Sylvain Billiard
email: sylvain.billiard@univ-lille.fr

Hélène Leman
email: helene.leman@inria.fr

Thomas Rey
email: thomas.rey@univ-lille.fr

Viet-Chi Chi Tran

Continuous

Keywords: ecological mutualistic community, birth and death process, interacting particles, limit theorem, kinetic limit, graphon, integro-dierential equation, stationary solution. AMS Codes: 92D40, 92D25, 05C90, 60J80, 60F17, 47G20

published or not. The documents may come

observations thus needs much eort, without being ensured that the inferred networks are representative of the studied communities (interactions can indeed depend on space and time, but also on the abundances of the species which makes them harder or not to observe). From a theoretical point of view, the study of the dynamics and stability of communities most often relies either on the numerical analysis of large systems of ordinary dierential equations (ODEs), or on the study of the system near equilibrium in simplied communities (see e.g. [START_REF] May | Will a large complex system be stable?[END_REF][START_REF] May | Qualitative stability in model ecosystems[END_REF][START_REF] Barbier | Generic assembly patterns in complex ecological communities[END_REF][START_REF] Bunin | Ecological communities with Lotka-Volterra dynamics[END_REF][START_REF] Cohen | When will a large complex system be stable[END_REF][START_REF] Fyodorov | Nonlinear analogue of the May-Wigner instability transition[END_REF][START_REF] Takeuchi | Global dynamical properties of Lotka-Volterra systems[END_REF][START_REF] Tang | Reactivity and stability of large ecosystems[END_REF][START_REF] Thébault | Stability of ecological communities and the architecture of mutualistic and trophic networks[END_REF][START_REF] Lever | The sudden collapse of pollinator communities[END_REF] and [START_REF] Akjouj | Complex systems in ecology: A guided tour with the lotka-volterra model[END_REF] for a review). This makes dicult the identication of general properties of ecological communities. Overall, despite being more than 50 years old, the question of how correctly modelling ecological networks and communities is still challenging. In this paper, we develop a new theoretical framework that could help in addressing the previous questions, based on the fact that networks are hierarchically structured.

Interactions can indeed be considered at dierent scales: between individuals, between species or at the level of the whole community [START_REF] Guimarães | The structure of ecological networks across levels of organization[END_REF]. At all scales however, several studies showed evidence that the structure of the interaction networks is due to traits variation between or within species' such as size, symmetry or time of activity, e.g. [START_REF] Chamberlain | Traits and phylogenetic history contribute to network structure across canadian plant-pollinator communities[END_REF][START_REF] Watts | The inuence of oral traits on specialization and modularity of plantpollinator networks in a biodiversity hotspot in the Peruvian Andes[END_REF][START_REF] Villalobos | Specialization in plantpollinator networks: insights from local-scale interactions in glenbow ranch provincial park in alberta, canada[END_REF][START_REF] Arroyo-Correa | Individual-based plantpollinator networks are structured by phenotypic and microsite plant traits[END_REF]. Starting from a stochastic individual-based model, our goal is to provide simplications of an ecological network deriving the continuous limits of both the population sizes and the interaction graph structured by a trait.

We focus on a particular ecological community: plant and pollinator species. The interactions between individuals of each species are modelled by a bipartite random network. Interactions aect the demographic rates of plant and pollinator individuals because of resources exchanges. On the one hand, plants benet from pollinators visitations to increase their reproduction rate. On the other hand, pollinators consume nectar, leaves, pollen, etc. which increases their reproduction or survival rates. However, producing such resources is costly for the plants, which can negatively aect plant demographic rates, proportionally to pollinators density [START_REF] Holland | Mutualism[END_REF][START_REF] Holland | A consumerresource approach to the density-dependent population dynamics of mutualism[END_REF].

The interaction rates between plant and pollinator individuals are modelled by a random graph where the species are the nodes of the graph. The existence of an edge between a plant and a pollinator species indicates that individuals of these two species can interact. The resulting graph is bipartite, since there is by denition no direct edges between insect species or between plant species. The topology of the graph depends on a trait that represents the degree of generalism of the species. Any pair of plant-pollinator species is connected independently from the other pairs with a probability that depends on the traits of the two considered species. This simple model corresponds to Erdös-Rényi graphs, where the probability of connection is the same for every pairs. Such a model can also generate particular structures such as nested or modular graphs which are commonly found in ecological networks [START_REF] Thébault | Stability of ecological communities and the architecture of mutualistic and trophic networks[END_REF][START_REF] Guimarães | The structure of ecological networks across levels of organization[END_REF].

In Section 1, we present the stochastic individual based model, how the random interactions graph is modelled, and how interactions aect demographic rates. Considering large populations but with a xed number of species, we show in Section 2 how the dynamics can be approximated by a system of ODEs (close to the ones commonly studied in the ecological literature). The uctuations between the approximated deterministic limit and the stochastic individual based process are established.

In Section 3, we derive a continuous approximation of the random graph when the numbers of plant and pollinator species, say n and m, are also large. When the graph is dense (i.e. the order of the number of edges is in O(n × m)), the complex random network can be replaced by a continuous object, namely a graphon [START_REF] Borgs | Limits of randomly grown graph sequences[END_REF][START_REF] Lovász | Large networks and graph limits[END_REF]. In this case, the high-dimensional system of ODEs can be replaced by two partial integro-dierential equations: one for the plants and the other for the pollinators. The latter equations fall into the broader class of kinetic equations, the most well known and studied being the seminal Boltzmann and Vlasov equations. The use of such mesoscopic scale, namely intermediate between a microscopic agent based approach [START_REF] Fournier | A microscopic probabilistic description of a locally regulated population and macroscopic approximations[END_REF], and an averaged macroscopic one (over space) such as [START_REF] Pouchol | Global stability with selection in integro-dierential lotka-volterra systems modelling trait-structured populations[END_REF], is not a novelty for modeling competitive interactions between species [START_REF] Desvillettes | On selection dynamics for continuous structured populations[END_REF][START_REF] Jabin | On selection dynamics for competitive interactions[END_REF][START_REF] Bellomo | On the interplay between mathematics and biology: Hallmarks toward a new systems biology[END_REF][START_REF] Berardo | Interactions between dierent predatorprey states: a method for the derivation of the functional and numerical response[END_REF], i.e. with a large but not innite number of species with trait structure. Indeed, such approach can be traced back as early as the book of [START_REF] Roughgarden | Theory of Population Genetics and Evolutionary Ecology: An Introduction[END_REF] for the mathematical modeling of evolutionary ecology. However, the kinetic treatment of large random networks and their approximations by integro-dierential equations is new to our knowledge. Finally, in Section 4, the large-time behaviors of the ODE system or of the integro-dierential equations are studied and simulations are produced to explore several situations. [START_REF] Abbe | Community detection and stochastic block models: recent development[END_REF]

Stochastic individual based model with species interactions

In this section, we introduce a stochastic individual-based model of plant and pollinator species. In this model, the number of species and the number of individuals in each species are nite. Each species is dened by a trait, for instance a morphological or functional trait, which determines its degree of generalism. This trait shapes the interaction network: generalist plant species can be visited by a large number of insect species and generalist pollinator species can visit a large number of plant species.

Even though several coevolved traits can be considered in the model (e.g. orchids have ower with a particular morphology, color, phenology, etc.), we will focus on a single trait for the sake of simplicity.

The dynamics is driven by the births and deaths of individuals at random times that depend 1) on their position in the plant-pollinator network, determined by their species' trait, 2) by the weights of the interactions between pairs of species, and 3) by the population sizes of the species. The evolution of these interacting populations is described by a stochastic dierential equation (SDE) involving Poisson point measures and an acceptance-rejection algorithm as in the Gillespie algorithm [START_REF] Gillespie | A general method for numerically simulating the stochastic time evolution of coupled chemical reactions[END_REF] (see Fournier and Méléard [START_REF] Fournier | A microscopic probabilistic description of a locally regulated population and macroscopic approximations[END_REF] or Ferrière and Tran [START_REF] Ferrière | Stochastic and deterministic models for age-structured populations with genetically variable traits[END_REF] for the mathematical formulation of the Gillespie algorithm by mean of SDEs in cases with interactions).

Description of the plant-pollinator community

We consider n and m plant and pollinator species, respectively. Each plant specie is characterized by a trait x ∈ [0, 1] and each pollinator specie by a trait y ∈

[0, 1]. For i ∈ {1, . . . , n} = [[1, n]] and j ∈ {1, . . . , m} = [[1, m]],
we respectively denote x i and y j the trait of the plant species i and of the pollinator species j. These traits can represent for instance their degree of generalism, i.e. their tendency to interact with a large number of other species: a species is considered as generalist when its trait (x i or y j ) is close to 1, or specialist when it is close to 0.

The plant-pollinator interaction network as a bipartite random graph

Plants and pollinators interact through a bipartite network where each species is a vertex, and an interaction is an edge. We denote i ∼ j or j ∼ i when individuals of the plant species i can interact with individuals of the pollinator species j. There are no edges between two plant or two pollinator species because the network only represents interactions between plants and pollinators. Yet, when specifying the birth and death rates, competition kernels within and between plant species, and within and between pollinator species, will be introduced in the model.

The bipartite graph can be represented by an n × m adjacency matrix G n,m , with

G n,m ij =
1 if the pollinator species j can interact with the plant species i,

0 otherwise.
Regarding the structure of the network, we consider a stochastic bipartite graph that generalizes the Erdös-Rényi random graph ( [START_REF] Bollobás | Random graphs[END_REF][START_REF] Durrett | Random graph dynamics[END_REF][START_REF] Van Der Hofstad | Random Graphs and Complex Networks, volume 1 of Cambridge Series in Statistical and Probabilistic Mathematics[END_REF]). The probability ϕ(x i , y j ) that there is an edge i ∼ j between plant species i and pollinator species j is assumed to depend only on the traits x i and y j of these two species. Dierent pairs are supposed to be connected independently, the interaction graph is thus built such that:

Assumption 1.1. We assume that there exists a continuous function ϕ : [1,m]] are independent and distributed as Bernoulli random variables with parameters (ϕ(x i , y j )) (i,j)∈[ [1,n]]×[ [1,m]] . Example 1.2. Case 1. If there exists ϕ 0 ∈ (0, 1) such that for all i and j, ϕ(x i , y j ) = ϕ 0 , then the stochastic network is a bipartite Erdös-Rényi graph. The degrees of nodes representing pollinators (resp. plants), i.e. the numbers of edges, do not depend on the species traits. Degrees are thus independent binomial r.v. with parameters (m, ϕ 0 ) (resp. (n, ϕ 0 )) and the total number of edges follows a binomial distribution with parameters (nm, ϕ 0 ). Case 2. The probability of an edge i ∼ j between two species i and j depends on their degree of generalism, for instance by assuming: ϕ(x i , y j ) = x i y j .

[0, 1] 2 → [0, 1], such that conditionally on (x i , y j ) i,j∈[[1,n]]×[[1,m]] , the random variables (G n,m ij ) (i,j)∈[[1,n]]×[
(1.1) Under this assumption, two generalist species (x i and y j close to 1) have a higher probability to be connected than two specialist species (x i and y j close to 0). Nested graphs result from such an assumption (see Fig. 1 a-left for an illustration of nested graphs denition). Case 3. When plant and pollinator species can be structured in groups, for instance because of pollination syndrome or spatio-temporal segregation, the probability of an edge i ∼ j ϕ(x i , y j ) = ϕ IJ depends only on classes I ∋ i and J ∋ j. The resulting random graph is a Stochastic Block Model (SBM [START_REF] Holland | Stochastic blockmodels: some rst steps[END_REF], see e.g. [START_REF] Abbe | Community detection and stochastic block models: recent development[END_REF] for a review), often called modular networks in community ecology (see Fig. 1 a-right for an illustration of modular graphs denition).

