizes [START_REF] Johnson | Evolution of dispersal: theoretical models and empirical tests using birds and mammals[END_REF].

In this part, we study the inuence of a change of the diusion coecient m u . Let us assume that the birth and death rates (b, d and c) are independent of the phenotypic traits. Let u, v ∈ U be two traits such that m u > m v . Since ḡu is the eigenvector associated to H u , the minimum in ( 4) is reached at ḡu and we deduce easily that H u ≤ H v . Moreover, for all w ∈ U,

κ wv = X c wv (y)ḡ v (y)dy = X c(y)ḡ v (y)dy =: κ v .
Hence, the sign of the invasion tness of the individuals with trait u in a resident population with trait v can be computed easily

H u κ vv -H v κ uv = (H u -H v )κ v ≤ 0, conversely, H v κ uu -H u κ vu ≥ 0.
As a conclusion, the individuals with the smallest diusion coecient will be selected. Thus, Theorem (2) justied the fact that dispersal is selected against in a bounded environment variable in space but constant in time. This evolutionary phenomenon has already been studied in particular cases [START_REF] Hastings | Dynamics of a single species in a spatially varying environment: the stabilizing role of high dispersal rates[END_REF][START_REF] Hastings | Can spatial variation alone lead to selection for dispersal?[END_REF][START_REF] Perthame | Rare mutations limit of a steady state dispersion trait model[END_REF]. Since there are more individuals in the good areas, the passive diusion leads individuals from the good locations to the bad locations [START_REF] Johnson | Evolution of dispersal: theoretical models and empirical tests using birds and mammals[END_REF]. Finally, spatial variation alone tends to reduce the diusion rate.

This phenomenon seems to hold for a more general competition kernel which depends on the locations of the two competing individuals as observed by means of numerical simulations presented in Figure 2. It has been computed using a competition kernel

∀(x, y) ∈ X := (0, 1), c(x, y) = 1 |x-y|≤0.1 , a birth rate b(x) = max{2 -20(x -0.8) 2 , 0}, a death rate d(x) = max{2 - 20(x -0.2) 2
, 0} and a mutation kernel identical to the previous simulation.

The diusion coecient depends on u as follows

m u = 0.003 1 + 10(u -0.3) 2 .
Here again, diusion is selected against. We observe several events of invasion and replacement, the individuals evolve to get a smaller and smaller diusion coecient until u ≈ 0.303 associated with the coecient of diusion m 0.303 ≈ 0.003. Fig. 2 Simulations with K = 100000, q K = 10 -5 ; K individuals start at location 0.2 with trait 0.8; on the gures, a black dot represents a population of more than 1000 individuals and a gray dot represents a population of less than 1000 individuals. We observe the evolution dynamics of the trait (left) and of the diusion coecient (right).

Survival probability for a branching diusion process

As explained at the end of Section 2, the dynamics of the ospring of any mutant individual will be compared with the dynamics of a branching diusion process. As a consequence, this section is devoted to the study of the survival probability of such a process.

Any individual is characterized by its location X i t ∈ X , solution to (2) with the diusion coecient m > 0. Moreover, each individual with location x ∈ X gives birth to a new individual at rate b(x) and dies at rate d(x). Let M t denote the number of individuals at time t. We describe the dynamics of the diusion process at each time by the nite measure

η t = Mt i=1 δ X i t .
We denote the probability measure under which η 0 = δ x by P δx . The next theorem concerns the survival probability of the population assuming that, initially, there is only one individual at location x. This probability is described by means of the solution to an elliptic dierential equation on X and the location x. The location x plays a key role since, if the rst individual appears in a place where the growth rate is low or negative, it has a high probability to die with no descendants. Let

Υ 0 = inf{t ≥ 0, M t = 0}.
Theorem 3 Let H be the principal eigenvalue of the elliptic operator m∆ x . + (b -d). with Neumann boundary conditions on X , see [START_REF] Champagnat | Adaptation in a stochastic multi-resources chemostat model[END_REF]. If H > 0, there exists a unique positive C 2 -solution φ * to the elliptic equation

0 = m∆ x φ * (x) + (b(x) -d(x))φ * (x) -b(x)φ * (x) 2 , ∀x ∈ X , ∂ n φ * (x) = 0, ∀x ∈ ∂X , (8)
and φ * (x) = lim t→∞ P δx [Υ 0 ≥ t] for all x ∈ X . If H ≤ 0, [START_REF] Dawson | Large deviations from the mc-kean-vlasov limit for weakly interacting diusions[END_REF] has no non negative solution, we set φ * ≡ 0, and

lim t→∞ P δx [Υ 0 ≥ t] = 0 for all x ∈ X .
The second result of this part estimates the probability that the population size is of order K after a logarithmic time log K. For all > 0 and K ∈ N * , we set

Υ K = inf{t ≥ 0, η t , 1 ≥ K}.
Theorem 4 Let > 0 and (t K ) K>0 be a sequence of times such that lim K→+∞ t K / log(K) = +∞. Then for all x ∈ X ,

lim K→+∞ P δx [Υ K < t K ] = φ * (x).
The end of this part is devoted to the proofs.

Proof (Proof of Theorem 3)

We rst study the probability P δx [Υ 0 ≤ t]. We denote the time of the rst event (birth or death) of the population by E 1 . The law of E 1 is given by

P δx [E 1 ≤ t] = E δx t 0 (d(X s ) + b(X s ))e -
P δx [Υ 0 ≤ t] = E δx 1 E1≤t 1 {M E 1 =0} + 1 E1≤t 1 {M E 1 =2} E 2δ X E 1 [1 M t-E 1 =0 ] = E x t 0 d(X s ) + b(X s )P δ Xs [Υ 0 ≤ t -s] 2 e -I(s) ds ,
where X under P x is solution to (2) with initial condition x and diusion coecient m. Thus g(x, t) = P δx [Υ 0 ≤ t] satises for all x ∈ X , and all t > 0,

     g(x, t) = E x t 0 d(X s ) + b(X s )g(X s , t -s) 2 e -I(s) ds , g(x, 0) = 0, ∀x ∈ X . (9)
Using Gronwall's Lemma for bounded functions, we deduce immediately that (9) has a unique bounded solution.

We now show that there exists a unique C 2 -solution to

     ∂ t f (x, t) = m∆ x f (x, t) -b(x) + d(x) f (x, t) + d(x) + b(x)f (x, t) 2 , ∂ n f (x, t) = 0, ∀(x, t) ∈ ∂X × R + f (x, 0) = 0, ∀x ∈ X , (10)
such that f belongs to C 2,1 (X × R + ), is positive and smaller than 1 by using super-and sub-solutions arguments. Indeed, let

F (x, f ) = -b(x) + d(x) f + d(x) + b(x)f 2 .
We easily see that f ≡ 0 and f ≡ 1 satisfy:

∂ t f ≤ m∆ x f + F (x, f ), ∀(x, t) ∈ X × R + , ∂ t f ≥ m∆ x f + F (x, f ), ∀(x, t) ∈ X × R + , f (x, 0) ≤ f (x, 0) ≤ f (x, 0), x ∈ X , ∂ n f (x, t) ≤ 0 ≤ ∂ n f (x, 0), x ∈ ∂X , t ∈ R + .
That is, f (resp. f ) is a sub-solution (resp. super-solution) to [START_REF] Desvillettes | Innite dimensional reaction-diusion for population dynamics[END_REF]. Moreover, F and ∂ f F belong to C(X × R) and F is a Lipschitz function with respect to

x by means of Assumptions 3. We apply Theorem 4 of Chapter III in [START_REF] Roques | Modèles de réaction-diusion pour l'écologie spatiale[END_REF] to deduce that [START_REF] Desvillettes | Innite dimensional reaction-diusion for population dynamics[END_REF]

admits a solution f ∈ C 2,1 (X × R + ) satisfying 0 ≤ f ≤ 1.
The uniqueness of the solution is a consequence of the maximum principle.

The next step is the use of a Feynman-Kac formula to deduce that f is also a solution to [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]. Let X be a solution to [START_REF] Berestycki | Analysis of periodically fragmented environment model: I-species persistence[END_REF] and for all t ≥ 0, and s ∈ [0, t],

we set

H(s, X s ) = f (X s , t -s)e -I(s) .
Applying Itô's formula to H(s, X s ), using ( 2), [START_REF] Desvillettes | Innite dimensional reaction-diusion for population dynamics[END_REF] and the fact that ∂ n f (x, t) = 0 for all x ∈ ∂X , we nd for all s ∈ [0, t[,

H(s, X s ) = H(0, X 0 ) - s 0 d(X σ ) + b(X σ )f (X σ , t -σ) 2 e -I(σ) dσ + s 0 √ 2m(∂ x f (X σ , t -σ))e -I(σ) dB σ . (11)
The expectation of the last term is equal to 0 as ( r 0 ∂ x f (X σ , t-σ)e -I(σ) dB σ ) r∈[0,s] is a martingale. In addition, E x [H(0, X 0 )] = f (x, t). Let us make s tend to t using the dominated convergence Theorem. As lim s→t E x [H(s, X s )] = E x [H(t, X t )] = 0, we deduce that f is a solution to [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]. Since (9) admits a unique bounded solution, both solutions are equal, i.e. P δx [Υ 0 ≤ t] = f (x, t).

