H Ibrahim
email: hassan.ibrahim@liu.edu.lb

R Monneau
email: monneau@cermics.enpc.fr

Reduced ODE dynamics as formal relativistic asymptotics of a Peierls-Nabarro model

Keywords: AMS subject classifications: 70H40, 45M05, 35D30 Peierls-Nabarro model, dislocation dynamics, relativitic dymamics, formal asymptotics, reduced model

In this paper, we consider a scalar Peierls-Nabarro model describing the motion of dislocations in the plane (x 1 , x 2 ), along the line x 2 = 0. Each dislocation can be seen as a phase transition and creates a scalar displacement field in the plane. This displacement field solves a simplified elasto-dynamics equation which is simply the linear wave equation. The total displacement field creates a stress which makes move the dislocation themselves. By symmetry, we can reduce the system to the wave equation in the half plane x 2 > 0 coupled with an equation for the dynamics of dislocations on the boundary of the half plane, i.e. on x 2 = 0. Our goal is to understand the dynamics of well-separated dislocations in the limit when the distance between dislocations is very large of order 1/ε. After rescaling, this corresponds to introduce a small parameter ε in the system. In the limit ε → 0, we are formally able to identify a reduced ODE model describing the dynamics of relativistic dislocations.

Introduction 1.Setting of the problem

In this paper, we consider a scalar Peiels-Nabarro model describing the dynamics of dislocations in the plane (x 1 , x 2 ), along the line x 2 = 0. This is a phase field model, where each dislocation can be seen as a phase transition essentially between two consecutive integers. We refer to [START_REF] Nabarro | Fifty-year study of the Peierls-Nabarro stress[END_REF] for an overview on the Peierls-Nabarro model. Our scalar Peiels-Nabarro model (see (1.1) below) can be seen as a scalar simplification of the vectorial Peierls-Nabarro Galerkin model introduced in [START_REF]Dynamic dislocation modeling by combining Peierls-Nabarro and Galerkin methods[END_REF] (see also [START_REF]Modeling dislocation by coupling Peierls-Nabarro and element-free Galerkin methods[END_REF]).

A dislocation is a defect in a crystal and creates a stress field in the material (see [START_REF] Hirth | Theory of dislocations[END_REF]). The total stress field creates a force acting on each dislocation, and makes those dislocations move on the line x 2 = 0. The whole model can be seen as a coupling between a dynamics on the line x 2 = 0 and a dynamics outside the line x 2 = 0.

In the model that we consider, the phase field is a scalar quantity which can be identified to a scalar displacement of atoms in the crystal. This displacement satisfies a scalar elasto-dynamics equation, which is simply the linear wave equation in the plane, outside the line x 2 = 0. By symmetry, we can reduce the problem to the wave equation on the half plane x 2 > 0 coupled with a Peierls-Nabarro type dynamics on the boundary of the half plane, i.e. on x 2 = 0. We are interested in the dynamics of well-separated dislocations and in the limit when the distance between dislocations is very large, of order 1/ε. After a suitable rescaling, it corresponds to introduce a small parameter ε > 0 in the model and then to study the limit ε → 0. More precisely we consider a phase field u ε (x 1 , x 2 , t) which is a real function solution of the following system

✷u ǫ = 0, x 2 > 0 N ε u ε = 0, x 2 = 0 (1.1)
where the two operators ✷ and N ε applied on a scalar function u = u(x, t), x = (x 1 , x 2 ) ∈ R 2 are defined as follows:

             ✷u := 1 c 2 0 u tt -∆u, x 2 > 0, N ε u := β 1 c 2 0 u tt -∂ 11 u + ku t - 1 ε ∂ 2 u - 1 ε W ′ (u) + σ(x 1 , t) , x 2 = 0, β = mc 2 0 , (1.2)
where c 0 ∈ (0, ∞) is the velocity of sound in the crystal, and m, k ∈ [0, ∞) are parameters. The quantity m can be interpreted as a kind of mass of the dislocation and k can be seen as a damping factor that is classical for the evolution Peierls-Nabarro model (see for instance [START_REF] Gonzalez | Slow motion of particle systems as a limit of a reactiondiffusion equation with half-Laplacian in dimension one[END_REF]). Here we use the notation

u t = ∂u ∂t , u tt = ∂ 2 u ∂t 2 , ∂ i u = ∂u ∂x i , ∂ ii u = ∂ 2 u ∂x 2 i for i = 1, 2 and ∆u = ∂ 11 u + ∂ 22 u.
In this model, the scalar-valued function W is a 1-periodic smooth potential mimicking the periodicity of the atoms in the crystal. We assume that W satisfies

         W (u + 1) = W (u) for any u ∈ R, W = 0 on Z, W > 0 on R \ Z, α 0 := W ′′ (0) > 0.
A dislocation will be naturally seen as a phase transition between a two consecutive minima of W . In this model, we consider the presence of a given exterior scalar stress field σ(x 1 , t) which has a contribution to the force acting on the dislocations on x 2 = 0. This contribution is taken into account in the definition of the operator N ε . The limit ε → 0 has been studied rigorously in [START_REF] Gonzalez | Slow motion of particle systems as a limit of a reactiondiffusion equation with half-Laplacian in dimension one[END_REF] in the particular case β = 0, c 0 = +∞ and k = 1, which corresponds to a quasistatic approximation.

In the present paper our goal is to study formally the limit ε → 0 for our more general model in the relativistic regime. For σ ≡ 0, the simplest situation is the case of a single and stationary dislocation

u ε (x, t) = φ x ε
where φ is a normalized phase transition between 0 and 1, solution of the following system:

     ∆φ = 0, x 2 > 0 β∂ 11 φ + ∂ 2 φ -W ′ (φ) = 0, x 2 = 0, φ(-∞, 0) = 0, φ(+∞, 0) = 1. (1.3)
We are interested in the dynamics of N ≥ 1 dislocations of positions X i (t) ∈ R for i = 1, ..., N on the axis x 2 = 0. Because we are considering a relativistic regime, it is natural to introduce the following relativistic coefficient:

γ i (t) = 1 1 - X ′ i (t) c 0 2 , (1.4)
where ( ) ′ denotes the time derivative. This coefficient γ i encodes the contraction of the fields in the x 1 direction. Then a natural ansatz for describing the phase transition associated to those dislocations is the following

ûε (x, t) = N i=1 φ γ i (t) x 1 -X i (t) ε , x 2 ε + εv ε (x, t) (1.5)
where v ε appears to be a correction term that will be precised later. Such an ansatz is compatible with the dynamics (1.1) only for suitable correction terms v ε which impose the following asymptotical dynamics:

m 0 (γ i X ′ i ) ′ + k 0 γ i X ′ i = -σ(X i , t) + 1 π j =i 1 γ j 1 (X i -X j ) , i = 1, ..., N (1.6)
where m 0 , k 0 are parameters that will be precised later. Our ODE dynamics (1.6) is similar to equation (1) in [START_REF] Pillon | Influence of inertia on the formation of dislocation dipoles[END_REF]. The term (γ i X ′ i ) ′ = γ 3 i X ′′ i is the natural relativistic acceleration and m 0 γ 3 i is the effective mass of the dislocation, which is coherent with the one computed in [START_REF] Hirth | Forces on high velocity dislocations[END_REF] for screw dislocations (see also (3.12) in [START_REF] Pillon | Modélisations du mouvement instationnaire et des interactions de dislocations[END_REF]).

The term k 0 γ i X ′ i can be seen as a friction term (viscous force) which will slow down the motion of the dislocations. This term is compatible with the one given in (2.21) and (2.18) in [START_REF] Pillon | Modélisations du mouvement instationnaire et des interactions de dislocations[END_REF], for the Eshelby approximation [START_REF] Eshelby | Uniformly moving dislocations[END_REF]. We will see that the damping factor k 0 vanishes when the coefficient k vanishes in (1.2). The precise statement of our result is given in Theorem 1.1.

Assumptions

We will choose

v ε (x, t) = σ(x 1 , t) W ′′ (0) - N i=1 α=1,2,3 a α i (t)ψ α γ i (t) x 1 -X i (t) ε , x 2 ε (1.7)
where the coefficients a α i are given by

a 1 i := 2γ ′ i X ′ i c 2 0 , a 2 i := γ ′ i X ′ i + (γ i X ′ i ) ′ c 2 0 and a 3 i := kγ i X ′ i for i = 1, ..., N. (1.8)
Here we assume the existence of corrector functions ψ α , α = 1, 2, 3 which satisfy:

   ∆ψ 1 = x 1 ∂ 11 φ, x 2 > 0 Aψ 1 = βx 1 ∂ 11 φ - β 0 2α 0 (W ′′ (φ) -W ′′ (0)), x 2 = 0, (1.9)    ∆ψ 2 = ∂ 1 φ, x 2 > 0 Aψ 2 = β∂ 1 φ + β 0 α 0 (W ′′ (φ) -W ′′ (0)), x 2 = 0, (1.10) and  


∆ψ 3 = 0, x 2 > 0 Aψ 3 = ∂ 1 φ + k 0 α 0 (W ′′ (φ) -W ′′ (0)), x 2 = 0, (1.11)
where the linearized operator A is defined by:

Aψ := β∂ 11 ψ + ∂ 2 ψ -W ′′ (φ)ψ.
It is possible to see (see Corollary 3.2) that those correctors can only exist if they satisfy a compatibility condition which forces the following values of the parameters:

k 0 = {x 2 =0} (∂ 1 φ) 2 and β 0 = {x 2 >0} (∂ 1 φ) 2 + {x 2 =0} β(∂ 1 φ) 2 .
(1.12)

We also define the following parameters

k 0 = k 0 k and m 0 = β 0 c 2 0 = 1 c 2 0 {x 2 >0} (∂ 1 φ) 2 + m {x 2 =0} (∂ 1 φ) 2 , (1.13)
where m 0 can be interpreted as a kind of effective mass of the dislocation. For the simplicity of notation, we call ψ 0 = φ and set the Heaviside function H(x 1 ) = 1 [0,+∞) (x 1 ). We consider the following assumptions on the profile functions ψ α , α = 0, 1, 2, 3,

(A)                                           |∇ψ α (x)| ≤ C 1 + |x 1 | α = 0, 1, 2, 3, |∂ 11 ψ α (x)| ≤ C 1 + |x 1 | 2 α = 0, 1, 2, 3, |ψ α (x 1 , 0)| ≤ C 1 + |x 1 | α = 1, 2, 3, φ(x 1 , 0) -H(x 1 ) + 1 α 0 πx 1 ≤ C 1 + |x 1 | 2 if |x 1 | ≥ 1.

