Marc Arnaudon
email: arnaudon@math.univ-poitiers.fr

AND Anton Thalmaier
email: anton.thalmaier@uni.lu

Feng-Yu Wang
email: wangfy@bnu.edu.cn

Wimcs

GRADIENT ESTIMATE AND HARNACK INEQUALITY ON NON-COMPACT RIEMANNIAN MANIFOLDS

Keywords: 1991 Mathematics Subject Classification. 58J65 58J35 60H30 Harnack inequality, heat equation, gradient estimate, diffusion semigroup

published or not. The documents may come

Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds

The main result

Let M be a non-compact complete Riemannian manifold, and P t be the Dirichlet diffusion semigroup generated by L = ∆ + ∇V for some C 2 function V . We intend to establish reasonable gradient estimates and Harnack type inequalities for P t . In case that Ric -Hess V is bounded below, a dimension-free Harnack inequality was established in [START_REF] Wang | Logarithmic Sobolev inequalities on noncompact Riemannian manifolds[END_REF], which according to [START_REF] Wang | Equivalence of dimension-free Harnack inequality and curvature condition[END_REF], is indeed equivalent to the corresponding curvature condition. See e.g. [START_REF] Bakry | On Sobolev and logarithmic Sobolev inequalities for Markov semigroups[END_REF] for equivalent statements on heat kernel functional inequalities; see also [START_REF] Li | On the parabolic kernel of the Schrödinger operator[END_REF][START_REF] Bakry | Harnack inequalities on a manifold with positive or negative Ricci curvature[END_REF][START_REF] Li | Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds[END_REF] for a parabolic Harnack inequality using the dimensioncurvature condition by shifting time, which goes back to the classical local parabolic Harnack inequality of Moser [START_REF] Moser | A Harnack inequality for parabolic differential equations[END_REF].

Recently, some sharp gradient estimates have been derived in [START_REF] Souplet | Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds[END_REF][START_REF] Zhang | Some gradient estimates for the heat equation on domains and for an equation by Perelman[END_REF] for the Dirichlet semigroup on relatively compact domains. More precisely, for V = 0 and a relatively compact open C 2 domain D, the Dirichlet heat semigroup P D t satisfies

(1.1)

|∇P D t f |(x) ≤ C(x, t) P D t f (x), x ∈ D, t > 0,
for some locally bounded function C : D × ]0, ∞[ → ]0, ∞[ and all f ∈ B + b , the space of bounded non-negative measurable functions on M . Obviously, this implies the Harnack inequality (1.2) P D t f (x) ≤ C(x, y, t) P D t f (y), t > 0, x, y ∈ D, f ∈ B + b , for some function C : M 2 ×]0, ∞[ → ]0, ∞[. The purpose of this paper is to establish inequalities analogous to (1.1) and (1.2) globally on the whole manifold M . On the other hand however, both (1.1) and (1.2) are in general wrong for P t in place of P D t . A simple counter-example is already the standard heat semigroup on R d . Hence, we turn to search for the following slightly weaker version of gradient estimate:

|∇P t f (x)| ≤ δ P t f log f -P t f log P t f (x) + C(δ, x) t ∧ 1 P t f (x), (1.3) x ∈ M, t > 0, δ > 0, f ∈ B + b , for some positive function C : ]0, ∞[ × M → ]0, ∞[. This kind of gradient estimate is new and, in particular, implies the Harnack inequality with power introduced in [START_REF] Wang | Logarithmic Sobolev inequalities on noncompact Riemannian manifolds[END_REF] (see Theorem 1.2 below).

Theorem 1.1. There exists a continuous positive function F on ]0, 1] × M such that

|∇P t f (x)| ≤ δ P t f log f -P t f log P t f (x) + F (δ ∧ 1, x) 1 δ(t ∧ 1)
+ 1 + 2δ e P t f (x), (1.4) δ > 0, x ∈ M, t > 0, f ∈ B + b . Theorem 1.2. There exists a positive function C ∈ C(]1, ∞[×M 2 ) such that (P t f (x)) α ≤ (P t f α (y)) exp 2(α -1) e + αC(α, x, y) αρ 2 (x, y) (α -1)(t ∧ 1) + ρ(x, y) , α > 1, t > 0, x, y ∈ M, f ∈ B + b , where ρ is the Riemannian distance on M . Consequently, for any δ > 2 there exists a positive function

C δ ∈ C([0, ∞[ × M
) such that the transition density p t (x, y) of P t with respect to µ(dx) := e V (x) dx, where dx is the volume measure, satisfies

p t (x, y) ≤ exp -ρ(x, y) 2 /(2δt) + C δ (t, x) + C δ (t, y) µ(B(x, √ 2t))µ(B(y, √ 2t)) , x, y ∈ M, t ∈ ]0, 1[ . Remark 1.1.
According to the Varadhan asymptotic formula for short time behavior, one has lim t→0 4t log p t (x, y) = -ρ(x, y) 2 , x = y. Hence, the above heat kernel upper bound is sharp for short time.

The paper is organized as follows: In Section 2 we provide a formula expressing P t in terms of P D t and the joint distribution of (τ, X τ ), where X t is the L-diffusion process and τ its hitting time to ∂D. Some necessary lemmas and technical results are collected. Proposition 2.5 is a refinement of a result in [START_REF] Zhang | Some gradient estimates for the heat equation on domains and for an equation by Perelman[END_REF] to make the coefficient of ρ(x, y)/t sharp and explicit. In Section 3 we use parallel coupling of diffusions together with Girsanov transformation to obtain a gradient estimate for Dirichlet heat semigroup. Finally, complete proofs of Theorems 1.1 and 1.2 are presented in Section 4.

To prove the indicated theorems, besides stochastic arguments, we make use of a local gradient estimate obtained in [START_REF] Souplet | Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds[END_REF] for V = 0. For the convenience of the reader, we include a brief proof for the case with drift in the Appendix.

Some Preparations

Let X s (x) be an L-diffusion process with starting point x and explosion time ξ(x). For any open C 2 domain D ⊂ M such that x ∈ D, let τ (x) be the first hitting time of X s (x) at the boundary ∂D. We have

P t f (x) = E f (X t (x)) 1 {t<ξ(x)} , P D t f (x) = E f (X t (x)) 1 {t<τ (x)} .
Let p D t (x, y) be the transition density of P D t with respect to µ. We first provide a formula for the density h x (t, z) of (τ (x), X τ (x) (x)) with respect to dt ⊗ ν(dz), where ν is the measure on ∂D induced by µ(dy) := e V (y) dy.

Lemma 2.1. Let K(z, x) be the Poisson kernel in D with respect to ν. Then

(2.1) h x (t, z) = D -∂ t p D t (x, y) K(z, y) µ(dy).
Consequently, the density s → ℓ x (s) of τ (x) satisfies the equation:

(2.2) ℓ x (s) = D -∂ t p D t (x, y) µ(dy).
Proof. Every bounded continuous function f : ∂D → R extends continuously to a function h on D which is harmonic in D and represented by

h(x) = ∂D K(z, x)f (z) ν(dz). Recall that z → K(z, x) is the density of X τ (x) (x). Hence E[f (X τ (x) (x))] = h(x) = ∂D K(z, x)f (z) ν(dz).
On the other hand, the identity

h(x) = E[h(X t∧τ (x) )] yields h(x) = D p D t (x, y)h(y) µ(dy) + ∂D ν(dz) t 0 h x (s, z)f (z)ds = D p D t (x, y) ∂D K(z, y)f (z)ν(dz) µ(dy) + ∂D ν(dz) t 0 h x (s, z)f (z)ds = ∂D f (z) D p D t (x, y)K(z, y) µ(dy) + t 0 h x (s, z)ds ν(dz),
which implies that

(2.3) K(z, x) = D p D t (x, y)K(z, y) µ(dy) + t 0 h x (s, z)ds.
Differentiating with respect to t gives (2.4)

h x (t, z) = -∂ t D p D t (x, y)K(z, y) µ(dy). Since ∂ t p D t (x, y) is bounded on [ε, ε -1 ] × D × D for any ε ∈ ]0,
P t f (x) = P D t f (x) + ]0,t]×∂D P t-s f (z)h x (s, z) dsν(dz) = P D t f (x) + ]0,t]×∂D P t-s f (z)P D s/2 h . (s/2, z)(x) dsν(dz).
Proof. By the strong Markov property we have

P t f (x) = E f (X t (x))1 {t<ξ(x)} = E f (X t (x))1 {t<τ (x)} + E f (X t (x))1 {τ (x)<t<ξ(x)} = P D t f (x) + E E f (X t (x))1 {τ (x)<t<ξ(x)} |(τ (x), X τ (x) (x)) = P D t f (x) +
]0,t]×∂D P t-s f (z)h x (s, z) ds ν(dz).

(2.5)

Next, since

∂ s p D s (x, y) = Lp D s (•, y)(x) = LP D s/2 p D s/2 (•, y)(x) = P D s/2 (Lp D s/2 (•, y))(x) = P D s/2 (∂ u p D u (•, y)| u=s/2 )(x), it follows from (2.1) that (2.6) h x (s, z) = P D s/2 h . (s/2, z)(x)
. This completes the proof.

We remark that formula (2.6) can also be derived from the strong Markov property without invoking Eq. (2.1). Indeed, for any u < s and any measurable set A ⊂ ∂D, the strong Markov property implies that

P τ (x) > s, X τ (x) (x) ∈ A = E 1 {u<τ (x)} P τ (x) > s, X τ (x) (x) ∈ A|F u = D p D u (x, y) P τ (y) > s -u, X τ (y) (y) ∈ A µ(dy),
and thus,

h x (s, z) = P D u h . (s -u, z)(x), s > u > 0, x ∈ D, z ∈ ∂D. Lemma 2.3.
Let D be a relatively compact open domain and ρ ∂D be the Riemannian distance to the boundary ∂D. Then there exists a constant C > 0 depending on D such that P{τ (x) ≤ t} ≤ Ce -ρ 2 ∂D (x)/16t , x ∈ D, t > 0.

Proof. For x ∈ D, let R := ρ ∂D (x) and ρ x the Riemannian distance function to x. Since D is relatively compact, there exists a constant c > 0 such that Lρ 2 x ≤ c holds on D outside the cut-locus of x. Let γ t := ρ x (X t (x)), t ≥ 0. By Itô's formula, according to Kendall [START_REF] Kendall | The radial part of Brownian motion on a manifold: a semimartingale property[END_REF], there exists a one-dimensional Brownian motion b t such that dγ 2 t ≤ 2 √ 2γ t db t + c dt, t ≤ τ (x). Thus, for fixed t > 0 and δ > 0,

Z s := exp δ t γ 2 s - δ t cs -4 δ 2 t 2 s 0 γ 2 u du , s ≤ τ (x)
is a supermartingale. Therefore,

P{τ (x) ≤ t} = P max s∈[0,t] γ s∧τ (x) ≥ R ≤ P max s∈[0,t] Z s∧τ (x) ≥ e δR 2 /t-δc-4δ 2 R 2 /t ≤ exp cδ - 1 t (δR 2 -4δ 2 R 2 ) .
The proof is completed by taking δ := 1/8.

