C Vernicos
email: constantin.vernicos@umontpellier.fr

CONVEX COMPACT SURFACES WITH NO BOUND ON THEIR SYNTHETIC RICCI CURVATURE

Convex compact surfaces with no bound on their synthetic Ricci curvature

Constantin Vernicos

Introduction and statement of results

Many notions of curvature adapted to a metric measure space have been defined to extend the ones existing in Riemanniann geometry. Most of them heavily rely on comparison to the Euclidean space and that is why they are quite restrictive. For instance a normed vectore space is CAT(0) if and only if it is a Euclidean space, as a consequence the only Finsler spaces wich can be CAT(0) are Riemanniann (see also [START_REF] Burago | Polyhedral finsler spaces with locally unique geodesics[END_REF]). The same thing happens with the Alexandroff spaces. It is even more restrictive in that case for an Alexandroff metric space is almost Riemanniann manifold (see [START_REF] Burago | A course in metric geometry[END_REF] for a precise statement).

Some older notion, such as the Busemann convexity, are less restrictive. However they might not pass to the Gromov-Hausdorff limit of a sequence of metric measured spaces. In the light of the current interest in understanding the limit spaces arising as limits of Riemanniann metric space, with Ricci curvature bounded from below for instance this is a huge flaw.

Following the work of Lott-Villani [START_REF] Lott | Ricci curvature for metric-measure spaces via optimal transport[END_REF] and Sturm [Stu06a,Stu06b], a new family of notion of "curved" spaces arised. They involve the convexity of an operator on the L p -Wasserstein space, which is a metrization of the space of probability measures with finite p-th moment. Among them one finds the space satisfying the curvature dimension condition CD(K, N ) or the measure contraction property M CP (K, N ). The later may be seen as a measure analog to the Busemann convexity. The former as a generalisation of having Ricci curvature bounded from below by (N -1)K and being of dimension less than N . We will refer to this last notion as synthetic Ricci curvature and describe such spaces as admitting a lower bound on their synthetic Ricci curvature. An example is given by normed vector space of dimension n which satisfy the curvature dimension condition CD(0, n) (This follows from proof of the interpolation inequality and concavity of det

1 n ).
Date: January 13, 2022.

1 Another point of view on curvature in metric spaces is based on analytical inequalities. For instance Cordero-Erausquin, McCann and Schmuckenschläger [START_REF] Cordero-Erausquin | A riemannian interpolation inequality à la borell, brascamp and lieb[END_REF] looked at the Brascamp-Lieb inequality which is a generalisation of the Prekopa-Leindler inequality that can be used to prove the Brunn-Minkoswki inequality in the Euclidean space.

The interesting aspect on which this paper is based on is that most notions of curvature deriving from the work of Lott-Villani and Sturm imply a Brunn-Minkowski inequality, hence our focus on this inequality.

Our main result is the following Theorem 1. There exist a compact C 1,1 convex surface in R 3 with the norm ||(x, y, z)|| = x 2 + y 2 + |z| which admits no lower bounds on its synthetic Ricci curvature.

The idea of that example came from the study of reflections and refraction in normed (not necessarily reflexive) vector spaces. Section 2 focuses on a specific example which allows us to obtain our convex set in Section 3.

Definitions and notations

Let (X, d, µ) be a metric measured space. For any pairs of point m 0 , m 1 ∈ X, we call m s ∈ X an s-intermediate point from m 0 to m 1 if and only if d(m 0 , m s ) = sd(m 0 , m 1 ) and d(m s , m 1 ) = (1 -s)d(m 0 , m 1 ).

Let K 0 and K 1 be two compact sets in X, the set of s-intermediate points from points of K 0 to points of K 1 will be denoted by

(1 -s)K 0 + sK 1 .
If (1 -s)K 0 + sK 1 is not measurable, we will still denote its outer measure by

µ (1 -s)K 0 + sK 1 .
Let us first start with the classical Brunn-Minkowski inequality:

Definition 2 (Classical Brunn-Minkowski inequality). Let N be greater than 1. We say that the Brunn-Minkowski inequality BM (0, N ) holds in the metric measured space (X, d, µ) if for every pair of compact set of non-zero measure K 0 and K 1 , the following inequality is satisfied

(1) µ 1/N (1 -s)K 0 + sK 1 ≥ (1 -s)µ 1/N (K 0 ) + sµ 1/N (K 1 ).
We also say that BM (0, +∞) holds if and only if

(2) µ (1 -s)K 0 + sK 1 ≥ µ 1-s (K 0 )µ s (K 1 ).
Remark 3. Notice that if for some n ∈ R * , and t,a and b ∈ R the inequality t ≥ (sa 1/n + (1 -s)b 1/n ) n holds, then from the concavity of the logarithm we have

ln t ≥ n ln sa 1/n + (1 -s)b 1/n ≥ s ln a + 1 -s ln b.
Hence any B(0, N ) implies B(0, ∞).

Now the general Brunn-Minkowski inequality BM (K, N ) requires the introduction of a family of functions depending on K, N and s ∈ [0, 1] denoted by τ Definition 4 (Generalised Brunn-Minkowski inequality). Let N be greater than 1. We say that the Brunn-Minkowski inequality BM (K, N ) holds in the metric measured space (X, d, µ) if for every pair of compact set of non-zero measure K 0 and K 1 , the following inequality is satisfied

(3) µ 1/N (1 -s)K 0 + sK 1 ≥ τ (1-s) K,N (ϑ)µ 1/N (K 0 ) + τ (s) K,N (ϑ)µ 1/N (K 1 ).
where ϑ is the minimal (resp. maximal) length of a geodesic between a point in K o and a point in

K 1 if K ≥ 0 (resp. K < 0).
We can also define the BM (K, +∞) as follows:

(4)

µ (1 -s)K 0 + sK 1 ≥ µ 1-s (K 0 )µ s (K 1 )e Ks(1-s)ϑ 2 /2 .
The curvature dimension property, denoted by CD(K, N ) is generalisation of the following sentence to metric measures spaces:

The space has dimension less than N and the ricci curvature is bigger than K. It is defined in term of a convexity property of the entropy along geodesics in the space of probability of the metric space (see [Stu06b] for more precise statements).

For our purpose we only need to know the following properties of a space satisfying a curvature dimension property (see K.T. Sturm [Stu06b]).

Property 5. Let (X, d, µ) be a metric measured space. The following implications are valid:

(1) Suppose CD(K, N ) holds. If K ≤ K, then CD(K , N ) holds as well. If N > N , then CD(K, N ) holds as well.

(2) Suppose CD(K, N ) holds. Then for any α, β > 0, the metric measured space (X, αd, βµ) satisfies the CD(K/α 2 , N ) condition.

(3) When N ≥ 1, CD(0, N ) implies BM (0, N ) and more generally CD(K, N ) implies BM (K, N ) . (4) When N > 1, then CD(K, N ) implies the Bishop-Gromov volume growth inequality with the Riemannian space of constant curvature K and dimension N .

Brunn-Minkowski inequality is not preserved in a two layers Banach space

In this section we are going to consider the vector space R 2 , and the hyperplane H = {(x, y) ∈ R 2 | y = 0}. We are going to put the classical Euclidean 2 norm ||(x, y)|| 2 = x 2 + y 2 on the half space y > 0 and the 1 norm ||(x, y)|| 1 = |x| + |y|, on the half space y < 0. We will denote by (R 2 , 2 , H, 1 ) the metric space obtained this way. Now let us specify the measures used here. On either half space we want a measure which is invariant by translation. This implies that in each half space it is a multiple of the Lebesgue measure λ. Let us take λ in the upper half space such that π is the measure of the standard Euclidean disk. Let α > 0 and take αλ in the lower half space (The Busemann normalisation would consist in taking α = π/2 for instance).

Properties 6. Let X 0 = (ρ, θ) be in the upper half plane in polar coordinates and X 1 = (0, y) be in the lower half plane in cartesian coordinates (y < 0), then

• the geodesic joining X 0 to X 1 is composed of the line segment from X 0 to the origin and the origin to X 1 . It is unique.

• The distance between X 0 to X 1 is equal to ρ -y. • Let X s be the s-intermediate point between X 0 to X 1 ,
(1) If s(ρ -y) < ρ, then X s belongs to the upper half plane and lies on the affine segment from X 0 to the origin, and X s = (1 -s)ρ + sy, θ in polar coordinates;

(2) If s(ρ -y) > ρ, then X s belongs to the lower half plane and lies in the line x = 0, and X s = 0, (1 -s)ρ + sy in cartesian coordinates.