The main result

We show the following theorem: Theorem 1.1 (Reduced ODE dynamics as an asymptotic of the PN model) Let us consider the assumptions of Subsection 1.2, and assume W and σ smooth enough. For i = 1, 2 . . . , N, let us consider particles

X i = X i (t) satisfying X i+1 (t) -X i (t) ≥ 2δ > 0 |X ′ i | ≤ c 0 (1 -δ), δ > 0 for t ∈ [0, T ], (1.14)
solution of the dynamics (1.6) for t ∈ (0, T ), namely

m 0 (γ i X ′ i ) ′ + k 0 γ i X ′ i = -σ(X i , t) + 1 π j =i 1 γ j 1 (X i -X j ) , i = 1, ..., N
with γ i is given by (1.4) and the parameters m 0 , k 0 are given in (1.13). Let us consider the ansatz function ûε given by (1.5) and the correction term v ε given by (1.7) with the coefficients a α i given by (1.8). Then for any fixed δ > 0, we have as ε → 0

✷û ε = O δ (1) uniformly in L ∞ ({x 2 > 0} × (0, T )) N ε ûε = O δ (1) uniformly in L ∞ ({x 2 = 0} × (0, T )) . (1.15)
Here we only get O δ (1) right hand sides of (1.15). Nevertheless this still means that the ansatz (1.5) is a good approximation of the solution because the operator N ε involves O(1/ε 2 )-terms and the boundary condition involving N ε has consequences on the first PDE of (1.15). The important consequence of this result is the identification of the limit dynamics (1.6). We leave open the rigorous justification of this limit dynamics. The question of long time existence of solutions of the reduced ODE dynamics (1.6) is also open.

In the case of finite ε for the Peierls-Nabarro Galerkin models, the effective dislocation dynamics can reveal retardation effects. We refer the reader for instance to [START_REF] Pellegrini | Dynamic Peierls-Nabarro equations for elastically isotropic crystals[END_REF][START_REF] Pillon | Inertial and retardation effects for dislocation interactions[END_REF][START_REF] Pillon | Equation of motion for dislocations with inertial effects[END_REF].

The interested reader can also consult [START_REF] Fino | The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model[END_REF] to see how the classical Peierls-Nabarro model can be rigorously derived from the Frenkel-Kontorova model at a smaller scale. We also refer the reader to [START_REF] Hajj | Dislocation dynamics: from microscopic models to macroscopic crystal plasticity[END_REF][START_REF] Monneau | Homogenization of the Peierls-Nabarro model for dislocation dynamics[END_REF] for the relation between the Peierls-Nabarro model and other models at larger scales.

Organization of the paper

In Section 2, we recall the physical derivation of our Peierls-Nabarro type model (1.1). In Section 3, we give some simple properties on the correctors including compatibility conditions. In order to do the proof of our main result, we start with preliminary computations on our ansatz ûε in Section 4. Finally in Section 5, we give the proof of our main result, namely Theorem 1.1.

Physical derivation of the model

We consider a two-dimensional crystal and call U(x 1 , x 2 , t) the horizontal displacement (along the axis x 1 ) of the atoms. A natural action of the system without dislocations describing waves of velocity c 0 , is the following

R 2 ×(0,T ) 1 2 |∇U| 2 - 1 2c 2 0 (U t ) 2 .
(2.1)

We now assume that dislocations are localized on the line x 2 = 0 and can only move along this line. We also assume the antisymmetry

U(x 1 , -x 2 , t) = -U(x 1 , x 2 , t)
but allow a jump of u when we cross the line x 2 = 0:

U(x 1 , 0 + , t) -U(x 1 , 0 -, t) =: η(x 1 , t).
Then the function η can describe a dislocation as a phase transition between two integers (if we normalise to one unit the Burgers vector which is here a scalar quantity because U is itself scalar).

Then the action (2.1) has to be modified as follows

A(U, η) := R 2 ×(0,T ) 1 2 |∇U -ηe 2 δ 0 (x 2 )| 2 - 1 2c 2 0 (U t ) 2 + {x 2 =0}×(0,T ) W (η).
Here (e 1 , e 2 ) the standard orthonormal basis and we had to substract a Dirac mass in order to compensate the jump of u. The last integral is an energy term created by the misfit of the upper and lower half crystal created by the presence of the dislocation. In particular W is 1-periodic, non negative and vanishes on integers (the case η ∈ Z corresponding to the case of no misfit in the crystal). Then the natural Peierls model is the following

A ′ U = 0, kη t = -A ′ η
where the first line is the first variation of the action with respect to u, and the second line is the gradient flow evolution of the field η where k is a damping factor. Then it is easy to check that

u(x 1 , x 2 , t) = 2U(x 1 , x 2 , t) if x 2 > 0, η(x 1 , t) if x 2 = 0, solves (1.1) for ε = 1, σ = 0, β = 0, k = k 4 and W = 1 4
W . More generally, we recover (1.1) if we consider the general action

A(U, η) := 1 ε R 2 ×(0,T ) 1 2 |∇U -ηe 2 δ 0 (x 2 )| 2 + (∇U -ηe 2 δ 0 (x 2 )) • Σ - 1 2c 2 0 (U t ) 2 + {x 2 =0}×(0,T ) 1 ε 2 W (η) + β |∂ 1 η| 2 - 1 c 2 0 (η t ) 2 (2.2)
assuming that div Σ = 0 and σ(x 1 , t) := 2e 2 • Σ(x 1 , 0, t) and β = β 4 . We have to emphasize the fact that model (2.2) in the special case β = 0, is a simplified scalar version of a more general model called the Peierls-Nabarro Galerkin model [START_REF]Dynamic dislocation modeling by combining Peierls-Nabarro and Galerkin methods[END_REF], where the displacement U is vectorial.

Some remarks on the correctors

For the special case of

W ′ (u) = - 1 2πa sin 2π(u - 1 2
) for some a > 0, we recall (see [START_REF] Gonzalez | Slow motion of particle systems as a limit of a reactiondiffusion equation with half-Laplacian in dimension one[END_REF]) that the solution φ 0 of (1.3) for β = 0 is

φ 0 (x) = 1 π arctan x 1 x 2 + a + 1 2 . ( 3.1)
We also refer to [START_REF] Cabré | Solà-Morales Layer solutions in a half-space for boundary reactions[END_REF] for properties of φ 0 in the case β = 0 for more general potentials W . From expression (3.1), it can be seen that assumption (A) on the profile functions, sounds reasonable (even if we have no proof of it, and the goal of the present paper is not to prove any rigorous results in this direction).

Let us consider a function Ψ solution of

∆Ψ = F on Ω := {x 2 > 0} , AΨ = G on ∂Ω = {x 2 = 0} . (3.2) Lemma 3.1 (Compatibility condition) If Ψ solves (3.
2) with sufficient decay at infinity, then we have

Ω F ζ + ∂Ω Gζ = 0 with ζ = ∂ 1 φ (3.3)
where φ is the solution of (1.3).

Proof of Lemma 3.1

Step 1: self-adjoint property For (F, G) and ( F , Ĝ), let us define the scalar product as

Ω F F + ∂Ω G Ĝ.
Then a simple computation (by integration by parts) shows that the operator Ψ → (∆Ψ, AΨ) is self-adjoint for this scalar product, i.e. for any Ψ, Φ, we have If ψ 1 , ψ 2 , ψ 2 solve respectively (1.9), (1.10), (1.11), then the values of the parameters k 0 and β 0 are given by (1.12).

Proof of Corollary 3.2 Applying Lemma 3.1 for Ψ = ψ 1 and using equation (1.9), we get

0 = Ω x 1 (∂ 11 φ)∂ 1 φ + ∂Ω (∂ 1 φ) βx 1 ∂ 11 φ - β 0 2α 0 (W ′′ (φ) -W ′′ (0)) = Ω - (∂ 1 φ) 2 2 + ∂Ω - β(∂ 1 φ) 2 2 + β 0 2 .
where we have used integration by parts and the fact that φ(-∞, 0) = 0, φ(+∞, 0) = 1, W ′ (0) = W ′ (1) and α 0 = W ′′ (0). This identifies the value of β 0 . The reasoning is similar when dealing with ψ 2 and ψ 3 . ✷

Preliminary computations

The goal of this section is to prove two technical results, namely Lemmata 4.1 and 4.2, that will be used in the next section to do the proof of our main result.

In order to simplify the presention, we will use the following notations.