Lemma 2.4. On a measurable space (E, F , μ) satisfying μ(E) < ∞, let f ∈ L 1 (μ) be non-negative with μ(f ) > 0. Then for every measurable function ψ such that ψf ∈ L 1 (μ), there holds:

(2.7)

E ψf dμ ≤ E f log f μ(f ) dμ + μ(f ) log E e ψ dμ.
Proof. This is a direct consequence of [START_REF] Stroock | An Introduction to the Analysis of Paths on a Riemannian Manifold[END_REF] Lemma 6.45. We give a proof for completeness. Multiplying f by a positive constant, we can assume that μ(f ) = 1.

If

E e ψ dμ = ∞, then (2.7) is clearly satisfied.
If E e ψ dμ < ∞, then since E e ψ dμ ≥ {f >0} e ψ dμ, we can assume that f > 0 everywhere. Now from the fact that e ψ 1 f ∈ L 1 (f μ), we can apply Jensen's inequality to obtain log

E e ψ dμ = log E e ψ 1 f f dμ ≥ E log e ψ 1 f f dμ (note the right-hand-side belongs to R ∪ {-∞}). To finish we remark that since ψf ∈ L 1 (μ), E log e ψ 1 f f dμ = E ψf dμ - E f log f dμ.
Finally, in order to obtain precise gradient estimate of the type (1.4), where the constant in front of ρ(x, y)/t is explicit and sharp, we establish the following revision of [19, Theorem 2.1].

Proposition 2.5. Let D be a relatively compact open C 2 domain in M and K a compact subset of D. For any ε > 0, there exists a constant C(ε) > 0 such that

|∇ log p D t (•, y)(x)| ≤ C(ε) log(1 + t -1 ) √ t + (1 + ε)ρ(x, y) 2t , t ∈ ]0, 1[, x ∈ K, y ∈ D. (2.8)
In addition, if D is convex, the above estimate holds for ε = 0 and some constant C(0) > 0.

Proof. Since δ := min K ρ ∂D > 0, it suffices to deal with the case where 0 < t ≤ 1∧δ. To this end, we combine the argument in [START_REF] Zhang | Some gradient estimates for the heat equation on domains and for an equation by Perelman[END_REF] with relevant results from [START_REF] Wang | Estimates of Dirichlet heat kernels[END_REF][START_REF] Wang | Estimates of the first Neumann eigenvalue and the log-Sobolev constant on non-convex manifolds[END_REF].

(a) Let t 0 = t/2 and y ∈ D be fixed. Take

f (x, s) = p D s+t0 (x, y), x ∈ D, s > 0. Applying Theorem 5.1 of the Appendix to the cube Q := B(x, ρ ∂D (x)) × [s -ρ ∂D (x) 2 /2, s] ⊂ D × [-t 0 , t 0 ], s ≤ t 0 , we obtain (2.9) |∇ log f (x, s)| ≤ c 0 ρ ∂D (x) 1 + log A f (x, s) , s ≤ t 0 ,
where A := sup Q f and c 0 > 0 is a constant depending on the dimension and curvature on D. By [9, Theorem 5.2],

(2.10)

A ≤ c 1 f x, s + ρ ∂D (x) 2 , s ∈ ]0, 1], x ∈ D,
holds for some constant c 1 > 0 depending on D and L. Moreover, by the boundary Harnack inequality of [START_REF] Fabes | A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations[END_REF] (which treats Z = 0 but generalizes easily to non-zero C 1 drift Z),

(2.11) f x, s + ρ ∂D (x) 2 ≤ c 2 f (x, s), s ∈ ]0, 1], x ∈ D,
for some constant c 2 > 0 depending on D and L. Combining (2.9), (2.10) and (2.11), there exists a constant c > 0 depending on D and L such that (2.12)

|∇ log f (x, s)| ≤ c √ s , x ∈ D, s ∈ ]0, t 0 ] with ρ ∂D (x) 2 ≤ s. (b) Let Ω = (x, s) : x ∈ D, s ∈ [0, t 0 ], ρ ∂D (x) 2 ≥ s and B = sup Ω f . Since ∂ s f = Lf , for any constant b ≥ 1, we have (L -∂ s ) f log bB f = - |∇f | 2 f .
Next, again by ∂ s f = Lf and the Bochner-Weizenböck formula,

(L -∂ s ) |∇f | 2 f ≥ -2k |∇f | 2 f ,
where k ≥ 0 is such that Ric -∇Z ≥ -k on D. Then the function

h := s|∇f | 2 (1 + 2ks)f -f log bB f satisfies (2.13) (L -∂ s )h ≥ 0 on D × ]0, ∞[.
Obviously h(•, 0) ≤ 0, and (2.12) yields h(x, s) ≤ 0 for s = ρ ∂D (x) 2 provided the constant b is large enough. Then the maximum principle and inequality (2.13) imply h ≤ 0 on Ω. Thus,

(2.14) |∇ log f (x, s)| 2 ≤ (2k + s -1 ) log bB f , (x, s) ∈ Ω. (c) If D is convex, by [16, Theorem 2.1] with δ = √ t and t = 2t 0 , we obtain (note the generator therein is 1 2 L) f (x, t 0 ) = p D 2t0 (x, y) = p D 2t0 (y, x) ≥ c 1 ϕ(y) t -d/2 0 e -ρ(x,y) 2 /8t0 , x ∈ K, y ∈ D
for some constant c 1 > 0, where ϕ > 0 is the first Dirichlet eigenfunction of L on D.

On the other hand, the intrinsic ultracontractivity for P D t implies (see e.g. [START_REF] Ouhabaz | Sharp estimates for intrinsic ultracontractivity on C 1,αdomains[END_REF])

f (z, s) = p D s+t0 (z, y) ≤ c 2 ϕ(y) t -(d+2)/2 0 , z, y ∈ D, s ≤ t 0 ,
for some constant c 2 > 0 depending on D, K and L. Combining these estimates we obtain

B f (x, s) ≤ c 3 t -1 0 e ρ(x,y) 2 /8t0 , x ∈ K, s ≤ t 0 ,
for some constant c 3 > 0 depending on D, K and L. Hence by (2.14) for s = t 0 we get the existence of a constant C > 0 such that

|∇ log p D 2t0 (•, y)| 2 ≤ (t -1 0 + 2k) C + log t -1 0 + ρ(x, y) 2 8t 0
for all y ∈ D, x ∈ K and t 0 ∈ ]0, 1[ with t 0 ≤ ρ ∂D (x) 2 . This completes the proof by noting that t = 2t 0 .

(d) Finally, if D is not convex, then there exists a constant σ > 0 such that

∇ N X, X ≥ -σ|X| 2 , X ∈ T ∂D,
where N is the outward unit normal vector field of ∂D. [START_REF] Wang | Estimates of the first Neumann eigenvalue and the log-Sobolev constant on non-convex manifolds[END_REF], ∂D is convex under the metric g := f -2 •, • . Let ∆, ∇ and ρ be respectively the Laplacian, the gradient and the Riemannian distance induced by g. By Lemma 2.2 in [START_REF] Wang | Estimates of the first Neumann eigenvalue and the log-Sobolev constant on non-convex manifolds[END_REF],

Let f ∈ C ∞ ( D) such that f = 1 for ρ ∂D ≥ ε, 1 ≤ f ≤ e 2εσ for ρ ∂D ≤ ε, and N log f | ∂D ≥ σ. By Lemma 2.1 in
L := ∆ + ∇V = f -2 ∆ + (d -2)f ∇f + ∇V.
Since D is convex under g, as explained in the first paragraph in Section 2 of [START_REF] Wang | Estimates of the first Neumann eigenvalue and the log-Sobolev constant on non-convex manifolds[END_REF],

g( ∇ρ(y, •), ∇ϕ)| ∂D < 0, so that σ(y) := sup D g( ∇ρ(y, •), ∇ϕ) < ∞, y ∈ D.
Hence, repeating the proof of Theorem 2.1 in [START_REF] Wang | Estimates of Dirichlet heat kernels[END_REF], but using ρ and ∇ in place of ρ and ∇ respectively, and taking into account that f → 1 uniformly as ε → 0, we obtain

p D 2t0 (x, y) ≥ C 1 (ε)ϕ(y)t -d/2 0 e -C2(ε) ρ(x,y) 2 /8t0 ≥ C 1 (ε)ϕ(y)t -d/2 0 e -C2(ε)C3(ε)ρ(x,y) 2 /8t0
for some constants

C 1 (ε), C 2 (ε), C 3 (ε) > 1 with C 2 (ε), C 3 (ε) → 1 as ε → 0.
Hence the proof is completed.

3. Gradient estimate for Dirichlet heat semigroup using coupling of diffusion processes Proposition 3.1. Let D be a relatively compact C 2 domain in M . For every compact subset K of D, there exists a constant C = C(K, D) > 0 such that for all δ > 0, t > 0, x 0 ∈ K and for all bounded positive functions f on M ,

|∇P D t f (x 0 )| ≤ δP D t f log f P D t f (x 0 ) (x 0 ) + C 1 δ(t ∧ 1) + 1 P D t f (x 0 ). (3.1)
Proof. We assume that t ∈]0, 1[, the other case will be treated at the very end of the proof.

We write ∇V = Z so that L = ∆+Z. Since P D t only depends on the Riemannian metric and the vector field Z on the domain D, by modifying the metric and Z outside of D we may assume that Ric -∇Z is bounded below (see e.g. [START_REF] Thalmaier | Gradient estimates for harmonic functions on regular domains in Riemannian manifolds[END_REF]); that is,

(3.2) Ric -∇Z ≥ -κ
for some constant κ ≥ 0.

Fix x 0 ∈ K. Let f be a positive bounded function on M and X s a diffusion with generator L, starting at x 0 . For fixed t ≤ 1, let

v = ∇P D t f (x 0 ) |∇P D t f (x 0 )| and denote by u → ϕ(u) the geodesics in M satisfying φ(0) = v. Then d du u=0 P D t f (ϕ(u)) = ∇P D t f (x 0 ) .
To formulate the coupling used in [START_REF] Arnaudon | Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below[END_REF], we introduce some notations.

If Y is a semimartingale in M , we denote by dY its Itô differential and by d m Y the martingale part of dY : in local coordinates,

dY = dY i + 1 2 Γ i jk (Y ) d Y j , Y k ∂ ∂x i
where Γ i jk are the Christoffel symbols of the Levi-Civita connection; if dY i = dM i + dA i where M i is a local martingale and A i a finite variation process, then

d m Y = dM i ∂ ∂x i . Alternatively, if Q(Y ) : T Y0 M → T Y. M is the parallel translation along Y , then dY t = Q(Y ) t d . 0 Q(Y ) -1 s • dY s t and d m Y t = Q(Y ) t dN t
where N t is the martingale part of the Stratonovich integral t 0 Q(Y ) -1 s • dY s . For x, y ∈ M , and y not in the cut-locus of x, let (3.3)

I(x, y) = d-1 i=1 ρ(x,y) 0 |∇ ė(x,y) J i | 2 + R( ė(x, y), J i )J i + ∇ ė(x,y) Z, ė(x, y) s ds
where ė(x, y) is the tangent vector of the unit speed minimal geodesic e(x, y) and (J i ) d i=1 are Jacobi fields along e(x, y) which together with ė(x, y) constitute an orthonormal basis of the tangent space at x and y: J i (ρ(x, y)) = P x,y J i (0), i = 1, . . . , d -1; here P x,y : T x M → T y M is the parallel translation along the geodesic e(x, y).