Finally, we deduce the survival probability lim t→+∞ (1 -f (x, t)) using results on Equation [START_REF] Desvillettes | Innite dimensional reaction-diusion for population dynamics[END_REF] obtained in [START_REF] Berestycki | Analysis of periodically fragmented environment model: I-species persistence[END_REF] and in Theorems 9 and 11 of Chapter III in [START_REF] Roques | Modèles de réaction-diusion pour l'écologie spatiale[END_REF]. Indeed, it is proved that if H > 0, there exists a unique positive solution φ * to the elliptic equation [START_REF] Dawson | Large deviations from the mc-kean-vlasov limit for weakly interacting diusions[END_REF], and that φ(t, x) = 1 -f (x, t) = P x (Υ 0 > t) tends to φ * (x) in C 2 (X ) as t → +∞. Moreover, if H ≤ 0, the unique solution to (8) is the zero function and φ(t, x) → 0 uniformly in X as t → +∞.

Proof (Proof of Theorem 4) First, we split the studied probability into three parts :

P δx (Υ K < t K ) = P δx (Υ K < t K , log log(K) < Υ 0 < +∞) + P δx (Υ K < t K , log log(K) ≥ Υ 0 ) + P δx (Υ K < t K , Υ 0 = +∞). ( 12
)
Let us start with the rst term of ( 12) :

P δx (Υ K < t K , log log(K) < Υ 0 < +∞) ≤ P δx (log log(K) < Υ 0 < +∞) → K→+∞ 0.
The second term of (12) will be treated using a comparison with a pure birth process. Let us consider a birth process with constant birth rate b and started with only one individual. Υ K denotes the rst time when the population size Ñ of the process is greater than K.

P δx (Υ K < t K , log log(K) ≥ Υ 0 ) ≤ P δx (Υ K ≤ log log(K)) ≤ P 1 ( Υ K ≤ log log(K)) ≤ P 1 ( Ñlog log K ≥ K) ≤ e b log log K /( K) → K→+∞ 0.
It remains to deal with the third term in [START_REF] Durrett | Spatial aspects of interspecic competition[END_REF]. Note that if H ≤ 0, Theorem 3 implies that the third term is equal to zero, and the proof is done.

From this point forward, we assume that H > 0. Let h be a positive eigenvector of the operator m∆ x . + (b -d). with Neumann boundary conditions on ∂X associated with the eigenvalue H. Thanks to Itô's formula, we nd

η t , e -Ht h = η 0 , h + t 0 η s , e -Hs m∆ x h + (b -d)he -Hs -Hhe -Hs ds + t 0 Ms i=1 √ 2m∇ x h(X i s )e -Hs dB i s ,
As m∆ x h + (b -d)h = Hh and ∇ x h is bounded on X , (e -Ht η t , h ) t≥0 is a martingale. Moreover it is positive, so, it converges a.s. to a non-negative random variable that will be denoted by W . Obviously, {Υ 0 < +∞} ⊂ {W = 0}. Our aim is to prove that this is an a.s. equality. As done in the previous proof, we denote the time of the rst event of the population by E 1 and we set I(s) := s 0 (b(X r )+d(X r ))dr. Using the Markov property and the independence between individuals, we nd an equation satised by P δx [W = 0]:

P δx [W = 0] = E x +∞ 0 e -I(s) d(X s ) + b(X s )P δ Xs [W = 0] 2 ds . (13) Finally, g(x) := P δx [W > 0] = 1 -P δx [W = 0] is solution to g(x) = E x +∞ 0 e -I(s) b(X s )g(X s )(2 -g(X s ))ds , ∀x ∈ X . ( 14
)
Let us show that there exists at most one non-zero solution with value in [0, 1] to Equation [START_REF] Fontbona | Uniqueness for a weak nonlinear evolution equation and large deviations for diusing particles with electrostatic repulsion[END_REF]. Let g 1 and g 2 be two such solutions. We dene

γ = sup{γ > 0, g 1 (x) -γg 2 (x) ≥ 0, ∀x ∈ X }.
Assume rst that γ < 1.

g 1 (x) -γg 2 (x) = E x +∞ 0 e -I(s) b(X s ) 2(g 1 -γg 2 )(X s ) -(g 2 1 -γg 2 2 )(X s ) ds . (15)
As γ < 1, g 2 1 -γg 2 2 ≤ (g 1 -γg 2 )(g 1 + γg 2 ) and we nd

g 1 (x) -γg 2 (x) ≥ E x +∞ 0 e -I(s) b(X s ) (g 1 -γg 2 )(2 -g 1 -γg 2 )(X s ) ds .
Moreover, by the denition of γ, there exists x 0 ∈ X such that g 1 (x 0 )γg 2 (x 0 ) = 0, so

0 = E x0 +∞ 0 e -I(s) b(X s ) g 1 (X s ) -γg 2 (X s ) 2 -g 1 (X s ) -γg 2 (X s ) ds .
Thus for a.e. s ∈ R

+ , P x0 -a.s., b(X s )[g 1 (X s )-γg 2 (X s )][2-g 1 (X s )-γg 2 (X s )] = 0.
Let us note that for all x ∈ X , for i = 1, 2, (13) implies that

1 -g i (x) ≥ E x +∞ 0 e -I(s) d(X s )ds = P δx [M E1 = 0] > 0, that is, for all x ∈ X , 2 -g 1 (x) -γg 2 (x) > 0. As b is positive, for a.e. s ∈ R + , P x0 -a.s. g 1 (X s ) -γg 2 (X s ) = 0. In addition with the fact that γg 2 2 -g 2 1 = γg 2 2 (1 -γ) -(g 1 -γg 2 )(g 1 + γg 2 ), ( 15
) implies that, 0 = g 1 (x 0 ) -γg 2 (x 0 ) = E x0 +∞ 0 e -I(s) b(X s )γg 2 (X s ) 2 (1 -γ)ds .
Using the same argument as before and that 1 -γ > 0, we deduce that for a.e. s ∈ R + , P x0 -a.s., g 2 (X s ) = 0. Moreover, under P x0 , the random variable X s has a density with respect to Lebesgue measure, that is, for Lebesgue-a.a.

x ∈ X , g 2 (x) = 0. This is a contradiction with the fact that g 2 is a non-zero solution.

Finally if γ ≥ 1, we dene instead γ = sup{γ > 0, g 2 (x) -γg 1 (x) ≥ 0, ∀x ∈ X } < 1 and we use symmetric arguments to reach a contradiction. Thus, there is at most one solution to [START_REF] Fontbona | Uniqueness for a weak nonlinear evolution equation and large deviations for diusing particles with electrostatic repulsion[END_REF] with values in [0, 1]. The next step is to show that φ * , solution to [START_REF] Dawson | Large deviations from the mc-kean-vlasov limit for weakly interacting diusions[END_REF], is also a solution to [START_REF] Fontbona | Uniqueness for a weak nonlinear evolution equation and large deviations for diusing particles with electrostatic repulsion[END_REF]. Let us write f * = 1 -φ * , it satises

0 = m∆ x f * -(b + d)f * + d + b(f * ) 2 , on X , ∂ n f * = 0, on ∂X . ( 16
)
We apply Itô's formula to f * (X t )e - R t 0 (b(Xr)+d(Xr))dr . Then taking the expec- tation and using Equation ( 16), we deduce

f * (x) =E x t 0 e -I(s) [d(X s ) + b(X s )f * (X s ) 2 ]ds + E x f * (X t )e -I(t) .
Our aim is now to let t tend to innity. Note that I(t) ≥ bt for all t ∈ R + and that f * is bounded by 1. Hence, we use the dominated convergence Theorem to nd those two convergences:

E x f * (X t )e -I(t) ≤ E x e -I(t) → t→+∞ 0 E x +∞ t e -I(s) [d + b(f * ) 2 ](X s )ds ≤ E x +∞ t e -I(s) ( b + d)ds → t→+∞ 0.
Thus, making t tend to innity, we nd for all x ∈ X , [START_REF] Fontbona | Uniqueness for a weak nonlinear evolution equation and large deviations for diusing particles with electrostatic repulsion[END_REF]. There exists at most one non-zero solution to ( 14), thus we have either P δx [W > 0] = φ * (x) for all x ∈ X , or P δx [W > 0] = 0 for all x ∈ X . Using Itô's formula, it is easy to check that in the case H > 0, ( η t , e -Ht h ) t≥0 is bounded in L 2 . So this martingale is uniformly bounded and it converges in

f * (x) = E x +∞ 0 e -I(s) [d(X s ) + b(X s )f * (X s ) 2 ]ds . Since φ * ≡ 1 -f * , φ * is a solution to
L 1 to W , hence E δx [W ] = h(x) > 0.
Finally,

P δx [W > 0] = φ * (x) = P δx [Υ 0 = +∞] ⇒ {W > 0} = {Υ 0 = +∞} a.s. ( 17
)
On {Υ 0 = +∞},

log( K) Υ K ≥ log( η Υ K , he -HΥ K . h -1 ∞ e HΥ K ) Υ K → K→+∞ H > 0 a.s., as η Υ K , h ≤ K h ∞ , Υ K → +∞ when K tends to innity and W > 0.
Hence,

lim K→+∞ Υ K log( K) < +∞ and lim K→+∞ t K log( K) = +∞,
and so, the third term in [START_REF] Durrett | Spatial aspects of interspecic competition[END_REF] satises

P δx (Υ K < t K , Υ 0 = +∞) = P δx Υ K log( K) < t K log( K) , Υ 0 = +∞ → K→∞ P δx (Υ 0 = +∞) . (18)
Finally, we have shown that the two rst terms in [START_REF] Durrett | Spatial aspects of interspecic competition[END_REF] tend to 0 and using additionally (18), we get

lim K→+∞ P δx (Υ K < t K ) = P δx (Υ 0 = +∞) = φ * (x).
That ends the proof in the case H > 0.