Abridged notations:

• ξ 1 i = γ i (t) x 1 -X i (t) ε , • ξ 2 i = x 2 ε , • ξ i = (ξ 1 i , ξ 2 i ), • φ i = φ(ξ i ), ψ α i = ψ α (ξ i ), φi = φ i -H(ξ 1 i ), • ∂ p φ i = (∂ p φ)(ξ i ), ∂ pq φ i = (∂ pq φ)(ξ i ), p, q = 1, 2, • ∂ p ψ α i = (∂ p ψ α )(ξ i ), ∂ pq ψ α i = (∂ pq ψ α )(ξ i ), p, q = 1, 2, • ∂ t ψ α i = d dt [ψ α (ξ i )], ∂ tt ψ α i = d 2 dt 2 [ψ α (ξ i )].
Remark that in regards of the above notations, the function ûε can be simply written

ûε = i φ i + ε σ α 0 - i=1,..,N α=1,2,3 a α i ψ α i . (4.1)
Then we have the following result:

Lemma 4.1 (Computation of W ′ (û ε ))
Assume (1.14) for some δ > 0 and assume W smooth enough. Given the point (x

1 , t) ∈ R × [0, T ],
there exists i 0 = i 0 (x 1 , t) ∈ {1, 2, . . . , N} such that we have with the previous abridged notations for x 2 = 0 and for ε small enough (depending on δ):

W ′ (û ε (x 1 , 0, t)) = W ′ ( φi 0 )+εW ′′ ( φi 0 )    σ(x 1 , t) α 0 - α=1,2,3 a α i 0 (t)ψ α i 0 - 1 ε i∈{1,...,N }\{i 0 } 1 α 0 πξ 1 i    +O δ (ε 2 ). (4.2)
Proof of Lemma 4.1 Using the expression (4.3) of ûε and the periodicity of W ′ , we can write

W ′ (û ε ) = W ′ i φi + ε σ α 0 - i α a α i ψ α i .
We recall from (1.14) that the values of the X i (t) are well separated, i.e.

X i+1 (t) -X i (t) ≥ 2δ > 0.
Then there exists an index i 0 = i 0 (x 1 t) (possibly non unique) such that

|x 1 -X i 0 (t)| = inf i |x 1 -X i (t)|.
Step 1:

computations for i = i 0 We have |x 1 -X i (t)| ≥ δ > 0, which implies |ξ 1 i | ≥ δ ε
, and we deduce from the last line of assumption (A) that

φ i -H(ξ 1 i ) + 1 α 0 πξ 1 i = O 1 1 + (δ/ε) 2 = O δ (ε 2 ), which shows that φi + 1 α 0 πξ 1 i = O δ (ε 2 ). ( 4.3)
Similarly, from the third line of (A), we deduce for α = 1, 2, 3

|ψ α i | ≤ C 1 + |ξ 1 i | ≤ C 1 + δ/ε = O δ (ε). (4.4)
Step 2: conclusion From (4.3) and (4.4), we obtain

i φi + ε σ α 0 - i α a α i ψ α i = φi 0 + ε σ α 0 - α a α i 0 ψ α i 0 + O δ (ε 2 ) - i =i 0 1 α 0 πξ 1 i , that yields W ′ (û ε ) = W ′ ( φi 0 ) + W ′′ ( φi 0 ) ε σ α 0 - α a α i 0 ψ α i 0 - i =i 0 1 α 0 πξ 1 i + O δ (ε 2 ),
where we have used a second order expansion of W ′ and the fact that

1 α 0 πξ 1 i = O δ (ε) for i = i 0 .

This is exactly (4.2). ✷

We also have the following result:

Lemma 4.

(Derivatives of the profile functions)

We recall that ψ 0 = φ, and for α = 0, 1, 2, 3, we set

Ψ α i (x, t) := ψ α γ i (t) x 1 -X i (t) ε , x 2 ε .
Under the assumptions and notations of Lemma 4.1, the following holds for (x,

t) ∈ R ×R + ×[0, T ]                      ∂ 1 Ψ α i 0 = γ i 0 ε ∂ 1 ψ α i 0 and ∂ 1 Ψ α i = O δ (1) if i = i 0 , ∂ 2 Ψ α i 0 = 1 ε ∂ 2 ψ α i 0 and ∂ 2 Ψ α i = O δ (1) if i = i 0 , ∂ t Ψ α i 0 = - γ i 0 X ′ i 0 ε ∂ 1 ψ α i 0 + O(1) and ∂ t Ψ α i = O δ (1) if i = i 0 , ∂ 11 Ψ α i 0 = γ 2 i 0 ε 2 ∂ 11 ψ α i 0 and ∂ 11 Ψ α i = O δ (1) if i = i 0 , (4.5)
and for all i = 1, ..., N

     ∂ 22 Ψ α i = 1 ε 2 ∂ 22 ψ α i ∂ tt Ψ α i = 1 ε 2 (γ i X ′ i ) 2 ∂ 11 ψ α i + J α i ε + O(1) with J α i := -2γ ′ i X ′ i ξ 1 i ∂ 11 ψ α i -(γ ′ i X ′ i + (γ i X ′ i ) ′ )∂ 1 ψ α i (4.6) with J α i 0 = O δ (1) and J α i = O δ (ε) if i = i 0 . (4.7)
Proof of Lemma 4.2

The computation of the space derivatives for i = i 0 are straightforward. For i = i 0 , the estimates like O δ (1) of the space derivatives ∂ p Ψ α i for p = 1, 2 and ∂ 11 Ψ α i follow from assumption (A). We have

∂ t Ψ α i = - γ i X ′ i ε + γ ′ i γ i ξ 1 i ∂ 1 ψ α i and ∂ tt Ψ α i = - γ i X ′ i ε + γ ′ i γ i ξ 1 i 2 ∂ 11 ψ α i + - (γ i X ′ i ) ′ ε + γ ′ i γ i ′ ξ 1 i + γ ′ i γ i - γ i X ′ i ε + γ ′ i γ i ξ 1 i ∂ 1 ψ α i .
We first notice that the second line of (1.14) implies that γ i is bounded, and then γ ′ i is also bounded as a consequence of (1.6). Using again assumptions (A), we immediately obtain the desired estimates for ∂ t Ψ α i and ∂ tt Ψ α i in each case i = i 0 and i = i 0 . ✷ 5 Proof of Theorem 1.1

The main result of this section is Proposition 5.1 below which will imply Theorem 1.1.

Proposition 5.1 (Plugging the ansatz in the equations) Let ûε given by (1.5) with v ε defined in (1.7) for general coefficients a α i (t) and for φ solution of (1.3) and for general ψ α , α = 1, 2, 3 such that assumption (A) holds. We assume moreover that

|∆ψ α (x)| ≤ C 1 + |x 1 | for α = 1, 2 , 3. (5.1)
We assume (1.14) and also that W, σ are smooth enough. Then for any (x, t) ∈ R × R + × [0, T ],

we have the following estimates with the index i 0 = i 0 (x 1 , t) defined in Lemma 4.1:

     ✷û ε = O δ (1) + 1 ε I 1 i 0 , N ε ûε = O δ (1) + 1 ε I 2 i 0 , (5.2)
where

                       I 1 i = - 1 c 2 0 2γ ′ i X ′ i ξ 1 i ∂ 11 φ i + (γ ′ i X ′ i + (γ i X ′ i ) ′ )∂ 1 φ i + α=1,2,3 a α i (t)∆ψ α i , I 2 i = 1 α 0 -σ(X i , t) + 1 π j =i 1 γ j (X i -X j ) (W ′′ (0) -W ′′ (φ i )) -2mγ ′ i X ′ i ξ 1 i ∂ 11 φ i + {kγ i X ′ i + m(γ ′ i X ′ i + (γ i X ′ i ) ′ )} ∂ 1 φ i + α=1,2,3 a α i (t)Aψ α i .
with

∆ψ α i := ∂ 11 ψ α i + ∂ 22 ψ α i , Aψ α i = β∂ 11 ψ α i + ∂ 2 ψ α i -W ′′ (φ i )ψ α i .
Proof of Theorem 1.1 Fom equations (1.9), (1.10) and (1.11) and assumption (A), we deduce that (5.1) holds. The same three equations also yield

         α=1,2,3 a α i ∆ψ α = a 1 i x 1 ∂ 11 φ + a 2 i ∂ 1 φ, x 2 > 0, α=1,2,3 a α i Aψ α = f α 0 (W ′′ (φ) -W ′′ (0)) + βa 1 i x 1 ∂ 11 φ + (βa 2 i + a 3 i )∂ 1 φ, x 2 = 0, (5.3)
with

f := β 0 - a 1 i 2 + a 2 i + k 0 a 3 i .
Then we see that

I 1 i = 0 if and only if        a 1 i = 2γ ′ i X i c 2 0 , a 2 i = γ ′ i X ′ i + (γ i X ′ i ) ′ c 2 0 , (5.4) and I 2 i = 0 if and only if            f = -σ(X i , t) + 1 π j =i 1 γ j (X i -X j ) , βa 1 i = 2mγ ′ i X ′ i , βa 2 i + a 3 i = kγ i X ′ i + m(γ ′ i X ′ i + (γ i X ′ i ) ′ ). (5.5)
We therefore see that (5.4) and (5.5) are satisfied if and only if β = mc 2 0 , the coefficients a α i are given by (1.8) and the ODE dynamics (1.6) for the coefficients m 0 , k 0 given by (1.13).

✷

We now turn to the proof of Proposition 5.1.

Proof of Proposition 5.1

The proof is made in several steps.

Step 1: computation of ✷û ε Using Lemma 4.2 (precisely we use (4.6) and the fourth line of (4.5)), we get

✷û ε = O δ (1) + i 1 ε 2 A 1 φ i + ε J 0 i c 2 0 - 1 ε α=1,2,3 a α i (t) A 1 ψ α i + ε J α i c 2 0 ,
where for α = 0, 1, 2, 3

A 1 ψ α i := γ i X ′ i c 0 2 ∂ 11 ψ α i -γ 2 i ∂ 11 ψ α i -∂ 22 ψ α i = -∆ψ α i
which reads explicitly for α = 0:

A 1 φ i = -∆φ i = 0.
Using moreover (4.7) and (5.1), we get

✷û ε = O δ (1) + 1 ε J 0 i 0 c 2 0 - α=1,2,3 a α i 0 (t)A 1 ψ α i 0 = O δ (1) + 1 ε I 1 i 0 .
Step 2: computation of N ε ûε

Step 2.1: computation Using Lemma 4.2, we get

N ε ûε = O δ (1) + W ′ (û ε ) ε 2 - σ ε + i 1 ε 2 (A 2 φ i + εmJ 0 i -εkγ i X ′ i ∂ 1 φ i ) - 1 ε α=1,2,3 a α i (A 2 ψ α i + εmJ α i -εkγ i X ′ i ∂ 1 ψ α i )
with for α = 0, 1, 2, 3

A 2 ψ α i := β (γ i X ′ i ) 2 c 2 0 ∂ 11 ψ α i -γ 2 i ∂ 11 ψ α i -∂ 2 ψ α i = -β∂ 11 ψ α i -∂ 2 ψ α i .
Using (4.7) and assumption (A), we deduce

N ε ûε = O δ (1)+ 1 ε 2 (W ′ (û ε ) + A 2 φ i 0 )+ 1 ε -σ + 1 ε i =i 0 A 2 φ i + mJ 0 i 0 -kγ i 0 X ′ i 0 ∂ 1 φ i 0 - α=1,2,3 a α i 0 A 2 ψ α i 0 .
From equation (1.3) we have, for all i = 1, ..., N A 2 φ i = -W ′ (φ i ) = -W ′ ( φi ).