If species i and j interact, i.e. if G n,m ij = 1, we denote by c n,m ij the weight of the interaction i ∼ j. The quantity c n,m ij describes the intensity and frequency of the relation between the pollinator species j and the plant species i. From the point of view of the plant, c n,m ij can be interpreted as a measure of the pollination services received from the pollinators. From the point of view of the pollinators, c n,m ij measures the quantity and quality of nutrients collected from the plants. Assumption 1.3. For all n, m ∈ N 2 , conditionally on [1,m]] is assumed to be a sequence of independent random variables such that the expected values only depend on the traits of the plants and pollinators through a function

(x i , y j ) (i,j)∈[[1,n]]×[[1,m]] , (c n,m ij ) (i,j)∈[[1,n]]×[
c n,m : [0, 1] 2 → R: for all i ∈ [[1, n]] , j ∈ [[1, m]]: E x,y c n,m ij = c n,m x i , y j , (1.2) with E x,y [A] = E[A|(x i , y j ) (i,j)∈[[1,n]]×[[1,m]] ],
and the variances decrease with n and m such that:

V max := sup n∈N,m∈N sup (i,j)∈[[1,n]]×[[1,m]] Var x,y ((m + n)c n,m i,j ) < ∞. (1.3)
Example 1.4. Depending on plants and pollinators' traits considered, dierent function c n,m i,j be considered:

The intensity and frequency of the interaction can be independent of the species traits, i.e. c n,m i,j = c n+m , which would mean that given a plant and a pollinator species interact, the eect of the interaction on their demographic rates is the same for all species;

x i and y j can represent the time of peak activity during a day or a season for plants i and pollinators j, and c n,m i,j a function measuring the overlap of activity for both species; Similarly, c n,m i,j can represent geographical rather than time overlap between both species;

x i and y j can virtually represent some aspects of the morphology of plants' owers and pollinators' body and the function c n,m i,j some morphological preference; For instance, assuming

(x i , y j ) ∈ [0, 1] 2 , c n,m i,j = x i y j
n+m would mean that pollinators and plants tend to interact more when the traits are both large, while if c n,m i,j = x i (1-y j ) n+m would mean that they will interact more when the plant has a large and the pollinator a low trait values.

Stochastic dynamics of the plant-pollinator community

The dynamics of the plant and pollinator populations within the community is ruled by random point birth and death events. We assume that the size of the population of plants and pollinators is scaled by a factor K > 0, called the carrying capacity in ecology. This carrying capacity K is a measure of the size of the system, in other words, it controls the total abundance of the whole community that can be sustained by the environment. A continuous limit of the stochastic dynamics will be obtained when the species abundances tend to innity, in other words when K → ∞.

Given the scaling factor K, we denote P K,i t and A K,j t the size of the plant and pollinator species i and j at time t. The plant and pollinator populations at time t can be represented by the following point measures:

P K,n,m t (dx) = 1 nK n i=1 P K,i t δ x i (dx), A K,n,m t (dy) = 1 mK m j=1
A K,j t δ y j (dy).

(1.4)

The interactions between plants and pollinators are characterized by an exchange of resources [START_REF] Holland | A consumerresource approach to the density-dependent population dynamics of mutualism[END_REF]. The quantity of resources gained by plants or pollinators R is modeled through a "mass-action model". At time t, in the population scaled by K, a single individual of the plant species i interacting with pollinator species j is supposed to gain a quantity of resources (here the pollination service) proportional to the abundance of pollinators A K,j t /K weighted by the interaction eciency c n,m ij , such that the total resource gained by a plant individual of species i through the pollination interactions is

R A,K,i t := j∼i c n,m ij A K,j t K .
(1.5)

Similarly, for a given pollinator of the species j, the resources gained from the interaction with the plant species at time t in the population parameterized by K is assumed to be:

R P,K,j t = i∼j c n,m ij P K,i t K .
(1.6)

The dynamics of the community is supposed to be governed by the birth and death rates of the plant and pollinator populations. We denote b P (R) and b A (R) the individual birth rate of plant and pollinator species, respectively, each depending on the quantity of resources exchanged R P or R A . Similarly, we denote d P (R) and d A (R) the individual death rates. Finally,

g P (R) := b P (R) -d P (R) and g A (R) := b A (R) -d A (R), (1.7)
are the component of the growth rate due to the interactions between plants and pollinators.

The plants and pollinators dynamics are also assumed to be aected by logistic competition among plants and among pollinators (within and between species competition). We suppose that competition strength depends on the traits x and y. This can represent the fact that plants or pollinators with similar traits tend to share similar ecological niche, or phenology, etc. A plant with trait x ∈ [0, 1]

suers an additional death rate term due to competition such that

k ⋆ P K,n,m t (x) := [0,1] k(x, x ′ )dP K,n,m t (dx ′ ) = 1 nK n i=1 k(x, x i )P K,i t , (1.8)
where k(x, x ′ ) quanties the competition pressure exerted by another plant of trait x ′ . Similarly, a pollinator with trait y ′ ∈ [0, 1] suers an additional death rate due to competition with pollinators of trait y is

h ⋆ A K,n,m t (y) := [0,1] h(y, y ′ )dA K,n,m t (dy ′ ) = 1 mK m j=1 h(y, y j )A K,j t , (1.9)
where h(y, y ′ ) quanties the competition of individuals with trait y ′ on individuals with trait y.

The following assumptions on the functions involved in the model will be needed, both for modeling and mathematical purposes.

Assumption 1.5. (i) The birth rates b P and b A are assumed to be bounded on R + by constants M P > 0 and M A > 0 respectively. Moreover, all the rate functions b P , b A , d P and d A are assumed to be locally Lipschitz continuous on [0, ∞).

(ii) The competition kernels k and h are assumed to be continuous on [0, 1] 2 .

Stochastic dierential equations

Following works by Méléard and co-authors [START_REF] Fournier | A microscopic probabilistic description of a locally regulated population and macroscopic approximations[END_REF][START_REF] Champagnat | Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models via timescale separation[END_REF], it is possible to describe the evolution of the population measures (P K,n,m t , A K,n,m t ) t∈R + dened in (1.4) by stochastic dierential equations (SDEs)

driven by Poisson point measures. Let us rst present these SDEs and then explain their heuristic meaning.

Denition 1.6. Suppose that assumptions 1.1, 1.3 and 1.5 hold.

Let Q P B (ds, dk, dθ), Q P D (ds, dk, dθ), (resp. Q A B (ds, dk, dθ) and Q A D (ds, dk, dθ)) be Poisson point measures on R + × E := R + × [[1, n]] × R + (resp. R + × F := R + × [[1, m]] × R + )
with intensity measure q(ds, dk, dθ) = ds n(dk) dθ where ds and dθ are Lebesgue measures on R + and where n(dk) is the counting measure on N * = {1, 2, . . . }. For i ∈ {1, . . . n} and j ∈ {1, . . . m}, one has

P K,i t =P K,i 0 + t 0 E 1l i=k 1l θ≤b P R A,K,i s- P K,i s- Q P B (ds, dk, dθ) - t 0 E 1l i=k 1l θ≤ d P R A,K,i s- +k⋆P K,n s-(x i ) P K,i s- Q P D (ds, dk, dθ) (1.10) A K,j t =A K,j 0 + t 0 F 1l j=k 1l θ≤b A R P,K,j s- A K,j s- Q A B (ds, dk, dθ) - t 0 F 1l j=k 1l θ≤ d A R P,K,j s- +h⋆A K,m s-(y j ) A K,j s- Q A D (ds, dk, dθ).
(1.11)

The above SDEs correspond to the mathematical formulation of the individual-based simulations classically used in ecology and originated with Gillespie's algorithm [START_REF] Gillespie | A general method for numerically simulating the stochastic time evolution of coupled chemical reactions[END_REF][START_REF] Gillespie | Exact stochastic simulation of coupled chemical reactions[END_REF]. The Poisson point mea-

sures Q P B and Q A B (resp. Q P D and Q A D )
give the random times of possible birth (resp. death) events. The indicators 1l i=k or 1l j=k in the integral ensure that the equations for P K,i or A K,j are indeed modied when the birth or death events aect the plant population i or pollinator population j. The indicators in θ correspond to an acceptance-rejection algorithm so that the birth and death times occur with the correct rates. Note that in these SDEs, the network is hidden in the denitions of the terms R A,K,i s and R P,K,j s (see (1.5) and (1.6)).

In order to approximate the individual-based model with ODEs or integro-dierential equations, we can reformulate the SDEs (1.10) and (1.11) by expressing the processes P K,i and A K,j as semimartingales. The following proposition also states the existence and uniqueness of a solution to (1.10)-

(1.11).
Proposition 1.7. Consider here n, m and K xed. Assume 1.5 and that

E ⟨P K,n,m 0 , 1⟩ 2 + ⟨A K,n,m 0 , 1⟩ 2 < +∞,
then the processes dened by (1.10) and (1.11) are well dened on R + . Moreover, for all T ≥ 0,

E sup t∈[0,T ] ⟨P K,n,m t , 1⟩ 2 + ⟨A K,n,m t , 1⟩ 2 < +∞.
(1.12)

For any f : [0, 1] → R measurable test function, we have:

⟨P K,n,m t , f ⟩ = 1 n n i=1 1 K P K,i t f (x i ) =⟨P K,n,m 0 , f ⟩ + t 0 1 nK n i=1 f (x i ) g P R A,K,i s - -k ⋆ P K,n,m s (x i ) P K,i s ds + 1 nK n i=1 f (x i )M K,i t , (1.13)
where (M K,i ) i∈{1,..,n} are square integrable martingales with predictable quadratic variation processes:

⟨M K,i ⟩ t = t 0 b P R A,K,i s + d P R A,K,i s + k ⋆ P K,n,m s (x i ) P K,i s ds. (1.14)
A similar expression holds for the pollinator populations. For any f : [0, 1] → R measurable test function, we have:

⟨A K,n,m t , f ⟩ = 1 m m j=1 1 K A K,j t f (y j ) =⟨A K,n,m 0 , f ⟩ + t 0 1 mK m j=1 f (y j ) g A R P,K,j s - -h ⋆ A K,n,m s (y j ) A K,j s ds + 1 mK m j=1 f (y j )M K,j t , (1.15)
where (M K,j ) j∈{1,..,m} are square integrable martingales with predictable quadratic variation processes:

⟨M K,j ⟩ t = t 0 b A R P,K,j s + d A R P,K,j s + h ⋆ A K,n,m s (y j ) A K,j s ds.
The proof follows from usual stochastic calculus with Poisson point processes (e.g. [START_REF] Ikeda | Stochastic Dierential Equations and Diusion Processes[END_REF]), as developed in [START_REF] Bansaye | Stochastic models for structured populations. Scaling limits and long time behavior[END_REF][START_REF] Fournier | A microscopic probabilistic description of a locally regulated population and macroscopic approximations[END_REF] for example. We however give a sketch of proof in Appendix A.1.

Thus the processes (P K,n,m t ) t∈R + and (A K,n,m t ) t∈R + are well-dened in the set D(R + , M F ([0, 1])) of right-continuous left-limited (càdlàg) processes with values in the set of nite measures on [0, 1]. The space M F ([0, 1]) embedded with the weak topology is a Polish space and the set of càdlàg functions is embedded with the Skorokhod topology which makes it Polish as well (e.g. [11, Th. 6.8 and 12.2]).

2 Community dynamics limit when abundances are large but the number of species is xed

In this section, we consider the numbers of species n and m as xed while the plant and pollinator populations size, represented by K, tend to +∞. In this section, since n and m are xed, these notations will be omitted in order to avoid cumbersomeness when there is no possible confusion.