Exponential deviations results

In this section, we are concerned by the upperbound of the large deviations from the large population limit (3) for the process (ν K t ) t∈[0,T ] when K tends to +∞ and q K tends to 0. Let us rst describe the rate function. It requires specic notation which will be only used in this subsection : let us x T > 0,

E = X × U × {1, 2}. ψ is the mapping such that for any function f ∈ C 2,0,1 (X × U × [0, T ]), for any (x, u, π, t) ∈ E × [0, T ], ψ(f )(x, u, t, π) = f (x, u, t) if π = 1, -f (x, u, t) if π = 2. For all ν = (ν t ) t∈[0,T ] ∈ D([0, T ], M F (X × U)), we dene the positive nite measure µ ν t (dx, du, dπ) = b(x, u)δ 1 (dπ) + (d(x, u) + c • ν t-(x, u))δ 2 (dπ) ν t-(dx, du).
Finally, we introduce the log-Laplace transform ρ of a centered Poisson distribution with parameter 1, ρ(x) = e x -x-1, and its Legendre transform ρ * , ρ * (y) = ((y + 1) log(y + 1) -y)

1 {y>-1} + 1 {y=-1} + ∞ • 1 {y<-1} .
We now dene the rate function : for all ξ 0 ∈ M F (X ×U) and ν ∈ D([0, T ], M F (X × U)),

I T ξ0 (ν) :=    sup f ∈C 2,0,1 n (X ×U ×[0,T ]) I f,T (ν), if ν 0 = ξ 0 + ∞, otherwise, (19)
where

I f,T (ν) := ν T , f T -ν 0 , f 0 - T 0 m∆ x f s + m|∇ x f s | 2 + ∂f s ∂s , ν s ds - T 0 E ψ(f )(x, u, s, π) + ρ(ψ(f )(x, u, s, π)) dµ ν s ds.
When there is no ambiguity, we will write I T (ν) instead of I T ν0 (ν). The next theorem describe the large deviations upperbound result.

Theorem 5 Suppose that Assumption 1 holds. For all α > 0,

ξ 0 ∈ M F (X × U), for all compact set C ⊂ B(ξ 0 , α), for all measurable subset A of D([0, T ], M F (X × U)) such that there exists M > 0 with A ⊂ {ν| sup t∈[0,T ] ν t , 1 ≤ M }, lim sup K→+∞ 1 K sup ν K 0 ∈C∩M K F log P ν K 0 (ν K ∈ A) ≤ -inf ξ∈C,ν∈ Ā I T ξ (ν), (20)
where

M K F = { 1 K N i=1 δ (xi,ui) , with N ∈ N, (x i , u i ) ∈ X × U}.
Proof We show the following upper bound lim sup

K→+∞ 1 K log P(ν K ∈ A) ≤ -inf ν∈ Ā I T ξ0 (ν). (21)
Equation ( 20) can be directly deduced from this bound by a similar reasoning as in the proof of Corollary 5.6.15 in Dembo and Zeitouni [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]. To prove (21), we need the exponential tightness of the process (ν K t ) t∈[0,T ] which is described by the following lemma whose proof can be easily adapted from [START_REF] Dawson | Large deviations from the mc-kean-vlasov limit for weakly interacting diusions[END_REF][START_REF] Graham | An upper bound of large deviations for a generalized star-shaped loss network[END_REF][START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF].

Lemma 1 Suppose that Assumption 1 holds, and that there exists C init > 0 such that sup K∈N ν K 0 , 1 < C init a.s.. Then for all L > 0, there exists a compact subset

C L of the Skorohod space D([0, T ], M F (X × U)) such that lim sup K→+∞ 1 K log P(ν K ∈ C L ) ≤ -L. Set τ K M = inf{t ≥ 0, ν K t , 1 ≥ M }.
Note that Lemma 1 is also true for (ν K t∧τ K M ) t≥0 . Using a proof similar to Theorem 4.4.2 of [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF], we deduce the inequality lim sup

K→+∞ 1 K log P(ν K ∈ A) = lim sup K→+∞ 1 K log P(ν K .∧τ K M ∈ A) ≤ -inf ν∈ Ā sup f ∈C 2,0,1 n (X ×U ×[0,T ]) (I f,T (ν) -H(I f,T )) ,
where

H(I f,T ) = lim sup K→+∞ 1 K log E[exp(KI f,T (ν K .∧τ K M )]. It remains to show that H(I f,T ) = 0. Let N T = exp KI f,T (ν K .∧τ K M ) -q K K× T ∧τ K M 0 ν K s , pb u (x) U φ(f s )(x, w)k(x, u, w)dw -φ(f s )(x, u) ds ,
where φ(x) = x + ρ(x) = e x -1. Itô's formula implies that (N T , T ≥ 0) is a local martingale. The denition of τ K M implies that N T is bounded. So it is a martingale of mean 1 and there exists a

constant C( f ∞ , M ) > 0 such that exp(-q K KC( f ∞ , M )) ≤ E exp KI f,T (ν K .∧τ K M ) ≤ exp(q K KC( f ∞ , M )).
We conclude easily, since q K tends to 0 when K → +∞.

The main point is now to write the rate function under a non-variational integral formulation which is more workable than [START_REF] Grant | Unpredictable evolution in a 30-year study of darwin's nches[END_REF]. This integral formulation is required to prove the results on the exit time in Section 6. It will be used to bound from above the distance between a solution to (3) and any ν, as proved below in Proposition 1.

Before writing the non-variational formulation, let us dene two functional spaces.

The Orlicz space associated with ρ * is L ρ * ,T the set of all bounded and measurable functions h on E × [0, T ] such that h ρ * ,T := inf α > 0,

E×[0,T ] ρ * |h| α dµ ν s ds ≤ 1 < +∞. (22)
The Orlicz space associated with ρ is dened on the same way.

L 2 T is the set of functions h ∈ L 2 (X × U × [0, T ], R d ) such that h L 2 ,T := T 0 2 ν s , m|h s | 2 ds 1/2 < ∞. (23)
Theorem 6 Suppose that Assumption 1 holds. Let T > 0 and ν ∈ D([0, T ], M F (X × U)), such that I T ν0 (ν) < +∞, then there exist two measurable functions

(h ν 1 , h ν 2 ) ∈ L ρ * ,T × L 2 T such that for all f ∈ C 2,0,1 n (X × U × [0, T ]), ν t , f t = ν 0 , f 0 + T 0 E (1 + h ν 1 (x, u, s, π))ψ(f )(x, u, s, π)dµ ν s ds+ T 0 X ×U ν s , m u ∆ x f s (x, u) + 2m u h ν 2 (x, u, s) • ∇ x f s (x, u) + ∂f s ∂s (x, u) ds, (24)
and the rate function can be written as follows

I T ν0 (ν) = T 0 E ρ * (h ν 1 )dµ ν s ds + T 0 m ν s , |h ν 2 | 2 ds < +∞. (25)
The proof of Theorem 6 uses convex analysis arguments which can be adapted from Leonard [START_REF] Leonard | Large deviations for long range interacting particle systems with jumps[END_REF][START_REF] Léonard | Convex conjugates of integral functionals[END_REF][START_REF] Léonard | Minimizers of energy functionals[END_REF]. We do not detail its proof but we give the main ideas.

For all ν ∈ D([0, T ], M F (X × U)), I T (ν) is equal to the Legendre transform Γ * of Γ : (ψ, ∇ x )(C 2,0,1 (X × U × [0, T ])) → R:

Γ : (g 1 , g 2 ) → T 0 E ρ(g 1 )dµ ν s ds + T 0 ν s , m|g 2 | 2 ds
at a well chosen point l ν . If l ν belongs to the interior of the set domΓ * of linear maps l with Γ * (l) < +∞, we can exhibit l ν by means of the derivative of Γ , the Legendre biconjugate of Γ . Studying directly Γ is dicult. The key point is thus to work on the product space L ρ,T × L 2

T . In this way, we can study the Legendre biconjugate of an extension of Γ on that space, in order to deal with the diusive part and the jumps part separately. The diusive part is treated using ideas of Dawson and Gartner [START_REF] Dawson | Large deviations from the mc-kean-vlasov limit for weakly interacting diusions[END_REF] and Fontbona [START_REF] Fontbona | Uniqueness for a weak nonlinear evolution equation and large deviations for diusing particles with electrostatic repulsion[END_REF] whereas the jumps part is treated using ideas of Leonard [START_REF] Léonard | Convex conjugates of integral functionals[END_REF][START_REF] Léonard | Minimizers of energy functionals[END_REF]. The next step is to deduce the Legendre biconjugate of Γ by restricting the denition domain by means of Proposition 3.3 in [START_REF] Léonard | Convex conjugates of integral functionals[END_REF]. Finally, to deal with points ν for which l ν does not belong to the interior of domΓ * , we use a continuity argument similar to that of Theorem 7.1's proof in [START_REF] Léonard | On large deviations for particle systems associated with spatially homogeneous boltzmann type equations[END_REF].