From assumption (A) we deduce that for i = i 0 ,

A 2 φ i = -W ′′ (0) φi + O( φ2 i ) = 1 πξ 1 i + O δ (ε 2 ), then N ε ûε = O δ (1)+ 1 ε 2 W ′ (û ε ) -W ′ ( φi 0 ) + 1 ε -σ + i =i 0 1 πεξ 1 i + mJ 0 i 0 -kγ i 0 X ′ i 0 ∂ 1 φ i 0 - α=1,2,3 a α i 0 A 2 ψ α i 0
.

Using now Lemma 4.1, we get

N ε ûε = O δ (1) + 1 ε          W ′′ ( φi 0 ) σ α 0 - α=1,2,3 a α i 0 ψ α i 0 - i =i 0 1 α 0 πεξ 1 i + i =i 0 1 πεξ 1 i -σ + mJ 0 i 0 -kγ i 0 X ′ i 0 ∂ 1 φ i 0 - α=1,2,3 a α i 0 A 2 ψ α i 0          .
Using the fact that A 2 ψ α i 0 = -Aψ α i 0 -W ′′ (φ i 0 )ψ α i 0 we get

N ε ûε = O δ (1) + 1 ε          (W ′′ ( φi 0 ) -W ′′ (0)) σ α 0 - i =i 0 1 α 0 πεξ 1 i +mJ 0 i 0 -kγ i 0 X ′ i 0 ∂ 1 φ i 0 + α=1,2,3 a α i 0 Aψ α i 0         
.

(5.6)

Step 2.2: evaluation We now write εξ 1 i = γ i (X i 0 -X i ) + γ i γ i 0 εξ 1 i 0 to obtain for i = i 0 :

1 α 0 πεξ 1 i = 1 α 0 πγ i (X i 0 -X i ) + O δ (εξ 1 i 0 ),
where we have used assumption (1.14). Therefore 1 ε (W ′′ ( φi 0 ) -W ′′ (0))

i =i 0 1 α 0 πεξ 1 i = 1 ε (W ′′ ( φi 0 ) -W ′′ (0)) i =i 0 1 α 0 πγ i (X i 0 -X i ) + O δ (ξ 1 i 0 φi 0 ).
Finally, by plugging this relation into (5.6) and using assumption (A) to see that O δ (ξ 1 i 0 φi 0 ) = O δ (1), we obtain the second equation of (5.2). ✷

Corollary 3 . 2 (

 32 Because φ solves (1.3), we deduce that ζ = ∂ 1 φ solves the linearized equation, i.e. Values of the parameters for the correctors)


André Ancel
email: andre.ancel@iphc.cnrs.fr

Michaël Beaulieu

Caroline Gilbert

Revue des différentes stratégies de reproduction des manchots The different breeding strategies of penguins: a review

Keywords: Southern hemisphere, breeding range, mate fidelity, egg formation, reproduction, vulnerability hémisphère Sud, aire de reproduction, fidélité, formation de l'oeuf, reproduction, vulnérabilité

Les 18 espèces de manchots vivent exclusivement dans l'hémisphère Sud, de l'Equateur jusqu'en Antarctique et subissent en conséquence différentes contraintes écologiques au cours de leur cycle reproducteur. Ceci se traduit par une forte variabilité dans tous les aspects de la biologie de la reproduction chez les différentes espèces. Bien que les manchots soient de prime abord adaptés à une vie marine, ils dépendent néanmoins d'une surface ferme pour se reproduire, élever leur progéniture et muer. Nous décrivons et comparons les cycles reproducteurs de toutes les espèces de manchots tout en mettant en avant les caractéristiques de chaque espèce que ce soit la durée de leur cycle reproducteur, le statut de la population et les menaces environnementales qui pèsent sur elle, la durée des -2 -différentes phases du cycle reproducteur, la fidélité du couple, la masse corporelle, la taille, la durée et les dimensions des oeufs. Nous nous sommes plus particulièrement étendus sur le cycle reproducteur du genre Aptenodytes qui diffère notablement de celui des autres espèces.

Introduction

Natural selection favours a breeding strategy which, in a given environment, is the most likely to ensure the production of the largest number of young which survive to breed, and the survival of parents until they breed again. The decision to breed is triggered by physiological changes set in motion by changes in the environment [START_REF] Groscolas | The endocrine control of reproduction and moult in male and female Emperor (Aptenodytes forsteri) and Adélie (Pygoscelis adeliae) penguins. I. Annual changes in plasma levels of gonadal steroids and luteinizing hormone[END_REF].

In birds, the timing of reproduction is determined by proximate factors and is often initiated before birds return to their breeding grounds [START_REF] Groscolas | The endocrine control of reproduction and moult in male and female Emperor (Aptenodytes forsteri) and Adélie (Pygoscelis adeliae) penguins. I. Annual changes in plasma levels of gonadal steroids and luteinizing hormone[END_REF]2]. Among these proximate factors, the photoperiod, the temperature, and the availability and the quality of the food required for successful breeding are frequently reported [START_REF] Charnov | On clutch size and fitness[END_REF][START_REF] Lack | Population studies of birds[END_REF][START_REF] Lambrechts | A single response mechanism is responsible for evolutionary adaptative variation in a bird's laying date[END_REF][START_REF] Sharp | Strategies in avian breeding cycles[END_REF].

The majority of birds lives in a non-uniform environment (spatially and temporarily) and must breed when the conditions are the most favourable. For instance, in temperate and Polar regions, most birds have to synchronize their breeding schedule with the time of the year that will give their offspring the best chances of survival [START_REF] Monaghan | Effect of changes in food availability on reproductive effort in Arctic Terns Sterna paradisaea[END_REF]. This restricted period of time dictated by resource availability leads individuals from the same population to breed synchronously. This breeding synchrony can also be associated with different advantages such as the reduction of the impact of predators and the maximisation of chick survival, by timing fledging when prey abundance is highest [START_REF] Camphuysen | Post-fledging dispersal of Common Guillemots Uria aalge guarding chicks in the North Sea: The effect of predator presence and prey availability at sea[END_REF][START_REF] Liddle | Interannual variation in the breeding biology of the Antarctic prion Pachyptila desolata at Bird Island, South Georgia[END_REF], and/or when climatic conditions are most favourable [START_REF] Croxall | Food, feeding ecology and ecological segregation of seabirds at South Georgia[END_REF].

Among seabirds, penguins are a distinctive group of flightless, long-lived pelagic seabirds. All the 18 species of penguins are grouped within a single family, the Spheniscidae. Within this family, there are six genera and each genus comprises one to eight species (Table 1). Several subspecies of little Eudyptula minor [START_REF] Sharp | Strategies in avian breeding cycles[END_REF] and gentoo penguins Pygoscelis papua (2) have been identified [START_REF] Marchant | Handbook of Australian[END_REF][START_REF] Williams | The Penguins: Sphenicidae[END_REF] but for the purpose of this review these penguins are considered as a single species. Some penguin species have been studied in detail, especially the Adélie penguin Pygoscelis adeliae (more than 400 papers, see ISI Web of Knowledge), but others, such as the Snares penguin Eudyptes robustus and erect-crested penguin Eudyptes sclateri have not (less than 10 publications, see ISI Web of Knowledge).

Consequently, there is less information available for some species. All penguins have a similar body shape and structure. The sexes are outwardly similar in all species, though males are usually heavier, at least at the onset of the breeding season, and larger than females (Table 1). They vary considerably in height, from about 40 to 130 cm, and in body mass, from 1 to 37 kg (little and emperor Aptenodytes forsteri penguins, respectively, Table 1). Such a range in body mass is only surpassed by another flightless family: the Struthioniformes, where the smallest species, the Little spotted kiwi Apteryx owenii, measures 40 cm and weighs 1.2 kg compared to the largest living ratite, the African ostrich Struthio camelus, that stands roughly 3 m tall and weighs 160 kg.

Penguins are highly adapted for marine life and some species spend as little as 20 % of the year on land (Fig. 1A). Nevertheless, this relatively short period on land represents one of the most important parts of their life cycle; during which penguins have to obtain a nest-site and mate, lay egg(s), rear chick(s), and moult.

