Assyr Abdulle
email: assyr.abdulle@epfl.ch

Gilles Vilmart
email: gilles.vilmart@epfl.ch

The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods

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Version française abrégée

Pour des problèmes elliptiques linéaires ou monotones, l'effet de l'intégration numérique sur la méthode des éléments finis est analysé dans [START_REF] Ciarlet | The combined effect of curved boundaries and numerical integration in isoparametric finite element method[END_REF][START_REF] Strang | Variational crimes in the finite element method[END_REF] et [START_REF] Feistauer | Finite element solution of nonlinear elliptic problems[END_REF]. Cependant, il n'existe à notre connaissance aucune analyse de vitesses de convergence pour des problèmes nonlinéaires de type nonmonotones. Dans [START_REF] Feistauer | An analysis of finite element variational crimes for a nonlinear elliptic problem of a nonmonotone type[END_REF], la convergence H 1 de la solution numérique est établie, mais sans vitesse de convergence et seulement pour des éléments finis linéaires par morceaux. L'objet de cet article est d'analyser l'influence des erreurs de quadrature pour la méthode des éléments finis appliquée à la classe d'équations elliptiques quasi-linéaires non-monotones [START_REF] Abdulle | On a-priori error analysis of Fully Discrete Heterogeneous Multiscale FEM[END_REF]. Sous des hypothèses usuelles sur le maillage pour des problèmes non-linéaires, sur la régularité des coefficients et des données, et sur les formules de quadrature (Q1),(Q2), également usuelles tant pour des problèmes avec intégration numérique (voir [START_REF] Ciarlet | The combined effect of curved boundaries and numerical integration in isoparametric finite element method[END_REF] ou [START_REF] Ciarlet | Basic error estimates for elliptic problems[END_REF]Sect. 29]) que pour des problèmes non-linéaires [START_REF] Feistauer | An analysis of finite element variational crimes for a nonlinear elliptic problem of a nonmonotone type[END_REF][START_REF] Xu | Two-grid discretization techniques for linear and nonlinear PDE[END_REF][START_REF] Ming | Analysis of the heterogeneous multiscale method for elliptic homogenization problems[END_REF], nous prouvons des estimations optimales d'erreur pour les normes H 1 et L 2 de la méthode d'éléments finis [START_REF] Abdulle | A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems[END_REF], pour des élements simpliciaux ou quadrilatéraux d'ordre arbitraire. Nous prouvons également l'unicité de la solution numérique.

Une application importante de notre étude est une analyse (avec discrétisation totale des échelles à la fois macroscopiques et microscopiques) d'une méthode d'homogénéisation numérique du type [START_REF] Abdulle | On a-priori error analysis of Fully Discrete Heterogeneous Multiscale FEM[END_REF][START_REF] Abdulle | The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs[END_REF][START_REF] Abdulle | A priori and a posteriori error analysis for numerical homogenization: a unified framework[END_REF][START_REF] Ming | Analysis of the heterogeneous multiscale method for elliptic homogenization problems[END_REF] pour une classe de problèmes non-linéaires d'homogénisation. La méthode d'homogénéisation numérique considérée peut être interprétée comme une méthode des éléments finis mise en oeuvre sur un schéma macroscopique, avec intégration numérique en la variable macroscopique, couplée à des schémas microscopiques mis en oeuvre sur des micro-cellules contenues dans le maillage macroscopique. Pour la classe de problèmes nonmonotones [START_REF] Hlaváček | On Galerkin approximations of a quasilinear nonpotential elliptic problem of a nonmonotone type[END_REF], il n'existait jusqu'alors qu'une analyse semi-discrete et pour les dimensions d ≤ 2. Notre analyse, avec discrétisation totale, permet de traiter la dimension d ≤ 3. De plus, nous proposons une analyse de convergence (optimale) dans la norme L 2 . Nous améliorons aussi l'estimation de l'erreur dite de résonnance et démontrons la convergence de la méthode de Newton utilisée pour calculer en pratique une solution du système non-linéaire. Plus de détails sur les résultats et l'analyse présentée ici sont donnés dans [START_REF] Abdulle | A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems[END_REF] (une seule échelle) et [START_REF] Abdulle | Fully discrete analysis of the finite element heterogeneous multiscale method for nonmonotone elliptic homogenization problems[END_REF] (problèmes multi-échelles).

Introduction

We study finite element (FE) discretizations of second-order quasilinear elliptic problems of the form

-∇ • (a(x, u(x))∇u(x)) = f (x) in Ω, u(x) = 0 on ∂Ω, (1)
where Ω is a bounded polyhedron in R d with d ≤ 3. We make the following assumptions on the tensor a(x, s) = (a mn (x, s)) 1≤m,n≤d :

-the coefficients a mn (x, s) are continuous functions on Ω × R which are uniformly Lipschitz continuous with respect to s, i.e.,

∃Λ 1 > 0, |a mn (x, s 1 ) -a mn (x, s 2 )| ≤ Λ 1 |s 1 -s 2 |, ∀x ∈ Ω, ∀s 1 , s 2 ∈ R, ∀ 1 ≤ m, n ≤ d. (2)
-a(x, s) is uniformly coercive and bounded, i.e.,

∃λ, Λ 0 > 0, λ ξ 2 ≤ a(x, s)ξ • ξ, a(x, s)ξ ≤ Λ 0 ξ , ∀ξ ∈ R d , ∀x ∈ Ω, ∀s ∈ R. (3)
Since ( 2)-( 3) hold, it is known [START_REF] Hlaváček | On Galerkin approximations of a quasilinear nonpotential elliptic problem of a nonmonotone type[END_REF] that (1) has a unique solution u ∈ H 1 0 (Ω) for all f ∈ L 2 (Ω). For linear or monotone elliptic problems, the effect of numerical quadrature in FEM has been analysed in [START_REF] Ciarlet | The combined effect of curved boundaries and numerical integration in isoparametric finite element method[END_REF][START_REF] Strang | Variational crimes in the finite element method[END_REF] and [START_REF] Feistauer | Finite element solution of nonlinear elliptic problems[END_REF]. To the best of our knowledge, there exist no analysis of the convergence rates for FEM with numerical quadrature applied to nonlinear problems of non-monotone type, as considered in this paper. In [START_REF] Feistauer | An analysis of finite element variational crimes for a nonlinear elliptic problem of a nonmonotone type[END_REF], the convergence in H 1 of the method is shown for piecewise finite elements, but without convergence rates. In the absence of numerical quadrature, optimal apriori error estimates in the H 1 and L 2 norms for FE methods (FEMs) were first given in [START_REF] Douglas | A Galerkin method for a nonlinear Dirichlet problem[END_REF].

Equations as [START_REF] Abdulle | On a-priori error analysis of Fully Discrete Heterogeneous Multiscale FEM[END_REF] enter in the modeling of many important problems, we mention the infiltration of water in porous medium, the study of electrical potential or thermal diffusion in materials. As exact integration in FEMs is rarely possible, it is important to quantify the effect of numerical quadrature. Optimal convergence rates in the H 1 and L 2 norms are proved in this case. The practical implementation of the non-linear FEM requires a Newton method. We also establish the convergence of this latter method (crucial in applications) and the uniqueness of the FE solution for a sufficiently fine FE mesh. If a(x, s) becomes independent of s, we recover the results of [START_REF] Ciarlet | The combined effect of curved boundaries and numerical integration in isoparametric finite element method[END_REF] on FEMs with numerical quadrature for linear problems (polyhedral domain case).

Application to numerical homogenization methods is then considered. In contrast to previous results [START_REF] Ming | Analysis of the heterogeneous multiscale method for elliptic homogenization problems[END_REF] obtained for nonmonotone homogenisation problems in dimension d ≤ 2 (based on 2d-Green function logarithmic estimates) for the H 1 norm and for a semi-discrete formulation, we obtain optimal convergence results for dimensions d ≤ 3 and for a fully discrete method, which takes into account the microscale FE discretization (see [START_REF] Abdulle | On a-priori error analysis of Fully Discrete Heterogeneous Multiscale FEM[END_REF][START_REF] Abdulle | The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs[END_REF][START_REF] Abdulle | A priori and a posteriori error analysis for numerical homogenization: a unified framework[END_REF] in the context of linear problems). In addition, our results are also valid for arbitrary high-order elements of simplicial or quadrilateral type, optimal error estimates are obtained for the L 2 norm, and improved estimates are obtained for the resonance error. More details on the results and the analysis presented here are given in [START_REF] Abdulle | A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems[END_REF] (one-scale problems) and [START_REF] Abdulle | Fully discrete analysis of the finite element heterogeneous multiscale method for nonmonotone elliptic homogenization problems[END_REF] (multi-scale problems) .

Finite element method with numerical quadrature

We consider a conformal shape regular family of partitions T h of Ω in simplicial or quadrilateral elements K of diameter h K and denote h := max K∈T h h K . We consider the family of FE spaces S 0 (Ω,

T h ) := {v h ∈ H 1 0 (Ω); v h | K ∈ R (K), ∀K ∈ T h },
where R (K) is the space P (K) of polynomials on K of total degree at most if K is a simplicial FE, or the space Q (K) of polynomials on K of degree at most in each variable if K is a quadrilateral FE. We define a quadrature formula {x j , ωj } J j=1 on a reference element K, where xj are integration points and ωj are quadrature weights. The quadrature formula {x Kj , ω Kj } J j=1 is then defined as usual on any element K of the triangulation using a C 1 -diffeomorphism. We make the following assumptions, which are similar to the case of linear elliptic problems (see [START_REF] Ciarlet | The combined effect of curved boundaries and numerical integration in isoparametric finite element method[END_REF] or [8, Sect. 29]):

(Q1) ωj > 0, j = 1, . . . , J, J j=1 ωj |∇p(x j )| 2 ≥ λ ∇p 2 L 2 ( K) , ∀p(x) ∈ R ( K), with λ > 0; (Q2) K p(x)dx = J j=1 ωj p(x j ), ∀p(x) ∈ R σ ( K), where σ = max(2 -2, ) if K is a simplicial FE, or σ = max(2 -1, + 1) if K is a rectangular FE.
Consider for v, w scalar or vector functions that are piecewise continuous with respect to the partition T h of Ω, the semi-definite inner product (u, v) h := K∈T h J j=1 ω Kj u(x Kj )v(x Kj ). The FE solution of (1) with numerical integration reads: find

u h ∈ S 0 (Ω, T h ) such that (a(•, u h )∇u h , ∇w h ) h = F h (w h ) ∀w h ∈ S 0 (Ω, T h ), (4)
where the linear form F h (w h ) is an approximation of Ω f (x)w h (x)dx obtained for example by using a quadrature formula. If f ∈ W ,r (Ω) with 1 ≤ r ≤ ∞ and > d/r, then f is continuous on Ω and one can take F h (w h ) := (f, w h ) h . The existence of the FE solution u h ∈ S 0 (Ω, T h ) in ( 4) can be shown for all h > 0 using the Brouwer fixed point theorem. Details can be found for example in [START_REF] Douglas | A Galerkin method for a nonlinear Dirichlet problem[END_REF].

