Sergueï Dachian
email: serguei.dachian@math.univ-bpclermont.fr

On Limiting Likelihood Ratio Processes of some Change-Point Type Statistical Models

Keywords: non-regularity, change-point, limiting likelihood ratio process, Bayesian estimators, maximum likelihood estimator, limiting distribution, limiting variance, asymptotic efficiency Mathematics Subject Classification (2000): 62F99, 62M99

Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. In this paper we consider two such likelihood ratios. The first one is an exponential functional of a two-sided Poisson process driven by some parameter, while the second one is an exponential functional of a two-sided Brownian motion. We establish that for sufficiently small values of the parameter, the Poisson type likelihood ratio can be approximated by the Brownian type one. As a consequence, several statistically interesting quantities (such as limiting variances of different estimators) related to the first likelihood ratio can also be approximated by those related to the second one. Finally, we discuss the asymptotics of the large values of the parameter and illustrate the results by numerical simulations.

Introduction

Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. In this paper we consider two of these processes. The first one is the random process Z ρ on Ê defined by

ln Z ρ (x) =    ρ Π + (x) -x, if x 0, -ρ Π -(-x) -x, if x 0, (1)
where ρ > 0, and Π + and Π -are two independent Poisson processes on Ê + with intensities 1/(e ρ -1) and 1/(1e -ρ ) respectively. We also consider the random variables

ζ ρ = Ê x Z ρ (x) dx Ê Z ρ (x) dx and ξ ρ = argsup x∈Ê Z ρ (x) (2)
related to this process, as well as to their second moments B ρ = Eζ 2 ρ and M ρ = Eξ 2 ρ . The process Z ρ (up to a linear time change) arises in some non-regular, namely change-point type, statistical models as the limiting likelihood ratio process, and the variables ζ ρ and ξ ρ (up to a multiplicative constant) as the limiting distributions of the Bayesian estimators and of the maximum likelihood estimator respectively. In particular, B ρ and M ρ (up to the square of the above multiplicative constant) are the limiting variances of these estimators, and the Bayesian estimators being asymptotically efficient, the ratio E ρ = B ρ /M ρ is the asymptotic efficiency of the maximum likelihood estimator in these models.

The main such model is the below detailed model of i.i.d. observations in the situation when their density has a jump (is discontinuous). Probably the first general result about this model goes back to Chernoff and Rubin [START_REF] Chernoff | The estimation of the location of a discontinuity in density[END_REF]. Later, it was exhaustively studied by Ibragimov and Khasminskii in [START_REF] Ibragimov | Statistical estimation. Asymptotic theory[END_REF]Chapter 5] see also their previous works [START_REF] Ibragimov | On the asymptotic behavior of generalized Bayes' estimator[END_REF] and [START_REF] Ibragimov | The asymptotic behavior of statistical estimates for samples with a discontinuous density[END_REF] .

Model 1. Consider the problem of estimation of the location parameter θ based on the observation X n = (X 1 , . . . , X n ) of the i.i.d. sample from the density f (xθ), where the known function f is smooth enough everywhere except at 0, and in 0 we have

0 = lim x↑0 f (x) = a = b = lim x↓0 f (x) = 0.
Denote P n θ the distribution (corresponding to the parameter θ) of the observation X n . As n → ∞, the normalized likelihood ratio process of this model defined by

Z n (u) = dP n θ+ u n dP n θ (X n ) = n i=1 f X i -θ -u n f (X i -θ)
converges weakly in the space D 0 (-∞, +∞) (the Skorohod space of functions on Ê without discontinuities of the second kind and vanishing at infinity) to the process Z a,b on Ê defined by

ln Z a,b (u) =        ln( a b ) Π b (u) -(a -b) u, if u 0, -ln( a b ) Π a (-u) -(a -b) u, if u 0,
where Π b and Π a are two independent Poisson processes on Ê + with intensities b and a respectively. The limiting distributions of the Bayesian estimators and of the maximum likelihood estimator are given by

ζ a,b = Ê u Z a,b (u) du Ê Z a,b (u)
ln Z a,b (u) =        ln( a b ) Π b ( x a-b ) -x, if x a-b 0, -ln( a b ) Π a (-x a-b ) -x, if x a-b 0, = ln Z ρ (x) = ln Z ρ (a -b) u .
So, we have

ζ a,b = ζ ρ a -b and ξ a,b = ξ ρ a -b ,
and hence

Eζ 2 a,b = B ρ (a -b) 2 , Eξ 2 a,b = M ρ (a -b) 2 and Eζ 2 a,b Eξ 2 a,b = E ρ .
Some other models where the process Z ρ arises occur in the statistical inference for inhomogeneous Poisson processes, in the situation when their intensity function has a jump (is discontinuous). In Kutoyants [START_REF] Kutoyants | Statistical Inference for Spatial Poisson Processes[END_REF]Chapter 5] see also his previous work [START_REF] Kutoyants | Parameter estimation for stochastic processes[END_REF] one can find several examples, one of which is detailed below.

Model 2. Consider the problem of estimation of the location parameter θ ∈ ]α, β[, 0 < α < β < τ , based on the observation X T on [0, T ] of the Poisson process with τ -periodic strictly positive intensity function S(t + θ), where the known function S is smooth enough everywhere except at points t * + τ k, k ∈ , with some t * ∈ [0, τ ], in which we have 0 = lim t↑t * S(t) = S -= S + = lim t↓t * S(t) = 0.

Denote P T θ the distribution (corresponding to the parameter θ) of the observation X T . As T → ∞, the normalized likelihood ratio process of this model defined by

Z T (u) = dP T θ+ u T dP T θ (X T ) = exp T 0 ln S θ+ u T (t) S θ (t) dX(t)- T 0 S θ+ u T (t) -S θ (t) dt
converges weakly in the space D 0 (-∞, +∞) to the process Z τ,S -,S + on Ê defined by

ln Z τ,S -,S + =        ln S + S -Π S - u τ -(S + -S -) u τ , if u 0, -ln S + S -Π S + -u τ -(S + -S -) u τ , if u 0,
where Π S -and Π S + are two independent Poisson processes on Ê + with in- tensities S -and S + respectively. The limiting distributions of the Bayesian estimators and of the maximum likelihood estimator are given by

ζ τ,S -,S + = Ê u Z τ,S -,S + (u) du Ê Z τ,S -,S + (u) du and ξ τ,S -,S + = argsup u∈Ê Z τ,S -,S + (u)
respectively. The convergence of moments also holds, and the Bayesian estimators are asymptotically efficient. So, Eζ 2 τ,S -,S + and Eξ 2 τ,S -,S + are the limiting variances of these estimators, and Eζ 2 τ,S -,S + /Eξ 2 τ,S -,S + is the asymptotic efficiency of the maximum likelihood estimator. Now let us note, that up to a linear time change, the process Z τ,S -,S + is nothing but the process Z ρ with ρ = ln S + S -. Indeed, by putting u = τ x S + -S - we get

Z τ,S -,S + (u) = Z ρ (x) = Z ρ S + -S - τ u .
So, we have

ζ τ,S -,S + = τ ζ ρ S + -S - and ζ τ,S -,S + = τ ξ ρ S + -S - ,
and hence

Eζ 2 τ,S -,S + = τ 2 B ρ (S + -S -) 2 , Eξ 2 τ,S -,S + = τ 2 M ρ (S + -S -) 2 and Eζ 2 τ,S -,S + Eξ 2 τ,S -,S + = E ρ .
The second limiting likelihood ratio process considered in this paper is the random process

Z 0 (x) = exp W (x) - 1 2 |x| , x ∈ Ê, (3)
where W is a standard two-sided Brownian motion. In this case, the limiting distributions of the Bayesian estimators and of the maximum likelihood estimator (up to a multiplicative constant) are given by

ζ 0 = Ê x Z 0 (x) dx Ê Z 0 (x) dx and ξ 0 = argsup x∈Ê Z 0 (x) (4)
respectively, and the limiting variances of these estimators (up to the square of the above multiplicative constant) are B 0 = Eζ 2 0 and M 0 = Eξ 2 0 . The models where the process Z 0 arises occur in various fields of statistical inference for stochastic processes. A well-known example is the below detailed model of a discontinuous signal in a white Gaussian noise exhaustively studied by Ibragimov and Khasminskii in [10, Chapter 7.2] see also their previous work [START_REF] Ibragimov | Estimation of a parameter of a discontinuous signal in a white Gaussian noise[END_REF] , but one can also cite change-point type models of dynamical systems with small noise see Kutoyants [START_REF] Kutoyants | Parameter estimation for stochastic processes[END_REF] and [START_REF] Kutoyants | Identification of dynamical systems with small noise[END_REF]Chapter 5] , those of ergodic diffusion processes see Kutoyants [START_REF] Kutoyants | Statistical inference for ergodic diffusion processes[END_REF]Chapter 3] , a changepoint type model of delay equations see Küchler and Kutoyants [START_REF] Küchler | Delay estimation for some stationary diffusion-type processes[END_REF] , an i.i.d. change-point type model see Deshayes and Picard [START_REF] Deshayes | Lois asymptotiques des tests et estimateurs de rupture dans un modèle statistique classique[END_REF] , a model of a discontinuous periodic signal in a time inhomogeneous diffusion see Höpfner and Kutoyants [START_REF] Höpfner | Estimating discontinuous periodic signals in a time inhomogeneous diffusion[END_REF] , and so on. Model 3. Consider the problem of estimation of the location parameter θ ∈ ]α, β[, 0 < α < β < 1, based on the observation X ε on [0, 1] of the random process satisfying the stochastic differential equation

dX ε (t) = 1 ε S(t -θ) dt + dW (t),
where W is a standard Brownian motion, and S is a known function having a bounded derivative on ]-1, 0[ ∪ ]0, 1[ and satisfying

lim t↑0 S(t) -lim t↓0 S(t) = r = 0.
Denote P ε θ the distribution (corresponding to the parameter θ) of the observation X ε . As ε → 0, the normalized likelihood ratio process of this model defined by

Z ε (u) = dP ε θ+ε 2 u dP ε θ (X ε ) = exp 1 ε 1 0 S(t -θ -ε 2 u) -S(t -θ) dW (t) - 1 2 ε 2 1 0 S(t -θ -ε 2 u) -S(t -θ) 2 dt
converges weakly in the space C 0 (-∞, +∞) (the space of continuous functions vanishing at infinity equipped with the supremum norm) to the process Z 0 (r 2 u), u ∈ Ê. The limiting distributions of the Bayesian estimators and of the maximum likelihood estimator are r -2 ζ 0 and r -2 ξ 0 respectively. The convergence of moments also holds, and the Bayesian estimators are asymptotically efficient. So, r -4 B 0 and r -4 M 0 are the limiting variances of these estimators, and E 0 is the asymptotic efficiency of the maximum likelihood estimator.