The last result of this part gives an upper bound on the distance between a solution to (3) and any ν, this bound is used in Subsection 6.2.

Proposition 1 Let T > 0 and M > 0. There exists C(T, M ) such that, for any ν satisfying sup t≤T ν t , 1 < M and for all (ξ t ) t≥0 solution to [START_REF] Champagnat | A microscopic interpretation for adaptive dynamics trait substitution sequence models[END_REF] with the initial condition ξ 0 = ν 0 ,

sup t∈[0,T ] W 1 (ν t , ξ t ) ≤ C(T, M ) I T (ν) + I T (ν) . Proof Let ν be such that ν t , 1 < M for all t ∈ [0, T ]. If I T (ν) = 0, i.e.
ν t = ξ t , or if I T (ν) = +∞, the result is obvious, so let us assume that 0 < I T (ν) < +∞. Let (ξ t ) t≥0 be the solution to (3) with initial condition ν 0 . The aim is to evaluate W 1 (ν t , ξ t ). Let (P u t ) t≥0 denote the semigroup of the reected diusion process which is the solution to (2) with initial condition

x and diusion coecient m u . Theorem 6 implies the existence of (h

1 , h 2 ) ∈ L ρ * ,T × L 2
T satisfying (24). Then, we nd the following mild formulation for (ν t ) t≥0 in a similar way to Lemma 4.5 in [START_REF] Champagnat | Invasion and adaptative evolution for individual-based spatially structured populations[END_REF] : for all f ∈ C Lip (X × U),

ν t , f = t 0 E ψ(P . t-s f )(1 + h 1 )dµ ν s ds + t 0 ν s , 2m∇ x P . t-s f • h 2 ds. ( 26
)
In addition with a mild equation for (ξ t ) t≥0 , we deduce that for all f ∈ C Lip (X × U) and for all t ≤ T ,

| ν t -ξ t , f | = t 0 E ψ(P . t-s f )(dµ ν s ds -dµ ξ s ds) + t 0 E ψ(P . t-s f )h 1 dµ ν s ds + t 0 ν s , 2m∇ x P . t-s f • h 2 ds ≤ C 1 t 0 sup r∈[0,s] W 1 (ν r , ξ r )ds + ψ(P . t-. f ) ρ,t h 1 ρ * ,t + ∇ x P . t-. f L 2 ,t h 2 L 2 ,t . (27)
The second line is a consequence of Hölder's inequality (see for example Theorem 6 of Chapter 1 in [START_REF] Rao | Applications of Orlicz spaces[END_REF] about Hölder's inequalities in Orlicz spaces).

The next step is to nd an upperbound on the two last terms. Theorem 6 implies that

I T (ν) = T 0 E ρ * (h 1 )dµ ν s ds + T 0 m ν s , |h 2 | 2 ds. ( 28
)
We easily deduce that for any t ≤ T ,

h 2 2 L 2 ,t ≤ h 2 2 L 2 ,T ≤ 2I T (ν). ( 29
)
Let us also nd an upper bound on h 1 ρ * ,T . Note that for all x ∈ R,

if α ≥ 1, ρ * (|x|/α) ≤ ρ * (|x|)/α ≤ ρ * (x)/α, if 0 < α ≤ 1, ρ * (|x|/α) ≤ ρ * (|x|)/α 2 ≤ ρ * (x)/α 2 . ( 30
)
Moreover, the non-variational formulation [START_REF] Leonard | Large deviations for long range interacting particle systems with jumps[END_REF] implies that I T (ν) ≥

T 0 E ρ * (h 1 )dµ ν s ds. Thus, using [START_REF] Jaj | Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction[END_REF] and the denition of the norm . ρ * ,T in [START_REF] Johnson | Evolution of dispersal: theoretical models and empirical tests using birds and mammals[END_REF], we obtain that if I T (ν) ≥ 1,

T 0 E ρ * |h 1 |/I T (ν) dµ ν s ds ≤ 1, i.e. h 1 ρ * ,T ≤ I T (ν), and if I T (ν) ≤ 1, T 0 E ρ * |h 1 |/ I T (ν) dµ ν s ds ≤ 1, i.e. h 1 ρ * ,T ≤ I T (ν). Thus, for any t ≤ T , h 1 ρ * ,t ≤ h 1 ρ * ,T ≤ I T (ν) + I T (ν) . ( 31
)
Let us now nd an upper bound on ψ(P . t-. f ) ρ,t . Since X is a convex set in R d and f ∈ C Lip (X ×U), it has been proved in Part 2 of [START_REF] Wang | Gradient estimate on convex domains and applications[END_REF]

that P u t f ∈ C Lip (X ). In addition with sup t∈[0,T ] ν t , 1 ≤ M , for all α > 0, t ≤ T t 0 E ρ |ψ(P . t-s f )| α dµ ν s ds ≤ T 0 E ρ 1 α dµ ν s ds ≤ T M [ b+ d+cM ]ρ 1 α , so, for all t ≤ T , ψ(P . t-. f ) ρ,t ≤ (ρ |R + ) -1 1 T M [ b + d + cM ] -1 := C 2 .
Furthermore, since P u t f ∈ C Lip (X ), we get that for any t ≤ T ,

∇ x P . t-. f 2 L 2 ,t = t 0 2 ν s , m|∇ x P . t-s f | 2 ds ≤ 2 mM t ≤ 2 mM T := C 3 .
Using the last two inequalities with ( 29), ( 31) and ( 27), we nd

sup r∈[0,T ] W 1 (ν r , ξ r ) ≤ C 1 T 0 sup r∈[0,s] W 1 (ν r , ξ r )ds+C 2 I T (ν)+(C 2 + 2C 3 ) I T (ν).
We use Gronwall's Lemma to conclude.

6 Lower bound on the exit time of a neighborhood of the stationary state

In this section, we assume that initially, two traits u and v are involved. The stochastic process starts in a state ν K

0 = ν K,u 0 + ν K,v 0 such that ν K,u 0 is close
to ξu and there exist only a few individuals with trait v. Since the considered initial state is close to the equilibrium ( ξu , 0) and according to Theorem 1, the dynamics of the stochastic process ν K is close to the equilibrium ( ξu , 0) on a nite interval time when K is large. Our aim is to control the exit time of the stochastic process ν K,u t from a neighborhood of the stationary solution ξu in M F (X ) when K is large and q K is small. We dene the exit time by : for all γ > 0,

R K γ = inf{t ≥ 0, W 1 (ν K,u t , ξu ) ≥ γ)}. ( 32
)
Theorem 7 gives a lower bound on R K γ . The lower bound involves the rst time when a new mutation occurs and the rst time when the v-population size is larger than a threshold:

S K 1 = inf{t ≥ 0, ∃w ∈ {u, v}, ν K t (X × {w}) = 0}, (33)
for all

> 0, T K = inf{t ≥ 0, ν K,v t , 1 ≥ }. ( 34
)
Theorem 7 Suppose that Assumption 1 holds and that H u > 0. Let γ > 0 such that γ < H u (κ uu ) -1 , and if

H u κ vv -H v κ uv > 0, γ satises also the assumption γ < | H u κ uu -H u κ vv -H v κ uv κ uu κ vv -κ vu κ uv | .
Then, there exist γ > 0, > 0, and

V > 0 such that, if ν K 0 = ν K,u 0 +ν K,v 0 with W 1 (ν K,u 0 , ξu ) < γ and ν K,v 0 , 1 < , then lim K→+∞ P ν K 0 (R K γ > e KV ∧ T K ∧ S K 1 ) = 1.
Thus, a well-established monomorphic population u is minimally aected during the emerging of a mutant population v.

The assumptions on the radius γ of the neighborhood ensure that there exists only one steady state in the neighborhood.

The result is proved using ideas similar to the ones of Freidlin and Wentzell [START_REF] Freidlin | Random Perturbations[END_REF]. In our framework, the diculties come from the continuous space motion. Firstly, our processes have values in an innite dimensional space, thus, the required deterministic results are much more involved, see Subsection 6.1.

Secondly, we deal with two kind of randomness : jump process and spatial diusion process. The end of the section is devoted to the proof of Theorem 7.

Stability for the weak topology

This subsection deals with the deterministic solution to (3). We denote by (ξ t ) t≥0 the solution to equation ( 3) with initial condition ξ 0 ∈ M F (X ×{u, v}).

In this case, ξ t ∈ M F (X × {u, v}) for all t ≥ 0. We prove that, as long as the size of the v-population density is small, the u-population density stays in a W 1 -neighborhood of its equilibrium ḡu .

Proposition 2 Suppose that Assumption 1 holds. Let γ > 0. There exist γ > 0 and > 0 such that for any

ξ 0 = ξ u 0 δ u + ξ v 0 δ v with W 1 (ξ u 0 , ξu ) < γ , for all t ≤ t = inf{t ≥ 0, ξ v t , 1 > }, W 1 (ξ u t , ξu ) < γ/2.
The proof of Proposition 2 implies two main diculties. First, using ideas similar to Part 3.3 of [START_REF] Leman | Inuence of a spatial structure on the long time behavior of a competitive lotka-volterra type system[END_REF], we can prove that the solution ξ t to (3) stays close to ξu if the initial condition admits a density which is close to the density ḡu of ξu for the L 2 -distance. However, this is not sucient since we will deal with discrete measures later. Thus, we need to enlarge the result for W 1 -distance. Secondly, we are concerned with the trajectories of the u-population process.