3.

Convergence rates for FEM with numerical quadrature for nonlinear problems Theorem 3.1 [START_REF] Abdulle | A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems[END_REF] Consider u the solution of problem (1). Let ≥ 1. Let d/ < r ≤ ∞. Let µ = 0 or 1. Assume (Q1), (Q2), that the family of triangulations is quasi-uniform, and

1 u ∈ H +1 (Ω) ∩ W 1,∞ (Ω), a ∈ (W +µ,∞ (Ω × R)) d×d , f ∈ W +µ,r (Ω).
In addition to (2), (3), assume that the operator

L * ϕ = -∇ • (a(•, u) T ∇ϕ) + ∂ u a(•, u)∇u • ∇ϕ satisfies 2 ϕ H 2 (Ω) ≤ C( L * ϕ L 2 (Ω) + ϕ H 1 (Ω) ), for all ϕ ∈ H 2 (Ω) ∩ H 1 0 (Ω). (5)
Assume further that ∂ u a mn ∈ W 1,∞ (Ω × R), and that the coefficients a mn (x, s) are twice differentiable with respect to s, with the first and second order derivatives continuous and bounded on Ω × R, for all m, n = 1 . . . d.

1. Except for the W 1,∞ assumption on u and the smoothness of s → a(x, s) assumed to treat the non-linearity (as in [START_REF] Douglas | A Galerkin method for a nonlinear Dirichlet problem[END_REF]), the smoothness assumptions of Thm. 3.1 are identical to those classically assumed for linear problems [START_REF] Ciarlet | The combined effect of curved boundaries and numerical integration in isoparametric finite element method[END_REF], [START_REF] Ciarlet | Basic error estimates for elliptic problems[END_REF]Sect. 29].

2.

The assumption (5) on the adjoint L * of the linearized operator L associated to (1) is also required for L 2 estimates in the case of linear problems [START_REF] Ciarlet | The combined effect of curved boundaries and numerical integration in isoparametric finite element method[END_REF]. Using classical H 2 regularity results, it is automatically satisfied -owing to the assumptions on the coefficients of L * -if the domain Ω is a convex polyhedron Then there exists h 0 > 0 such that for all h ≤ h 0 , the solution u h of (4) is unique and the following H 1 and L 2 error estimates hold,

if µ = 0, 1, u -u h H 1 (Ω) ≤ Ch for all h ≤ h 0 , (6)
if µ = 1, u -u h L 2 (Ω) ≤ Ch +1 for all h ≤ h 0 , (7)
where the constant C is independent of h.

Inspired by [START_REF] Douglas | A Galerkin method for a nonlinear Dirichlet problem[END_REF], the proof of Theorem 3.1 is conducted in three main steps.

Step 1. Using the compact injection H 1 (Ω) ⊂ L 2 (Ω), the boundedness of a numerical solution in H 1 0 (Ω) and the uniqueness in H 1 0 (Ω) of the exact solution of (1), we show,

u -u h L 2 (Ω) → 0 for h → 0. ( 8
)
Step 2. We derive the following H 1 a-priori error bound

u -u h H 1 (Ω) ≤ C(h + u -u h L 2 (Ω) ), for all h > 0. ( 9
)
The additional term u -u h L 2 (Ω) in the right-hand side is due to the non-monotonicity of the differential operator of (1). The proof of ( 9) relies on an estimate for (a(u h )∇u h , ∇w h )-(a(u h )∇u h , ∇w h ) h (obtained by using the Bramble-Hilbert lemma), uniform bounds for the semi-definite inner product (v, w) T h := K∈T h J j=1 ω Kj v(x Kj ) • w(x Kj ) (defined for piecewise continuous functions v, w) and the use of the Gagliardo-Niremberg inequality

v 2 L 3 (Ω) ≤ C v L 2 (Ω) v H 1 (Ω) for all v ∈ H 1 (Ω), for d ≤ 3.
Step 3. Using an Aubin-Nitsche duality argument and (5), we show that there exists

h 1 > 0 such that u -u h L 2 (Ω) ≤ C(h +µ + u -u h 2 H 1 (Ω) ), for all h ≤ h 1 . (10)
We consider the FEM solution with numerical quadrature associated to the indefinite linear elliptic problem L * . We first show that L * is an isomorphism and then derive error estimates generalizing a compactness result of Schatz [START_REF] Schatz | An observation concerning Ritz-Galerkin methods with indefinite bilinear forms[END_REF] to FEM with numerical quadrature.

Proof of the H 1 and L 2 estimates. Substituting ( 9) into (10) (with µ = 0), we obtain

u -u h H 1 (Ω) ≤ C(h + u -u h 2 H 1 (Ω) ), for all h ≤ h 1 .
Substituting (8) into (9), we obtain u -u h H 1 (Ω) → 0 for h → 0. We deduce in the above inequality 1 -C u -u h H 1 (Ω) ≥ δ > 0 for all h ≤ h 2 , with h 2 small enough (but independent of the particular solution u h ) hence, ( 6) is established for all h ≤ min{h 1 , h 2 }. The estimate [START_REF] Ciarlet | The combined effect of curved boundaries and numerical integration in isoparametric finite element method[END_REF] is deduced by substituting (6) into [START_REF] Ming | Analysis of the heterogeneous multiscale method for elliptic homogenization problems[END_REF] with µ = 1. The uniqueness of the FEM solution follows from Theorem 3.2. 2 Theorem 3.2 Consider u h a solution of (4). Under the assumptions of Theorem 3.1, there exist h 0 , δ > 0 such that if h ≤ h 0 and σ h z h 0 -u h H 1 (Ω) ≤ δ, then the sequence {z h k } for the Newton method3

N h (z h k ; z h k+1 -z h k , v h ) = F h (v h ) -(a(z h k )∇z h k , ∇v h ) h , ∀v h ∈ S 0 (Ω, T h ), (11)
is well defined, and

z h k+1 -u h H 1 (Ω) ≤ Cσ h z h k -u h 2 H 1 (Ω) , ( 12
)
where C is a constant independent of h, k.

In the above theorem,

σ h := sup v h ∈S 0 (Ω,T h ) v h L ∞ (Ω) / v h H 1 (Ω)
. Using the quasi-uniformity of the family of triangulations, one can show the standard estimates σ h ≤ C(1 + | ln h|) 1/2 for d = 2, and σ h ≤ Ch -1/2 for d = 3, where C is independent of h.

Remark 1 Notice that the requirement of a quasi-uniform mesh for the family of triangulations is often assumed for the analysis of FEM for nonlinear problems [START_REF] Feistauer | An analysis of finite element variational crimes for a nonlinear elliptic problem of a nonmonotone type[END_REF][START_REF] Xu | Two-grid discretization techniques for linear and nonlinear PDE[END_REF][START_REF] Ming | Analysis of the heterogeneous multiscale method for elliptic homogenization problems[END_REF]. In our proof, we need it in Step 3 to have an a-priori estimate in W 1,6 (Ω) for the FEM solution (with numerical quadrature) associated to L * . We further need this assumption in the uniqueness result below. However, if u H 2 (Ω) or the Lipschitz constant are small enough such that Cλ -1 Λ 1 u H 2 (Ω) < 1, where C depends only on Ω and the polynomial degree of the FE space, then (5) and u ∈ W 1,∞ (Ω) are not required to prove the uniqueness result, and removing in addition the assumptions of quasi-uniform meshes and h ≤ h 0 , the H 1 estimate (6) still holds.

Application to numerical homogenization.

We consider a class of nonlinear nonmonotone multiscale problems

-∇ • (a ε (x, u ε (x)))∇u ε (x)) = f (x) in Ω, u ε (x) = 0 on ∂Ω, (13)
with a d×d tensor a ε (x, x) satisfying ( 2), (3) uniformly in ε. Here ε represent a small scale in the problem.

The following homogenization result is shown in [6, Theorem 3.6]: there exists a subsequence of {a (•, s)} (again indexed by ε) such that the corresponding sequence of solutions {u ε } converges weakly to u 0 in H 1 (Ω), where u 0 is solution of the so-called homogenized problem

-∇ • a 0 (x, u 0 (x))∇u 0 (x) = f (x) in Ω, u 0 (x) = 0 on ∂Ω, (14)
with a homogenized tensor a 0 (x, s) which can be shown to have similar properties as assumed for a ε (x, s).