Proof. The fact that this path is a geodesic is an easy computation in that case. The only thing we need to prove is uniqueness. Any geodesic between these points has to pass through the origin. Hence on the upper half plane, as there is only one geodesic between any two points we don't have any choice. Now on the lower half plane, let γ : [0, 1] → R 2 be a piecewise C 1 path between the origin and the point (0, y), if γ(t) = x(t), y(t) , then t → g(t) = (0, y(t)) is also a piecewise C 1 path between the origin and the point (0, y). Now we have almost everywhere

| ẏ(t)| ≤ | ẋ(t)| + | ẏ(t)|
therefore the length of γ is bigger that the length of g, except if x(t) = 0 almost everywhere. This implies that the only geodesic between (0, 0) and (0, y) is the segment between these two points. The other properties are easy to check. Proposition 7. In the metric space (R 2 , 2 , H, 1 ) no Brunn-Minkowski inequality holds, i.e., for any K ∈ R and N ∈ N ∪ {+∞}, BM (K, N ) does not hold.

Proof. First one can notice that for N < +∞ the space (R 2 , 2 , H, 1 ) is invariant under linear dilations. This implies that if it is BM (K, N ) then it is BM (0, N ).

Let (ρ, θ) be the polar coordinates in R 2 . Consider the annulus

K 0 = {(ρ, θ) | 6 ≤ ρ ≤ 8, π/3 ≤ θ ≤ 2π/3},
and the affine segment

I = {(x, y) ∈ R 2 | -101 ≤ y ≤ -100, x = 0}.
Now let X 0 = (ρ 0 , θ) be in K 1 , and X I = (0, -100-t) in I. Following the previous section, there is a unique geodesic from X 0 to X I , and is composed of the affine segment joining X 0 to the origin O = (0, 0) and of the affine segment joining the origin to X I . We therefore have ||X 0 || 2 = ρ 0 and ||X I || 1 = 100 + t, from which we deduce that the distance between these two points is ρ 0 + 100 + t. Now following the Properties 6, as

(ρ 0 + 100 + t)/2 > 106/2 = 53 > 8 ≥ ρ 0 ,
for s ≥ 1/2 the point X s = 0, (1 -s)ρ 0 + s(-100 -t) , is the sintermediate point on the geodesic from X 0 to X I , From this we easily deduce that the 1/2-intermediate set from X 0 to I is 1 2

K 0 + 1 2 I = (x, y) | x = 0, -47, 5 ≤ y ≤ -46 .
This suffices to prove that BM (0, N ) is not satisfied as (αλ)

1 N 1 2 K 0 + 1 2 I = 0 < 1 2 λ 1 N (K 0 )
(Actually, we have that the space is not M CP (0, N )). Now let us prove that BM (K, +∞) is never satified. For s > 1/2 the s-intermediate set from X 0 to I is easily seen to be

(1-s)K 0 +sI = (0, y) ∈ R 2 | -101s+6(1-s) ≤ y ≤ -100s+8(1-s) .
We start by considering some 0 < ε < 1, whose value will be chosen at the end, and replace I with

K 1 = (x, y) ∈ R 2 | |x| ≤ ε, |y + 100, 5| ≤ 0, 5 .
The next step is to introduce the slices of K 1 :

I α = {(x, y) ∈ R 2 | -101 ≤ y ≤ -100, x = α}
for 0 < α ≤ ε < ρ, and to identify their intermediate sets (1 -s)K 0 + sI α . In order to do this we compute the distance between X 0 and (α, 0), which gives

ρ α = ρ 2 0 -2αρ 0 cos θ + α 2 ,
and it is now easy to check that for X 0 in K 0 we have

(6 -α) ≤ (ρ 0 -α) < ρ 2 0 -2αρ 0 + α 2 ≤ ρ α ≤ ρ 2 0 + 2αρ 0 + α 2 < (ρ 0 + α) ≤ (8 + α).
(5)

The description we were seeking is therefore (recall that s > 1/2)

(1-s)K 0 + sI α = (α, y) ∈ R 2 | -101s + (1 -s) √ 6 2 -6α + α 2 ≤ y ≤ -100s + √ 8 2 + 8α + α 2 (1 -s) . (6)
To obtain an upper bound on its area we notice that it can be seen as a subset as follows

(1 -s)K 0 + sI α ⊂ (α, y) ∈ R 2 | -101s + (1 -s)(6 -α) ≤ y ≤ -100s + (1 -s)(8 + α) , (7)
Therefore the area of the intermediate set

K s = (1 -s)K 0 + sK 1 is less than ε • 16 -15s
up to some multiplicative constant C, depending on the normalisation chosen for the Lebesgue measure. This also tells us (depending on the sign of K) that (see definition 3 for the definition of ϑ)

(8) 105 ≤ 106 -ε ≤ ϑ(ε) ≤ 108 + ε ≤ 109.
The area of K 1 is exactly 2ε. Hence, for some fixed constant C we have

(9) µ (1 -s)K 0 + sK 1 µ s (K 1 ) ≤ ε 1-s • C 1-s • 8 - 15 2 s
We need now to compare, as s → 1, the right hand part of (9) with

µ 1-s (K 0 )e Ks(1-s)ϑ(ε) 2 /2 , which is the same as comparing ε • C • 8 -15 2 s 1/(1-s) with µ(K 0 )e Ksϑ(ε) 2 /2 .
This last term converges towards µ(K 0 )e Kϑ(ε) 2 /2 , while the first to ε • C • e 15/2 . To conclude, as ϑ(ε) stays bounded, we can find and fix an ε small enough such that

ε • C • e 15/2 < 1 2 µ(K 0 )e Kθ(ε) 2 /2 .
Then, for values of s close enough to 1, we will obtain

(10) µ 1-s (K 0 )e Ks(1-s)ϑ(ε) 2 /2 > µ (1 -s)K 0 + K 1 µ s (K 1 ) ,
which contradicts BM (K, +∞).

Proposition 8. There exists a Minkowski norm f on R 2 such that that BM (-1, +∞) does not hold in the metric space (R 2 , 2 , H, f ).

Proof. Recall that a Minkowski norm f is twice differentiable on R 2 \ {0}, with a definite positive Hessian. Let (f n ) n∈N be a sequence of Minkowski norms, converging towards the 1 norm. Up to a rescaling we can suppose that the intersection of their unit ball with H coincide with the intersection of the unit ball of the 1 norm. In any case we will consider the measures µ n such that µ n = λ on the upper half plane, and µ n = α n λ on the lower half plane where

α n = π λ({f n ≤ 1})
.

(Observe also that we can chose the norms f n such that their tangents at their point of intersection with H is orthogonal to H. This will be useful in the last section of this paper.)

Then the sequence of metric spaces (R 2 , 2 , H, f n ) converges in the Gromov-Haussdorf measured topology towards (R 2 , 2 , H, 1 ).

Consider again the sets K 0 and K 1 and the intermediate set K s as in the proof of Proposition 7. Then for any n we would get another intermediate set K s (n), and another function θ(n) which is the maximum (resp. Minimum) between two points from K 0 to K 1 or from K 1 to K 0 . Following our assumption we have that θ(n) converges towards the θ of the limit, µ(K 0 ) does not change and µ n (K 1 ) converges towards µ(K 1 ) thanks to the gromov-hausdorff measured convergence.

We suppose that s is close enough to 1 to be on the lower half plane. We need to prove that lim µ n (K s (n)) ≤ µ(K s ) as n goes to infinity. First notice that K s is a compact closed set, and so are the sets K s (n). Secondly, the geodesics from a point on the upper half space to the lower half space are unique, because both norms are strictly convex. Hence the geodesics are converging to the geodesics, thus K s (n) converges to a subset K s of K s .

Therefore we get lim µ n (K s (n)) ≤ µ(K s ) ≤ µ(K s ). Now let us take K 0 , K 1 as in the proof of Proposition 7 and s close enough to one such that

λ 1-s (K 0 )e -s(1-s)ϑ(ε) 2 /2 > α 1-s λ (1 -s)K 0 + K 1 λ s (K 1 ) ,
then for any n large enough we would also get

λ 1-s (K 0 )e -s(1-s)ϑ(ε) 2 /2 > α 1-s n λ K s (n)) λ s (K 1 ) ,
which concludes our proof, because any f n for n large enough can be chosen.

3.

A compact Finsler surface with no lower ricci bound embbeded in a Minkowski space 3.1. First example. Let us consider in the Euclidean three-dimensional space, the two-dimensional disk

S = (x, y, z) | z = 0, x 2 + y 2 ≤ 1 ,
and let B be the conve hull of S ∪ (0, 0, 1), (0, 0, -1) .