The 18 penguins' species share some breeding characteristics such as breeding synchrony (except for the Galápagos penguins Spheniscus mendiculus, [START_REF] Vargas | Biological effects of El Niño on the Galápagos penguin[END_REF]) for annual reproduction on land, and alternation of sojourns on land and at sea between partners during the chick-rearing period. An important characteristic of all penguin species is their ability to withstand prolonged period of fasting, on land or sea-ice, during breeding. With the exception of the yellow-eyed penguin Megadyptes antipodes, all penguin species breed colonially [START_REF] Cockrem | Annual cycle of sex steroids in the yellow-eyed penguin (Megadyptes antipodes) on south-island, New-Zealand[END_REF]. In these colonies, birds often breed in high densities, for instance, 1.4 nest.m -2 in Adélie penguins [START_REF] Beaulieu | Alloparental feeding in Adélie penguins: why is it uncommon?[END_REF] and up to 10 birds.m -2 in huddling emperor penguins [START_REF] Prévost | Ecologie du manchot empereur, Expéditions polaires françaises[END_REF][START_REF] Gilbert | How do weather conditions affect the huddling behaviour of emperor penguins?[END_REF], which highly increases the level of social stimulations and interactions [START_REF] Viblanc | Coping with social stress: heart rate responses to agonistic interactions in king penguins[END_REF]. High densities of breeding birds have led to the evolution of a varied and complex repertoire of visual and vocal displays (e.g. [START_REF] Aubin | How to vocally identify kin in a crowd: the penguin model[END_REF]). As penguins are widely distributed in the Southern hemisphere, from the Equator to the Antarctic continent, they are submitted to various ecological constraints that will impact the timing of reproduction. For example, for a high latitude species such as the emperor penguin, two proximate factors controlling the onset of breeding are exogenous factors such as day-length [START_REF] Groscolas | The endocrine control of reproduction and moult in male and female Emperor (Aptenodytes forsteri) and Adélie (Pygoscelis adeliae) penguins. I. Annual changes in plasma levels of gonadal steroids and luteinizing hormone[END_REF][START_REF] Groscolas | The endocrine control of reproduction and molt in male and female Emperor (Aptenodytes forsteri) and Adélie (Pygoscelis adeliae) penguins. II. Annual changes in plasma levels of thyroxine and triiodothyronine[END_REF] and the peak of ocean productivity preceding breeding [START_REF] Ancel | The long engagement of the emperor penguin[END_REF]. For the equatorial Galápagos penguin, the onset of breeding is closely linked to mean sea surface temperature [START_REF] Boersma | Breeding patterns of Galápagos penguins as an indicator of oceanographic conditions[END_REF]. Therefore, because of the variety of environments where penguins live, the examination of the breeding cycle observed across the Sphenisciformes order can give some insights about how environmental conditions can shape a variety of reproductive strategies.

The different aspects of the biology of penguins are relatively sparse in the literature although some books attempted to gather these data (e.g. [START_REF] Marchant | Handbook of Australian[END_REF][START_REF] Williams | The Penguins: Sphenicidae[END_REF][START_REF] Dann | The Penguins: Ecology and Management[END_REF][START_REF] Reilly | Penguins of the World, S. Dobney[END_REF]). Therefore, the aim of the present study is to provide researchers working on seabirds with a concise and clear overview of the breeding cycle of the 18 penguin species.

In this review, the breeding cycle of penguins was divided into three stages:

(1) the pairing period, when breeders come ashore for courtship and mating, (2) the incubation period, when mates generally take turns to incubate the egg(s), and (3) the rearing period, from chick(s) guard to fledging. We defined the courtship period as the shortest period of time between the early arrival of both sexes on the reproductive site and the laying of the first egg because females may occasionally go to sea before the full clutch is completed (e.g. gentoo penguins, [START_REF] Marchant | Handbook of Australian[END_REF], magellanic penguins Spheniscus magellanicus, [START_REF] Yorio | Causes of nest desertion during incubation in the magellanic penguin (Spheniscus-magellanicus)[END_REF]). The body mass of the adult birds considered was the mean body mass at the onset of the breeding period, i.e. at their arrival on the breeding site.

Geographical range and population status

Penguins are widely distributed in the Southern hemisphere, mainly between 45 and 60°S (Fig. 2) and four penguin species are endemic to their breeding island: Fiordland (or crested) Eudyptes pachyrhynchus, Galápagos, royal Eudyptes schlegeli and Snares penguins. Penguins represent roughly 90 % of bird biomass in the Southern Ocean [START_REF] Williams | The Penguins: Sphenicidae[END_REF] and Pygoscelis species (chinstrap P. antarctica, Adélie and gentoo penguins) represent 70 % of Antarctic avian biomass [START_REF] Trivelpiece | Ecological segregation of Adélie, gentoo and chinstrap penguins at King George Island, Antarctica[END_REF]. Penguins feed in pelagic cold waters, rich in zooplankton and biomass, where they consume approximately two millions tons of carbon per year [START_REF] Woehler | Consumption of Southern Ocean marine resources by penguins[END_REF]. However, they occupy a wide variety of habitats while breeding on land, ranging from the burrows in the volcanic rocks of the hot Galápagos Islands (Galápagos penguin), the bushes of south Australia and New-Zealand (little penguin), to the ice of the border of the Antarctic continent (emperor, Adélie, chinstrap and gentoo penguins). This results in a high variability in all aspects of the breeding biology of the different penguins' species (Fig. 1A). While most species breed annually, some species, such as African (or jackass) and Galápagos penguins, can have no distinct annual breeding season (i.e. breeding season occurs year-round). Because of the short Antarctic summer, breeding is generally much synchronised in Antarctic species, but in temperate climates, it can be spread over a longer period of time. For example, the timing of egg-laying greatly varies between years for the little penguin [START_REF] Robinson | The effect of body condition on the timing and success of breeding in Little Penguins Eudyptula minor[END_REF].

Penguins exclusively rely on marine food sources that are spatially and temporarily unpredictable. Penguins make use of wide geographical areas in the ocean while foraging and during migrations. Penguins, which are central place foragers during the breeding season, are thus particularly sensitive to variations in ecosystem structure and processes. Of the 18 penguins' species, 13 are considered endangered or threatened (Table 1) and some species are now at their lowest recorded populations: Galápagos, yellow-eyed, and Fiordland have less than 3,000 pairs; Humboldt (Spheniscus humboldti), Snares and African have less than 30,000 pairs. Even abundant species like the macaroni (Eudyptes chrysolophus) and the rockhopper (Eudyptes chrysocome and E. moseleyi) are in steep decline [START_REF]IUCN 2012, IUCN Red List of Threatened Species[END_REF]. On the contrary, the global population trend is stable for Snares and Adélie penguins [START_REF]IUCN 2012, IUCN Red List of Threatened Species[END_REF] or is increasing for gentoo [START_REF] Clausen | Status and numerical trends of King, Gentoo, and Rockhopper Penguins breeding in the Falkland Islands[END_REF][START_REF] Ghys | Population Trend and Breeding Biology of Gentoo Penguin in Martillo Island, Tierra Del Fuego, Argentina[END_REF] and king penguins [START_REF]IUCN 2012, IUCN Red List of Threatened Species[END_REF]. The status of the emperor penguin might change in the near future. Indeed, thanks to satellite images showing faecal stains on ice, Fretwell and Trathan [START_REF] Fretwell | Penguins from space: faecal stains reveal the location of emperor penguin colonies[END_REF] discovered 10 new emperor penguins colonies [START_REF] Fretwell | An emperor penguin population estimate: the first global, synoptic survey of a species from space[END_REF]. Around 80 % of the threatened species live on islands, increasing their vulnerability to threats such as introduced predators. Many penguin species face the same four key threats: global climate change, marine pollution, fisheries mismanagement, and introduced mammalian predators [START_REF] Barbraud | Effects of climate change and fisheries bycatch on Southern Ocean seabirds: a review[END_REF].

Relative importance of each phase of the breeding cycle: focus on pairing period

Among the 18 penguin species, the duration of the breeding season ranges from 4 to 15 months (Fig. 1A). It is especially long for the two largest penguins (who need more time for their chick to grow) and particularly for the king penguin Aptenodytes patagonicus. Emperor penguins' cycle lasts 9 months including a 1.5 month pairing period that is long relative to other penguin species (Fig. 1A; [START_REF] Ancel | The long engagement of the emperor penguin[END_REF]). Indeed, while the courtship period ranges from less than 1 week in little, gentoo, Jackass or African Spheniscus demersus and magellanic penguins, it lasts up to 6 weeks in emperor penguins (Fig. 1A). The ratio courtship-breeding duration accounts for 16 % in emperor penguin (1.5 month vs. 9 months) whereas it accounts for only 3 % in its closest relative, the king penguin (1.5 week vs. 15 months). Considering the long duration of the total breeding cycle in emperor penguins, this ratio is, however, almost of the same magnitude (14 -18 %) as with other species breeding in Antarctica (Adélie, chinstrap and gentoo penguins; Fig. 1B). Contrary to the emperor penguin, the king penguin does not exhibit a long courtship period but an extended chick-rearing period (84 % of the breeding cycle while it represents about 57 % in other species; Fig. 1B). In king penguins, breeding overlaps two years (Fig. 1A) which consequently results in a maximum of two chicks being reared within a three year period.

Fidelity, sex ratio, and courtship

All penguins are monogamous, mating with only a single partner each year. In addition, most penguin species are territorial and show a moderate to high interannual fidelity in breeding (59 -89 %; Table 1), partners reuniting from year to year on their nest site. Pair bonds can therefore be long-lasting (Table 1) with many birds returning to meet their previous partners at the same breeding site each year. Two Aptenodytes species and especially the emperor penguin, however, show a low interannual fidelity (15 %; Table 1), due to the fact that they do not use a nest, and incubate their egg on their feet [START_REF] Bried | Why do Aptenodytes penguins have high divorce rates?[END_REF][START_REF] Jouventin | Comportement et structure sociale chez le manchot empereur[END_REF]. Partners cannot therefore reunite themselves from year to year on the nest site and one could hypothesize that these penguins need more time than other species to find and reunite with their previous mate. King penguins, however, show a low inter-annual fidelity (29 %; Table 1) despite being highly territorial and occupying distinct nest sites. These birds could easily reunite with their partners provided they return on time but it does not seem to be the case for 70 % of them.

