Samy Skander Bahoura

SUP × INF INEQUALITY FOR AN ELLIPTIC EQUATION

We give a sup × inf inequality for an equation with nonlinear perturbation with

INTRODUCTION AND MAIN RESULTS

We are on Riemannian manifold (M, g) of dimension n ≥ 3. In this paper we denote ∆ g = -∇ j (∇ j ) the Laplace-Beltrami operator and N = 2n n-2 . We consider the following equation [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF] ∆ g u = V u N -1 + W u n/(n-2) , u > 0.

Where V is a function. For a, b, c, d, (A i ), (B i ) > 0, we consider a sequence (u i , V i , W i ) i of solutions of the previous equation with the following conditions:

0 < a ≤ V i ≤ b < +∞, ||∇V i || ∞ ≤ A i → 0. 0 < c ≤ W i ≤ d < +∞, ||∇W i || ∞ ≤ B i .
Here we study some properties of this nonlinear elliptic equation. We try to find some estimates of type sup × inf. We denote by S g the scalar curvature.

There are many existence and compactness results which concern this type of equations, see for example . In particluar in [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF], we can find some results about the Yamabe equation and the Prescribed scalar curvature equation. Many methods where used to solve these problems, as a variationnal approach and some other topological methods. Note that the problems come from the nonlinearity of the critical Sobolev exponent. We can find in [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF] some uniform estimates for various equations on the unit sphere or for the Monge-Ampere equation. Note that Tian and Siu proved uniform upper and lower bounds for the sup + inf for the Monge-Ampere equation under some condition on the Chern class, see [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF]. In the case of the Scalar curvature equation and in dimension 2 Shafrir used the isoperimetric inequality of Alexandrov to prove an inequality of type sup + inf with only L ∞ assumption on the prescribed curvature, see [START_REF] Shafrir | A sup+inf inequality for the equation -∆u = V e u[END_REF]. The result of Shafrir is an extention of a result of Brezis and Merle, see [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF] and later, Brezis-Li-Shafrir proved a sharp sup + inf inequality for the same equation with Lipschitzian assumption on the prescribed scalar curvature, see [START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF]. Li in [START_REF] Yy | Harnack Type Inequality: the Method of Moving Planes[END_REF] extend the previous last result to compact Riemannian surfaces. In the higher dimensional case, we can find in [START_REF] Korevaar | Refined asymptotics for constant scalar curvature metrics with isolated singularities[END_REF] a proof of the sup × inf inequality in the constant case for the scalar curvature equation on open set of R n . We have various estimates in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF] when we consider the nonconstant case. To prove our result, we use a blow-up analysis and the moving-plane method, based on the maximum principle and the Hopf Lemma as showed in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF][START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF][START_REF] Korevaar | Refined asymptotics for constant scalar curvature metrics with isolated singularities[END_REF][START_REF] Yy | Harnack Type Inequality: the Method of Moving Planes[END_REF], and a condition on the scalar curvature is sufficient to prove the estimate.

Our main result is:

Theorem 1.1. Assume S g > 0 on M , then, for every compact K of M , there exist a positive constant c ′ = c ′ (a, b, c, d, (A i ), (B i ), K, M, n, g) such that:

sup K u i × inf M u i ≤ c.
Remark: in the case where (M, g) = (Ω ⊂ R n , δ) an open set of the euclidean space with the flat metric, we have the same inequality on compact sets of Ω in this case the scalar curvature S δ ≡ 0, see a previous print.

If we consider the Green function G of the Laplacian with Dirichlet condition on small balls of M , we can have a positive lower bound for G and we have the following corollary: Corollary 1.2. Assume S g > 0 on M , then, for every compact K of M , there exist a positive constant c ′′ = c ′′ (a, b, c, d, (A i ), (B i ), K, M, n, g) such that:

K u 2n n-2 i dv g ≤ c ′′ .

PROOF OF THE THEOREM.

Let us consider x 0 ∈ M , by a conformal change of the metric g = ϕ 4/(n-2) g with ϕ > 0 we can consider an equation of type:

(2)

∆ g u + R g u = V u N -1 + W ϕ -2/(n-2) u n/(n-2) + R g ϕ 2-N u, u > 0.
with, Ricci g(x 0 ) = 0.

Here

; R g = n -2 4(n -1) S g and R g = n -2 4(n -1)
S g

Part I: The metric in polar coordinates.

Let (M, g) a Riemannian manifold. We note g x,ij the local expression of the metric g in the exponential map centered in x.

We are concerning by the polar coordinates expression of the metric. Using Gauss lemma, we can write: g = ds 2 = dt 2 + g k ij (r, θ)dθ i dθ j = dt 2 + r 2 gk ij (r, θ)dθ i dθ j = g x,ij dx i dx j , in a polar chart with origin x", ]0, ǫ 0 [×U k , with (U k , ψ) a chart of S n-1 . We can write the element volume:

dV g = r n-1 |g k |drdθ 1 . . . dθ n-1 = [det(g x,ij )]dx 1 . . . dx n , then, dV g = r n-1 [det(g x,ij )][exp x (rθ)]α k (θ)drdθ 1 . . . dθ n-1 , where, α k is such that, dσ Sn-1 = α k (θ)dθ 1 . . . dθ n-1 . (Riemannian volume element of the sphere in the chart (U k , ψ) ).
Then,

|g k | = α k (θ) [det(g x,ij )].
Clearly, we have the following proposition: Proposition 2.1. Let x 0 ∈ M , there exist ǫ 1 > 0 and if we reduce U k , we have:

|∂ r gk ij (x, r, θ)| + |∂ r ∂ θ m gk ij (x, r, θ)| ≤ Cr, ∀ x ∈ B(x 0 , ǫ 1 ) ∀ r ∈ [0, ǫ 1 ], ∀ θ ∈ U k . and, |∂ r |g k |(x, r, θ)| + ∂ r ∂ θ m |g k |(x, r, θ) ≤ Cr, ∀ x ∈ B(x 0 , ǫ 1 ) ∀ r ∈ [0, ǫ 1 ], ∀ θ ∈ U k . Remark: ∂ r [log |g k |]
is a local function of θ, and the restriction of the global function on the sphere S n-1 , ∂ r [log det(g x,ij )]. We will note, J(x, r, θ) = det(g x,ij ).

Part II: The laplacian in polar coordinates

Let's write the laplacian in [0,

ǫ 1 ] × U k , -∆ = ∂ rr + n -1 r ∂ r + ∂ r [log |g k |]∂ r + 1 r 2 |g k | ∂ θ i (g θ i θ j |g k |∂ θ j ).
We have,

-∆ = ∂ rr + n -1 r ∂ r + ∂ r log J(x, r, θ)∂ r + 1 r 2 |g k | ∂ θ i (g θ i θ j |g k |∂ θ j ).
We write the laplacian ( radial and angular decomposition),

-∆ = ∂ rr + n -1 r ∂ r + ∂ r [log J(x, r, θ)]∂ r -∆ Sr(x) ,
where ∆ Sr (x) is the laplacian on the sphere S r (x).

We set

L θ (x, r)(...) = r 2 ∆ Sr (x) (...)[exp x (rθ)],
clearly, this operator is a laplacian on S n-1 for particular metric. We write,

L θ (x, r) = ∆ gx,r, S n-1 ,
and,

∆ = ∂ rr + n -1 r ∂ r + ∂ r [J(x, r, θ)]∂ r - 1 r 2 L θ (x, r). If, u is function on M , then, ū(r, θ) = u[exp x (rθ)]
is the corresponding function in polar coordinates centered in x. We have,

(3) -∆u = ∂ rr ū + n -1 r ∂ r ū + ∂ r [J(x, r, θ)]∂ r ū - 1 r 2 L θ (x, r)ū.
Part III: "Blow-up" and "Moving-plane" methods

The "blow-up" analysis Let, (u i ) i a sequence of functions on M such that, (4)

∆ g u i + R gu i = V i u N -1 i + W i ϕ -2/(n-2) u n/(n-2) i + R g ϕ 2-N u i , u i > 0, N = 2n n -2 ,
We argue by contradiction and we suppose that sup × inf is not bounded.