Let c ∈ ]0, 1[. For h > 0 but smaller than the injectivity radius of D, and t > 0, let X h be the semimartingale satisfying X h 0 = ϕ(h) and (3.4) dX h s = P Xs,X h s d m X s + Z(X h s ) ds + ξ h s ds, where

ξ h s := h ct + κh n(X h s , X s )
with n(X h s , X s ) the derivative at time 0 of the unit speed geodesic from X h s to X s , and P Xs,X h s : T Xs M → T X h s M the parallel transport along the minimal geodesic from X s to X h s . By convention, we put n(x, x) = 0 and P x,x = Id for all x ∈ M .

By the second variational formula and (3.2) (cf. [START_REF] Arnaudon | Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below[END_REF]), we have

dρ(X s , X h s ) ≤ I(X s , X h s ) - h ct -κh ds ≤ - h ct ds, s ≤ T h ,
where T h := inf{s ≥ 0 : X s = X h s }. Thus, (X s , X h s ) never reaches the cut-locus. In particular, T h ≤ ct and (3.5)

X s = X h s , s ≥ ct. Moreover, we have ρ(X s , X h s ) ≤ h and

(3.6) |ξ h s | 2 ≤ h 2 κ + 1 ct 2 .
We want to compensate the additional drift of X h by a change of probability. To this end, let

M h s = - s∧ct 0 ξ h r , P Xr ,X h r d m X r , and
R h s = exp M h s - 1 2 [M h ] s .
Clearly R h is a martingale, and under Q h = R h • P, the process X h is a diffusion with generator L.

Letting τ (x 0 ) (resp. τ h ) be the hitting time of ∂D by X (resp. by X h ), we have

1 {t<τ h } ≤ 1 {t<τ (x0)} + 1 {τ (x0)≤t<τ h } .
But, since X h s = X s for s ≥ ct, we obtain

1 {τ (x0)≤t<τ h } = 1 {τ (x0)≤ct} 1 {t<τ h } .
Consequently,

1 h P D t f (ϕ(h)) -P D t f (x 0 ) = 1 h E f (X h t )R h t 1 {t<τ h } -f (X t (0))1 {t<τ (x0)} ≤ 1 h E f (X h t )R h t 1 {t<τ (x0)} -f (X t (0))1 {t<τ (x0)} + 1 h E f (X h t )R h t 1 {τ (x0)≤ct} 1 {t<τ h } ,
and since

X h t = X t this yields 1 h P D t f (ϕ(h)) -P D t f (x 0 ) ≤ E f (X t )1 {t<τ (x0)} 1 h (R h t -1) + 1 h E f (X h t )R h t 1 {τ (x0)≤ct} 1 {t<τ h } . (3.7)
The left hand side converges to the quantity to be evaluated as h goes to 0. Hence, it is enough to find appropriate lim sup's for the two terms of the right hand side. We begin with the first term. Letting

Y h s = M h s - 1 2 [M h ] s
and noting that n(X h r , X r ), P Xr ,X h r d m X r = √ 2 db r up to the coupling time T h for some one-dimensional Brownian motion b r , we have

R h t = exp M h t - 1 2 [M h ] t ≤ 1 + M h t - 1 2 [M h ] t + (Y h t ) 2 exp(Y h t ) = 1 + M h t - t 0 |ξ h s | 2 ds + (Y h t ) 2 exp(Y h t ).
From the assumptions, exp(Y h t ) and Y h t /h have all their moments bounded, uniformly in h > 0. Consequently, since f is bounded,

lim sup h→0 E f (X t )1 {t<τ (x0)} 1 h t 0 |ξ h r | 2 dr + (Y h t ) 2 exp(Y h t ) = 0, which implies lim sup h→0 E f (X t )1 {t<τ (x0)} 1 h (R h t -1) ≤ lim sup h→0 E f (X t )1 {t<τ (x0)} 1 h s 0 ξ h r , P Xr ,X h r d m X r .
Using Lemma 2.4 and estimate (3.6), we have for δ > 0

E f (X t )1 {t<τ (x0)} 1 h s 0 ξ h r , P Xr ,X h r d m X r ≤ δP D t f log f P D t f (x 0 ) (x 0 ) + δP D t f (x 0 ) log E 1 {t<τ (x0)} exp 1 δh ct 0 ξ h s , P Xs,X h s d m X s ≤ δP D t f log f P D t f (x 0 ) (x 0 ) + δP D t f (x 0 ) log E exp 1 δ 2 h 2 ct 0 ξ h s 2 ds ≤ δP D t f log f P D t f (x 0 ) (x 0 ) + δP D t f (x 0 ) ct δ 2 1 c 2 t 2 + κ 2 ≤ δP D t f log f P D t f (x 0 ) (x 0 ) + C ′ cδt P D t f (x 0 ),
where C ′ = 1 + (cκ) 2 (recall that t ≤ 1). Since the last expression is independent of h, this proves that

lim sup h→0 E f (X t )1 {t<τ (x0)} 1 h (R h t -1) ≤ δP D t f log f P D t f (x 0 ) (x 0 ) + C ′ cδt P D t f (x 0 ). (3.8)
We are now going to estimate lim sup of the second term in (3.7). By the strong Markov property, we have

E f (X h t )R h t 1 {τ (x0)≤ct} 1 {t<τ h } = E Q h P D t-ct f (X h ct )1 {τ (x0)≤ct<τ h } ≤ P D t-ct f ∞ Q h τ (x 0 ) ≤ ct < τ h . (3.9) Since ρ(X h s , X s ) ≤ h ct -s ct for s ∈ [0, ct], we have on {τ (x 0 ) ≤ ct < τ h }: ρ ∂D (X h τ (x0) ) ≤ h ct -τ (x 0 ) ct .
For s ∈ [0, τ h -τ (x 0 )], define

Y ′ s = ρ(X h τ (x0)+s , ∂D),
and for fixed small ε > 0 (but

ε > h), let S ′ = inf{s ≥ 0, Y ′ s = ε or Y ′ s = 0}. Since under Q h the process X h
s is generated by L, the drift of ρ(X h s , ∂D) is Lρ(•, ∂D) which is bounded in a neighborhood of ∂D. Thus, for a sufficiently small ε > 0, there exists a Q h -Brownian motion β started at 0, and a constant N > 0 such that

Y s := h ct -τ (x 0 ) ct + √ 2β s + N s ≥ Y ′ s , s ∈ [0, S ′ ]. Let S = inf u ≥ 0, Y u = ε or Y u = 0 .
Taking into account that on {τ (x 0 ) = u},

{Y ′ S ′ = ε} ∪ {S ′ > ct -u} ⊂ {Y S = ε} ∪ {S > ct -u},
we have for u ∈ [0, ct],

Q h ct < τ h |τ (x 0 ) = u ≤ Q h Y S ′ = ε|τ (x 0 ) = u + Q h S ′ ≥ ct -u|τ (x 0 ) = u ≤ Q h Y S = ε|τ (x 0 ) = u + Q h S ≥ ct -u|τ (x 0 ) = u ≤ Q h Y S = ε|τ (x 0 ) = u + 1 ct -u E Q h S|τ (x 0 ) = u .
Now using the fact that e -N Ys is a martingale and Y 2 s -2s a submartingale, we get

Q h {Y S = ε|τ (x 0 ) = u} = 1 -e -N h ct-u ct 1 -e -N ε ≤ C 1 h and E Q h S|τ (x 0 ) = u ≤ E Q h Y 2 S |τ (x 0 ) = u ≤ ε 2 Q h Y S = ε|τ (x 0 ) = u = ε 2 1 -e -N h ct-u ct 1 -e -N ε ≤ C 2 h(ct -u) ct for some constants C 1 , C 2 > 0. Thus, Q h ct < τ h |τ (x 0 ) = u ≤ C 1 h + 1 ct -u C 2 h(ct -u) ct ≤ C 1 h + C 3 h ct ≤ C 4 h t
for some constants C 3 , C 4 > 0 (recall that t ≤ 1). Denoting by ℓ h the density of τ (x 0 ) under Q h , this implies

Q h τ (x 0 ) ≤ ct < τ h = ct 0 ℓ h (u) Q h {ct < τ h |σ h = u} du ≤ C 4 h t ct 0 ℓ h (u) du = C 4 h t Q h τ (x 0 ) ≤ ct .
In terms of

D -h = {x ∈ D, ρ ∂D (x) > h} and σ h = inf{s > 0, X h s ∈ ∂D -h }, we have σ h ≤ τ (x 0 ) a.s. Hence, by Lemma 2.3, Q h τ (x 0 ) ≤ ct ≤ Q h σ h ≤ ct ≤ C exp - ρ ∂D -h (ϕ(h)) 16ct ,
where we used that X h s is generated by L under Q h . This implies

(3.10) Q h τ (x 0 ) ≤ ct < τ h ≤ C 5 h t exp - ρ ∂D -h (ϕ(h)) 16ct . Since 1 h P D t (ϕ(h)) -P D t (x 0 ) converges to |∇P D t f (x 0 )|, we obtain from (3.7), (3.8)
, (3.9) and (3.10),

|∇P D t f (x 0 )| ≤ δP D t f log f P D t f (x 0 ) (x 0 ) + C ′ cδt P D t f (x 0 ) + C 5 P D t-ct f ∞ 1 t exp - ρ ∂D (x 0 ) 16ct . (3.11)
Finally, as explained in steps c) and d) of the proof of Proposition 2.5, for any compact set K ⊂ D, there exists a constant C(K, D) > 0 such that

P D t-ct f ∞ ≤ e C(K,D)/t P D t f (x 0 ), c ∈ [0, 1/2], x 0 ∈ K, t ∈ ]0, 1]
. Combining this with (3.11), we arrive at

|∇P D t f (x 0 )| ≤ δP D t f log f P D t f (x 0 ) (x 0 ) + C ′ cδt P D t f (x 0 ) + C 5 1 t exp - ρ ∂D (x 0 ) 16ct exp C(K, D) t P D t f (x 0 ). (3.12)
Finally, choosing c such that

0 < c < 1 2 ∧ dist(K, ∂D) 16C(K, D) ,
we get for some constant C > 0,

|∇P D t f (x 0 )| ≤ δP D t f log f P D t f (x 0 ) (x 0 ) + C 1 δt + 1 P D t f (x 0 ), (3.13)
x 0 ∈ K, δ > 0, which implies the desired inequality.

To finish we consider the case t > 1. From the semigroup property, we have P D t f = P D 1 (P D t-1 f ). So letting g = P D t-1 f and applying (3.13) to g at time 1, we obtain

|∇P D t f (x 0 )| ≤ δP D 1 g log g P D 1 g(x 0 ) (x 0 ) + C 1 δ + 1 P D 1 g(x 0 ). Now using P D 1 g = P D t f , we get |∇P D t f (x 0 )| ≤ δP D 1 (g log g)(x 0 ) -P D t f (x 0 ) log P D t f (x 0 ) + C 1 δ + 1 P D t f (x 0 ). Letting ϕ(x) = x log x, we have for z ∈ D g log g(z) = ϕ E f (X t-1 (z))1 {t-1<τ (z)} ≤ E ϕ f (X t-1 (z))1 {t-1<τ (z)} = E ϕ(f )(X t-1 (z))1 {t-1<τ (z)} = P D t-1 (f log f )(z)
, where we successively used convexity of ϕ and the fact that ϕ(0) = 0. This implies

|∇P D t f (x 0 )| ≤ δP D t f log f P D t f (x 0 ) (x 0 ) + C 1 δ + 1 P D t f (x 0 ),
which is the desired inequality for t > 1.