Law of large numbers

Proposition 2.1. We consider a sequence (P K,n,m , A K,n,m ) K∈N of processes as in Denition 1.6, with initial conditions such that

sup K∈N E ⟨P K,n,m 0 , 1⟩ 3 + ⟨A K,n,m 0 , 1⟩ 3 < +∞, (2.1)
and such that there exist ( P 1 0 , . . . , P n 0 ) and ( A 1 0 , . . . A m 0 ) satisfying the following convergences almost surely:

lim K→+∞ P K,i 0 K = P i 0 , lim K→+∞ A K,j 0 K = A j 0 .
Then, for all T ≥ 0, the following convergence holds almost surely for all

i ∈ [[1, n]] and j ∈ [[1, m]]: lim K→∞ sup t≤T sup i,j P K,i t K -P i t , A K,j t K -A j t = 0, (2.2)
where ( P 1 t , . . . , P n t ) t≥0 and ( A 1 t , . . . A m t ) t≥0 are the unique solution of the system, which exists on R n+m :

∀1 ≤ i ≤ n, d P i t dt =   g P j∼i R A,ij t - 1 n n ℓ=1 k(x i , x ℓ ) P ℓ t   P i t ∀1 ≤ j ≤ m, d A j t dt =   g A i∼j R P,ij t - 1 m m ℓ=1 h(y j , y ℓ ) A ℓ t   A j t , (2.3)
where, for all

t ∈ R + , R P,ij t = c n,m ij P i t and R A,ij t = c n,m ij A j t .
Eq. (2.3) is similar to a classical Lotka-Volterra system applied to mutualistic interactions with competition. In our case, the species community is structured by the plants traits x i and the pollinators traits y j which determine the probability and strength of the interactions. In addition, how interactions translate into births and deaths (the so-called numerical response in ecological terms) is embedded in functions g A and g P (demographic growth) and k and h (competitive kernels). Those functions can take any form (see section 4 for some examples). As a consequence, Eqs. (2.3) can capture a large variety of ecological situations. In particular, many ODE models published in the ecological literature are special cases of Eqs. (2.3) (e.g. [START_REF] Thébault | Stability of ecological communities and the architecture of mutualistic and trophic networks[END_REF][START_REF] Bascompte | The nested assembly of plant-animal mutualistic networks[END_REF][START_REF] Lever | The sudden collapse of pollinator communities[END_REF]). Our analysis thus shows that ODE models can be commonly seen as a limits of stochastic individual-based models as dened in Section 1. Notice that coecients c n,m ij are random coecients as explained in the previous section. However, these coecients remain identical with time.

Proof. Since the numbers n and m of species are supposed constant, working with the vector processes

(P K,i . , A K,j . ) i∈[[1,n]],j∈[[1,m]] with values in R n+m +
(and not the measure-valued processes) is here sucient.

Notice that under the assumption (2.1), we can obtain by computations similar to the Step 1 of the proof of Proposition 1.7 that:

sup K∈N E sup t∈[0,T ] ⟨P K,n,m t , 1⟩ 3 + ⟨A K,n,m t , 1⟩ 3 < +∞.
(2.4)

The global existence and uniqueness of solution to (2.3) can be proved by classical results for ordinary dierential equations: it follows from the local boundedness and Lipschitz property of the functions on the right-hand side. Then this proposition is a direct application of Theorem 2.1 p.456 in the book by Ethier and Kurtz [START_REF] Ethier | Markov Processes, Characterization and Convergence[END_REF].

■

As a consequence:

Corollary 2.2. Under the assumptions of Proposition 2.1, and for any T > 0, the sequence of measurevalued processes (P K,n,m , A K,n,m ) K∈N converges a.s. and uniformly in the Skorohod space D [0, T ], M F ([0, 1]) 2 to the process ( P n , A m ) dened by:

P n t (dx) = 1 n n i=1 P i t δ x i (dx), A m t (dy) = 1 m m j=1
A j t δ y j (dy).

In this corollary, the space M F ([0, 1]) 2 can be embedded with the total variation topology which is stronger than the weak topology: as n, m, the sequences (x 1 , . . . x n ) and (y 1 , . . . y m ) remain unchanged, all the measures are absolutely continuous with respect to the same counting measures n i=1 δ x i or m j=1 δ y j . Also, the uniform convergence in the Skorohod space for any T > 0 yields the convergence for the Skorohod topology on R + (see Theorem 16.2 in [START_REF] Billingsley | Convergence of Probability Measures[END_REF]).

For the remainder, let us denote by Φ P,n,m = (Φ P,n,m

1 , • • • Φ P,n,m n ) and Φ A,n,m = (Φ A,n,m 1 , • • • Φ A,n,m m )
the applications from R n+m + into R n and R m respectively such that (2.3) rewrites for all i ∈ {1, . . . n} and j ∈ {1, . . . m}:

d P i t dt = Φ P,n,m i ( P t , A t ) and d A j t dt = Φ A,n,m j ( P t , A t ).
(2.5)

Central limit theorem

The individual based model (mathematically described by the SDE of Denition 1.6) is very commonly encountered in simulations in Biology for simulation, while the ODEs (2.3) are often used in the modelling papers. The law of large number (Proposition 2.1) shows that both point of view are related in the limit K → +∞. In this section, we quantify the speed at which the convergence holds by establishing a central limit theorem. This can be useful for instance to compute condence intervals or study the statistical properties of parameter estimators.

Let us introduce the uctuation processes:

η K,P t = √ K    P K,1 t /K -P 1 t . . . P K,n t /K -P n t    , and η K,A t = √ K    A K,1 t /K -A 1 t . . . A K,m t /K -A m t    . ( 2
(η K,P t , η K,A t ) t≥0 ⇒ K→∞ ( η P t , η A t ) t≥0 ,
where the converge holds in law in the Skorohod space D(R + , R n+m ) and where the processes ( η P . , η A . ) = ( η P,i . , η A,j . ) i∈{1,...n},j∈{1,...m} are solutions of the following SDEs driven by n + m independent standard Brownian motions (W P,i , W A,j ) 1≤i≤n,1≤j≤m in R: for all

1 ≤ i ≤ n and 1 ≤ j ≤ m, η P,i t = η P,i 0 + t 0 b P R A,i s + d P R A,i s + 1 n n ℓ=1 k(x i , x ℓ ) P ℓ s P i s dW P,i s + t 0 n ℓ=1 ∂Φ P,n,m i ∂p ℓ ( P s , A s ) η P,ℓ s + m ℓ=1 ∂Φ P,n,m i ∂a ℓ ( P s , A s ) η A,ℓ s ds, η A,j t = η A,j 0 + t 0 b A R P,j s + d A R P,j s + 1 m m ℓ=1 h(y j , y ℓ ) A ℓ s A j s dW A,j s + t 0 n ℓ=1 ∂Φ A,n,m j ∂p ℓ ( P s , A s ) η P,ℓ s + m ℓ=1 ∂Φ A,n,m j ∂a ℓ ( P s , A s ) η A,ℓ s ds.
(2.7)

Notice that the limiting uctuation process is an Ornstein-Uhlenbeck process that is centered. Systems of SDEs have already been introduced in the literature to describe the evolution of communities, but they are to our knowledge of a dierent nature. In (2.7) the noise relates to the uctuation of the stochastic individual-based model around its deterministic limit (2.5) for K → +∞. In other works, such as in [START_REF] Bunin | Ecological communities with Lotka-Volterra dynamics[END_REF] for instance, the white noise corresponds to a diusive limit obtained when considering a dierent longer time-scale, as in the Donsker theorem (see e.g. [START_REF] Champagnat | Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models via timescale separation[END_REF]Section 4.2]): the random noise comes from the rapid successions of birth and death events in this accelerated time-scale. In recent works, following the steps of May [START_REF] May | Will a large complex system be stable?[END_REF], [START_REF] Ben Arous | Counting equilibria of large complex systems by instability index[END_REF][START_REF] Fyodorov | Nonlinear analogue of the May-Wigner instability transition[END_REF]introduce a system of equations, coupled via a smooth random vector eld, which describes the approximated dynamical system around an equilibrium state: in this case, the noise models the complexity and nonlinearity of interactions.

Proof of Proposition 2.3. It is a direct application of Theorem 2.3 from Chapter 11 of [START_REF] Ethier | Markov Processes, Characterization and Convergence[END_REF]. The proof can also be carried from the semi-martingale expressions of P K,i t and A K,j t (1.13)-(1.15) and (2.5). To understand (2.7), consider for example the ith component of η K,P t , η K,P,i t . The other terms can be treated similarly. We sketch here this alternative proof:

η K,P,i t =η K,P,i 0 + √ K t 0 1 n n ℓ=1 k(x ℓ , x i ) P ℓ s P i s - P K,ℓ s K P K,i s K ds + √ K M K,i t K + √ K t 0 g P j∼i c n,m ij A K,j s K P K,i s K -g P j∼i c n,m ij A j s P i s ds =η K,P,i 0 + 1 n n ℓ=1 k(x ℓ , x i ) t 0 η K,P,ℓ s P K,i s K + P ℓ s η K,P,i s ds + √ K M K,i t K (2.8) + t 0 g P j∼i c n,m ij A K,j s K η K,P,i s + √ K P i s g P j∼i c n,m ij A K,j s K -g P j∼i c n,m ij A j s ds,
for the martingale M K,i appearing in (1.13) with quadratic variation (1.14). Because the birth and death rates are assumed to be of class C 1 , so is g P and a Taylor expansion can be used for the last term:

√ K g P j∼i c n,m ij A K,j s K -g P j∼i c n,m ij A j s =(g P ) ′ j∼i c n,m ij A j s × j∼i c n,m ij √ K A K,j s K -A j s + ε K =(g P ) ′ j∼i c n,m ij A j s × j∼i c n,m ij η K,A,j s + ε K , (2.9)
where ε K is a remainder term. From (2.8) and (2.9), we recognize that:

η K,P,i t = η K,P,i 0 + √ K M K,i t K + t 0 n ℓ=1 ∂Φ P,n,m i ∂p ℓ ( P s , A s ) η P,ℓ s + m ℓ=1 ∂Φ P,n,m i ∂a ℓ ( P s , A s ) η A,ℓ s ds + ε K .
Using the Aldous-Rebolledo criterion (e.g. [START_REF] Joe | Weak convergence of sequences of semimartingales with applications to multitype branching processes[END_REF]) it is possible to prove that the distributions of the processes (η K,P , η K,A ) K form a tight family with a unique limiting value that solves the SDEs (2.7).

■

3 Continuous limits when abundances and the number of species are large

We now consider that the numbers of plant and pollinator species in the network tend to innity.

Taking the limit n, m → +∞, we obtain equations describing the evolution of a population consisting in a continuum of species.

Let the traits of plants and pollinators be chosen according to i.i.d. random variables with cumulative distribution functions F P and F A respectively. To order species according to their respective trait, we proceed as follows: let (ũ i ) i≥1 and (ṽ j ) j≥1 be two sequences of i.i.d. random variables with uniform distribution in [0, 1]; for any n, m ∈ N 2 , let (u i,n ) i=1..n be the ordered n th rst values of (ũ i ) i≥1 and (v j,m ) j=1...m be the ordered m th rst values of (ṽ j ) j≥1 , then for any i ∈ {1, .., n} and j ∈ {1, .., m},

x i,n = F -1 P (u i,n ) and y j,m = F -1 A (v j,m ).
The indices n and m will be dropped when no confusion is possible.

Recall that the plant-pollinator network is dened by its adjacency matrix (G n,m ij ) (i,j)∈[ [1,n]]×[ [1,m]] and the weight of c n,m ij of the interaction i ∼ j whenever G n,m ij = 1. The random variables G n,m ij are supposed to satisfy Assumption 1.1 for all n and m ∈ N: the entries of the adjacency matrix are here independent Bernoulli with parameters ϕ(x i , y j ) for the term (i,

j) ∈ [[1, n]] × [[1, m]].
For the harvesting coecients c n,m ij , the following assumption is made: Assumption 3.1. We assume that there exists for any n, m ∈ N 2 a continuous function c n,m satisfying the condition (1.2) of Assumption 1.3. Additionally, we assume that there exists a function c : [0, 1] 2 → R to which the sequences of functions nc n,m (., .) and mc n,m (., .) converge uniformly, in L ∞ ([0, 1] 2 ), when n and m tend to +∞. Notice also that Assumption 3.1 implies that n and m grow to innity with a similar speed. To state this idea in specic terms, we assume that there exists a sequence (α n ) n≥1 such that m = α n n and lim n→∞ α n = 1.

(3.1)
In what follows, we write n → ∞, m → ∞ or n, m → ∞ indistinctly. Proposition 3.2. Let us assume Assumptions 1.1, 1.5 and 3.1 hold and that there exist deterministic continuous bounded densities p0 and ā0 such that the following weak convergences hold:

lim n→+∞ 1 n n i=1 P i 0 δ x i w = p0 (x)dx, lim m→+∞ 1 m m j=1
A j 0 δ y j w = ā0 (y)dy a.s.