We assume that:

∀ c, R > 0 ∃ u c,R solution of (E) such that: R n-2 sup B(x0,R) u c,R × inf M u c,R ≥ c. (H) Proposition 2.2.
There exist a sequence of points (y i ) i , y i → x 0 and two sequences of positive real number

(l i ) i , (L i ) i , l i → 0, L i → +∞, such that if we consider v i (y) = u i [exp yi (y)] u i (y i ) , we have: i) 0 < v i (y) ≤ β i ≤ 2 (n-2)/2 , β i → 1. ii) v i (y) → 1 1 + |y| 2 (n-2)/2
, uniformly on every compact set of R n .

iii) l

(n-2)/2 i [u i (y i )] × inf M u i → +∞ Proof:
Without loss of generality, we can assume that:

V (x 0 ) = n(n -2).
We use the hypothesis (H). We can take two sequences

R i > 0, R i → 0 and c i → +∞, such that, R i (n-2) sup B(x0,Ri) u i × inf M u i ≥ c i → +∞. Let, x i ∈ B(x 0 , R i ), such that sup B(x0,Ri) u i = u i (x i ) and s i (x) = [R i -d(x, x i )] (n-2)/2 u i (x), x ∈ B(x i , R i ). Then, x i → x 0 . We have, max B(xi,Ri) s i (x) = s i (y i ) ≥ s i (x i ) = R i (n-2)/2 u i (x i ) ≥ √ c i → +∞.
Set :

l i = R i -d(y i , x i ), ūi (y) = u i [exp yi (y)], v i (z) = u i [exp yi z/[u i (y i )] 2/(n-2) ] u i (y i ) .
Clearly, y i → x 0 . We obtain:

L i = l i (c i ) 1/2(n-2) [u i (y i )] 2/(n-2) = [s i (y i )] 2/(n-2) c 1/2(n-2) i ≥ c 1/(n-2) i c 1/2(n-2) i = c 1/2(n-2) i → +∞. If |z| ≤ L i , then y = exp yi [z/[u i (y i )] 2/(n-2) ] ∈ B(y i , δ i l i ) with δ i = 1 (c i ) 1/2(n-2) and d(y, y i ) < R i -d(y i , x i ), thus, d(y, x i ) < R i and, s i (y) ≤ s i (y i ), we can write, u i (y)[R i -d(y, y i )] (n-2)/2 ≤ u i (y i )(l i ) (n-2)/2 . But, d(y, y i ) ≤ δ i l i , R i > l i and R i -d(y, y i ) ≥ R i -δ i l i > l i -δ i l i = l i (1 -δ i ), we obtain, 0 < v i (z) = u i (y) u i (y i ) ≤ l i l i (1 -δ i ) (n-2)/2 ≤ 2 (n-2)/2 .
We set,

β i = 1 1 -δ i (n-2)/2 , clearly β i → 1.
The function v i is solution of:

-g jk [exp yi (y)]∂ jk v i -∂ k g jk |g| [exp yi (y)]∂ j v i = W i ϕ -2/(n-2) [u i (y i )] 2/(n-2) v n/(n-2) i + µϕ -4/(n-2) [u i (y i )] 4/(n-2) v i +V i v i N -1 ,
By elliptic estimates and Ascoli, Ladyzenskaya theorems, (v i ) i converge uniformely on each compact to the function v solution on R n of, ( 5)

∆v = n(n -2)v N -1 , v(0) = 1, 0 ≤ v ≤ 1 ≤ 2 (n-2)/2 ,
By using maximum principle, we have v > 0 on R n , the result of Caffarelli-Gidas-Spruck ( see [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF]) give, v(y

) = 1 1 + |y| 2 (n-2)/2
. We have the same properties for v i in the previous paper [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF].

Polar coordinates and "moving-plane" method Let, w i (t, θ) = e (n-2)/2 ūi (e t , θ) = e (n-2)t/2 u i o exp yi (e t θ), et a(y i , t, θ) = log J(y i , e t , θ).

Lemma 2.3. The function w i is solution of:

(6) -∂ tt w i -∂ t a∂ t w i -L θ (y i , e t )+cw i = V i w N -1 i +e t W i ϕ -2/(n-2) w n/(n-2) i +µe 2t ϕ -4/(n-2) w i , with, c = c(y i , t, θ) = n -2 2 2 + n -2 2 ∂ t a + R g e 2t .
Proof:

We write:

∂ t w i = e nt/2 ∂ r ūi + n -2 2 w i , ∂ tt w i = e (n+2)t/2 ∂ rr ūi + n -1 e t ∂ r ūi + n -2 2 2 w i . ∂ t a = e t ∂ r log J(y i , e t , θ), ∂ t a∂ t w i = e (n+2)t/2 [∂ r log J∂ r ūi ] + n -2 2 ∂ t aw i .
the lemma is proved.

Now we have,

∂ t a = ∂ t b 1 b 1 , b 1 (y i , t, θ) = J(y i , e t , θ) > 0,
We can write,

- 1 √ b 1 ∂ tt ( b 1 w i )-L θ (y i , e t )w i +[c(t)+b -1/2 1 b 2 (t, θ)]w i = V i w N -1 i +e t W i ϕ -2/(n-2) w n/(n-2) i +µe 2t ϕ -4/(n-2) w i , where, b 2 (t, θ) = ∂ tt ( √ b 1 ) = 1 2 √ b 1 ∂ tt b 1 - 1 4(b 1 ) 3/2 (∂ t b 1 ) 2 . Let, wi = b 1 w i , Lemma 2.4. The function wi is solution of: -∂ tt wi + ∆ g y i ,e t , S n-1 ( wi ) + 2∇ θ ( wi ).∇ θ log( b 1 ) + (c + b -1/2 1 b 2 -c 2 ) wi = (7) = V i 1 b 1 (N -2)/2 wN-1 i + e t W i ϕ -2/(n-2) 1 b 1 1/(n-2)
wn/(n-2)

i + µe 2t ϕ -4/(n-2) wi ,
where, c 2 = [ Proof:

We have:

-∂ tt wi -b 1 ∆ θ w i + (c + b 2 ) wi = V i 1 b 1 (N -2)/2 wN-1 i + +e t W i ϕ -2/(n-2) 1 b 1 1/(n-2)
wn/(n-2)

i + µe 2t ϕ -4/(n-2) wi .

But,

∆ θ ( b 1 w i ) = b 1 ∆ g y i ,e t , S n-1 w i -2∇ θ w i .∇ θ b 1 + w i ∆ g y i ,e t , S n-1 ( b 1 ),
and,

∇ θ ( b 1 w i ) = w i ∇ θ b 1 + b 1 ∇ θ w i , we deduce than, b 1 ∆ θ w i = ∆ θ ( wi ) + 2∇ θ ( wi ).∇ θ log( b 1 ) -c 2 wi , with c 2 = [ 1 √ b 1 ∆ g y i ,e t , S n-1 ( √ b 1 ) + |∇ θ log( √ b 1 )| 2 ].
The lemma is proved.

The "moving-plane" method:

Let ξ i a real number, and suppose ξ i ≤ t. We set t ξi = 2ξ i -t and wξi i (t, θ) = wi (t ξi , θ).

We have,

-∂ tt wξi i + ∆ θ ( wi ) + 2∇ θ ( wξi i ).∇ θ log( b 1 ) wξi i + [c(t ξi ) + b -1/2 1 (t ξi , .)b 2 (t ξi ) -c ξi 2 ] wξi i = = V i 1 b ξi 1 (N -2)/2 ( wξi i ) N -1 + +e t W i ϕ -2/(n-2) 1 b 1 1/(n-2)
wn/(n-2)

i + µe 2t ϕ -4/(n-2) wi .

By using the same arguments than in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF], we have:

Proposition 2.5. We have:

1) wi (λ i , θ) -wi (λ i + 4, θ) ≥ k > 0, ∀ θ ∈ S n-1 .
For all β > 0, there exist c β > 0 such that:

2) 1 c β e (n-2)t/2 ≤ wi (λ i + t, θ) ≤ c β e (n-2)t/2 , ∀ t ≤ β, ∀ θ ∈ S n-1 .
We set,

Zi = -∂ tt (...) + ∆ θ (...) + 2∇ θ (...).∇ θ log( b 1 ) + (c + b -1/2 1 b 2 -c 2 )(...)
Remark: In the operator Zi , by using the proposition 3, the coeficient c + b

-1/2 1 b 2 -c 2 satisfies: c + b -1/2 1 b 2 -c 2 ≥ k ′ > 0, pour t << 0,
it is fundamental if we want to apply Hopf maximum principle.