Let us also note that Terent'yev in [START_REF] Terent'yev | Probability distribution of a time location of an absolute maximum at the output of a synchronized filter[END_REF] determined explicitly the distribution of ξ 0 and calculated the constant M 0 = 26. These results were taken up by Ibragimov and Khasminskii in [START_REF] Ibragimov | Statistical estimation. Asymptotic theory[END_REF]Chapter 7.3], where by means of numerical simulation they equally showed that B 0 = 19.5 ± 0.5, and so E 0 = 0.73 ± 0.03. Later in [START_REF] Golubev | Computation of the efficiency of the maximumlikelihood estimator when observing a discontinuous signal in white noise[END_REF], Golubev expressed B 0 in terms of the second derivative (with respect to a parameter) of an improper integral of a composite function of modified Hankel and Bessel functions. Finally in [START_REF] Rubin | Exact computation of the asymptotic efficiency of maximum likelihood estimators of a discontinuous signal in a Gaussian white noise[END_REF]

ζ(s) = ∞ n=1 1 n s .
The random variables ζ ρ and ξ ρ and the quantities B ρ , M ρ and E ρ , ρ > 0, are much less studied. One can cite Pflug [START_REF] Pflug | On an argmax-distribution connected to the Poisson process[END_REF] for some results about the distribution of the random variables

argsup x∈Ê + Z ρ (x) and argsup x∈Ê - Z ρ (x) related to ξ ρ .
In this paper we establish that the limiting likelihood ratio processes Z ρ and Z 0 are related. More precisely, we show that as ρ → 0, the process Z ρ (y/ρ), y ∈ Ê, converges weakly in the space D 0 (-∞, +∞) to the process Z 0 . So, the random variables ρ ζ ρ and ρ ξ ρ converge weakly to the random variables ζ 0 and ξ 0 respectively. We show equally that the convergence of moments of these random variables holds, that is,

ρ 2 B ρ → 16 ζ(3), ρ 2 M ρ → 26 and E ρ → 8 ζ(3)/13.
These are the main results of the present paper, and they are presented in Section 2, where we also briefly discuss the second possible asymptotics ρ → +∞. The necessary lemmas are proved in Section 3. Finally, some numerical simulations of the quantities B ρ , M ρ and E ρ for ρ ∈ ]0, ∞[ are presented in Section 4.

Main results

Consider the process X ρ (y) = Z ρ (y/ρ), y ∈ Ê, where ρ > 0 and Z ρ is defined by [START_REF] Chernoff | The estimation of the location of a discontinuity in density[END_REF]. Note that

Ê y X ρ (y) dy Ê X ρ (y) dy = ρ ζ ρ and argsup y∈Ê X ρ (y) = ρ ξ ρ ,
where the random variables ζ ρ and ξ ρ are defined by [START_REF] Cramér | On some questions connected with mathematical risk[END_REF]. Remind also the process Z 0 on Ê defined by (3) and the random variables ζ 0 and ξ 0 defined by [START_REF] Gihman | The theory of stochastic processes I[END_REF]. Recall finally the quantities

B ρ = Eζ 2 ρ , M ρ = Eξ 2 ρ , E ρ = B ρ /M ρ , B 0 = Eζ 2 0 = 16 ζ(3), M 0 = Eξ 2 0 = 26 and E 0 = B 0 /M 0 = 8 ζ(3)/13
. Now we can state the main result of the present paper.

Theorem 1 The process X ρ converges weakly in the space D 0 (-∞, +∞) to the process Z 0 as ρ → 0. In particular, the random variables ρ ζ ρ and ρ ξ ρ converge weakly to the random variables ζ 0 and ξ 0 respectively. Moreover, for any k > 0 we have

ρ k Eζ k ρ → Eζ k 0 and ρ k Eξ k ρ → Eξ k 0 ,
and in particular

ρ 2 B ρ → 16 ζ(3), ρ 2 M ρ → 26 and E ρ → 8 ζ(3)/13.
The results concerning the random variable ζ ρ are direct consequence of Ibragimov and Khasminskii [10, Theorem 1.10.2] and the following three lemmas.

Lemma 2

The finite-dimensional distributions of the process X ρ converge to those of Z 0 as ρ → 0.

Lemma 3 For all ρ > 0 and all y 1 , y 2 ∈ Ê we have

E X 1/2 ρ (y 1 ) -X 1/2 ρ (y 2 ) 2 1 4 |y 1 -y 2 | .
Lemma 4 For any c ∈ ] 0 , 1/8 [ we have

EX 1/2 ρ (y) exp -c |y|
for all sufficiently small ρ and all y ∈ Ê.

Note that these lemmas are not sufficient to establish the weak convergence of the process X ρ in the space D 0 (-∞, +∞) and the results concerning the random variable ξ ρ . However, the increments of the process ln X ρ being independent, the convergence of its restrictions (and hence of those of X ρ ) for all sufficiently small ρ and all A > 0.

on finite intervals [A, B] ⊂ Ê that is,
All the above lemmas will be proved in the next section, but before let us discuss the second possible asymptotics ρ → +∞. One can show that in this case, the process Z ρ converges weakly in the space D 0 (-∞, +∞) to the process Z ∞ (u) = e -u ½ {u>η} , u ∈ Ê, where η is a negative exponential random variable with P{η < t} = e t , t 0. So, the random variables ζ ρ and ξ ρ converge weakly to the random variables

ζ ∞ = Ê u Z ∞ (u) du Ê Z ∞ (u) du = η + 1 and ξ ∞ = argsup u∈Ê Z ∞ (u) = η
respectively. One can equally show that, moreover, for any k > 0 we have

Eζ k ρ → Eζ k ∞ and Eξ k ρ → Eξ k ∞ ,
and in particular, denoting

B ∞ = Eζ 2 ∞ , M ∞ = Eξ 2 ∞ and E ∞ = B ∞ /M ∞ , we finally have B ρ → B ∞ = E(η + 1) 2 = 1, M ρ → M ∞ = Eη 2 = 2 and E ρ → E ∞ = 1/2.
Let us note that these convergences are natural, since the process Z ∞ can be considered as a particular case of the process Z ρ with ρ = +∞ if one admits the convention +∞ • 0 = 0.

Note also that the process Z ∞ (up to a linear time change) is the limiting likelihood ratio process of Model 1 (Model 2) in the situation when a • b = 0 (S -• S + =0). In this case, the variables ζ ∞ = η + 1 and ξ ∞ = η (up to a multiplicative constant) are the limiting distributions of the Bayesian estimators and of the maximum likelihood estimator respectively. In particular, B ∞ = 1 and M ∞ = 2 (up to the square of the above multiplicative constant) are the limiting variances of these estimators, and the Bayesian estimators being asymptotically efficient, E ∞ = 1/2 is the asymptotic efficiency of the maximum likelihood estimator.

Proofs of the lemmas

First we prove Lemma 2. Note that the restrictions of the process ln X ρ (as well as those of the process ln Z 0 ) on Ê + and on Ê -are mutually independent processes with stationary and independent increments. So, to obtain the convergence of all the finite-dimensional distributions, it is sufficient to show the convergence of one-dimensional distributions only, that is,

ln X ρ (y) ⇒ ln Z 0 (y) = W (y) - |y| 2 = N - |y| 2 , |y|
for all y ∈ Ê. Here and in the sequel "⇒" denotes the weak convergence of the random variables, and N (m, V ) denotes a "generic" random variable distributed according to the normal law with mean m and variance V .

Let y > 0. Then, noting that Π + y ρ is a Poisson random variable of

parameter λ = y ρ (e ρ -1) → ∞, we have ln X ρ (y) = ρ Π + y ρ - y ρ = ρ y ρ (e ρ -1) Π + y ρ -λ √ λ + y e ρ -1 - y ρ = √ y ρ e ρ -1 Π + y ρ -λ √ λ -y e ρ -1 -ρ ρ (e ρ -1) ⇒ N - y 2 , y , since ρ e ρ -1 = ρ ρ + o(ρ) → 1, e ρ -1 -ρ ρ (e ρ -1) = ρ 2 /2 + o(ρ 2 ) ρ ρ + o(ρ) → 1 2 and Π + y ρ -λ √ λ ⇒ N (0, 1).
Similarly, for y < 0 we have

ln X ρ (y) = -ρ Π - -y ρ - y ρ = ρ -y ρ (1 -e -ρ ) λ ′ -Π - -y ρ √ λ ′ - -y 1 -e -ρ - y ρ = √ -y ρ 1 -e -ρ λ ′ -Π - -y ρ √ λ ′ + y e -ρ -1 + ρ ρ (1 -e -ρ ) ⇒ N y 2 , -y ,
and so, Lemma 2 is proved. Now we turn to the proof of Lemma 4 (we will prove Lemma 3 just after). For y > 0 we can write

EX 1/2 ρ (y) = E exp ρ 2 Π + y ρ - y 2ρ = exp - y 2ρ E exp ρ 2 Π + y ρ .
Note that Π + y ρ is a Poisson random variable of parameter λ = y ρ (e ρ -1) with moment generating function M(t) = exp λ (e t -1) . So, we get

EX 1/2 ρ (y) = exp - y 2ρ exp y ρ (e ρ -1) (e ρ/2 -1) = exp - y 2ρ + y ρ (e ρ/2 + 1)
= exp -y e ρ/2 -1 2ρ (e ρ/2 + 1) = exp -y e ρ/4e -ρ/4 2ρ (e ρ/4 + e -ρ/4 ) = exp -y tanh(ρ/4) 2ρ .

For y < 0 we obtain similarly

EX 1/2 ρ (y) = E exp - ρ 2 Π - -y ρ - y 2ρ = exp - y 2ρ exp -y ρ (1 -e -ρ ) (e -ρ/2 -1) = exp - y 2ρ + y ρ (1 + e -ρ/2 ) = exp y 1 -e -ρ/2 2ρ (1 + e -ρ/2 ) = exp y tanh(ρ/4) 2ρ .
Thus, for all y ∈ Ê we have Further we verify Lemma 3. We first consider the case y 1 , y 2 ∈ Ê + (say y 1 y 2 ). Using ( 5) and taking into account the stationarity and the independence of the increments of the process ln X ρ on Ê + , we can write

EX 1/2 ρ (y) = exp -|y| tanh(ρ/4) 2ρ , (5)
E X 1/2 ρ (y 1 ) -X 1/2 ρ (y 2 ) 2 = EX ρ (y 1 ) + EX ρ (y 2 ) -2 EX 1/2 ρ (y 1 )X 1/2 ρ (y 2 ) = 2 -2 EX ρ (y 2 ) E X 1/2 ρ (y 1 ) X 1/2 ρ (y 2 ) = 2 -2 EX 1/2 ρ (y 1 -y 2 ) = 2 -2 exp -|y 1 -y 2 | tanh(ρ/4) 2ρ |y 1 -y 2 | tanh(ρ/4) ρ 1 4 |y 1 -y 2 | .
The case y 1 , y 2 ∈ Ê -can be treated similarly.