The FE-HMM method for computing a numerical approximation u H of u 0 , essentially similar to the method proposed in [START_REF] Ming | Analysis of the heterogeneous multiscale method for elliptic homogenization problems[END_REF] 4 reads as follows. It is based on a macroscopic FEM defined on QF with a macro FE space S 0 (Ω, T H ) (defined as in Sect. 2), and microscopic FEMs recovering the missing macroscopic tensor at the macroscopic quadrature points. For each macro element K ∈ T H and each integration point x Kj ∈ K, j = 1, . . . , J, we define the sampling domains K δj = x Kj + (-δ, δ) d , (δ ≥ ε). For each K δj , we then define a micro FE space S q (K δj , T h ) ⊂ W (K δj ) with simplicial or quadrilateral FEs and a conformal and shape regular family of triangulations T h . The space W (K δj ) is either the Sobolev space W (K δj ) = W 1 per (K δj ) = {z ∈ H 1 per (K δj ); K δ j zdx = 0} for a periodic coupling or W (K δj ) = H 1 0 (K δj ) for a coupling through Dirichlet boundary conditions. ) denotes the solution of the following micro problem [START_REF] Xu | Two-grid discretization techniques for linear and nonlinear PDE[END_REF] with parameter s = u H (x Kj ). Find w h,s Kj such that w h,s Kj -(w H (x Kj )+(x-x Kj )•∇w H (x Kj )) ∈ S q (K δj , T h ) and K δ j a ε (x, s)∇w h,s Kj (x) • ∇z h (x)dx = 0 ∀z h ∈ S q (K δj , T h ). ( 16)

FE-HMM

We make the following smoothness and structure assumptions on the tensor.

4 .

 4 In[START_REF] Ming | Analysis of the heterogeneous multiscale method for elliptic homogenization problems[END_REF] [START_REF] Strang | Variational crimes in the finite element method[END_REF] is based on exact micro functions v K j , w K j instead of the FE micro functions v h,s K j , w h,s K j and the microproblems are nonlinear (see[10, equs. (5.3)-(5.4)]).

  Find u H ∈ S 0 (Ω, T H ) such that B H (u H ; u H , w H ) = F H (w H ), ∀w H ∈ S 0 (Ω, T H ), where B H (u H ; v H , w H ) := Kj |K δj | K δ j a ε (x, u H (x Kj ))∇v

		J	ω h,u H (x K j ) Kj	(x) • ∇w	h,u H (x K j ) Kj	(x)dx,	(15)
	K∈T H	j=1
	and w

h,u H (x K j ) Kj (and similarly for v h,u H (x K j ) Kj

We define N h (z h ; v h , w h ) := (a(•, z h )∇v h , ∇w h ) h + (v h ∂ua(•, z h )∇z h , ∇w h ) h .

(H1) Given q ∈ N, the cell functions ψ i,s Kj ∈ W (K δj ) such that K δ j a ε (x, s)∇ψ i,s Kj (x) • ∇z(x)dx = -K δ j a ε (x, s)e i • ∇z(x)dx, ∀z ∈ W (K δj ).

(17) satisfy the bound |ψ i,s Kj | H q+1 (K δ j ) ≤ Cε -q |K δj |, with C independent of ε, the quadrature point x Kj , the domain K δj , and the parameter s for all i = 1 . . . d. Here, e 1 , . . . , e d denotes the canonical basis of R d . The same assumption also holds with the tensor a ε replaced by (a ε ) T in (17). (H2) for all m, n = 1, . . . , d, we assume a ε mn (x, s) = a mn (x, x/ε, s), where a mn (x, y, s) is y-periodic in Y , and the map (x, s) → a mn (x, •, s) is Lipschitz continuous and bounded from Ω × R into W 1,∞ per (Y ). Following the framework presented in Sect. 3, we obtain the following H 1 and L 2 a-priori estimates. Theorem 4.1 [START_REF] Abdulle | Fully discrete analysis of the finite element heterogeneous multiscale method for nonmonotone elliptic homogenization problems[END_REF] Let ≥ 1, q ≥ 1 and µ = 0 or 1. In addition to the assumptions of Theorem 3.1 on problem [START_REF] Schatz | An observation concerning Ritz-Galerkin methods with indefinite bilinear forms[END_REF], assume (H1), (H2), and assume that a ε satisfies (2), (3). Then, there exist H 0 > 0 and r 0 > 0 such at if H ≤ H 0 and h/ε ≤ r 0 then

and δ ε ∈ N, and a ε (x, s) is replaced by a(x Kj , x/ε, s) in ( 15),( 16),(17),

where we also assume δ ≤ r 0 or δ + ε/δ ≤ r 0 in the first and third cases, respectively. We use the notation

with first and second derivatives bounded by Cε -1 , then for sufficiently fine meshes and modeling errors (e.g. in the second case of Theorem 4.1, for (h/ε) 2q ≤ H ≤ H 1 ), one can show the convergence of a Newton method, and the uniqueness of the numerical solution u H . Notice that in the third case with non-perodic boundary conditions for the micro-macro coupling (i.e. W (K δj ) = H 1 0 (K δj )) we obtain the resonance error estimate r M OD ≤ C(δ + ε/δ) (similar as for linear problems), whereas r M OD ≤ C(δ + (ε/δ) 1/2 ) has been obtain in [START_REF] Ming | Analysis of the heterogeneous multiscale method for elliptic homogenization problems[END_REF]Thm. 5.5].


Georges Stora
email: stora@com-univ-mrs.fr

Michel Bourcier

Andre Arnoux

Magali Gerino

Joel Le Campion

Franck Gilbert

Jean Pierre Durbec

The deep-sea macrobenthos on the continental slope of the northwestern Mediterranean Sea: a quantitative approach

As part of the ECOMARGE operation (J.G.O.F.S. France), macrobenthic assemblages in the Toulon Canyon were described and quanti"ed on the basis of sampling carried out between 250 and 2000 m depth on the Mediterranean continental slope. Results show that Mediterranean bathyal assemblages are made up mainly of continental shelf eurybathic species. The qualitative and quantitative composition of populations varies with depth on the slope and also varies with station position at equivalent depth, whether on the #anks or in the canyon channel. Various analyses have provided evidence on the factors responsible for this population distribution pattern. No single factor emerges as predominant, but rather a group of factors, which are related to the nature and origin of sediments and more particularly their grain size distribution, geochemical composition and mode of transportation and sedimentation (benthic nepheloid or originating from the water column), act in conjunction to determine the pattern. Comparison with ocean continental slopes shows that in the Mediterranean Sea the absence of tidal current modi"es the trophic structure of the macrobenthic assemblages, which are characterized by a dominance of surface and subsurface deposit feeders as compared to a dominance of suspension feeders and carnivores in the upper and median part of the slope in the ocean. Surface dumping of dredge spoil at the canyon head and channelling of waste induces an increase of organic matter and pollutant concentrations in sediment from the upper part of the canyon channel but does not give rise to any marked population degradation.

Introduction

The continental slope and more particularly the canyons cutting across it have long been recognized as zones of matter and energy transfer between the continental shelf and the deep ocean [START_REF] Griggs | Deep-sea sedimentation and sediment fauna interaction in Cascadia Channel and on Cascadia Abyssal Plain[END_REF][START_REF] Gardner | Baltimore Canyon as a modern conduit of sediment to the deep sea[END_REF]. Since 1983, the ECOMARGE operation, which is part of the program Joint Global Ocean Flux Study France, has involved a series of multidisciplinary investigations to study the functioning of continental margin ecosystems [START_REF] Monaco | Particles #uxes and ecosystem response on a continental margin: the 1985}1988 Mediterranean ECOMARGE experiment[END_REF]. Research carried out on the continental slope has focused mainly on the quali"cation and the quanti"cation of advective and convective particle #uxes and their variation with time, and on the impact of matter and energy transfer to the water}sediment interface. It is in an attempt to shed light on the latter question that studies have been carried out on the macrobenthic assemblages of a Mediterranean canyon, the Toulon Canyon.

Transfers of matter and energy from the continental shelf and the water column to the continental slope have a strong impact on the distribution and composition of benthic assemblages, which constitute the biological testimony of the environmental conditions. From the quantitative and qualitative points of view, deep macrobenthic assemblages have been studied in many regions of the world's oceans, notably those of the continental slope [START_REF] Sanders | An introduction to the study of deep-sea benthic faunal assemblages along the Gay Head-Bermuda transect[END_REF][START_REF] Griggs | Deep-sea sedimentation and sediment fauna interaction in Cascadia Channel and on Cascadia Abyssal Plain[END_REF][START_REF] Grassle | Faunal changes with depth in the deep-sea benthos[END_REF]Rowe and Headrich, 1979;[START_REF] Pearce | Benthic fauna and heavy metals burdens in marine organisms and sediments of a continental slope dumpsite o! the northeast coast of the United States (Deepwater Dumpsite 106)[END_REF][START_REF] Rowe | The deep-sea macrobenthos on the continental margin of the N.W. Atlantic Ocean[END_REF][START_REF] Blake | Infaunal-sediment relationships at the shelf -slope break[END_REF]Pfannuche et al., 1983;[START_REF] Romero-Wetzel | Abundance, biomass, size-distribution and bioturbation potential of deep-sea macrozoobenthos on the Voring Plateau (1200}1500 m, Norwegian Sea)[END_REF][START_REF] Blake | Benthic community structure on the U.S. South Atlantic slope o! the Carolinas: spatial heterogeneity in a current-dominated system[END_REF][START_REF] Blake | Dense infaunal assemblages on the continental slope o! Cape Hatteras, North Carolina[END_REF], but the lack of quantitative data on the deep Mediterranean benthos has been pointed out by several authors (Peres, 1982;[START_REF] Fredj | The deep mediterranean benthos[END_REF][START_REF] Bouchet | The Mediterranean deep-sea fauna: pseudopopulations of Atlantic species? Deep[END_REF][START_REF] Laubier | The mediterranean deep-sea benthic fauna[END_REF]. Our present knowledge is limited to qualitative and descriptive aspects of the deep sea fauna. How the deep Mediterranean ecosystem functions, compared with other ecosystems of the world's oceans, is largely unknown [START_REF] Fredj | The deep mediterranean benthos[END_REF]. Macrobenthic assemblages at the limit of the continental shelf and the upper part of the slope have been described mainly on the basis of dredgings rarely exceeding 1000 m depth in the northwestern region of the Mediterranean Sea by [START_REF] Picard | Recherches qualitatives sur les bioceH noses marines des substrats meubles dragables de la reH gion marseillaise[END_REF], [START_REF] Carpine | Ecologie de l'eH tage bathyal dans la MeH diteH rraneH e occidentale[END_REF], [START_REF] Reyss | Bionomie benthique de 2 canyons sous marin de la mer Catalane: le rech du Cap et le rech Lacaze-Duthiers[END_REF], [START_REF] Bourcier | Ecoulement des boues rouges dans le canyon de Cassidaigne (DeH cembre 1968)[END_REF], [START_REF] Falconetti | Bionomie benthique des fonds situeH s a`la limite du plateau continental du banc du Magaud (Iles d'Hye`res) et de la reH gion de Calvi (Corse)[END_REF], and in the northeastern region by [START_REF] Vamvakas | Peuplements benthiques des substrats meubles du sud de la mer EgeH e[END_REF] and Chardy et al. (1973). The rare quantitative data obtained by sediment cores have been reported by Tchukhchin (1964), [START_REF] Cattaneo | Macrobenthos del fondi batiali liguri[END_REF], Tahey et al. (1992), Tselepides and Eleftheriou (1992), [START_REF] Bourcier | ReH ponse du macrobenthos d'un canyon sous marin meH diterra-neH en a`des apports particulaires telluriques et anthropiques[END_REF] and Gerino et al. (1994).