A male-biased sex ratio appears to be a characteristic of breeding populations of several penguin species such as yellow-eyed, gentoo, little, Adélie, macaroni penguins [START_REF] Marchant | Handbook of Australian[END_REF]. In Adélie and macaroni penguins, males return first [START_REF] Marchant | Handbook of Australian[END_REF] to previous year's nest-site to maximize their chances of reuniting with their mates from the previous year. This suggests that the primary "aim" of male penguins returning to the breeding colony would be to retain their old nest-sites and, only secondarily, to reunite with their previous mates. Because in many species there are more males in the population than females, male-male competition for nest-sites, rather than for mates, is an important determinant of breeding opportunity. In contrast, at least for the Pointe Géologie colony (Adélie Land, 140°E-67°S), sex ratio favours female emperor penguins that outnumber males by about 10 % [START_REF] Isenmann | Contribution à l'éthologie et à l'écologie du manchot empereur (Aptenodytes forsteri Gray) à la colonie de Pointe Géologie (Terre Adélie)[END_REF][START_REF] Bried | Why do Aptenodytes penguins have high divorce rates?[END_REF][START_REF] Jouventin | Comportement et structure sociale chez le manchot empereur[END_REF][START_REF] Jenouvrier | Long-term contrasted responses to climate of two Antarctic seabirds species[END_REF]. This might be due to a higher mortality of male breeders after their long winter fast [START_REF] Barbraud | Emperor penguins and climate change[END_REF]. Because of this unbalanced sex-ratio and because mate fidelity is particularly low in this species, competition between females is high to find a male and the earlier a female returns at the onset of the breeding cycle, the higher the probability she will get a mate [START_REF] Isenmann | Contribution à l'éthologie et à l'écologie du manchot empereur (Aptenodytes forsteri Gray) à la colonie de Pointe Géologie (Terre Adélie)[END_REF],

and the lower the probability that her previous mate will already be paired with another female. Furthermore, the number of male partners available per unpaired female decreases as time passes. Consequently, an early arrival enhances the likelihood for the females to preserve their breeding status by finding an unpaired mate. Unbalanced sex ratio in emperor penguins probably also explains occurrence of polygynous trios (one male with two females) but these groups are temporary, one female usually ejecting the other after a few hours [START_REF] Prévost | Ecologie du manchot empereur, Expéditions polaires françaises[END_REF][START_REF] Isenmann | Contribution à l'éthologie et à l'écologie du manchot empereur (Aptenodytes forsteri Gray) à la colonie de Pointe Géologie (Terre Adélie)[END_REF].

From yolk development to fledging

The period of yolk development that precedes ovulation is given by the following equation for non-procellariiform seabirds: log t = 0.396 + 0.283 log egg mass [START_REF] Astheimer | A comparison of yolk growth rates in seabird eggs[END_REF] where t stands for time (day) and egg mass is expressed in grams. Once the yolk is fully developed, it is retained within the ovarian follicle for about 6 days before ovulation [START_REF] Astheimer | The timing and energetic consequences of egg formation in the Adélie penguin[END_REF]. The albumen and shell are then added over about 24 hours, following ovulation. In species with two eggs [START_REF] Scholten | Choice of nest-site and mate in Humboldt penguins (Spheniscus humboldti)[END_REF][START_REF] Spurr | Behaviour of the Adélie penguin chick[END_REF][START_REF] Williams | The clutch size of macaroni and rockhopper penguins[END_REF][START_REF] Williams | The laying interval and incubation period of rockhopper and macaroni penguins[END_REF], development of the second egg (B-egg) is initiated 4 days after the first one (A-egg). Thus, in order to calculate the time elapsed from yolk formation until egg-laying, we added 7 (6+1) and 11 days (4+6+1) for the A-egg and the B-egg, respectively (Table 1).

Incubation refers to the process by which birds lay their egg(s), and to the development of the embryo within the egg. The most vital factors of incubation are the temperature required for embryo development over a specific period, humidity (which could be problematic in dry environments like the Galápagos Islands or Antarctica), and egg rotation rate [START_REF] Beaulieu | When sea-ice clock is ahead of Adélie penguins' clock[END_REF]. If the air is too dry, the egg will lose too much water, which can make hatching difficult or impossible. As incubation progresses, an avian egg becomes lighter, and the air space within the egg becomes larger owing to the evaporation from the egg. In most penguin species, incubation is divided differently between parents with the male and the female taking turns incubating the egg(s). The emperor penguin represents an exception as in this species, only the male incubates. Compared with other birds, penguins lay small eggs (and small egg clutches) relative to their body weight (2.9 ± 0.9 %, n = 16, mean ± SD, Table 1), and eggs of smaller penguin species are proportionately larger (Table 1). The majority of penguins lays two white eggs (except genus Aptenodytes laying only one egg), weighing from 55 to 445 g (little and emperor penguins, respectively, Table 1). Within the genus Eudyptes, the second egg or B-egg is 20 -78 % larger than the first one (Table 1) and consequently the first chick is smaller than the second one and generally fails to survive [START_REF] Marchant | Handbook of Australian[END_REF]. Erect-crested penguins are obligate brood reducers [START_REF] St | Incubation behaviour, brood patch formation and obligate brood reduction in Fiorland crested penguins[END_REF] and exhibit the most extreme egg-size dimorphism of any bird: the second egg is up to 100 % the size of the first and is the only one to be incubated. It remains unclear why these birds should lay two eggs but only ever rear one chick. The total time necessary for yolk deposition is proportional to female's body mass [START_REF] Astheimer | The timing and energetic consequences of egg formation in the Adélie penguin[END_REF] but full clutch synthesis has the same duration among the 18 penguin species, ca. 3 weeks (Table 1; 20.6 ± 0.7 days, n = 16, mean ± SD). By subtracting the duration of the full clutch completion from the duration of the pairing period, we can report that egg formation begins while females are still at sea (in 13 species), while they arrive at the breeding site (in 1 species), and while they already are on the breeding site (in 3 species; Fig. 3). Among the latter, emperor penguins' egg formation begins late, three weeks after their arrival on the colony. Thus, egg formation is not responsible for the long duration of the pairing period, representing only one half of the courtship duration.

Incubation ranges from ca. 35 to 65 days (Eudyptes and Pygoscelis species, emperor penguins, respectively, Fig. 1B). It is an energetically demanding process, especially in male emperor penguins that can lose up to 45 % of their initial body weight during this time. Some species begin incubation with the first egg, causing the young to hatch asynchronously; others begin after laying the second egg, thus decreasing hatching asynchrony between siblings [START_REF] Moreno | Hatching asynchrony, sibling hierarchies and brood reduction in the chinstrap penguin Pygoscelis Antarctica[END_REF].

Both parents are involved in parental care. From the moment the egg is hatched, one parent cares for the newly hatched chick(s) while the other forages for food. Penguin chicks are semi-altricial, i.e. they need parental care (food, warmth, protection) before becoming independent. At the start of the rearing period, the chicks either sit on their parents feet (emperor and king penguins) or under their bellies, to be kept warm and dry. Young are guarded (guarding stage) by both parents for varying periods of time before forming crèches. Both parents feed the chicks by regurgitation. Nestlings beg for food by pecking adult's bill and/or by singing. Guarding stage may be affected by environmental conditions [START_REF] Beaulieu | When sea-ice clock is ahead of Adélie penguins' clock[END_REF] and when they are not being protected by the adult, the chicks form crèches to keep warm and stay protected. As the chicks grow, their feeding requirements quickly increase, making it difficult for just one of the parents to obtain enough food. Eventually, the chicks are large enough so that both parents can go to sea to gather food for their chick simultaneously. When parental provisioning is low, alloparental feeding (feeding of offspring by adults other than their own parents) sometimes occurred in little [START_REF] Wienecke | Adoption of chicks by Little Penguins Eudyptula minor on Penguin Island[END_REF], emperor [START_REF] Jouventin | Adoption in the emperor penguin, Aptenodytes forsteri[END_REF], king [START_REF] Lecomte | Alloparental feeding in the king penguin[END_REF] and Adélie penguins [START_REF] Beaulieu | Alloparental feeding in Adélie penguins: why is it uncommon?[END_REF]. In most species the chicks gather together in crèches to provide protection both from predators and from the elements. In some species, such as king penguins, crèches can be large with many hundreds of chicks tightly packed together. In other species such as African penguins, the crèches are smaller (with up to 10 chicks coming together) and less dense. Age at fledging ranges from one to eleven months (Fig. 1A) and fledglings are a bit smaller and lighter than adults, except in emperor penguins for which body mass of fledglings is about half of adults' [START_REF] Prévost | Ecologie du manchot empereur, Expéditions polaires françaises[END_REF]. While they are still on their breeding grounds, chicks have to moult to adorn themselves with a waterproof plumage, which will allow them in turn to go hunting offshore and acquire their food independently.

Adults also have to moult like other birds and all penguins replace their feathers each year on their breeding grounds, except for the emperor penguins for which moulting often takes place far away from their breeding grounds [START_REF] Kooyman | Moult of the emperor penguin: travel, location, and habitat selection[END_REF][START_REF] Wienecke | Pre-moult foraging trips and moult locations of Emperor penguins at the Mawson Coast[END_REF][START_REF] Zimmer | Foraging movements of emperor penguins at Pointe Géologie, Antarctica[END_REF]. Although other birds lose some feathers individually at a time and replace them over a few months, penguin moulting is fast, taking only two (e.g. Pygoscelis antarctica, [START_REF] Ainley | Bird life at Cape Crozier, Ross Island[END_REF]) to five weeks (e.g. genus Aptenodytes, [START_REF] Prévost | Ecologie du manchot empereur, Expéditions polaires françaises[END_REF][START_REF] Groscolas | Study of molt fasting followed by an experimental forced fasting in emperor penguin Aptenodytes forsteri -relationship between feather growth, bodyweight loss, body-temperature and plasma fuel levels[END_REF][START_REF] Le Maho | Thermoregulation in fasting emperor penguins under natural conditions[END_REF]). Moulting is extremely important to penguins as they need to maintain their feathers in perfect condition at all times to insulate their body from environmental conditions at sea or on land. Adults generally moult after the breeding season. Once their chicks have moulted into their own juvenile plumage the adults return to sea for a few weeks to build up their own fat reserves and then come back ashore to moult. During the moult, penguins are no longer waterproof and cannot enter the sea, they can lose up to 50 % of their body mass, they are not well insulated and they are vulnerable to predation. Therefore, this is a critical period of time for penguins during which they have to face the elements and starve until their new set of feathers is ready.