Even though the v-population size is small, it does have an impact on the death rate of individuals u which we cannot ignore.

The proof is divided into three steps. Firstly, we study how fast a solution with initial condition close to ξu moves away from ξu in W 1 -distance during a small time interval [0, t 0 ]. Then, as t 0 > 0, ξ u t0 admits a density and so, we can compare the W 1 -distance and the L 2 -distance of the densities between ξ u t0 and ξu . We nally prove a L 2 -stability result adapted from Part 3.3 in [START_REF] Leman | Inuence of a spatial structure on the long time behavior of a competitive lotka-volterra type system[END_REF].

Proof (Proof of Proposition 2) First, we may assume that ≤ 1 and γ < γ.

Hence, there exists M > 0 such that any considered initial state satises ξ 0 , 1 < M . We x t 0 > 0 and we start with the rst step. On the one hand, we can nd an upper bound to sup r∈[0,t] ξ u r , 1 . Indeed

ξ u t , 1 ≤ ξ u 0 , 1 + b t 0 ξ u s , 1 ds, (35)
and using Gronwall's Lemma, we deduce that sup r∈[0,t] ξ u r , 1 ≤ M e bt , for all t ≥ 0. On the other hand, using (3) with U = {u, v} and a mild formulation similar to Lemma 4.5 in [START_REF] Champagnat | Invasion and adaptative evolution for individual-based spatially structured populations[END_REF], we nd that ξ u t satises: for any f ∈ C Lip (X ),

ξ u t -ξu , f = ξ u 0 -ξu , P u t f + t 0 ξ u s -ξu , (b u -d u -c uu • ξu )P u t-s f ds + t 0 c uu • ( ξu -ξ u s ) ξ u s , P u t-s f ds - t 0 (c uv • ξ v s ) ξ u s , P u t-s f ds.
For any g Lipschitz-continuous, we denote by g Lip the smallest constant such that g/ g Lip ∈ C Lip (X ). Since sup t∈[0,t ] ξ v t , 1 ≤ , P u t f ∈ C Lip (X ) and using the denition of distance W 1 , we obtain that, for all t ≤ t ,

| ξ u t -ξu , f | ≤W 1 (ξ u 0 , ξu ) + ( b + d + c ξu , 1 ) t 0 W 1 (ξ s , ξ)ds + c Lip sup r∈[0,t] ξ u r , 1 t 0 W 1 (ξ u s , ξu )ds + . (36)
Finally, ( 35), ( 36) and Gronwall's Lemma imply that there exist C 1 , C 2 independent of and γ such that

sup r∈[0,t0∧t ] W 1 (ξ u r , ξu ) ≤ (W 1 (ξ u 0 , ξu ) + C 2 )e C1t0∧t ≤ (γ + C 2 )e C1t0 . ( 37
)
According to [START_REF] Wang | Gradient estimate on convex domains and applications[END_REF], we have to choose γ and such that (γ + C 2 )e C1t0 < γ/2. Note that if for all ξ v 0 ∈ M f (X ), t ≤ t 0 , the proof of Proposition 2 is complete. In what follows, let us assume that t > t 0 for the initial state ξ v 0 under consideration. The next step is to compare the L 2 -distance and the W 1 -distance between ξ u t0 and ξu . According to Theorem 1, for any t 0 > 0, ξ u t0 has a Lipschitz- continuous density with respect to Lebesgue measure on X that we denote by g u t0 . In addition with the fact that ḡu ∈ C 1 (X ), we obtain

g u t0 -ḡu 2 L 2 = X (g u t0 (x)-ḡu (x)) 2 dx ≤ W 1 (ξ u t0 , ξu )( g u t0 Lip + ḡu Lip ). ( 38
)
Let us bound g u t0 Lip from above. For any t > 0, we dene

h u t (x) = g u t (x) exp( t0 0 (c uu • g u s + c uv • ξ v s ) ds).
The exponent of the exponential term is positive and independent of x, thus g u t0 Lip ≤ h u t0 Lip . Furthermore, accord- ing to Part 4 of Chapter 5 in [START_REF] Friedman | Partial Dierential Equations of Parabolic Type[END_REF], h u t0 (x) = X Γ t0 (x, y)ξ 0 (dy) where Γ is the fundamental solution to the system

     ∂ t Γ = m u ∆Γ + (b u (x) -d u (x))Γ on X × R + , ∂ n Γ = 0 on ∂X × R + , Γ (0, dx) = ξ 0 (dx).
As t 0 > 0, Γ t0 L ∞ (X ) and ∇Γ t0 L ∞ (X ) are bounded from above and there exists C 3 such that

g u t0 Lip ≤ h u t0 Lip ≤ C 3 ξ u 0 , 1 ≤ C 3 M, (39)
where M has been dened in the beginning of the proof. Combining ( 37), ( 38), (39) and the fact that W 1 (ξ u 0 , ξu ) < γ , we nd C 4 ( , γ ) > 0 such that

g u t0 -ḡu L 2 ≤ (W 1 (ξ u 0 , ξu ) + C 2 )e C1t0 (C 3 M + ḡu Lip ) 1/2 ≤ (γ + C 2 )e C1t0 (C 3 M + ḡu Lip ) 1/2 := C 4 ( , γ ).
We now deal with the last step of the proof. Let H u 2 < H u be the second eigenvalue of the operator m u ∆ x . + (b u -d u ). with Neumann boundary condition. Following ideas of Part 3.3 in [START_REF] Leman | Inuence of a spatial structure on the long time behavior of a competitive lotka-volterra type system[END_REF] but including the v-population process with size lower than , we prove that if and γ satisfy c+3

c uu L 2 C 4 ( , γ ) < (H u -H u 2 )/2, then for all t 0 ≤ t ≤ t , g u t -ḡu L 2 ≤ ḡu L 2 H u ( c + 3C 4 ( , γ ) c uu L 2 ).
Finally, if and γ satisfy max 1, ḡu

L 2 H u • ( c + 3 c uu L 2 C 4 ( , γ )) < min H u -H u 2 2 , γ 2 1 L 2 , then, for all t ≤ t , W 1 (g u t , ḡu ) ≤ g u t -ḡu L 2 1 L 2 ≤ γ 2 .
That ends the proof of Proposition 2.

Exit time

This subsection is devoted to the proof of Theorem 7. We split the proof into three lemmas similar to the ones in Dembo and Zeitouni [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF].

Let γ > 0 satisfying the assumptions of Theorem 7. We consider and γ as in Proposition 2 and set = 2 and ρ = γ 3 < γ. R K γ , S K 1 and T K have been dened by ( 32), ( 33) and ( 34) and let us dene

τ = inf{t ≥ 0, W 1 (ν K,u t , ξu ) ∈]ρ, γ[}.
Lemma 2 Under Assumption 1, we have

lim t→+∞ lim sup K→+∞ 1 K log sup ν K 0 ∈(B( ξu ,γ)×B(0, ))∩M K F P ν K 0 (τ ∧ T K ∧ S K 1 > t) = -∞.
That is, the probability that the process ν K,u stays a long time in the ring B( ξu , γ) \ B( ξu , ρ) is exponentially small. The proof requires a comparison with the deterministic paths of Equation ( 3), the diculty is to prove that there exists a nite time after which all deterministic paths starting in the ring are out of the ring. The fact that the probability is exponentially small is a consequence of Theorem 5.

Lemma 3 Under Assumption 1, there exists V > 0 such that

lim sup K→+∞ 1 K log sup ν K 0 ∈C∩M K F P ν K 0 (W 1 (ν K,u τ , ξu ) ≥ γ, τ ≤ T K ∧ S K 1 ) ≤ -V,
where C = B( ξu , γ ) × B(0, ).

Once again, the proof is based on a comparison with the deterministic paths and application of Proposition 2 and Theorem 5.

Lemma 4 Under Assumption 1, for all C > 0, there exists T (C, ρ) > 0 such that,

lim sup K→+∞ 1 K log sup ν K,u 0 ∈B( ξu ,γ)∩M K F P ν K 0 sup t∈[0,T (C,ρ)] W 1 (ν K,u t , ν K,u 0 ) ≥ ρ < -C. (40)
This lemma means that there exists a small time interval during which any process stays close from its starting point with an exponentially high probability. Since the stochastic process includes a jump part and a diusive part, we study not only the size of the population process during a small time interval, but also a sum of reected diusion processes.

Theorem 7 is proved using the two last lemmas 3 and 4 and we do not give details as it can be adapted from [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]. The main change is that the proof has to be done on {t ≤ T K ∧ S K 1 } to ensure that the v-process size ν K,v , 1 is small and that no other mutation appears, but Lemma 3 allows to circumvent this diculty.

There remains to prove the three lemmas.