In view of this gap in our knowledge of the Mediterranean Sea, the primary aim of this study in the Toulon Canyon was to qualify and quantify the macrobenthic assemblages of the continental slope, to determine the factors responsible for the distribution and composition of these assemblages, to analyze the impact of the channelled inputs in the canyon axis and to compare the results obtained with those from the Atlantic margin.

In the northwestern Mediterranean Sea, the coast of Provence (from Marseilles to the Italian border), where the Toulon Canyon is located (Fig. 1), is characterized by a narrow continental margin that extends between 20 and 30 km o!shore. O! Toulon, the continental shelf represents one-half or one-third of the continental slope and presents an average slope of the order of 2}3%. The Toulon Canyon [START_REF] Poydenot | Le canyon de Toulon: Morphologie et sedimentation (MeH diterraneH e Nord -Occidentale)[END_REF] is a linear valley lying north}south, perpendicular to the base of the slope. It is 18 km long and 9 km wide, with a height di!erential of 2100 m. The canyon is not an extension of an existing river on the continent. The canyon walls are asymmetrical. The west #ank represents 43% of the canyon's surface and is strongly sloping (26.7%). The east #ank is more eroded, represents 57% of the canyon's surface and slopes more gently (21.1%). The width of the channel increases from about 200 m at the canyon head to more than 1000 m at the deep fan outlet. Water circulation in the canyon follows a complex pattern. It is in#uenced by the general #ow of the Liguro-Provenial current #owing from east to west [START_REF] Millot | The gulf of Lions'hydrodynamics[END_REF], and by meteorological conditions inducing upwelling currents in the canyon channel with north}northwesterly winds, or downwelling currents with east, south and southwesterly winds [START_REF] Bourcier | Ecoulement des boues rouges dans le canyon de Cassidaigne (DeH cembre 1968)[END_REF][START_REF] Millot | The e!ect of Mistral wind on the ligurian current near Provence[END_REF][START_REF] Poydenot | Le canyon de Toulon: Morphologie et sedimentation (MeH diterraneH e Nord -Occidentale)[END_REF]. In the absence of a large coastal river, material contributions to the continental slope mainly originate from the continental shelf in the Toulon area and from the Liguro-Provenial current [START_REF] Poydenot | Le canyon de Toulon: Morphologie et sedimentation (MeH diterraneH e Nord -Occidentale)[END_REF]. Surface dumping of dredge spoil from Toulon harbour at the canyon head is added to these inputs on the sea #oor between 250 and 500 m depth. The frequency of these deposits, in tonnages that are con"dential (military zone), varies from once every six months to once per year.

Materials and methods

Sampling

Eleven stations were studied in Toulon Canyon, distributed along three transects (canyon channel, east #ank, west #ank) between 200 and 2000 m depth (Fig. 1). Because of the presence of rocky cli!s on the west #ank, only two stations, located at 1000 and 1500 m depth, could be sampled. Most of the samples were taken between May and July 1988. At each station, with coordinates speci"ed in Table 1 (Toran navigation system), samples were randomly collected within a circle not exceeding 1/10 mile from the reference point. The box core used is the FLUCHA corer, which retrieves relatively undisturbed sediment samples with a surface area of 6.25 dm (2.5;2.5 dm) and a height of 30}40 cm. Six replicate cores were taken at each station. In these cores, only the upper 12 cm of the sediment was collected, corresponding to a sediment volume of 7.5 dm per core. A total of 66 samples were sieved through a 500 m mesh and then sorted. In most cases, organisms were identi"ed to species level.

Since the surface layer is more representative of recent inputs to the canyon, chemical analysis of sediment at each station was carried out on an average sample obtained by the mixing and homogenization of part of the "rst cm of six replicate cores.

Geochemical analysis

Analyses were carried out on the sediment fraction smaller than 63 m sifted by wet sieving. Loss of ignition (LOI) was obtained by heating in a furnace at 5503C. Organic carbon was measured by oxidation by the chromic acid method [START_REF] Fao | Manual of methods in aquatic environment research. Part 1. Methods for detection, measurment and monitoring of water pollution[END_REF].

Determination of the total content of heavy metals (Cu, Pb, Cd, Zn, Cr, Ni, CO, Mn and Fe) was carried out after digestion of sediments by adding 1 g of sediment to 10 ml of the mixture HCL#HNO (3#1 V/V) at 1203C for 2 h with a re#ux condenser [START_REF] Arnoux | Comparaison de quelques meH thodes d'attaque des seH diments marins pour l'analyse des meH taux lourds[END_REF]. Each metal assay was carried out by #ame atomic absorption spectroscopy in air}acetylene #ame for Cu, Pb, Cd, Zn, Ni, Co, Mn, in N O}acetylene #ame for Cr and by spectrophotometry of the complex given with the o-phenantroline for Fe [START_REF] Arnoux | Comparaison de quelques meH thodes d'attaque des seH diments marins pour l'analyse des meH taux lourds[END_REF].

Biological analysis and data processing

Classical analytical and synthetic methods (Legendre and Legendre, 1984a, b) were used to assess the spatial dynamics of the assemblages. Interpretation of results is based on faunal data expressed as species richness, species density, dominance and biomass. After CaCO elimination with HCl, the biomass was determined by drying the organisms at 603C in an oven until the sample weight remained constant. [START_REF] Rowe | The deep-sea macrobenthos on the continental margin of the N.W. Atlantic Ocean[END_REF] worked in an environment similar to the Toulon Canyon (Hudson Canyon, western atlantic), and in order to facilitate comparison with their results, when possible we applied the same processing methods they used. Homogeneous groups of stations were singled out by cluster analysis [START_REF] Legendre | Le progiciel R-Analyse multidimensionelle, analyse spatiale[END_REF] applied to a similarity matrix obtained by the coe$cients of Sorensen [START_REF] Sorensen | A method of establishing groups of equal amplitude in plant sociology based on similarity of species content and its application to analysis of the vegetation on Danish commons[END_REF] and Whittaker (Whittaker and Fairbanks, 1958), which are, respectively, a qualitative coe$cient based on the presence}absence of the species and a quantitative coe$cient based on their dominance.

In order to give evidence of rare species e!ect on the clusterings we used a third similarity coe$cient, the Normalized Expected Shared Species (NESS) [START_REF] Grassle | A similarity measure sensitive to the contribution of rare species and its use in investigation of variation in marine benthic communities[END_REF]. NESS was used for analyses of stations with combined replicates, and the number of individuals (m) was set at 10 and 30.

The study of species association was carried out using, in R mode, the Dice index [START_REF] Dice | Measures of the amount of ecologic association between species[END_REF], whose formulation is identical to the Sorensen index (Legendre and Legendre, 1984). Species represented by a single individual at a station were eliminated from this procedure. The diversity measurements were carried out using two widely used methods [START_REF] Gage | Deep-Sea Biology: A Natural History of Organisms at the Deep-Sea Floor[END_REF], the rarefaction curves (Sanders, 1978) modi"ed by Hurlbert (1971) and the Shannon}Wiener information function (H) [START_REF] Shannon | The Mathematical Theory of Communication[END_REF] and its associated eveness value (J) [START_REF] Pielou | The measurement of diversity in di!erent types of biological collections[END_REF]. Each H and J station value was obtained by pooling data from the six replicate cores collected at a station. The Shannon Wiener index was calculated using the base log .

The relation between density or biomass and depth was investigated by means of generalized linear models (McCullagh and Nelder, 1989). In order to determine the main factors responsible for the composition and the structure of the assemblages, a factorial correspondence analysis (FCA) was carried out [START_REF] Benzecri | L'analyse de donneH es. Tome II: L'analyse de correspondance[END_REF]. The advantage of this method is that both the lines and columns of a contingency table can be represented on the same projection planes. Given the large number of collected species, of which few contribute strongly to the inertia of the "rst axes, species represented by a single individual at a station were eliminated. From the results obtained by the FCA, correlations (Coe$cient of Spearman) were calculated between the ordination of variables on the adopted axes and the di!erent chemical and physical parameters analysed in sediments of the surface layer [START_REF] Vincent | Profondeur, vase et courant, facteurs de micro-reH partition transversale du benthos dans l'estuaire d'eau douce du Saint-Laurent Quebec[END_REF]. Interpretation of results was based on the presence of biological indicators [START_REF] Picard | Recherches qualitatives sur les bioceH noses marines des substrats meubles dragables de la reH gion marseillaise[END_REF][START_REF] Carpine | Ecologie de l'eH tage bathyal dans la MeH diteH rraneH e occidentale[END_REF]Peres and Picard, 1964;Peres 1982;[START_REF] Bourcier | Economie benthique d'une baie meH diterraneH enne largement ouverte et des reH gions voisines en fonction des in#uences naturelles et humaines[END_REF]). These indicators can be either speci"c to an original assemblage (characteristic species) or to one or several grain size fractions (silts, sands, gravels, etc.). In other cases, they can mark particular environmental conditions such as sediment instability or pollution, or they can have no particular ecological requirements (wide ecological range species). Finally, the assemblages were characterized by their nutrition mode using indications provided notably by [START_REF] Coull | Ecology of marine benthos[END_REF], [START_REF] Fauchald | The diet of worms: a study of polychaete feeding guilds[END_REF], [START_REF] Jumars | Deep sea deposit-feeding strategies suggested by environmental and feeding constraints[END_REF] and [START_REF] Gage | Deep-Sea Biology: A Natural History of Organisms at the Deep-Sea Floor[END_REF].

