Anne De Bouard

Arnaud Debussche

SOLITON DYNAMICS FOR THE KORTEWEG-DE VRIES EQUATION WITH MULTIPLICATIVE HOMOGENEOUS NOISE

Keywords: 1991 Mathematics Subject Classification. 35Q53, 60H15, 76B25, 76B35 Korteweg-de Vries equation, stochastic partial differential equations, white noise, central limit theorem, solitary waves

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Soliton dynamics for the Korteweg-de Vries equation with multiplicative homogeneous noise

Introduction

Our aim is to describe the dynamics of a soliton solution of the Korteweg-de Vries equation in the presence of a random potential, depending both on space and time and which is white in time. After the first paper [START_REF] Wadati | Stochastic Korteweg-de Vries equations[END_REF] showing "superdiffusion" of the soliton of the KdV equation in the presence of an external force which is a white noise in time (see also [START_REF] Bass | Dynamics of solitons under random perturbations[END_REF], [START_REF] Herman | The stochastic, damped Korteweg-de Vries equation[END_REF]), the interest in such questions of soliton dynamics in the presence of either deterministic or random perturbations has recently increased in the mathematical community. In [START_REF] Garnier | Long-time dynamics of Korteweg-de Vries solitons driven by random perturbations[END_REF], e.g. the question is investigated with the help of inverse scattering methods, for different types of time-white noise perturbations, still for the KdV equation, while in [START_REF] Fröhlich | Solitary wave dynamics in an external potential[END_REF], [START_REF] Fröhlich | Long time motion of NLS solitary waves in a confining potential[END_REF], the case of a soliton of the NLS equation is studied, with the presence of a slowly varying deterministic external potential. Random potential perturbations for NLS equations have also been considered in [START_REF] Garnier | Asymptotic transmission of solitons through random media[END_REF] and [START_REF] De Bouard | Modulation analysis for a stochastic NLS equation arising in Bose-Einstein condensation[END_REF]. The diffusion of solitons of the KdV equation in the presence of additive noise was numerically investigated in [START_REF] Printems | Aspects Théoriques et numériques de l'équation de Korteweg-de Vries stochastique[END_REF]. Also, in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF], we studied the soliton dynamics for a KdV equation with an additive space-time noise. Our aim here is to reproduce the analysis of [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF] in the case of a random potential, which is stationary in space : the solution of the stochastic equation starting from a soliton at initial time will then stay close to a modulated soliton up to times small compared to ε -2 where ε is the amplitude of the random perturbation (see below). In the present case, where the noise is multiplicative (the random potential) we are then able to analyze more precisely the modulation equations for the soliton parameters and the linearized equation for the remaining (dispersive) part of the solution, and especially its asymptotic behavior in time.

We consider a stochastic KdV equation which may be written in Itô form as

(1.1) du + (∂ 3 x u + 1 2 ∂ x (u 2 ))dt = εudW
where ε > 0 is a small parameter, u is a random process defined on (t, x) ∈ R + × R, W is a Wiener process on L 2 (R) whose covariance operator φφ * is such that φ is a convolution operator on L 2 (R) defined by

φf (x) = R k(x -y)f (y)dy, for f ∈ L 2 (R).
The convolution kernel k satisfies (1.2)

k 1 := R (k 2 + (k ′ ) 2 )dx < +∞.
Considering a complete orthonormal system (e i ) i∈N in L 2 (R), we may alternatively write W as

(1.3) W (t, x) = i∈N β i (t)φe i (x),
(β i ) i∈N being an independent family of real valued Brownian motions. The correlation function of the process W is then given by

E(W (t, x)W (s, y)) = c(x -y)(s ∧ t), x, y ∈ R, s, t > 0,
where

c(z) = R k(z + u)k(u)du.
The existence and uniqueness of solutions for stochastic KdV equations of the type (1.1) but with an additive noise have been studied in [START_REF] De Bouard | On the stochastic Korteweg-de Vries equation[END_REF], [START_REF] De Bouard | White noise driven Korteweg-de Vries equation[END_REF], [START_REF] De Bouard | Periodic solutions of the Korteweg-de Vries equation driven by white noise[END_REF]. The multiplicative case with homogeneous noise as described above was considered in [START_REF] De Bouard | The Korteweg-de Vries equation with multiplicative homogeneous noise[END_REF]: assuming, together with the above condition, that k is an integrable function of x ∈ R allowed us to prove the global existence and uniqueness of solutions to equation (1.1) in the energy space H 1 (R), that is in the space where both the mass

(1.4) m(u) = 1 2 R u 2 (x)dx
and the energy

(1.5) H(u) = 1 2 R (∂ x u) 2 dx - 1 6 R u 3 dx
are well defined. Note that m and H are conserved for the equation without noise, that is

(1.6) ∂ t u + ∂ 3 x u + 1 2 ∂ x (u 2 ) = 0.
Under the above conditions on k, it was then proved in [START_REF] De Bouard | The Korteweg-de Vries equation with multiplicative homogeneous noise[END_REF] that for any given initial data u 0 ∈ H 1 (R), there is a unique solution u of (1.1) with paths a.s. continuous for t ∈ R + with values in H 1 (R).

Our aim in this article is to analyze the qualitative influence of a noise on a soliton solution of the deterministic equation. More precisely, we study the qualitative behavior of solutions of (1.1) in the limit ε tends to zero, assuming that the initial state of the solution is a soliton of equation (1.6). We recall indeed that equation (1.6) possesses a two-parameter family of solitary waves (or soliton) solutions, propagating with a constant velocity c > 0, with the expression

u c,x 0 (t, x) = ϕ c (x -ct + x 0 ), x 0 ∈ R, where (1.7) ϕ c (x) = 3c 2 cosh 2 ( √ c x 2 ) satisfies the equation (1.8) ϕ ′′ c -cϕ c + 1 2 ϕ 2 c = 0.
We do not recall here the well-known results concerning the stability of the soliton solutions u c,x 0 in equation (1.6), but we refer to [START_REF] Benjamin | The stability of solitary waves[END_REF], [START_REF] Bona | Stability and instability of solitary waves of Korteweg-de Vries type[END_REF], [START_REF] Martel | Asymptotic stability of solitons of the subcritical gKdV equations revisited[END_REF] or [START_REF] Pego | Asymptotic stability of solitary waves[END_REF] for a review of the stability questions using PDE methods, or to [START_REF] Gardner | Method for solving the Korteweg-de Vries equation[END_REF] and [START_REF] Schuur | Asymptotic analysis of soliton problems[END_REF] for a review of the stability of the solitons with the help of the inverse scattering transform.

Let us consider as in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF] the solution u ε (t, x) of equation (1.1) which is such that u ε (0, x) = ϕ 0 (x) where c 0 > 0 is fixed. Then, in Section 2, we show, as we did in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF] for the additive equation that up to times Cε -2 , where C is a constant, we may write the solution u ε as (1.9)

u ε (t, x) = ϕ c ε (t) (x -x ε (t)) + εη ε (x -x ε (t))
where the modulation parameters c ε (t) and x ε (t) satisfy a system of stochastic differential equations and the remaining term εη ε is small in H 1 (R). We then prove in Section 3 that the process η ε converges as ε goes to zero, in quadratic mean, to a centered Gaussian process η which satisfies an additively driven linear equation, with a conservative deterministic part; we also investigate the behavior of the process η as t goes to infinity and prove that η is in some sense a Ornstein-Uhlenbeck process, with a unique Gaussian invariant measure. In addition, the parameters x ε (t) and c ε (t) may be developed up to order one in ε and we get

dx ε = c 0 dt + εB 1 dt + εdB 2 + o(ε) dc ε = εdB 1 + o(ε),
where B 1 and B 2 are correlated real valued Brownian motions; keeping only the order one terms in those modulation parameters, we then obtain a diffusion result on the modulated soliton similar to the result obtained by Wadati in [START_REF] Wadati | Stochastic Korteweg-de Vries equations[END_REF], but with a different time exponent (see Section 4).

In all what follows, (., .) will denote the inner product in L 2 (R),

(u, v) = R u(x)v(x)dx
and we denote by T x 0 the translation operator defined for ϕ ∈ C(R) by (T x 0 ϕ)(x) = ϕ(x + x 0 ). Note that since the process W is stationary in space, for any x 0 ∈ R the process T x 0 W is still a Wiener process with covariance φφ * . Indeed by (1.3),

T x 0 W (t, x) = k∈N (φe k )(x + x 0 )β k (t) = k∈N (φẽ k )(x)β k (t),
with ẽk (x) = T x 0 e k .

Modulation and estimate on the exit time

In this section, we prove the following theorem.

Theorem 2.1. Assume that the kernel k of the noise satisfies (1.2) together with k ∈ L 1 (R) and let c 0 be fixed. For ε > 0, let u ε (t, x), as defined above, be the solution of (1.1) with u(0, x) = ϕ c 0 (x). Then there exists α 0 > 0 such that, for each α, 0 < α ≤ α 0 , there is a stopping time τ ε α > 0 a.s. and there are semi-martingale processes c ε (t) and x ε (t), defined a.s. for t ≤ τ ε α , with values respectively in R + * and R, so that if we set εη ε (t) = u ε (t, .+x ε (t))-ϕ c ε (t) , then a.s. for t ≤ τ ε α , εη ε (t) 1 ≤ α and |c ε (t)c 0 | ≤ α. In addition, for α 0 sufficiently small, and any α ≤ α 0 , there is a constant C > 0, depending only on α and c 0 , such that for any T > 0, there is an ε 0 > 0, with, for each ε < ε 0 , (2.1)

P(τ ε α ≤ T ) ≤ exp - C(α, c 0 ) ε 2 T k 2 H 1 .
It was noticed heuristically in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF], and proved in [START_REF] De Bouard | Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise[END_REF] that in the additive case, the use of the modulation parameters x ε (t) and c ε (t) was necessary in order to get the estimate (2.1). Indeed, it was proved in [START_REF] De Bouard | Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise[END_REF] that if we denote by τ ε,n α = inf{t > 0, u ε,n (t, .)ϕ c 0 1 > α}, where u ε,n is here the solution of equation (1.1), but with an additive noise that becomes stationary in space as n goes to infinity (see [START_REF] De Bouard | Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise[END_REF] for a precise statement) then there exists a constant C(α, c 0 ) which depends on α and c 0 but not on T such that

(2.2) lim n→∞ lim ε→0 ε 2 log P (τ n,ε α ≤ T ) ≥ - C(α, c 0 ) T 3 . It is not clear that (2.
2) is still true in the present multiplicative case, because the proof involves a controlability problem with a potential which -up to now -is open.