We now endow R 3 with the norm || • || B whose unit ball is B. In other words for any (x, y, z

) ∈ R 3 , ||(x, y, z)|| B = x 2 + y 2 + |z|.
The affine planes normal to the vector (0, 0, 1) endowed with the norm induced by || • || B are all isometric to the two dimensional Euclidean plane. In the same way, the affines planes containing the direction (0, 0, 1) are isometric to the 1 -plane (i.e., that is the manhattan distance).

In this normed vector space, we will consider the cube C ρ obtained as the convex hull of the eight points (±ρ, ±ρ, ±ρ) .

The cube C ρ admits two faces which are Euclidean, and four faces which are 1 . The measures considered are the Hausdorff measures. In other words, λ(B) = 4 3 π and for any linear subspace L of dimension 2, the measure is the Lebesgue measure λ L normalised such that Proof. Let us denote by d ρ the distance induced on C ρ by || • || B and µ ρ the induced Lebesgue measure. Focus on two adjacent faces of C ρ , one Euclidean and the second one 1 . Then we are locally exactly as in section 2, and therefore the same computations as in section 2 show that for any ρ ∈ R * , the Brunn-Minkowski BM (K, N ) inequality does not hold for any N ∈ N ∪ {+∞} and any K ∈ R.

λ L (B ∩ L) = π.
Therefore in C ρ , d ρ , λ ρ the curvarture dimension CD(K, N ) does not hold for any K and any N .

Corollary 10. There exists a C 1,1 compact and convex surface in (R 3 , ||•|| B ) such that for any N ∈ N∪{+∞} and any K ∈ R , CD(K, N ) does not hold.

Remark here that in our example both the C 1,1 assumption and the fact that the norm is not smooth restricted to the surface are important. If the objects are too smooth, there is always some K and N for which it is CD(K, N ).

Proof. let B(ε) be the Euclidean ball of radius ε. Consider the Minkowski sum of the cube and this ball, that is, Actually C(ε) is obtained by translating the faces of the cube C 1 outward at a Euclidean distance ε and then closing by rolling the Euclidean ball of radius ε along the edges, from the inside.

C(ε) = B(ε) + C 1 = {x + y | x ∈ B(ε), y ∈ C 1 },
Hence the difference is on the surface obtained along these curved edges. On the flat section we have the same distance than in C 1 .

Fix some K = -1 and N = +∞. We can use the annulus K 0 and the rectangle K 1 from the proof of Proposition 7, the only thing that will change is the s-intermediate set from K 0 to K 1 , denoted by K s (ε).

Fix an s such that we get the inequality (10) as in proof of Proposition 7 for K 0 , K 1 and K s .

Then as ε goes to zero the corresponding sequence of s-intermediate sets K s (ε) converges towards a subset of K s and thus

lim →0 λ (K s (ε)) ≤ λ(K s ),
where K s is the same as in the proof of Proposition 7. Hence for some ε small enough, we would get the same contradiction.

Let us now fix such an ε for K = -1. Let h ρ be the dilation of ratio ρ of center the origin. Consider the images of K 0 , K 1 and K s (ε) by h ρ , they all lie on C ρ +B(ρ•ε). Furthermore the image of K s (ε) by h ρ is the s-intermediate set from h ρ (K 0 ) to h ρ (K 1 ) on C ρ + B(ρ • ε). Therefore we still get the inequality 10 which is invariant by dilations, which proves that C ρ + B(ρ • ε) is not BM (-1, +∞) as well.

Hence for any ρ > 0, C ρ + B(ρ • ε) is not BM (-1, +∞) (and not CD(-1, +∞)). Now let us suppose that C 1 + B(ε) is CD(K, N ) for some K < -1. Then C 1 + B(ε), ρd, ρ 2 λ is CD(K/ρ 2 , N ). Observe now that h ρ is an isometry between C 1 + B(ε), ρd, ρ 2 λ and C ρ + B(ρ • ε), because,

d(h ρ (x), h ρ (y)) = ρ • d(x, y) but then for ρ 2 > -K, we get that C ρ + B(ρ • ε) is C(-1, N ), which contradicts the choice of ε.
The question I am often asked with this example is why C(ε) does not satisfies some CD(K, N ) with K → -∞ as ε → 0 ? In the proof one can see that this is due to the very nature of all the objects defined here which behave nicely with respect to dilation on one side, and the translation on the other side. That is to say that the very specificity of the Lebesgue measure, that its homogeneity by dilation and invariance by translation are important here.

Another point of view should be from the point of view of optics, as was explained to me a long time ago. The laws of refraction are an approximation, that is to say that in realiy there is no discontinuity of the differential of a ray of light, but to our eyes it looks like that. In other words the intersection between two media behaves as C(ε) for ε small, but our CD(K, N )-eyes see C(0) = C 1 .

3.2. Second example. This second example is to justify that one can get an example with a smoother norm.

Let H in R 3 be the x-axis (that is the line z = 0 an y = 0). Consider f a norm in the plane y = 0 such that (y = 0, 2 , H, f ) does not satisfy CD(-1, +∞) as in proposition 8. Then consider B f the convex obtained by rotating the norm f around the z-axis.

Then let us denote by || • || f the norm whose unit ball coincide with B f . Proposition 11. There exists a C 1,1 compact and convex surface in (R 3 , || • || f ) such that for any N ∈ N ∪ {+∞} and any K ∈ R , CD(K, N ) does not hold.

Proof. Again, let us consider the family of cubes C ρ with our two sets K 0 and K 1 . Then for any ρ, C ρ with the induced metric in not BM (-1, +∞).

Then let us also consider C 1 + B(ε), then for some ε small enough it will not be BM (-1, +∞) as in the previous example. And again, by homotating the sets contradicting BM (-1, +∞), we obtain that for any ρ > 0,

C ρ + B(ε • ρ) is not BM (-1, +∞).
Again the same reasoning by contradiction as in the proof of Corollary 10 shows that C 1 + B(ε) can not satisfy any CD(K, N ), for any K and any N .

Concluding remarks

The current work has been the subject of various talk and discutions with many colleagues having their own idea of what is a good notion of curvature in metric measured spaces.

The first main problem which forbids the notion of synthetic Ricci curvature to apply in our first example is the branching occuring when one passes from one media to another. It is also related to the Finslerian nature of our spaces.

Both these problems excludes all the notions of curvatures that have been presented to us by our various colleagues. For instance one could decide to work with spaces admitting a Gromov-Bishop comparison theorem, as some nice theorem and results in Riemannian geometry are actully based on the fact that manifolds with Ricci curvature bounded from below admits such a comparison. An easy computation shows that the metric space (R 2 , 2 , H, 1 ) does not satisfy such a comparison with the standard hyperbolic plane.

Notice that by smoothing our norm, we still got a surface without synthetic Ricci curvature bounded from below, but without branching. This illustrate the fact that by being close to a branching space is also problematic.

Aknowledgment

The author is strongly indebted to Juan Carlos Álvarez Paiva who introduced him to the reflections in normed spaces with strictly convex unit balls, and shared many hours in Lille on this subject and many other fruitfull mathematical insights.

I therefore have to present my apologies to him to have use this very beautiful geometric object for my own selfish interest in synthetic Ricci curvature.

I am also grateful to Nicolas Juillet whose work [START_REF] Juillet | Ricci curvature bounds and geometric inequalities in the heisenberg group[END_REF] and advice have been of great help and the two anonymous referees who made helpful suggestions.

  : R + → R + . For a fixed s ∈ [0, 1] and θ ∈ R + , τ (s) K,N (θ)is continuous, nonincreasing in N and nondecreasing in K. Its exact definition is not important for our applications, refer to [Stu06b].

Proposition 9 .

 9 Let R 3 be endowed with the norm || • || B . Then the cube C 1 with the metric induced by ||•|| B does not satisfy any curvature dimension.

  with the unduced metric by || • || B . Then C(ε) is C 1,1 , and as ε goes to zero, it converges in the Gromov-Hausdorff measured topology towards C 1 .

Funding

This work was partially funded by the ANR Finsler grant (projet-ANR-12-BS01-0009) of the Blanc -Simi 1 program.