Genus Aptenodytes

Birds of the genus Aptenodytes (A. forsteri and A. patagonicus) are bigger and taller than other penguin species (Table 1). They do not build a nest and only lay one large egg (on average 445 and 307 g, respectively; Table 1), which is kept on the top of the incubating parents' feet at least for 54 days (Fig. 1B). As for other penguins, parents recognise their chick by voice and young also recognise parents by call [START_REF] Marchant | Handbook of Australian[END_REF].

The breeding cycle of the genus Aptenodytes largely differs from the breeding cycle of most of the other penguin species. In most penguin species, it takes from 8 to 15 weeks to raise a chick to the juvenile stage (Fig. 1B) but it can take 10 to 13 months for king penguins to fledge their chick [START_REF] Stonehouse | The king penguin Aptenodytes patagonicus of South Georgia I. Breeding behaviour and development[END_REF]. Because of this long chickrearing period, king penguins only produce two chicks every three years. As a result, 12-month-old chicks cohabit with incubating birds in king penguin colonies. Every year, emperor penguins manage to raise their large chicks more quickly (five months), using a different strategy. Indeed the chicks moult into juvenile plumage while they are still much smaller than their parents. The juveniles then continue to grow out at sea. The emperor penguin also seems to be an exception among penguins as they begin their breeding cycle when other Antarctic birds have finished theirs. Each year, from late December to March (i.e. late summer), emperor penguins disperse into the oceans, travelling and foraging into the waters surrounding the Antarctic continent [START_REF] Zimmer | Foraging movements of emperor penguins at Pointe Géologie, Antarctica[END_REF][START_REF] Kooyman | Penguin dispersal after fledging[END_REF]. In March-April, as winter approaches and fast-ice extent grows, all mature emperor penguins move south towards their colonial breeding areas at the border of the Antarctic continent. The breeding cycle of emperors stands in contrast with that of other penguin species (except for the king penguin) by its long duration and by the fact that it takes place in the midst of the severe Antarctic winter, whereas other penguins breed during the short and milder summer season. Indeed, the emperor penguin is one of the few birds for which gonadal growth is coincident with short days, when birds are still at sea. Gonadal steroids are several fold above basal level at the time of arrival on the breeding area suggesting that environmental cues, especially decreasing daylength, decreasing air temperature, and sea-ice formation, stimulate gonadal development and reproduction [START_REF] Groscolas | The endocrine control of reproduction and moult in male and female Emperor (Aptenodytes forsteri) and Adélie (Pygoscelis adeliae) penguins. I. Annual changes in plasma levels of gonadal steroids and luteinizing hormone[END_REF]. Before breeding, emperor penguins forage far away from their breeding grounds [START_REF] Zimmer | Foraging movements of emperor penguins at Pointe Géologie, Antarctica[END_REF][START_REF] Peron | Projected poleward shift of king penguins' (Aptenodytes patagonicus) foraging range at the Crozet Islands, southern Indian Ocean, P. Roy[END_REF][START_REF] Scheffer | Frontal zones, temperature gradient and depth characterize the foraging habitat of king penguins at South Georgia[END_REF]. In the Southern Ocean, a period of high productivity occurs during summer, from October to April, and is followed by a period of low productivity during winter, from May to September [START_REF] Foxton | The distribution of the standing crop of zooplankton in the southern ocean[END_REF][START_REF] Hart | Phytoplancton periodicity in Antarctic surface waters[END_REF][START_REF] Kopczyńska | Spring phytoplankton assemblages in the Southern Ocean between Australia and Antarctica[END_REF]. Because emperor penguins breed in winter, they have to anticipate their breeding season by accumulating body reserves during high ocean productivity in the previous summer [START_REF] Ancel | The long engagement of the emperor penguin[END_REF]. To our knowledge, this breeding strategy is unique among animals. Furthermore, emperors breed on sea-ice in a few favourable zones that may be a hundred kilometres distant from the open sea or polynias where they exclusively feed [START_REF] Ancel | Foraging behaviour of emperor penguins as a resource detector in winter and summer[END_REF]. As a consequence of the distance from the feeding grounds, and because breeding activity competes with feeding, female and male emperors fast for as long as 1.5 and 4 months, respectively [START_REF] Prévost | Ecologie du manchot empereur, Expéditions polaires françaises[END_REF][START_REF] Isenmann | Contribution à l'éthologie et à l'écologie du manchot empereur (Aptenodytes forsteri Gray) à la colonie de Pointe Géologie (Terre Adélie)[END_REF]. For females, the breeding fast comprises only the courtship period, since they leave their single egg to their mate as soon as it is laid and then go back to sea for building up their reserves. For males, the period of fasting includes the courtship and the whole incubation period. To face the austral winter, emperor penguins have to exploit in an optimal way their limited body fuels in order to succeed in their breeding [START_REF] Prévost | Ecologie du manchot empereur, Expéditions polaires françaises[END_REF][START_REF] Ancel | Energy saving in huddling penguins[END_REF][START_REF] Gilbert | Body temperature changes induced by huddling in breeding male emperor penguins[END_REF]. This is possible only thanks to their huddling behaviour, which allows them to decrease energy expenditure [START_REF] Ancel | Energy saving in huddling penguins[END_REF][START_REF] Gilbert | Body temperature changes induced by huddling in breeding male emperor penguins[END_REF].

Which environmental changes might affect the breeding cycle of the different penguin' species?

The Earth's climate is undergoing rapid warming which is driving shifts in the distribution and phenology of many plants and animals [START_REF] Hansen | Climate Events Synchronize the Dynamics of a Resident Vertebrate Community in the High Arctic[END_REF][START_REF] Hindell | Long-term breeding phenology shift in royal penguins[END_REF]. Among animals, penguins are adapted to live in extreme environments (Fig. 2), but, because each species is restricted to a limited latitudinal range (Table 1), they can be highly sensitive to climate change [START_REF] Forcada | Penguin responses to climate change in the Southern Ocean[END_REF]. Environmental changes are not uniform across -16 -regions, with resource increasing in the subantarctic areas and decreasing in Antarctica [START_REF] Hindell | Long-term breeding phenology shift in royal penguins[END_REF]. Quantifying changes in breeding phenology is important for understanding how populations respond to these changes, especially those resulting from human activities [START_REF] Hindell | Long-term breeding phenology shift in royal penguins[END_REF].

Climatic changes and resource availability: Detecting and predicting how populations respond to environmental variability are crucial challenges in management and conservation research. This is particularly true for populations at high latitudes, many of which demonstrate changes in population dynamics associated with global warming [START_REF] Baylis | Climate as a driver of population variability in breeding Gentoo Penguins Pygoscelis papua at the Falkland Islands[END_REF]. Some seabird populations of the Southern Ocean have been responding to climate change for the last three decades and demographic models suggest that projected warming will cause dramatic population changes over the next century [START_REF] Peron | Projected poleward shift of king penguins' (Aptenodytes patagonicus) foraging range at the Crozet Islands, southern Indian Ocean, P. Roy[END_REF]. In the Antarctic ecosystem, population dynamics of top predators like penguins may yield important information about how the environment is changing [START_REF] Barbosa | Population decline of chinstrap penguins (Pygoscelis antarctica) on Deception Island, South Shetlands, Antarctica[END_REF]. The phenotypic plasticity of penguins may allow them to continue to exploit their transformed ecological niche and maintain their current distributional ranges. For instance, penguins may vary the timing of breeding in response to changes in environmental conditions [START_REF] Hinke | Flexible reproductive timing can buffer reproductive success of Pygoscelis spp. penguins in the Antarctic Peninsula region[END_REF]. However, paleoecological records suggest that penguins are more likely to respond by dispersal rather than adaptation [START_REF] Forcada | Penguin responses to climate change in the Southern Ocean[END_REF]. Thus shift in species distribution is likely to be one of the major possible adaptations to changing environmental conditions [START_REF] Peron | Projected poleward shift of king penguins' (Aptenodytes patagonicus) foraging range at the Crozet Islands, southern Indian Ocean, P. Roy[END_REF]. This is exemplified by the distributional range of chinstrap [START_REF] Forcada | Penguin responses to climate change in the Southern Ocean[END_REF][START_REF] Barbosa | Population decline of chinstrap penguins (Pygoscelis antarctica) on Deception Island, South Shetlands, Antarctica[END_REF][START_REF] Lynch | Spatially integrated assessment reveals widespread changes in penguin populations on the Antarctic Peninsula[END_REF], gentoo and Adélie penguins [START_REF] Forcada | Penguin responses to climate change in the Southern Ocean[END_REF][START_REF] Carlini | Responses of Pygoscelis adeliae and P. papua populations to environmental changes at Isla 25 de Mayo (King George Island)[END_REF][START_REF] Lynch | Population trends and reproductive success at a frequently visited penguin colony on the western Antarctic Peninsula[END_REF] that has shifted southwards around the Antarctic Peninsula. However, as each species is limited to a specific latitudinal range, a latitudinal shift may be very limited. Thus, emperor penguins' colonies north of 70°S are projected to decrease or disappear, and limited growth might occur south of 73°S -17 - [START_REF] Ainley | Antarctic penguin response to habitat change as Earth's troposphere reaches 2 degrees C above preindustrial levels[END_REF]. These population trends are likely to be related to sea ice conditions [START_REF] Massom | Fast ice distribution in Adélie Land, East Antarctica: interannual variability and implications for emperor penguins Aptenodytes forsteri[END_REF]. For example, at Pointe Géologie (Adélie Land), distance to the fast ice edge and its extent are major determinants of emperor penguin breeding success [START_REF] Massom | Fast ice distribution in Adélie Land, East Antarctica: interannual variability and implications for emperor penguins Aptenodytes forsteri[END_REF].