Note also that the decomposition given in Theorem 2.1 is not unique, and is determined by the choice of specific orthogonality conditions (see the proof below). In particular, contrary to the additive case, we will be able here to investigate the asymptotic behavior in time of the limit process by choosing one particular decomposition of the form given in Theorem 2.1. This is the object of Section 3.3. Proof of Theorem 2.1 The proof follows closely the proof of Theorem 2.1 in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF] and we refer to [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF] for more details. The parameters x ε (t) and c ε (t) are obtained thanks to the use of the implicit function Theorem. These are then local semi-martingales defined as long as

|c ε (t) -c 0 | < α and u ε (t, . + x ε (t)) -ϕ c 0 1 < α, and setting εη ε (t) = u ε (t, . + x ε (t)) -ϕ c ε (t) , one has for each ε > 0, almost surely, (2.3) (η ε , ϕ c 0 ) = (η ε , ∂ x ϕ c 0 ) = 0.
In order to estimate the exit time

τ ε α = inf{t ≥ 0, |c ε (t) -c 0 | > α or εη ε (t) 1 > α},
we make use , as in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF], of the functional defined for u ∈ H 1 (R),

(2.4)

Q c 0 (u) := H(u) + c 0 m(u)
where H and m are defined respectively in (1.4) and (1.5). Note that ϕ c 0 is a critical point of Q c 0 . We denote by L c 0 the linearized operator around ϕ c 0 , that is

(2.5) L c 0 = -∂ 2 x + c 0 -2ϕ c 0 .
The next lemma, which is proved with the use of the Itô Formula, using the same regularization procedure as in [START_REF] De Bouard | On the stochastic Korteweg-de Vries equation[END_REF], gives the evolution of H and m for the solution u ε of (1.1) with u ε (0) = ϕ c 0 :

Lemma 2.2. For any stopping time τ < +∞ a.s, one has

m(u ε (τ )) = m(ϕ c 0 ) -ε τ 0 ((u ε )(s), dW (s)) + ε 2 |k| 2 L 2 τ 0 m(u ε (s))ds and
H(u ε (τ )) = H(ϕ c 0 ) + ε τ 0 (∂ x u ε , ∂ x (u ε dW (s))) - ε 2 τ 0 ((u ε ) 3 , dW (s)) (2.6) + ε 2 2 τ 0 |k| 2 L 2 |∂ x u ε | 2 L 2 + |k ′ | 2 L 2 |u ε | 2 L 2 ds (2.7) - ε 2 2 k τ 0 R (u ε ) 3 |φe k | 2 dxds. (2.8) Consider ν > 0 such that (Q ′′ c 0 (ϕ c 0 )v, v) ≥ ν v 2 1 for any v ∈ H 1 satisfying (v, ϕ c 0 ) = (v, ∂ x ϕ c 0 ) = 0.
The existence of such a constant is a classical result (see [START_REF] Benjamin | The stability of solitary waves[END_REF] or [START_REF] Bona | Stability and instability of solitary waves of Korteweg-de Vries type[END_REF]). Then it is easy to show (see [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF]) that there is a constant C(α 0 ) > 0 such that for any t < τ ε α , (2.9)

Q c 0 (u ε (t, . + x ε (t))) -Q c 0 (ϕ c ε (t) ) ≥ ν 4 εη ε (t) 2 1 -C|c ε (t) -c 0 | 2 .
Now, if τ = τ ε α ∧ t, then by (2.9), the translation invariance of Q c 0 , and Lemma 2.2

(2.10)

εη ε (τ ) 2 1 ≤ 4 ν Q c 0 (ϕ c 0 ) -Q c 0 (ϕ c ε (τ ) ) + ε τ 0 (∂ x u ε (s), ∂ x (u ε dW (s))) - ε 2 τ 0 ((u ε ) 3 (s), dW (s)) + ε 2 2 τ 0 (|k| 2 L 2 |∂ x u ε | 2 L 2 + |k ′ | 2 L 2 |u ε | 2 L 2 )ds - ε 2 2 k τ 0 R (u ε ) 3 (s)|φe k | 2 dxds -c 0 ε τ 0 ((u ε ) 2 , dW (s)) +c 0 ε 2 |k| 2 L 2 τ 0 m(u ε (s))ds + C|c ε (τ ) -c 0 | 2 .
The term |c ε (τ )c 0 | is then estimated thanks to the orthogonality condition (η ε , ϕ c 0 ) = 0 and the evolution of m(u ε (τ )) given in Lemma 2.2; one obtains, for some constants µ > 0 and C > 0, depending only on c 0 and α 0 (with α ≤ α 0 )

µ|c ε (τ ) -c 0 | ≤ |ϕ c 0 | 2 L 2 -|ϕ c ε (τ ) | 2 L 2 ≤ |εη ε (τ )| 2 L 2 + Cα|c ε (τ ) -c 0 | + 2ε τ 0 ((u ε ) 2 , dW (s)) +2ε 2 |k| 2 L 2 τ 0 |u ε (s)| 2 L 2 ds.
Hence, choosing α 0 sufficiently small one gets

(2.11) |c ε (τ ) -c 0 | 2 ≤ C |εη ε (τ )| 4 L 2 + 4ε 2 τ 0 ((u ε ) 2 , dW (s)) 2 +4ε 4 |k| 4 L 2 τ 0 |u ε (s)| 2 L 2 ds 2
which, once inserted into (2.10) leads to

εη ε (τ ) 2 1 ≤ C |εη ε (τ )| 4 L 2 + ε τ 0 (∂ x u ε , ∂ x (u ε dW (s))) +ε τ 0 ((u ε ) 3 , dW (s)) + c 0 ε τ 0 ((u ε ) 2 , dW (s)) +4ε 2 τ 0 ((u ε ) 2 , dW (s)) 2 + ε 2 k 2 1 τ 0 u ε (s) 2 1 ds +ε 2 |k| 2 L 2 τ 0 u ε (s) 3 1 ds + ε 4 |k| 4 L 2 τ 0 |u ε (s)| 2 L 2 ds 2 .
With this estimate in hand, together with (2.11), the conclusion of Theorem 2.1 follows with the same arguments as in the proof of Proposition 3.1 in [START_REF] De Bouard | Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise[END_REF]. These arguments rely on classical exponential tail estimates for stochastic integrals, after noticing that u ε (s) 1 ≤ C, a.s. for s ∈ [0, τ ε α ∧ T ] and α ≤ α 0 , so that the quadratic variation of each of the integrals involved in the above estimates are bounded above by CT .

A central limit theorem

This section is devoted to the proof of the next theorem: Theorem 3.1. Under the assumptions of Theorem 2.1, let α < α 0 be fixed. Then we can find cε (t) and xε (t) satisfying the conclusion of Theorem 2.1 such that if ηε is defined as in Theorem 2.1, for any T > 0, the process

(η ε (t)) t∈[0,T ] converges in L 2 (Ω; L ∞ (0, τ ε α ∧ T ; L 2 (R)
)) to a Gaussian process η satisfying the additive linear equation

(3.1) dη = ∂ x L c 0 η dt + Qϕ c 0 d W ,
with η(0) = 0, where W is the Wiener process with covariance φφ * given by W = T c 0 t W , and Q is a projection operator. Moreover, for a > 0 sufficiently small compared to c 0 , the process w(t, x) = e ax η(t, x) is a well defined H 1 valued process, of Ornstein-Uhlenbeck type, which converges in law to an H 1 -valued Gaussian random variable as t goes to infinity.

The conclusion of Theorem 3.1 will be obtained in three steps. The first step consists in estimating the modulation parameters obtained in Theorem 2.1, in terms of η ε , using the equations for those parameters; then the convergence of η ε as ε tends to zero is proved, and finally in the third step, a slight change in the modulation parameters is performed, in order that the limit process η may be written as an Ornstein-Uhlenbeck process.

From now on, we assume that α is fixed and sufficiently small, so that the conclusion of Theorem 2.1 holds, and we denote τ ε α by τ ε .