4. Proof of Theorems 1.1 and Theorem 1.2 of Theorem 1.1. We assume that t ∈]0, 1[ and refer to the end of the proof of Proposition 3.1 for the case t > 1. Fixing δ > 0 and x 0 ∈ M , we take R = 160/(δ ∧ 1). Let D be a relatively compact open domain with C 2 boundary containing B(x 0 , 2R) and contained in B(x 0 , 2R + ε) for some small ε > 0. By the countable compactness of M , it suffices to prove that there exists a constant C = C(D) such that (1.4) holds on B(x 0 , R) with C in place of F (δ ∧ 1, x 0 ). We now fix x ∈ B(x 0 , R), t ∈ ]0, 1] and f ∈ B + b . Without loss of generality, we may and will assume that P t f (x) = 1.

(a) Let P s (x, dy) be the transition kernel of the L-diffusion process, and for x ∈ D, z ∈ M , let ν s (x, dz) = ∂D h x (s/2, y) P t-s (y, dz) ν(dy), where ν is the measure on ∂D induced by µ(dy) = e V (y) dy. By Lemma 2.2 we have

P t f (x) = P D t f (x) + ]0,t]×D×M p D s/2 (x, y) f (z) dsµ(dy)ν s (y, dz).
Then

|∇P t f (x)| ≤ |∇P D t f (x)| + ]0,t]×D×M |∇ log p D s/2 (•, y)(x)| p D s/2 (x,
I 2 ≤ δ E (f log f )(X t (x))1 {τ (x)≤t<ξ(x)} + δ e + δE f (X t (x))1 {τ (x)≤t<ξ(x)} × log ]0,t]×M×D exp C log(e + s -1 ) δ √ s + 2ρ(x, y) sδ ds p D s/2 (x, y)ν s (y, dz) µ(dy) ≤ δE (f log f )(X t (x))1 {τ (x)≤t<ξ(x)} + δ e + δE f (X t (x))1 {τ (x)≤t<ξ(x)} × log ]0,t]×M×D exp A δ + 9R sδ ds p D s/2 (x, y)ν s (y, dz) µ(dy), (4.4)
where

A := sup r>0 C √ r log(e + r) -r < ∞.
We get

I 2 ≤ δE (f log f )(X t (x))1 {τ (x)≤t<ξ(x)} + δ e + δE f (X t (x))1 {τ (x)≤t<ξ(x)} log E exp (9R/δτ (x)) + A δ ≤ δE (f log f )(X t (x))1 {τ (x)≤t<ξ(x)} + δ e + δ log E exp (9R/δτ (x)) + A ≤ δE (f log f )(X t (x))1 {τ (x)≤t<ξ(x)} + δ log E exp 9R (δ ∧ 1)τ (x) δ∧1 δ + A + δ e = δE (f log f )(X t (x))1 {τ (x)≤t<ξ(x)} + (δ ∧ 1) log E exp 9R (δ ∧ 1)τ (x) + A + δ e . ( 4
.5) By Lemma 2.3 and noting that ρ ∂ (x) ≥ R, we have

E exp 9R (δ ∧ 1)τ (x) ≤ 1 + E 9R (δ ∧ 1)τ (x) exp 9R (δ ∧ 1)τ (x) = 1 + ∞ 0 9Rs (δ ∧ 1) exp 9Rs (δ ∧ 1) d ds -P{τ (x) ≤ s -1 } ds = 1 + 9R (δ ∧ 1) ∞ 0 9R (δ ∧ 1) s + 1 exp 9Rs (δ ∧ 1) P{τ (x) ≤ s -1 } ds ≤ 1 + 9R (δ ∧ 1) ∞ 0 9R (δ ∧ 1) s + 1 exp 9Rs (δ ∧ 1) exp -R 2 s 16 ds = 1 + 9R (δ ∧ 1) ∞ 0 9R (δ ∧ 1) s + 1 exp -Rs (δ ∧ 1) ds = 1 + 9 ∞ 0 (9u + 1) exp (-u) du =: A ′ ,
since R = 160/(δ ∧ 1). This along with (4.5) yields (4.6)

I 2 ≤ δ E (f log f )(X t (x))1 {τ (x)≤t<ξ(x)} + log A ′ + A + δ e .
The proof is completed by combining (4.6) with (4.1), (4.2) and (4.4).

of Theorem 1.2. By Theorem 1.1,

|∇P t f (x)| ≤ δ P t (f log f )(x) -(P t f )(x) log P t f (x) + F (δ ∧ 1, x) 1 δ(t ∧ 1) + 1 + 2δ e P t f (x), δ > 0, x ∈ M. (4.7)
For α > 1 and x = y, let β(s) = 1 + s(α -1) and let γ : [0, 1] → M be the minimal geodesic from x to y. Then | γ| = ρ(x, y). Applying (4.7) with δ = α-1 αρ(x,y) , we obtain d ds log Then choosing T > t such that q := p/2(p -1)

(P t f β(s) ) α/β(s) (γ s ) = α(α -1) β(s) 2 P t (f β(s) log f β(s) ) -(P t f β(s) ) log P t f β(s) P t f β(s) (γ s ) + α β(s) ∇P t f β(s) , γs P t f β(s) (γ s ) ≥ αρ(x, y) β(s)P t f β(s) (γ s ) α -1 αρ(x, y) P t (f β(s) log f β(s) ) -(P t f β(s) ) log P t f β(s) (γ s ) -|∇P t f β(s) (γ s )| ≥ -F α -1 αρ(x, y) ∧ 1, γ s α 2 ρ 2 (x, y) β(s)(α -1)(t ∧ 1) + αρ(x, y) β(s) - 2(α -1) eβ(s) ≥ -C(α, x, y) αρ 2 (x, y) (α -1)(t ∧ 1) + ρ(x, y) - 2(α -
< T /t, µ B(x, √ 2t) exp - 2(2 -p) e -2K(α, t, x) 2α α -1 + √ 2t - t T -qt (P t f (x)) 2 ≤ B(x, √ 2t
)

(P t f α (y)) p exp - ρ(x, y) 2 2(T -qt) µ(dy).
Similarly to the proof of [1, Corollary 3], we obtain that for any δ > 2, choosing α = 2δ 2+δ ∈]1, 2[ such that δ > 2 2-α = p p-1 > 2, there is a constant c(δ) > 0 such that the following estimate holds:

Appendix

The aim of the Appendix is to explain that the arguments in Souplet-Zhang [START_REF] Souplet | Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds[END_REF] and Zhang [START_REF] Zhang | Some gradient estimates for the heat equation on domains and for an equation by Perelman[END_REF] for gradient estimates of solutions to heat equations work as well in the case with drift. (1-f ) 2 . By (5.2) we have

Lf + |∇f | 2 -∂ t f = 0 so that ∂ t ω = 2 ∇f, ∇∂ t f (1 -f ) 2 + 2 |∇f | 2 ∂ t f (1 -f ) 3 = 2 ∇f, ∇(Lf + |∇f | 2 ) (1 -f ) 2 + 2 |∇f | 2 (Lf + |∇f | 2 ) (1 -f ) 3 = 2 ∇f, ∇(∆f + |∇f | 2 ) (1 -f ) 2 + 2 |∇f | 2 (∆f + |∇f | 2 ) (1 -f ) 3 + 2 ∇ ∇f Z, ∇f + 2Hess f (∇f, Z) (1 -f ) 2 + 2 |∇f | 2 Z, ∇f (1 -f ) 3 .
(5.3)

Moreover,

Lω = ∆ω + Z, ∇|f | 2 (1 -f ) 2 + 2|∇f | 2 Z, ∇f (1 -f ) 3 = ∆ω + 2 Hess f (∇f, Z) (1 -f ) 2 + 2 |∇f | 2 Z, ∇f (1 -f ) 3 .
(5.4)

Finally, by the proof of [13, (2.9)] with -k replaced by Ric(∇f, ∇f )/|∇f | 2 , we obtain ∆ω -2 ∇f, ∇(∆f

+ |∇f | 2 ) (1 -f ) 2 + 2 |∇f | 2 (∆f + |∇f | 2 ) (1 -f ) 3 ≥ 2f 1 -f ∇f, ∇ω + 2(1 -f )ω 2 + 2ω Ric(∇f, ∇f ) |∇f | 2 .
(5.5)

Combining (5.1), (5.3), (5.4) and (5.5), we arrive at

Lω -∂ t ω ≥ 2f 1 -f ∇f, ∇ω + 2(1 -f )ω 2 -2Kω.
This implies the desired estimate by the Li-Yau cut-off argument as in [START_REF] Souplet | Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds[END_REF]; the only difference is, using the notation in [START_REF] Souplet | Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds[END_REF], in the calculation of -(∆ψ)ω after Eq. (2.13) in [START_REF] Souplet | Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds[END_REF]. By The remainder of the proof is the same as in the proof of [13, Theorem 1.1], using Lψ in place of ∆ψ.

+ 1 2

 12 y)f (z) dsµ(dy)ν s (y, dz) =: I 1 + I 2 . (4.1) (b) By Proposition 3.1, we have (4.2)I 1 ≤ δP D t (f log f )(x) + , x ∈ B(x 0 , R), t ∈]0, 1[, δ > 0 for some C = C(D) > 0.(c) By Proposition 2.5 with ε = 1, we have (4.3) I 2 ≤ ]0,t]×M×D C log(e + s -1 ) (x, y) f (z) dsν s (y, dz)µ(dy) for some C = C(D) > 0 and all t ∈ ]0, 1]. Applying Lemma 2.4 to the measure μ := p D s/2 (x, y) ds ν s (y, dz)µ(dy) on E := ]0, t] × M × D so that μ(E) = P(τ (x) ≤ t < ξ(x)) ≤ 1, we obtain

EBy [ 6 ,

 6 δ (x, t) := M p t (x, y) 2 exp ρ(x, y) 2 δt µ(dy) ≤ exp c(δ)K(α, t, x)Eq. (3.4)],this implies the desired heat kernel upper bound for C δ (t, x) := c(δ)K(α, t, x)(1 + √ 2t).

Theorem 5 . 1 .

 51 Let L = ∆+Z for a C 1 vector field Z. Fix x 0 ∈ M and R, T, t 0 > 0 such that B(x 0 , R) ⊂ M . Assume that (5.1) Ric -∇Z ≥ -K on B(x 0 , R). There exists a constant c depending only on d, the dimension of the manifold, such that for any positive solution u of (5.2)∂ t u = Lu on Q R,T := B(x 0 , R) × [t 0 -T, t 0 ], the estimate |∇ log u| ≤ c 1 R + T -1/2 + √ K 1 + log sup QR,T u u holds on Q R/2,T /2 .Proof. Without loss of generality, let N := sup QT,R u = 1; otherwise replace u by u/N . Let f = log u and ω = |∇f | 2

( 5 . 1 )

 51 and the generalized Laplacian comparison theorem (see [3, Theorem 4.2]), we have )ω = -(∂ 2 r ψ + (∂ r ψ)Lr)ω ≤ |∂ r ψ| 2 + |∂ r ψ| d r + √ Kd |∂ r ψ| ω.