Goal:

Like in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF], we have elliptic second order operator. Here it is Zi , the goal is to use the "movingplane" method to have a contradiction. For this, we must have:

Zi ( wξi i -wi ) ≤ 0, if wξi i -wi ≤ 0. We write: Zi ( wξi i -wi ) = (∆ ξi θ -∆ θ ) wξi i )+ +2(∇ θ,e t ξ i -∇ θ,e t )(w ξi i ).∇ θ,e t ξ i log( b ξi 1 ) + 2∇ θ,e t ( wξi i ).∇ θ,e t ξ i [log( b ξi 1 ) -log b 1 ]+ +2∇ θ,e t w ξi i .(∇ θ,e t ξ i -∇ θ,e t ) log b 1 -[(c + b -1/2 1 b 2 -c 2 ) ξi -(c + b -1/2 1 b 2 -c 2 )] wξi i + +V ξi i 1 b ξi 1 (N -2)/2 ( wξi i ) N -1 -V i 1 b 1 (N -2)/2 wN-1 i + +e t ξ i (W i ϕ -2/(n-2) ) ξi b ξi 1 -1/(n-2) ( wξi i ) n/(n-2) -e t (W i ϕ -2/(n-2) )b -1/(n-2) 1 ( wi ) n/(n-2) + +µ(e 2t ξ i (ϕ -4/(n-2) ) ξi wξi i -e 2t ϕ -4/(n-2) wi ) ( * * * 1)
Clearly, we have:

Lemma 2.6. b 1 (y i , t, θ) = 1 - 1 3 Ricci yi (θ, θ)e 2t + . . . , R g (e t θ) = R g (y i )+ < ∇R g (y i )|θ > e t + . . . .
According to proposition 1 and lemma 3, Proposition 2.7.

Zi ( wξi

i -wi ) ≤ Ãi (e t -e t ξ )( wξi i ) N -1 + (1/2)(e t ξ i -e t )( wξi i ) n/(n-2) + +C|e 2t -e 2t ξ i | |∇ θ wξi i | + |∇ 2 θ ( wξi i )| + o(1)[ wξi i + ( wξi i ) N -1 ] + ϕ -2/(n-2) ( wξi i ) n/(n-2) + µ wξi i .
Proof:

We use proposition 1, we have:

a(y i , t, θ) = log J(y i , e t , θ) = log b 1 , |∂ t b 1 (t)| + |∂ tt b 1 (t)| + |∂ tt a(t)| ≤ Ce 2t ,
and,

|∂ θj b 1 | + |∂ θj,θ k b 1 | + ∂ t,θj b 1 | + |∂ t,θj,θ k b 1 | ≤ Ce 2t , then, |∂ t b 1 (t ξi ) -∂ t b 1 (t)| ≤ C ′ |e 2t -e 2t ξ i |, on ] -∞, log ǫ 1 ] × S n-1 , ∀ x ∈ B(x 0 , ǫ 1 )
Locally,

∆ θ = L θ (y i , e t ) = - 1 |g k (e t , θ)| ∂ θ l [g θ l θ j (e t , θ) |g k (e t , θ)|∂ θ j ].
Thus, in [0, ǫ 1 ] × U k , we have,

A i =   1 |g k | ∂ θ l (g θ l θ j |g k |∂ θ j ) ξi - 1
|g k | ∂ θ l (g θ l θ j |g k |∂ θ j )   ( wξi i )
then, A i = B i + D i with, B i = gθ l θ j (e t ξ i , θ) -gθ l θ j (e t , θ) ∂ θ l θ j wξi i (t, θ), and,

D i = 1 |g k |(e t ξ i , θ) ∂ θ l [g θ l θ j (e t ξ i , θ) |g k |(e t ξ i , θ)] - 1 |g k |(e t , θ) ∂ θ l [g θ l θ j (e t , θ) |g k |(e t , θ)] ∂ θ j wξi i (t, θ),
we deduce,

A i ≤ C k |e 2t -e 2t ξ i | |∇ θ wξi i | + |∇ 2 θ ( wξi i )| , We take C = max{C i , 1 ≤ i ≤ q}
and if we use ( * * * 1), we obtain proposition 4. We have:

∂ θj w λ i (t, θ) w λ i = e (n-2)[(λ-λi)+(ξi-t)]/2 e [(λ-λi)+(ξi-t)] (∂ θj v i )(e [(λ-λi)+(λ-t)] θ) e (n-2)[(λ-λi)+(λ-t)]/2 v i [e (λ-λi)+(λ-t) θ] ≤ Ci ,
and,

∂ θj ,θ l w λ i (t, θ) w λ i = e (n-2)[(λ-λi)+(ξi-t)]/2 e 2[(λ-λi)+(ξi-t)] (∂ θj ,θ l v i )(e [(λ-λi)+(λ-t)] θ) e (n-2)[(λ-λi)+(λ-t)]/2 v i [e (λ-λi)+(λ-t) θ] ≤ Ci ,
with Ci tending to 0 and does not depend on λ ≤ λ i + 2.

We have,

c(y i , t, θ) = n -2 2 2 + n -2 2 ∂ t a + R g e 2t , (α 1 ) b 2 (t, θ) = ∂ tt ( b 1 ) = 1 2 √ b 1 ∂ tt b 1 - 1 4(b 1 ) 3/2 (∂ t b 1 ) 2 , ( α 2 )
c 2 = [ 1 √ b 1 ∆ g y i ,e t , S n-1 ( b 1 ) + |∇ θ log( b 1 )| 2 ], ( α 3 )
Then,

∂ t c(y i , t, θ) = (n -2) 2 ∂ tt a, by proposition 1, |∂ t c 2 | + |∂ t b 1 | + |∂ t b 2 | + |∂ t c| ≤ K 1 e 2t .
We have:

w i (2ξ i -t, θ) = w i [(ξ i -t + ξ i -λ i -2) + (λ i + 2)],
Thus,

w i (2ξ i -t, θ) = e [(n-2)(ξi-t+ξi-λi-2)]/2 e n-2 v i [θe 2 e (ξi-t)+(ξi-λi-2) ] ≤ 2 (n-2)/2 e n-2 = c. We set δ = (n + 2) -(n -2)α 2 .
The left right side are denoted Z 1 et Z 2 , we can write:

Z 1 = ( V ξi i -Vi )( wξi i ) N -1 + Vi [( wξi i ) N -1 -wN-1 i ],
and, 2) ]+e t ξ i ( wξi i ) n/(n-2) [(W i ϕ -2/(n-2) ) ξi -W i ϕ -2/(n-2) ]+ +ϕ -2/(n-2) ( wξi i ) n/(n-2) (e t ξ i -e t ). We can write the part with nonlinear terms as: 2) [(A i w ξi i 2/(n-2) + o( 1))(e t -e t ξ i ) + c(e t ξ i -e t )].

Z 2 = e t ϕ -2/(n-2) [( wξi i ) n/(n-2) -( wi ) n/(n-
( wξi i ) n/(n-
where o(1) = e t ξ i → 0 come from the difference [W i ϕ -2/(n-2) ] ξi -[W i ϕ -2/(n-2) ].

Because w ξi i ≤ c, we have:

-Zi (( wi ) ξi -( wi )) ≤ ( wξi i ) n/(n-2) [(A i c2/(n-2) +o( 1))(e t -e t ξ i )+c(e t ξ i -e t )]+( wi ) ξi (µ/2)(e 2t ξ i -e 2t )]

where o(1) = e t ξ i → 0. Finaly:

-Zi (( wi ) ξi -( wi )) ≤ ( wξi i ) n/(n-2) [-1+A i c2/(n-2) +o( 1)](e t -e t ξ i )+( wi ) ξi (µ/2)(e 2t ξ i -e 2t )

We apply proposition 3. We take t i = log √ l i with l i like in proposition 2. The fact We define ξ i by: ξ i = sup{λ ≤ λ i + 2, wi (2λ -t, θ) -wi (t, θ) ≤ 0 on [λ, t i ] × S n-1 }.