3.1. Modulation equations. Since we know that the modulation parameters x ε (t) and c ε (t) are semi-martingale processes adapted to the filtration generated by (W (t)) t≥0 , we may a priori write the stochastic evolution equations for those parameters in the form

(3.2) dx ε = c ε dt + εy ε dt + ε(z ε , dW ) dc ε = εa ε dt + ε(b ε , dW )
where y ε and a ε are real valued adapted processes with a.s. locally integrable paths on [0, τ ε ), and b ε , z ε are predictable processes with paths a.s. in L 2 loc (0, τ ε ; L 2 (R)). We then proceed as in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF] : the Itô-Wentzell Formula applied to u ε (t, x + x ε (t)), together with equation (1.1) for u ε and the first equation of (3.2) for x ε give a stochastic evolution equation for u ε (t, x + x ε ). On the other hand, the standard Itô Formula together with the second equation of (3.2) for c ε give an equation for the evolution of ϕ c ε (t) . Replacing then ϕ c ε (t) + εη ε (t, x) for u ε (t, x + x ε (t)) in the first equation leads to the following stochastic equation for the evolution of η ε (t) :

(3.3) dη ε = ∂ x L c 0 η ε dt + (y ε ∂ x ϕ c ε -a ε ∂ c ϕ c ε )dt -∂ x ((ϕ c ε -ϕ c 0 )η ε )dt +(c ε -c 0 + εy ε )∂ x η ε dt -ε 2 ∂ x ((η ε ) 2 )dt + ϕ c ε T x ε dW +∂ x ϕ c ε (z ε , dW ) -∂ c ϕ c ε (b ε , dW ) + εη ε T x ε dW + ε∂ x η ε (z ε , dW ) + ε 2 ∂ 2 x ϕ c ε |φ * z ε | 2 L 2 dt -ε 2 ∂ 2 c ϕ c ε |φ * b ε | 2 L 2 dt + ε l∈N ∂ x (ϕ c ε T x ε φe l )(z ε , φe l )dt + 1 2 ε 2 ∂ 2 x η ε |φ * z ε | 2 L 2 dt + ε 2 l∈N ∂ x (η ε T x ε φe l )(z ε , φe l )dt
where L c 0 is defined in (2.5). Now, taking the L 2 -inner product of equation (3.3) with ϕ c 0 , on the one hand, and with ∂ x ϕ c 0 on the other hand, then using the orthogonality conditions (2.3) and the fact that L c 0 ∂ x ϕ c 0 = 0, and finally identifying the drift parts and the martingale parts of each of the resulting equations lead to the same kind of system that we previously obtained in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF]; namely, setting

Y ε (t) = y ε (t) a ε (t) and Z ε l (t) = (z ε , φe l ) (b ε , φe l )
then one gets for the drift parts

(3.4) A ε (t)Y ε (t) = G ε (t)
where

(3.5) A ε (t) = (∂ x ϕ c ε + ε∂ x η ε , ∂ x ϕ c 0 ) -(∂ c ϕ c ε , ∂ x ϕ c 0 ) -(∂ x ϕ c ε , ϕ c 0 ) (∂ c ϕ c ε , ϕ c 0 )
and

G ε (t) = G ε 1 (t) G ε 2 (t) , with (3.6)
G ε 1 (t) = (η ε , L c 0 ∂ 2 x ϕ c 0 ) + (c ε -c 0 )(η ε , ∂ 2 x ϕ c 0 ) + ε 2 (∂ x (η ε ) 2 , ∂ x ϕ c 0 ) +(∂ x ((ϕ c ε -ϕ c 0 )η ε ), ∂ x ϕ c 0 ) -ε 2 (∂ 2 x ϕ c ε , ∂ x ϕ c 0 )|φ * z ε | 2 L 2 + ε 2 (∂ 2 c ϕ c ε , ∂ x ϕ c 0 )|φ * b ε | 2 L 2 -ε l∈N (z ε , φe l )(∂ x (ϕ c ε T x ε φe l ), ∂ x ϕ c 0 ) + 1 2 ε 2 (η ε , ∂ 3 x ϕ c 0 )|φ * z ε | 2 L 2 -ε 2 l∈N (∂ x (η ε T x ε φe l ), ∂ x ϕ c 0 )(z ε , φe l ) and (3.7)
G ε 2 (t) = -ε 2 (∂ x (η ε ) 2 , ϕ c 0 ) -(∂ x ((ϕ c ε -ϕ c 0 )η ε ), ϕ c 0 ) + ε 2 (∂ 2 x ϕ c ε , ϕ c 0 )|φ * z ε | 2 L 2 -ε 2 (∂ 2 c ϕ c ε , ϕ c 0 )|φ * b ε | 2 L 2 + ε (z ε , φe l )(∂ x (ϕ c ε T x ε φe l ), ϕ c 0 ) + ε 2 2 (η ε , ∂ 2 x ϕ c 0 )|φ * z ε | 2 L 2 + ε 2 l∈N (∂ x (η ε T x ε φe l ), ϕ c 0 )(z ε , φe l ); note that A ε (t) = A 0 + O(|c ε -c 0 | + εη ε 1 ), a.s. for t ≤ τ ε with A 0 = |∂ x ϕ c 0 | 2 L 2 0 0 (ϕ c 0 , ∂ c ϕ c 0 )
and O(|c εc 0 | + η ε 1 ) is uniform in ε, t and ω as long as t ≤ τ ε . Concerning the martingale parts, one gets the equation

(3.8) A ε (t)Z ε l (t) = F ε l (t), ∀l ∈ N with (3.9) F ε (t) = -((ϕ c ε + εη ε )T x ε φe l , ∂ x ϕ c 0 ) ((ϕ c ε + εη ε )T x ε φe l , ϕ c 0 ).
Proposition 3.2. Under the above assumptions, there is a constant

α 1 > 0, such that if α ≤ α 1 , then (3.10) |φ * z ε (t)| L 2 + |φ * b ε | L 2 ≤ C 1 |k| L 2 , a.s. for t ≤ τ ε and (3.11) |a ε (t)| + |y ε (t)| ≤ C 2 |η ε (t)| L 2 + εC 3 , a.s. for t ≤ τ ε
for some constants C 1 , C 2 , C 3 , depending only on α and c 0 , and for any ε ≤ ε 0 .

Proof The proof is exactly the same as the proof of Corollary 4.3 in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF], once noticed that, a.s.

for t ≤ τ ε , l∈N |F ε l (t)| 2 ≤ C l∈N |(ϕ c ε + εη ε )T x ε φe l | 2 L 2 ≤ C l R (ϕ c ε + εη ε ) 2 (x)[(T x ε k) * e l ] 2 (x)dx ≤ R (ϕ c ε + εη ε ) 2 (x) l (T x ε k(x -.), e l ) 2 dx ≤ C R (ϕ c ε + εη ε ) 2 (x)|T x ε k(x -.)| 2 L 2 dx ≤ C|k| 2 L 2 |ϕ c ε + εη ε | 2 L 2 ≤ C|k| 2 L 2
where we have used the Parseval equality in the fourth line.

3.2.

Convergence of η ε . Let us first assume that η ε has a limit as ε goes to zero, and take formally the limit as ε goes to zero in the preceding equations. Then, as was noticed above,

lim ε→0 A ε = A 0 = |∂ x ϕ c 0 | 2 L 2 0 0 (ϕ c 0 , ∂ c ϕ c 0 ) hence (3.12) lim ε→0 φ * z ε = - 1 |∂ x ϕ c 0 | 2 L 2 (T c 0 t φ) * (ϕ c 0 ∂ x ϕ c 0 ) := z (3.13) lim ε→0 φ * b ε = 1 (ϕ c 0 , ∂ c ϕ c 0 ) (T c 0 t φ * )(ϕ 2 c 0 ) := b (3.14) lim ε→0 y ε = 1 |∂ x ϕ c 0 | 2 L 2 (η, L c 0 ∂ 2 x ϕ
dη = ∂ x L c 0 ηdt + 1 |∂xϕc 0 | 2 L 2 (η, L c 0 ∂ 2 x ϕ c 0 )∂ x ϕ c 0 dt +ϕ c 0 T c 0 t dW - 1 2|∂xϕc 0 | 2 L 2 (∂ x (ϕ 2 c 0 ), T c 0 t dW )∂ x ϕ c 0 - 1 (ϕc 0 ,∂cϕc 0 ) (ϕ 2 c 0 , T c 0 t dW )∂ c ϕ c 0 .
It is easy to show that (3.16) has a unique adapted solution η with paths a.s. in C(R + , H 1 ) satisfying η(0) = 0. Moreover using the fact that (∂ c ϕ c 0 , ∂ x ϕ c 0 ) = 0, one easily gets from the above equation that (η,

ϕ c 0 ) = (η, ∂ x ϕ c 0 ) = 0, ∀t > 0.
Next, we make use of the following lemmas, whose proofs are obtained in the same way as the corresponding Lemmas in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF]. Lemma 3.3. Let η be the solution of (3.16) with η(0) = 0. Then, for any T > 0, there is a constant C depending only on c 0 , T and k 1 such that E η(t) 4 1 ≤ C, ∀t ≤ T.

Lemma 3.4. Let η ε be the solution of (3.3), defined for t ∈ [0, τ ε [, obtained thanks to the modulation procedure of Section 2. Then, for any T > 0,

E sup t≤τ ε ∧T |η ε (t)| 4 L 2 ≤ C(T, α, c 0 , k 1 ).
The above lemmas show that

(3.17) ∀T > 0, ∀q ≥ 2, lim ε→0 E sup t≤T ∧τ ε |c ε (t) -c 0 | q = 0.
Indeed, the expression of c ε (t)c 0 given by (3.2) together with (3.10) and (3.11) imply easily

E sup t≤T ∧τ ε |c ε (t) -c 0 | 2 ≤ Cε 2 [1 + E T ∧τ ε 0 |η ε (s)| 2 L 2 ds]
with C = C(α, c 0 , T, k 1 ). Then, (3.17) is deduced form Lemma 3.4 for q = 2, and follows for all other values of q from the uniform boundedness of

|c ε (t) -c 0 | on [0, T ∧ τ ε ].
Note that an immediate consequence of (3.17) is the fact that

(3.18) ∀T > 0, ∀q ≥ 2, lim ε→0 E sup t≤T ∧τ ε ϕ c ε (t) -ϕ c 0 2 2 = 0.
We will finally need the next lemma.