(3.2)

For any T ≥ 0, and for n, m → +∞, the sequence of measure-valued processes

P n,m t (dx), A n,m t (dy) t≥0 := 1 n n i=1 P i t δ x i , 1 m m j=1 A j t δ y j t≥0 , n, m ≥ 1 converge in law in C([0, T ], M F ([0, 1]) 2 ) to a deterministic process ( P, Ā) in C([0, T ], M F ([0, 1]) 2 ),
such that: (i) for all t ≥ 0, Pt and Āt admit densities pt and āt with respect to the Lebesgue measure on [0, 1],

(ii

) ( P, Ā) is the unique solution, for f ∈ C([0, 1], R), of 1 0 f (x)d Pt (x) = 1 0 f (x)p 0 (x)dx + t 0 1 0 f (x) g P 1 0
c(x, y)ϕ(x, y)ā s (y)dy -k ⋆ ps (x) ps (x)dx ds,

1 0 f (y)d Āt (y) = 1 0 f (y)ā 0 (y)dy + t 0 1 0 f (y) g A 1 0
c(x, y)ϕ(x, y)p s (x)dx -h ⋆ ās (y) ās (y)dy ds,

where k ⋆ ν denotes in this case the convolution on two functions and ϕ was introduced in Assumption 1.1. In all this statement, the space M 2 F ([0, 1]) is endowed with its weak topology.

The proof of this Proposition is given in Appendix A.2 Equation (3.3) is analogous to the ODE system given in Equation (2.3). However, here species are not considered as discrete but continuously distributed along a continuous trait. Modelling the dynamics of a plant-pollinator community with the PDEs given in Equation (3.3) is thus a functional rather than a species representation of the same system. The connexions are modelled in (3.3) by ϕ. This function ϕ : [0, 1] 2 → [0, 1] is a graphon (see [START_REF] Lovász | Large networks and graph limits[END_REF]): it can be understood (in this case) as a graph on node sets [0, 1] for the plants and [0, 1] for the pollinators, where ϕ(x, y) describes the density of connexions between plants x and pollinator y.The term cϕ reects both the topology (ϕ) and the intensity (c) of plant-pollinator interactions throughout the community depending on the traits values

x and y involved.

Remark 3.3. Note that one can formally write a strong form of (3.3). This leads to the following system of integral equations on the densities pt and āt :

∂ t pt (x) = g P 1 0 c(x, y)ϕ(x, y)ā t (y)dy -k ⋆ pt (x) pt (x), ∂ t āt (y) = g A 1 0 c(x, y)ϕ(x, y)p t (x)dx -h ⋆ āt (y) āt (y), (3.4)
with initial conditions p0 and ā0 .

Study of the limiting dynamical systems

Let us now study the behaviour of the limiting dynamical systems that have been obtained: the ODE system (2.3) when K → +∞, and the kinetic PDEs (3.3) when additionally n, m → +∞.

Stationary states of the ODE system (2.3)

In this section, we give some results about the dynamics of solutions to Systems (2.3), for general forms of g P and g A . Notice that any stationary points ( P 1 ∞ , . . .

P n ∞ , A 1 ∞ , . . . A m ∞ ) of (2.
3) are solutions to the following system:

∀1 ≤ i ≤ n, P i ∞ = 0 or g P j∼i c n,m ij A j ∞ = 1 n n ℓ=1 k(x i , x ℓ ) P ℓ ∞ , ∀1 ≤ j ≤ m, A j ∞ = 0 or g A i∼j c n,m ij P i ∞ = 1 m m ℓ=1 h(y j , y ℓ ) A ℓ ∞ . (4.1)
The community with no individuals is an obvious stationary point called the null equilibrium in the rest of the paper:

P i ∞ = 0 for all i ∈ {1, ..n} and A j ∞ = 0 for all j ∈ {1, ..m}, is a stationary state of (2.3). The local stability of this stationary state depends only on the sign of g P (0) and g A (0).

Remark 4.1. According to the expression of the non trivial equilibria, a plant-pollinator community can exist, i.e. P i ∞ > 0 and A j ∞ > 0, as soon as the interactions between plants and pollinators translate into positive growth rate for both plants and pollinators. Despite this condition is little restrictive, a nontrivial equilibrium is possible only if trajectories have to converge to it, (i.e. a non-trivial equilibrium has to be stable). The stability of an equilibrium can be more tricky to study and obtained in general. Lemma 4.2. If g P (0) ∨ g A (0) < 0, the null equilibrium is locally stable. If g P (0) ∨ g A (0) > 0, the null equilibrium is locally unstable.

Proof. From the HartmanGrobman theorem, we know that local stability is given by the signs of the Jacobian matrix eigenvalues. The Jacobian matrix around the null equilibrium can be directly computed:

      g P (0) 0 0 ... 0 0 0 g P (0) 0 ... 0 0 ... ... ... ... ... ... 0 0 ... 0 g A (0) 0 0 0 ... 0 0 g A (0)      
, whose eigenvalues are g P (0) with multiplicity n and g A (0) with multiplicity m. This ends the proof. ■

Study of a plant-pollinator interaction with a trade-o for plants

As explained in Remark 4.1, more precise functional forms for the birth and death rates are needed to carry further computation.

In this Section, we give results for the following particular forms for the individual growth rate functions:

g P (R A ) =b P (R A ) -d P (R A ) = α P R A β P + γ P R A -(d P + δ P R A ), g A (R P ) =b A (R P ) -d A (R P ) = α A R P β A + γ A R P -d A , (4.2)
where R A and R P are the total resources collected by respectively plants and pollinators (Eqs. (1.5) and (1.6)). Parameters α P , α A , β P , β A , γ P , γ A , d A , d P and δ P are assumed positive. Note that according to Lemma 4.2, the null equilibrium is stable in this particular case.

On the right-hand side of Equation (4.2), the death rate is an increasing function of the resources exchanged R for plants. This reects an interaction trade-o for the plant, i.e. it is supposed that there is a cost for interacting with pollinators due to nectar production, leaves consumption, etc. Graphical representations of g P and g A are given in Figure 2. The form given here to the growth rates (or Figure 2: Graphical representation of g A and g P numerical responses) g P and g A can be found in [START_REF] Holland | Mutualism[END_REF]. It assumes that there is no competition between plants for collecting resources from pollinators, and that there is no competition between pollinators for collecting resources from plants.

We state an additional assumption on the parameters of both functions to avoid the case where one growth rate is never positive. Assumption 4.3. Assume that there exist r P > 0 and r A > 0 such that g P (r P ) > 0 and g A (r A ) > 0.

Case n = m = 1

Let us rst consider the particular case of a single plant species interacting with a single pollinator species. In other words, we consider the system (2.3) with n = m = 1 and omit the indices and exponents i = 1 and j = 1 for the sake of simplicity (so that P 1 t becomes P t and c 1,1 11 becomes c for instance). We also assume that the two species interact, i.e. G 11 = 1. Otherwise, computations are trivial: the community goes to extinction as only death events occur. We have then:

dP t dt = g P cA t -kP t P t dA t dt = g A cP t -hA t A t (4.3)
(see [START_REF] Holland | A consumerresource approach to the density-dependent population dynamics of mutualism[END_REF] for a similar ODE model in the ecology literature).

As discussed in Lemma 4.2, the null equilibrium is a locally stable equilibrium of the system (4.3).

Let us search for positive equilibrium. To help the study, we may draw some phase plan of the system, for the choice of parameters α A = 25, α P = 9, β A = β P = γ A = γ P = 1 and δ P = 3, and some various d A and d P . (see Figure 3).

Precisely, we prove the following lemma that gives the number of equilibria and their stability.

max x∈(C - 0 /c,C + 0 /c) f (x) > 0.
In this case, the equilibria are

(P - ∞ , A - ∞ ) and (P + ∞ , A + ∞ ) with A - ∞ < A +
∞ and are the two positive solutions to Proof of Lemma 4.4. Any stationary point (P ∞ , A ∞ ) is solution to

     P ∞ = 1 k g P (cA ∞ ) A ∞ = 1 h g A c k g P (cA ∞ ) ,
g P cA ∞ = kP ∞ or P ∞ = 0 g A cP ∞ = hA ∞ or A ∞ = 0. (4.5)
Recall that either both or no coordinates are null. In view of (4.5), the existence of a positive equilibrium requires at least that sup x∈R + g P (x) > 0 and sup y∈R + g A (y) > 0. This is true under Assumption 4.3. Recall that C - 0 and C + 0 are the two zeros of g P . From (4.5), we deduce that, if there exists a positive stationary state, it satises System (4.4).

It remains to nd the number of zeros of f (

x) = 1 h g A c k g P (cx) -x on C - 0 c , C + 0 c . A rapid dierentiation gives f ′′ (x) = c 3 hk (g P ) ′′ (cx)(g A ) ′ c k g P (cx) + c 3 (hk) 2 (g P ) ′ (cx) 2 (g A ) ′′ c k g P (cx) < 0,
since g A is increasing and g P and g A are concave functions. Thus, f ′ is decreasing and

f ′ (x) = c 2 hk (g P ) ′ (cx)(g A ) ′ c k g P (cx) -1,
converges to +∞ (resp. -∞) when x goes to C - 0 /c (resp. C + 0 /c). Finally, f is increasing then decreasing and admits a unique maximum. Since f converges to -∞ when x goes to C - 0 /c and C + 0 /c, it admits Finally, in the case of 2 positive equilibrium, which will be denoted by (P - ∞ , A - ∞ ) and (P

2 zeros if max x∈(C - 0 /c,C + 0 /c) f (x) > 0,
+ ∞ , A + ∞ ) with A - ∞ < A +
∞ , the Jacobian matrix can be computed in order to study their stability (see the HartmanGrobman theorem):

J ± = -kP ± ∞ cP ± ∞ (g P ) ′ (cA ± ∞ ) cA ± ∞ (g A ) ′ (cP ± ∞ ) -hA ± ∞ .
The trace is negative, thus the stability of the stationary states depend on the sign of the determinant.

det(J ± ) = P ± ∞ A ± ∞ hk -c 2 (g P ) ′ (cA ± ∞ )(g A ) ′ (cP ± ∞ ) = -P ± ∞ A ± ∞ hkf ′ (A ± ∞ ),
and its sign depends only on the sign of f ′ (A ± ∞ ). According to the previous study on function f , we deduce that det(J + ) is negative, and det(J -) is positive. In other words, System (4.5) admits 1 stable positive equilibrium and 1 unstable positive equilibrium.

In the case when System (4.5) admits a unique positive equilibrium, the same study implies that this equilibrium is a non-hyperbolic equilibrium. ■ 4.2.2 Behaviour of the kinetic equations (3.3) Recall that we are still working with the growth rates dened in (4.2).

The stationary solutions of (3.3) are couples of measures P ∞ (dx) and Ā∞ (dy) in M F ([0, 1]) such that, for all positive, bounded and continuous function f on [0, 1]: The null measures constitute a trivial solution to (4.6)-(4.7), which is stable in our particular case (4.2). Let us discuss non-zero solutions. Proposition 4.5. Assume that the competitive kernels k and h are constant functions; for all x 0 , y 0 ∈ [0, 1], y → ψ(x 0 , y) and x → ψ(x, y 0 ) are increasing and continuous functions; then, System (3.3) does not admit non-null stationary state with densities w.r.t Lebesgue measure. Moreover, any non-null stationary state in L 1 ([0, 1] 2 ) is a couple of measures ( P ∞ , Ā∞ ) such that

0 = 1 0 f (x) g P 1 0 ψ(x, y) Ā∞ (dy) -k ⋆ P ∞ (x) P ∞ (dx), (4.6) 0 = 1 0 f (y) g A 1 0 ψ(x, y) P ∞ (dx) -h ⋆ Ā∞ (y) Ā∞ (dy) (4.
∃ā 0 ,p 1 ∈ R * + , p2 ∈ R + , x1 , x2 , ȳ0 ∈ [0, 1], P ∞ = p1 δ x1 + p2 δ x2 Ā∞ = ā0 δ ȳ0 with    g P ā0 ψ(x 1 , ȳ0 ) = g P ā0 ψ(x 2 , ȳ0 ) = k(p 1 + p2 ) g A p1 ψ(x 1 , ȳ0 ) + p2 ψ(x 2 , ȳ0 ) = hā 0 . (4.9)
All these stationary states are unstable, except the state

       P ∞ = max R + g P k δ x 0 Ā∞ = arg max R + g P ψ(x 0 , 1) δ 1 (4.10)
if x 0 , solution to

g A max R + g P k ψ(x 0 , 1) ψ(x 0 , 1) = h • arg max R + g P , (4.11)
exists and is unique. Finally, assuming that, for all initial conditions with positive densities w.r.t Lebesgue measure, the quantities 1 0 ψ(x, .)p t (

., y)ā t (y)dy and 1 0 āt (y)dy converge when t grow to innity, then the trajectory converges to equilibrium (4.10).