V Lahaye
email: vlahaye@univ-lr.fr

P Bustamante

J Spitz

W Dabin

K Das

G J Pierce

F Caurant

Long-term dietary segregation of short-beaked common dolphins (Delphinus delphis) in the Bay of Biscay determined using cadmium as an ecological tracer

Keywords: Trace elements, Marine mammals, Diet, Cephalopods, Exposure, North Eastern Atlantic

Dietary studies in marine mammals are traditionally performed by stomach contents analyses, which may be insufficient to determine long-term dietary preferences of these upper-level predators. The primary objective of this study was to test the efficiency of trace metal measurements as complementary tools in dietary studies. Variations in cadmium (Cd) exposure through the diet and its effective renal levels in the short-beaked common dolphin Delphinus delphis were investigated to study the long-term feeding ecology of this predator in the neritic and oceanic waters of the Bay of Biscay. Based upon previous stomach contents analyses, the main prey occurring in the diet of common dolphins were analysed for their Cd contents. Results showed that cephalopods, and especially oceanic Cranchids and Histioteuthids, constituted a major source of Cd for common dolphins. Estimated Cd intake would therefore be 12 times higher in oceanic common dolphins (1400 ± 65 µg day -1 ) compared to neritic ones (120 ± 30 µg day -1

). Accumulation of renal Cd concentrations with age was 5 times higher in by-caught oceanic dolphins than neritic ones (p < 0.0001). At the neritic area scale, renal Cd accumulation rate was 2 times more important in by-caught individuals compared to strandings (p = 0.002). Thus, the use of Cd concentrations in by-caught individuals constituted an efficient complementary tool to assess the existence of dietary segregation between neritic and oceanic common dolphins from the Bay of Biscay. On the contrary, the use of Cd data in stranded animals to infer about their feeding ecology would have to be considered cautiously.

INTRODUCTION

The short-beaked common dolphin Delphinus delphis is generally considered to be the most common small cetacean inhabiting the temperate waters of the north eastern Atlantic. Indeed, its distribution covers a large area, which extends from about 35°N to 60°N, with most sightings off south western England, southern Ireland, Brittany, and in the Bay of Biscay [START_REF] Forcada | Distribution of common and striped dolphins in the temperate waters of the Eastern North Atlantic[END_REF][START_REF] Evans | Common Dolphin, White-bellied Porpoise -Delphinus delphis Linnaeus, 1758[END_REF][START_REF] Hammond | Distribution and abundance of the harbour porpoise and other small cetaceans in the North Sea and adjacent waters[END_REF]. No overall population data are available, but the MICA (Mesure de l'Impact des Captures Accessoires, 1993) and SCANS (Small Cetacean Abundance in the North and adjacent Seas, 1994) surveys provided an estimate for the summer period of about 120 000 individuals in the Celtic Sea and south west of this area, from 6 to 20°W and 43 to 51°N [START_REF] Goujon | Captures accidentelles du filet maillant dérivant et dynamique des populations de dauphins au large du Golfe de Gascogne[END_REF]. The short-beaked common dolphin is a pelagic species that mainly occurs over the continental shelf, notably around the 100-200m depth contour, or over areas with prominent underwater topographic features [START_REF] Evans | Distribution and differentiation of stocks of Delphinus delphis Linnaeus in the northeastern Pacific[END_REF]. The presence of this species in offshore waters has also been reported, e.g. in northern Scotland [START_REF] Weir | Cetaceans of the Atlantic Frontier, north and west Scotland[END_REF]) and in the northern Pacific Ocean [START_REF] Ferrero | Growth and reproduction of the common dolphin, Delphinus delphis Linnaeus, in the offshore waters of the Pacific Ocean[END_REF]. In the Bay of Biscay, short-beaked common dolphins show a wide distribution since they are observed both over the continental shelf, the shelf-edge and in deep oceanic waters [START_REF] Forcada | Distribution of common and striped dolphins in the temperate waters of the Eastern North Atlantic[END_REF][START_REF] Brereton | Distribution, and relative abundance of the common dolphin (Delphinus delphis) in the Bay of Biscay[END_REF]). In addition, [START_REF] Forcada | Distribution of common and striped dolphins in the temperate waters of the Eastern North Atlantic[END_REF] suggested a bimodality in distribution by depth for the North part of the Bay of Biscay and proposed the occurrence of two different common dolphin populations in the north-east Atlantic, one neritic and one oceanic.

Although distributions of cetacean species can often be related to physical oceanographic features such as water depth, it is likely that prey abundance and movements are the most important factors in determining the occurrence and movements of cetaceans [START_REF] Gaskin | The ecology of whales and dolphins, 1 st edn[END_REF][START_REF] Evans | The Natural History of Whales and Dolphins[END_REF][START_REF] Selzer | The distribution of white-sided (Lagenorhynchus acutus) and common dolphins (Delphinus delphis) vs. environmental features of the continental shelf of the northeastern United States[END_REF].

Furthermore, varied diet exhibited by short-beaked common dolphins in different geographic regions (Klinowska 1991[START_REF] Young | Diet of common dolphins (Delphinus delphis) off the south-east coast of southern Africa: opportunism or specialization ?[END_REF][START_REF] Santos | Diets of small cetaceans stranded NW Spain[END_REF] has been taken to imply that they are opportunistic, i.e. that their diet is likely to reflect the fish and cephalopod occurrence in the area where they feed. Hence, dietary studies could also be used to monitor the habitat preferences of these long-lived upper-level predators. Consequently, if the spatial segregation of common dolphins previously proposed by [START_REF] Forcada | Distribution of common and striped dolphins in the temperate waters of the Eastern North Atlantic[END_REF] exists in the Bay of Biscay, oceanic dolphins may exhibit different diet composition than neritic ones, without yet excluding gene flows between populations. Such dietary differences are expected to be emphasised since the neritic and oceanic areas constitute two different ecosystems.

The present study aimed at comparing dietary preferences of common dolphins from the oceanic and the neritic areas. The "oceanic" samples originated from fishery bycatch in the deep waters off the Bay of Biscay (summer 1992Biscay (summer -1993) ) while the "neritic" samples comprised by-caught animals from the continental shelf (winter 2004-2005) plus stranded animals from the French Atlantic coast (2001)(2002)(2003). Some previous stomach content analyses were carried out on the samples collected from 1992 to 2003 [START_REF] Pierce | Bioaccumulation of persistent organic pollutants in small cetaceans in European waters: transport pathways and impact on reproduction[END_REF]. Results showed that the prey species present in the stomachs of stranded and "oceanic" common dolphins were characteristic of each habitat. In addition, "oceanic" common dolphins would consume 10 times more cephalopods than the "neritic" ones. Although stomach content analyses can effectively provide valuable data about the food consumed, results are likely to reflect the diet only on a short timescale, i.e. a few days. Such a "snapshot" of the diet can be problematic since these dietary studies relied on stomach contents from stranded and by-caught animals. The difficulty of ensuring that samples are representative of the population is all the more important for strandings, which can include a high number of sick animals. Unhealthy animals may have fed on prey or prey sizes that do not represent the diet of the healthy specimens [START_REF] Pierce | A review of methods for diet analysis in piscivorous marine mammals[END_REF]. Furthermore, by-catches could be biased towards animals feeding on commercially important fish in areas frequented by fishermen [START_REF] Santos | Assessing the importance of cephalopods in the diets of marine mammals and other top predators: problems and solutions[END_REF]. Moreover, food preferences evaluated by this method can vary with both season and year [START_REF] Santos | Variability in the diet of harbor porpoises (Phocoena phocoena) in scottish waters 1992-2003[END_REF], which can limit valuable long-term insights into dietary preferences. The use of both stranded and by-caught dolphins, combined with the fact that the "oceanic" sample was collected over a limited period (summer months) compared to the "neritic" one (around the year), means that we cannot be sure that the animals sampled offshore did not move to inshore area in other seasons. For those reasons, additional analytical techniques were required to provide a longer timescale view of a cetacean's diet and to identify potential dietary segregation between neritic and oceanic common dolphins.

Analyses of the fatty constituents of blubber and of stable isotopes of carbon and nitrogen in tissues have recently received considerable attention since these techniques may potentially overcome some of the difficulties that arise using stomach content analyses, providing insight into food preferences (e.g. [START_REF] Rau | 15 N/ 14 N and 13 C/ 12 C in Weddel sea birds, seals and fish: implication for diet and trophic structure[END_REF][START_REF] Borobia | Blubber fatty acids of finback and humpback whales from the Gulf of St. Lawrence[END_REF], Abend & Smith1997, Iverson et al. 1997[START_REF] Hobson | Investigating trophic relationships of pinnipeds in Alaska and Washington using stable isotope rations of nitrogen and carbon[END_REF][START_REF] Guitart | Comparative study on the fatty acid composition of two marine vertebrates: striped dolphins and loggerhead turtles[END_REF][START_REF] Dahl | Fatty acid composition of the blubber in the white whales (Delphinapterus leucas)[END_REF], Das et al. 2000a[START_REF] Walton | Use of blubber fatty acid profiles to distinguish dietary differences between grey seals Halichoerus grypus from two UK breeding colonies[END_REF][START_REF] Hooker | Diet of northern bottlenose whales inferred from fatty-acid and stable isotope analyses of biopsy samples[END_REF][START_REF] Lea | Antarctic fur seals foraging in the polar frontal one: inter-annual shifts in diet as shown from fecal and fatty acid analyses[END_REF][START_REF] Born | Population substructure of North Atlantic minke whales (Balaenoptera acutorostrata) inferred from regional variation of element and stable isotopic signatures in tissues[END_REF]. However, the integration of these elements generally occurs over some weeks in the case of fatty acids in the blubber (Thiemann et al. 2003) and 1-2 months for stable isotopes in the muscle [START_REF] Hobson | Assessing avian diets using stable isotopes: I. Turnover of carbon-13[END_REF], which remains insufficient to assess the existence of a long-term dietary segregation between habitat for these long-lived predators; common dolphins typically reaching 20-30 years old [START_REF] Ferrero | Growth and reproduction of the common dolphin, Delphinus delphis Linnaeus, in the offshore waters of the Pacific Ocean[END_REF]. Stable isotope analyses in hard tissues (teeth and bone) can yet provide information about diet on a much longer time-scale (e.g. Tieszen et al. 1983, Hobson 1993) but no hard tissues from the "oceanic" sample was available for dietary analyses.