Lemma 3.5. For any T > 0, and any q ≥ 1,

lim ε→0 E sup t≤T ∧τ ε l∈N |Z ε l (t) -Z l (t)| 2 q = 0
where we have set for l ∈ N

Z l (t) = (z, φe l ) (b, φe l ) ,
z and b being given by (3.12) and (3.13), respectively.

Proof Here again, it is sufficient to consider the case q = 1. We recall that Z ε l satisfies equation (3.8). First, it is clear that

lim ε→0 E sup t≤T ∧τ ε (A ε (t)) -1 -(A 0 (t)) -1 2q = 0, ∀q ≥ 1.
On the other hand, in view of (3.9), denoting F 0 l (t) the formal limit of F ε l (t), one has

E sup t≤T ∧τ ε l |F ε l (t) -F 0 l (t)| 2 ≤ CE sup t≤T ∧τ ε l |∂ x ϕ c 0 (T x ε φ -T c 0 t φ)e l | 2 L 2 + CE sup t≤T ∧τ ε ϕ c ε (t) -ϕ c 0 2 1 and E sup t≤T ∧τ ε l |∂ x ϕ c 0 (T x ε φ -T c 0 t φ)e l | 2 L 2 ≤ ϕ c 0 2 1 E sup t≤T ∧τ ε |k(. + x ε (t) -c 0 t) -k| 2 L 2 .
Then, the Itô Formula applied to the function

K ε (t, x) = (k(x + x ε (t) -c 0 t) -k(x)) 2
using equation (3.2) for dx ε (t), together with (3.10), (3.11), and (3.17) lead to the conclusion of Lemma 3.5. Now, in order to prove that

(3.19) lim ε→0 E sup t≤T ∧τ ε |η ε (t) -η(t)| 2 L 2 = 0,
where η is the solution of (3.16) with η(0) = 0, it suffices to set v ε = η εη, to deduce from (3.16) and (3.3) the equation for dv ε and to apply the Itô Formula to get the evolution of |v ε | 2 L 2 . We do not give the details of those tedious, but easy computations. Finally, the use of the following estimates :

ε|(v ε , ∂ x ((η ε ) 2 ))| = ε|(∂ x η, (η ε ) 2 )| ≤ ε η 1 |η ε | 2 L 4 ≤ Cε η 1 |η ε | 3/2 L 2 |∂ x η ε | 1/2 L 2 ≤ C √ ε η 1 |η ε | 3/2 L 2
on the one hand, and

|y ε -y| + |a ε | ≤ C(|v ε | L 2 + |c ε -c 0 ||η ε | L 2 + ε|η ε | 2 L 2 + |η ε | L 2 ϕ c ε -ϕ c 0 1 + ε)
which is obtained as in the proof of Lemma 3.5 on the other hand, together with Lemma 3.3 to 3.5 allow to get the conclusion, that is the convergence of η ε to η in L 2 (Ω, L ∞ (0, τ ε ∧T ; L 2 (R))).

3.3. Complements on the limit equation. First of all, we note that the modulation equations may be written at order one in ε as

dx ε = c 0 dt + εydt + εW 1 dt + εdW 2 + o(ε) dc ε = εdW 1 + o(ε) where y = |∂ x ϕ c 0 | -2 L 2 (η, L c 0 ∂ 2 x ϕ c 0 ), W 1 (t) = (ϕ c 0 , ∂ c ϕ c 0 ) -1 (ϕ 2 c 0 , W (t)) and W 2 (t) = - 1 2 |∂ x ϕ c 0 | -2 L 2 (∂ x (ϕ c 0 2 ), W (t)).
Note that W 1 and W 2 are real valued Brownian motions, which are independent since

E(W 1 (t)W 2 (s)) = - 1 2 |∂ x ϕ c 0 | -2 L 2 (ϕ c 0 , ∂ c ϕ c 0 ) -1 (φ * (∂ x (ϕ 2 c 0 )), φ * (ϕ 2 c 0 ))(t ∧ s) = 0
because the operator φ * commutes with spatial derivation. Now, we want to investigate the asymptotic behavior in time of the process η. However, in the present form, the process η does not converge in law as t goes to infinity; this is due to the fact that the preceding modulation does not exactly correspond to the projection of the solution u ε on the (two-dimensional) center manifold, in which case the remaining term would belong to the stable manifold around the soliton trajectory. We now show that by slightly changing the modulation parameters, we can get a new decomposition of the solution u ε which is defined on the same time interval as before, but which fits with the preceding requirements. For that purpose, we first need to recall a few facts from [START_REF] Pego | Asymptotic stability of solitary waves[END_REF].

The generalized nullspace of the operator ∂ x L c 0 (that is the operator arising in the linearized evolution equation in the soliton reference frame) is spanned by the functions ∂ x ϕ c 0 and ∂ c ϕ c 0 , with the equality

∂ x L c 0 ∂ c ϕ c 0 = -∂ x ϕ c 0
and there are constants θ 1 and θ 2 (with

θ 1 = (ϕ c 0 , ∂ c ϕ c 0 )) such that if we set g1 (x) = -θ 1 x -∞ ∂ c ϕ c 0 (y)dy + θ 2 ϕ c 0 and g2 (x) = θ 1 ϕ c 0
then the generalized nullspace of -L c 0 ∂ x is spanned by g1 and g2 and

(g 1 , ∂ x ϕ c 0 ) = 1, (g 1 , ∂ c ϕ c 0 ) = 0, (g 2 , ∂ x ϕ c 0 ) = 0, (g 2 , ∂ c ϕ c 0 ) = 1.
We also set, for a > 0,

f a 1 (x) = e ax ∂ x ϕ c 0 , f a 2 (x) = e ax ∂ c ϕ c 0 , g a 1 (x) = e -ax g1 (x), g a 2 (
x) = e -ax g2 (x), so that (f a i , g a j ) = δ ij . Then the operator A a defined for a > 0 by A a = e ax ∂ x L c 0 e -ax has a well defined generalized nullspace spanned by f a 1 , f a 2 and the spectral projection on this nullspace is given by P w = 2 k=1 (w, g a k )f a k where w = e ax v, and v is an L 2 function. Moreover, if Q = I -P , then Q is the spectral projection on the stable manifold of A a , and under the condition 0 < a < c 0 /3, there are constants C > 0 and b > 0 such that (3.20) e Aat Qw 1 ≤ Ce -bt w 1 , ∀t > 0, ∀w ∈ H 1 , where e Aat is the C 0 -semi-group generated by A a (see Theorem 4.2 in [START_REF] Pego | Asymptotic stability of solitary waves[END_REF]). Now, let η be the solution of (3.16) with η(0) = 0, and consider w(t, x) = e ax η(t, x). Note that the orthogonality condition (η, ϕ c 0 ) = 0 implies (w, g a 2 ) = 0, so that P w = λ(t)f a 1 with λ(t) = (w(t), g a 1 ) a real valued stochastic process whose evolution is given by (3.21)

λ(t) = t 0 |∂ x ϕ c 0 | -2 L 2 (η(s), L c 0 ∂ 2 x ϕ c 0 )ds - t 0 |∂ x ϕ c 0 | -2 L 2 (ϕ c 0 ∂ x ϕ c 0 , d W (s)) + t 0 (e ax ϕ c 0 d W (s), g a 1 )
where we have used (3.16) and the fact that A a P w = 0 and λ(0

) = 0. Hence, λ(t) is bounded in L 4 (Ω; L ∞ (0, T ∧ τ ε )) by Lemma 3.3. Let us set xε (t) = x ε (t) -ελ(t) for t ∈ [0, τ ε [. Then (3.22) u ε (t, x + xε (t)) = ϕ c ε (t) (x) + εη ε (t, x) with ηε (t, x) = 1 ε (ϕ c ε (t) (x -ελ(t)) -ϕ c ε (t) (x)) + η ε (t, x -ελ(t)).
Note that, a.s. for t ≤ τ ε :

|ϕ c ε (t) (. -ελ(t)) -ϕ c ε (t) -ελ(t)∂ x ϕ c ε (t) | L 2 ≤ ε 2 λ 2 (t)C(c 0 , α).
Hence, it follows from Lemma 3. Moreover, this covariance operator converges as t goes to infinity and it follows that w 2 converges in law in H 1 to a Gaussian random variable. The end of the statement of Theorem 3.1 follows, setting Qv = e -ax Qe ax v.

A remark on the soliton diffusion

Let us go back to the stochastic evolution equations for the new modulation parameters, that we may write as If we keep only the order one terms in ε i.e. we consider the solution (X ε (t), C ε (t)) of the system of SDEs dX ε = c 0 dt + εB 1 dt + εdB 2 dC ε = εdB 1 , then (X ε (t)c 0 t, C ε (t)c 0 ) is a centered Gaussian vector, and it is easy to compute its covariance matrix. Let us denote by µ ε t the law of (X ε (t)c 0 t, C ε (t)c 0 ); we may compute

E ϕ C ε (t) (x -X ε (t)) ≤ K 0 ε -1/2 t -5/4
for t large enough. This inequality has to be compared to the result of [START_REF] Wadati | Stochastic Korteweg-de Vries equations[END_REF] where an additive equation with a white noise in time was considered. An inequality of the form (4.3) was obtained, but with a power t -3/2 instead of t -5/4 .

t 0 e 2 1 dσ ≤ C t 0 e -bσ dσ l e ax ϕ c 0 φe l 2 1

 0202 Aa(t-σ) Q[e ax ϕ c 0 d W (σ)]; the trace of the covariance operator of the Gaussian process w 2 in H 1 may be easily computed and estimated thanks to (3.20) as t 0 l e Aaσ Qe ax ϕ c 0 φe l dσ ≤ C k 2 1 e ax ϕ c 0 2 1 .