Finally, if y 1 y 2 0 (say y 2 0 y 1 ), we have

E X 1/2 ρ (y 1 ) -X 1/2 ρ (y 2 ) 2 = 2 -2 EX 1/2 ρ (y 1 ) EX 1/2 ρ (y 2 ) = 2 -2 exp -|y 1 | tanh(ρ/4) 2ρ -|y 2 | tanh(ρ/4) 2ρ = 2 -2 exp -|y 1 -y 2 | tanh(ρ/4) 2ρ 1 4 |y 1 -y 2 | ,
and so, Lemma 3 is proved. Now let us check Lemma 5. First let y 1 , y 2 ∈ Ê + (say y 1 y 2 ) such that

∆ = |y 1 -y 2 | < h. Then P ln X ρ (y 1 ) -ln X ρ (y 2 ) > ε 1 ε 2 E ln X ρ (y 1 ) -ln X ρ (y 2 ) 2 = 1 ε 2 E ln X ρ (∆) 2 = 1 ε 2 E ρ Π + ∆ ρ - ∆ ρ 2 = 1 ε 2 ρ 2 (λ + λ 2 ) + ∆ 2 ρ 2 -2λ∆ = 1 ε 2 β(ρ) ∆ + γ(ρ) ∆ 2 < 1 ε 2 β(ρ) h + γ(ρ) h 2 ,
where λ = ∆ ρ (e ρ -1) is the parameter of the Poisson random variable Π + ∆ ρ ,

β(ρ) = ρ (e ρ -1) = ρ ρ + o(ρ) → 1 and γ(ρ) = 1 (e ρ -1) 2 + 1 ρ 2 - 2 ρ (e ρ -1) = 1 ρ - 1 e ρ -1 2 = e ρ -1 -ρ ρ (e ρ -1) 2 = ρ 2 /2 + o(ρ 2 ) ρ ρ + o(ρ) 2 → 1 4
as ρ → 0. So, we have

lim ρ→0 sup |y 1 -y 2 |<h P ln X ρ (y 1 ) -ln X ρ (y 2 ) > ε lim ρ→0 1 ε 2 β(ρ) h + γ(ρ) h 2 = 1 ε 2 h + h 2 4 ,
and hence

lim h→0 lim ρ→0 sup |y 1 -y 2 |<h P ln X ρ (y 1 ) -ln X ρ (y 2 ) > ε = 0,
where the supremum is taken only over y 1 , y 2 ∈ Ê + .

For

y 1 , y 2 ∈ Ê -such that ∆ = |y 1 -y 2 | < h one can obtain similarly P ln X ρ (y 1 ) -ln X ρ (y 2 ) > ε 1 ε 2 E ln X ρ (y 1 ) -ln X ρ (y 2 ) 2 = 1 ε 2 β ′ (ρ) ∆ + γ ′ (ρ) ∆ 2 < 1 ε 2 β ′ (ρ) h + γ ′ (ρ) h 2 ,
where

β ′ (ρ) = ρ (1 -e -ρ ) = ρ ρ + o(ρ) → 1 and γ ′ (ρ) = e -ρ -1 + ρ ρ (1 -e ρ ) 2 = ρ 2 /2 + o(ρ 2 ) ρ ρ + o(ρ) 2 → 1 4
as ρ → 0, which will yield the same conclusion as above, but with the supremum taken over y 1 , y 2 ∈ Ê -.

Finally, for y 1 y 2 0 (say y 2 0 y 1 ) such that |y 1y 2 | < h, using the elementary inequality (ab) 2 2(a 2 + b 2 ) we get

P ln X ρ (y 1 ) -ln X ρ (y 2 ) > ε 1 ε 2 E ln X ρ (y 1 ) -ln X ρ (y 2 ) 2 13 ε 2 E ln X ρ (y 1 ) 2 + E ln X ρ (y 2 ) 2 = 2 ε 2 β(ρ)y 1 +γ(ρ)y 2 1 +β ′ (ρ)|y 2 |+γ ′ (ρ)|y 2 | 2 2 ε 2 β(ρ) + β ′ (ρ) h + γ(ρ) + γ ′ (ρ) h 2 ,
which again will yield the desired conclusion. Lemma 5 is proved.

It remains to verify Lemma 6. Clearly,

P sup |y|>A X ρ (y) > e -bA P sup y>A X ρ (y) > e -bA + P sup y<-A X ρ (y) > e -bA .
In order to estimate the first term, we need two auxiliary results.

Lemma 7 For any c ∈ ] 0 , 3/32 [ we have

EX 1/4 ρ (y) exp -c |y|
for all sufficiently small ρ and all y ∈ Ê.

Lemma 8 For all ρ > 0 the random variable

η ρ = sup t∈Ê + Π λ (t) -t ,
where Π λ is a Poisson process on Ê + with intensity λ = ρ/(e ρ -1) ∈ ]0, 1[,

verifies E exp ρ 4 η ρ 2.
The first result can be easily obtained following the proof of Lemma 4, so we prove the second one only. For this, let us remind that according to Shorack and Wellner [19, Proposition 1 on page 392] see also Pyke [START_REF] Pyke | The supremum and infimum of the Poisson process[END_REF] , the distribution function

F ρ (x) = P{η ρ < x} of η ρ is given by 1 -F ρ (x) = P{η ρ x} = (1 -λ) e λx n>x (n -x) n n! λ e -λ n
for x > 0, and is zero for x 0. Hence, for x > 0 we have

1 -F ρ (x) (1 -λ) e λx n>x (n -x) n √ 2πn n n e -n λ e -λ n = 1 -λ √ 2π e λx n>x 1 √ n 1 - x n n λ e 1-λ n 1 -λ √ 2π e λx n>x e -x λ e 1-λ n √ n 1 -λ √ 2π e (λ-1)x λ e 1-λ x n>x λ e 1-λ n-x √ n -x = 1 -λ √ 2π λ x k>0 λ e 1-λ k √ k 1 -λ √ 2π λ x Ê + λ e 1-λ t √ t dt = 1 -λ √ 2π λ x Γ(1/2) -ln λ e 1-λ = 1 -λ -2 ln λ e 1-λ ρ e ρ -1 x ρ e -ρ/2 e ρ/2 -e -ρ/2 x = ρ e -ρ/2 2 sinh(ρ/2) x e -ρx/2 ,
where we used Stirling inequality and the inequality 1-λ -2 ln λ e 1-λ , which is easily reduced to the elementary inequality ln(1µ) -µµ 2 /2 by putting µ = 1λ. So, we can finish the proof of Lemma 8 by writing

E exp ρ 4 η ρ = Ê e ρx/4 dF ρ (x) = e ρx/4 F ρ (x) -1 +∞ -∞ - ρ 4 Ê e ρx/4 F ρ (x) -1 dx = ρ 4 Ê - e ρx/4 dx + ρ 4 Ê + e ρx/4 1 -F ρ (x) dx 1 + ρ 4 Ê + e -ρx/4 dx = 2.
Now, let us get back to the proof of Lemma 6. Using Lemma 8 and taking into account the stationarity and the independence of the increments of the process ln X ρ on Ê + , we obtain for all sufficiently small ρ and all A > 0, and so the first term is estimated.

P sup y>A X ρ (y) > e -bA e bA/4 E sup y>A X 1/4 ρ (y) = e bA/4 EX 1/4 ρ (A) E sup y>A X 1/4 ρ (y) X 1/4 ρ (A) = e bA/4 EX 1/4 ρ (A) E sup z>0 X 1/4 ρ (z) = e bA/4 EX 1/4 ρ (A) E sup z>0 exp ρ 4 Π + (z/ρ) - z 4ρ = e bA/4 EX 1/4 ρ (A) E exp sup t>0 ρ 4 Π ρ e ρ -1 (
The second term can be estimated in the same way, if we show that for all ρ > 0 the random variable

η ′ ρ = sup t∈Ê + -Π λ ′ (t) + t = -inf t∈Ê + Π λ ′ (t) -t ,
where Π λ ′ is a Poisson process on Ê + with intensity λ

′ = ρ/(1 -e -ρ ) ∈ ]0, 1[, verifies E exp ρ 4 η ′ ρ 2.
For this, let us remind that according to Pyke [17] see also Cramér [START_REF] Cramér | On some questions connected with mathematical risk[END_REF] , η ′ ρ is an exponential random variable with parameter r, where r is the unique positive solution of the equation λ ′ (e -r -1) + r = 0.

In our case, this equation becomes ρ 1e -ρ (e -r -1) + r = 0, and r = ρ is clearly its solution. Hence η ′ ρ is an exponential random variable with parameter ρ, which yields

E exp ρ 4 η ′ ρ = 4

Numerical simulations

In this section we present some numerical simulations of the quantities B ρ , M ρ and E ρ for ρ ∈ ]0, ∞[. Besides giving approximate values of these quantities, the simulation results illustrate both the asymptotics

B ρ ∼ B 0 ρ 2 , M ρ ∼
B ρ → B ∞ , M ρ → M ∞ and E ρ → E ∞ as ρ → ∞,
with B ∞ = 1, M ∞ = 2 and E ∞ = 0.5. First, we simulate the events x 1 , x 2 , . . . of the Poisson process Π + with the intensity 1/(e ρ -1) , and the events x ′ 1 , x ′ 2 , . . . of the Poisson process Π - with the intensity 1/(1e -ρ ) .

Then we calculate

ζ ρ = Ê x Z ρ (x) dx Ê Z ρ (x) dx = ∞ i=1
x i e ρi-x i + Finally, repeating these simulations 10 7 times (for each value of ρ), we approximate B ρ = Eζ 2 ρ and M ρ = Eξ 2 ρ by the empirical second moments, and E ρ = B ρ /M ρ by their ratio. The results of the numerical simulations are presented in Figures 1 and2. The ρ → 0 asymptotics of B ρ and M ρ can be observed in Figure 1, where besides these functions we also plotted the functions ρ 2 B ρ and ρ 2 M ρ , making apparent the constants B 0 ≈ 19.2329 and M 0 = 26. In Figure 2 we use a different scale on the vertical axis to better illustrate the ρ → ∞ asymptotics of B ρ and M ρ , as well as both the asymptotics of E ρ . Note that the function E ρ appear to be decreasing, so we can conjecture that bigger is ρ, smaller is the efficiency of the maximum likelihood estimator, and so, this efficiency is always between E ∞ = 0.5 and E 0 ≈ 0.7397.

  du and ξ a,b = argsup u∈Ê Z a,b (u) respectively. The convergence of moments also holds, and the Bayesian estimators are asymptotically efficient. So, Eζ 2 a,b and Eξ 2 a,b are the limiting variances of these estimators, and Eζ 2 a,b /Eξ 2 a,b is the asymptotic efficiency of the maximum likelihood estimator. Now let us note, that up to a linear time change, the process Z a,b is nothing but the process Z ρ with ρ = ln( a b ) . Indeed, by putting u = x a-b we get

  , Rubin and Song obtained the exact values B 0 = 16 ζ(3) and E 0 = 8 ζ(3)/13, where ζ is Riemann's zeta function defined by

Lemma 5 Lemma 6

 56 convergence in the Skorohod space D[A, B] of functions on [A, B] without discontinuities of the second kind follows from Gihman and Skorohod [4, Theorem 6.5.5], Lemma 2 and the following lemma. For any ε > 0 we have lim h→0 lim ρ→0 sup |y 1 -y 2 |<h P ln X ρ (y 1 )ln X ρ (y 2 ) > ε = 0. Now, Theorem 1 follows from the following estimate on the tails of the process X ρ by standard argument. For any b ∈ ] 0 , 3/40 [ we have P sup |y|>A X ρ (y) > e -bA 2 e -bA

2 ρ

 2 ρ → 0, for any c ∈ ] 0 , 1/8 [ we have EX 1/(y)exp -c |y| for all sufficiently small ρ and all y ∈ Ê. Lemma 4 is proved.