√ l i [u i (y i )]
If we use proposition 4 and the similar technics that in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF] we can deduce by Hopf maximum principle, min Sn-1 wi (t i , θ) ≤ max Sn-1 wi (2ξ i -t i , θ), which implies,

l i (n-2)/2 u i (y i ) × min M u i ≤ c.
It is in contradiction with proposition 2.

Then we have,

u

  ≤ c = c(a, b, A, K, M, g, n).

  2/(n-2) → +∞ ( see proposition 2), implies t i = log √ l i > 2 n -2 log u i (y i ) + 2 = λ i + 2.Finaly, we can work on ] -∞, t i ].


El Aziz Djoudi
email: djoudizz@gmail.com

Alexia Marie

Angélique Mangenot

Camille Puech

Stéphanie Aviron

Manuel Plantegenest

Julien Pétillon

Farming system and landscape characteristics differentially affect two dominant taxa of predatory arthropods

Keywords: Organic farming, semi-natural habitats, spiders, carabids, Western France

come

Introduction

During the second half of the 20th century, agriculture experienced widespread industrialization worldwide, resulting in a strong increase in crop yield and in an intensification of farming practices [START_REF] Stoate | Ecological impacts of arable intensification in Europe[END_REF]. Semi-natural habitats (SNH), such as woodlots, hedgerows covered by perennial vegetation had especially suffered from the mechanization and fields' expansion [START_REF] Tscharntke | Landscape perspectives on agricultural intensification and biodiversity -Ecosystem service management[END_REF]. Agricultural practices such as the use of chemical fertilizers and pesticides had devastating effects, both direct and indirect, on not-target animal and plant species [START_REF] Stoate | Ecological impacts of arable intensification in Europe[END_REF]. Indeed the intensification of farming practices and landscapes simplification have been identified as the main drivers of biodiversity loss in arable lands (Schmidt and Tscharntke, 2005), with strong consequences for agroecosystem functioning [START_REF] Tscharntke | Landscape perspectives on agricultural intensification and biodiversity -Ecosystem service management[END_REF].

Biodiversity provides many ecosystem services crucial for agroecosystem functioning (Tscharntke et al., 2012a), biological regulation of pests by predatory arthropods being one of the most important [START_REF] Benton | Farmland biodiversity: Is habitat heterogeneity the key?[END_REF][START_REF] Tscharntke | Landscape perspectives on agricultural intensification and biodiversity -Ecosystem service management[END_REF]. At the field or farm scale, it has been shown that organic farming positively affects the abundance and species richness of predatory arthropods, but this depends on the studied taxa [START_REF] Bengtsson | The effects of organic agriculture on biodiversity and abundance: A meta-analysis[END_REF][START_REF] Gabriel | Scale matters: The impact of organic farming on biodiversity at different spatial scales[END_REF]. Recent studies have also shown a positive effect of the proportion of organic farming in the landscape for beneficial arthropods [START_REF] Rundlöf | Local and landscape effects of organic farming on butterfly species richness and abundance[END_REF][START_REF] Gabriel | Scale matters: The impact of organic farming on biodiversity at different spatial scales[END_REF].

The landscape heterogeneity, related to the composition and spatial configuration of SNH around the cropland, further influences both the structure (abundance and species richness) and species composition of beneficial arthropods' communities [START_REF] Batáry | Landscape-moderated biodiversity effects of agri-environmental management: a meta-analysis[END_REF]. The effect of field farming system might also strongly vary depending on the heterogeneity of surrounding landscapes. Indeed, according to "the intermediate landscape-complexity hypothesis" (Tscharntke et al., 2012b), fields in moderately complex landscapes often host higher species diversity compared to homogeneous landscapes where croplands dominate [START_REF] Rundlöf | The effect of organic farming on butterfly diversity depends on landscape context[END_REF],. However, SNH may also constitute barriers to long-distance dispersal for arthropods [START_REF] Larrivée | Diversity of canopy and understorey spiders in northtemperate hardwood forests[END_REF][START_REF] Gauffre | Spatial heterogeneity in landscape structure influences dispersal and genetic structure: Empirical evidence from a grasshopper in an agricultural landscape[END_REF]. Overall, the effects of interactions between field farming system at local and landscape scales and landscape heterogeneity on beneficial arthropods' communities remain little studied (but see [START_REF] Rundlöf | The effect of organic farming on butterfly diversity depends on landscape context[END_REF][START_REF] Flohre | Agricultural intensification and biodiversity partitioning in European landscapes comparing plants, carabids, and birds[END_REF][START_REF] Winqvist | Mixed effects of organic farming and landscape complexity on farmland biodiversity and biological control potential across Europe[END_REF].

The effects of predator diversity on pest regulation are usually enhanced when they act at different spatiotemporal scales [START_REF] Schmidt | Relative importance of predators and parasitoids for cereal aphid control[END_REF][START_REF] Straub | Are the conservation of natural enemy biodiversity and biological control compatible goals?[END_REF]. The effectiveness of biological control is strongly influenced by the structure and composition of predator communities [START_REF] Riechert | Test for predation effects of single versus multiple species of generalist predators: Spiders and their insect prey[END_REF][START_REF] Menalled | Manipulating carabid beetle abundance alters prey removal rates in corn fields[END_REF][START_REF] Griffin | Effects of predator richness on prey suppression: A meta-analysis[END_REF][START_REF] Rusch | Predator body sizes and habitat preferences predict predation rates in an agroecosystem[END_REF]. In addition, investigating both community structure and species composition allows to better understand agroecosystem functioning [START_REF] Bommarco | Ecological intensification: Harnessing ecosystem services for food security[END_REF], which argues to use these parameters as key response variables.

This study investigated the relative effects of farming systems and other environmental characteristics perceived to operate at the local (field) and landscape scales on grounddwelling arthropod predators in agroecosystems. Indeed, these generalist and polyphagous predators strongly contribute to biological regulation [START_REF] Thies | The relationship between agricultural intensification and biological control: experimental tests across Europe[END_REF][START_REF] Cardinale | Biodiversity loss and its impact on humanity[END_REF]. Among them, spiders and carabid beetles received special attention because i) they are abundant arthropods largely contributing to the local diversity of agroecosystems and ii) they have recognized bio-indicator values in the way they quickly react to changes in habitat structure (for spiders see [START_REF] Bell | The implications of grassland and heathland management for the conservation of spider communities: a review[END_REF]; for carabid beetles see [START_REF] Luff | Provisional atlas of the ground beetles (Coleoptera, Carabidae) of Britain[END_REF]. We were more specifically interested in assessing the influence of field farming systems at both local and landscape scales (Organic Farming vs. Conventional Farming further abbreviated as OF and CF respectively), habitat and landscape characteristics and their interactions on the structure and species composition of spider and carabid assemblages. We tested the following hypotheses: i) Field farming systems (OF vs. CF) locally drive the structure and composition of arthropod communities. We expect that both abundance and species richness of arthropods are significantly higher in OF fields [START_REF] Bengtsson | The effects of organic agriculture on biodiversity and abundance: A meta-analysis[END_REF], mainly because of less disturbing agricultural practices (e.g. compared to the use of pesticides and chemical inputs in CF). We also expect clear differences in species composition between farming systems for both spiders and carabids. Large species and higher diversity of diets are expected in OF fields and more open habitat and carnivorous species are expected in CF, because of differences in local habitat conditions like vegetation structure and density (including weeds) [START_REF] Tuck | Landuse intensity and the effects of organic farming on biodiversity: A hierarchical metaanalysis[END_REF][START_REF] Henckel | Organic fields sustain weed metacommunity dynamics in farmland landscapes[END_REF] and prey availability [START_REF] Roubinet | Diet of generalist predators reflects effects of cropping period and farming system on extra-and intraguild prey[END_REF].

ii) At the landscape scale, elements surrounding fields modulate the structure and composition of predator communities for both spiders and carabids. We expect a positive effect of the proportion of SNH and organic farming in the landscape on arthropod abundance and species richness (Tscharntke et al., 2012b). We also expect species composition to be influenced by the proportion of SNH only because they provide refuges, habitat for overwintering and alternative food resources [START_REF] Holland | Contrasting the farm-scale spatio-temporal dynamics of boundary and field overwintering predatory beetles in arable crops[END_REF].

iii) There are interacting effects of local, field farming system and landscape elements on the structure and composition of predator communities. We expect the effects of SNH to depend on farming system (Tscharntke et al., 2012b), because a high proportion of SNH can buffer the impact of farming practices (e.g. pesticides in CF) acting as source habitats from which individuals disperse in arable fields [START_REF] Chaplin-Kramer | A meta-analysis of crop pest and natural enemy response to landscape complexity[END_REF].