(4. 1 )

 1 dx ε = c 0 dt + εB 1 dt + εdB 2 + o(ε) dc ε = εdB 1 + o(ε) with B 1 = W 1 and B 2 = -(e ax ϕ c 0 W (t), g a 1 ) = -( W (t), ϕ c 0 g1). Note that B 1 and B 2 are now correlated Brownian motions. We denote by σ = (σ ij ) i,j = cov(B 1 , B 2 ).

E

  ϕ C ε (t) (x -X ε (t)) = max x∈R ϕ c+c 0 (xc 0 ty)µ ε t (dy, dc)

  3, 3.4 and the above bound on λ that

	(3.23)	lim ε→0	E sup t≤T ∧τ ε	|η ε (t) -η(t)| 2 L 2 = 0
	with η(t) = η(t) -λ(t)∂ x ϕ c 0 . So now, with this new decomposition, we clearly have, setting w(t, x) = e ax η(t, x) :
			P w = 0, Q w = Qw.
	Also, if w 2 = Qw, then the equation (3.16) implies
	(3.24)	dw 2 = A a w 2 dt + Qe ax ϕ c 0 d	W
	hence
		w 2 (t) =

  1/2 ϕ c+c 0 (xc 0 ty) exp -the covariance matrix of (X ε (t)c 0 t, C ε (t)c 0 ), given by Σ = ε 2 σ 11 t σ 12 t + σ 11Inserting this inequality in (4.2), using the fact that ϕ c (x) = cϕ 1 ( √ cx) and integrating in y

	where Σ is t 2
					σ 12 t + σ 11	t 2 2	2 σ 22 t + σ 12 t 2 + σ 11	3 t 3	.
	It is not difficult to see that
	exp -	1 2	Σ -1 c y	.	c y	≤ exp -	1 2	ε 2 det Σ	σ 11	t 3 12	+ (σ 22 -	σ 2 12 σ 11	t) c 2 .
	give the bound
	E ϕ C ε (t) (x -X ε (t)) ≤	K (det Σ) 1/2	0	+∞	√	c + c 0 e -1 2	ε 2 det Σ [σ 11	t 3 12 +(σ 22 -	σ 2 12 σ 11	t)]c 2	dc
	where K is a constant, and since
					0	+∞	√ ce -c 2 2α 2 dc ≤ Kα 3/2
	for another constant K, it follows
	(4.3)		max x∈R
														1 2	Σ -1 c y	.	c y	dcdy


Panayiota Pissaridou

Marco Cantonati

Agnes Bouchez

Iakovos Tziortzis

Gerald Dörflinger

Marlen I Vasquez
email: marlen.vasquez@cut.ac.cy

Daniel Hering

How can integrated morphotaxonomy-and metabarcoding-based diatom assemblage analyses best contribute to the ecological assessment of streams?

Keywords: DNA metabarcoding, morpho-taxonomy, river biomonitoring, Cyprus rivers

Environmental conditions, such as nutrient concentrations, salinity, elevation etc., shape diatom assemblages of periphytic biofilms. These assemblages respond rapidly to environmental changes, a fact which makes diatoms valuable bioindicators. Hence, freshwater biomonitoring programmes currently use diatom indices (e.g. EU Water Framework Directive -WFD). To date, microscopy-based assessments require high taxonomic expertise for diatom identification at the species level. High-throughput technologies now provide cost-effective identification approaches that are promising, complementary or alternative tools for bioassessment. The suitability of the metabarcoding method is evaluated for the first time in the Cyprus streams WFD monitoring network, an eastern Mediterranean country with many endemic species and results are compared to the results acquired from the morphotaxonomic analysis. Morphotaxonomic identification was conducted microscopically, using the most updated taxonomic concepts, literature and online resources. At the same time, DNA metabarcoding involved the use of the rbcL 312 bp barcode, high-throughput sequencing and bioinformatic analysis. The ecological status was calculated using the IPS Index. Results show a positive correlation between morpho-taxonomic and molecular IPS scores. Discrepancies between the two methodologies are related to the limitations of both techniques. This study confirmed that Fistulifera saprophila can have a crucial role in key differences observed, as it negatively influences IPS scores and microscopy methods frequently overlook it. Importantly, gaps in the DNA barcoding reference databases lead to a positive overestimation in IPS scores. Overall, we conclude that DNA metabarcoding offsets the morphotaxonomic methodology for the ecological quality assessment of freshwaters.

Introduction

Freshwater ecosystems are transient environments, featuring hotspots of biodiversity and supporting about 10% of the total known species on the earth [START_REF] Strayer | Freshwater biodiversity conservation: Recent progress and future challenges[END_REF]. Due to the extensive urbanisation and advancement in human activities in recent decades, these ecosystems are subjected to environmental degradation and impacted by anthropogenic pressures [START_REF] Hupało | Fresh insights into Mediterranean biodiversity: Environmental DNA reveals spatio-temporal patterns of stream invertebrate communities[END_REF]. As a result, these freshwater biodiversity hotspots are declining. The Mediterranean region fosters a unique combination of geography, geological history and climate, making it an excellent example of a freshwater biodiversity hotspot. This is estimated to result from a distinctly cool and wet season, followed by a warm and dry season influenced by a sequence of regular and often extreme flooding and drying periods and relatively rare snowfall. Additionally, it is characterised by intense human activities. Mediterranean islands are no exception, being the most threatened ecosystems within the region, having intermittent and harsh-intermittent rivers highly impacted by human activities [START_REF] Hupało | Fresh insights into Mediterranean biodiversity: Environmental DNA reveals spatio-temporal patterns of stream invertebrate communities[END_REF].

Although the Mediterranean is one of the most studied and well-known regions globally, little is known about the organisms inhabiting freshwater ecosystems. Studies focus mainly on vertebrates, macroinvertebrate orders and macrophytes of the northern and north-western domain [START_REF] Figueroa Jmt De | Freshwater biodiversity in the rivers of the Mediterranean Basin[END_REF].

Diatoms are unicellular photosynthetic autotrophs (microalgae) inhabiting a huge variety of habitats, such as oceans, rivers, wetlands and even the soil environment. They are the chief primary producers and key contributors in food webs, biogeochemical cycles and carbon fixation [START_REF] Pérez-Burillo | Evaluation and sensitivity analysis of diatom DNA metabarcoding for WFD bioassessment of Mediterranean rivers[END_REF]. Interestingly, an estimated 100,000 species of diatoms are believed to exist, with newly-discovered diatom species present in unique areas worldwide [START_REF] Mann | An inordinate fondness? The number, distributions, and origins of diatom species[END_REF]. Diatom assemblages have selective growth preferences. They bloom at specific environmental conditions (salinity, temperature etc.) with short-generation periods (Smol and Stoermer 2010). Therefore, they respond rapidly to environmental changes that can influence their habitat, such as nutrient concentration, organic matter and water contamination [START_REF] Hering | Assessment of European streams with diatoms, macrophytes, macroinvertebrates and fish: a comparative metric-based analysis of organism response to stress[END_REF].

For several decades, scientists have assessed the effect of pollution on diatom assemblages. Sixty years ago, scientists started assessing the influence of pollution on diatom assemblages and provided occurrence probabilities of microalgal species in water quality classes. During the late 20 th century, scientists managed to incorporate the long-term monitoring of the diatom assemblages as a means of bioassessment [START_REF] Butcher | Studies in the ecology of rivers. IV. The algae of organically enriched water[END_REF][START_REF] Zelinka | Zur Präzisierung der biologischen Klassifikation der Reinheit fliessender Gewässer[END_REF][START_REF] Mccormick | Algae as indicators of environmental change[END_REF]. In 2000, the Water Framework Directive (WFD) incorporated the concept of reference conditions and the adoption of phytobenthos as one of the obligatory Biological Quality Elements (BQEs) (European Comission 2000) considering that changes in diatom assemblages can be a valuable monitoring and assessment tool. Therefore, diatoms are used by the Member States to meet their legislative obligations towards the WFD (European Comission 2000).

To date, the quality of water-bodies worldwide is assessed by applying different diatom indices [START_REF] Dalu | Assessment of water quality based on diatom indices in a small temperate river system, Kowie River, South Africa[END_REF]. Such indices include the Biological Diatom Index (BDI), the Eutrophication Pollution Index (EPI), the Trophic Diatom Index (TDI), the Specific Polluosensitivity Index (IPS), the Sladecek Index (SLA) and the Saprobic Index of Rott (ROT), each of which has different sensitivities and investigate various environmental drives towards pollution [START_REF] Rimet | Response of diatom indices to simulated water quality improvements in a river[END_REF][START_REF] Dalu | Assessment of water quality based on diatom indices in a small temperate river system, Kowie River, South Africa[END_REF][START_REF] Pawlowski | The future of biotic indices in the ecogenomic era: Integrating (e)DNA metabarcoding in biological assessment of aquatic ecosystems[END_REF].

In Cyprus, the IPS Index [START_REF] Cemagref | Etude des méthodes biologiques d'appréciation quantitative de la qualité des eaux[END_REF]) is used since it was more suitable for the pressures that could affect the water quality in Cyprus' streams (WDD 2008) and the official assessment method's class boundary values, based on the morphotaxonomy. The IPS Index correlates parameters related to organic pollution, ionic strength and eutrophication and, therefore, provides an overall estimate of water quality and is used by several European countries, such as Spain, Belgium, Estonia, Luxembourg, Sweden, Bulgaria, Greece and Portugal [START_REF] Wdd | Report on the classification of water status (rivers, natural lakes, water reservoirs)[END_REF]European Comission 2018;[START_REF] Masouras | Benthic diatoms in river biomonitoring-present and future perspectives within the Water Framework Directive[END_REF]. It is worth mentioning that most of these indices and assessment concepts were developed elsewhere in Europe. Therefore their transfer to Cyprus and the eastern Mediterranean countries should be done with caution due to the combination of harsh climate and geographical isolation (Cantonati et al. 2020). In 2014, several Mediterranean countries performed an intercallibration exercise to provide a common framework for the successful comparison of diatom assessments of river quality across the Mediterranean European region [START_REF] Almeida | Water quality assessment of rivers using diatom metrics across Mediterranean Europe: A methods intercalibration exercise[END_REF].