M 0 ρ 2

 2 and E ρ → E 0 as ρ → 0, with B 0 = 16 ζ(3) ≈ 19.2329, M 0 = 26 and E 0 = 8 ζ(3)/13 ≈ 0.7397, and

= argmax i 1 (ρi -x i ) and ℓ = argmax i 1 (

 11 x∈Ê Z ρ (x) = x k , if ρkx k > ρρℓ + x ′ ℓ , -x ′ ℓ , otherwise,wherek ρρi + x ′ i ), so that x k = argsup x∈Ê + Z ρ (x) andx ′ ℓ = argsup x∈Ê - Z ρ (x).

Figure 1 :

 1 Figure 1: B ρ and M ρ (ρ → 0 asymptotics)

Figure 2 :

 2 Figure 2: B ρ and M ρ (ρ → ∞ asymptotics) E ρ (both asymptotics)

< 2, and so, Lemma 6 is proved.


Mathieu-Bégné Eglantine
email: eglantine.mathieu-begne@unibas.ch

Simon Blanchet

Rey Olivier

Toulza Eve

Veyssière Charlotte

Manzi Sophie

Lefort Maxim

Scelsi Orlane

A longitudinal survey in the wild reveals major shifts in fish host Microbiota after parasite infection

Keywords: Host-parasite interaction, Co-infection, Bacteria community, Individual survey, Metabarcoding

Recent studies have highlighted associations between diseases and host microbiota. It is yet extremely challenging -especially under natural conditions-to clarify whether the host microbiota promotes future infections, or whether changes in host microbiota result from infections. Nonetheless, deciphering between these two processes is essential for highlighting the role of microbes in disease progression. We longitudinally surveyed, in the wild, the microbiota of individual fish hosts (Leuciscus burdigalensis) both before and after infection by a crustacean ectoparasite (Tracheliastes polycolpus). We found a striking association between parasite infection and the host microbiota composition restricted to the fins the parasite anchored. We clearly demonstrated that infections by the parasite induced a shift in (and did not result from) the host fin microbiota. Fin microbiota further got similar to that of the adult stage, and the free-living infective stage of the parasite during infection with a predominance of the Burkholderiaceae bacteria family. This suggests that Burkholderiaceae bacteria is involved in a co-infection process and possibly facilitate T. polycolpus infection. We reveal novel mechanistic insights for understanding the role of the microbiota in host-parasite interactions, which has implications for predicting the progression of diseases in natural host populations.

Introduction

Understanding the factors shaping host susceptibility to pathogens allow anticipating disease risks. Infection has often been described as the result of tri-partite interaction between parasites virulence, host resistance and environmental factors [START_REF] Wolinska | Environment can alter selection in host-parasite interactions[END_REF]. However, recent studies demonstrated that infected hosts are associated with specific microbiota [START_REF] Aivelo | Parasite-microbiota interactions potentially affect intestinal communities in wild mammals[END_REF][START_REF] Bär | The interplay of host microbiota and parasitic protozoans at mucosal interfaces: Implications for the outcomes of infections and diseases[END_REF][START_REF] Portet | Infection of the Biomphalaria glabrata vector snail by Schistosoma mansoni parasites drives snail microbiota dysbiosis[END_REF][START_REF] Reynolds | Cohabitation in the intestine: interactions among helminth parasites, bacterial microbiota, and host immunity[END_REF][START_REF] Speer | Microbiomes mediate host-parasite interactions[END_REF]. Differences in microbiota among individuals hence provide a novel mechanism to explain infection risk within host populations. Parasitism could be seen as a multi-faceted interaction between the environment, the host, the parasite and their associated microbiota [START_REF] Bordenstein | Host biology in light of the microbiome: ten principles of holobionts and hologenomes[END_REF][START_REF] Dheilly | Holobiont-holobiont interactions: Redefining host-parasite interactions[END_REF].

A key question recently raised by [START_REF] Speer | Microbiomes mediate host-parasite interactions[END_REF] [START_REF] Speer | Microbiomes mediate host-parasite interactions[END_REF] is whether "the microbiome differentially attract parasites or […] reflect the infection status of its host" (i.e., the causal role of host microbiota during infection). Differences in host microbiota before infection may modulate host susceptibility to parasite infection, making the host microbiota central to predict individual infection risks [START_REF] Chavez | Skin-penetrating nematodes exhibit life-stage-specific interactions with host-associated and environmental bacteria[END_REF][START_REF] Speer | Microbiomes mediate host-parasite interactions[END_REF]. For instance, host microbiota can provide optimal conditions for parasite development [START_REF] Britton | Role of the intestinal microbiota in resistance to colonization by Clostridium difficile[END_REF][START_REF] Buffie | Microbiota-mediated colonization resistance against intestinal pathogens[END_REF][START_REF] Hayes | Exploitation of the intestinal microflora by the parasitic nematode Trichuris muris[END_REF] or constitute a cue for the parasite to locate its hosts [START_REF] Chavez | Skin-penetrating nematodes exhibit life-stage-specific interactions with host-associated and environmental bacteria[END_REF][START_REF] Speer | Microbiomes mediate host-parasite interactions[END_REF]. Different microbiota between infected and uninfected hosts can also occur after parasite infection. For instance, parasites while disrupting host immune system can modulate host microbiota composition [START_REF] Beatty | Giardia duodenalis induces pathogenic dysbiosis of human intestinal microbiota biofilms[END_REF][START_REF] Lee | Helminth colonization is associated with increased diversity of the gut microbiota[END_REF]. Parasites can also indirectly facilitate colonization by opportunist pathogenic bacteria or fungi from the surrounding environment by weakening host resistance [START_REF] Mooney | Inflammation-associated alterations to the intestinal microbiota reduce colonization resistance against non-typhoidal Salmonella during concurrent malaria parasite infection[END_REF]. Finally parasites can directly carry new microbes that co-infect the host, while benefiting parasite fitness [START_REF] Dheilly | Biological warfare: Microorganisms as drivers of hostparasite interactions[END_REF][START_REF] Reynolds | Cohabitation in the intestine: interactions among helminth parasites, bacterial microbiota, and host immunity[END_REF]. Specific association between microbiota communities and parasites have been described in the wild in several plant and animal species [START_REF] Andre | The role of endosymbiotic wolbachia bacteria in the pathogenesis of river blindness[END_REF][START_REF] Boissière | Midgut microbiota of the malaria mosquito vector Anopheles gambiae and interactions with Plasmodium falciparum infection[END_REF][START_REF] Stensvold | Associations between gut microbiota and common luminal intestinal parasites[END_REF]. Nonetheless, a pressing challenge is to distinguish the mechanisms leading to these associations in wild populations.

A powerful approach to decipher causality in parasite-microbiota-host interactions is to survey the microbiota of individual hosts before and after infection [START_REF] Afrin | Sequential changes in the host gut microbiota during infection with the intestinal parasitic nematode Strongyloides venezuelensis[END_REF][START_REF] Blanchet | Reciprocal effects between host phenotype and pathogens: New insights from an old problem[END_REF][START_REF] Gaulke | A longitudinal assessment of host-microbe-parasite interactions resolves the zebrafish gut microbiome's link to Pseudocapillaria tomentosa infection and pathology[END_REF][START_REF] Jenkins | Schistosoma mansoni infection is associated with quantitative and qualitative modifications of the mammalian intestinal microbiota[END_REF]. Most host microbiota surveys have focused on changes occurring during or after parasite exposure, which limits our understanding of the mechanisms sustaining relationships between parasites, hosts and the microbiota (but see, [START_REF] Jenkins | Schistosoma mansoni infection is associated with quantitative and qualitative modifications of the mammalian intestinal microbiota[END_REF][START_REF] White | Manipulation of host and parasite microbiotas: Survival strategies during chronic nematode infection[END_REF]. Also, to investigate the mechanisms leading to changes in host microbiota after parasite infection, both host-associated microbiota and microbiota from any potential colonization sources must be investigated. Understanding the role of host-associated microbiota during parasitic infection is therefore a challenging task, especially in the wild, where controlling confounding factors is difficult. However, investigating the role of host-associated microbiota during parasitic infection constitutes a promising avenue for realistic and novel insights into the complex links between parasites, hosts, the environment, and their microbiota.

Here, we tested whether variation in the host microbiota composition is associated with parasitic infection, and we questioned whether this variation pre-exist to infection or is induced by the infection. We focused on an ectoparasite (the freshwater Crustacean Tracheliastes polycolpus) that has emerged in Western Europe in the 1920's and that infects several native fish species [START_REF] Lootvoet | Patterns and processes of alternative host use in a generalist parasite: Insights from a natural host-parasite interaction[END_REF][START_REF] Rey | Elucidating the spatio-temporal dynamics of an emerging wildlife pathogen using approximate Bayesian computation[END_REF]. The parasite anchors and feeds on fins where secondary infections by bacteria and fungi are often observed [START_REF] Blanchet | Evidence of parasite-mediated disruptive selection on genetic diversity in a wild fish population[END_REF]. We tracked the microbiota of the fin mucus of its main fish host species, the rostrum dace (Leuciscus burdigalensis), both before and after infection at the individual host (fin) level. We first tested whether infected hosts displayed different microbiota compared to non-infected ones. We then tested whether fin microbiota may facilitate the infection by T. polycolpus by comparing fin microbiota before parasite infection between hosts that became infected or remained noninfected. We assumed that, if fin microbiota is a cause of infection, specific bacterial communities before infection should be associated with those of infected hosts. We further investigated the temporal dynamics of microbiota during infection to test whether the infection by T. polycolpus resulted in changes in fin bacteria community. If so, we expected that, during infection, the microbiota of infected hosts will shift away from those of non-infected hosts and we tested whether these changes resulted from colonisation of opportunistic environmental bacteria and/or from colonisation from the parasite.

Material and Method Ethics approval

The experiment was performed in agreement with the national legislation (Arrêté Préfectoral 22/05/2018) and did not require ethical approval.

Biological history of the interaction

Tracheliastes polycolpus is a copepod that feed on fish fins causing degradations and inflammations [START_REF] Loot | The effects of the ectoparasite Tracheliastes polycolpus (Copepoda: Lernaeopodidae) on the fins of rostrum dace (Leuciscus leuciscus burdigalensis)[END_REF]. Only females are parasitic, whereas males are microscopic free-living forms. Once anchored on a fin, the parasite develops two egg sacs [START_REF] Mazé-Guilmo | Etude du potentiel de colonisation des parasites: Une approche intégrative[END_REF]. After a few weeks, eggs hatch and release free living larvae that rapidly develop into a copepodite infective stage [START_REF] Mazé-Guilmo | Etude du potentiel de colonisation des parasites: Une approche intégrative[END_REF]. Larvae are not in direct contact with the environment until hatching. Therefore, the microbiota of T. polycolpus larvae -until hatching-is mainly transmitted from the mother, which makes this model particularly powerful to study parasite-associated microbiota.