According to [START_REF] Blake | Benthic community structure on the U.S. South Atlantic slope o! the Carolinas: spatial heterogeneity in a current-dominated system[END_REF], organisms for which the identi"cation was uncertain (e.g. juveniles, anterior fragments, etc.) were used only for density and biomass measurements and were not included in calculations of similarity, diversity indexes and multivariate analysis.

Results

Geochemical analysis

Table 2 presents percentages and concentrations of various elements analyzed in the surface layer of Toulon sediments. The geochemical characteristics of continental slope sediment resulting from the particle inputs, either transported along the bottom by the nepheloid benthic layer or originating from the water column, depends on several geological, oceanic and anthropogenic factors [START_REF] Karl | Factors that in#uence sediment transport at the shelfbreak[END_REF]McGrail and Carnes, 1983). With the exception of stations at the head of the canyon that are in#uenced by currents linked to the shelfbreak, "ne fraction percentages are greater than 70% for all the canyon stations. The presence in canyon sediments of Posidonia "bers originating from the continental shelf [START_REF] Poydenot | Le canyon de Toulon: Morphologie et sedimentation (MeH diterraneH e Nord -Occidentale)[END_REF] results in high percentages of loss of ignition in both channel and #ank stations below 500 m depth. However, stations located in the channel of the Canyon are characterized by higher organic carbon percentages than the other stations, associated with strong concentrations of lead, zinc and copper, notably between 250 and 1100 m depth. These high-concentrations mark the preferential deposition area of the dredge spoils originating from Toulon roadstead, which are rich in organic elements and heavy metals [START_REF] Romana | Apports en micro-polluants organiques et mineH raux. La mer et les rejets urbains[END_REF] dumped between stations CTL1 and CTL2. With the exception of lead and zinc, the other heavy metal concentrations, notably at the canyon #anks, increase with increasing depth as a result of natural sedimentation of "ne particles with high metal content [START_REF] Fok Rstner | Metal Pollution in the Aquatic Environment[END_REF] transported by the general or local circulation. Manganese, and to a lesser degree nickel and chromium are more particularly carried by the Liguro-Provenial current [START_REF] Arnoux | AlteH ration des peuplements benthiques de roches profondes en MeH diteH ranneH e nord-occidentale: quelques aspects biologiques et molysmologiques[END_REF].

Biological analysis

Species accumulation

We "rst veri"ed whether the sampling carried out at each station collected the majority of species present. Species accumulation curves were constructed by plotting the addition of new species as a variable dependent on an increasing number of cores for each station. The species accumulation curves in our assemblages (Fig. 2) indicate that, except for stations located at 250 m depth (CTL1, CTL6), where we could have bene"tted from more samples, the area sampled at the other stations was more than enough for capturing the majority of the species present and for estimating various parameters of the benthic assemblage structure. In fact, it appears (Fig. 2) that three samples would have been su$cient to assess variation.

Species richness and density

There is a strong decrease in the species richness with depth, ranging in the canyon channel from 124 species at station CTL1 (250 m) to 31 species at station CTL5 (2000 m) (Table 3). The decrease in species is in fact particularly marked between 250 and 1000 m depth. Below 1000 m, a certain stabilization of the species number is apparent.

The variations in fauna density (numbers m\) for individual samples as a function of depth (X) were described as usual [START_REF] Rowe | The deep-sea macrobenthos on the continental margin of the N.W. Atlantic Ocean[END_REF][START_REF] Gage | Deep-Sea Biology: A Natural History of Organisms at the Deep-Sea Floor[END_REF]) by a log-linear equation: log (density)"a#b (X) (1) and are shown in Fig. 3. Firstly, to compare the mean densities at the di!erent stations, the equality of variances was veri"ed. However, since the homoscedasticity hypothesis cannot be held, we used generalized linear models (Glim 4, 1993;McCullagh and Nelder, 1989), with density at each station weighted by the inverse of the corresponding standard deviation. In this way, we estimated the parameters of the regression equation ( 1) by the maximum likelihood method. The statistical signi"cance of parameters was assessed by considering the increase of the log-likelihood statistic between two hypotheses (with and without this parameter) (McCullagh and Nelder, 1989). The di!erence is distributed as a chi square in the absence of the e!ect of the considered variable.

The estimated regression equation is Log (density)"6.9!0.0011X

(Depth e!ect: Chi"48.30, 1 df, p(0.001). Species per 300 individuals 84.9

Sample size was too small to allow calculation of the parameter. The means of log-density were compared according to the considered depth either as a "ve-level factor or according to the localization of stations at each depth on #anks or in the channel. For this, the generalized linear models and Tuckey's multiple comparison method (GLIM 4, 1994) were used. Comparison of mean densities at the di!erent depths (homogeneity test chi"69.32, 4 df, p(0.001) shows that only the stations at 250 m depth have a mean log-density signi"cantly higher than the others (chi'5.6, 1 df, p(0.02); from 500 to 2000 m depth, the log-densities are apparently not di!erent (chi(3.8, 1 df, p'0.05). The position of the stations has no apparent e!ect (chi"8.29, 10 df, p"0.60).

Biomass

The same method of analysis was carried out on dry weight biomass (biomass m\). The variation of biomass as a function of depth for individual samples is shown in Fig. 3. The estimated regression equation of the biomass as a function of depth is Log (biomass m\)"6.87!0.0018X (Fig. 3)

(Depth e!ect, chi"89.62, 1 df, p(0.001).

Comparison of the mean biomass at di!erent depths (homogeneity test chi" 109.0, df 4, p(0.0001) shows that the di!erence of the means of log-biomass between any two levels of this factor is signi"cantly di!erent from zero (chi'5.44, 1 df, p(0.02) except between 1500 and 2000 m (chi"0.53, 1 df, p"0.47). For stations (chi"156.2, 10 df, p(0.01), it is only at 500 m depth (CTL 2, CTL7) that no signi"cant di!erence can be detected (chi(1.54, 1 df, p'0.215). For the other stations at each depth, log-biomass are di!erent between stations (chi'5.25, 1 df, p(0.022) depending on their location, whether on channel or #anks, but no systematic relation was apparent.

Diversity

Rarefaction curves (Fig. 4) indicate that the shallower stations are the most diverse in Toulon Canyon. At the same depth, the number of expected species is di!erent in the channel and the #ank stations. Nevertheless, the general tendency is a decrease of the diversity according to the depth. Shannon Wiener indexes, ranging from 4.09 bits in the canyon axis at 250 m to 2.71 bits at 2000 m depth, con"rm this tendency (Table 3). The evenness is very strong, always greater than 0.80 except for station 10, which gives an evenness of 0.68. There is no particular correlation between the variation of evennesses and increasing depth.

Aznity of the assemblages

Clusters obtained from NESS index (m"10) show that clustering depends on depth and location of stations in the channel and on the #ank (Fig. 5). Three groups of stations can be distinguished: (i) stations in the upper part of the channel canyon and the east #ank (CTL1, CTL2, CTL6, CTL7); (ii) channel stations from 1500 to 2000 m depth (CTL4, CTL5); (iii) stations located between 1000 and 1500 m depth on the east and west #anks (CTL3, CTL8, CTL9, CTL10, CTL11). With m"30 (not represented), the clustering remains similar except for the channel station CTL1, CTL2, CTL3 separated from the others stations.

The cluster patterns resulting from Sorensen and Wittaker analysis are similar to those from NESS with m"10, except for station CTL3, completely separated from the other stations for both indexes, and station CTL7, which is grouped with station CTL10 in Wittaker analysis.

For all the similarity indexes, the a$nities between stations are far weaker and are always lower than 0.5.

NESS when used at low &&m'' values is dependent on the dominant species (Graslle and [START_REF] Grassle | A similarity measure sensitive to the contribution of rare species and its use in investigation of variation in marine benthic communities[END_REF]. The weak values obtained with this index and the two others do not result from the presence of rare species at each station but re#ect qualitatively and quantitatively the existence of strongly individualized assemblages related to the spatial distribution of the stations.

Species association

The clusters obtained from the analysis in R mode are presented in Fig. 6. Taking into account associations of species with a$nities most often close to or greater than 50%, nine major groups can be distinguished.

Analysis of the species distribution of each group at the di!erent stations shows that the "rst group, made up of 17 taxa, associates species that are found most of the time in the whole studied area.

Group 2, subdivided into three sub-groups, is made up of 48 species, most of them colonizing the east #ank and the upper slope.

Sub-group 2a ("ve species) corresponds to species collected on the east #ank between 250 and 1500 m depth. Sub-group 2b (19 species) is made up of species colonizing the upper east #ank (CTL6), and sub-group 2c (24 species) is made up of species colonizing the upper slope (canyon channel and east #ank between 250 and 500 m depth).

Group 3 (39 species) corresponds to species that are limited to the upper part of the canyon channel between 250 and 500 m depth (sub-group 3a, 12 species) and at 250 m depth (sub-group 3b, 27 species).

Groups 4, 5, 6, 7, 8 and 9 are made up, respectively, of the association of 9, 5, 6, 7, 7 and 4 taxa corresponding to species more particularly limited to station CTL 7 at 500 m depth on the east #ank for group 4 and group 5, to species limited to deep zones of the canyon channel and west #ank for group 6, to species colonizing the west #ank stations for group 7 (CTL 11) and group 8 (CTL 10) and to species mainly limited to station CTL 3 for group 9.