Maxime Logez

Bernadette Bounket

Pierre Gibert

Georges Carrel

Virginie Diouloufet

Xavier Colombet

Fabien Morat

José Maria Santos

Jeyaraj Antony Johnson

The Interannual Variability of the Fall Size of Young-of-The-Year Chub (Squalius cephalus): Influence of Phenology, Growth Patterns and Abiotic Factors

Keywords: Leuciscidae, logistic model, temperature, hatching, otoliths, large regulated river

The Interannual Variability of the Fall Size of Young-of-The-Year Chub (Squalius cephalus): Influence of Phenology, Growth Patterns and Abiotic Factors

INTRODUCTION

Organism size has been widely studied, because of its impact on organism internal functioning (metabolism, thermoregulation of ectotherms) [START_REF] West | The Origin of Allometric Scaling Laws in Biology from Genomes to Ecosystems: Towards a Quantitative Unifying Theory of Biological Structure and Organization[END_REF], on the life-history traits (survival, longevity, fecundity) [START_REF] Roff | The Evolution of Life Histories: Theory and Analysis[END_REF][START_REF] Winemiller | Patterns of Life-History Diversification in North American Fishes: Implications for Population Regulation[END_REF], on ecology (ontogenetic shift in habitat use) [START_REF] Werner | The Ontogenetic Niche and Species Interactions in Size-Structured Populations[END_REF], and more broadly because of the consequences of organism size on ecosystem functioning (food web) [START_REF] Barnes | Global Patterns in Predator-Prey Size Relationships Reveal Size Dependency of Trophic Transfer Efficiency[END_REF].

The "bigger is better hypothesis" suggest the body size is determinant for young fish survival [START_REF] Miller | Larval Size and Recruitment Mechanisms in Fishes: Toward a Conceptual Framework[END_REF] and particularly for the winter survival of fish larvae and juveniles [START_REF] Sogard | Size-selective Mortality in the Juvenile Stage of Teleost Fishes: A Review[END_REF]. Indeed, the acquisition of a "refuge size" allows to escape to predation and to be less sensible at environmental stressors (e.g. temperature, starvation) [START_REF] Sogard | Size-selective Mortality in the Juvenile Stage of Teleost Fishes: A Review[END_REF][START_REF] Hurst | Causes and Consequences of Winter Mortality in Fishes[END_REF]. For youngof-the-year (YOY) fish, the recruitment depends to first winter survival [START_REF] Hurst | Causes and Consequences of Winter Mortality in Fishes[END_REF]. However, YOY generally show a large variability in size at the end of the growing period (e.g. [START_REF] Nunn | Is Water Temperature an Adequate Predictor of Recruitment Success in Cyprinid Fish Populations in Lowland Rivers? Freshw[END_REF]. The knowledge of causes of this size variability (differences in spawning date or in the growth capacity, for example) is crucial as the survival of YOY at the first winter influence the population dynamics.

Long-term surveys have revealed high variation of freshwater fish body size at the same location, but between years [START_REF] Nunn | Is Water Temperature an Adequate Predictor of Recruitment Success in Cyprinid Fish Populations in Lowland Rivers? Freshw[END_REF][START_REF] Dudgeon | Prospects for Sustaining Freshwater Biodiversity in the 21st Century: Linking Ecosystem Structure and Function[END_REF]. Between 1984[START_REF] Gamito | Growth Models and Their Use in Ecological Modelling: An Application to a Fish Population[END_REF][START_REF] Nunn | Is Water Temperature an Adequate Predictor of Recruitment Success in Cyprinid Fish Populations in Lowland Rivers? Freshw[END_REF] have observed important variation in roach (Rutilus rutilus (Linnaeus, 1758)), dace (Leuciscus leuciscus (Linnaeus, 1758) and chub (Squalius cephalus (Linnaeus, 1758)) sizes, especially for young of the year. Abiotic factors such as discharge, water temperature and climate regime [START_REF] Konečná | River Discharge Drives Recruitment Success of the European Bitterling Rhodeus amarus in a Regulated River in Central Europe[END_REF][START_REF] Nunn | Is There Evidence for a Shift in Fish Growth and Recruitment Success Linked to Climate Change[END_REF] and biotic factors such as species phenology have been suggested to influence YOY size [START_REF] Nunn | Is There Evidence for a Shift in Fish Growth and Recruitment Success Linked to Climate Change[END_REF]. Indeed, temperature can influence the growth of YOY (through the metabolism) but also fish species phenology. In warmer conditions, fish would spawn earlier (e.g. [START_REF] Nunn | Is There Evidence for a Shift in Fish Growth and Recruitment Success Linked to Climate Change[END_REF] which could potentially lead to a longer growing period than during cooler years, leading to larger YOY in early winter.

While numerous studies have focused on endangered species and their potential loss regarding global changes [START_REF] Ceballos | Vertebrates on the Brink as Indicators of Biological Annihilation and the Sixth Mass Extinction[END_REF] and/or the multiple anthropogenic pressures impacting temperate rivers [START_REF] Dudgeon | Prospects for Sustaining Freshwater Biodiversity in the 21st Century: Linking Ecosystem Structure and Function[END_REF][START_REF] Harrison | The Freshwater Biodiversity Crisis[END_REF], it is also important to focus on "common species." Indeed, the analysis of long-term surveys confirms the rapidity of environmental and biological changes, that also concernced relatively "tolerant" species, especially rheophilic species [START_REF] Mueller | Comprehensive Analysis of >30 Years of Data on Stream Fish Population Trends and Conservation Status in Bavaria, Germany[END_REF][START_REF] Santos | Life-history Traits Correlate with Temporal Trends in Freshwater Fish Populations for Common European Species[END_REF]. This is the case of the European chub, a widespread species in Europe and a dominant leuciscid in French rivers. Chub is a potamodromous, lithophilic species that can live in a wide variety of habitats (lentic or lotic), even altered [START_REF] Philippart | Atlas des poissons de Wallonie. Distribution, écologie, éthologie, pêche, conservation[END_REF][START_REF] Cowx | Flow and Level Criteria for Coarse Fish and Conservation Species[END_REF], and that can move on great distances during the spawning period to find gravel-bed spawning grounds in lotic areas [START_REF] Fredrich | Spawning Migrations of the Chub in the River Spree, Germany[END_REF][START_REF] Benitez | The Influence of Environmental Factors on the Upstream Movements of Rheophilic Cyprinids According to Their Position in a River Basin[END_REF]. Chub larvae and YOY are mostly observed in groups in shallow shoreline habitats of lotic channels [START_REF] Mann | Observations on the Age, Growth, Reproduction and Food of the Chub Squalius cephalus (L.) in the River Stour, Dorset[END_REF][START_REF] Copp | Ecology of Fish Spawning and Nursery Zones in the Flood Plain, Using a New Sampling Approach[END_REF].

Our current knowledge on the spawning periodicity, growth patterns and size of YOY is mostly inferred from surveys of 0+ year size distributions (e.g. [START_REF] Mills | Environmentally-induced Fluctuations in Year-Class Strength and Their Implications for Management[END_REF][START_REF] Nunn | Variations in the Spawning Periodicity of Eight Fish Species in Three English Lowland Rivers over a 6 Year Period, Inferred from 0+ Year Fish Length Distributions[END_REF], but only rare studies have compared the growth pattern of YOY under different hydrological regimes and thermal contexts. Using otoliths collected on YOY chub, we aimed to assess the phenology and the growth pattern of YOY in three nearby sections of the same reach that are typical of the run-of-river hydropower development along the Rhône River: bypassed section, tailrace and reservoir. Fish were studied during two thermally contrasted years (2013: cool and 2015: warm). Our objectives were to estimate the hatching date of eggs and to model the fish daily growth curves (with a Bayesian hierarchical nonlinear model), in order to assess how YOY growth could be related to phenological shifts and/or to environmental conditions (temperature).

MATERIALS AND METHODS

Study Area

The reach of Caderousse is located in the middle of the lower Rhône and combines the hydraulic components of two hydropower schemes: Donzère-Mondragon and Caderousse. The headrace of the Donzère-Mondragon hydroelectric power plant has been receiving cooling water from the Tricastin nuclear power plant since 1980. After complete mixing of the heated effluent downstream of the hydroelectric plant, the mean daily temperature increment of the tailwater was +1.6 °C for the period 2002[START_REF] Gillet | Effect of Temperature Changes on the Reproductive Cycle of Roach in Lake Geneva from 1983 to 2001[END_REF][START_REF] Hartmann | Centrales nucléaires et environnement-Prélèvements d'eau et rejets[END_REF].