Experimental design

We tracked the microbiota of each individual fin of captivity-raised rostrum dace (L. burdigalensis) in a semi-wild caging situation before and after infection. Twenty cages (30x50x80 cm, mesh size=1 cm), each containing four dace (80 dace in total) were set up within a 200-m long section in the Arize River in southwestern France (X=366422, Y=4770813, WGS 84/UTM zone 31N) during summer 2018. Cages were spaced one from the other of a one meter minimal distance in microhabitats previously identified as hotspots of infections (deep whirlpool areas, [START_REF] Mathieu-Bégné | A finescale analysis reveals microgeographic hotspots maximizing infection rate between a parasite and its fish host[END_REF]) so as to maximize the chance of infection. Cages were upstream-downstream orientated so as to favor a natural flow in each cage. Water temperature ranges from ~13°C to ~17 °C throughout the experiment and was constant over the entire river section (no substantial differences among cages). Dace were F2 individuals from wild ancestors bred at the INRAe hatchery facility in Rennes (U3E, France). All dace shared a similar genetic background (i.e., they hatched from the same pool of genitors), were of the same age (2-years old fish, sub-adult to adult stage) and were similar in weight and body size (15±3 g and 110±8 mm) to limit confounding factors. They were brought back from Rennes in early spring, raised in groups of 10 fish in height large outdoor tanks at the SETE laboratory (Moulis, France). The tanks were filled with dechlorinated tap water and included enrichment elements such as gravel at the bottom of each tank and artificial refugees for the wellbeing of the fish. Dace were fed ad libitum before the onset of the experiment. None of them were previously exposed to T. polycolpus (but their ancestors were in the wild). Fish were introduced in their cages in situ the 9 th of July 2018. Once every two days, cages were brushed (to avoid mesh clogging and to let natural particles including food and infective larvae circulate) and dace were visually inspected for infections (expected to occur after four weeks of caging [START_REF] Mathieu-Bégné | A finescale analysis reveals microgeographic hotspots maximizing infection rate between a parasite and its fish host[END_REF]) using an Aqua Scope II TM (i.e., underwater viewing device). After ~3 weeks of acclimatization, dace were collected from their cages and fin mucus was sampled to characterize fin microbiota before infection. All individuals were anesthetized (using benzocaine), we verified that no infection has occurred yet, and the mucus was sampled on each of the seven fins (anal, caudal, dorsal, pelvic, and pectoral) and on the body using sterile cotton swabs (Copan Diagnostics®). Fish were weighted, measured, and individually tagged using colored elastomers (Northwest Marine Technology®) injected at fin insertions. Fish were then gently replaced in their cage. From that period, cages were checked daily for T. polycolpus infection, and a second swab sampling was performed on surviving daces one to three weeks after the first swabbing. All fish were extracted from their cages, measured, weighted, checked for T. polycolpus infection, and mucus was sampled on each fin and on the body to characterize fin microbiota after infection. Suspicious aspects of the fish fins such as those caused by other fin ectoparasites (e.g., Gyrodactylus sp.) were not observed, and hence only T. polycolpus infections were recorded. However, eight dace were not considered in the final analyses either because they were found dead or either because they displayed abnormal behavior (difficulty to maintain a vertical position in the cage or a horizontal position in the water column). This resulted in 57 dace at the end of the experiment including 13 infected dace. When a parasite was found (34 parasites in total), it was gently removed and stored in a 50-ml vial containing water from the river for transporting them alive to our lab facilities. Water samples were also collected at each cage to characterize the environmental bacteria. All swab samples were stored at -40°C until DNA extraction.

To sum up, host microbiota was sampled on all host fins and body before and after infections.

Four main fin categories can be defined according to their status after the infections: (i) microbiota from infected fins (n = 18 fins from 13 fish), (ii) microbiota from healthy (uninfected) fins sampled on an infected fish (i.e., T. polycolpus generally infects 1 or 2 fins out of the seven fins, letting the others uninfected -healthy-, n = 60 fins from 13 fish), (iii) microbiota from healthy (uninfected) fins sampled on healthy fish in a cage where at least one infected fish was observed (n = 144 fins from 24 fish), and (iv) microbiota from healthy fins sampled on healthy fish in cages where no infected fish was observed (n = 66 fins from 11 fish).

Parasite microbiota

The microbiota of all parasites was characterized from the trunk of adult parasites and from infective larvae reared in the laboratory. When parasites carried mature egg sacs (7 parasites, corresponding to 14 egg sacs), they were brought alive to the laboratory. The two egg sacs were separated from the parasite trunk using sterile scalpels. Trunks of adult parasites were stored at -40°C until DNA extraction, and each egg sac was placed in a single 2mL well from a sterile 24-well plate (as in [START_REF] Mazé-Guilmo | Etude du potentiel de colonisation des parasites: Une approche intégrative[END_REF]. Wells were filled with 2 mL natural mineral water (renewed every day) until hatching. Well plates were stored in a thermostat incubator at a temperature matching the one measured in the field during sampling (16°C ± 1 °C). The thermostat incubator was enlightened according to a 12:12 day-night cycle. Once egg sacs hatched, larvae were collected, stored in 2mL Eppendorf® and kept at -40°C until extraction (~225 larvae, pooled by egg sac).

DNA extraction and 16S rRNA gene amplicon sequencing

We followed the DNA extraction and amplification protocol described in [START_REF] Bestion | Climate warming reduces gut microbiota diversity in a vertebrate ectotherm[END_REF]. Bacteria DNA of each sample were extracted using a standard protocol for gram-positive bacteria based on the Qiagen DNeasy Blood & Tissue Kit (Qiagen GmbH, Hilden, Germany). After extraction, the concentration of the resulting gDNA was 10-20 ng/μl. DNA was amplified using PCRs realized in a final volume of 20 μL including 2 μL of DNA, 10 μL of AmpliTaq Gold TM 360 Master Mix (ThermoFisher Scientific), 0.5 µM of each primer and 3,2 μg of bovine serum albumin. Initial denaturation was realized at 95°C for 10 minutes followed by 30 cycles of denaturation (95°C for 30 s), annealing (57°C for 30 s) and elongation (72°C for 90 s) and finally followed by a 7 min final elongation step at 72°C. We amplified the V5-V6 region of the 16S rRNA gene (295 bp) using a pair of universal bacteria primers (BACTB-F: GGATTAGATACCCTGGTAGT; and BACTB-R: CACGACACGAGCTGACG, [START_REF] Fliegerova | Effect of DNA extraction and sample preservation method on rumen bacterial population[END_REF]. Samples were identified with unique combination of 8-base tags added in 5' end of each primer. PCR products were purified (QIAquick PCR purification Kit, Qiagen GmbH, Hilden, Germany) and pooled. A single library of the amplicon multiplex was prepared accordingly to the MetaFast protocol. Sequencing was performed at the GeT-PlaGe facility (Toulouse, France) using the sequencer Illumina MiSeq and 2x250 bp paired-end sequencing technology. Positive controls (Zymobiomics Bacteria Community DNA Stan) and negative controls (blanks, and air samples realized during the sampling campaign and during extraction) were included in the sequenced multiplex to investigate extraction and amplification success as well as marker universality, and potential contamination, respectively. Overall, 759 samples were sequenced.

Bioinformatic analysis and filtering

Illumina sequencing data were demultiplexed using the OBITools suite of scripts [START_REF] Boyer | OBITOOLS: A UNIXinspired software package for DNA metabarcoding[END_REF]. Data were then processed using FROGS [START_REF] Escudié | FROGS: Find Rapidly OTU with Galaxy Solution[END_REF]. FROGS first includes a preprocessing step that de-replicates and filters amplicons according to their length. We set a range of possible amplicon size of 200-350 bp, which fits the size of amplicons for the V5-V6 region of the bacterial 16S rRNA gene. Clustering in Operation Taxonomic Units (OTUs) was then realized using the swarm algorithm implemented in FROGS with aggregation distance set to three [START_REF] Mahé | Swarm: Robust and fast clustering method for amplicon-based studies[END_REF]. Chimera and singletons were removed. The taxonomic affiliation of each OTU was done using BLAST on the Silva database up to the species level and equally good affiliation were denoted as multiaffiliation (v132, [START_REF] Quast | The SILVA ribosomal RNA gene database project: Improved data processing and web-based tools[END_REF]. Finally, using the R package Phyloseq [START_REF] Mcmurdie | phyloseq: An R package for reproducible interactive analysis and graphics of microbiome census data[END_REF], OTU composition of each sample was rarefied to 2 000 sequences per OTU to allow samples comparisons and we further filtered out OTUs occurring in less than 3 samples and under a frequency of 1‰ (excluding all the negative and blank controls) and 4 603 OTUs were retained.

Statistical analysis Association between parasitic infection and host microbiota

We first tested whether fin microbiota of host differ when the host is infected by comparing the microbiota among the four categories of fin samples after infection (see above, categories i to iv). We measured beta dissimilarity based on the Bray-Curtis distance to test for differences in community composition and assessed significance of through a Permutational Multivariate Analysis of Variance (adonis2 function, from the R package vegan, [START_REF] Oksanen | vegan: Community Ecology Package[END_REF] followed by pairwise comparisons (pairwise.adonis2 function from the R package pairwiseAdonis, [START_REF] Martinez | PairwiseAdonis: Pairwise multilevel comparison using adonis[END_REF]. Source of beta diversity (balanced variation in species abundances or abundance gradient) was assessed using the R package betapart [START_REF] Baselga | Separating the two components of abundance-based dissimilarity: Balanced changes in abundance vs. abundance gradients[END_REF]. We also measured alpha-diversity based on the Shannon index and tested differences of fin microbiota richness through a Kruskal-Wallis test followed by pairwise comparisons.

Fin microbiota before parasite infection

To test whether differences in microbiota were associated with fin mucus existed prior to parasite infection, we focused on the fin microbiota sampled before infection. We relied on the same approach as previously, based on measures of alpha-diversity and beta dissimilarity to assess community differences in term of richness and composition.

Fin microbiota after parasite infection

To test whether parasite infection triggers changes in host microbiota, we investigated the temporal dynamics of fin bacteria communities during infection. We expected that temporal shifts (from before to after infection) in fin microbiota should differ between infected and healthy fins, and that shifts should be stronger in magnitude in infected-than in healthy fins. We conducted a Community Trajectory Analysis (CTA, De Cáceres et al., 2019) based on a Discriminant Analysis of Principal Component ran on OTU relative abundances using the R package adegenet [START_REF] Jombart | adegenet: A R package for the multivariate analysis of genetic markers[END_REF]. The four categories of fins microbiota, both before and after infection, were used as grouping variables. Trajectories through time for each fin were drawn using the R package vegclust (De Cáceres & De Cáceres, 2010) and using the DAPC axis as a projection plan. For each trajectory we tested differences in trajectory length using a KruskallWallis test combined with a post-hoc test correcting for false discovery rate. Similarly, we tested for differences in trajectory direction (relying on a comparison of pairwise Euclidean distances between trajectories) using a MANOVA (multivariate analysis of variance) followed by pairwise permutation tests (1000 permutations, corrections for false discovery rate) using the R packages vegan and RVaideMemoire [START_REF] Hervé | Package 'RVAideMemoire[END_REF][START_REF] Oksanen | vegan: Community Ecology Package[END_REF].