It should be pointed out that most of the collected species, except characteristic deep species of the bathyal or abyssal zones (Fig. 6), are eurybathic species commonly occurring in the di!erent assemblages of the Mediterranean continental shelf. Fig. 7 shows the dominance within the di!erent groups and subgroups, as de"ned by the analysis in R mode, of eurybathic and deep species. First of all, it can be observed that eurybathic species are dominant in Toulon Canyon compared to deep species. The dominance of these eurybathic species exceeds 75% at all stations except Fig. 6. Dendrogram of slope species measured with Dice's index and species making up the di!erent groups. Group 1: species occurring at every depth of the continental slope; Group 2: species mainly restricted to the east #ank (2a), the upper east #ank (2b) and the upper slope (2c); Group 3: species limited to the upper part of the canyon channel between 250 and 500 m depth (3a) and to 250 m depth (3b); Group 4 and group 5: species mainly limited to station CTL7; Group 6: species mainly limited to the lower slope; Groups 7,8, 9: species, respectively, limited to station CTL11, CTL10 and CTL3. Species in bold are deep species which are characteristic of bathyal deep mud or abyssal assemblages. for station CTL 7 (60.80%) and station CTL 10 (38.52%), presenting a facies of benthic foraminifera, Cyclammina cancellata, that is characteristic of the Mediterranean deep mud assemblage [START_REF] Picard | Recherches qualitatives sur les bioceH noses marines des substrats meubles dragables de la reH gion marseillaise[END_REF]. At the other stations, the dominance of strictly deep species is always less than 25%.

Relation between macrofauna and environmental factors

The factorial correspondence analysis (FCA) was carried out on a contingency table crossing 142 species against 11 stations. The "rst three axes explain 44.90% of the total data variability (axis 1"16.72%, axis 2"14.60%, axis 3"13.58%). The results are given in Fig. 8, where the plotted projection plane is illustrated by the factorial planes I}II and I}III, of all the stations and species presenting the strongest absolute and relative contributions.

The "rst axis separates channel stations located between 250 and 500 m depth and the shallowest station on the east slope from all the other stations. These shallower stations are characterized by species that are not related to particular sediment grain size such as Ophiothrix fragilis and Aponuphis bilineata (wide ecological range species), ¹imoclea ovata and Syllis cornuta (mixticolous species living in substrates presenting a mixing of silt, sand and gravel fractions). On the other hand, species associated with the deeper stations on the negative side of the "rst axis require substrates with strict granulometric compositions such as Capitellides giardi and Onchnesoma steenstrupii (characteristic species of the muddy bottom bathyal assemblage). One notes a signi"cant negative correlation between the ordination of the variables along axis 1 and the variation in the fraction smaller than 63 m at the di!erent stations (r"!0.76, p"0.02). The variables and species position structuring this axis as well as the obtained correlation demonstrate that the factor responsible for the creation of the main axis is in#uenced by the increasing gradient of silting from the upper to the lower part of the slope, resulting from hydrodynamic and bathymetric conditions.

Axis 2 contrasts station CTL3 from the other stations. With regard to the species, Fauveliopsis brevis, ¸evinsenia sp, which are present only at CTL 3, and Chaetozone cf. setosa, contribute strongly to the formation of axis 2. As shown in Table 2, CTL1, CTL2 and CTL3, on the positive side of axis 2, are characterized by high levels of organic carbon compared to the other stations. If station 11 is discarded, we observe a positive correlation (r"0.63, p"0.03) between the ordination of the variables and the organic carbon concentrations measured on this axis. The factor responsible for the formation of axis 2, giving rise to the individualization of the upper stations of the axis and their particular enrichment in organic matter, can only be anthropogenic input in the form of the erratic dumping of spoils between stations CTL1 and CTL2.

Except for stations CTL1 and CTL2, axis 3 contrasts the stations above 1000 m depth (CTL6, CTL7, CTL8, CTL3 and CTL10), located on the negative side of the axis, with the deepest stations below 1000 m depth (CTL9, CTL11, CTL4 and CTL5), on the positive side. Among the species responsible for the creation of axis 3, ¸aonice cirrata and Maldane glebifex, respectively characteristic and preferential species of the terrigenous mud shelf assemblage, occur and are associated with the deeper stations of the channel. This assemblage normally occurs at outlets of rivers at depths ranging from 30 to 110 m in the Mediterranean Sea [START_REF] Picard | Recherches qualitatives sur les bioceH noses marines des substrats meubles dragables de la reH gion marseillaise[END_REF][START_REF] Salen | Contribution a`l'eH tude dynamique des peuplements marins de substrats meubles: les peuplements macrobenthiques circalittoraux soumis a`l'envasement dans la reH gion provencale[END_REF]. The settlement of these species as far down as 2000 m depth is evidence of rapid sedimentation at these depths of "ne particles of continental origin [START_REF] Bourcier | ReH ponse du macrobenthos d'un canyon sous marin meH diterra-neH en a`des apports particulaires telluriques et anthropiques[END_REF]. With regard to the sediment composition, the depth of 1000 m is recognized in the Mediterranean Sea as the boundary separating an upper zone, with dominance of benthic advective contributions transported by the benthic nepheloid layer, and a lower zone, with convective particulate supplies originating from the water column (De Bovee et al., 1990;[START_REF] Buscail | Le cycle du carbone sur une marge continentale: aspect biogeochimiques du transfert de la matie`re organique a`l'interface eau-seH diment[END_REF][START_REF] Bourcier | ReH ponse du macrobenthos d'un canyon sous marin meH diterra-neH en a`des apports particulaires telluriques et anthropiques[END_REF][START_REF] Poydenot | Le canyon de Toulon: Morphologie et sedimentation (MeH diterraneH e Nord -Occidentale)[END_REF]. The presence of circalittoral biocalcareous sand in the isolated fraction of sieved sediment from Toulon Canyon clearly demonstrates the contribution of the continental shelf over 1000 m depth. Deeper still, one observes mainly pelagic tests such as Pteropoda shells. On the other hand, although not signi"cant at level 0.05, a positive correlation (r"0.50, p"0.1) is apparent between the ordination of variables along the third axis and manganese concentrations measured in the surface sediments. The zones, respectively, in#uenced by contributions originating from the continental shelf and contributions from the water column presenting high concentrations of heavy metals such as manganese, can be recognized as the third factor responsible for the composition and the structure of the canyon assemblages.

The position of the upper channel stations (CTL1 and CTL2) on the positive side of axis 3 compared to the other upper #ank stations could correspond to the channel population's response to the arti"cial hypersedimentation induced by the dumping of erratic surface dredging spoils at the canyon head.

Distribution of trophic groups

Fig. 9 shows the percentages of the various trophic groups occurring at the studied stations. At most stations, surface deposit feeders predominate and represent between 36 and 73% of the assemblages. The second most strongly represented trophic group is the subsurface deposit feeders, which account for 9}45% of the populations; this is the major feeding mode at stations CTL3 and CTL11. Carnivores (2}30% of the populations) are well represented in the upper stations between 250 and 500 m depth. Suspension feeders have the lowest percentages and represent between 0 and 8% of the assemblages and do not occur in the lower channel.

Discussion

Depth-related patterns

In spite of the relatively homogeneous conditions of temperature and salinity that prevail in the Mediterranean Sea below 300 m depth, patterns of species replacement with depth are similar to these observed in the Atlantic Ocean by [START_REF] Haedrich | The megabenthic fauna in the deep sea south of New England, USA[END_REF] and [START_REF] Hecker | Photographic evidence for the rapid #ux of particles to the sea #oor and their transport down the continental slope[END_REF] for megafaunal and bentho-pelagic fauna or by [START_REF] Rowe | The deep-sea macrobenthos on the continental margin of the N.W. Atlantic Ocean[END_REF] for the macrofauna. However, [START_REF] Gage | Deep-Sea Biology: A Natural History of Organisms at the Deep-Sea Floor[END_REF] have noted, on the basis of work by various authors in the Atlantic Ocean, a more gradual replacement of assemblages according to depth for macrofauna than for megafauna. In the Mediterranean Sea, the weak a$nity percentages obtained between stations (Fig. 5) are evidence of the sharp change in macrofaunal assemblages according to the depth. Among the species present in the Mediterranean bathyal zone, [START_REF] Reyss | Bionomie benthique de 2 canyons sous marin de la mer Catalane: le rech du Cap et le rech Lacaze-Duthiers[END_REF] and [START_REF] Carpine | Ecologie de l'eH tage bathyal dans la MeH diteH rraneH e occidentale[END_REF] distinguish between eurybathic species with restricted or wide distribution and deep species. Restricted and widely distributed eurybathic species correspond to species found on the continental shelf and slope at depths not exceeding 500 and 2000 m, respectively. Deep species are species strictly limited to the bathyal zone such as characteristic species of the muddy bottom bathyal assemblage or abyssal species that may settle on the continental slope. Most authors who have worked with anchor dredges in the upper part of the Mediterranean slope have stressed the progressive enhancement, with increasing depth, of dominance by characteristic species of the bathyal zone and the decrease of dominance by eurybathic species [START_REF] Picard | Recherches qualitatives sur les bioceH noses marines des substrats meubles dragables de la reH gion marseillaise[END_REF][START_REF] Carpine | Ecologie de l'eH tage bathyal dans la MeH diteH rraneH e occidentale[END_REF][START_REF] Fredj | The deep mediterranean benthos[END_REF][START_REF] Laubier | The mediterranean deep-sea benthic fauna[END_REF]. In Toulon Canyon, eurybathic species predominate from 250 to 2000 m depth, independently of station location (Fig 7).

Table 4 shows the average dominance of deep species present in various bathyal assemblages of the northwestern Mediterranean Sea. Given that the sampling devices, sieve mesh size and sites studied are di!erent, how can we explain the variations in the dominance of these species? With regard to the in#uence of the site, in spite of the low proportion of deep species observed for the Lacaze Duthiers and Toulon Canyon sites, respectively sampled with anchor dredge and corer, the possibility of a canyon e!ect, favoring the settlement of eurybathic species to the detriment of deep species, must be rejected. Cores collected in an area that is beyond the in#uence of the canyon, on the Cassis slope near the Toulon area [START_REF] Stora | Etude de la macrofaune benthique dans le secteur du canyon de Cassidaigne[END_REF], show that dominance of characteristic bathyal species does not exceed 10%. On the other hand, at the di!erent sites studied, regardless of the sampling devices used, the decrease of bathyal species dominance is directly related to the decrease of the sieve mesh size. It would seem that the strong dominance of deep species reported up till now in the Mediterranean bathyal zone is, in fact, more closely linked to the large mesh size of the sieve, which lets small eurybathic specimens pass through that are usually retained on 500 or 250 m mesh size (Chardy et al., 1973), than to any real increase of bathyal populations with depth.