The reach of Caderousse includes three main contrasting sections (Figure 1). First, the tailrace of Bollène (TAIL), an artificialized deep lotic channel with a trapezoidal cross-section with alternating concrete and riprap riverbanks. Its flow is controlled by the Bollène hydroelectric power plant. The bypassed section (BPS) corresponds to the old river channelized during the 19th century with low submersible dikes. It has both lentic and lotic areas and is characterised by the presence of a riparian forest, residual backwaters. The third section is the reservoir (RES) created by the Caderousse dam. This is a large area of lentic, deep waters and the shoreline habitats mainly consist of ripraps and large macrophyte beds.

The Compagnie Nationale du Rhône provided us with the daily flow data of the Rhône River recorded near Pont-Saint-Esprit for the BPS and near Chusclan for the RES. The daily discharge in the tailrace was obtained by subtracting from the total river flow measured in the upstream gauging station near Viviers, the instream flow of the bypassed section near Saint-Montan (Figure 1). The daily average temperature of the Rhône waters was obtained from data recorded every 15 min by HOBO ® thermometers set up in the three sections.

Fish Sampling

YOY chub were collected from Sept. 24 to 26, 2013 and from Sept. 14 to 17, 2015. Fish were sampled by electrofishing using the point abundance sampling method [START_REF] Nelva | Une nouvelle méthode d'étude des peuplements ichtyologiques dans les grands cours d'eau par échantillonnage ponctuel d'abondance[END_REF], a strategy suitable to take into account the habitat diversity on both sides of each section. Following eugenol anaesthesia, they were put on ice, and kept frozen until dissection. Prior to dissection, size (total length -TL) was measured to the nearest millimetre. In 2013 and 2015, 139 YOY (81 in the BPS, 41 in the RES, and 17 in the TAIL) and 124 YOY (44 for BPS, 45 in the RES, and 35 in the TAIL) were recorded respectively. The size classes (TL) of the collected individuals were ranged between 20 and 80 mm. The lapilli, the most appropriate otoliths in cyprinid species to assess fish growth [START_REF] Bounket | Otolith Shape Analysis and Daily Increment Validation During Ontogeny of Larval and Juvenile European Chub Squalius cephalus[END_REF], were extracted, immediately cleaned in distilled water and placed to dry in plastic tubes.

Research Involving Animals

All relevant international, national, and/or institutional guidelines for the care and use of animals were followed. Fish were sacrificed using anaesthetic overdose (eugenol) in accordance with the European Directive 2010/63/EU. Fish were collected in accordance with the authorisation numbers: 2013-203-0005 and 2015-229-DDTSE01 delivered by the prefecture.

Hatching Date and Daily Otolith Growth

Otoliths were embedded in CrystalbondTM resin and polished on the frontal plane with abrasive paper and diamond solutions (Ø 10, 3, 0.25 μm, Escil, Chassieu, France). Each otolith was observed and digitized at ×1,000 magnification using a light microscope (Leica DM 2500, Solms, Germany) combined with a camera (Leica MC 170HD). Age was estimated by counting the daily increments [START_REF] Bounket | Otolith Shape Analysis and Daily Increment Validation During Ontogeny of Larval and Juvenile European Chub Squalius cephalus[END_REF] between the hatching check and the edge of the otolith along the maximum growth axis from the core to the anterior face. The hatching check was similar to those previously observed in other cyprinids [START_REF] Smith | Validation of the Ageing of 0+ Carp (Cyprinus carpio L.)[END_REF][START_REF] Li | Otolith Characteristics and Age Determination of an Endemic Ptychobarbus dipogon (Regan, 1905) (Cyprinidae: Schizothoracinae) in the Yarlung Tsangpo River, Tibet[END_REF]. Three independent readers performed the age estimations three times. The degree of agreement between readings was assessed by determining the coefficient of variation (3.3 ± 1.0% and 3.1 ± 0.9% in mean for 2013 and 2015, respectively) [START_REF] Panfili | Manuel de sclérochronologie des poissons[END_REF]. The distance between the nucleus and each daily growth increment was measured using the software ImageJ. Hatching dates were estimated by subtracting the age of YOY from the date of capture.

Streamflow and Growing Degree-Day (GDD)

To characterise the hydrological conditions faced by YOY during their growth period, we computed the median of the daily discharge ( Q) between hatching and sampling dates for each fish as a proxy of the flows experienced by fish.

The growing degree-day is a thermal integral used to calculate the heat accumulation above a minimum temperature threshold. It is a good predictor of fish growth [START_REF] Neuheimer | The Growing Degree-Day and Fish Size-At-Age: The Overlooked Metric[END_REF] mainly used to characterize the thermal regime experienced by fish during their growing period. GDDs were computed for each fish as the sum of the degree-days above the threshold of 12 °C [START_REF] Mills | Environmentally-induced Fluctuations in Year-Class Strength and Their Implications for Management[END_REF] from the estimated hatching date to the day of capture.

We assessed the link between Q and GDD in each section and each year by computing Spearman's rank correlation coefficients (ρ).

Back-Calculation of Fish Daily Growth

The daily otolith growth increments were back-transformed into daily sizes to reconstruct the growth pattern for individual fish. In line with [START_REF] Vigliola | The Back-Calculation of Fish Growth from Otoliths[END_REF], we used the Modified Fry model (MF) to infer fish size-at-age from otolith growth. We followed the MF procedure, which is composed of five steps. The first two steps consist in estimating the size of the fish at the first increment, L 0p , and the size of the radius at the first increment, R 0p . These two parameters are biological intercepts and were determined from a controlled experiment [START_REF] Bounket | Otolith Shape Analysis and Daily Increment Validation During Ontogeny of Larval and Juvenile European Chub Squalius cephalus[END_REF]. The third and fourth steps consist in determining whether the relationship between otolith radii and fish sizes is allometric or isometric and were performed independently for each section/ year. To do this, we fitted the nonlinear regression:

L cpt L 0p -b • R c 0p + b • R c cpt
With L cpt the size at capture and R cpt the otolith radius at capture. The six coefficient c were tested against 1 and we concluded that allometry was required, since all the tests were significant (α 5%) excepted for the TAIL. For each section/year the backcalculation was performed with the formula:

L i a + exp ⎛ ⎝ ln L 0p -a + ln L cpt -a -ln L 0p -a ln(R i ) -ln R 0p ln R cpt -ln R 0p ⎞ ⎠
With L i the size at age i, R i the radius at age i, and a derived from the formula a L 0pb • R c 0p and computed for each section/ year. For more details, see "Box 1" from [START_REF] Vigliola | The Back-Calculation of Fish Growth from Otoliths[END_REF].

Relationship Between GDD and Fish Size

We fitted exponential response curves between length at capture and GDD using multiple linear regression (MLR) on the log-scale. To assess the relative influence of GDD, year (taken as a qualitative variable) and section, we first considered in the "complete model" all variables, the two-way interactions and the three-way interaction. Then, we tested the significance of the three-way interaction by testing (F-test) the complete model against a model without this interaction. If it was not significant, the interaction was removed and we tested models with and without the two-way interactions. If the two-way interactions were not significant, they were removed from the MLR and the MLR integrated only of the additional effect of the three explanatory variables.

We checked for the homoscedasticity. If it was not verified, then Generalised Least Square models (GLS) were used to account for the heteroscedasticity. GLS extend the linear models by modelling the heterogeneity with covariates [START_REF] Zuur | Mixed Effects Models and Extensions in Ecology with R[END_REF].

Fish Growth Curves

To model the growth of fish we used a logistic growth model [START_REF] Ricker | Growth Rates and Models[END_REF] with four parameters, of the form:

L i α + L ∞ -α 1 + e T-i δ
With L i the size at age i, α the minimal asymptotic size, L ∞ the asymptotic size, T the inflexion point (in days), and δ a scale factor (Figure 2) inversely related to the rate of maximum growth (the slope at the inflexion point is steepest for the lower value of δ). Indeed, the first derivate of this function is:

dL i di (L ∞ -α) • e T-i δ δ • 1 + e T-i δ 2
At the inflexion point T (when i T), it simplifies to:

K (L ∞ -α) 4δ
The minimal asymptomatic size, α, is not biologically meaningful (i.e., it is not an estimation of the size at hatching), but is important to fit constituent growth curves.