Thus, the present work explores the utility of trace element measurements to provide additional information on diet. Some previous studies underlined the potential efficiency of trace elements to provide long-term insights into both feeding and habitat preferences of such upper-level predators (e.g. [START_REF] Noda | Distribution of heavy metals in muscle, liver and kidney of northern fur seal (Callorhinus ursinus) caught off Sanriku, Japan and from the Pribilof islands, Alaska[END_REF][START_REF] Sanpera | Heavy metals in two populations of North Atlantic fin whales (Balaenoptera physalus)[END_REF][START_REF] Monaci | Trace elements in striped dolphins (Stenella coeruleoalba) from the western Mediterranean[END_REF], Das et al. 2000a[START_REF] Dietz | Geographical differences of zinc, cadmium, mercury and selenium in polar bears (Ursus maritimus) from Greenland[END_REF][START_REF] Kunito | Using trace elements in skin to discriminate the populations of minke whales in southern hemisphere[END_REF][START_REF] Born | Population substructure of North Atlantic minke whales (Balaenoptera acutorostrata) inferred from regional variation of element and stable isotopic signatures in tissues[END_REF], Das et al. 2004a). Indeed, the natural origin of trace elements [START_REF] Nriagu | A history of global metal pollution[END_REF] has involved adaptations of marine organisms facing the presence of toxic metals in their environment. Thus, marine mammals have developed capacities to accumulate high levels of some of these metals during their life [START_REF] Wagemann | Concentrations of heavy metals and organochlorines in marine mammals of northern waters : overview and evaluation[END_REF][START_REF] Aguilar | Biological factors affecting variability of persistent pollutant levels in cetaceans[END_REF] with the ability to detoxify them through different mechanisms (see reviews by [START_REF] Cuvin-Aralar | Mercury and selenium interaction : a review Ecotoxicol[END_REF][START_REF] Cuvin-Aralar | Mercury and selenium interaction : a review Ecotoxicol[END_REF]Das et al. 2000b). Thus, the concentrations of cadmium (Cd) can be particularly elevated in the kidneys of marine mammals [START_REF] Dietz | Heavy metals[END_REF], in which the biological half-life of Cd is more than 10 years for mammals [START_REF] Elinder | Cadmium and health: a survey[END_REF][START_REF] Stoeppler | Metal and their compounds in the environment[END_REF]. Given that the main pathway of exposure to this metal for marine mammals is diet [START_REF] Aguilar | Biological factors affecting variability of persistent pollutant levels in cetaceans[END_REF] and that the major source of Cd is feeding on cephalopods [START_REF] Bustamante | Cephalopods as a vector for the transfer of cadmium to top marine predators in the north-east Atlantic Ocean[END_REF](Bustamante et al. , 2002a)), Cd concentrations in the kidneys of these top predators may constitute a signature of the consumption of cephalopods over a long time-scale. Indeed, the use of Cd as tracer of diet has already been shown when applied on grey seals from the Faroe Islands [START_REF] Bustamante | Trace element bioaccumulation in grey seals Halichoerus grypus from the Faroe Islands[END_REF]). The author highlighted the fact that Cd levels in seals were not in accordance with the stomach contents analyses (limited to the summer months), emphasising also that the use of stomach contents data alone is not sufficient when considering feeding over a longer time-scale.

Therefore, the present study employs a relatively new approach to investigate common dolphin feeding preferences by examining both short-term exposure to cadmium through the diet and effective levels in common dolphin tissues. The approach is based on the hypothesis that if common dolphins have a long-term affinity to specific feeding areas, which are characterised by different biotic or/and chemical environments, significant differences in long-term metal signatures might allow us to distinguish common dolphins from each area. The objectives of the paper were: (1) to provide baseline data on levels of cadmium concentrations in the prey previously found in stomach contents; (2) to estimate a short-term value of the exposure to Cd through the diet for each area, based upon the proportion of prey found in stomachs; (3) to compare renal Cd levels in common dolphins from each area. Since common dolphins from nertic and oceanic areas differ in their consumption of cephalopods on a short-time scale, the hypothesis that Cd concentrations in their kidneys could discriminate dietary preferences on a long-time scale can be tested for these two Atlantic marine food webs.

MATERIAL AND METHODS

Study area

The Bay of Biscay is a sector of the northeastern Atlantic between 1 to 10°W and 43 to 48°N. This area is characterised by a variable distance of the shelf-edge from the coastline (Fig. 1). Indeed, continental shelf width varies from 60 to 100 nautical miles (NM) in the northern part of the bay (up to 45°N) and is reduced to 25-30 NM in the southern part, reaching 3 NM at the latitude of the Capbreton trough. In this study, the oceanic and neritic areas have been distinguished as potential habitats for common dolphins for the north part of the Bay of Biscay, i.e. up to 45°N.

Dolphin sampling

Between 2001 and 2003, 37 stranded common dolphins (17 males and 20 females) were sampled by the Réseau National Echouage (RNE), co-ordinated by the Centre de Recherche sur les Mammifères Marins (CRMM) from La Rochelle (France), along the North French Atlantic coast, i.e. from 45 to 48°N (Fig. 1). The state of decomposition was noted as: 1) very fresh, less than 48 hours after death (blood still separates serum);

2) fresh (blood does not separate serum, cloudy eyes) or 3) slightly decomposed (skin peeling, moderate smell of decomposition). Given that the continental shelf covers a large area at these latitudes and that the sampled animals were relatively fresh, these dolphins were assumed to have been feeding in the neritic area. After the necropsies, all the collected tissues (i.e. kidney for Cd analyses and teeth for age determination) were stored at -20°C until processed in the laboratory. In addition, this "neritic" sample was completed by 10 by-caught animals (2 males and 8 females) which were collected over the continental shelf-edge in winter 2004-2005. The "oceanic" sample comprised 10 individuals (5 males and 5 females) by-caught during albacore drifnet fisheries in oceanic waters from May to September 1993, between 46° and50°N and9° and20°W (Fig. 1). This collection was conducted by the Institut Français de Recherche pour l'Exploitation de la Mer (IFREMER) from Brest (France) as part of a study on the impact of this fishery on small cetaceans (GERDAU program, see [START_REF] Goujon | Incidental catches of cetaceans by the French albacore tuna fishery: preliminary results[END_REF]).

Tissue samples were taken at the fishing site and frozen immediately in liquid nitrogen and then kept in a freezer (-20°C) until metal analyses in the Oceanology Laboratory (Liege, Belgium; Das et al. 2000a).

Other parameters determined during the sampling included total length (cm) and gender.

Age was determined at the CRMM of La Rochelle. The procedure consisted of counting Growth Layer Groups (GLGs) from tooth sections, assuming that one GLG equals one year (as described by [START_REF] Perrin | Age determination of toothed whales and sirenians[END_REF].

Prey sampling

Results from previous stomach content analyses allowed the collection of the main prey of common dolphins, aiming to cover the range of sizes found in stomach contents. This prey sampling was composed of 7 species of cephalopod and 11 species of fish (see Table 1). These prey were collected during IFREMER groundfish surveys in the Bay of Biscay, from the continental shelf to the shelf-edge in autumn 2001 to 2003 (EVHOE cruises survey). During these campaigns, oceanic species were also collected by pelagic trawling over the shelf-edge in the early night. Prey samples were immediately frozen at -20°C on board and then stored at the laboratory until use. After species determination, each individual was weighed and measured and, owing to relatively small sample sizes, prey of similar size from a same trawling for several species were pooled. Metal analyses were carried out on whole prey in order to reflect the exposure to predators.

Heavy metal analyses

All equipment used in the sample processing was cleaned, and subsequently decontaminated for 24 h in a solution comprised of 35 ml HNO 3 (65%) and 50 ml HCl (36%) to 1 L of Milli-Ro quality water. Fresh samples were freeze-dried and ground to powder. Each sample was then treated in duplicate.

Prior to Cd analyses, 2 aliquots of approximately 200 mg of each homogenised dry sample were digested with 3.5 ml of 65% HNO 3 at 60°C for 3 days. The digested contents were then diluted to 10 ml in milli-Q quality water. Then Cd contents were assayed using a flame (Varian 250 Plus) with deuterium background correction, and graphite furnace (Hitachi Z-5000) atomic-absorption spectrophotometer with Zeeman background correction. Metal concentrations in tissues are reported in µg g -1 wet weight (wet wt).

Quality control was assessed using dogfish liver DOLT-2 and DOLT-3 (NRCC). These standards were treated and analysed under the same conditions as the samples. Our results were in agreement with the certified values (respectively 20.6 ± 0.4 vs 20.8 ± 0.5 for DOLT-2 and 19.2 ± 0.4 vs 19.4 ± 0.6 µg g -1 dry wt for DOLT-3). Detection limits were 0.002 and 0.4 µg g -1 dry wt, for furnace and flame AAS, respectively.

Data treatment

Statistical analyses were performed using XL-STAT. All the concentrations which were below the detection limit were replaced with "dummy values" that were half of the detection limit [START_REF] Gibbons | Statistical methods for detection and quantification of environmental contamination[END_REF].

Calculation of exposure to metal through the diet

Cd levels in whole prey were used to calculate the exposure to this metal through the diet. This calculation took into account the basal metabolic rate of common dolphins (BMR; eq.1), the field metabolic rate (FMR; eq.2) and the daily consumption rate of prey (DCR; eq.3).

(1) BMR = 70 .W 0.75

(2) FMR = BMR . AE . AM (3) DCR i = (FMR . F i ) / EV i (4) Exposure i to j = Σ i to j DCR i . [Cd] i
The basal metabolic rate (BMR, in kJ day -1

) was estimated according to the standard mammalian metabolic model of [START_REF] Kleiber | The fire of life, an introduction to animal energetic[END_REF], re-investigated for marine mammals by [START_REF] Hunter | Estimates of basal metabolic and feeding rates for marine mammals from measurements of maximum body length[END_REF]. An average body mass (W, in kg) of 90 kg has been used for common dolphins [START_REF] Collet | Biologie du dauphin commun Delphinus delphis L. en[END_REF]. BMR was converted to field metabolic rate (FMR, in kJ day -1

) by multiplying BMR by factors accounting for assimilation efficiency (AE) and active metabolism (AM). Assimilation efficiency was assumed to be 80% following [START_REF] Hinga | The requirements of whales in the southern hemisphere[END_REF], resulting in a multiplication factor of 1.25. Active metabolism in cetaceans has been established at approximately 2 to 5 times BMR [START_REF] Hinga | The requirements of whales in the southern hemisphere[END_REF][START_REF] Santos | Feeding ecology of harbour porpoises, common and bottlenose dolphins, and sperm whales[END_REF]. A value of 2.5 times BMR was chosen at near the bottom of that range [START_REF] Kenney | Estimates of prey consumption and trophic impacts of cetaceans in the USA northeast continental shelf ecosystem[END_REF]). The daily consumption rate (DCR, in g day -1

) was obtained by multiplying the reconstituted biomass of each prey (f i ; see [START_REF] Pierce | Bioaccumulation of persistent organic pollutants in small cetaceans in European waters: transport pathways and impact on reproduction[END_REF]) in the diet by the FMR, divided by the energetic value of each prey (EV i , in kJ g -1 ; see Table 1) according to the values of [START_REF] Evans | The Natural History of Whales and Dolphins[END_REF] and Desmercières (unpublished data, see Table 1). Finally the exposure to Cd (eq.4, in µg day

Variations of metal concentrations in common dolphins

The variations of Cd concentrations in dolphins were investigated using analysis of covariance (ANCOVA, Quinn & Keough 2002). As the age is an important factor influencing the accumulation of metals [START_REF] Aguilar | Biological factors affecting variability of persistent pollutant levels in cetaceans[END_REF], this parameter was used as the co-variate in the ANCOVA. The factors tested were successively the parameters state of decomposition, gender and origin of samples and their first-order interaction with age, depending of the suitability of each sampling. When significant, the ANCOVA was followed by Fisher multi-comparison tests.

RESULTS

Exposure to Cd through the diet

Cd levels of the prey species examined in this study are given in Table 1, along with the mean length of the individuals sampled, their energetic value and their proportion and size in the diet of common dolphins from neritic and oceanic areas.

Cephalopods exhibited much higher levels of Cd than fishes, with respective mean concentrations of 0.291 ± 0.302 and 0.022 ± 0.016 µg g -1 wet wt. The high variability exhibited among cephalopods (104%) is partly due to the specimen of Teuthowenia megalops, which exhibited the highest concentration of 1.32 µg g -1 wet wt (Table 1).

Moreover, a comparison between families of cephalopod revealed decreasing levels of Cd with Cranchids > Histioteuthids > Ommastrephids > Loliginids (Fig. 2). In the case of fish, the maximal levels were obtained in Clupeids, Scombrids and Carangids, in which concentrations were at least twice those of the other fishes (Table 1).