Potential sources for host microbiota changes after infection

To identify potential source of host microbiota changes after infection, we developed an approach inspired by [START_REF] Roguet | Fecal source identification using random forest[END_REF]. First, we aimed to identify OTUs discriminating healthy and infected fins after infection. We built a Random Forest model (hereafter RF model) classifying healthy and infected fins according to their respective OTU relative abundances. To simplify the model and the interpretation, we only considered the microbiota sampled on healthy fins from fish in a cage where no infection occurred compared to infected fins that we contrasted to the microbiota sampled from infected fins (i.e., the two most contrasted modalities). We set the number of trees to 10 000 and estimated the best mtry value to 134. Because we had an unbalanced experimental design, we constrained the number of samples used to achieve classification to two third of the minimum sample size (i.e., 12) and the remaining third was used to estimate the model error referred as to Out Of Bag (OOB) error. We then identified OTUs that contributed the most to the discrimination between infected and healthy fins using the rfPermute R package to compute p-values through permutation tests (1 000 permutations) (Archer, 2019). The most contributing OTUs were those with Gini-index p-values lower than 5% [START_REF] Breiman | Random forests[END_REF]. We further evaluated the ability of these candidate OTUs to discriminate fins by building a refined RF model (number of trees=4000 and mtry=26) and compared the OBB error rate of the refined RF models to that of the RF model including all OTUs. Secondly, we searched for potential sources of OTUs characterizing the microbiota of healthy and infected fins by screening the microbiota composition of the environment (i.e., water samples after infections), the adult parasites and the parasite larvae. We hypothesised that if one of these sources contributes significantly to either the healthy or infected fin microbiota, the relative abundances of candidate OTUs found in the potential source(s) should significantly match those found in either healthy or infected fins. We used the refined RF model described above and classified the samples from the three potential sources as either healthy or infected fins. The most likely source(s) should be classified as either healthy or infected fins consistently through samples. Significance of the final classification was tested with a Chi-squared test.

Results

Association between parasitic infection and host microbiota

We compared the host microbiota after infection among the four fin categories: infected fins, healthy fins from an infected fish, healthy fins from a healthy fish in a cage where at least one infected fish was observed, and healthy fins from a healthy fish in cages where no infected fish was observed. We identified significant differences in microbiota composition between infected fins and healthy fins (either from infected fish, healthy fish or healthy cage) (F=4.395; df=3,277 and p =0.001). Pairwise comparisons revealed differences between each fin categories except for the comparison between healthy fin from a healthy fish in an infected cage and healthy fin from an infected fish (Table 1). This indicates that infected fins differed in their microbiota composition compared to all other types of healthy fins, but also that some differences existed among some healthy fins from the different categories. Nonetheless, the dissimilarity estimates were twice higher for comparisons involving infected fins and healthy fins (mean Bray-Curtis=0.353) than for comparisons involving two types of healthy fins (mean BrayCurtis=0.186, Table S1), showing that differences between infected fins and healthy fins were higher than among categories of healthy fins. This was confirmed through a visual inspection of the relative abundance of the most predominant bacteria families, as the strongest differences were observed for infected fins, with a noticeable predominance of bacteria from the Burkholderiaceae family in infected fins, while healthy fins (any categories) were characterized by a predominance of Neisseriaceae (Fig. 1). The partitioning of dissimilarity revealed that all the dissimilarity measured among fin categories was attributed to balanced variation in species abundances, which highlighted changes in the relative abundance of bacteria OTUs among fin categories (nestedness), rather than changes in the composition of bacteria OTUs (turnover) (Table S1). We did not find a consistent association between OTU richness and parasite infection. We indeed identify significant alpha-diversity differences (chi-squared = 16.809, df = 3, p-value<0.001) but pairwise comparisons revealed significant differences only between healthy fins of fish kept in healthy cage and other categories of healthy fins (on fish kept in a parasitized cage, or on a fish that have been parasitized on another fin, see Table S2). At the contrary no significant alpha diversity differences were observed between healthy fins and parasitized fins (see supplementary Fig S1 andTable S2).

Fin microbiota before parasite infection

We did not identify significant differences in microbiota composition of host fins before infections occurred (F=1.371; df=3,214 and p-value=0.074). The microbiota of fins that turned out to be infected later on were similar at the onset of the experiment to those of fins that stayed un-infected (irrespectively of the category of fins). The most abundant bacteria families were the same in the different categories of fins before infection (Fig. 2). Similarly, no significant differences in term of alpha diversity were observed between the different categories of fins (chi-squared = 0.62159, df = 3, p-value = 0.892).

Host microbiota composition after parasite infection

We found significant differences between the trajectories of bacteria communities associated with infected fins compared to those associated with healthy fins. Specifically, we found that changes -from before to after infection-in bacteria communities of infected fins were significantly stronger in magnitude (i.e., difference in term of trajectory lengths) than changes measured in all other healthy fin categories (Kruskal-Wallis chi-squared = 69.168, df = 3, pvalue <0.001; Fig. 3, Table S3). Similarly, the trajectory direction observed for the bacteria communities of infected fins was significantly different from that of all other categories of healthy fins (Manova F=131.8;df=3,206;Fig. 3,Table S3). Overall, these results suggest that infections strikingly modified the microbiota of fins on which the parasite anchors, and that these changes were more important than those naturally occurring over time in fins that remains healthy (even when these fins were sampled on an infected fish) (Fig. 3).

Potential source for host microbiota changes after infection

The first RF model classifier (based on all OTUs) achieved satisfying assignment success (OOB was 13.41% and class errors were 12.50% and 16.67% for healthy and infected fins respectively). Up to 188 OTUs significantly contributed to the decrease in the Gini index, and hence to the purity of the RF model (Table S4). The significance of these candidate OTUs was further validated by the refined RF model relying only on these 188 candidate OTUs, since the refined model achieved an even lower OOB error (OOB was 7.32 % and class errors were 6.25 % and 11.11% for healthy and infected fins respectively). We were hence able to classify infected vs. healthy fins after infection based on a few highly discriminating OTUs (188) with a high accuracy.

When considering these candidate OTUs, we found that the Burkholderiaceae family was abundant in the microbiota of infected fin, adult parasites and parasite larvae (16%, 15% and 5% respectively), whereas this family was rare in the environment (0.8%) and in the microbiota of healthy fins (0.5%) (Fig. 4). We further investigated the similarity of potential sources of bacteria (i.e., environment, adult parasite, infective parasite larvae) either with healthy fins or infected fins using the refined RF classifier model. Attributions resulting from the RF classifier were significantly different from randomness df=2,. Specifically, we found that the microbiota associated with parasite adult and larvae were classified more often than by chance in the same class as infected fins (Chi-squared = 30.118, df = 1, p-value < 0.001 and Chi-squared = 13, df = 1, p-value = 0.001 respectively, Fig. 5). For instance, two of the four OTUs that mostly contributed to the purity of the RF classifier (i.e., OTU-15 and OTU-10, Fig. S2-a) showed higher relative abundances in infected fins, in adult parasites and more strikingly in parasite larvae, and both belonged to the Burkholderiaceae family (Table S4 and Fig. S2-b). For the two other OTUs (1 and 3), their abundances were on the contrary lower in the infected fins, adult parasites and parasite larvae than in the environment and on the healthy fins. These two OTUs belonged to the Neisseriaceae and the Actinomycetaceae families respectively (Table S4).

Discussion

We tested whether infection by an ectoparasite of freshwater fish was associated with a particular host microbiota, and assessed the causal role of host microbiota in parasitic infection. We identified striking differences in the microbiota composition of host fins associated with parasite infection. Host microbiota differences did not exist prior to infection but were rather triggered during infection and were restricted to the fin where the parasite anchored (i.e., they were not generalized across the host). The microbiota of infected fins also shared strong similarities with parasite microbiota (in particular the larvae), which likely suggests a coinfection dynamic.

The composition of host-associated microbiota changed when hosts were infected by T. polycolpus. We compared the host microbiota of infected fins from that of healthy fins from various categories, which allowed testing whether the effects of parasite infection on the host microbiota are local (only the infected fin) or more general (the infected host and/or neighbor fish). Microbiota differences existed between healthy fins after infections, but the strongest community dissimilarity was observed between infected fins and all type of healthy fins. This suggests that the effects of parasite infection on host microbiota is local. We further showed that these dissimilarities resulted from balance variation in species abundance (nestedness-like patterns) [START_REF] Baselga | Separating the two components of abundance-based dissimilarity: Balanced changes in abundance vs. abundance gradients[END_REF] which is congruent with the lack of differences in term of OTU richness. Infected fins were characterized by a higher abundance of Burkholderiaceae bacteria, whereas healthy fins were mostly characterized by Neisseriaceae bacteria. These conclusions were confirmed by the Community Trajectory Analysis and the Random Forest models. The former analysis established a specific community trajectory over time of infected fins compared to that of healthy fins that were relatively homogeneous among healthy fins from the different categories. The RF models accurately discriminated healthy fins (from healthy fish in healthy cages) and infected fins based on the relative abundance of a set of candidate OTUs that included OTUs from the Burkholderiaceae and Neisseriaceae families. All these lines of evidence strongly suggest that the fin microbiota of infected dace differ from that of healthy dace (and fins), which aligns with previous studies having investigated the association between parasitism and host microbiota [START_REF] Aivelo | Parasite-microbiota interactions potentially affect intestinal communities in wild mammals[END_REF][START_REF] Bär | The interplay of host microbiota and parasitic protozoans at mucosal interfaces: Implications for the outcomes of infections and diseases[END_REF][START_REF] Portet | Infection of the Biomphalaria glabrata vector snail by Schistosoma mansoni parasites drives snail microbiota dysbiosis[END_REF][START_REF] Reynolds | Cohabitation in the intestine: interactions among helminth parasites, bacterial microbiota, and host immunity[END_REF].

We further showed that the differences in fin microbiota observed between infected fins and healthy fins resulted from infection rather than pre-existed to it. When comparing the same host fins before and after the infection period, we indeed identified important changes in host microbiota that occurred only for fins that became infected. On the contrary, prior to infection, no significant differences were noticeable between fins that became infected and those that remained healthy, and the most abundant families were stunningly the same (Fig. 3). Previous studies suggested that some host microbiota can promote parasite infection, by providing suitable conditions for parasite development [START_REF] Hayes | Exploitation of the intestinal microflora by the parasitic nematode Trichuris muris[END_REF], by lowering host resistance to parasite [START_REF] Britton | Role of the intestinal microbiota in resistance to colonization by Clostridium difficile[END_REF][START_REF] Buffie | Microbiota-mediated colonization resistance against intestinal pathogens[END_REF] or by serving as cues used by the parasite to track their host [START_REF] Chavez | Skin-penetrating nematodes exhibit life-stage-specific interactions with host-associated and environmental bacteria[END_REF][START_REF] Speer | Microbiomes mediate host-parasite interactions[END_REF]. In our study, both host and environmental factors were experimentally controlled. Thus, if some microbiota may favor future parasitic infection, this microbiota might be built upon multiple factors, notably prior differences between host in term of genetic background and/or life history traits, and/or differences in the environment in which hosts are living.