To model fish growth curves we used a Bayesian hierarchical framework, considering that each combination of section and year k, (i.e. BPS-2013, BPS-2015, TAIL-2013, TAIL-2015, RES-2013, RES-2015) had its own set of hyperparameters: α k , L ∞k , T k and δ k . Then each fish, j, was supposed to have its own set of parameters sampled from normal distributions with mean the hyperparemeters and with a given variance σ 2 xk (x being a hyperparameter):

α kj ∼ N α k , σ 2 α k L ∞kj ∼ N L ∞k , σ 2 L αk T kj ∼ N T k , σ 2 T k δ kj ∼ N δ k , σ 2 δ k
Therefore "hyperparameters" represented the average coefficients of the growth model in each section/year, while "parameters" represented the coefficients of the individual growth curves. For each fish (j) of each section/year (k) we computed the average size at age i (µ kji ) such as:

µ kji α kj + L ∞ kj -α kj 1 + e T kj -i δ kj
Finally, the observed size at age i, L kji , was supposed to be normally distributed:

L kji ∼ N μ kji , σ 2 j
This hierarchical model is also heteroscedastic as each fish has its own dispersion parameters (σ 2 j ). For each hyperparameters we used a normal non-informative prior with mean 0 and for variance the inverse of the precision (τ) for which we used a gamma non-informative prior:

α k ∼ N 0, σ 2 k σ 2 k 1 τ k τ k ∼ Γ(0.01, 0.01)
To compare the hyperparameters of each section/year, we computed the median and the associated credible interval (CI, 95%) of their posterior distributions that were obtained from 10 Monte-Carlo Markov Chains with 100,000 iterations. After an initial burning of 50,000 values, one value on 10 was conserved (to limit the autocorrelation between estimations). The convergence was checked through the Gelman and Rubin's convergence diagnostic that had to be close to one for all estimated parameters [START_REF] Congdon | Bayesian Statistical Modelling[END_REF].

The growth patterns were compared using the average growth curves of each section/year and their associated CI. To do so, the hyperparameters posterior distributions served to predict the average sizes at age for each section/year (50,000 µ ki values), for which we computed the mean value (average pattern of growth) and the quantiles 2.5 and 97.5% (limits of the CI).

All these computations were performed using the Bayesian software JAGS, the statistical software R (v. 4.1.0) and the libraries emmeans (1.6.1), R2jags (0.7.1), rjags (4.10), coda (0.19.4), car (3.0.10), dplyr (1.0.6), doParallel (1.0.16), gridExtra (2.3) ggplot2 (3.3.3), itertools (0.1.3), nlme (3.1.152), nls2 (0.2), and tidyr (1.1.3).

The R script is available in Supplementary Material.

RESULTS

Hatching and Growth Period

The hatching dates varied from the end of April to the end of July in 2013, and from the beginning of April to the end of July in 2015. Hatching dates differed significantly between years (F 1,259 289.40, p-value < 0.001) and between sections (F 2,259 4.46, p 0.012), but without interaction between years and sections (F 2,257 2.44, p 0.090) (Figure 3). On average in 2015, the YOY hatched 28.24 days earlier (average hatching day 137.67 Julian days) than in 2013 (average hatching day 165.91 Julian days). In mean, the hatching dates were not significantly different in the BPS and in the RES (t -0.029, p 0.999), whereas YOY hatched 6.12 days (t 2.765, p 0.017) and 6.17 days (t 2.684, p 0.021) earlier in the TAIL than in the BPS and the RES, respectively.

The growth period was on average 16.55 days longer in 2015 than in 2013 (F 1,261 104.81, p < 0.001).

Stream-Flow and GDD

All ρ-values were close to one (≥0.96) and highly significant (p-values < 0.001). These correlations were explained by the highest discharge values in early spring (Figure 3). Therefore, fish that hatched earlier experienced higher discharge, had a longer growth period and thus greater GDD values than fish that hatched later in the year. Owing to the correlations observed between discharge and temperatures, we only presented and discussed the effect of temperature on fish growth.

GDD and Size at Capture

The sum of degree day (GDD) varied between years and between sections (GLS, p-values <0.001), without interaction (F 2,257 0.094, p 0.910). On average in 2015, GDD exceeded by 379.4 °C those experienced by fish in 2013 (t 259 44.1, p < 0.001). As for the hatching dates, GDD in the BPS and RES were not significantly different (t 0.172, p 0.984), whereas the TAIL exhibited higher values of GDD than the BPS and the RES, with differences of 42.4 and 40.6 °C, respectively (t-tests, p-values < 0.001).

The effect of GDD on size at capture (L cpt ) varied between sections and years, as revealed by the significant interaction between these three variables (F 2,251 3.616, p 0.029). The slope of the relationship between GDD and the logarithm of L cpt was different but always positive whatever the couple of sectionyear (Figure 4). The goodness of fit of these relationships, measured by the root mean square errors (RMSE; Table 1), was variable and greater for the BPS or the TAIL (depending of the year), followed by the RES. RMSEs were higher in 2015 than in 2013 (Table 1).

Fish Growth Curve

Gelman and Rubin's convergence diagnostics were all close to 1 revealing that convergence was achieved. The average RMSEs were low whatever the sections or years, ranging between 0.298 and 0.556 mm (Table 2). Therefore, the logistic growth model seemed well adapted to model the daily growth of YOY chub.

Minimal Asymptotic Size, α

The α varied between years and sections (Figure 5). The lower median value was observed for the TAIL in 2013, while RES had always the highest values (even higher in 2015). There was no difference between the 2 years for the BPS (Figure 5).

Asymptotic Size, L '

The fitted L ∞ varied between sections and years, as revealed by the CI displayed on Figure 5. TAIL displayed the highest L ∞ , but conversely to the BPS and RES, they were consistent between years (77.8 mm). BPS had the lowest L ∞ values and RES intermediate values. BPS and RES displayed greater asymptotic size during 2015 than in 2013, but the difference was magnified in the reservoir (△L ∞ 13.32 mm for RES and 9.93 mm for BPS) (Figure 5).

Inflexion Point, T

Similar patterns were observed for T and L ∞ values, except that TAIL displayed intermediate values and not the greatest ones. For BPS and RES, the inflexion points appeared later in the growing period during the warmest year, but no significant differences were observed for the TAIL (Figure 5).

Maximum Growth Rate, K

The maximum growth rates (K, slope at the inflexion point) were stronger in 2015 than in 2013 in all sections, but the difference was only significant in RES. The greatest values were observed in RES in 2015. The lowest K were observed in the BPS. TAIL had a greater median value than RES in 2013 (but CIs overlapped) and shared with RES fairly similar values in 2015 (Figure 5).

Fish Growth

Growth patterns were highly variable between sections and between years (Figure 6) with all average daily growth curves different except for the BPS (Figure 6). Up to the 75th day of growth, the curves of BPS were almost similar. Then in 2015 the YOY tended to have a slightly faster growth (but overlapping CIs) to reach a larger asymptotic size, also due to a longer growing period. For the TAIL, all along their growing period YOY were larger in 2015 than in 2013, the deviations increased with time (but overlapping CIs). For the RES, until the 70th day of growth, YOY of 2013 were slightly larger. Then a shift occurred with a steeper growth for YOY of 2015, which became increasingly larger than those of 2013, with magnitude of deviation not observed in the two other sections. When comparing growth curves within each year, we observed that in the colder year (2013) YOY of TAIL had the fastest growth, and that YOY of BPS and RES had close growth patterns. In 2015, YOY of TAIL became larger sooner than all the others. After the 50th day of growth YOY of TAIL were significantly larger than YOY of the BPS. Up to the 85th day of growth, this was also true for RES, but due to the steeper growth in this section, the deviation between the sizes of TAIL and RES decreased and became not significant (CIs overlapping). The growth curve in the RES was partly similar to that of the BPS (during very early life) and that of the TAIL (after day 75).

Individual Variability

Even not observable in Figure 5 (but see Supplementary Material S1), the variability between YOY collected at the same location, and during the same year, was also important as revealed by the posterior distributions of the dispersion parameters (τ) expressed as standard deviations (Table 3). They were higher for the parameters with the greatest values (L ∞ and T), but standard deviations also varied between sections and years without clear patterns.

DISCUSSION

First, our results showed that YOY chub hatched earlier and had a longer hatching period in 2015 than in 2013 resulting in a longer growing period during the warmer year. Second, the YOY sizes at capture were positively related to growing degree-days. Third, the growth patterns were highly variable between sections and years, with an increase in size-at-age in all sections when GDD increased.