Therefore, the increased frequency of warm events associated with projected decreases in sea ice extent is likely to reduce population viability [START_REF] Jenouvrier | Demographic models and IPCC climate projections predict the decline of an emperor penguin population[END_REF][START_REF] Jenouvrier | Effects of climate change on an emperor penguin population: analysis of coupled demographic and climate models[END_REF].

Other physical factors than sea-ice can also affect penguin populations. For instance, sea surface temperature consistently drives the foraging behaviour of king penguins, and, according to climate models, the projected warming of surface waters could lead to a gradual southward shift of their more profitable foraging zones [START_REF] Peron | Projected poleward shift of king penguins' (Aptenodytes patagonicus) foraging range at the Crozet Islands, southern Indian Ocean, P. Roy[END_REF].

Such a shift would negatively affect the king penguin population, unless penguins develop alternative foraging strategies [START_REF] Peron | Projected poleward shift of king penguins' (Aptenodytes patagonicus) foraging range at the Crozet Islands, southern Indian Ocean, P. Roy[END_REF] as to modify their timing of breeding [START_REF] Hinke | Flexible reproductive timing can buffer reproductive success of Pygoscelis spp. penguins in the Antarctic Peninsula region[END_REF].

The Antarctic Peninsula is among the fastest-warming areas on the Earth, with 5-6°C increases in mean winter air temperatures and associated decreases in winter sea-ice cover [START_REF] Trivelpiece | Variability in krill biomass links harvesting and climate warming to penguin population changes in Antarctica[END_REF]. These perturbations have affected the ecosystem profoundly [START_REF] Trivelpiece | Variability in krill biomass links harvesting and climate warming to penguin population changes in Antarctica[END_REF]. To respond to these climatic changes, varying the timing of reproduction in response to local environmental conditions is a key factor influencing reproductive success [START_REF] Hinke | Flexible reproductive timing can buffer reproductive success of Pygoscelis spp. penguins in the Antarctic Peninsula region[END_REF]. For example, clutch initiation and hatching dates of royal, Adélie and gentoo penguins occur earlier with warmer temperatures [START_REF] Hindell | Long-term breeding phenology shift in royal penguins[END_REF][START_REF] Hinke | Flexible reproductive timing can buffer reproductive success of Pygoscelis spp. penguins in the Antarctic Peninsula region[END_REF]. However, these behavioural adjustments may not be sufficient to prevent populations from declining.

The "sea-ice hypothesis" proposing that ice-obligate species directly decline because of sea-ice reduction, does not appear to be sufficient to explain why populations of both ice-loving and ice-avoiding penguins have declined significantly [START_REF] Trivelpiece | Variability in krill biomass links harvesting and climate warming to penguin population changes in Antarctica[END_REF]. Some researchers argue in favour of an alternative, more robust hypothesis, that attributes both increases and decreases in penguin populations to changes in the abundance of their main prey, Antarctic krill [START_REF] Trivelpiece | Variability in krill biomass links harvesting and climate warming to penguin population changes in Antarctica[END_REF]. Indeed, decline of chinstrap penguin populations has been suggested as being related to climate change through a reduction in sea-ice extent during winter and a consequent decline in the abundance of krill in summer during the breeding season [START_REF] Barbosa | Population decline of chinstrap penguins (Pygoscelis antarctica) on Deception Island, South Shetlands, Antarctica[END_REF].

Climate changes can also have more subtle consequences on the foraging behaviour of penguins. For instance, mixed water regimes resulting from storms, result in the dispersion of prey items in the water column. This lack of prey stratification has been described as resulting in reduced foraging efficiency and poor breeding success in the little penguin [START_REF] Ropert-Coudert | Impact of small-scale environmental perturbations on local marine food resources: a case study of a predator, the little penguin[END_REF]. Mixed water regimes are currently unusual during the breeding period of little penguins, but are expected to become more frequent due to climate change and may therefore represent an important threat for this species [START_REF] Ropert-Coudert | Impact of small-scale environmental perturbations on local marine food resources: a case study of a predator, the little penguin[END_REF].

Tourism: Antarctica now fuels one of the fastest growing tourism markets in the world with over 30,000 visitors annually travelling to the continent [START_REF] Powell | From awe to satisfaction: immediate affective responses to the Antarctic tourism experience[END_REF]. Increasing ecotourism activity has led to concerns about the effects of ecotourism on wildlife populations. While some species of penguins habituate to human visits, others exhibit negative effects due to disturbance [START_REF] Villanueva | A matter of history: effects of tourism on physiology, behaviour and breeding parameters in Magellanic Penguins (Spheniscus magellanicus) at two colonies in Argentina[END_REF]. Behavioural, physiological, and reproductive parameters might thereby be affected. For example, human presence at the nest site is physiologically stressful for breeding Magellanic penguins that are not accustomed to seeing humans [START_REF] Fowler | Behavioral and hormonal responses of Magellanic penguins (Spheniscus magellanicus) to tourism and nest site visitation[END_REF]. Indeed, Magellanic penguins in visited areas have higher corticosterone stress responses than penguins in undisturbed areas [START_REF] Villanueva | A matter of history: effects of tourism on physiology, behaviour and breeding parameters in Magellanic Penguins (Spheniscus magellanicus) at two colonies in Argentina[END_REF][START_REF] Walker | Physiological and behavioral differences in Magellanic Penguin chicks in undisturbed and tourist-visited locations of a colony[END_REF]. Moreover, birds exposed to moderate levels of disturbance do not show evidence of habituation over a period of a few years [START_REF] Fowler | Behavioral and hormonal responses of Magellanic penguins (Spheniscus magellanicus) to tourism and nest site visitation[END_REF]. However, penguins may habituate to Humans, as birds that have been exposed to very high levels of human visitation do not respond anymore to human presence as a stressor. Furthermore, Magellanic chicks from tourist-visited colonies do not flee anymore when approached by humans [START_REF] Walker | Physiological and behavioral differences in Magellanic Penguin chicks in undisturbed and tourist-visited locations of a colony[END_REF], and breeding success is not affected by visitation levels in this species [START_REF] Villanueva | A matter of history: effects of tourism on physiology, behaviour and breeding parameters in Magellanic Penguins (Spheniscus magellanicus) at two colonies in Argentina[END_REF]. However, penguin species differ in their sensitivity to human presence.

For instance, in contrast to Magellanic and Adélie penguins, yellow-eyed and gentoo penguins show significantly lower breeding success at sites exposed to unregulated tourism compared to areas visited infrequently [START_REF] Lynch | Population trends and reproductive success at a frequently visited penguin colony on the western Antarctic Peninsula[END_REF][START_REF] Ellenberg | Elevated hormonal stress response and reduced reproductive output in yellow-eyed penguins exposed to unregulated tourism[END_REF]. This may be attributed to the presence of people on beaches that delays post-foraging landing by penguins provisioning their chicks, which may in turn affect the amount of food delivered to the young. Indeed, yellow-eyed chicks from nesting areas with high numbers of tourists have significantly lower fledging weights than chicks from areas with no tourist visitors [142]. Taking into account that the probability of survival is positively associated with mass at fledging, lower fledging weights may have long-term population consequences [142].

Conclusion

The present article shows that breeding strategies are diverse and differ between penguin species. However, breeding behaviour can also exhibit some plasticity within each penguin species and particularly when environmental conditions vary (e.g. [START_REF] Beaulieu | When sea-ice clock is ahead of Adélie penguins' clock[END_REF]). More studies simultaneously conducted (1) in several penguin species breeding in the same location and (2) on the same species in different locations/environmental conditions would be useful to highlight how environmental conditions influence breeding strategies in penguins and how penguins can adapt to environmental perturbations.

Many penguin species face the same threats [START_REF] Barbraud | Effects of climate change and fisheries bycatch on Southern Ocean seabirds: a review[END_REF]. Marine and coastal ecosystems are undergoing unprecedented alterations in their processes and structure. Penguins are sensitive species impacted by these phenomena. As top predators, they are key constituents of marine ecosystems, and are indicators of the oceanic and coastal ecosystem health. Larger scale ecosystem-based conservation planning and more focused local efforts are needed for the successful conservation of many penguin species. fasting on land, foraging at sea and feeding, chick rearing, respectively. Lettering stands for species (see Table 1).

Figures and Table caption

A/ Breeding cycle of the 18 penguin species and for both sexes (M stands for males and F for females). B/ Relative breeding cycle of the 18 penguin species. Mean courtship period is defined as the shortest time between pair formation and the laying of the first egg.

Breeding cycle is defined as the time elapsed from the arrival of the penguins on their breeding site to the fledging of the chick(s).

  Figures and Table caption:

Figure 1 .

 1 Figure 1. Breeding cycle of the penguins. White, dark and dashed bars denote for

Figure 2 .

 2 Figure 2. Repartition of the 18 species of breeding penguins (%) according to the

Figure 3 .

 3 Figure 3. Time elapsed from arrival at the breeding site (dashed line) and onset of

  Body mass is the mean body mass of birds at the onset of the breeding season.Mate fidelity is from one year to the consecutive year. For egg formation values, see the Material and Methods section. The B-egg formation column represents the time elapsed since the onset of the first yolk development. Abbreviations of the different species are given in brackets after the scientific name of each species. Each reference is indicated by a number in brackets. IUCN 2012 red list codes: EN=endangered, LC=least concern, NT=near threatened, VU=vulnerable.

Table 1 .

 1 ). Some biological characteristics of the 18 penguin species.

Acknowledgements

We thank Drs C.-A. Bost, A. Chiaradia, P. Dee Boersma, C. Hull, M. Massaro, K.

Pütz, Y. Ropert-Coudert, A. Steinfurth and P. Trathan for providing data on the different species of penguins. We also thank Dr. S. Gallon for her help in revising the language. The manuscript benefited from critical comments by anonymous reviewers.