Hosts microbiota changed locally at the infected fin level rather than at the infected host level. Changes in the host microbiota over the time of infection have already been reported in other hostparasite interactions [START_REF] Afrin | Sequential changes in the host gut microbiota during infection with the intestinal parasitic nematode Strongyloides venezuelensis[END_REF][START_REF] Gaulke | A longitudinal assessment of host-microbe-parasite interactions resolves the zebrafish gut microbiome's link to Pseudocapillaria tomentosa infection and pathology[END_REF]). Yet, these changes in microbiota can result from different scenarios [START_REF] Mooney | Inflammation-associated alterations to the intestinal microbiota reduce colonization resistance against non-typhoidal Salmonella during concurrent malaria parasite infection[END_REF][START_REF] Reynolds | Cohabitation in the intestine: interactions among helminth parasites, bacterial microbiota, and host immunity[END_REF][START_REF] Swe | Parasitic scabies mites and associated bacteria joining forces against host complement defence[END_REF]. Tracheliastes polycolpus infection could have for instance weakened the host immune response [START_REF] Lochmiller | Trade-offs in evolutionary immunology: Just what is the cost of immunity?[END_REF][START_REF] Sandland | Costs of immune defense: An enigma wrapped in an environmental cloak[END_REF], which may favor fin colonization by opportunistic bacteria from the environment [START_REF] Mooney | Inflammation-associated alterations to the intestinal microbiota reduce colonization resistance against non-typhoidal Salmonella during concurrent malaria parasite infection[END_REF]. In that case, the general health status of the host is impacted, and opportunistic colonization by bacteria is expected on the entire host (i.e., a "generalized" response). Alternatively, T. polycolpus could have disrupted the fin bacteria communities according to a weapon strategy (i.e., a "localized" response, eg., [START_REF] Swe | Parasitic scabies mites and associated bacteria joining forces against host complement defence[END_REF]). In that case, microbiota disruption is directly caused by the parasite (not by the infection process) that can bring new bacteria taxa. Dace microbiota changes occurred on the fin where T. polycolpus is anchoring, which suggests that T. polycolpus is following a weapon strategy.

Finally, we investigated the source of host microbiota after infection and deduced that host microbiota was probably transmitted (at least partly) directly by the parasite. The microbiota of infected fins was highly similar to that of both adult parasite and the free-living infective parasite larvae. We can suppose that some specific bacteria strains, including from the Burkholderiaceae family are intimately linked to the host-parasite couple and are maintained via some transfers from the infective parasite larvae and the adult parasite to the host. We believe that these bacteria are transmitted and maintained in the parasite larvae since those examined in this study were hatched and reared in a common and controlled environment before their microbiota were analysed (with no contacts with the host nor natural environment). Those shared bacteria between infected fin and T. polycolpus may result from a co-infection dynamics between the parasite and its associated microbiota [START_REF] Dheilly | Biological warfare: Microorganisms as drivers of hostparasite interactions[END_REF][START_REF] Reynolds | Cohabitation in the intestine: interactions among helminth parasites, bacterial microbiota, and host immunity[END_REF]. A taxonomic investigation further revealed that the Burkholderiaceae family, which was characteristic of infected fin, was also quite prevalent in parasite microbiota (adult and larvae, 15% and 5% respectively). Contrastingly, this family was extremely rare in the microbiota of healthy fins and of the environment (<1%). In agreement with a weapon strategy [START_REF] Reynolds | Cohabitation in the intestine: interactions among helminth parasites, bacterial microbiota, and host immunity[END_REF], we suggest that bacteria from the Burkholderiaceae family are carried by the parasite and transmitted directly to the host during infection, which results in a major modification of the microbiota of infected fins. Those Burkholderiacea bacteria could be commensal of T. polycolpus or beneficial for the parasite. For instance, in [START_REF] Flórez | Antibiotic-producing symbionts dynamically between plant pathogenicity and insect-defensive mutualism[END_REF] the authors shown that the symbiotic Burkholderia gladioli bacteria, provided two antifungal compounds to its beetle host while being also vertically transmitted to the larvae stages. However, the functional role of a bacteria species remains very heterogeneous within bacteria families, and it is difficult to draw clear conclusions on the role of Burkholderiacea bacteria for T. polycolpus. Future work avenues hence relate to i) specifying the candidate bacteria species that are facilitating T. polycolpus infection and ii) their functional role in the infection process.

Abundant literature regarding host microbiota related to parasitic infections provide evidence of correlative links between host microbiota and parasite infections [START_REF] Jenkins | Schistosoma mansoni infection is associated with quantitative and qualitative modifications of the mammalian intestinal microbiota[END_REF][START_REF] Lutz | Associations between Afrotropical bats, eukaryotic parasites, and microbial symbionts[END_REF][START_REF] Speer | Microbiomes mediate host-parasite interactions[END_REF][START_REF] Stensvold | Associations between gut microbiota and common luminal intestinal parasites[END_REF]. Our results provide one of the few evidences that it is possible to dissect more accurately the way the microbiota participates to the host-parasite interaction in natural settings. In particular, the host microbiota seems to be more than the extended phenotype of the host whose changes are triggered by parasite infection. We are notably emphasising that parasite associated bacteria are very likely to be shared with the host during infection. Hence, microbiota (in particular that of the parasites) appears to be an important intermediary in host-parasite interactions, if not an active actor of these interactions. Our study adds upon other empirical evidences showing that some components of host-parasite interactions are actually directly mediated by the microbiota. For instance [START_REF] Chavez | Skin-penetrating nematodes exhibit life-stage-specific interactions with host-associated and environmental bacteria[END_REF] showed that host cue used by the some skin-penetrating nematodes to localize their host are actually emitted by host microbiota rather than by the host per se. Hence, we argue that host but also parasite microbiota are critical components of host-parasite interactions and notably of their dynamics. For instance, one could question the role of host microbiota during repeated infections. If a parasite such as T. polycolpus actively participate to host skin microbiota changes, infected host with these altered microbiotas could be more prone to future infections. Our results hence pave the way for future research avenues integrating host and parasite microbiota as a critical actors or host-parasite interaction.

To conclude, we conducted one of the first individual-based surveys describing changes in host microbiota over the course of a parasite infection under natural conditions. We unambiguously demonstrated that the parasite T. polycolpus induces shifts of host microbiota composition at the fin level. We suggest a possible co-infection dynamic between parasite and its associated bacteria and that a complex interaction between host, parasite, environment, and respective microbiota is shaping the infection process. We argue for further investigations of these complex interactions in natural setting to improve our knowledge of the relative role of parasiteassociated microbiota in infection success.

Data accessibility and benefit sharing statement

Dataset necessary to interpret and replicate the analyses conducted in this study along with raw sequencing data and a description of fish used in the experiment have been released in a figshare repository (https://doi.org/10. 6084/m9.figshare.20073005, Mathieu-Bégné et al. 2023) No benefit is to be reported for the present study.

Authors' contributions

EMB, SB, OR, ET and GL developed the original idea, supervised and coordinated the study. EMB, SB, ML, OS and GL conducted the caging experiment and realized the sampling. CV, SM and GL prepared the libraries for sequencing. EMB and ET defined the bioinformatic protocol. EMB ran the statistical analysis. EMB, SB, OR, ET and GL wrote the manuscript. All authors approved the final version of the manuscript, and all agree to be held accountable for the content therein.

Competing interest statement

We have no competing interest. For each category of fin (infected fin, healthy fin from an infected fish, healthy fin from an healthy fish in an infected cage and healthy fin from an healthy fish in a healthy cage) microbiota composition is drawn at the family level for the seven most prevalent families. Relative abundances over the total bacteria community are displayed.

Figure captions

Figure 1 :

 1 Figure1: Microbiota composition of host after infection. For each category of fin (infected fin, healthy fin from an infected fish, healthy fin from an healthy fish in an infected cage and healthy fin from an healthy fish in a healthy cage) microbiota composition is drawn at the family level for the seven most prevalent families. Relative abundances over the total bacteria community are displayed.

Figure 2 :Figure 3 :

 23 Figure2: Microbiota composition of host before infection. For each category of fin (infected fin, healthy fin from an infected fish, healthy fin from an healthy fish in an infected cage and healthy fin from an healthy fish in a healthy cage) microbiota composition is drawn at the family level for the seven most prevalent families. Relative abundances over the total bacteria community are displayed. a)

Figure 4 :

 4 Figure 4: Microbiota composition, associated with infected fins, healthy fins, adult parasite, parasite larvae and surrounding environment. Microbiota composition is drawn at the family level and forOTUs significantly discriminating healthy fin associated microbiota from infected fin associated microbiota. Relative abundances over the total community are displayed.

Figure 5 :

 5 Figure 5: Barplot showing the percentage of samples belonging to each potential source (environment, adult parasite and parasite larvae) attributed either to the same class as microbiota associated with healthy fin or as microbiota associated with infected fin. Attributions are based on the random forest model built on candidate OTUs and classifying microbiota associated with healthy or infected fins.

Table 1 : Permutational Multivariate Analysis of Variance pairwise comparisons.

 1 For each comparison (a-f) degree of freedom (Df), test statistic (F) and the p-value are given.

	Df	F	Pr(>F)

Infected fin vs healthy fin from a healthy fish in an infected cage

	1,155	6.501	0.001
	c)

Infected fin vs healthy fin from a healthy fish in a healthy cage

	1,810	6.943	0.001
	d)

Healthy fin from an infected fish vs healthy fin from a healthy fish in an infected cage

	1,198	1.776	0.051
	e) Healthy

fin from an infected fish vs healthy fin from a healthy fish in a healthy cage

	1,123	5.245	0.001
	f) Healthy

fin from a healthy fish in an infected cage vs healthy fin from a healthy fish in a healthy cage

	1,202	3.943	0.002

Acknowledgements

We sincerely thank the people who dedicated their time to this project. We are very grateful to Kéoni Saint-Pé, Murielle Richard, Alexandre Garreau and Eloise Duval who contributed to the experiment and data collection. We also thank Thierry Paulin who gently allowed us to conduct our caging experiment on his property and Antoine Gallard who provided the dace along with instructions to take good care to them. We thank the reviewers for insightful comments.

This work was founded by the Labex Tulip Project 2019, Parasite-microbiota interaction, Variability in fish fins-associated microbiota: a serious side effect or a real chance for parasites? EMB was supported by a PhD grant from the French Ministry for Education and Sciences. This study was supported by the « Laboratoire d'Excellence (LabEx) » TULIP (ANR-10-LABX-41).