Material and methods

Study site

The study site was located in Brittany,Western France (48° 06' 53'' N,1° 40' 46'' O). It is characterized by a dense hedgerow network and dominated by mixed crop-livestock farming systems. The landscape is dominated by meadow (~ 40%) followed by corn (30%) and wheat (20%).

Forty fields (twenty pairs of organic and conventional fields) cultivated with winter wheat (Triticum aestivum) were selected in an area of about 200,000 ha along two landscape gradients: the first was made up by the proportion of OF around the sampled fields (radius of 500m; ranging from 3.5 to 30 %) and the second by the proportion of SNH around the sampled fields (from 6.5 to 65.5 %). A Moran's I test showed that the sites and gradients in landscape elements were not spatially auto-correlated (see [START_REF] Puech | Do farming practices affect natural enemies at the landscape scale?[END_REF] for details in sites selection).

Characterization of local and landscape variables

In each field, vegetation height, wheat density, and percentage of wheat and weeds cover (using the Braun-Blanquet index) were measured in four quadrats (1m²) at a 3 m distance from pitfall traps in June 2013. Landscape metrics were computed to characterize the composition and spatial organization of land covers in the surrounding of each focal field. The landscape was characterized in a radius of 500m around each field (Table 1). This distance was chosen because it is relevant to describe landscape for both spiders (Schmidt and Tscharntke, 2005) and carabids [START_REF] Batáry | Effects of local and landscape scale and cattle grazing intensity on Orthoptera assemblages of the Hungarian Great Plain[END_REF]. In total sixteen variables were calculated, five local variables (field scale) and eleven landscape variables (see Table 1).

Arthropod sampling

Arthropods were continuously sampled using pitfall traps from May to July 2013. Traps were filled with preservative solution (50 % mono-propylene glycol, 50 % water). In each field, two sampling stations with two pitfall traps each were set up 10 m apart and placed 10 m away from field edges to avoid potential edge effects. Traps were left open continuously and collected every two weeks, for a total of six sampling periods. Catches by pitfall traps are dependent on the activity (mobility) and local density of arthropods, and lead to the calculation of 'activity-density'.

Statistical analyses

Variable selection

To avoid multicollinearity between variables, a procedure of variable selection was first applied. A method of 'Clustering of variables' that arranges variables into groups of variables strongly related to each other was performed. This method allows handling both quantitative and qualitative variables at the same time. A bootstrap approach was then used to assess the stability of the variable partition and to determine the best number of clusters (R package 'ClustOfVar' v.0.8: Chavent, 2013). At the end of this selection seven variables (Fig. A.1) were selected for subsequent analyses (see below).

Influence of field farming system and landscape context on community structure

To investigate the relationships between the selected predictor variables (local and landscape variables) and activity-density and species richness of spiders and carabids, General Linear Mixed-effect Models (GLMM) [START_REF] Zuur | Mixed Effects Models and Extensions in Ecology with R[END_REF], model averaging [START_REF] Burnham | Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach[END_REF] and multi-model inference [START_REF] Grueber | Multimodel inference in ecology and evolution: Challenges and solutions[END_REF] were used. In each model, a nested spatial random effect was added to account for the spatial dependence between observations: site (landscape around each pair of organic/conventional fields) and field within sites. A temporal random effect, date sampling, was also added to account for a temporal change in assemblages. Full models for spider and carabid activity-density and species richness included the following predictor variables: farming system as fixed effect, height of vegetation, length of hedgerows, proportion of organic farming, wood, meadow and grass strips, as well as the interactions between farming system and all landscape variables. Full models were first fitted using the 'glmer' function in the R package 'lme4 ' (v.1.1-12;[START_REF] Bates | Fitting Linear Mixed-Effects Models Using lme4[END_REF]. To assess the appropriate distribution, statistics on residuals were first calculated with Poisson distribution. When over-dispersion was observed (assessed by comparing the sum of squared Pearson residuals to the residual degrees of freedom), a negative binomial distribution was used. Each model was fitted using the appropriate distribution type and link function. Poisson distribution was selected for spider and carabid species richness. Negative binomial distribution was selected for spider and carabid activitydensity.

Model selection procedure started with the standardization of all predictor variables [START_REF] Grueber | Multimodel inference in ecology and evolution: Challenges and solutions[END_REF] using the 'standardize' function of the R package 'arm ' (v. 19-3;[START_REF] Gelman | arm: Data Analysis Using Regression and Multilevel/Hierarchical Models[END_REF]. The 'dredge' function of the R package 'MuMin ' (v. 1.16.4;[START_REF] Barton | MuMIn: Multi-Model Inference[END_REF] was then used to fit all possible combinations of models, and their associated Akaike's Information Criteria corrected for small sample sizes were calculated (AICc. A list of the best fitted models with Δ AICc < 2 was generated with the 'get.models' function of the R package 'MuMin' and was used for model averaging with the 'model.avg' function in the 'MuMIn' package. The final averaged models provided model-averaged coefficients for each retained predictor variable and variable importance (i.e., the sum of the model weights within the set of models that included this variable). Finally, 95% confidence intervals (CI) were calculated for the model-averaged parameter estimates with the 'confint' function in the 'MuMIn' package.

Influence of local and landscape variables on community composition

Because of differences in scales (units) the range of values exhibited by predictor variables was high. Predictor variables were then mean-centered and standardized. To avoid the effect of rare species in analyses of community composition, spider and carabid species with less than 5 individuals in the total catches were omitted.

To assess possible relationships between local and landscape variables and arthropod assemblage composition, constrained analyses were used. Following [START_REF] Borcard | Numerical Ecology with R[END_REF], species activity-densities were transformed to a Hellinger distance matrix prior to analyses.

The Hellinger distance offers a better compromise between linearity and resolution than the chi-square metric and the chi-square distance [START_REF] Legendre | Numerical Ecology[END_REF].

The choice between redundancy analysis (RDA) and constrained correspondence analysis (CCA) was made according to the first axis length of a detrended correspondence analysis (DCA), (lengths of gradient for the first axis <3.0 and >4 for RDA and CCA respectively: [START_REF] Ter Braak | CANOCO reference manual and CanoDraw for Windows user's guide: software for canonical community ordination (version 4.5)[END_REF]. In analyses, the distance matrix was the response variable and the environmental (landscape and local) variables were the predictors. A stepwise forward selection was used to select variables which best explained variation in the response data.

Monte Carlo tests with 999 permutations were carried out to assess the significance of the selected environmental factors and constrained analyses axes, using the R package vegan (V 2.4-2; [START_REF] Oksanen | vegan: Community Ecology Package. R package version 2[END_REF].

To better observe the patterns of community composition between OF and CF, a non-metric multidimensional scaling (NMDS) ordination was performed using the Bray-Curtis similarity index. A two-dimensional solution was selected because it consistently maintained a low stress (<0.2) across multiple runs [START_REF] Faith | Compsitional dissimilarity as a robust measure of ecogical distance[END_REF]. Analyses of similarity (ANOSIM) and Permutational Multivariate Analysis of Variance (PERMANOVA) using Bray-Curtis distance matrices were used to test differences in species composition between assemblages in OF and CF for both spiders and carabids, using the R package 'vegan' (V 2.4-2; [START_REF] Oksanen | vegan: Community Ecology Package. R package version 2[END_REF].

All analyses were conducted using R software (R Core Team 2017).