Proof (Proof of Lemma 2) Note that if W 1 (ν K,u 0 , ξu ) < ρ for all K ≥ 1, the result is obvious. Otherwise, let us dene the following set:

A(T ) = adh{ν ∈ D([0, T ], M F (X × {u, v})), ∀t ∈ [0, T ], W 1 (ν u t , ξu ) ∈]ρ, γ[ and ν v t , 1 < }, ( 41
)
where adhS is the closure of the set S. Observe that {ν K ∈ A(T )} = {τ ∧T K ∧ S K 1 > T } a.s. and that the set C = adh(B( ξu , γ) × B(0, )) is a compact set of M F (X × {u, v}) as X is bounded. By applying Theorem 5 with the closed set A(T ) and the initial compact set C, we nd lim sup

K→+∞ 1 K log sup ν K 0 ∈C∩M K F P ν K 0 (ν ∈ A(T )) ≤ -inf ν∈A(T ) I T (ν).
Thus, the proof is complete if we show that inf ν∈A(T ) I T (ν)→ + ∞ as T tends to +∞. To this aim, we will rst show that any solution to (3) cannot belong to A(T 0 ) if T 0 is large enough. Precisely, we set δ ∈]0, ρ/2[, and prove that there exists T 0 > 0, such that any (ξ t ) t≥0 solution to (3) with an initial condition satisfying

W 1 (ξ u 0 , ξu ) ∈]ρ, γ[ and ξ v 0 , 1 < satises inf ν∈A(T0) sup t∈[0,T0] W 1 (ξ t , ν t ) ≥ δ. (42)
Assume that (42) holds. Since sup ν∈A(T0) sup t∈[0,T0] ν, 1 is bounded, we can use Proposition 1 to deduce that there exists C > 0 such that for any ν ∈ A(T 0 ), δ ≤ C(I T0 (ν) + I T0 (ν)). As x → x + √ x is a bijective function from R + to R + , we nd a constant C(δ) > 0 which is a lower bound on I T0 (ν). Finally, for T > T 0 and ν ∈ A(T ), we decompose ν as a sum of n := [T /T 0 ] terms : using the non-variational formulation (25) of I T > 0, we nd a sequence (ν i ) i=1..n ∈ A(T 0 ) such that

I T (ν) ≥ n i=1 I T0 (ν i ) ≥ C(δ)n -→ T →+∞

+∞.

There remains to prove that (42) holds. Let W 1 (ξ u 0 , ξu ) ∈]ρ, γ[ and ξ v 0 , 1 < , and let (ξ t ) t≥0 be the solution to (3) with initial condition ξ 0 = ξ u 0 δ u + ξ v 0 δ v . Using Theorems 1.2 and 1.4 in [START_REF] Leman | Inuence of a spatial structure on the long time behavior of a competitive lotka-volterra type system[END_REF], we know that ξ t ∈ M F (X × {u, v}) converges to a stationary state which is either ξu δ u or ξv δ v or a state with coexistence. The assumptions on γ insure that none of those stationary states belong to (B( ξu , γ) \ B( ξu , ρ)) × B(0, ). Therefore, there exists T ξ0 such that

W 1 (ξ u T ξ 0 , ξu ) ∈ [ρ -2δ, γ + 2δ] or ξ v T ξ 0 , 1 ≥ + 2δ. ( 43
)
However T ξ0 depends on ξ 0 . Thus, we will use a compactness argument to nd a uniform time and conclude. First, using Gronwall's Lemma, we obtain : for

M ≥ ξu , 1 + 2γ + 2 , sup t∈[0,T ξ 0 ] ξ t , 1 ≤ ξ 0 , 1 e bT ξ 0 ≤ M e bT ξ 0 . ( 44
)
Then two solutions (ξ t ) t≥0 and (ζ t ) t≥0 to (3) which are initially close stay close during a short time. Indeed, using a mild equation, for any f ∈ C Lip (X ×{u, v})

ξ t -ζ t , f = ξ 0 -ζ 0 , P t f + t 0 ξ s -ζ s , (b -d -c • ξ s )P t-s f ds - t 0 ξ s , c • (ξ s -ζ s )P t-s f ds.
Since P u t f ∈ C Lip (X ) and according to Assumption 1, we nd a constant C 1 such that (b -d -c • ξ t )P t-s f /C 1 and c/C 1 belong to C Lip (X × {u, v}). We deduce that for all t ≤ T ξ0 ,

| ξ t -ζ t , ϕ | ≤ W 1 (ξ 0 , ζ 0 ) + C 1 t 0 sup r∈[0,s] W 1 (ξ r , ζ r )ds + C 1 sup r∈[0,T ξ 0 ] ξ r , 1 t 0 sup r∈[0,s] W 1 (ξ r , ζ r )ds. (45)
Using Gronwall's Lemma and (44), we conclude that there exists a constant

C(T ξ0 ) such that sup r∈[0,T ξ 0 ] W 1 (ξ r , ζ r ) ≤ C(T ξ0 )W 1 (ξ 0 , ζ 0 ). Choosing α ξ0 = δ/C(T ξ0 ), we nd for all ζ 0 with W 1 (ξ 0 , ζ 0 ) < α ξ0 , sup r∈[0,T ξ 0 ] W 1 (ξ r , ζ r ) < δ.
In addition with (43) and (41), we obtain for all ζ 0

with W 1 (ζ 0 , ξ 0 ) < α ξ0 , inf ν∈A(T ξ 0 ) sup t∈[0,T ξ 0 ] W 1 (ζ t , ν t ) ≥ δ.
Note that adh((B( ξu , γ)\B( ξu , ρ))×B(0, )) is a compact set of M F (X ×{u, v}) as X is bounded. It is covered by ∪ ξ0∈(B( ξu ,γ)\B( ξu ,ρ)×B(0, )) B(ξ 0 , α ξ0 ). We extract a nite cover ∪ n i=1 B(ξ i 0 , α ξ i 0 ). Finally, dening T 0 = max i=1..n T ξ i 0 , we conclude : for any ξ 0 with W 1 (ξ u 0 , ξu ) ∈]ρ, γ[ and ξ v 0 , 1 < , we have inf

ν∈A(T0) sup t∈[0,T0] W 1 (ξ t , ν t ) ≥ δ.
Proof of Lemma 2 is now complete.

Proof (Proof of Lemma 3) Lemma 2 gives T 1 such that lim sup

K→+∞ 1 K log sup ν K 0 ∈(B( ξu ,γ)×B(0, ))∩M K F P ν K 0 (τ ∧ T K ∧ S K 1 > T 1 ) ≤ -1. ( 46
)
Thus we limit our study to the time interval [0, T 1 ]. Since the initial states under study satisfy ν K 0 , 1 ≤ ( ξu , 1 + 2γ + 2 ) and using Lemma 1, we nd N > 0 such that lim sup

K→+∞ 1 K log sup ν K 0 ∈(B( ξu ,γ)×B(0, ))∩M K F P ν K 0 ( sup t∈[0,T1] ν K t , 1 ≥ N ) ≤ -1. (47) Let M ≥ ( ξu , 1 + 2γ + 2 ) ∨ N and A = ν ∈ D([0, T 1 ], M F (X × {u, v})) | ∃t ∈ [0, T 1 ], W 1 (ν u t , ξu ) ≥ γ, sup t∈[0,T1] ν v t , 1 < and sup t∈[0,T1] ν t , 1 < M . (48)
For all K and ν 0 ∈ M K F ,

P ν0 (W 1 (ν K,u τ , ξu ) ≥ γ, τ ≤ T K ∧ S K 1 ) ≤ P ν0 τ ≤ T 1 , sup t∈[0,T1] ν K t , 1 ≥ M + P ν0 (τ ≤ T 1 , ν K ∈ A) + P ν K 0 (τ > T 1 , τ ≤ T K ∧ S K 1 ) ≤ P ν0 sup t∈[0,T1] ν K t , 1 ≥ M + P ν0 (ν K ∈ A) + P ν K 0 (τ ∧ T K ∧ S K 1 > T 1 )
.

Then, we use Theorem 5, the denition of A (48), (46), and (47) to nd lim sup

K→+∞ 1 K log sup ν K 0 ∈C∩M K F P ν K 0 (ν K τ ∈ B c ( ξu , γ), τ ≤ T K ∧ S K 1 ) ≤ -min 1, inf ν∈A,ν0∈C I T1 (ν) ,
where C = adh(B( ξu , 3ρ) × B(0, )).

ξ v t , 1 < = 2 , then sup t∈[0,T1] W 1 (ξ u t , ξu ) < γ/2.
We deduce immediately that for any ν ∈ A, with ν 0 ∈ C, if ξ is the solution to (3) with ξ 0 := ν 0 ,

sup t∈[0,T1] W 1 (ν t , ξ t ) ≥ sup t∈[0,T1] max{W 1 (ν u t , ξ u t ), W 1 (ν v t , ξ v t )} ≥ γ 2 ∧ .
Finally Proposition 1 ensures that there exists C > 0 such that for any ν ∈ A, T T1 (ν) ≥ C.That ends the proof of Lemma 3.