Yannick Dongmo Zangue

Yannick Dongmozangue

Romain Melot

Philippe Martin

RESERVING ENVIRONMENT THROUGH FARMLAND MANAGEMENT PRACTICES (FMP)? A REVIEW

Slide n°P

reserving environment in agriculture: the importance of farmland management practices.

Cropping (farming) system design as the mainstream agronomic approach  crop diversification  optimization of agricultural practices  extension of crop rotation  Etc.

…. Under the assumption that farm territories are stable

A typology of FMP is an important output of our study, as far as it seems to be relevant in many contexts (example: Type 1.1: land temporary renting found in Canada, France, USA, China, etc.)

Discussion and Conclusion

13

FMP could be a real help for a better design of farming systems

There is still a need of a methodology to assess the environmental consequences of FMP

-Four

  Actions implying changes in land use AND in use rights words: «Farmland and environment OR Cropland tenure OR farmland tenure OR farm size and land consolidation» References found= 8,879 papers (selection factor: relavance by keywords in i) title (Papers selected after title reading = 820) and ii) abstract) FMP implemented by public policies (39 papers) -Agri-environmental impact of FMP (13 papers) -Other aspects of FMP e.i: farmland tenure, land reform, etc. (21 papers) under farmers' strategies without explicit or apparent collective coordination Temporary lease of plots by farmers to meet phytosanitary controls or to impede the growing cycle of weed species (see: Amblard and Colin 2009; Choumert and Phélinas 2015) Purchase of scattered plots in different agroecological zones to avoid local drought (see Gedefaw et al. 2019) New farm plots created by forest clearing (strategies based on local farmers' initiatives in search of adaptive solutions to adjust the overall functioning of their farms (2of land use rights through the use of agricultural work companies 2. FMP related to collective farmer initiatives A farmer authorizes a herder to let the livestock graze on his plots after the crop harvest or during inter-cropping (see Poinsot and Faure 2000). In France, joint crop rotation is a means of putting plots of land into a single one to gain productivity and save working time (see Gabriel et al. 2019) Land owners or tenants decide to entrust an agricultural contractor to carry out agricultural work on their land (See Horst 2019 or Magnan 2015) a defined perimeter where agrienvironmental regulations are edicted.e.i: Natura 2000 network:(see:Salsi et al. 2010; Zmihorski et al. 2016; Gałecka-Drozda et al. 2019; Santana et al. 2014 etc.) These are instruments or agreements carried out by institutional actors which affect land use rights. e.i: Land tenure reform (see:HuiXu et al. 2014; Sallaku et al.2016) Slide n°FMP typologyNatura 2000 network: Contributes to improving biodiversity (Zmihorski et al. 2016; Santana et al 2013) Environmental consequences of FMP possible, but in-depth studies required… Few papers explicitly address the environmental consequences of FMP (only 12 papers out of 95) Land lease: Reduction of chemical inputs (Liu et al. 2019); Land clearing: destruction of forests; Increased urbanization (Lipscomb et Prabakaran 2020; King et Balogh 2001) 1.3 Land use rights access through land conquest 1.1 Purchase of land use rights to improve the farming technical system or structural

  acquisition to improve technical system or structural farmland organisation grouping of plots with levelling of hedges or purchase of neighbouring plots Marie et al. (2009); Francart and Pivot (1998); Janovska et al. (2017); Barbottin et al. (2018); Saint-Cyr et al. (2019); Puech et al. (2020) Temporary exchanges of plots (Marie et al. (2009); Teijeiro et al. (2020) Environmental constraints + Constraints linked to the farm topology Land use rights acquisition to adapt to external constraints (climatic, urban pressure with competition on land) plot leasing Amblard and Colin 2009; Jarrige et al. (2003); Rotz, Fraser and Martin 2019; Wästfelt and Zhang 2018) Plot fragmentation (acquisition of dispersed plots) Sklenicka (2016a); Latruffe and Piet (2014); Gedefaw et al. (2019) Purchase or sale of plots Latruffe et Piet (2014) Environmental constraints Land use rights access through land conquest or land cleared Dynamics of occupation of still vacant land (pioneer front in Africa and the Amazon) Valette et al. (2013); Dounias (1998); Lipscomb and Prabakaran (2020); Choumert and Phélinas (2015); Tayeb (2019) Sharing land use rights between multiple farmers The vain pasture Sabourin et al. (1995) Socio-institutional constraints Community land use rights (communal property) Hu (1997); Binot and Karsenty (2007) Land use agreements between landowners and farmers Horst (2019); Poinsot and Faure 2000; Clément et al. (2019) Environmental constraints + Constraints linked to the farm topology Pooling of land use rights to meet a common project Voluntary pooling of land use (collective land use) Gabriel et al. (2019) Environmental constraints Delegation of land use rights through the use of agricultural work companies Use of agricultural contractors Horst (2019); Amblard and Colin (2009); Anzalone and Purseigle (2014) FMP induced by national or local public policies that impact the management of property or use rights National or local public policies implementation Natura 2000 Network Salsi et al. (2010); Zmihorski et al. (2016); Gałecka-Drozda et al. (2019); Santana et al. (2014) Environmental fallow land Toivonen et al. (2013); Xie and Jin (2019) Long terms exchanges of plots Pauchard et al.(2016); Teijeiro et al. (2020) Rural Environmental Leases / land trust) Léger-Bosch (2019); Wästfelt and Zhang (2018); Xie and Jin (2019); Bodiguel (2015) Ecological compensation through supply Etrillard and Pech (2015) Land consolidation Nilsson (2019); Djanibekov and Finger (2018); Asiama et al. (2019); Grammatikopoulou et al. (2013); Yaslioglu et al. (2009); Wang et al. (2002)

The papers consulted are spread over about 40 countries on all continents. China ranges first (30 papers), followed by France (26 papers), Spain (11 papers), the United States (9 papers) , Poland (4 papers), Czech Republic and Canada (3 papers each), etc. About 60% of the papers are less than 5 years old.


Keywords: fibre optique Valeur politique : ex, budgets participatifs Valeur sociale : ex, réseaux sociaux de solidarité Valeur stratégique : ex, hackathons Valeur qualité de vie : ex, civic techs 2020) Ressources financières Ressources humaines Performance organisationnelle

Valeur idéologique : ex. jumeaux numériques Valeur d'intendance : ex. outils de mesure de la satisfaction Valeur économique : ex.

Territoires numériques

Territoire connecté

Ressources disponibles

La transformation des organisations associée au déploiement des outils numériques puis l'animation des dynamiques de coconstruction sont consommatrices de ressources humaines et financières

Gouvernance en réseau

Territoire Territoire Intelligent Territoire serviciel Smart City Territoire augmenté

	Pourquoi parler
	de « Territoires
	numériques » ?
	Plan France Très Haut Débit (THD)	Contraintes réglementaires
		Pressions / incitations financières
	Conflits de valeurs entre « cité politique » et « ville algorithmique »	Mutations sociétales entre « culture numérique » et « capitalisme de surveillance »

Un enjeu de valeur publique Une recherche qualitative longitudinale Des politiques publiques des usages du numérique Une gouvernance proactive et ouverte Le défi de la co- création de valeur Conclusion Environnement de soutien et de légitimité Capacités opérationnelles Propositions de valeur publique

	Pourquoi parler
	de « Territoires
	numériques » ?
	❑ 4 collectivités représentatives de la diversité des territoires français et
	des multiples « alliances territoriales » constituées autour de la
	Types de valeur publique création de valeur publique par le numérique (2021 -2023)
			(Harrison et al., 2012)
	Collectivité A	Collectivité B	Collectivité C	Collectivité D
	• 150 000	• 110 000	• 25 000	• 680 000
	habitants « Triangle stratégique » habitants de la valeur publique (Ville centre) (Ville centre)	habitants (Ville étudiée)	habitants (Département)
	(Moore, 1995) • 300 000	• 275 000	• 310 000
	habitants (EPCI)	habitants (EPCI)	habitants (EPCI)	• Projet Syndicat mixte
	• Projet Ville	• Projet Ville	• Projet Ville	Département
	centre et EPCI	centre	terrain d'étude	et EPCI
	• 2018-2032	• 2014-2026	• 2020-2026	• Depuis 2018
	• 20 entretiens	• 10 entretiens	• Observation	• Observation
	semi-directifs	semi-directifs	participante	non
				participante

Un enjeu de valeur publique Une recherche qualitative longitudinale Des politiques publiques des usages du numérique Une gouvernance proactive et ouverte Le défi de la co- création de valeur Conclusion Politiques publiques des usages numériques Autres politiques publiques locales

	Autres politiques
	publiques locales
	• Adaptés au contexte
	• Multi-dimensionnels
	• Déclinés sous forme de cas d'usage

Des objectifs de création de valeur publique •

  Déploiement des socles technologiques et des outils • Définition et mise en oeuvre de stratégies « Datas » • Transformation des organisations • Accompagnement au changement et communication

Un ensemble d'actions et outils Pourquoi parler de « Territoires numériques » ? Un enjeu de valeur publique Une recherche qualitative longitudinale Des politiques publiques des usages du numérique Une gouvernance proactive et ouverte Le défi de la co- création de valeur Conclusion

	Pourquoi parler
	de « Territoires
	numériques » ?

6 Source : Frucquet et al. (2021)

Un enjeu de valeur publique Une recherche qualitative longitudinale Des politiques publiques des usages du numérique Une gouvernance proactive et ouverte Le défi de la co- création de valeur Conclusion

❑ Malgré les ambitions, beaucoup

de difficultés pour mettre en oeuvre des pratiques systématiques de co-création de valeur publique Phasage des projets

	Le déploiement et la stabilisation du socle
	technologique constituent des prérequis à
	l'amplification des usages

Un enjeu de valeur publique Une recherche qualitative longitudinale Des politiques publiques des usages du numérique Une gouvernance proactive et ouverte Le défi de la co- création de valeur Conclusion ❑ Pour des Territoires numériques créateurs de valeur publique :

	Pourquoi parler
	de « Territoires
	numériques » ?
	• Concevoir des politiques publiques des usages du numérique
	Le dépassement des expérimentations et • Qui dialoguent à rang égal avec les autres politiques publiques
	conduites dans les territoires	le passage à l'échelle impliquent de faire
		travailler ensemble des acteurs aux
		intérêts parfois contradictoires

• Qui soit dotées d'une gouvernance proactive et ouverte

• En veillant à penser la dimension « ouverture » dès la phase d'initialisation de ces politiques et à

mobiliser les ressources ad hoc

	Pourquoi parler
	de « Territoires
	numériques » ?
	• En dépassant les limites administratives, les enjeux de pouvoirs et…
	les égos dans une logique de « leadership de la chaîne de valeur
	publique » (Walravens, 2015) autour des technologies numériques

Un enjeu de valeur publique Une recherche qualitative longitudinale Des politiques publiques des usages du numérique Une gouvernance proactive et ouverte Le défi de la co- création de valeur Conclusion Merci pour votre attention Contact :

  Pascal Frucquet pfrucquet@univ-pau.fr

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