Results

A total of 24241 spiders (OF: 14217, CF: 10024) and 27767 carabids (OF: 18355, CF=9412) belonging to 120 (OF: 104, CF: 95) and 75 (OF: 69, CF: 57) species respectively (Tables A.1 and A.2) were collected.

Influence of field farming system and landscape context on community structure

Field farming system had strong effects on activity-density and species richness of spiders which were significantly higher in OF fields than in CF fields (Fig. 1). Both activity-density and species richness of spiders were influenced by vegetation height (-0.22 ± 0.16 and -0.08 ± 0.07, respectively; Table 2). Spider activity-density was significantly influenced by the interaction between field farming system and the proportion of wood (500 m) (Table 2), the effect being positive in OF fields and negative in CF fields (Fig. 2a). It also decreased with the proportion of meadow in the landscape (-025 ± 0.17; Table 2, Fig. A.2). It was marginally influenced by the interaction between the percentage of grass strips (500 m) and field farming system (average estimate ± CI: 0.37 ± 0.33; Table 2), the effect being positive in CF fields and non-significant in OF fields (Fig. 2b).

As for spiders, field farming system had a strong effect on carabid activity-density and species richness (0.56 ± 0.31 and 0.12 ± 0.09, respectively; Table 2), that were significantly higher in OF fields (Fig. 3). Carabid activity-density and species richness were positively influenced by the proportion of grass strips in the landscape (0.42 ± 0.37 and 0.15 ± 0.09, respectively; Table 2; Fig. A.2). On the contrary, the proportion of meadow and wood in interaction with field farming system had negative effects on carabid species richness (-0.09 ± 0.09 and -0.21 ± 0.1, respectively; Table 2, Fig. A.2). The proportion of meadow had a negative impact on carabid species richness in OF fields and no effect in CF fields (Table 2, Fig. 4a), whereas woods had a negative effect (Table 2) in both OF and CF fields (Fig. 4b). In contrast, the proportion of grass strips in the landscape interacted with farming system (Table 2), and had a positive and significant effect on carabid species richness only in CF field (Fig. 4c).

Regarding farming systems at the landscape scale, the results showed that the proportion of OF in the landscape had a positive effect on carabid species richness (Table 2; Fig. 5).

Influence of local and landscape variables on community composition

The first two axes of the RDA performed on spider assemblages were significant (p=0.001 and p=0.007, respectively) and explained 30.2 % of the total variance. The main variables explaining spider species composition were field farming system and the proportion of wood in the landscape (F=5.03, p=0.001 and F=3.3, p=0.006, respectively) (Fig. 6). The first axis of the RDA tended to oppose spider assemblages observed in organic fields to spider assemblages observed in conventional fields. Fields under OF were characterized by high abundances of species belonging to the lycosid family (Pardosa proxima, Pardosa palustris) whereas fields under CF were characterized by high abundances of species belonging to the linyphiid family (Oedotorax apicatus, Erigone atra and Tenuiphantes tenuis). The second axis segregated spider assemblages observed in fields situated in landscapes dominated by high proportions of woody SNH in the negative part of the axis (with high abundances of the lycosids Pardosa prativaga and Pardosa saltans and of the linyphiid Diplosyla concolor).

The first three axes of the RDA performed on carabid assemblages were significant (p=0.001, p=0.004 and p=0.039, respectively) and explained 29.83 % of the total variance. As for spiders, the main variables explaining carabid species composition were field farming system and the proportion of wood in the surrounding landscape (F=4.91, p=0.005 and F=2.56, p=0.01, respectively) (Fig. 7). The first axis tended to contrast carabid assemblages according to the field farming system. The species Brachinus sclopeta and Poecilus cupreus were especially abundant in OF fields whereas Pterosticus melanarius and Trechus gr. quadristriatus were especially abundant in fields under CF. On the second axis, landscapes dominated by high proportions of woody SNH in the negative part were associated with high abundances of Nebria salina, Carabus nemoralis and Carabus granulatus.

Both in carabids and spiders, the non-metric multidimensional scaling (NMDS) ordination tended to cluster species assemblages observed in fields according to the farming system applied (OF vs. CF) (Figs. A.3 and A.4; ANOSIM tests: spiders: R²=0.11, p=0.001 carabids: R²=0.12, p=0.001). Results obtained using NMDS analysis were consistent with those obtained using RDA. NMDS also showed more heterogeneity of both spider and carabid assemblages in OF compared to CF.

Discussion

In this study we investigated the relative influence of (i) the farming system applied at local and landscape scales, (ii) the landscape characteristics and (iii) their interaction on spider and carabid assemblages. Confirming our first and our second hypotheses, we found consistent evidence that community structure and composition of both taxa are strongly affected by field farming system, with positive effect of OF for both spiders and carabids. Surrounding landscape characteristics and their interaction with field farming systems were also strong drivers of community structure, which validates our last hypothesis. In this study we highlighted the impact of the interaction between local and landscape variables is shown on community structure for both spiders and carabids, in agreement with very few studies on other taxa [START_REF] Diekötter | Landscape and management effects on structure and function of soil arthropod communities in winter wheat[END_REF][START_REF] Feber | Organic farming: Biodiversity impacts can depend on dispersal characteristics and landscape context[END_REF][START_REF] Flohre | Agricultural intensification and biodiversity partitioning in European landscapes comparing plants, carabids, and birds[END_REF][START_REF] Hawro | Landscape complexity is not a major trigger of species richness and food web structure of European cereal aphid parasitoids[END_REF].

Our results also show that spider and carabid assemblages respond differently to changes in their environment (see also [START_REF] Lafage | Relative importance of management and natural flooding on spider, carabid and plant assemblages in extensively used grasslands along the Loire[END_REF], both locally and at landscape scale.

Community structure

Effect of field farming system and local environmental conditions on community structure

Our results showed a highly significant and positive effect of organic farming at field scale on both the activity-density and the species richness of spiders and carabids. Our results confirm several previous studies [START_REF] Bengtsson | The effects of organic agriculture on biodiversity and abundance: A meta-analysis[END_REF][START_REF] Hole | Does organic farming benefit biodiversity?[END_REF]Schmidt et al., 2005;[START_REF] Gabriel | Insect pollinated plants benefit from organic farming[END_REF][START_REF] Rundlöf | Local and landscape effects of organic farming on butterfly species richness and abundance[END_REF][START_REF] Puech | Organic vs. conventional farming dichotomy: Does it make sense for natural enemies?[END_REF][START_REF] Rusch | Management intensity at field and landscape levels affects the structure of generalist predator communities[END_REF]. This important effect of field farming system can be mainly explained by two factors. The first factor is related to the practices associated with the two farming systems. Chemicals (e.g. pesticides, synthetic fertilizers) applied in CF fields are indeed recognized for their harmful effects on biodiversity [START_REF] Köhler | Population Level and Beyond? Wildlife Ecotoxicology of Pesticides: Can We Track Effects to the Wildlife Ecotoxicology of Pesticides: Can We Track Effects to the Population Level and Beyond[END_REF]. Other practices such as organic fertilization, mainly in OF fields, are known to be favorable to arthropod communities [START_REF] Garratt | The effects of farming system and fertilisers on pests and natural enemies: A synthesis of current research[END_REF] by improving soil quality and the availability of potential prey (Schmidt et al., 2005). Secondly, habitat quality differs between the two farming systems. Indeed, the cropping practices applied in OF fields, and especially the lack of herbicides, result in denser vegetation and higher weed diversity [START_REF] Tuck | Landuse intensity and the effects of organic farming on biodiversity: A hierarchical metaanalysis[END_REF][START_REF] Henckel | Organic fields sustain weed metacommunity dynamics in farmland landscapes[END_REF]. This is likely to provide particular microclimatic conditions (e.g. temperature, moisture) and a higher spatial stratification, resulting in an increase in abundance and diversity of food resources in OF fields. This can in turn promote the abundance and diversity of spiders and carabid beetles [START_REF] Lövei | Ecology and Behavior of Ground Beetles (Coleoptera: Carabidae)[END_REF][START_REF] Langellotto | Responses of invertebrate natural enemies to complexstructured habitats: A meta-analytical synthesis[END_REF].