Hatching Date

In the Caderousse reach, the YOY chub hatched from mid-April to the end of July. Most of the estimated hatching dates were observed when the water temperature exceeded 12 °C and followed the last spring high. Our observations indicated an extended and longer hatching period compared to former results from European temperate rivers: the Ourthe River, Belgium (April), the Spree River, Germany (May), the rivers Jilhava and Rokytna, Czech Republic (mid-May to mid-June), rivers of Great Britain such as the Stour (May to June), the Lugg (May/June), the Trent (mid-June to mid-July), or in the Upper Rhône (June) [START_REF] Hellawell | The Autecology of the Chub, Squalius cephalus (L.), of the River Lugg and the Afon Llynfi[END_REF][START_REF] Mann | Observations on the Age, Growth, Reproduction and Food of the Chub Squalius cephalus (L.) in the River Stour, Dorset[END_REF][START_REF] Peňáz | Fish Fry Community of the Jilhava River Near Mohelno[END_REF][START_REF] Prokeš | Early Development of the Chub. Leuciscus cephalus[END_REF][START_REF] Philippart | Atlas des poissons de Wallonie. Distribution, écologie, éthologie, pêche, conservation[END_REF][START_REF] Carrel | Caractérisation physico-chimique du Haut-Rhône français et de ses annexes ; incidences sur la croissance des populations d'alevins[END_REF][START_REF] Nunn | Recruitment Patterns of Six Species of Cyprinid Fishes in the Lower River Trent, England[END_REF][START_REF] Fredrich | Spawning Migrations of the Chub in the River Spree, Germany[END_REF][START_REF] Nunn | Variations in the Spawning Periodicity of Eight Fish Species in Three English Lowland Rivers over a 6 Year Period, Inferred from 0+ Year Fish Length Distributions[END_REF]. These differences could be linked to the elapsed time between these studies and our observations, and the possible temperature shift along this period. The extension of the breeding period could result from multiple or protracted spawning events of chub in a new climate context [START_REF] Dayon | Impacts of Climate Change on the Hydrological Cycle over France and Associated Uncertainties[END_REF][START_REF] Ouellet | River Temperature Research and Practice: Recent Challenges and Emerging Opportunities for Managing thermal Habitat Conditions in Stream Ecosystems[END_REF]. Temperature played a major role in hatching date variability, both between years and between sections. Indeed, the hatching occurred on average 1 month earlier (≈28 days) during the warmest year and about 1 week earlier (≈6 days) in the warmer section (within the same year). Such variability in hatching date is consistent with previous observations on the aforementioned chub and other species such as roach and common bream (Abramis brama (Linnaeus, 1758)) [START_REF] Nõges | Climate Driven Changes in the Spawning of Roach (Rutilus rutilus (L.)) and Bream (Abramis brama (L.)) in the Estonian Part of the Narva River basin[END_REF], the roach [START_REF] Gillet | Effect of Temperature Changes on the Reproductive Cycle of Roach in Lake Geneva from 1983 to 2001[END_REF], the Selincuo naked carp (Gymnocypris selincuoensis), a cyprinid from the Tibetan Plateau [START_REF] Tao | Strong Evidence for Changing Fish Reproductive Phenology under Climate Warming on the Tibetan Plateau[END_REF], or the threespined stickleback (Gasterosteus aculeatus (Linnaeus, 1758)) [START_REF] Hovel | Climate Change Alters the Reproductive Phenology and Investment of a Lacustrine Fish, the Three-spine Stickleback[END_REF]. [START_REF] Gillet | Effect of Temperature Changes on the Reproductive Cycle of Roach in Lake Geneva from 1983 to 2001[END_REF] showed that an increase of 1 °C in Lake Geneva led to earlier development of the ovaries and spawning date of roach, suggesting that spawning would be reached earlier with global warming. In a historical review, [START_REF] Thorpe | Morphology, Physiology, Behavior, and Ecology of Perca fluviatilis L. And P. flavescens Mitchill[END_REF] showed a positive relationship between spawning date and latitude for the Eurasian perch (Perca fluviatilis (Linnaeus, 1758)), which generally spawn at a lower temperature and later date as latitude increases. However, current work using long-term surveys showed significant phenological trends associated with climate change. Spring spawning of Eurasian perch of the North Basin and the South Basin of Windermere (UK) advanced by 2.5-5.0 days decade -1 from 1968 to 2009 [START_REF] Thackeray | Food Web De-synchronization in England's Largest lake: An Assessment Based on Multiple Phenological Metrics[END_REF]. Furthermore, shifts in the annual thermal regime to an earlier spring warming and later autumn cooling now affect both spring and autumn spawners. In the Laurentian Great Lakes, [START_REF] Lyons | Trends in the Reproductive Phenology of Two Great Lakes Fishes[END_REF] suggested that the yellow perch (P. flavescens (Mitchill, 1814)) was spawning earlier to maintain a consistent spawning temperature. The phenological patterns observed for lake trout (Salvelinus namaycush (Walbaum, 1792)) indicated a possible tendency for later spawning in autumn, but the trend was less explicit for the period studied . In the current context of global warming, multi-spawning species as chub and thermophilous fish species could benefit from an extended spawning period, a longer first year growing period and progressively dominate fish assemblages [START_REF] Mueller | Comprehensive Analysis of >30 Years of Data on Stream Fish Population Trends and Conservation Status in Bavaria, Germany[END_REF][START_REF] Maire | Poleward Shift in Large-river Fish Communities Detected with a Novel Meta-analysis Framework[END_REF].

Temperature is already known to affect numerous aspects of fish reproduction and early life stages, such as gonad development, hatching date [START_REF] Pankhurst | Effects of Climate Change on Fish Reproduction and Early Life History Stages[END_REF], and the number of annual spawning events for multiple or protracted spawning species [START_REF] Nunn | Variations in the Spawning Periodicity of Eight Fish Species in Three English Lowland Rivers over a 6 Year Period, Inferred from 0+ Year Fish Length Distributions[END_REF]. Multiple spawners such as bleak and chub [START_REF] Peňáz | Fish Fry Community of the Jilhava River Near Mohelno[END_REF][START_REF] Fredrich | Spawning Migrations of the Chub in the River Spree, Germany[END_REF] experienced more spawning events in warm compared with cold environmental conditions [START_REF] Mann | Geographical Variation in the Life-History Tactics of Some Species of Freshwater Fish[END_REF]. While these breeding strategies were known to increase the number of yearly sub-cohorts and the extent of YOY size range during warmer and dryer years [START_REF] Bolland | Lifetime Consequences of Variable 0 Year Group Length in Riverine Populations of Chub Leuciscus cephalus (L.)[END_REF][START_REF] Nunn | Variations in the Spawning Periodicity of Eight Fish Species in Three English Lowland Rivers over a 6 Year Period, Inferred from 0+ Year Fish Length Distributions[END_REF], current change in the thermal regime of large rivers such as the Rhône River could explain the observed recent success of species with multiple spawning such as chub, bleak (Alburnus alburnus (Linnaeus, 1758)), gudgeon (Gobio gobio (Linnaeus, 1758)), barbel (Barbus barbus (Linnaeus, 1758)), schneider (Alburnoides bipunctatus (Bloch, 1782)), bitterling (Rhodeus amarus (Bloch, 1792)), and topmouth gudgeon (Pseudorasbora parva (Temminck and Schlegel, 1846)) [START_REF] Maire | Poleward Shift in Large-river Fish Communities Detected with a Novel Meta-analysis Framework[END_REF]. In addition to these changes in fish assemblages, current monitoring surveys showed an increase in the variability and extent of YOY size range in autumn, with sampling of small surviving individuals in following year (G. Carrel, pers. obs.).

Growth

Degree-days are known to be a reliable predictor of fish growth [START_REF] Neuheimer | The Growing Degree-Day and Fish Size-At-Age: The Overlooked Metric[END_REF]. In our study, the size at capture of YOY chub had a clear positive exponential relationship with GDD regardless of section or year. The thermal integral has already been used to explain the inter-annual variation of the mean size of dominant cyprinids such as roach, dace, and chub in British rivers [START_REF] Mills | Environmentally-induced Fluctuations in Year-Class Strength and Their Implications for Management[END_REF][START_REF] Nunn | Is Water Temperature an Adequate Predictor of Recruitment Success in Cyprinid Fish Populations in Lowland Rivers? Freshw[END_REF] and of the most common species of fish assemblage in the lower Oder River [START_REF] Wolter | Temperature Influence on the Fish Assemblage Structure in a Large lowland River, the Lower Oder River, Germany[END_REF]. Nonetheless, our results suggest that the variation 1.37 (1.12-1.74) 15.38 (12.63-19.34) 10.84 (8.91-13.60) 3.95 (3.23-4.95) Precision parameters were transformed into standard deviations (sdk 1/τk ), averaged and the credible intervals (95%) added in parenthesis. BPS, bypassed section; RES, reservoir of Caderousse; TAIL, tailrace of the Bollène power plant; L ∞ , asymptotic size; α, the minimal asymptotic size; T, the time at the inflexion point; δ, scale factor.

in GDD experienced individually by YOY during their first growth period could also explain the variability in autumnal size between individuals. Therefore, GDD is a useful thermal parameter to explain both inter-annual and intra-annual variability of size. As proposed by [START_REF] Spurgeon | Ecosystem-specific Growth Responses to Climate Pattern by a Temperate Freshwater Fish[END_REF], GDD could be more widely used as a predictor of fish growth.