Taking into account the contribution by weight of each prey to the diet, a 12 times more elevated exposure to Cd was obtained for oceanic common dolphins compared to neritic ones (Fig. 3). This particularly elevated estimation for offshore (1400 ± 65 µg day -1 ) is linked to the important contribution of cephalopods in the calculation (98%, see Fig. 3), and especially that of Teuthowenia megalops (86%). On the contrary, the relative contribution of fish in the calculation for inshore reached 88% (Fig. 3), which accounts for the low exposure obtained. Furthermore, it should be underlined that the calculation of exposure through the prey analysed is representative of the diet for neritic dolphins but the exposure may be under-estimated for oceanic dolphins since the total proportions in mass of the prey in diet analysed were 94 % for inshore and 53.6% for offshore (Table 1). The lower percentage of the prey analysed for the oceanic area is linked to the difficulty of collecting all the oceanic species for which both occurrence and diel vertical migration pattern are poorly known.

Given that the specimen of Teuthowenia megalops was the prey which showed the highest contribution in the calculation for oceanic dolphins, the same approach was conducted without this species. The under-estimation of the resulted exposure was also emphasised since a further 17% of prey were not used (Table 1). However, this second calculation indicated a 2 times higher Cd exposure for oceanic dolphins compared to neritic ones (Fig. 4).

Renal Cd levels in predators

Table 2 shows the results of ANCOVA on Cd concentrations found in the kidney of common dolphins from both the neritic and oceanic areas of the Bay of Biscay. Firstly, ANCOVA confirmed the significant influence of age on Cd concentrations (p < 0.0001). The state of decomposition of stranded dolphins did not significantly influence Cd concentrations (p = 0.938). In addition, there were no significant differences in Cd bioaccumulation between male and female for by-caught "oceanic" dolphins and stranded animals (p > 0.05), which enabled us to pool all individuals from each area for the further comparison.

Thus, the origin of samples and the age were the most important factors explaining renal Cd variability in dolphins (p < 0.0001) with 52.5% of the observed variation being explained by age and 43.2% by the interaction of this factor and the origin of samples. Indeed, Cd concentrations in "oceanic" by-caught dolphins were about 4 times higher than "neritic" ones (p < 0.0001), with respective mean levels of 16.3 ± 14.0 and 4.35 ± 2.89 µg g -1 wet wt. The high standard deviations are the consequence of the increase of concentrations with age, with a 5 times higher rate of Cd bioaccumulation in "oceanic" by-caught dolphins than in "neritic" ones (Fig. 5). Moreover, among "neritic" samples, stranded animals displayed 2 times less Cd levels (and 1.66 ± 1.64 µg g -1 wet wt) than by-caught ones (p = 0.002 ; Table 2) although the range of concentrations was similar for the two samples (i.e. 0.15 -9.34 and 0.34 -9.08 µg g -1 wet wt. Respectively for stranding and by-catch).

DISCUSSION

Elevated concentrations of Cd in marine mammals are usually ascribed to a diet in which cephalopods and/or crustaceans predominate [START_REF] Hanamaka | Age-related change and distribution of cadmium and zinc concentrations in the steller sea lion (Eumetopias jubata) from the coast of Hokkaido, Japan[END_REF][START_REF] Honda | Distribution of cadmium and zinc in tissues and organs, and their age-related changes in striped dolphins, Stenella coeruleoalba[END_REF][START_REF] Noda | Distribution of heavy metals in muscle, liver and kidney of northern fur seal (Callorhinus ursinus) caught off Sanriku, Japan and from the Pribilof islands, Alaska[END_REF][START_REF] Sanpera | Heavy metals in two populations of North Atlantic fin whales (Balaenoptera physalus)[END_REF], Das et al. 2000a[START_REF] Szefer | Intercomparison studies on distribution and coassociations of heavy metals in liver, kidney and muscle, of harbor porpoise, Phocoena phocoena, from southern Baltic and coastal waters of Denmark and Greenland[END_REF][START_REF] Kunito | Using trace elements in skin to discriminate the populations of minke whales in southern hemisphere[END_REF], Bustamante et al., 2004). Yet, attempts to compare the intake through the diet with effective levels in predators remain scarce (Caurant & Amiart-Triquet 1995, Bustamante et al. 1998). Hence, this study aimed at calculating a global Cd exposure for common dolphins from the Bay of Biscay examining a combination of prey species which occur in a high mass proportion in the diet. Although the selection of these prey species was based upon stomach analysis from stranded and by-caught individuals, the reliability of the prey chosen as representative of the short-term diet can be supported by the relatively high number of stomach contents analysed (n > 60 for each area), which may avoid too high an influence of intra-individual variation. Note that the individuals analysed for metals constituted a sub-sampling of the one used for stomach content analyses.

Dietary sources of Cd for common dolphins in the Bay of Biscay

As expected, cephalopods exhibited 13 times higher Cd levels than fish (Table 1), which is in good agreement with previous studies [START_REF] Hanamaka | Age-related change and distribution of cadmium and zinc concentrations in the steller sea lion (Eumetopias jubata) from the coast of Hokkaido, Japan[END_REF][START_REF] Sapunar | Total mercury and cadmium in some cephalopods and fish from the Adriatic Sea[END_REF][START_REF] Bustamante | Bioaccumulation des éléments traces (métaux et terres rares) chez les mollusques céphalopodes et bivalves pectinidés. Implication de leur biodisponibilité pour le transfert vers les prédateurs[END_REF]. Cd is mainly bioaccumulated in the digestive gland of cephalopods reaching up to 98% of the total body Cd in some species (e.g. [START_REF] Miramand | Concentration and distribution of heavy metals in tissues of two cephalopods, Eledone cirrhosa and Sepia officinalis, from the french coast of the English Channel[END_REF], Bustamante et al. 2002b). The high bioavailability of Cd in the digestive gland indicates a high potential for the trophic transfer of this metal to top predators such as marine mammals (Bustamante et al. 2002a). Thus, given that crustaceans occurred at less than 1% in mass in stomach [START_REF] Pierce | Bioaccumulation of persistent organic pollutants in small cetaceans in European waters: transport pathways and impact on reproduction[END_REF], cephalopods may constitute a major source of Cd for common dolphins in the Bay of Biscay.

Among cephalopods, the highest Cd level was found for a single specimen of the Cranchid squid Teuthowenia megalops with a concentration of 1.32 µg g -1 wet wt. Even if the number of the prey analysed was low and did not allow any robust statistical analyses, Cd levels in squids would be different between families with a decrease of Cd concentrations in the following pattern: Cranchids > Histioteuthids > Ommastrephids > Loliginids (Fig. 2). Minimal data exists for Cranchids and Histioteuthids, but the Ommastrephids have also higher burdens of Cd than the Loliginids for the Bay of Biscay, the Celtic Sea and around Faroe Islands [START_REF] Bustamante | Cephalopods as a vector for the transfer of cadmium to top marine predators in the north-east Atlantic Ocean[END_REF]. Loliginids are considered as a neritic family while Histioteuthids and Cranchids are rather found in the oceanic area [START_REF] Clarke | Cephalopods in the diet of cetaceans and seals[END_REF][START_REF] Nesis | Cephalopods of the world: squid, cuttlefishes, octopuses, and allies[END_REF]. Given that Ommastrephids occur in both inshore and offshore waters, with migrations over long distances between feeding and breeding grounds during their life cycle [START_REF] Nesis | Cephalopods of the world: squid, cuttlefishes, octopuses, and allies[END_REF], they are likely to integrate the exposure to Cd from both areas and to exhibit also an intermediate burden compared to the strictly oceanic and neritic cephalopods.

Both Teuthowenia megalops and Histioteuthis sp. are found mostly in the productive waters of northern temperate Atlantic Ocean [START_REF] Voss | Systematics, biology and biogeography of the cranchiid cephalopod genus Teuthowenia (Oegopsida)[END_REF][START_REF] Voss | The cephalopod family Histioteuthidae (Oegopsida): Systematics, biology, and biogeography[END_REF]. Likewise, it is now well-established that the geographical distribution of Cd in offshore waters is similar to that of the labile micronutrient phosphorus [START_REF] Boyle | On the marine geochemistry of cadmium[END_REF][START_REF] Frew | Influence of Southern Ocean waters on the cadmiumphosphate properties of the global ocean[END_REF]. Like phosphate, Cd is enriched in the surface waters of upwelling regions relative to areas of lower productivity and is depleted in the surface ocean relative to the deeper waters because of its uptake by organisms at the surface and regeneration from sinking biological debris deeper in the water column [START_REF] Boyle | On the marine geochemistry of cadmium[END_REF]). As the main Cd source for cephalopods is dietary [START_REF] Koyama | Bioaccumulation of waterborne and dietary cadmium by oval squid, Sepioteuthis lessoniana, and its distribution among organs[END_REF], Bustamante et al. 2002b) and the oceanic squids of this study were collected over the shelf-edge, the elevated concentrations of Cd in Histioteuthids and Cranchids may be due to the consumption of prey highly contaminated in Cd in the mesopelagic waters. However, the question arises about the validity of using prey caught over the shelf-edge while dolphins have been sampled over deeper waters. As the vertical water column distribution of Cd over the shelf-edge did not differ than that from the open ocean [START_REF] Cotté-Krief | Trace metal (Cd, Cu, Ni and Pb) cycling in the upper water column near the shelf edge of the European continental margin (Celtic Sea)[END_REF], cephalopods from the oceanic part of the Bay of Biscay may also feed in productive waters which may expose them to high levels of Cd. Given that Cd has a long biological half-life in cephalopods (Bustamante et al., 2002b), it is likely that most of the Cd assimilated through the diet is stored during the short-life of squid, leading to the very high levels observed. This hypothesis is supported by other studies on oceanic cephalopods which also reported very high Cd concentrations in their tissues (e.g. [START_REF] Martin | High copper concentrations in squid livers in association with elevated levels of silver, cadmium and zinc[END_REF].

Dietary segregation between oceanic and neritic common dolphins

The exposure estimated for common dolphins in the oceanic area reached 1400 ± 65 µg day -1 with 98% of this intake owing to the consumption of cephalopods (Fig. 3), and especially T. megalops (compare Fig. 3 and Fig. 4). This estimation of exposure to Cd in the oceanic area may be biased since only one specimen of T. megalops could be collected. However, this estimation could be expected to remain under-estimated since only 53.4% of the prey occurring in the diet of "oceanic" dolphins were analysed (Table 1) and that the uncollected prey comprised cephalopods (Gonatidae and Onychoteuthidae, i.e. 22% of the overall diet). Despite uncertainties of the estimation, the exposure to Cd calculated for the oceanic area was 12 times higher than the one obtained for the neritic area (Fig. 3).

In contrast to the oceanic dolphins, the exposure estimated for neritic mammals took into account a larger proportion of their diet, with 94% (by mass) of the prey present in stomachs being analysed (Table 1). Although the contribution of fish for neritic dolphins was elevated (88%), cadmium levels were nevertheless increased (Fig. 4) by the important consumption of the fatty fishes Sardina pilchardius (35.6%), Trachurus trachurus (18.3%) and Engraulis encrasicolus (11.7%), which exhibited the highest levels of Cd among fishes (Table 1).