Diatom taxonomic classification involves morphological identification under the microscope by categorising the siliceous frustules of diatoms [START_REF] Pandey | The use of diatoms in ecotoxicology and bioassessment: Insights, advances and challenges[END_REF]. This is a time-consuming procedure that requires an accurate morphological identification at species or sub-species level. Therefore, it requires high taxonomic expertise [START_REF] Kahlert | Harmonization is more important than experience-results of the first Nordic-Baltic diatom intercalibration exercise 2007 (stream monitoring)[END_REF]. At the same time, such practices are subjected to biases and are susceptible to human error. New strategies have been examined to develop high-throughput, cost-effective approaches [START_REF] Hering | Implementation options for DNA-based identification into ecological status assessment under the European Water Framework Directive[END_REF][START_REF] Kelly | Environment Agency A DNA based diatom metabarcoding approach for Water Framework Directive classification of rivers[END_REF][START_REF] Vasselon | Avoiding quantification bias in metabarcoding: Application of a cell biovolume correction factor in diatom molecular biomonitoring. Mahon A[END_REF]. Recently, DNA metabarcoding, combined with high throughput sequencing and bioinformatics, is an alternative approach to morpho-taxonomic identification [START_REF] Hering | Implementation options for DNA-based identification into ecological status assessment under the European Water Framework Directive[END_REF][START_REF] Pawlowski | The future of biotic indices in the ecogenomic era: Integrating (e)DNA metabarcoding in biological assessment of aquatic ecosystems[END_REF][START_REF] Rivera | Metabarcoding of lake benthic diatoms: from structure assemblages to ecological assessment[END_REF]Vasselon et al. 2018;[START_REF] Tapolczai | The impact of OTU sequence similarity threshold on diatom-based bioassessment: A case study of the rivers of Mayotte (France, Indian Ocean)[END_REF]. DNA metabarcoding involves identifying the microbial diversity from environmental samples by analysing relatively conserved small gene sequences (i.e. barcodes).

The start of Cyprus streams' bioassessment using diatoms dates back to 2005 (Cantonati et al. 2020) and the Water Development Department (WDD) of Cyprus has incorporated it into routine assessments since 2010. The routine assessment involves the use of the morpho-taxonomic identification method for the calculation of IPS. To date, DNA metabarcoding has been assessed mainly by the west-northern Mediterranean and northern European countries (Vasselon et al. 2017b;[START_REF] Kelly | Environment Agency A DNA based diatom metabarcoding approach for Water Framework Directive classification of rivers[END_REF][START_REF] Bailet | Molecular versus morphological data for benthic diatoms biomonitoring in Northern Europe freshwater and consequences for ecological status[END_REF][START_REF] Feio | A taxonomy-free approach based on machine learning to assess the quality of rivers with diatoms[END_REF][START_REF] Pérez-Burillo | Evaluation and sensitivity analysis of diatom DNA metabarcoding for WFD bioassessment of Mediterranean rivers[END_REF]. Additionally, extensive studies have been performed throughout Mediterranean marine environments regarding diatom biodiversity using DNA metabarcoding [START_REF] Malviya | Insights into global diatom distribution and diversity in the world's ocean[END_REF][START_REF] Kaleli | Taxonomy and distribution of diatoms on the Turkish Mediterranean Coast, Dalyan (Mugla)[END_REF][START_REF] Vijver B Van De | Geographical variation in the diatom communities associated with loggerhead sea turtles (Caretta caretta)[END_REF]. Therefore, this study compares, for the first time, the diatom diversity and IPS scores obtained using high-throughput DNA sequencing and the morpho-taxonomic identification approach for an eastern Mediterranean country, like Cyprus, with a known population of endemic diatom species [START_REF] Cantonati | A new Navicula (Bacillariophyta) species from low-elevation carbonate springs affected by anthropogenic disturbance[END_REF][START_REF] Cantonati | Taxonomic and ecological characterization of two ulnaria species (Bacillariophyta) from streams in Cyprus[END_REF]. Here, the study uses the results of the metabarcoding analysis that utilised a small 312 bp fragment of the rbcL, which encodes the Ribulose-1,5-bisphosphate carboxylase/oxygenase. This gene shows alternating highly conserved and polymorphic regions, which are key requirements for a successful genetic identification to the species level (Kermarrec et al. 2013). We compared both methods to: a) assess whether the two methodologies are complementary or they can replace one another and b) test which methodologies are more suitable within Cyprus' routine bioassessment protocol using the IPS Index.

Materials and methods

Study area and sampling methodology.

Sampling stations were selected following the annual schedule of the WDD during May and June 2018. Biofilm samples were collected from 32 river sites in 11 catchments. Permanently submerged cobbles making up an area of about 100 cm 2 were randomly selected from the main stream channel in moderate flow conditions and a minimum of 5 cm water depth. The cobbles were gently rubbed with a brush at the parts exposed to the river flow. The dislodged material was preserved in formaldehyde (3%) for morphological or ethanol (70%) for the molecular analysis (CEN 2018). Samples were stored in a cool, dark area until laboratory analysis, as previously described by Pissaridou et al. (2021).

DNA extraction and molecular analysis

DNA extraction was performed using a commercial kit (NucleoSpin Soil Kit, MACHEREY-NAGEL) according to the manufacturer's instructions. Two technical samples were evaluated. Briefly, 2 ml of suspended diatom biofilm were centrifuged for 30 min at 13,000 rpm. The pellet was then lysed and the DNA was purified following the Kit's protocol. The DNA quality was assessed spectrophotometrically (NanoDrop 1000, Thermo Fisher Scientific). The rbcL barcode was amplified by PCR using diatom-specific rbcL primers and the HiFi KAPA Taq Mix (KAPA Biosciences) to a final volume of 25 ml. Samples were then sent to GenoToul Genomics and Transcriptomics platform (GeT-PlaGe, Auzeville, France) for Illumina MiSeq sequencing.

Bioinformatic analysis

Raw Fastq sequences were analysed using Mothur software (version 1.41.3) [START_REF] Schloss | Introducing Mothur: Open-source, platform-independent, community-supported software for describing and comparing microbial communities[END_REF]. Sequences were filtered for quality and length (criteria: minimum length 280 bp, one mismatch allowed and homopolymers less than 8 bp), dereplicated and aligned to the Diat. barcode database version 7.1 (Rimet et al. 2019[START_REF] Rimet | Diat.barcode, an open-access barcode library for diatoms[END_REF]. Then, chimera detection and removal were performed. Taxonomic affiliation of reads was achieved using the Wang Method at a 75% confidence threshold. Lastly, a distance matrix was created to allow the clustering of reads into operational taxonomical units (OTUs). The clustering method was the Furthest-Neighbour, where sequences with more than or equal to 95% similarity were clustered in one OTU. OTUs singletons were removed. Then all samples were normalised to the lowest read abundance. The method is described in detail in Pissaridou et al. (2021).

Morpho-taxonomic analysis

The diatom biofilm samples underwent processing using the hydrogen peroxide and potassium dichromate method, following the European standard EN 13946 [START_REF] Din En | Directive of the European Parliament and of the Council 2000/60/EC. Establishing a Framework for Community Action in the Field of Water Policy[END_REF], before permanent slide mounting. Briefly, 5 to 10 ml of diatom samples were centrifuged. Samples were then cleaned by repeated rinsing with 30% hydrogen peroxide (H 2 O 2 ) and hydrochloric acid (HCl) following final decantation with distilled water. Next, samples were air-dried and mounted on microscope glass slides using Naphrax (Brunel Microscopes Ltd). At least 400 valves were identified for each sample and counted under an optical microscope Zeiss Axioskop 2, equipped with phase contrast and a digital camera Axiocam (Carl Zeiss JSC, Milan, Italy). The species identification was performed to the lowest taxonomic level possible (species or intraspecific rank) using various keys, with [START_REF] Cantonati | qKmxSTKw7QCaZjNLBHZ-CLIyBbZXUlETzpBTwbmknJ6PgKGNXb-MoxHpZYEsLaGgp5ETB4nT4wIWxBajhYc7MwGQMuq7BMtwQtG3D0gRY-nYKawJCDdeB80HEiQEOQ6g03QQYW0F4CIQam1DxhBjanfG-BLrFKEQccN2MfNKAe28ooUkoCFfkZ[END_REF] as the primary reference and recent scientific publications for specific taxa. The results were harmonised according to the "Final harmonisation of diatom names" list [START_REF] Feio | WFD Intercalibration Phase 2: Milestone 6 report -River[END_REF]. A complete species list and number of valves per species were created for each sample, along with each species' relative abundance.

Data analysis

Correction factor: The Correction Factor (CF) was applied to account for the rbcL copy number in relation to the biovolume of cells. The application was performed as described by [START_REF] Vasselon | Avoiding quantification bias in metabarcoding: Application of a cell biovolume correction factor in diatom molecular biomonitoring. Mahon A[END_REF] according to the information supplied in the Diat.barcode database version 7.1 [START_REF] Rimet | Diat.barcode, an open-access barcode library for diatoms[END_REF](Rimet et al. , 2019)).