This proposition ensures the fact that the only possible stable equilibrium is composed of only one plant species and one pollinator species. Simulations below will illustrate this Proposition.

Remark 4.6. The proof of Proposition 4.5 is not restricted to the specied forms of g P and g A , (4.2). The proposition is still true as soon as shapes (successions of increases and decreases) of g P and g A are the same as the ones of specied functions (4.2). Otherwise, in any case, System (3.3) has no stationary state with densities, all stationary states will be composed of Dirac measures. Moreover, the maximal number of Dirac measures found in such a stationary state corresponds to the maximal number of points that can be found in an inverse image of a positive real for functions g P and g A respectively. The stability of these stationary states can then be deduced using the same kind of arguments as these of the following proof. From a biological point of view, it means that that the number of pollinator species and plant species is reduced to 1 when considering all type of growths given in Figure 2 of [START_REF] Holland | Mutualism. In Encyclopedia of Ecology, Five-Volume Set[END_REF]. This result shows that when the plant-pollinator network is nested and the competition among plants and among pollinators is constant, then the plant-pollinator community collapses to a single plantpollinator species pair. At the stationary state, the value of the trait for the pollinators is 1, where the probability of interactions with plant is maximized. The trait for plants is <1 ( (4.11)), because of the trade-os for plants due to the cost of the interaction.This is in line with the numerical analysis of a system of ODEs by [START_REF] Lever | The sudden collapse of pollinator communities[END_REF]. Our results is a formal demonstration of this necessary collapse. It is also more general since we show that it does not depend on the specic form of g P and g A . This raises the question whether it is possible, in this type of ecological model, to maintain a stable coexistence of many plant and pollinator species in a single community by modifying the structure of the interaction graph (i.e. with other assumptions for the function ψ), or the structure of the competition graph (i.e. with non-constant competition functions k and h). ′ ) and 1 0 h Ā∞ (dy ′ ) are nite. Then, according to the continuity and the variations of the functions (ψ is increasing w.r.t each variable, g A is increasing and g P is increasing then decreasing), we deduce (4.9).

Proof. Any stationary state in

L 1 ([0, 1] 2 ) satises that ( P ∞ , Ā∞ )          g P 1 0 ψ(x, y) Ā∞ (dy) = 1 0 k P ∞ (dx ′ ), for all x ∈supp P ∞ , g A 1 0 ψ(x, y) P ∞ (dx) = 1 0 h Ā∞ (dy ′ ), for all y ∈supp Ā∞ (4.12) Since ( P ∞ , Ā∞ ) ∈ L 1 ([0, 1] 2 ), 1 0 k P ∞ (dx
Let us denote by S the set of stationary states given by (4.9) excluding equilibrium (4.10) Let us prove that all states in S are unstable. To this aim, we develop an argument by contradiction, assuming that it is not true and that there exists at least a stationary state ( P ∞ , Ā∞ ) ∈ S. We deal with the case where x1 < x 0 < x2 , ȳ0 being any value in [0

, 1]. Other cases ((2) x2 < x 0 < x1 , ȳ0 ∈ [0, 1] or (3) x1 = x2 = x 0 , ȳ0 < 1) can be treated similarly.
Let consider an initial state that is close (in Wasserstein distance) to ( P ∞ , Ā∞ ) and that has positive densities w.r.t Lebesgue measures. Thus, p t exists and is positive for all t ≥ 0 and we can study log(p t (x)) for all t ∈ R + and all x ∈ [0, 1] 2 :

d dt log(p t (x)) = g P 1 0 ψ(x, y)ā t (y)dy - 1 0 k, hel p0 t(x ′ )dx ′ .
Under our assumption, S admits at least a stable state and since the initial state is close to this set, we deduce that (p t , āt ) will converge to some ( P ∞ , Â∞ ) ∈ S, which is close to ( P ∞ , Ā∞ ) (if not it), suciently close to have a similar form. Then for all x ∈ [0, 1], ψ(x 1 , y)â ∞ (y)dy .

The latter quantity is positive as soon as x ∈ (x 1 , x 0 ), which contradicts the fact that pt (x) converges to 0 for all x ̸ ∈ {x 1 , x2 }.

The last point of Proposition (4.5) can be obtained using the same argument, once the convergences of the four quantities detailed in the statement are assumed.

■

Let us illustrate numerically Proposition 4.5. Let (p i (t)) i , resp. (a j (t)) j be an approximation of (p t (x i )) i , resp. (ā(y j )) j , for x i = i/N , y j = j/N , 0 ≤ i, j ≤ N . Then the continuous model (3.4) can be approximated using the rectangular rule by the following system of coupled ODEs:

dp i dt =   g P   1 N N j=0 c ij a j   - 1 N N j=0 k i,j p j   p i , da j dt = g A 1 N N i=0 c ij p i - 1 N N i=0 h i,j a i a j , (4.13)
where the initial data has been dened as p i (0) = p0 (x i ), p i (0) = p0 (x i ), a j (0) = ā0 (x j ). We recognize the ODE system (2.3) obtained when the numbers of plant and pollinator species are nite with n = m = N . We shall consider the case of growth functions (4.2) with α A = 3, α P = 25, β A = β P = γ P = 1, γ A = 0.3, d P = 1, d A = 3 and δ P = 3, with N = 500 species of plants and pollinators. In order to t the hypotheses of Proposition 4.5 and insure the convergence towards an explicit equilibrium of the dynamics, the interaction matrix (c ij ) ij is nondecreasing in each index i and j, and the competition kernels will be chosen constant. We consider the simple case

c ij = (i + 1)(j + 1) 2N 2 .
We represent in Figure 4 (in logarithmic scale) the convergence towards the equilibrium state reached when no evolution occurs anymore on the discrete dynamics, for a random initial datum supported in [0, 1]. We observe that the conclusion of proposition 4.5 are in agreement, namely the system converges towards an equilibrium which consists of a Dirac delta centered inside the domain for the plants, and on the right border for the pollinators: the last remaining plant specie is moderately specialist (x ≃ 0.6) whereas the remaining pollinator is genuinely generalist (y = 1).

Conclusion

Two of the main goals of ecology are i) to explain how emerged the size, composition and structure of communities, for example plant-pollinator communities, and ii) predict the dynamics and stability of a given community. Such questions involve dierent hierarchical scales, from the individuals which eectively interact (the microscopic scale), to the species and the whole community (the macroscopic scale).

Theoretical ecology mostly address these questions by studying a system of ODEs where one equation refers to a given species. This approach has several methodological and conceptual drawbacks.

First, the system is discretized and structured by species, which precludes the possibility of withinspecies variability regarding the rate and intensity of interactions, in particular between-species overlaps. Second, the interaction graphs underlying the system of ODEs are most often arbitrarily given without specied mechanisms. Finally, due to the high-dimensionality of the ODEs system, it is dicult to obtain general properties of communities.

Here, we modelled an individual-based plant-pollinator network, where interactions are structured by an individual trait. We found continuous limits of the microscopic system and nally showed that it could be approximated by PDEs. We nally studied the dynamics, stationary state and stability of this continuous limits. Our approach allows to address several limits exposed before: the relationships between the individuals, species and community scales are explicit; interactions variability within and between-species is taken into account; the interaction graph is based on individuals' traits; PDEs approximations allows an explicit and analytical study of the property of a community. We showed in particular that a nested plant-pollinator network is expected to collapse, a phenomenon already observed by previous works, but to our knowledge for the rst time formally demonstrated under general conditions. Our approach can thus provide a new and original theoretical framework for ecologists to address long-standing questions.

One of the actual limit of our model is that interactions take place through a mass-action model", in particular with the use of the resources (1.5)- (1.6) which are already at a macroscopic scale. Returning to an event-based modelling for establishing these functional responses (e.g. [START_REF] Bansaye | Rejuvenating functional responses avec renewal theory[END_REF]) and showing how interactions at the individual level would translate into an interaction graph and into the dynamics of the whole community is an open question.

Code

The code for simulations is available here: https://gitlab.com/thoma.rey/PlantPollinatorsNetwork/

A Proofs

A.1 Bounds on the microscopic representation

Here, we give a sketch of proof of Proposition 1.7. The proof follows from usual stochastic calculus with Poisson point processes, as developed in [START_REF] Fournier | A microscopic probabilistic description of a locally regulated population and macroscopic approximations[END_REF] for example.

Proof of Proposition 1.7.

Step 1: let us rst prove some moment estimates, including (1.12). First, notice that the process can be stochastically bounded by a process where only births occur at the maximal birth rate, denoted by M P := sup r∈R + b P (r):

⟨P K,n t , 1⟩ = 1 nK n i=1 P K,i t ≤ ⟨ PK,n t , 1⟩ = ⟨P K,n 0 , 1⟩ + t 0 E 1 nK n i=1 1l i=k 1l θ≤M P P K,i s- Q P B (ds, dk, dθ). Since E sup s≤t∧τ K N ⟨P K,n s , 1⟩ ≤ 1 + E sup s≤t∧τ K N ⟨P K,n s , 1⟩ 2 ≤ 1 + E sup s≤t∧τ K N ⟨ PK,n s , 1⟩ 2 ,
we only have to nd a bound for the second moment of ⟨ PK,n s , 1⟩. For a constant N > 0, we introduce the stopping time

τ K N = inf t ≥ 0, ⟨ PK,n t , 1⟩ ≥ N or ⟨ ÃK,m t , 1⟩ ≥ N .
Using Itô's formula for jump processes (e.g. [34, Th.5.1 P.66]), we have:

⟨ PK,n t , 1⟩ 2 =⟨P K,n 0 , 1⟩ 2 + t 0 E n i=1 1l k=i ⟨ PK,n s-, 1⟩ + 1 nK 2 -⟨ PK,n s-, 1⟩ 2 1l θ≤M P P K,i s- Q P B (ds, dk, dθ) =⟨P K,n 0 , 1⟩ 2 + t 0 E n i=1 1l k=i 2 nK sup u≤s- ⟨ PK,n u , 1⟩ + 1 n 2 K 2 1l θ≤M P P K,i s- Q P B (ds, dk, dθ).
Since the right hand side is an increasing process, we can again replace the left hand side by sup s≤t∧τ K N ⟨ PK,n s , 1⟩.

Then, taking the expectation gives:

E sup s≤t∧τ K N ⟨ PK,n s , 1⟩ 2 ≤E ⟨P K,n 0 , 1⟩ 2 + E t∧τ K N 0 n i=1 2 nK sup u≤s ⟨ PK,n u , 1⟩ + 1 n 2 K 2 M P P K,i s ds ≤E ⟨P K,n 0 , 1⟩ 2 + t 0 M P (2 + 1 nK )E 1 + sup u≤s∧τ K N ⟨ PK,n u , 1⟩ 2 ds.
Using Gronwall's lemma, we deduce

E sup s≤t∧τ K N ⟨ PK,n s , 1⟩ 2 ≤ E ⟨P K,n 0 , 1⟩ 2 + 3M P e 3M P t < +∞.
Since the bound does not depend on N , we make N grow to ∞ and obtain

E sup s≤t ⟨ PK,n s , 1⟩ 2 ≤ E ⟨P K,n 0 , 1⟩ 2 + 3M P e 3M P t < +∞.
Similarly,

E sup s≤t ⟨ ÃK,m s , 1⟩ 2 ≤ E ⟨A K,m 0 , 1⟩ 2 + 3M A e 3M A t < +∞. (A.1)
Notice that the upper bound that we obtain do not depend on K. Once these estimates have been obtained, existence and uniqueness of a strong solution to the stochastic dierential equation is given by [34, Th. IV.9.1].