Such an elevated global exposure for offshore dolphins compared to inshore ones is the consequence of a far higher importance of cephalopods in the diet of dolphins from the oceanic area compared to the neritic one, i.e. 45.7% vs 4.8% [START_REF] Pierce | Bioaccumulation of persistent organic pollutants in small cetaceans in European waters: transport pathways and impact on reproduction[END_REF]). Yet, the difference in Cd exposure through the consumption of cephalopods may be emphasised by the higher impregnation of Cranchids, Histioteuthids and Ommastrephids consumed by oceanic dolphins compared with that of Loliginids, being the main source of cephalopods for neritic dolphins (Table 1). The consideration of variations of the exposure between gender, age classes and/or seasons would be necessary to better understand the transfer of Cd through the diet at the population scale.

When looking at effective renal Cd concentrations in common dolphins, age and origin of samples were the two factors explaining the Cd variations in dolphins (p < 0.0001; Table 2). At the neritic scale, stranded animals displayed 2 times less Cd levels than bycaught ones (p = 0.002). The Cd levels found in the kidney of the stranded common dolphins collected between 2001 and 2003 were yet similar to those previously obtained by Holsbeek et al. (1998) for common dolphins stranded between 1977 and 1990 in the same area (Fig. 6). Therefore, the observed differences of renal Cd levels between stranded and by-caught "neritic" animals may also rather be the result of the sample source than inter-annual variations. Indeed, stranded animals may not be really representative of a healthy population, which is likely to affect metal concentrations for this biological material.

However, the Cd levels in by-caught common dolphins displayed a similar trend to that obtained by the calculation of the exposure, the mean Cd levels being 5 times higher in "oceanic" dolphins compared to "neritic" ones. Such differences in Cd levels between areas could be attributed to temporal variations of Cd in the environment since "oceanic" dolphins were collected in 1992-93 whereas "neritic" ones were sampled in [2004][2005]. The combination of our data with those of Holsbeek et al. (1998) would suggest that Cd levels in the kidney of common dolphins from the Bay of Biscay would have not changed during these last 20 years (Fig. 6). As well, Cd levels in the neritic prey from the present study are similar to those previously reported (e.g. Amiart-Triquet et al. 1983;[START_REF] Bustamante | Cephalopods as a vector for the transfer of cadmium to top marine predators in the north-east Atlantic Ocean[END_REF]). Thus, the observed difference of renal Cd levels between "neritic" and "oceanic" common dolphins may also be the consequence of a long-term integration of Cd in two distinct food webs.

Such differences in Cd levels between different areas have already been reported and were attributed to variations in feeding behaviour among areas [START_REF] Hanamaka | Age-related change and distribution of cadmium and zinc concentrations in the steller sea lion (Eumetopias jubata) from the coast of Hokkaido, Japan[END_REF][START_REF] Szefer | Intercomparison studies on distribution and coassociations of heavy metals in liver, kidney and muscle, of harbor porpoise, Phocoena phocoena, from southern Baltic and coastal waters of Denmark and Greenland[END_REF][START_REF] Born | Population substructure of North Atlantic minke whales (Balaenoptera acutorostrata) inferred from regional variation of element and stable isotopic signatures in tissues[END_REF], chemical water properties [START_REF] André | Cadmium contamination of tissues and organs of delphinids species (Stenella attenuata) -Influence of biological and ecological factors[END_REF][START_REF] Noda | Distribution of heavy metals in muscle, liver and kidney of northern fur seal (Callorhinus ursinus) caught off Sanriku, Japan and from the Pribilof islands, Alaska[END_REF][START_REF] Monaci | Trace elements in striped dolphins (Stenella coeruleoalba) from the western Mediterranean[END_REF], or metabolism [START_REF] Sanpera | Heavy metals in two populations of North Atlantic fin whales (Balaenoptera physalus)[END_REF]. Concerning common dolphins from the Bay of Biscay, the variations of renal Cd observed may indeed be the consequence of different dietary preferences between areas, but the high concentrations exhibited by "oceanic" dolphins may also reflect a signature of the particular geochemical conditions that would occur offshore. As well, the low carbon stable isotope "delta" reported for muscle of "oceanic" common dolphins were consistent with an oceanic feeding behaviour on a mid-time scale (i.e. some months; Das et al. 2000a). Therefore, we can assess the existence of a long-term dietary segregation between common dolphins from inshore and offshore waters of the Bay of Biscay. The higher rate of Cd bioaccumulation in the kidney of "oceanic" dolphins compared to "neritic" ones (p < 0.0001 ; Fig. 5) could be linked to the clear difference in bioavailability of Cd in the prey (Bustamante et al. 2002a) and to their different feeding behaviour. However, the eventuality of a difference in metabolism of dolphins between areas could also be envisaged. In this context, the analysis of both morphometric data and growth curves will enable the investigation of biological characteristics for these groups of dolphins. More than that, the combination of dietary studies, biological characteristics and genetic will allow us to determine the degree of differences between these two groups of common dolphins and to assess the existence of two populations in the Bay of Biscay, as previously suggested by [START_REF] Forcada | Distribution of common and striped dolphins in the temperate waters of the Eastern North Atlantic[END_REF]. The identification of common dolphin stocks in the north-eastern Atlantic would also be relevant in term of conservation and management of these populations.

CONCLUSIONS

This investigation showed that the consumption of cephalopods was a major Cd exposure route to common dolphins, providing insights in the feeding behaviour of both prey and predators. At the prey scale, the elevated levels of Cd obtained in oceanic cephalopods would be the consequence of their feeding behaviour and the chemical particularity of the area where they feed. The collection of more samples of oceanic squids is however mandatory to confirm the different burdens seen in coastal and oceanic cephalopods, and to determine the Cd bioavaibility in oceanic cephalopods in order to evaluate its potential for the transfer to top predators.

The concordance of the estimation of the Cd exposure through the diet and effective Cd levels in predators constitute a preliminary stage for the modelling of the bioaccumulation of this metal in marine mammals. In the future, the estimation of the exposure to Cd through the diet would have to take into account the variability of each parameter involved in the calculation of the FMR, as well as variations linked to age classes and gender.

In addition, this approach appeared to be an efficient tool to get a better knowledge of the ecology of common dolphins in the Bay of Biscay since the use of Cd as a metallic tracer confirmed the existence of a dietary segregation between oceanic and neritic dolphins on a long-term scale. Genetic techniques will have to be used to improve knowledge in the state of the two identified groups of common dolphins in the Bay of Biscay. Finally, the differences of Cd levels between stranded and by-caught animals would have to be confirmed with a larger sampling, but the obtained results make metals in stranded animals to be considered cautiously. ) of neritic and oceanic dolphins, with respective percent of explanation through the diet of 94% and 53.6%. ) of oceanic and neritic dolphins with respective percent of explanation through the diet of 94% and 36.6%. Length refers to standard length for fish and to mantle length for cephalopods ; n : number of samples analysed ; n* : when samples were pooled ; EV, energetic value.

CAPTION TO FIGURES

  diet was obtained by multiplying the DCR by the level of Cd in whole prey ([Cd], in µg g -1 wet wt) for all prey occurring in the diet of common dolphins. Estimates of uncertainty are based upon Cd concentrations standard deviation for each prey.

Fig. 1 .

 1 Fig. 1. Delphinus delphis. Locations of sampling of oceanic (n = 10) and neritic (n = 37)

Fig. 2 .

 2 Fig. 2. Mean ± SD Cd concentrations (µg g -1

Fig. 3 .

 3 Fig. 3. Delphinus delphis. Contribution of fish (in white) and cephalopod (all families,

Fig. 4 .

 4 Fig. 4. Delphinus delphis. Contribution of fish (in white) and cephalopod (without

Fig. 5 .

 5 Fig. 5. Delphinus delphis. Renal Cd (µg g -1

Fig

  Fig.1.

Fig

  Fig.3. Fig.4.

Table 1 .

 1 Characteristics and Cd levels of the prey occurring in the diet of common dolphins from the bay of Biscay.

		PREY IN DIET		PREY ANALYSED FOR Cd
	TAXON
	Family	Mass frequency Length (cm) Length (cm)	Cd levels (µg g -1 w. wt)	n	EV
	Species	Inshore Offshore Mean ± SD	Mean ± SD	Mean ± SD	Range		(kJ g -1 )
	FISH
	Carangidae
	Trachurus trachurus	18.3	11.7 ± 3.5	14.1 ± 0.03	0.039 ± 0.007	0.034 -0.046	3*	5.8
	Clupeidae
	Sardina pilchardius	35.6	19.2 ± 3.2	19.8 ± 0.2	0.048 ± 0.017	0.029 -0.060	3	10.5
	Sprattus sprattus	4.7	11.4 ± 2.1	7.5 ± 0.04	0.028 ± 0.001	0.027 -0.029	3*	8.3
	Engraulidae
	Engraulis encrasicolus	11.7	12.1 ± 2.6	11.7 ± 0.2	0.019 ± 0.004	0.015 -0.022	3*	6.7
	Gadidae
	Merlangius merlangus	3.1	19.7 ± 4.1	14.1 ± 0.3	0.005 ± 0.000		3*	4.1
	Micromesistius poutassou 6.9	12.2 ± 4.2	14.1 ± 0.2	0.012 ± 0.003	0.009 -0.014	3*	4.5
	Trisopterus minutus	3.7	8.0 ± 3.6	8.5 ± 0.1	0.016 ± 0.003	0.013 -0.019	3*	5.7
	Gobididae	1.0
	Pomatochistus minutus		4.5 ± 0.5	6.4 ± 0.11	< 0.002	< 0.002	3*	4.8
	Myctophidae
	Notoscopelus kroeyeri	29.5	4.6 ± 2.0	7.6 ± 0.9	0.017 ± 0.010	0.009 -0.028	3	6.5
	Paralepididae
	Arctozenus risso	0.6	13.0 ± 2.7	13.1 ± 1.8	0.020 ± 0.008	0.011 -0.026	3	5
	Scombridae
	Scomber scombrus	6.2	29.0 ± 7.7	27.5 ± 1.1	0.041 ± 0.006	0.034 -0.046	3	9.2
	CEPHALOPOD
	Cranchiidae
	Teuthowenia megalops	17.0	11.6 ± 2.7	18.1	1.32		1	5
	Histioteuthidae
	Histioteuthis sp	6.3	2.3 ± 8.0	2.63 ± 0.55	0.500 ± 0.146	0.385 -0.665	3	5
	Loliginidae	2.8	18.5 ± 11.5
	Loligo forbesi			19.1 ± 3.2	0.077 ± 0.015	0.060 -0.090	3	5
	L. vulgari			16.8 ± 6.3	0.113 ± 0.010	0.102 -0.121	3	5
	Ommastrephidae	0.2	14.3
	Illex coindetii			13.7 ± 4.8	0.220 ± 0.026	0.200 -0.250	3	4.9
	Todarodes sagitattus			13.7	0.28	0.230 -0.320	2	5
	Todaropsis eblanae			14.9 ± 1.3	0.216 ± 0.147	0.121 -0.385	3	5.5

Table 2 .

 2 Delphinus delphis. Significance of the factors influencing Cd concentrations in the kidneys of neritic and oceanic dolphins (analyses by ANCOVA).Significant values are in bold characters and the percent of the variability explained by each factor variable is indicated in brackets.

	Factor

Acknowledgements. This work was supported by the European program BIOCET (EC: EVK3-CT-2000-00027) and by research grant from the Conseil Régional de Poitou-Charentes of V.L. We are very grateful to all the participants of the French national stranding network (RNE) for the collection of dolphin samples and to IFREMER for facilitating the prey sampling. We wish to thank A. Guerra and A. González of the Instituto de Investigaciones Marinas for the species determination of oceanic squids.