IPS score calculation:

1 1 1 1 n n i i i j j j i i n n i i j j i i IPSV IPSS IPSV IPSS IPS IPSV IPSV α α α α = = = =     × × + × × +         =     × + × +         ∑ ∑ ∑ ∑  

IPSV = pollution tolerance values IPSS = pollution sensitivity values

The IPSV and the IPSS values for each species were acquired from the Diat.barcode database version 7.1 [START_REF] Rimet | Diat.barcode, an open-access barcode library for diatoms[END_REF](Rimet et al. , 2019)). IPS scores were transformed to water quality estimates where IPS ≥ 17 -"High"; IPS (13-17) -"Good"; IPS (9-13) -"Moderate"; IPS (5-9) -"Poor"; and IPS (1-5) -"Bad" class, respectively (CE-MAGREF 1982).

Statistical analysis:

Pearson's Correlation was performed using GraphPad PRISM version 9 for Windows, GraphPad Software, (La Jolla, California, USA). Pie charts and Venn Diagrams were performed in Excel (Microsoft). Plots were created using R (R Core Team 2020).

Results

DNA metabarcoding diatom identification

The DNA metabarcoding analysis identified a total of four classes, 13 orders, 28 families, 53 genera, 106 species and 136 strains amongst 1,553,538 reads for the 32 stations studied. [START_REF] Vasselon | Avoiding quantification bias in metabarcoding: Application of a cell biovolume correction factor in diatom molecular biomonitoring. Mahon A[END_REF] observed that the molecular approach results overestimate the presence of high biovolume species, which may have higher rbcL copies per cell. The metabarcoding approach performed in this study was also able to observe such changes in abundance, as expressed as OTU copy numbers (Fig. 1). Consequently, a Correction Factor (CF) was applied to take into account the biovolume of diatom frustules to make DNA metabarcoding counts fit better with those estimated by microscopic counts [START_REF] Charles | Benthic algae assessments in the EU and the US: Striving for consistency in the face of great ecological diversity[END_REF]. This removed species with high biovolume from the top ten most abundant species, like Ulnaria ulna, Cocconeis euglypta and Cymbella cymbiformis and, instead, intro-duced species like Fistulifera saprophila, Planothidium frequentissimum, Planothidium lanceolatum and Mayamaea permitis with low biovolume (Fig. 1). This correction allows for a better alignment of the diatom assemblages from the molecular approach and subsequently a higher correlation between the two methodologies.

The ten most abundant species of the morpho-taxonomic analysis are represented in Figure 2. Cocconeis euglypta, Achnanthidium jackii and A. minutissimum are the most abundant diatoms. This was also the case in the molecular analysis (Fig. 1), as A. minutissimum was detected second in line. A. jackii was not detected in the molecular data as, in the reference database, it is considered a variant of A. minutissimum and is, therefore, represented by the same barcode (Rimet et al. 2019).

Comparison of the morpho-taxonomic and molecular identification of diatoms

To evaluate the performance of the two methods, we compared the species identified in both cases. Figure 3A shows that both methodologies identified 39 common species, corresponding to 18% (Suppl. material 1: Table S1). These 39 species represent a large part of the community in terms of biomass since the 39 species include several of the most frequent and abundant taxa in the Cyprus diatom assemblages. Interestingly, 99 species were identified only by the morpho-taxonomic identification and 67 species only by the molecular identification method (Fig. 3A).

The number of species observed with each technique was used to calculate the IPS index for the ecological status evaluation of each station [START_REF] Cemagref | Etude des méthodes biologiques d'appréciation quantitative de la qualité des eaux[END_REF]. The IPS scores, obtained with the two methodologies, show significant positive correlation (p < 0.0001, r = 0.6745, Fig. 3B). More than half of the samples shared the same quality class.

At the same time, some discrepancies between the two methodologies were observed. Twelve stations show substantial differences between the two approaches (Fig. 3B, Fig. 4, Suppl. material 2). The molecular approach calculated lower ecological status values than the morpho-taxonomic approach for six stations (Fig. 3B). Interestingly, four out of six stations have Fistulifera saprophila as the most abundant species. As detected by the molecular analysis, one station has Amphora pediculus as the most abundant with Fistulifera saprophila following and one station has Achnanthidium minutissimum and Ulnaria ulna as the most abundant species.

On the other hand, the molecular approach calculated higher IPS values than the morpho-taxonomic approach for six more stations (Suppl. material 2). These differences could be due to the lack of barcoding data for diatom species at these stations, creating gaps in comparing the two methodologies. The molecular approach identifies a higher abundance of Planothidium victorii, Hantzschia unclassified and Achnanthes unclassified. For the same stations, higher abundance of Achnanthidium jackii, Achnanthidium minutissimum, Cocconeis euglypta, Cyclostephanos cf. invisitatus, Amphora pediculus and Amphora micra have been identified S1 gives the number of valves and number of reads recorded for the 39 common species; B) IPS correlation of the morpho-taxonomic and molecular identification methodologies. Pearson Correlation was performed indicating a significant correlation (r = 0.6745, p < 0.0001. The boundaries of the ecological status classes are marked: red (Bad), orange (Poor), yellow (Moderate), green (Good), blue (High). Dashed lines: Good/moderate boundary, Red dashed circle: morphological classifies as good status, while molecular classifies below good status. Blue cirlce: morphological classifies as below good status, while molecular classifies good status.

using the morpho-taxonomic approach (data not shown) (Pérez-Burillo et al. 2020). As previously noted, Cyclostephanos cf. invisitatus and Amphora micra are diatoms species whose barcode is not yet available in online barcode databases.

Discussion

Thus far, several studies have shown disagreements between molecular and morphological datasets due to the inherent biases of both methods [START_REF] Keck | Meta-analysis shows both congruence and complementarity of DNA metabarcoding to traditional methods for biological community assessment[END_REF]). The most apparent reason for these discrepancies is the incompleteness of molecular reference databases since barcoding studies focused so far on particular river types and geographical regions, including mostly north-west Mediterranean countries and perennial rivers. [START_REF] Pawlowski | The future of biotic indices in the ecogenomic era: Integrating (e)DNA metabarcoding in biological assessment of aquatic ecosystems[END_REF].

Here, as presented by recent studies, the identification of more species by the morphological approach than by DNA metabarcoding, in many cases, does not significantly impact the IPS assignment (Vasselon et al. 2017a;[START_REF] Keck | Boosting DNA metabarcoding for biomonitoring with phylogenetic estimation of operational taxonomic units' ecological profiles[END_REF][START_REF] Rivera | Metabarcoding of lake benthic diatoms: from structure assemblages to ecological assessment[END_REF][START_REF] Mortágua | Applicability of DNA metabarcoding approach in the bioassessment of Portuguese rivers using diatoms[END_REF]. Additionally, we have demonstrated how the CF's application to the molecular approach is important in comparing the two methodologies. The CF accounts for the rbcL copy number concerning the biovolume of cells and, therefore, is considered to obtain DNA sequence proportions closer to cell proportions obtained through microscopy and account for large diatom frustules having several rbcL copies [START_REF] Vasselon | Assessing pollution of aquatic environments with diatoms' DNA metabarcoding: Experience and developments from France Water Framework Directive networks[END_REF]). Both methodologies have their limitations, including the classification biases in species complexes and oversimplification requiring high scientific expertise for morphology-based approaches and the gaps of DNA barcoding data in existing references databases in molecular-based approaches [START_REF] Jahn | Planothidium lanceolatum and Planothidium frequentissimum reinvestigated with molecular methods and morphology: four new species and the taxonomic importance of the sinus and cavum[END_REF][START_REF] Pinseel | Molecular and morphological characterization of the Achnanthidium minutissimum complex (Bacillariophyta) in Petuniabukta (Spitsbergen, High Arctic) including the description of A. digitatum sp. nov[END_REF]. [START_REF] Kelly | Development of a novel metric for evaluating diatom assemblages in rivers using DNA metabarcoding[END_REF] suggested that, as microscopy data also have some biases and limitations, it would be better to treat molecular data at their face value to integrate them into indicator metrics. Indeed, the differences in size reflected by the rbcL data [START_REF] Vasselon | Avoiding quantification bias in metabarcoding: Application of a cell biovolume correction factor in diatom molecular biomonitoring. Mahon A[END_REF]) may give a more accurate indication of the relative contribution of each taxon to diatom productivity than the microscopy data, which could lead to novel bioindicator metrics, better reflecting the ecosystem functioning.