Step

M K,i t = t 0 E 1l i=k 1l θ≤b P j∼i R A,K,ij s- P K,i s- Q P B (ds, dk, dθ) - t 0 b P j∼i R K,ij s P K,i s ds - t 0 E 1l i=k 1l θ≤ d P j∼i R A,K,ij s- -k⋆P K,n s-(x i ) P K,i s- Q P D (ds, dk, dθ) + t 0 d P j∼i R A,K,ij s -k ⋆ P K,n s (x i ) P K,i s ds (A.2)
is a square integrable martingale with bracket given by (1.14). A similar computation can be done for A K,m to obtain (1.15). ■

A.2 Large number of species limit Proof of Proposition 3.2. Recall that for m ≥ 1, we have:

1 0 f (x)d P n,m t (x) = 1 0 f (x)d P n,m 0 (x)dx + t 0 1 n n i=1 f (x i ) g P 1 m m j=1 mc n,m i,j G ij A j s - 1 n n ℓ=1 k(x i , x ℓ ) P ℓ s P i s ds, 1 0 f (y)d A n,m t (y) = 1 0 f (y)d A n,m 0 (y)dy + t 0 1 m m j=1 f (y j ) g A 1 n n i=1 nc n,m i,j G ij P i s - 1 m m ℓ=1
h(y j , y ℓ ) A ℓ s A j s ds. where conditionally on (x i , y j ) 1≤i≤n,1≤j≤m , G ij ∼ Bern(ϕ(x i , y j )) and are independent random variables, and c n,m i,j are independent random variables satisfying Equations (1.2) and (1.3).

Step 1: Processes are bounded.

C(T ) = sup n,m ⟨ P n,m 0 , 1⟩e M P T + ⟨ A n,m 0 , 1⟩e M A T < +∞ a.s.. (A.5)
Thus, the processes P n,m t (dx), A n,m t (dy) t∈[0,T ] take their values in (M ≤C(T ) ([0, 1])) 2 which is a com- pact set.

Step Same computations can be done for the sequence of processes ( A n,m ) m∈N * . And these are sucient to conclude point (ii).

2: Relative compactness in C [0, T ], M ≤C(T ) ([0, 1])
|⟨ A n,m s , f ⟩ -⟨ A n,m t , f ⟩| ≥ ε = 0 Point (i) follows from Equation (A.5). Concerning point (ii), notice rst that E[ R A,n,m ] = E   1 m m j=1 mc n,m i,j G ij A j s   ≤ 1 m m j=1 E[mc n,m i,j ] C(T ) ≤ max n,m∈N * ∥mc n,m ∥ ∞ C(T ) < ∞,
Step 3: Density of the limiting values. Let us then prove that every limiting values of P n,m , A n,m , denoted here by ( P, Ā), is a process whose time marginals at t > 0 admit densities pt and āt with respect to the Lebesgue measure on [0, 1]. To achieve this, we dominate the measures P n,m t (dx) and A n,m t (dy) by measures with densities. For all positive continuous function f on [0, 1], with arguments similar to these of (A.4), we prove that ⟨ Pt , f ⟩ ≤ ⟨ P0 , f ⟩e M P t = 1 0 f (x) p 0 (x)e M P t dx, a.s.

⟨ Āt , f ⟩ ≤ ⟨ Ā0 , f ⟩e M A t = 1 0 f (y) a 0 (y)e M A t dy, a.s. which is sucient to deduce the existence of densities with respect to Lebesgue measure. A j s G ij m c n,m i,j -c n,m (x i , y j ) + A j s c(x i , y j ) G ij -ϕ(x i , y j ) ,

with C := ∥f ∥ ∞ C(T )L P and L P is the Lipschitz constant of function g P . Then ∥mc n,m -c∥ ∞ converges to 0 when n, m grow to ∞, and it will be suciently small for n, m large enough. Finally, from Cauchy-Schwarz inequality, Assumptions 3.1 and 1.1, and denoting by V ar x,y (A) = V ar(A|(x i , y j ) i≤n,j≤m ), we nd for n, m suciently large, (A. ϕ(x i , y j )(1 -ϕ(x i , y j ))

  1/2 ≤ ε + CT C(T )(V 1/2 max + ∥c∥ ∞ ∥ϕ(1 -ϕ)∥ 1/2
∞ ) m 1/2 ≤ 2ε for n, m suciently large.

The second term (A.8) can be treated similarly. This nally proves that E ∥ψ( P n,m , A n,m )∥ 1 converges to 0 when n, m go to ∞. In addition with (A.6), we deduce that ψ( P, Ā) = (0, 0) a.s. In other words, the limiting value satises Equation (3.3) a.s.

Step 5: Uniqueness of the solution of (3.3). It remains to prove the uniqueness of the solution to (3.3) to conclude. To this aim, let us take (P 1 , A 1 ) and (P 2 , A 2 ) two deterministic measures, weak solutions to (3.3) and with identical initial conditions. Let us denote by ρ the Radon metric between two measures ν and µ on M F ([0, 1]) 2 : ρ(ν, µ) = sup We conclude with Gronwall inequality that ρ(P 1 t , P 2 t ) + ρ(A 1 t , A 2 t ) = 0 for all t ∈ [0, T ]. In other words, (P 1 , A 1 ) and (P 2 , A 2 ) are identical, and the solution to (3.3) is unique. This ends the proof. ■

Figure 1 :

 1 Figure 1: (a): Nested (left) or modular (right) bipartite networks, from Fontaine et al. [24]. (b) Pollinationnetwork for diurnal and nocturnal insect species, from Knop et al.[START_REF] Knop | Articial light at night as a new threat to pollination[END_REF]

  The idea in this assumption is that the function c(., .) is a harvesting function underlying the matrix (c n,m ij ) i∈[[1,n]],j∈[[1,m]] of harvesting coecients. For instance, Examples 1.4 satisfy the previous assumptions.

Figure 3 :

 3 Figure 3: Phase plan and nullclines of the system of ODE (4.3): nullclines for the pollinators dynamics in blue; nullclines for the plants dynamics in cyan. Right: d A = 2 and d P = 1, the system has 3 stationary states: the null equilibrium, 1 stable positive equilibrium and 1 unstable positive equilibrium. Left: d A = 3 and d P = 1.2, the unique equilibrium of the system is the null equilibrium.

  and cA ∞ ∈ (C - 0 , C + 0 ).

∞

  ) is an unstable positive equilibrium and (P + ∞ , A + ∞ ) is a stable positive equilibrium.To conclude on the dynamics of the trajectories of System (4.3), we performed simulations. In summary, it appeared that, depending on the parameters values, either all trajectories are attracted by the null equilibrium (Fig 3(left)), or trajectories converge to 0 or some positive equilibrium, depending on the initial conditions (Fig 3(right)). We never observed cycles. Moreover, the competitive terms ensure that trajectories remain bounded.

  and System (4.5) has 2 positive equilibrium,1 zero if max x∈(C - 0 /c,C + 0 /c) f (x) = 0,and System (4.5) has 1 positive equilibrium, no zero otherwise, and System (4.5) has no positive equilibrium.

  [START_REF] Barbier | Generic assembly patterns in complex ecological communities[END_REF] where ψ(x, y) := c(x, y)ϕ(x, y).

  , y)â ∞ (y)dy -g P 1 0

Figure 4 :

 4 Figure 4: Solutions to (4.13) at time 100 (blue), 500 (orange) and 1500 (green). The left panel represents the distribution of plant species and the right panel the distribution of pollinator species. Parameters are α A = 3, α P = 25, β A = β P = γ P = 1, γ A = 0.3, d P = 1, d A = 3 and δ P = 3, with N = 500

which is nite from Assumption 3 . 1 .

 31 Thus, there exists C > 0 such that P( R A,n,m > C) is suciently small. Then g P is bounded on [0, C] by some constant M (since it is a Lypschitz function). Thus, on { RA,n,m ≤ C}, for any positive continuous function f on [0, 1] and any 0 ≤ s ≤ t ≤ T ,|⟨ P n,m t , f ⟩ -⟨ P n,m s , f ⟩| ≤ t s ( M + ∥k∥ ∞ C(T ))∥f ∥ ∞ C(T )du.

Step 4 : 1 0 1 0 1 0f(A. 6 )GG 5 ,(

 411165 Equation satised by the limiting value. Let us x t > 0 and f a continuous positive function on [0, 1]. We dene the map ψ on C([0, 1], M ≤C(T ) ([0, 1]) 2 ), endowed with the uniform convergence, dened byψ(P, A) = ⟨P t , f ⟩ -⟨P 0 , f ⟩ -t 0 f (x) g P ((cϕ) ⋆ A s (x)) -k ⋆ P s (x) dP s (x)ds ⟨A t , f ⟩ -⟨A 0 , f ⟩ -t 0 f (x) g A ((cϕ) ⋆ P s (x)) -h ⋆ A s (x) dA s (x)ds . The continuity of map ψ is straightforward once noticing that the weak convergence on M 2 ≤C(T ) ([0, 1])is equivalent to convergence of measures in the KantorovichRubinstein distance (see for example Theorem 6.9 of[START_REF] Villani | Optimal transport: old and new[END_REF])W 1 (ν, µ) = sup (x)d(ν -µ)(x) f : [0, 1] → R + Lipschitz , Lip(f ) ≤ 1 , since [0, 1] is a compact space.The function ψ is also bounded since all measures are bounded by C(T ).Then, as ( P, Ā) is a weak limiting value of P n,m , A n,m , we obtain by denition of the weak convergence thatE ∥ψ( P n,m , A n,m )∥ 1 -→ n,m→+∞ E ∥ψ( P, Ā)∥ 1 .On the other hand, E ∥ψ( P n,m , A n,m )∥ 1 ij mc n,m i,j A j s -g P (cϕ) ⋆ A n,m d P n,m s (x)ds ij nc n,m i,j P j s -g A (cϕ) ⋆ P n,m d A the rst term (A.7) of the r.h.s can be bounded from above by mc n,m i,j -c(x i , y j ) + c(x i , y j ) G ij -ϕ(x i , y j )   ds ≤ C T C(T )∥mc n,m -c∥ ∞

7 )

 7

1 0f

 1 (x)d(ν -µ)(x) f : [0, 1] → [-1, 1] continuous .With straightforward computations, for any continuous function f : [0, 1] → [-1, 1], we can nd some nite constants C 1 , C

  Since g P and g A are bounded from above respectively by M P and M A and since the competition terms for the pollinators are non-positive, choosing f ≡ 1 in the

	previous equation (A.3), we nd that a.s.,
		⟨ P n,m t	, 1⟩ ≤ ⟨ P n,m 0 , 1⟩e M P t and ⟨ A n,m t	, 1⟩ ≤ ⟨ A n,m 0 , 1⟩e M A t ,	(A.4)
	for all n and m. Moreover, from Assumption (3.2), we have that
	lim n,m→+∞	⟨ P n,m 0 , 1⟩ =	[0,1]	p0 (x)dx and	lim n,m→+∞	⟨ A n,m 0 , 1⟩ =	[0,1]	ā0 (x)dx a.s..
	In other words, sup n,m (⟨ P n,m 0 , 1⟩ + ⟨ A n,m 0 , 1⟩) is bounded by a nite constant a.s. and for T > 0, we
	dene

  2 , independent from t and f , such that x → f (x)(g P (cϕ ⋆ A 1 s (x)) -k ⋆ P 1 s (x)) are continuous bounded functions on [0, 1] for all s ∈ [0, T ], and k and cϕ are continuous bounded functions on [0, 1] 2 . Same computations can be done with A 1 and A 2 , then taking the supremum over all f , we nd a constant C 3 > 0 such that

	t	1
	|⟨P 1 t -P 2 t , f ⟩| =		f (x) g P (cϕ ⋆ A 1 s (x)) -k ⋆ P 1 s dP 1 s (x)ds
	0	0
		t	1
	-		f (x) g P (cϕ ⋆ A 2 s (x)) -k ⋆ P 2 s dP 2 s (x)ds
		0	0
		t	t	1
	≤ C 1	ρ(P 1 s , P 2 s )ds +	L P cϕ ⋆ (A 1 s -A 2 s (x)) + k ⋆ (P 1 s -P 2 s )(x) dP 2 s (x)ds
	0		0	0
		t
	≤ C 2	(ρ(P 1 s , P 2 s ) + ρ(A 1 s , A 2 s ))ds,
	0
				t
	ρ(P 1 t , P 2 t ) + ρ(A 1 t , A 2 t ) ≤ C 2	(ρ(P 1 s , P 2 s ) + ρ(A 1 s , A 2 s ))ds.
				0

since

thank Mathilde Dufa¸and Pauline Latte-Godillon who participated to the early discussions on this project. They also thank Emmanuel Grenier for his help in the proof of Proposition 4.5. S.B., H.L. and V.C.T. acknowledge support of the Chaire Modélisation Mathématique et Biodiversité (MMB-Museum National d'Histoire Naturelle-Fondation X. T.R. is partially funded by Labex CEMPI (ANR-11-LABX-0007-01) and ANR Project MoHyCon (ANR-17-CE40-0027-01). V.C.T is supported by ANR EcoNet (ANR-18-CE02-0010), ANR CADENCE (ANR-16-CE32-0007) and Labex Bézout (ANR-10-LABX-0058).