The R mode analysis shows that Toulon Canyon assemblages are formed in part by a group of ubiquitous eurybathic species occurring in the whole set of stations, associated to individualized species groups that dominate in di!erent parts of the canyon. Since the species forming these groups are also for the most part eurybathic, their settlement pattern does not seem to be linked to optimal life conditions according to the depth [START_REF] Hecker | Photographic evidence for the rapid #ux of particles to the sea #oor and their transport down the continental slope[END_REF], but is rather dependent on the in#uence exerted in the di!erent parts of the canyon of variable anthropogenic or natural environmental conditions.

Zonation pattern

&&Determining the causes of the pattern of species replacement with depth remains one of the most di$cult and elusive problems faced by deep sea ecologists'' [START_REF] Gage | Deep-Sea Biology: A Natural History of Organisms at the Deep-Sea Floor[END_REF]. The surveys by [START_REF] Rex | Community structure in the deep-sea benthos[END_REF], Peres (1982), [START_REF] Carney | Zonation of fauna in the deep-sea[END_REF], [START_REF] Gage | Deep-Sea Biology: A Natural History of Organisms at the Deep-Sea Floor[END_REF], [START_REF] Blake | Benthic community structure on the U.S. South Atlantic slope o! the Carolinas: spatial heterogeneity in a current-dominated system[END_REF] and [START_REF] Blake | Dense infaunal assemblages on the continental slope o! Cape Hatteras, North Carolina[END_REF] show that the main factors that in#uence species distribution may be of physical origin (temperature, pressure, particle supplies that control grain size distribution, geochemical characteristics and trophic availability of the sediments and currents favoring larval dispersal) or of biological origin (life-history tactics, interspeci"c competition and predation, trophic strategy).

For the Mediterranean bathyal macrofauna, according to [START_REF] Picard | Recherches qualitatives sur les bioceH noses marines des substrats meubles dragables de la reH gion marseillaise[END_REF], the main factor governing the distribution of species is the texture of the sediment, while for [START_REF] Carpine | Ecologie de l'eH tage bathyal dans la MeH diteH rraneH e occidentale[END_REF] it is the depth and not the nature of the substrate that is signi"cant. REYSS (1970) mentions grain size distribution, depth and the constancy of T/S as the essential factors governing community distribution in the Lacaze Duthiers Canyon. On the other hand, in the eastern Mediteranean sea, food availability is the principal regulating factor according to Tselepides and Eleftheriou (1992).

The factorial correspondence analysis carried out with Toulon Canyon samples shows in planes I}II and I}III a &&V'' pattern for variables. This pattern indicates a Gutman e!ect, which suggests the dominant in#uence of a common factor responsible for the positioning of the descriptor along axes I, II and III [START_REF] Benzecri | L'analyse de donneH es. Tome II: L'analyse de correspondance[END_REF].

The analysis demonstrates that the common factor governing the distribution and structure of populations is related to the nature and the origin of sediments and more particularly their grain size distribution (fraction smaller than 63 m: factor 1), geo-chemical composition (factor 2) and mode of transport and sedimentation (benthic nepheloid or originating from the water column) (factor 3). The role of sediment particle size on the composition and the diversity of deep sea macrobenthic communities and the deep species' sensitivity to small di!erences in grain size has been demonstrated by [START_REF] Etter | Pattern of species diversity in the deep sea as a function of sediment particle size diversity[END_REF]. It is quite evident, however, that the chemical and physical nature of sediments results from the general environmental conditions. The factors identi"ed can have a direct or indirect impact on population distribution. In addition, environmental conditions governing the composition of sediments regulate other factors, notably trophic supply. [START_REF] Rex | Community structure in the deep-sea benthos[END_REF] emphasizes that a multiplicity of physical factors must in part underlie patterns of faunal change, but direct evidence on the relative importance of these factors is entirely lacking. This di$culty may be explained by the absence of particular predominant factors, as we can see from the results obtained in the Toulon Canyon. The very close inertia percentages obtained for the three "rst axes emphasize the absence of hierarchy among the factors in#uencing the zonation pattern of populations.

The ewect of environmental factors on community structure

Species richness, number of individuals, diversity and evenness

Surveys on this topic (see [START_REF] Rex | Community structure in the deep-sea benthos[END_REF][START_REF] Gage | Deep-Sea Biology: A Natural History of Organisms at the Deep-Sea Floor[END_REF] show that the species richness pattern along a depth gradient follows a parabolic curve. In the Atlantic Ocean, on the continental slope o! Cape Hatteras between 500 and 2000 m depth, species richness tends to increase with increasing depth [START_REF] Blake | Dense infaunal assemblages on the continental slope o! Cape Hatteras, North Carolina[END_REF]. Table 5 shows the main parameters of macrobenthic communities recorded in and in the neighbourhood of the Hudson canyon by [START_REF] Rowe | The deep-sea macrobenthos on the continental margin of the N.W. Atlantic Ocean[END_REF] and in the Toulon Canyon. While the total number of species between the two sites cannot be compared due to the di!erence in the surface area sampled, it is nonetheless apparent that in the Mediterranean Sea species richness on the slope decreases regularly according to depth, while it tends to increase in the Hudson canyon. This decrease in species richness gives rise to a similar decrease in the diversity index according to depth in the Mediterranean Sea, while diversity increases in the Hudson canyon. While the evenness is very similar in Atlantic and Mediterranean Sea from 1500 m depth, in the Mediterranean it is particularly high on the upper slope, demonstrating that most of the species composing the communities occurring between 250 and 2000 m are represented by a very similar number of individuals. It is well known that Mediterranean population density and biomass are particularly low compared to those of the world's oceans (Peres, 1982). The biomass expressed in dry weight (Table 3) cannot be directly compared at the two sites in the same way as the density; however, the decline of these parameters according to depth observed in Toulon Canyon is consistent with the general tendency widely reported in the literature.

Canyon funneling ewects

Canyons induce particular environmental conditions giving rise to the settlement of di!erent macrofaunal and megafaunal populations in the channel and on adjacent slopes [START_REF] Rowe | Observations on bottom currents and epibenthic populations in Hatteras submarine canyon[END_REF][START_REF] Haedrich | Zonation and faunal composition of epibenthic populations on the continental slope south of New England[END_REF][START_REF] Haedrich | The megabenthic fauna in the deep sea south of New England, USA[END_REF][START_REF] Rowe | The deep-sea macrobenthos on the continental margin of the N.W. Atlantic Ocean[END_REF]Peres, 1982;[START_REF] Gage | Deep-Sea Biology: A Natural History of Organisms at the Deep-Sea Floor[END_REF][START_REF] Bourcier | ReH ponse du macrobenthos d'un canyon sous marin meH diterra-neH en a`des apports particulaires telluriques et anthropiques[END_REF]. The channel of Toulon Canyon is characterized, especially between 250 and 1000 m depth, by marked sediment enrichment in organic matter (Table 2). Similar enrichment observed in the upper part of the Hudson Canyon is attributed by [START_REF] Keller | Bottom currents in the Hudson canyon[END_REF] to a shelfward source. In Toulon Canyon, as previously mentioned, the association of strong concentrations in heavy metals and high rates of organic matter shows that the observed enrichment is of anthropogenic origin and is largely the result of dredge spoil dumping at the canyon head.

From the qualitative point of view, the presence of these spoils and deposit channelling constitute both favorable conditions for groups of species exclusively occurring in the channel and, conversely, unfavorable conditions for groups restricted to the #anks. In the canyon axis, the occurrence may be noted of the polychaetes Capitella capitata, Chone ,licaudata, Platynereis dumerili, ¸eiocapitella dollfusi, Pectinaria auricoma, the crustaceans Cirolana neglecta, ¸eptocheirus bispinosus, Pagurus prideauxi, the Mollusc Myrthea spinifera and the echinodermata Ophiothrix fragilis which are eurybathic species proliferating in the Toulon harbours [START_REF] Bourcier | ReH partition des bioceH noses benthiques en fonction des substrats seH dimentaires de la rade de Toulon (France)[END_REF]. These species can "nd in the deposit area conditions similar to those of the Toulon harbours, allowing the larvae settlement. Such observations have been made by [START_REF] Blake | Dense infaunal assemblages on the continental slope o! Cape Hatteras, North Carolina[END_REF] on the continental slope o! Cape Hatteras. In this area, the topography and the circulation favor the sedimentation of inner shelf and estuarine sediments on the slope involving a large dominance of shelf species in a bathyal environment. Observations made at Toulon Canyon channel support the [START_REF] Blake | Dense infaunal assemblages on the continental slope o! Cape Hatteras, North Carolina[END_REF] hypotheses about the recolonization of near shore dredge material or other type of sediment dumped onto a continental slope by non slope faunal assemblage.

It is also conceivable at Toulon Canyon that some species, such as molluscs protected by their shells, may have been deposited directly with dredging spoil and be found in survival condition. Moreover, the survival capability of circalittoral species could explain the presence of Idotea balthica, sampled at 1000 m at station CTL 3. This species normally proliferates in the littoral zone in plant debris. During planktonic trawls in the canyon area, one of the present authors (J. Le Campion) has collected fragments of Posidonia leaves with Idotea clinging to them in plankton nets. If dredge spoils and channelled deposits favor the presence of some coastal species in the bathyal zone, in contrast upwellings induced by winds can cause the dispersion and settlement of abyssal species larvae. The polychaete Fauvielopsis brevis, collected at 1000 m depth at the canyon axis, has so far never been reported at depths shallower than 2000 m in the Mediterranean Sea [START_REF] Laubier | The mediterranean deep-sea benthic fauna[END_REF].