In this paper, some discrepancies in IPS scores appear when comparing both approaches. The microscopy can sometimes underestimate small diatom species with low sensitivity values, thereby having a significant negative impact on the IPS scores estimated with DNA metabarcoding [START_REF] Jahn | Planothidium lanceolatum and Planothidium frequentissimum reinvestigated with molecular methods and morphology: four new species and the taxonomic importance of the sinus and cavum[END_REF][START_REF] Pinseel | Molecular and morphological characterization of the Achnanthidium minutissimum complex (Bacillariophyta) in Petuniabukta (Spitsbergen, High Arctic) including the description of A. digitatum sp. nov[END_REF]). Interestingly, literature suggests that small species like Fistulifera saprophila and Mayamaea permitis are often overlooked/ underestimated by the morphological approach [START_REF] Kahlert | Harmonization is more important than experience-results of the first Nordic-Baltic diatom intercalibration exercise 2007 (stream monitoring)[END_REF][START_REF] Pérez-Burillo | Evaluation and sensitivity analysis of diatom DNA metabarcoding for WFD bioassessment of Mediterranean rivers[END_REF]. In this study, only F. saprophila was detected exclusively by the molecular approach, whereas M. permitis was detected by both methods (Suppl. material 1). Both species become prominent in eutrophic and polluted environments and consequently lower the IPS Index scores [START_REF] Rott | Monitoring organic pollution and eutrophication in the Grand River, Ontario, by means of diatoms[END_REF][START_REF] Rimet | Response of diatom indices to simulated water quality improvements in a river[END_REF][START_REF] Kahlert | Harmonization is more important than experience-results of the first Nordic-Baltic diatom intercalibration exercise 2007 (stream monitoring)[END_REF][START_REF] Pérez-Burillo | Evaluation and sensitivity analysis of diatom DNA metabarcoding for WFD bioassessment of Mediterranean rivers[END_REF]. This explains why six of the twelve differing stations have lower IPS scores when estimated through the molecular approach (Suppl. material 2). F. saprophila valves are known to be prone to dissolution during laboratory preparation and this could explain why this species is usually overlooked in the morphological identification procedures [START_REF] Zgrundo | Morphological and molecular phylogenetic studies on Fistulifera saprophila[END_REF][START_REF] Cantonati | qKmxSTKw7QCaZjNLBHZ-CLIyBbZXUlETzpBTwbmknJ6PgKGNXb-MoxHpZYEsLaGgp5ETB4nT4wIWxBajhYc7MwGQMuq7BMtwQtG3D0gRY-nYKawJCDdeB80HEiQEOQ6g03QQYW0F4CIQam1DxhBjanfG-BLrFKEQccN2MfNKAe28ooUkoCFfkZ[END_REF][START_REF] Cantonati | Taxonomic and ecological characterization of two ulnaria species (Bacillariophyta) from streams in Cyprus[END_REF][START_REF] Kelly | Development of a novel metric for evaluating diatom assemblages in rivers using DNA metabarcoding[END_REF]. Our results agree with the observations of [START_REF] Pérez-Burillo | Evaluation and sensitivity analysis of diatom DNA metabarcoding for WFD bioassessment of Mediterranean rivers[END_REF] that Fistulifera saprophila was often the species responsible for sites becoming critical, i.e. having a low IPS score. The same study identified A. minutissimum as the diatom species contributing positively to the IPS scores [START_REF] Pérez-Burillo | Evaluation and sensitivity analysis of diatom DNA metabarcoding for WFD bioassessment of Mediterranean rivers[END_REF].

On the other hand, gaps in the reference databases may be unable to identify some other taxa of low sensitivity value, with a significant positive impact on the IPS scores estimated with DNA metabarcoding [START_REF] Vasselon | Assessing pollution of aquatic environments with diatoms' DNA metabarcoding: Experience and developments from France Water Framework Directive networks[END_REF][START_REF] Bailet | Diatom DNA metabarcoding for ecological assessment: Comparison among bioinformatics pipelines used in six European countries reveals the need for standardization[END_REF]. Regarding the eastern Mediterranean countries, there is a lack of sufficient DNA barcoding studies to date. Most recent DNA barcoding and metabarcoding studies focus on the west-northern Mediterranean and northern European countries (Vasselon et al. 2017b;[START_REF] Kelly | Environment Agency A DNA based diatom metabarcoding approach for Water Framework Directive classification of rivers[END_REF][START_REF] Bailet | Molecular versus morphological data for benthic diatoms biomonitoring in Northern Europe freshwater and consequences for ecological status[END_REF][START_REF] Feio | A taxonomy-free approach based on machine learning to assess the quality of rivers with diatoms[END_REF][START_REF] Pérez-Burillo | Evaluation and sensitivity analysis of diatom DNA metabarcoding for WFD bioassessment of Mediterranean rivers[END_REF]. When considering the eastern Mediterranean region, most studies focus on the marine environment [START_REF] Malviya | Insights into global diatom distribution and diversity in the world's ocean[END_REF][START_REF] Kaleli | Taxonomy and distribution of diatoms on the Turkish Mediterranean Coast, Dalyan (Mugla)[END_REF][START_REF] Vijver B Van De | Geographical variation in the diatom communities associated with loggerhead sea turtles (Caretta caretta)[END_REF]) and morphological identification of endemic species, such as Navicula veronensis sp. nov. found in low elevation carbonate springs [START_REF] Cantonati | A new Navicula (Bacillariophyta) species from low-elevation carbonate springs affected by anthropogenic disturbance[END_REF], Ulnaria monodii found in streams with relatively warm waters and Ulnaria acuscypriacus sp. nov. having tolerance to brackish conditions [START_REF] Cantonati | Taxonomic and ecological characterization of two ulnaria species (Bacillariophyta) from streams in Cyprus[END_REF]. Therefore, the gaps, observed here in the comparison of the two methodologies, are more or less expected.

Even though the morpho-taxonomic approach is able to identify more species than the DNA metabarcoding method, the DNA metabarcoding approach is still considered an effective method for diatom assemblage identification as it offers high-throughput, cost-effective and accurate analysis provided that the online barcode databases are up to date (Vasselon et al. 2017a;[START_REF] Keck | Boosting DNA metabarcoding for biomonitoring with phylogenetic estimation of operational taxonomic units' ecological profiles[END_REF][START_REF] Rivera | Metabarcoding of lake benthic diatoms: from structure assemblages to ecological assessment[END_REF][START_REF] Mortágua | Applicability of DNA metabarcoding approach in the bioassessment of Portuguese rivers using diatoms[END_REF].

Conclusions

To conclude, this study has investigated diatom samples from the Cyprus stream monitoring network and the correlation of the two methodologies currently in play for stream-quality bioassessments, based on phytobenthos/diatoms. For the first time, we compared and contrasted DNA metabarcoding to the traditional morphological, ecological assessment for river biomonitoring in Cyprus, a typical eastern Mediterranean country. Our results are in line with what has already been shown in literature. The limitations of both techniques can offset each other nicely when using spatial datasets. Despite the differences between the two methods and the taxonomic inventories they produced, the IPS scores from the molecular data were well correlated with those from morpho-taxonomic data when a CF is applied to bring molecular data in line with the expectations, based on microscopy data. As [START_REF] Kelly | Development of a novel metric for evaluating diatom assemblages in rivers using DNA metabarcoding[END_REF] stated: "HTS is not a simple transaction in which LM is replaced by a better technology, but requires a deeper level of institutional transformation than had hitherto been anticipated".

Future work can investigate how the two methodologies could contribute to an integrated assessment of different Mediterranean rivers with different hydrological impacts and pollution [START_REF] Charles | Benthic algae assessments in the EU and the US: Striving for consistency in the face of great ecological diversity[END_REF]. Additionally, it is worth studying the temporal observation of rivers to examine whether a potential shift to metabarcoding alters perceptions of status at an individual site level, as previously demonstrated by [START_REF] Kelly | Development of a novel metric for evaluating diatom assemblages in rivers using DNA metabarcoding[END_REF]. Lastly, there is a parallel need of: 1) an enrichment of DNA reference databases for the diatom species that are primarily present and endemic to southern-eastern Mediterranean rivers and 2) development of so-called taxonomy-free approaches to overcome these reference gaps, as recently tested on diatoms [START_REF] Apothéloz-Perret-Gentil | Monitoring the ecological status of rivers with diatom eDNA metabarcoding: A comparison of taxonomic markers and analytical approaches for the inference of a molecular diatom index[END_REF][START_REF] Tapolczai | The potential of exact sequence variants (ESVs) to interpret and assess the impact of agricultural pressure on stream diatom assemblages revealed by DNA metabarcoding[END_REF]).

Figure 1 .

 1 Figure 1. Correction factor application. Ten most abundant species identified by the molecular approach before and after the CF application.

Figure 2 .

 2 Figure 2. Ten most abundant species identified by the morpho-taxonomic approach.

Figure 3 .

 3 Figure 3. Comparison of morpho-taxonomic and molecular identification at the species level. A) Venn Diagram demonstrating the number of species identified by both methodologies, by morpho-taxonomic identification only (99), by molecular identification only (67) and by both methods (39). Suppl. material 1: TableS1gives the number of valves and number of reads recorded for the 39 common species; B) IPS correlation of the morpho-taxonomic and molecular identification methodologies. Pearson Correlation was performed indicating a significant correlation (r = 0.6745, p < 0.0001. The boundaries of the ecological status classes are marked: red (Bad), orange (Poor), yellow (Moderate), green (Good), blue (High). Dashed lines: Good/moderate boundary, Red dashed circle: morphological classifies as good status, while molecular classifies below good status. Blue cirlce: morphological classifies as below good status, while molecular classifies good status.

Figure 4 .

 4 Figure 4. Map representation and comparison of the IPS values obtained using the morpho-taxonomic and molecular approach. The detailed values are provided in Suppl. material 2: Fig. S1.

https://mbmg.pensoft.net

Acknowledgements

The research was funded by the COST Action DN-Aqua-Net (CA15219), the Cyprus University of Technology and the UMR CARRTEL -INRAE (National Research Institute for Agriculture, Food and Environment). Additionally, the research was partially funded by the Research and Innovation Foundation and the European Regional Development Funds, in the context of the WAT-DIMON project (EUROSTARS/0519/0005) -Novel DNA based test for the identification of benthic diatoms of European freshwater bodies. The sample collection was partly performed within Contract No. WDD15/2017 "Sampling, Sample Analysis and Evaluation of Biological Quality Elements. Implementation of Article 8 of Directive 2000/60/EC". Contractor: I.A.CO Environmental & Water Consultants Ltd. Contracting Authority: Water Development Department, Ministry of Agriculture, Rural Development and Environment, Republic of Cyprus. MC is grateful to Dr. Ionel Ciugulea, who provided digital micrographs, counts and useful discussion of the morpho-taxonomical species identifications.

Supplementary material 1