In our study, the average vegetation height in wheat fields negatively influenced the richness and abundance of spider communities. Our results, on one hand, are in contradiction with those of previous studies that have shown that the height of vegetation increases the structural heterogeneity of the habitat, thus offering a greater diversity of niches [START_REF] Greenstone | Determinants of web spider species diversity : Determinants vegetation structural diversity vs . prey availability[END_REF][START_REF] Uetz | Habitat Structure and Spider Foraging[END_REF][START_REF] Langellotto | Responses of invertebrate natural enemies to complexstructured habitats: A meta-analytical synthesis[END_REF][START_REF] Prieto-Benítez | Effects of land management on the abundance and richness of spiders (Araneae): A meta-analysis[END_REF]. On the other hand, our results are in accordance with the results of few studies (Štokmane andSpuņģis, 2014, 2016), which can be explained by two main factors. The first one is related to changes in the microclimatic (e.g. temperature, moisture) and structural conditions of habitats modified by vegetation height [START_REF] Langellotto | Responses of invertebrate natural enemies to complexstructured habitats: A meta-analytical synthesis[END_REF]. These changes in habitat directly impact the mobility of individuals, and therefore their catchability, which results in an effect of the sampling method (already mentioned by e.g. [START_REF] Topping | Limitations to the Use of Pitfall Traps in Ecological Studies Exemplified by a Study of Spiders in a Field of Winter Wheat[END_REF].

The second explanation is related to the "nature" of the habitat (i.e. simply structured). Wheat field indeed forms a very homogeneous habitat, in all CF and in some OF fields, with the total or partial absence of other plant species and a reduced spatial heterogeneity. As a result, CF creates a more open habitat with reduced ecological niches, which negatively impacts the structure of spider communities. Overall, most studies were conducted on vegetation-dwelling spiders, which make them hardly comparable to our results on ground-dwelling spiders. It is thus hard to know whether the negative effect of vegetation height on spider abundance and species richness is due to the sampling method or to spatial heterogeneity, but the fact that spiders were caught in higher numbers in OF fields despite they have higher vegetation plaids in favor of the second hypothesis.

Effects of landscape characteristics on community structure

The proportion of grass strips in the landscape had a positive effect on carabid activity-density irrespective of field farming system and on carabid species richness in CF fields only. Grass strips are known to play an important role in the functioning of agroecosystems [START_REF] Marshall | Field margins in northern Europe: Integrating agricultural, environmental and biodiversity functions[END_REF]. They can represent an alternative habitat for carabids that use them for food, shelter and refuges, but also for overwintering [START_REF] Holland | Contrasting the farm-scale spatio-temporal dynamics of boundary and field overwintering predatory beetles in arable crops[END_REF]. Grass strips may also favor the dispersal of arthropods in the landscape matrix [START_REF] Holzschuh | Grass strip corridors in agricultural landscapes enhance nest-site colonization by solitary wasps[END_REF]. The influence of grass strips in the landscape on carabid species richness was significantly lower in OF fields than in CF fields. This may result from OF fields being more similar to grass strips in their plant structure and diversity than CF fields.

The proportion of meadow in the landscape had a negative effect on spider activity-density in OF and CF fields. It also interacted with farming system, with a negative effect on carabid species richness in OF fields only. This is partly in contradiction with some previous studies (see Purtauf et al., 2005 for carabids). On the other hand, [START_REF] Caro | Multiscale effects of agri-environment schemes on carabid beetles in intensive farmland[END_REF] suggested that meadows may be poor habitats for carabids. Other studies also reported a negative effect of the percentage of meadow in the landscape on the structure of spider assemblages, as summarized in the review of [START_REF] Prieto-Benítez | Effects of land management on the abundance and richness of spiders (Araneae): A meta-analysis[END_REF]. Meadow exploitation in the last decades, has been found to have drastic effects on both spiders and carabids (see [START_REF] Bell | The implications of grassland and heathland management for the conservation of spider communities: a review[END_REF]Morris, 2000 respectively). Mowing of meadows, especially, leads to a strong destruction of the habitat, which negatively affects arthropod communities [START_REF] Dennis | The response of spider (Araneae) assemblages to structural heterogeneity and prey abundance in sub-montane vegetation modified by conservation grazing[END_REF]. Intensive grazing is also generally harmful to biodiversity [START_REF] Benton | Farmland biodiversity: Is habitat heterogeneity the key?[END_REF][START_REF] Donald | Further evidence of continent-wide impacts of agricultural intensification on European farmland birds, 1990-2000[END_REF]. In our study area, meadows are intensively managed through grazing by cattle and mowing, which may explain the negative impact of percentage of meadow on both spider activity-density and on carabid species richness. The influence of the proportion of meadow in the landscape on carabid species richness differed according to the field farming system, and was significant only in OF fields. Carabid assemblages are modulated by the structure and diversity of vegetation [START_REF] Lövei | Ecology and Behavior of Ground Beetles (Coleoptera: Carabidae)[END_REF][START_REF] Rainio | Ground beetles ( Coleoptera : Carabidae ) as bioindicators[END_REF]. Carabid assemblages in meadows are then assumed to be more similar to those of CF fields, with both environments disturbed by local practices (e.g. synthetic fertilization, mowing, pesticides), explaining the lack of effect of meadows in CF. Conversely, carabid assemblages in OF fields differed from those in meadows mainly because of the relative stability of OF fields and their higher floristic diversity.

The proportion of OF in the landscape positively impacted carabid species richness irrespective of the field farming system. Our results are consistent with the few studies that addressed this issue [START_REF] Rundlöf | Local and landscape effects of organic farming on butterfly species richness and abundance[END_REF][START_REF] Gabriel | Scale matters: The impact of organic farming on biodiversity at different spatial scales[END_REF], which found an overall positive effect of the proportion of OF at landscape scale on the diversity of several taxa (e.g. butterflies, birds, plants). [START_REF] Gabriel | Scale matters: The impact of organic farming on biodiversity at different spatial scales[END_REF] found positive effects of OF at the landscape scale, but not for all the studied taxa, whereas [START_REF] Gosme | Local and neighbourhood effects of organic and conventional wheat management on aphids, weeds, and foliar diseases[END_REF] found a negative effect of the proportion of OF at the landscape scale on crop pests (aphids) and diseases (leaf blotch).

Our result suggest that OF fields could act as sources and favor carabid species richness by spreading individuals into the landscape matrix [START_REF] Meyer | Contrasting resource-dependent responses of hoverfly richness and density to landscape structure[END_REF].

The influence of the proportion of wood in the landscape on spider activity-density and carabid species richness depended on the field farming system. For spiders, this resulted in an increasing activity-density with the proportion of wood in the surrounding landscape in OF fields, and in a decreasing activity-density with the proportion of wood in the surrounding landscape in CF fields. OF fields can be considered an attractive environment for spiders (e.g. more food resources and habitat quality). Woods can provide alternative food resources, and shelters for species escaping disturbances from crops, [START_REF] Tscharntke | Landscape perspectives on agricultural intensification and biodiversity -Ecosystem service management[END_REF][START_REF] Bianchi | Sustainable pest regulation in agricultural landscapes: a review on landscape composition, biodiversity and natural pest control[END_REF]. They constitute sources from which individuals can disperse in the landscape matrix [START_REF] Chaplin-Kramer | A meta-analysis of crop pest and natural enemy response to landscape complexity[END_REF]. Carabid species richness decreased with the proportion of wood in the surrounding landscape. This can be explained by the fact that dominant species in both farming systems have low dispersal capacities and are mainly species of cultivated and / or open areas. As a result, this negative effect suggests that woods might act as barriers to carabid dispersal (see Lafage et al., 2015b). The fact that CF fields host more small-sized and mobile species could explain the reason why the negative effect of woods is stronger under CF fields.

Community composition

Farming system and community composition

Although spiders and carabids have different community drivers, we found a major difference in species composition according to the farming systems probably resulting from differences in habitat structure and quality between the two farming systems.