Thomas Blanchet

Guillaume Laffont

Romain Cotillard

Emmanuel Marin

Adriana Morana

Aziz Boukenter

Youcef Ouerdane

Sylvain Girard

Radiation effects on type I fiber Bragg gratings: influence of recoating

Keywords: fiber Bragg gratings, radiation, coating, annealing

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Since the last decades, Fiber Bragg Gratings (FBGs) have been studied for nuclear industries to measure external parameters such as temperature or mechanical strain. FBGs present many advantages like small size, light weight, electromagnetic immunity, short interrogation time response (compared with other fiber sensors), and wavelength multiplexing capabilities. A FBG consists in a periodic modulation of the refractive index along the core which causes the reflection of a narrow range of wavelengths, centered around the so-called Bragg wavelength (λB). This resonance wavelength is defined by two parameters via the relation: λB = 2 neff Λ (1) with neff, the effective index and Λ the grating period. The spectral position of the Bragg wavelength is shifted when temperature or mechanical strain are applied to the fiber and can be used to monitor these changes. For example, for temperature sensing, in the range 20 -100°C, λB linearly shifts with the temperature, according to the following relation:

λB (T) = λB (T0) + α (T -T0) ( 2
)
where T0 is a reference temperature and α is the temperature sensitivity coefficient, whose value is typically around 10 pm/°C at 1550 nm for silica fibers. Radiation can modify the grating response through two main processes: the creation of point defects and the compaction. The first process is responsible for an increase of the optical attenuation [START_REF] Girard | Radiation Effects on Silica-Based Optical Fibers: Recent Advances and Future Challenges[END_REF]. As a consequence, the signal intensity at the fiber output decreases as the signal-to-noise ratio. In some cases, the signal can be totally lost. Both the radiation-induced attenuation and the compaction can cause a modification of the refractive index and the grating period, which induce a shift of the Bragg wavelength peak [START_REF] Gusarov | Radiation Effects on Fiber Gratings[END_REF][START_REF] Morana | Radiation Vulnerability of Fiber Bragg Gratings in Harsh Environments[END_REF] hence causing a direct error on the temperature or strain measurements. Furthermore, other effects have to be considered as the possible influence of the fiber recoating on the Radiation Induced BWS (RI-BWS). Until now, only a few papers discussed the coating influence on the FBG radiation response. [START_REF] Gusarov | Influence of the Coating Type on the Radiation Sensitivity of FBGs[END_REF] In this work, we characterize the X-ray induced BWS in Type I FBGs written in Telecom-grade fiber with and without acrylate coating at different dose-rates up to a total dose of 100 kGy that covers the environments associated with space, high energy physics and some nuclear industry needs.

*thomas.blanchet@cea.fr // thomas.blanchet@univ-st-etienne.fr , phone (+33)46966326

EXPERIMENTAL PROCEDURE

The 5 mm long FBGs have been written at Laboratory H. Curien in Saint Etienne (France), using a cw Argon Laser at 244 nm and Lloyd's mirror interferometer in the Ge-doped SMF-28 manufactured by Corning. In order to increase the fiber photosensitivity, H2 loading was performed at 150 bars and room temperature for one week [START_REF] Hill | Fiber Bragg Grating Technology Fundamentals and Overview[END_REF]. Three type I FBGs have been written per fiber at different Bragg wavelengths (1548, 1554 and 1559 nm) to study the repeatability in the FBG radiation response. The gratings are separated spatially by a few millimeters.

After the inscription, a thermal treatment was applied to the gratings. The gratings were separated on two groups, the first one was annealed at 100 °C during 4 days, and the second one at 300°C for 1 h [START_REF] Baker | Thermal Decay of Fiber Bragg Gratings Written in Boron and Germanium Codoped Silica Fiber[END_REF]. After the thermal treatment, half of all the gratings were recoated with acrylate No-61 (from Norland Products Incorporated) with a mini-coater from Nyfors. The irradiation was performed using an X-rays machine (MOPERIX -at the Lab. H. Curien) up to a total dose of 100 kGy(SiO2) at three dose-rates (0.5, 5 and 50 Gy/s) at room temperature (RT). As shown in figure 1, an Aluminium plate with bridles maintains the gratings inside the X-rays machine without any constraint. A thermocouple was integrated on the plate in order to measure the temperature variations and to separate the temperature effects on the BWS from the RI-BWS. The reflection spectra of the FBGs were recorded with a NI PMA-1115 optical sensor interrogator. This system has four optical channels operating between 1510 and 1590 nm, with a resolution of 30 pm at a 10 Hz frequency rate.

RESULTS AND DISCUSSION

In Figure 2, we report the radiation test results obtained for a group of 3 FBGs at the three studied wavelengths (1548, 1554, and 1559 nm). Those FBGs were uncoated and annealed at 100°C before irradiation at 5 Gy/s and RT. All the three FBGs show the same behavior during and after irradiation within a variation of 5%. In the following figures we will report only the average of the three curves obtained from the 3 FBGs of each group.

Figure 2. BWS induced at 5 Gy/s up to a total dose of 100 kGy, on the FBGs annealed at 100°C and non-recoated for 3 different wavelengths: full black dots: 1548nm; red dots: 1554 nm and blue lozenges 1559 nm. On the right the y axis corresponds to the error induced by the radiation on the temperature measurement when the FBGs are used as temperature sensors.

Figure 3 shows the results obtained on the four types of FBGs during X-rays irradiation at a dose rate of 5 Gy/s to a total of 100 kGy. The FBGs annealed at 100°C were more stable than those annealed at 300°C. This can be due to the different annealing durations. More studies will be performed for the conference. Moreover, the recoated FBGs show a smaller BWS than the uncoated gratings for the 2 annealing temperatures, with a difference at the end of the irradiation of 70 pm for the FBGs annealed at 300°C and 20 pm for the FBGs annealed at 100°C. The results reported in the literature by Gusarov et al [START_REF] Gusarov | Influence of the Coating Type on the Radiation Sensitivity of FBGs[END_REF] are however different. They suggest that the acrylate coating induces stress on the grating during gamma-rays irradiations. As a consequence, the FBGs without coating are less influenced by radiation than the ones with the acrylate coating. The difference might come to the radiation type: we used X-rays in our case. Since the γ-rays are more energetic than X-rays, we hypothesize that a part of the X-rays can be stopped by the coating. More tests and simulation will be performed to verify this hypothesis. Finally, figure 4 highlights the dose-rate effect by reporting the BWS induced by X-rays on the 3 uncoated FBGs annealed at 100°C at the three dose-rates, in order to reach 100 kGy accumulated dose. All the gratings exhibit a red shift as already observed on other type I FBGs [START_REF] Gusarov | Radiation Effects on Fiber Gratings[END_REF]. Moreover, the BWS induced at 100 kGy is about 300 pm at 50 Gy/s and 120 pm at 0.5Gy/s. The higher is the dose-rate the larger is the RI-BWS, indeed at lower dose-rate the recovery process is more efficient. These results agree with those reported in literature [START_REF] Fernandez | Dose-Rate Dependencies in Gamma-Irradiated Fiber Bragg Grating Filters[END_REF].

CONCLUSION

In this work, we have studied the behavior of different type I FBGs under X-rays at different irradiation conditions. Preliminary results suggest that FBGs with acrylate recoating maybe slightly more tolerant to X-rays. Additional experiments will be done to support this hypothesis for the conference and further results on polyimide coating impact will be presented too as for some applications, FBGs have to operate in mixed high temperature/high radiation levels environments, eg. in nuclear power plants.

Figure 1 .

 1 Figure 1. Experimental setup

Figure 3 .

 3 Figure 3. BWS at 5Gy/s up to 100kGy for the 4 types of FBGs full black dots: FBG 100, red dots: FBG 100 Co, full blue lozenges: FBG 300, pink lozenges: FBG 300 Co. On the right the y axis corresponds to the error induced by the radiation on the temperature measurement when the FBGs are used as temperature sensors.

Figure 4 . 3

 43 Figure 4. 3 FBGs annealed at 100°C without recoating at 3 dose rates 0.5, 5 and 50 Gy/s. On the right the y axis corresponds to the error induced by the radiation on the temperature measurement when the FBGs are used as temperature sensors.

Table 1

 1 resumes all the treatments performed on the FBGs.

		Annealing		Acrylate re-coating
	FBG	Temperature (°C)	Time
	FBG 100	100	1 hour	No
	FBG 300	300	4 days 15 hours	No
	FBG 100 Co	100	1 hour	Yes
	FBG 300 Co	300	4 days 15 hours	Yes

Table 1 .

 1 Parameters of the FBGs studied under X-rays


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Using high throughput phenotyping of growth and feed intake to improve adaptation of chickens to sustainable diets

Sandrine Mignon-Grasteau, Quentin Berger, Elodie Guettier, Elisabeth Le

Bihan-Duval

To cite this version:

Sandrine Mignon-Grasteau, Quentin Berger, Elodie Guettier, Elisabeth Le Bihan-Duval. Using high throughput phenotyping of growth and feed intake to improve adaptation of chickens to sustainable diets. 7. Mediterranean Poultry Summit, Jun 2022, Cordoba, Spain. hal-03702770


Alina Muravitskaya
email: alina.muravitskaya@gmail.com

Anna Rumyantseva
email: banna.rumyantseva@utt.fr

Atse Julien

Eric Ndohi

Camille Sonneville
email: camille.sonneville@insa-lyon.fr

Dominique Planson
email: edominique.planson@insa-lyon.fr

Pierre-Michel Adam
email: fpierre_michel.adam@utt.fr

Sylvain Potiron

Mihai Lazar
email: mihai.lazar@utt.fr

Enhanced Resonant Raman scattering of GaN functional layers using Al thin films -a versatile tool for multilayer structure analysis

Keywords: SERS, GaN, SiC, plasmonic nanoparticles, power device, Raman scattering, nanostructures, thin layers

In order to selectively analyze active thin layers close to surface in power devices structures, Raman scattering is necessary with UV excitation. However, the Raman spectra of GaN are usually affected by the direct bandgap photoluminescence of the material, which interferes with the Raman measurements and decreases the quality and resolution of the Raman spectra. In this work, we demonstrate experimentally that nanostructured aluminum films deposed on GaN epitaxial layers decrease the influence of the photoluminescence on the resonant Raman spectra and increase its overall spatial resolution under UV illumination.

Introduction

The resonant Raman scattering is a well-established non destructive method for analyzing the surface properties of the GaN determining the crystalline quality, stress, doping, free carrier concentration for process and growth monitoring [1,2]. In GaN power devices one needs to selectively analyze thin layers in the surface (active layers as emitters, channels, junction termination extensions created by ion implantation, metal-contact interfaces) with thickness that does not exceed micrometer at most and can go to ~100 nm and even less [3]. To gain the correct values, the influence of under-layers present in the vertical device (hetero)structure (substrate, buffer, drift layers, etc.) should be eliminated. Thus, for GaN thin layer analyses, UV excitation is preferred to the classical visible or IR excitation wavelengths, since it allows to examine a surface layer of about 40 nm thick (under 325 nm excitation) [4]. However, as GaN has a direct bandgap, strong photoluminescence (PL) contribution can interfere with the measurements and decrease the quality and resolution of the obtained Raman spectra under UV excitation.

One of the approaches to decrease the PL background and increase the Raman signal is to place plasmonic nanoparticles on the top of a layer of interest, which results in a quenching of the PL due to the high non-radiative transfer of the excitation energy to metal, and in the same time, Raman scattering benefits from the local field enhancement and increase in radiative decay rates near resonant plasmonic nanoparticles. In this work, we propose to use structured Al films to diminish the influence of the PL on the resonant Raman spectra of GaN and to increase its overall resolution.