From the quantitative point of view, conditions prevailing at the canyon channel, in particular, the sediment enrichment in organic matter, do not entail any particular di!erence at each depth in comparison with the canyon #anks in the species richness, density and biomass of the resident assemblages. It is of course evident that this does not automatically lead to the conclusion that high rates of organic matter have no in#uence on these assemblage parameters. Either organic matter is composed of refractory elements that are not usable by the populations [START_REF] Kripouno | Les #ux organiques particulaires et ses transformations a`l'interface eau-seH diment[END_REF] and therefore has no impact, or the heavy input of organic matter associated with pollutant discharge results in impoverishment of populations in the upper and middle part of the canyon channel [START_REF] Pearson | Macrobenthic succession in relation to organic enrichment and pollution of the marine environment[END_REF], which in the absence of dredge spoils might have higher densities. Although the di!erences are not statistically signi"cant, this last hypothesis is especially supported by the low average densities at station CTL2 of the channel in comparison with the other stations, and conversely by the higher densities at stations CTL4 and CTL5 that are not directly exposed to the e!ects of the dredge spoils.

Trophic group distribution

The distribution of the major trophic groups of bottom invertebrates depends on the availability of their speci"c food resources, which in turn depends on both the bottom pro"le and the degree of eutrophy (Peres, 1982). From observations of the epibenthic fauna in the Atlantic Ocean, it appears that generally carnivore/scavengers and suspension feeders predominate on the upper slope, suspension feeders alone predominate in mid-slope and deposit feeders on the lower slope [START_REF] Gage | Deep-Sea Biology: A Natural History of Organisms at the Deep-Sea Floor[END_REF]. At the shelf-slope break, macroinfauna assemblages are dominated by suspension feeders on the shelf side of the shelf break and by deposit feeders on the slope [START_REF] Blake | Infaunal-sediment relationships at the shelf -slope break[END_REF]. In Toulon Canyon, independently of the depth and location of the stations, depositivore species are largely dominant. Below 500 m depth, suspension feeders disappear in the canyon axis, while low percentages of this trophic group remain on the slopes. Similarly, while carnivores are well represented in stations located at 250 m in the canyon channel and on the east #ank, a higher percentage of individuals are found in the channel between 500 and 1000 m depth than on the slopes. The trophic structure of the assemblages found in this part of the canyon channel, marked by the disappearance of the "lter-feeding species and the maintenance of carnivores, is characteristic of perturbed areas exposed to strong organic inputs [START_REF] Pearson | Macrobenthic succession in relation to organic enrichment and pollution of the marine environment[END_REF]. This trophic structure can only re#ect the impact of dredge spoils at the canyon head. On the other hand, the weak dominance of suspension feeders on the upper and median parts of the slope may be related to the low tidal range of the Mediterranean Sea, with the consequent absence of strong, regular currents at the shelf break.

While at most of the stations surface deposit feeders are more numerous than subsurface deposit feeders, the contrary is apparent for the assemblages of stations CTL3 and CTL11, respectively located at 1000 m depth in the canyon channel and at 1500 m depth on the west #ank, where subsurface deposit feeders predominate. This pattern of distribution has been described in coastal environments [START_REF] Rice | Early diagenesis in bioadvective sediments: relationships between the diagenesis of beryllium-7, sediment reworking rates, and the abundance of conveyor-belt deposit-feeders[END_REF][START_REF] Craig | The e!ects of varying population density on feeding and irrigation rates of Clymenella torquata[END_REF]), but has rarely been observed on the continental slope. It would seem that at station CTL3, this predominance of subsurface deposit feeders, mentioned for samples collected at di!erent periods by Gerino et al. (1994), persists. The rapid burying of the organic matter as a result of sedimentary slipping (presence of turbidites) or the biological activity of the megafaunal populations (in particular, burrows of Calocaris macandreae) have been suggested as possible explanations of the development of this feeding strategy at station CTL3. It is likely that the same phenomena in#uence the percentage of subsurface deposit feeders at station CTL11, this below-surface feeding mode being the best strategy for exploiting organic matter resources buried in the sediment column.

Conclusions

We are aware that Toulon area, with regard to the Mediterranean Sea as a whole, presents an example of one particular form of continental margin, characterised by a narrow continental shelf and a steep slope. Further studies will have to be undertaken on other types of continental margin presenting di!erent characteristics in order to identify more fully the assemblages of the Mediterranean continental slope. Nevertheless, whether it is a matter here of a speci"c case or a general pattern, these assemblages present di!erent characteristics from those described in previous studies.

The results show especially that Mediterranean bathyal populations are composed mainly of continental shelf eurybathic species. If characteristic species of the bathyal zone are present, their increasing predominance in the assemblages of the slope as a function of depth has not been con"rmed. A marked variability is apparent in the composition of the assemblages according to depth. Analyses have provided some evidence on the factors responsible for the distribution of the populations. With regard to population composition as a function of depth, there is not a single major factor but rather a group of factors acting together, including the sediment grain size, organic matter and pollutant content and mode of sediment transport (benthic nepheloid or by the water column).

The absence of tidal currents in#uences the trophic structure of the assemblages, which are characterized by a dominance of surface and subsurface deposit feeders compared to suspension feeders and carnivores at the upper and median part of the slope in the Atlantic. Dredge spoil inputs also have an impact on the assemblages of the canyon channel. This impact is marked in the upper part of the canyon. However, it should be pointed out that these inputs do not result in marked degradation of the assemblages as a function of the levels of pollutant and organic matter present, which could indicate a particular resistance or resilience of the bathyal assemblages. Further studies carried out in other perturbed Mediterranean bathyal sites are required to con"rm or invalidate this hypothesis.

Fig. 1 .

 1 Fig. 1. Locations of the stations in the Toulon Canyon.

Fig. 2 .

 2 Fig. 2. Species recruitment curves.

Fig. 3 .

 3 Fig. 3. Density and dry-weight biomass per m vs. depth.

Fig. 4 .

 4 Fig. 4. Hurlbert rarefaction curves for 11 stations sampled at Toulon Canyon. Number of individuals limited to 100.

Fig. 5 .

 5 Fig. 5. Dendrogram of Toulon samples clustered with NESS at 10 individuals (A), Sorensen's index (B), Wittaker's index (C) and group average sorting.

Fig. 7 .

 7 Fig. 7. Dominance at the sample stations of eurybathic and deep species of the di!erent groups de"ned by analysis in mode R (Total slope: group 1. Upper slope: group 2c. Lower slope: group 6. Upper channel: group 3. Middle Channel: group 9. East #ank: groups 2a, 2b, 4 and 5. West #ank: groups 7 and 8).

Fig. 8 .

 8 Fig. 8. Results of the correspondence analysis in factorial planes I}II and I}III.

Fig. 9 .

 9 Fig. 9. Percentages of trophic groups at the di!erent stations of Toulon Canyon (C: carnivores; S: suspensivores; SSDF: subsurface deposit feeders; SDF: surface deposit feeders; U: undetermined).

Table 3

 3 Infauna } assemblage parameters in the Toulon canyon stations

	West #ank	CTL 9 CTL 10 CTL 11	1500 1570 1000	22 24 26	181.3 237.3 397.3	157.1 85.91 161.2	0.01 0.11 0.60	0.01 0.19 0.50	2.80 2.63 2.25	0.91 0.83 0.68	8.0 7.0 5.7	13.5 12.0 9.2	17.8 15.0 12.0	18.2
	East #ank	CTL 8 CTL 7	1000 500	21 51	202.7 552	137.1 338.5	0.16 0.54	.24 0.45	2.74 3.24	0.90 0.82	7.72 7.8	12.59 13.4	16.32 18.2	29.27
		CTL 6	250	93	1272	477.9	0.70	0.41	3.88	0.86	8.8	15.6	21.3	34.5	47.2	68.9
		CTL 5	2000	21	210.7	144.2	0.05	0.02	2.71	0.92	7.7	12.3	15.5	18.1
		CTL 4	1500	23	306.7	220.1	0.26	0.17	2.52	0.80	6.8	10.9	14.1	20.6
	Channel	CTL 3	1100	25	149.3	80.66	0.01	0.01	2.95	0.92	8.4	14.2	18.9
		CTL 2	500	36	176	123.9	0.32	0.33	3.37	0.94	9.0	16.2	22.5
		CTL 1	250	98	1000	157.44	2.14	1.57	4.09	0.89	9.0	16.5	23.0	38.6	54.4	81.9
			Depth (m)	No. species	X specimens m\	($CL)	Biomass g m\	($CL)	H	J	Species per 10 individuals	Species per 20 individuals	Species per 30 individuals	Species per 60 individuals	Species per 100 individuals	Species per 200 individuals

Table 4

 4 Dominance of characteristic species of deep mud bathyal assemblages at various Mediterranean sites

					Rech Lacaze
		Marseille	Monaco	Canal de Corse Duthiers Reyss	Toulon
	Sites	Picard 1965	Carpine 1970	Carpine 1970	1970	present study
	Samplers	Anchor dredge Anchor dredge Anchor dredge Anchor dredge	Box corer
	Sieve mesh size	1.6 mm	2 mm	2 mm	1 mm	0.5 mm
	% deep sp	63.31	60.77	67.36	34.00	16.04

Table 5

 5 Comparison of infauna assemblage parameters of the Atlantic slope (Hudson Canyon,[START_REF] Rowe | The deep-sea macrobenthos on the continental margin of the N.W. Atlantic Ocean[END_REF] and the Mediterranean slope(Toulon Canyon)

			Hudson Canyon		Toulon Canyon
	Depth (m)	203}570 1141}1437 1707}1815 2351}2673 250}500 1000}1500 2000
	Total area of
	samples (m)	0.26	0.11	0.26	0.28	1.5	2.24	0.37
	No. of collections	12	5	12	8	24	36	6
	No. Species	118	70	105	128	188	60	22
	X specimens m\	9280	4760	1880	1790	750	246	211
	H	3.35	3.28	3.99	4.28	4.47	3.63	2.71
	J	0.7	0.77	0.86	0.88	0.85	0.81	0.92
	Biomass g m\	46.2	10.4	9.2	8.2	0.93	0.19	0.05
	Biomass is respectively wet weight biomass for the Atlantic and dry weight biomass without shell for the
	Mediterranean Sea.

Acknowledgements

We thank the captain and crew of the R.V. Prof. Georges Petit for all their cooperation. This research was supported by the ECOMARGE operation.