CF fields provide a more homogeneous and simplified habitat structure, regularly disturbed by practices harmful for arthropods (e.g. pesticides, synthetic fertilizers: [START_REF] Köhler | Population Level and Beyond? Wildlife Ecotoxicology of Pesticides: Can We Track Effects to the Wildlife Ecotoxicology of Pesticides: Can We Track Effects to the Population Level and Beyond[END_REF]. Assemblages are consequently primarily determined by species' ability to resist disturbances [START_REF] Hendrickx | How landscape structure, land-use intensity and habitat diversity affect components of total arthropod diversity in agricultural landscapes[END_REF] and prey availability [START_REF] Roubinet | Diet of generalist predators reflects effects of cropping period and farming system on extra-and intraguild prey[END_REF]. This results in spider assemblages dominated by small and highly dispersive species (mainly Linyphiidae) in CF fields [START_REF] Downie | The Impact of Different Agricultural Land-uses on Epigeal Spider\r Diversity in Scotland[END_REF][START_REF] Cole | Comparing the effects of farming practices on ground beetle (Coleoptera: Carabidae) and spider (Araneae) assemblages of Scottish farmland[END_REF][START_REF] Feber | Organic farming: Biodiversity impacts can depend on dispersal characteristics and landscape context[END_REF]. For carabids, assemblages were dominated in CF fields by predatory species, as a consequence of little diversity of food resources and therefore greater direct and / or indirect competition (Niemelä 1993), and mostly nocturnal species, a strategy to escape from (other) predators in these open environments (Erikstad et al., 1989).

In contrast, OF fields provide more favorable environments with no synthetic inputs, higher spatial heterogeneity and stratification, thicker litter layer, and a greater diversity of cultivated plants and weeds [START_REF] Tuck | Landuse intensity and the effects of organic farming on biodiversity: A hierarchical metaanalysis[END_REF][START_REF] Henckel | Organic fields sustain weed metacommunity dynamics in farmland landscapes[END_REF]. These conditions increase ecological niches diversity and result in higher food resources and lowered competition, which could explain spider assemblages dominated by larger species of lycosids, species known to hunt on the ground and finding refuges in the litter [START_REF] Harvey | Provisional Atlas of British spiders (Arachnida, Araneae)[END_REF]. Habitat in OF fields may provide protection against other predators [START_REF] Uetz | Habitat Structure and Spider Foraging[END_REF][START_REF] Castro | Influence of fine woody debris on spider diversity and community structure in forest leaf litter[END_REF].

Carabid assemblages were characterized by medium to large species, feeding on a larger diversity of diets and including both diurnal and nocturnal species, mainly due to an enhanced diversity of cultivated vegetation and weeds. It must finally be stressed that a large part of the variance in species assemblages remains unexplained for both spiders and carabids, suggesting that variables other than those mentioned above (either not measured or not retained for the statistical analysis) play an important role in shaping arthropod communities.

Landscape variables and community composition

Spider and carabid community compositions were also modulated by the surrounding landscape characteristics. The second RDA axes indeed opposed communities of landscapes dominated by cropped elements to communities of landscapes dominated by uncultivated, woody elements. Our results are in agreement with previous studies that showed the importance of landscape in modulating arthropod community composition (e.g. [START_REF] Öberg | Influence of habitat type and surrounding landscape on spider diversity in Swedish agroecosystems[END_REF][START_REF] Batáry | Are spiders reacting to local or landscape scale effects in Hungarian pastures?[END_REF][START_REF] Schmidt | Contrasting responses of arable spiders to the landscape matrix at different spatial scales[END_REF]. In landscapes where the cultivated area is dominant, small and medium sized species, with high dispersal abilities, were found (for spiders, see [START_REF] Blandenier | Ballooning of spiders (Araneae) in Switzerland: General Results from an Eleven-Year Survey[END_REF][START_REF] Simonneau | Phenological and meteorological determinants of spider ballooning in an agricultural landscape[END_REF] for carabids, see Luff, 1988;[START_REF] Desender | Een nieuwe verspreidingsatlas van de loopkevers en zandloopkevers (Carabidae) in België[END_REF].

In landscapes dominated by woods, we found medium and large sized species, mostly forestspecialist species. Woody areas can actually constitute sources from which individuals will disperse in the landscape [START_REF] Chaplin-Kramer | A meta-analysis of crop pest and natural enemy response to landscape complexity[END_REF]. This is reflected for spiders by the abundance of the following species: the lycosids Pardosa prativaga and Pardosa saltans, and the linyphiid Diplosyla concolor. All are forest species, except for P. prativaga usually found in permanent meadows [START_REF] Harvey | Provisional Atlas of British spiders (Arachnida, Araneae)[END_REF], but known to be very mobile. For carabids, characteristic species were Carabus nemoralis, Carabus granulatus and Nebria salina, all medium to large species, occurring in forests and having low dispersal capacities [START_REF] Desender | Een nieuwe verspreidingsatlas van de loopkevers en zandloopkevers (Carabidae) in België[END_REF][START_REF] Luff | Provisional atlas of the ground beetles (Coleoptera, Carabidae) of Britain[END_REF].

Conclusion

Understanding the interactions between cultivated and uncultivated environments at local and landscape scales can help the design of new agricultural systems [START_REF] Marshall | Field margins in northern Europe: Integrating agricultural, environmental and biodiversity functions[END_REF]. Our study highlights the importance of distinguishing the various types of semi-natural elements and not considering them as a homogeneous entity, because their effects vary depending on the studied taxa and on the nature of the semi-natural element. Different taxa respond to agricultural practices at different spatial scales and often at multiple spatial scales [START_REF] Gabriel | Scale matters: The impact of organic farming on biodiversity at different spatial scales[END_REF]. This suggests that considering both local and a landscape condition in conservation strategy is required, and that no single indicator group is appropriate to represent biodiversity. Our study indeed shows that spiders and carabids differentially react to changes in local and landscape structure (see also [START_REF] Pétillon | Influence of abiotic factors on spider and ground beetle communities in different salt-marsh systems[END_REF][START_REF] Varet | Age-dependent colonization of urban habitats: a diachronic approach using carabid beetles and spiders[END_REF] in other habitat types), and can be considered as complementary ecological models in agroecosystems. variable "wood" and the factorial variable "Farming system". Variables and species are represented by their abbreviations (see Tables 1 andA .1, respectively). Fields under organic farming are represented by the letter "O" and those under conventional ones by the letter "C". variable "wood" and the factorial variable "Farming system". Variables and species are represented by their abbreviations (see Tables 1 andA .2, respectively). Fields under organic farming are represented by the letter "O" and those under conventional ones by the letter "C".

FiguresFig. 1 .

 1 Figures

Fig. 2 .

 2 Fig. 2. Activity-density of spiders depending on the percentage of wood (a) and grass strip

Fig. 3 .

 3 Fig. 3. Comparison of activity-densities and species richness of carabids between organic and

Fig. 4 .

 4 Fig. 4. Activity-density of carabids depending on the percentage of wood (a), meadow (b) and

Fig. 5 .

 5 Fig. 5. Species richness of carabids depending on the percentage of the organic farming in the

Fig. 6 .

 6 Fig. 6. Redundancy analysis ordinations (RDA) diagram representing the effects of local and

Fig. 7 .

 7 Fig. 7. Redundancy analysis ordinations (RDA) diagram representing the effects of local and

  Fig. 1

  Fig.

Acknowledgments

E.A.D. PH.D was granted by the "Région Bretagne" (Founding ARED 8960) and the French Foundation Research on Biodiversity (funding the SOLUTION project). This study was partially supported by the FP7-PEOPLE-2013-IRSES fund (project APHIWEB, grant no. 611810). We thank the farmers who allowed us access to their farms. Many thanks to Sylvain Poggi and Maxime Hervé for some statistical advices. We thank two anonymous referees for helpful comments, Cyril Courtial for some help in spider identification and Jean-Luc Roger in carabid identification.

Supplementary materials Table A.1

Taxonomic list of spider species, and their abbreviation.

Table A.2

Taxonomic list of carabid species, and their abbreviation. Clustering method on all the qualitative and quantitative, local and landscape variables (see Table 1 for details). Points to the left of the median line are negative relationships, to the right positive. Black points and whiskers indicate variables with statistically significant effect, gray are those with no significant effect. The effect was considered statistically significant if the 95% CI did not overlap zero.