Proof (Proof of Lemma 4) Let us x f ∈ C Lip (X ) and study the following dierence, using the construction of the process ν K,u :

| ν K,u t , f -ν K,u 0 , f | ≤ 1 K   i∈N notdead t |f (X i t ) -f (X i 0 )| + i∈N dead t |f (X i 0 )| + i∈N born t |f (X i t )|   ,
where N notdead

f ∈ C Lip (X ), | ν K,u t , f -ν K,u 0 , f | ≤ 1 K   i∈N notdead t |X i t -X i 0 |   + N dead t + N born t K . (49)
(49) is true for any f ∈ C Lip (X ) so we nd the same upper bound for W

1 (ν K,u t , ν K,u 0
). We use now the stopping time

τ K N = inf{t > 0, | ν K,u t , 1 | > N }. On {τ K N ≥ t}, W 1 (ν K,u t , ν K,u 0
) is stochastically bounded by 1

K KN i |X i t -X i 0 |+ P(t)
K , where {(X i t ) t≥0 } i∈{1..KN } are KN independent reected diusion processes driven by Equation ( 2) with the diusion coecient m u and (P(t)) t≥0 is a Poisson process with intensity ( b + d + N c)KN . Finally,

P sup t∈[0,T ] W 1 (ν K,u t , ν K,u 0 ) ≥ ρ ≤ P(τ K N ≤ T ) + P τ K N ≥ T, sup t∈[0,T ] W 1 (ν K,u t , ν K,u 0 ) ≥ ρ , ≤ P(τ K N ≤ T ) + P 1 K KN i=0 sup t∈[0,T ] |X i t -X i 0 | ≥ ρ 2 + P sup t∈[0,T ] P(t) K ≥ ρ 2 , (50)
Using Lemma 1, we can x N ∈ N such that lim sup

K→+∞ 1 K log P(τ K N ≤ T ) ≤ -C. (51)
Let us now consider the second term of (50). For x = (x 1 , .., x KN ) ∈ X KN , P x denote the probability of (X 1 , .., X KN ) under which (X 1 0 , .., X KN 0

) is equal to x. Let Υ be the stopping time Υ = inf{s ≥ 0, KN i=0 |X i s -X i 0 | ≥ ρK/2}. Using
the Markov property, we nd

P x sup t∈[0,T ] KN i=0 |X i t -X i 0 | ≥ ρK 2 = P x(Υ ≤ T ) ≤ P x Υ ≤ T, KN i=0 |X i T -X i 0 | ≥ ρK 4 + P x Υ ≤ T, KN i=1 |X i T -X i Υ | ≥ ρK 4 ≤ P x KN i=0 |X i T -X i 0 | ≥ ρK 4 + E x T 0 P Xs KN i=1 |X i T -s -X i 0 | ≥ ρK 4 1 Υ ∈ds ≤ 2 sup ȳ∈X KN ,s∈[0,T ] P ȳ KN i=0 |X i s -X i 0 | ≥ ρK 4 . ( 52
)
The aim is thus to nd an upper bound on the last term for any x ∈ X KN and any s ∈ [0, T ]. Using Markov's inequality,

P x KN i=0 |X i s -X i 0 | ≥ ρK 4 ≤ e -Kρ 4 √ T KN i=1 E x i e |X i s -x i | √ s . ( 53
)
If we denote the kernel of the semigroup P u s of the reected diusion process by p u s (x, y), Part 3 in [START_REF] Wang | Gradient estimate on convex domains and applications[END_REF] and the fact that X is compact imply that there exist two positive constants C 1 , C 2 such that for any x, y ∈ X ,

p u s (x, y) ≤ C 1 s d/2 e -|x-y| 2 C 2 s . (54)
Thus, using (54) and a change of variables, we nd that there exists C 3 > 0 independent from s such that

E x i e |X i s -x i | √ s ≤ R d C 1 s d/2 e -|x-y| 2 C 2 s e |y-x| √ s dy ≤ R d C 1 e -|z| 2 C 2 e |z| dz = C 3 < +∞. (55)
We deduce with the last line in (52), ( 53) and (55) that

P x sup t∈[0,T ] KN i=0 |X i t -X i 0 | ≥ ρK 2 ≤ 2e -K " ρ 4 √ T -ln(C3)N " , (56)
and there exists T 1 such that for all T ≤ T 1 , ρ We are now ready to prove the main theorem of this paper. The structure of Theorem 2's proof is similar to the one of [START_REF] Champagnat | A microscopic interpretation for adaptive dynamics trait substitution sequence models[END_REF]. Thus, we do not repeat all the details but only focus on the points that are dierent.

P(t) K ≥ ρ 2 ≤ P e rP(T ) ≥ e r Kρ 2 ≤ e -ρrK 2 E[e rP(T ) ] ≤ e (-K( ρr 2 -( b+ d+cN )N T (e r -1))) , ≤ e " -K " ρ 2 h log " ρ 2( b+ d+cN )N T " -1 i +( b+ d+cN )N T ""
The rst proposition concerns the behavior of the rst mutation occurence S K 1 , when the initial state is monomorphic. Proposition 3 Suppose that Assumptions 1 and (7) hold. Let u ∈ U and C u a compact subset of M F (X × {u}) such that 0

∈ C u . If ν K 0 ∈ C u ∪ M K F (X ),
then for any γ > 0,

lim K→+∞ P ν K 0 S K 1 > log K, sup t∈[log K,S K 1 ]
W 1 (ν K t , ξu δ u ) ≥ γ = 0, and, lim

K→+∞ P ν K 0 S K 1 > t Kq K
= exp -t X pb u (x) ξu (dx) .

Proposition 3 is proved using similar arguments as those of the proof of Lemma 2 in [START_REF] Champagnat | A microscopic interpretation for adaptive dynamics trait substitution sequence models[END_REF]. It is a consequence of the following lemma.

Lemma 5 For any α > 0, there exists T α > 0 such that for any ξ 0 ∈ C u , for any t ≥ T α , W 1 (ξ t , ξu ) < α, where (ξ t ) t≥0 is the solution to Equation (3) with initial state ξ 0 .

Proof On the one hand, Theorem 1.4 in [START_REF] Leman | Inuence of a spatial structure on the long time behavior of a competitive lotka-volterra type system[END_REF] implies that the density of ξu is a stable monomorphic equilibrium for the L 2 -distance. Using a proof in three steps as that of Proposition 2, we prove a W 1 -stability : there exists α such that for any ξ 0 ∈ B( ξu , α ) and for any t ≥ 0, W 1 (ξ t , ξu ) < α.

On the other hand, for any ξ 0 ∈ C u , ξ t converges towards ξu . There exists T ξ0 such that W 1 (ξ T ξ 0 , ξu ) < α /2. Using that P u t f ∈ C Lip (X ) and inequalities similar to [START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF] and [START_REF] Wang | Gradient estimate on convex domains and applications[END_REF], we show that for any t ≥ 0 and ζ 0 ∈ M F (X ), sup r∈[0,t] W 1 (ζ t , ξ t ) ≤ C(t)W 1 (ζ 0 , ξ 0 ), where (ζ t ) t≥0 is the solution to (3) with initial state ζ 0 ∈ M F (X × {u}). Consequently, there exists δ ξ0 > 0 such that for any ζ 0 ∈ B(ξ 0 , δ ξ0 ), W 1 (ζ T ξ 0 , ξ T ξ 0 ) < α /2. Thus, for any ζ 0 such that W 1 (ζ 0 , ξ 0 ) < δ ξ0 , for any t ≥ T ξ0 , W 1 (ζ t , ξu ) < α. Finally, as C u is a compact set, there exists a nite number of balls such that C u ⊂ ∪ n i=1 B(ξ i 0 , δ ξ i 0 ). Dening T α = max i=1..n T ξ i 0 , we deduce the lemma.

Moreover, for all η > 0 and all γ > 0, lim K→∞ P θ 0 ≤ S K 1 ∧ η Kq K = 1, and lim K→∞ P W 1 (ν K θ0 , ξV0 ) < γ = 1.

Proof We set γ > 0 small enough to use Theorem 7 : there exist γ , , V such that sup ν K,u 0 ∈B( ξu ,γ ),ν K,v 0 ∈B(0, )

P ν K 0 R K γ ≥ T K ∧ S K 1 ∧ e KV → K→∞ 1, (59)
where R K γ , T K have been dened by [START_REF] Polechová | Speciation through competition: a critical review[END_REF] and [START_REF] Roques | Modèles de réaction-diusion pour l'écologie spatiale[END_REF]. Let assume that ≤ γ and that K is large enough such that q K ≤ γ. Then on {t ≤ R K γ ∧ T K ∧ S K 1 }, the process (ν K,v t ) t≥0 is stochastically bounded: Using same kind of computations as in Lemma 3 in [START_REF] Champagnat | A microscopic interpretation for adaptive dynamics trait substitution sequence models[END_REF], we deduce that for all η > 0,

1 K Z inf ν K,v 1 K Z sup , Z inf
P ν K 0 θ 0 ≤ S K 1 ∧ η q K K , V 0 = u, W 1 (ν K θ0 , ξu ) < γ ≥ P δx 0 Υ sup 0 ≤ η q K K ∧ Υ sup K -P ν K 0 η q K K ≥ S K 1 -P ν K 0 η q K K ∧ S K 1 ∧ T K ≥ R K γ . (60)
(59) implies that the last term tends to 0 under Assumption [START_REF] Coville | Convergence to equilibrium for positive solutions of some mutation-selection model[END_REF]. Since the number of individuals is stochastically bounded from above by a birth and death process with birth rate b and competition rate c, Lemma 2 in [START_REF] Champagnat | A microscopic interpretation for adaptive dynamics trait substitution sequence models[END_REF] implies that for any α > 0, there exists η > 0 such that lim sup K→+∞ P(S K 1 ≤ η Kq K ) ≤ α/3. The main diculty is to evaluate the rst term. Theorem 3 implies that Let us show that this solution is close to φ vu . Theorem 3 implies that φ γ,vu is positive if and only if H v -c vu • ξu + cγ = H v -H u κ vu κ uu + cγ > 0.