Methods

GaN thin epilayers (between 500 nm and 1µm thick MOCVD) on sapphire substrates from CNRS-CRHEA with a doping concentration in the range of 10 16 to 10 17 cm -3 were covered with Al nanoparticles and Al thin films, which were deposited by means of the vacuum evaporation technique (e-beam PlassysMU400). The samples are presented in Figure 1. The Al nanoparticles film was obtained for a thickness of the deposition of 6 nm. The Al thin film was obtained after 20 nm deposition. In both cases samples were heated up to 400°C during the deposition. The Al film had a reflection which confirms the formation of the rough continuous film, while Al nanoparticles film was non-reflecting. From the area covered with the Al a FEG-SEM (Field emission Scanning Electron Microscope) image is also presented in the right side of Figure 1. On these parts of the samples, photoluminescence (PL) and µRaman measurements were done under UV (λ ex =325 nm) illumination and in back-scattering geometry with LabRAM spectrometer of Horiba Scientific (scheme in Fig. 2). Three Raman modes are allowed in GaN layers in the backscattering configuration [5]: 732 cm -1 -A 1 (LO), 568 cm -1 -E 2 high and 140 cm -1 -E 2 low

In order to compare and analyze PL and µRaman signals similar commercial 4H-SiC samples (substrate with n-type epilayers) have been prepared with the same experimental Al deposition setup.

Results and discussion

As shown in figure 3a, the deposition of the Al on top of the GaN resulted in a drastic fall of the photoluminescence intensity. In the case of the Al nanoparticles film the PL has decreased 10 times, while for the Al thin film it has decreased 100 times.

Then we observed in figure 3b the revealing of the three Raman modes in the spectra: 732 cm -1 -A 1 (LO), 568 cm -1 -E 2 high and 140 cm -1 -E 2 low . A schematic presentation of this SERS (surfaceenhanced Raman spectroscopy) effect is highlighted in Figure 2 and measurements are presented in Figure 3.

Important to note that we do not observe the contribution of the sapphire (GaN epilayer substrate in our samples) modes, which often change the shape of the spectrum of GaN. Indeed the light penetration depth in GaN is less than the epilayer thickness. Compared to visible non resonant Raman spectra (e.g at 532 nm) in which the E 2 (H) peak is the more intense, in UV Raman enhanced by Al the A 1 (LO) peak dominates the resonant Raman spectra. This effect is more visible with the Al film than with Al nanoparticles. More over A 1 (LO) peak is more intense compared to the PL background in the case of the Al thin film, than for the Al nanoparticles film. However, for the Al nanoparticles latter the lines width is sharper and the E 2 (H) peak is more pronounced. For both films the absorption of the GaN emission by Al layers and consequent fall of the PL play an important role in the Raman lines appearing. Also, the plasmonic enhancement of the Raman scattering may occur, especially as Al sustains localized surface plasmon resonances in the UV region [5,7].

4H-SiC epilayers and substrates have been also tested without and with Al layer deposited during the same run with the GaN samples. Even though the PL signal was reduced 10 times (Figure 4a) the enhancement of the Raman scattering was not observed (Figure 4b). We note that SiC compared to GaN has an indirect bandgap, and then the PL signal is not so high. In the PL spectrum one can even see the Raman scattering contribution around 330 nm (Figure 4a). Therefore, doping check or other stress and defects identification could be done by direct UV Raman measurements without Al nanoparticles or nanolayers on the surface.

Conclusions

As for SiC layers µRaman can be utilized on GaN layers as a versatile and nondestructive tool to investigate critical layers as those close to two-dimensional electron gas in AlGaN/GaN heterostructures. We show that Al thin layers deposited on top of the GaN epitaxial layers may improve the quality and intensity of the resonant Raman scattering of the GaN functional layers and therefore provide control of a doping, defects and various kinds of stress in thin layers of actual structures of power devices. In this case, a thin Al layer is needed to increase Raman scattering signal and decrease the photoluminescence. This metallic layer can be kept or removed after test.

Fig. 1 .

 1 Fig. 1. Optical photos of the GaN epilayers partially covered with Al nanoparticles and films (in the right) and a scanning-electron microscopy image of the area covered with Al nanoparticles

Fig. 2 .

 2 Fig. 2. PL and Raman signals obtained from different GaN epilayers surfaces covered or not with Al nanoparticles and films, schematically highlighting the SERS effect. The length of the arrows corresponds to the relative intensity.

Fig. 3 .

 3 Fig. 3. Photoluminescence (a) and Raman scattering spectra (b) (λ ex =325 nm) of the GaN epilayers covered with layer of Al nanoparticles and covered with thin Al film layer.

Fig. 4 .

 4 Fig. 4. Photoluminescence (a) and Raman scattering spectra (b) (λ ex =325 nm) of the SiC epilayers covered with thin Al film.

Acknowledgments

The authors acknowledge our colleagues from CRHEA-CNRS for the GaN/sapphire samples. This work has been made within the framework of the Graduate School NANO-PHOT (École Universitaire de Recherche, contract ANR-18-EURE-0013). The authors acknowledge the project RETOR.


Nicolas Bercu
email: nicolas.bercu@univ-reims.fr

Mihai Lazar
email: bmihai.lazar@utt.fr

Olivier Simonetti
email: colivier.simonetti@univ-reims.fr

Pierre-Michel Adam
email: pierre_michel.adam@utt.fr

Mélanie Brouillard
email: emelanie.brouillard@univ-reims.fr

Louis Giraudet
email: flouis.giraudet@univ-reims.fr

KPFM -Raman spectroscopy coupled technique for the characterization of wide bandgap semiconductor devices

Keywords: SiC device characterization, Raman spectroscopy, AFM, KPFM

A non-destructive technique for the characterization of the doped regions inside wide bandgap (WBG) semiconductor structures of power devices is presented. It consists in local measurements of the surface potential by Kelvin Probe Force Microscopy (KPFM) coupled to micro-Raman spectroscopy. The combined experiments allow to visualize the space charge extent of the doped region using the near-field mapping and to estimate its dopant concentration using the Raman spectroscopy. The technique has been successfully applied for the characterization of a WBG SiC (silicon carbide) device.

Introduction

Electronic devices based on wide bandgap semiconductors such as 4H-SiC are now widely used in power electronics. The reliability of these power devices is determined by how the strong electric fields present in the semiconductor are handled by device design. Precisely localized doped areas are necessary in order to minimize local electrical field spikes. We present here a non-destructive technique for the characterization of the doped regions in these power devices coupling local measurements of the surface potential by Kelvin Probe Force Microscopy (KPFM) and micro-Raman spectroscopy. The KPFM measurements allow to: i) localize and visualize the extension of the doped regions, ii) identify the location of electric field spikes and iii) characterize the depletion regions at the junction edges or due to doping variations in the semiconductor [1]. Micro-Raman spectroscopy measurements provide additional information on the type the dopants and estimates their concentrations [2].

Experimental details

The near-field KPFM technique is an electrical mode of the atomic force microscope. The operating principle is illustrated in Fig. 1 and it uses an original method [3], [4] which allows the simultaneous measurement of the surface electric potential and the second derivative of the tip/air/surface capacitance. This experimental near-field technique uses a two-step scan (dual-pass mode) for each line in order to allow quantitative surface potential measurements on biased devices [4]. While the first line scan (round trip) is used for the surface topography measurement, during the second scan the tip is lifted-up from the surface at a user-defined height of typically a few tenths of nanometers and follows the previous recorded surface profile while measuring the potential surface profile. An external electronic device (Keysight E5270B) applies the bias voltages to both the sample under test and the AFM tip during the lift scan, the synchronization with the AFM raster scan being performed using the AFM end-of-line TTL signals [3].

An external digital phase control loop [4] is implemented, enabling to quantitatively measure the surface potential in phase modulation-KPFM mode. The phase shift control loop is based on the measurement of the phase difference ∆𝜑 between the cantilever excitation drive signal and the actual cantilever oscillation signal. Within a limited tip-to-surface voltage difference (𝑉 𝑡𝑖𝑝 -𝑉 𝑠𝑢𝑟𝑓 ), the phase shift describes a parabola following [5]:

∆𝜑 = - 𝑄 2𝑘 𝑑 2 𝐶 𝑑𝑧 2 (𝑉 𝑡𝑖𝑝 -𝑉 𝑠𝑢𝑟𝑓 ) 2 (1)
where 𝑉 𝑠𝑢𝑟𝑓 is the sample under test surface potential at the tip location and 𝑉 𝑡𝑖𝑝 the bias applied to the tip. The term

𝑄 2𝑘 𝑑 2 𝐶
𝑑𝑧 2 is the phase parabola opening with Q the quality factor, k the spring constant of the cantilever and 𝐶 the tip/air/surface capacitance.

Fig. 1. Schematics of the KPFM setup and illustration of the lift mode (A) and measured phase shift versus tip bias at various lift heights (B).

In this study, the area of interest is the lateral emitter junction periphery of a high voltage (5 kV) 4H-SiC PiN diode. This area, called the Junction Termination Extension (JTE), is designed to avoid the abrupt increase of the electric field at the P + emitter edge. The fabrication process and static electrical characteristics of these 5 kV 4H-SiC PiN diodes are detailed in [6,7]. The diodes crosssection is shown in Fig. 2A and optical images are presented in Fig 2B . The identification of this area of interest is obtained using KPFM thanks to the variation of the second derivative of the capacitance which gives information on the location of the charges [4].

Fig. 2. Structure of the sample with the area of interest of the vertical bipolar PiN diode highlighted (A); Identification of the different device parts on the optical microscope image (B).

Raman spectroscopy was used to measure dopant concentration in specific areas [8]. Compared to conventional techniques such as C-V, SIMS (Secondary Ion Mass Spectrometry) or Hall effect, Raman spectroscopy is a non-contact, non-destructive and easy to implement alternative to investigate doped areas.

Due to its high photon energy, a 325 nm near-UV was selected in order to limit the investigated volume, since the JTE depth is less than 1 µm. The JTE Al doping profile is presented in Fig 3 . A doping concentration plateau between 2 and 3 10 17 cm -3 is obtained from Monte-Carlo simulations and confirmed by SIMS measurements [7].

Results and discussion

Using the local capacitance second derivative mapping of the KPFM measurements, the edge of the extension of the JTE implanted junction in reverse bias [6,7] was identified and precisely located as illustrated in Fig. 4. The interface between the JTE (P-doped at 2.5x10 17 cm -3 ) and the N-doped epitaxial layer (with a concentration of 1.1x10 15 cm -3 ) is visible 250 µm away from the emitter, which corresponds to the expected JTE length. As shown in Fig 4B, the JTE edge detection is performed using a significant variation of approximately 37% of the measured capacitance second derivative in the KPFM scan lines.

Fig. 4. Mapping of the second derivative of the local capacitance highlighting the edge of the JTE in reverse bias at -45V (A) and capacitive second derivative cross-section showing the step marking the edge of the JTE (B).

Once the location and extent of the JTE are identified, the intensity and the position in Raman shift of the characteristic line of the longitudinal optical plasmons-phonons coupling (LOPC) are used to determine the type and the concentration of dopants in 4H-SiC device structure [7]. A systematic shift and a decrease of the LOPC peak intensity are found by focusing alternatively at the JTE and at the N-type epilayer surface (Fig 5), confirming the junction interface detected by KPFM. Therefore, by coupling the near-field KPFM measurements with Raman spectroscopy we obtained a non-destructive technique able to identify the exact extent of the JTE region of a SiC diode and we were able to quantify its doping level. Compared to other near-field techniques (such as SCM) the KPFM technique has the advantage of allowing in-situ quantitative surface potential measurements during device operation which can be used for extensive device characterization. This coupled technique can be very useful in reverse engineering of 4H-SiC and other WBG power devices.

Conclusion

A non-destructive technique coupling local measurements of the surface potential by KPFM and micro-Raman spectroscopy was presented. The method was successfully applied to the characterization of the dopants in wide bandgap semiconductor structures. The space charge extent of the JTE region of a wide bandgap SiC diodes could be visualized using the near-field potential mapping, while its dopant concentration could be estimated using Raman spectroscopy.

Fig. 3 .

 3 Fig. 3. Doping profile as a function of the JTE depth.

Fig. 5 .

 5 Fig. 5. Raman spectroscopic measurements of the LOPC line (4H-SiC) showing a variation of its position inRaman shift and of its intensity as a function of the doping.

Acknowledgments

This work was financially supported by the Région Grand-Est and the European Community through the FEDER project ATRAKTOR, by the ESR network "site champardennais" through the programs NELO and NOTOR and by the Graduate School NanoPhot. The experiments were carried out within the Nanomat platform (www.nanomat.eu).