P δx 0 Υ sup 0 ≤ η q K K ∧ Υ sup
(62)

First case: H v κ uu -H u κ vu < 0 (Point 1 in Assumption 2)

We can nd γ small enough such that (62) is not satised, thus φ γ,vu ≡ φ vu ≡ 0. In addition with (60) and (61), we deduce that for all α > 0, lim

K→+∞ P ν K 0 θ 0 ≤ S K 1 ∧ η q K K , V 0 = u, W 1 (ν K θ0 , ξu ) < γ ≥ 1 -α.
Proposition 4 is proved in this case.

Second case: H v κ uu -H u κ vu > 0 (Point 2 in Assumption 2)

Hence ( 62) is satised for all γ > 0. Let C be cγ inf y∈X b v (y)φ vu (y) and set L γ (f ) =

m v ∆ x f + (b v (x) -d v (x) -c vu • ξu + cγ)f -b v (x)f 2 . We have L γ (φ vu ) = cγφ vu ≥ 0, L γ ((1 + C)φ vu ) = (1 + C)φ vu [cγ -Cb v φ vu ] ≤ 0.
As φ γ,vu is the unique solution to L γ (f ) = 0, we deduce the following inequalities from a comparison theorem (see for example Theorem III.5 in [START_REF] Roques | Modèles de réaction-diusion pour l'écologie spatiale[END_REF]) : for any x ∈ X , (1 + C) φ vu (x) ≥ φ γ,vu (x) ≥ φ vu (x).

(63)

We split the end of the proof into three steps regarding as the proof of Lemma 3 in [START_REF] Champagnat | A microscopic interpretation for adaptive dynamics trait substitution sequence models[END_REF].

Let us x α > 0. (60), ( 61) and (63) imply that if K is large enough,

P ν K 0 T K 0 ≤ η q K K ∧ S K 1 ∧ R K γ ∧ T K ≥ 1 -φ vu (x 0 ) -α.
On the other hand, if K is large enough :

P ν K 0 T K ≤ η q K K ∧ S K 1 ∧ R K γ ≥ P δx 0 T inf K ≤ η q K K ∧ T inf 0 -P ν K 0 η q K K ≥ S K 1 -P ν K 0 η q K K ∧ S K 1 ∧ T K ≥ R K γ .
We use Theorem 4 to bound from below the rst term of the r.h.s and we deal with the two other terms as in (60). Thus,

P ν K 0 T K ≤ η q K K ∧ S K 1 ∧ R K γ ≥ φ vu (x 0 ) -α.
Thus, the v-process ν K,v reaches a non-negligible size K , before η q K K ∧ S K 1 ∧ R K γ , with a probability that tends to φ vu (x 0 ).

  , we observe the evolution of the trait associated with a change of spatial niches and spatial patterns over time. After a short time (a), the spatial distribution of the monomorphic population with trait u = 0.4 stabilizes. Then, in Figures (b), (c) and (d), we observe a phenomenon of invasion and replacement : some individuals with trait u = 0.499 appear, invade and nally replace the previous population with trait u = 0.4. Note the change of spatial niche, see Figure (c). The locations of the population with trait u = 0.499 are slightly larger than the one of trait u = 0.4. Other phenomena of invasion and replacement with a displacement of the spatial niche are detected until the time t = 4500 (Figure (e)). In a second phase, the population evolves to become more and more generalist (Figure (f )) : the length of the spatial nicheis increasing at each event of invasion and replacement. Indeed, it is protable to become generalist and feed oneself on a larger range of resources since the maximal encounter rate of a more generalist individual does not decrease[START_REF] Futuyma | The evolution of ecological specialization[END_REF].

  Those rates are assumed to have the following properties : Assumption 3 b, d are two Lipschitz functions, b is a positive function and there exists b, b, d such that for all x ∈ X , b < b(x) ≤ b and 0 ≤ d(x) ≤ d. Moreover, d is a non-zero function.

R s 0

 0 (b(Xr)+d(Xr))dr ds . We set I(s) := s 0 (b(X r ) + d(X r ))dr. Using the Markov property of η, we obtain

t

  is the set of indices of the individuals with trait u alive at time 0 and not dead during [0, t]; N dead t represents the set of indices of the individuals with trait u alive at time 0 and dead during [0, t]; and N born t is the set of indices of the individuals with trait u born during ]0, t]. Since

4 √T

 4 -ln(C 4 )N ≥ C. Finally, concerning the third term of (50), let r = log ρ 2( b+ d+cN )N T , P sup t∈[0,T ]

, and there exists T 2 ≤, ρ 2 log ρ 2 (

 22 T 1 such that for all T ≤ T 2 b+ d+cN )N T -1 + ( b + d + cN )N T ≥ C. In addition with (12), (51) and (56), we deduce (40) with T (C, ρ) = T 2 . 7 Proof of Theorem 2

, 1 ≥ K }, Υ sup 0 =

 10 , Z sup are two branching diusion processes starting with one individual at location x 0 , their birth rates are respectively b v (x)(1 -γ) and b v (x), and their death rates ared v (x) + c vu • ξu + 2cγ and d v (x) + c vu • ξu -cγ. Let us set Υ sup K = inf{t ≥ 0, Z sup t inf{t ≥ 0, Z sup t , 1 = 0},and respectively Υ inf K , Υ inf 0 associated with Z inf .

K → K→+∞ P δx 0 (Υ sup 0 <

 0 +∞) = 1 -φ γ,vu (x 0 ), (61)where φ γ,vu is the solution to the following elliptic equation on X with Neumann boundary conditionm v ∆ x φ γ,vu (x) + [b v (x) -d v (x) -c vu • ξu + cγ]φ γ,vu (x) -b v (x)φ γ,vu (x) 2 = 0.

  It remains to prove that the r.h.s. is strictly negative. Proposition 2 implies that any solution (ξ t ) t≥0 to (3) with ξ 0 ∈ C satises the property :

	if	sup
		t∈[0,T1]

if H v κ uu -H u κ vu ≤ 0, φ vu (x) = 0 for all x ∈ X ,

Acknowledgements: I would like to thank S. Méléard for her guidance during my work. I also thank C. Léonard, G. Raoul, C. Tran and A. Veber for their help. I acknowledge partial support by the Chaire Modélisation Mathématique et Biodiversité of VEOLIA -École Polytechnique -MNHN -F.X.

Proof (Proof of Proposition 3) First, note that the rst probability of Proposition 3 is non-increasing with γ. Thus, it is sucient to prove the property for any small γ > 0. Let us assume that γ satises the assumptions of Theorem 7. Therefore, there exist γ > 0, V > 0, such that sup

R K γ , S K 1 are dened by [START_REF] Polechová | Speciation through competition: a critical review[END_REF] and [START_REF] Rao | Applications of Orlicz spaces[END_REF] respectively. We set 2α = γ , then Lemma 5 and Theorem 5 imply that

Using the Markov property, we deduce if K is suciently large such that log(K) > T α ,

.

The second term of the l.h.s tends to 0 when K tends to +∞ according to (58) and the rst term of the r.h.s tends to 0 according to (57). It remains to deal with the second term of the r.h.s. On {t ≤ R K γ ≤ S K 1 }, the number of mutations M t is stochastically bounded from below by a Poisson process with parameter Kq K ( b u p, ξu -γ b u p Lip ) which is positive if γ is small enough. We conclude the proof of the rst point with the fact that, under Assumption [START_REF] Coville | Convergence to equilibrium for positive solutions of some mutation-selection model[END_REF],

The second point of Proposition 3 is easily deduced from this rst point and Lemma 2 in [START_REF] Champagnat | A microscopic interpretation for adaptive dynamics trait substitution sequence models[END_REF].

The second proposition studies the process with a dimorphic initial state.

Let us dene : θ 0 is the rst time when the population becomes monomorphic, V 0 is the phenotypic trait of the population at time θ 0 .

Proposition 4 Suppose Assumptions 1, 2 and (7) hold, and that the initial

Once the mutant population has reached a non-negligible size, we can compare the stochastic process and the deterministic limiting process. Under point 2 in Assumption 2, there exist T > 0 and γ 2 > 0 such that for any ξ 0 ∈ adh(B( ξu , γ) × (B(0, 2 ) \ B(0, )), for any t ≥ T ,

where ξ is the solution to (3) with a dimorphic initial state ξ 0 ∈ M F (X × {u, v}). This can be proved using similar arguments than those of the proof of Lemma 5. Moreover, using Theorem 5 and Proposition 1,

with ξ t,ν K 0 the solution to (3) with initial state ν K 0 . The two previous results and the Markov property imply that, if K is large enough,

Finally, we use the Markov property at time T K + T and we conclude as in Lemma 3 in [START_REF] Champagnat | A microscopic interpretation for adaptive dynamics trait substitution sequence models[END_REF]. If γ 2 is suciently small, we prove that, with a probability that tends to 1, after time T K + T , the u-population process ν K,u will become extinct before its size reaches the threshold √ γ 2 and before the v-process ν K,v moves away from a neighborhood of the equilibrium ξv .

That concludes the proof of Proposition 4 in the second case.

Theorem 2 is deduced from Propositions 3 and 4 in a similar way to Theorem 1 in [START_REF] Champagnat | A microscopic interpretation for adaptive dynamics trait substitution sequence models[END_REF] by using the transition probabilities of the jump process (Λ t ) t≥0 .