Several models were developed to model individual growth and were successfully used with fish [START_REF] Gamito | Growth Models and Their Use in Ecological Modelling: An Application to a Fish Population[END_REF][START_REF] Blanck | Large-scale Intraspecific Variation in Life-History Traits of European Freshwater Fish[END_REF]. Even if the Von Bertalanffy model (one of the most used model) was successfully used by some authors to model daily growth (e.g., [START_REF] Loisel | Variation of thermal Plasticity in Growth and Reproduction Patterns: Importance of Ancestral and Developmental Temperatures[END_REF], it performed poorly on our data (unpublished results) in accordance with other studies (e.g. [START_REF] Lester | Interpreting the von Bertalanffy model of Somatic Growth in Fishes: The Cost of Reproduction[END_REF]. By contrast, the logistic growth model with four parameters (minimal asymptotic size, maximal asymptotic size time at the point of inflexion, and the rate of maximum growth) was well adapted to model YOY daily growth from size back-calculation, as demonstrated by the RMSE values.

Growth patterns showed an increase in fish growth with the increase in temperature for the three sections. Overall, when temperature increased, we observed an increase in the asymptotic size (L ∞ ), the growth rate (K), and a delayed inflexion point (≈10 days later than in 2013). Therefore, fish were larger due to the fact that they grew faster and for a longer period of time in warmer conditions. However, the magnitude of the difference between average growth curves varied among sections and was most marked for the reservoir. For the BPS, growth curves diverged lately but led to very different asymptotic size, mainly due to the delayed inflexion point. This reveals the importance of the early hatching and thus of the longer growing period on the autumn size of the YOY inhabiting the BPS. Fish in the tailrace also exhibited magnified growth in warmer conditions, with fast deviation of the average growth curves (but overlapping credible intervals). Surprisingly, the estimated L ∞ were relatively similar. Several hypotheses could explain the high L ∞ values observed in the TAIL: a selection of the most robust individuals according to the hydraulic characteristics of this section; the achievement of maximal fish growth in relation to the artificial thermal context; a greatest food availability or a higher food quality [START_REF] Jonsson | Effects of Temperature and Food Quality on Age and Size at Maturity in Ectotherms: An Experimental Test with Atlantic salmon[END_REF][START_REF] Uphoff | Degree-day Accumulation Influences Annual Variability in Growth of Age-0 Walleye[END_REF]; or a lower density of fish for example [START_REF] Grant | Patterns of Density-dependent Growth in Juvenile Stream-Dwelling Salmonids[END_REF]. In a previous study, [START_REF] Morat | Spatial Distribution, Total Length Frequencies and Otolith Morphometry as Tools to Analyse the Effects of a Flash Flood on Populations of Roach (Rutilus rutilus)[END_REF] also showed that roach were more abundant in the BPS than in the TAIL. By reducing competition, this lower fish density could explain the higher fish size observed in the TAIL. The relative stability of TAIL abiotic conditions (oxygen saturation and temperature) compared with other sections could also explain this pattern as variations in oxygen saturation was shown to reduce fish growth [START_REF] Mustafa | Hypoxiainduced Oxidative DNA Damage Links with Higher Level Biological Effects Including Specific Growth Rate in Common Carp, Cyprinus carpio[END_REF]. Although these parameter values are estimated and not observed size at the end of the first growing season, they were consistent with YOY total size already recorded in this section during early winter [START_REF] Meynard | Etude thermique du Rhône Phase IV -Lot 2(Aix-en-Provence: U.R. Hydrobiologie, Irstea. Facteurs hydroclimatiques et taille des poissons juvéniles[END_REF]; but other physicochemical characteristics could also have been of major importance. Therefore, the mechanisms explaining the different growth patterns observed between the two studied years were different for the three sections, as revealed by the distributions of the model parameters.

Globally, our results are consistent with the general theory that attempts to explain the variation in individual size with temperature. Indeed, the "Temperature-Size Rule" (TSR) theory [START_REF] Atkinson | Temperature and Organism Size-A Biological Law for Ectotherms?[END_REF] states that, in warmer conditions, growth at young stages is magnified compared with the growth of individuals reared in cooler temperatures. While working with individuals sampled in situ, we obtained results consistent with those obtained in controlled conditions [START_REF] Loisel | Variation of thermal Plasticity in Growth and Reproduction Patterns: Importance of Ancestral and Developmental Temperatures[END_REF]: larger fish and greater growth rates in warmer conditions. This was observed when comparing the growth pattern of chub YOY for each section: as GDD increased, sizeat-age also increased. As other large river-floodplain hydrosystems in Europe, the composition of Rhône fish assemblages had been historically modified following major engineering works during the 19th and 20th centuries [START_REF] Olivier | The Rhône River Basin[END_REF]. [START_REF] Daufresne | Fish Community Dynamics (1985-2010) in Multiple Reaches of a Large River Subjected to Flow Restoration and Other Environmental Changes[END_REF] showed that global warming increased the rapid turnover of the fish communities in the Rhône River, with "southern" and smaller-size species increasingly represented. Along the Rhône River, three open-cycle cooling nuclear power plants (Bugey, Saint-Alban, and Tricastin) contribute to an artificial increase in water temperature along the river [START_REF] Wawrzyniak | Longitudinal and Temporal Thermal Patterns of the French Rhône River Using Landsat ETM+ thermal Infrared Images[END_REF]. Current climate change exacerbates the ecological impacts of rising temperatures [START_REF] Daufresne | Long-term Changes within the Invertebrate and Fish Communities of the Upper Rhône River: Effects of Climatic Factors[END_REF]. The TSR predicts that adults will be smaller and their size at first reproduction will likewise be smaller [START_REF] Kuparinen | Fish Age at Maturation Is Influenced by Temperature Independently of Growth[END_REF][START_REF] Loisel | Variation of thermal Plasticity in Growth and Reproduction Patterns: Importance of Ancestral and Developmental Temperatures[END_REF], leading to the rejuvenation of fish populations (e.g. Daufresne and Boet, 2007;[START_REF] Piffady | Quantifying the Effects of Temperature and Flow Regime on the Abundance of 0+ Cyprinids in the Upper River Rhone Using Bayesian Hierarchical Modelling[END_REF]. The success of small thermophile species such as the bitterling or the topmouth gudgeon in the lower Rhône River, as in other large regulated European rivers, strengthens the general decrease in fish size within the overall fish assemblage. Nonetheless, even if these results were consistent with the TSR, the patterns observed in the different sections and their variations between years demonstrated that other factors such as the water velocity, phenology, or the oxygen availability are able to modulate the effect of temperature on the growth of chub YOY.

In addition, even if significant differences in GDD were observed between years, only fish in the RES displayed large differences in their growth pattern, although growth was magnified for the three sections during the warmer year. This is consistent with the study of [START_REF] Richard | Water Body Type Influences Climate-Growth Relationships of Freshwater Drum[END_REF], which showed that temperature effect on growth rate is magnified in lentic systems. Temperature effect varies with fish size and/or life stages. Indeed, the effect of warming on fish growth was positive but stronger for small individuals than for larger fish; this was observed for flatfishes [START_REF] Imsland | The Interaction of Temperature and Fish Size on Growth of Juvenile Turbot[END_REF], Gadids [START_REF] Pedersen | Growth Rates of Large, Sexually Mature Cod Gadus morhua, in Relation to Condition and Temperature During an Annual Cycle[END_REF], Salmonids [START_REF] Beauchamp | Bioenergetic Ontogeny: Linking Climate and Massspecific Feeding to Life-Cycle Growth and Survival of salmon[END_REF], and Percids [START_REF] Huss | Experimental Evidence of Gradual Size-dependent Shifts in Body Size and Growth of Fish in Response to Warming[END_REF]. This difference seems linked to thermal optimum that decrease with fish size. On two long time series, [START_REF] Huss | Experimental Evidence of Gradual Size-dependent Shifts in Body Size and Growth of Fish in Response to Warming[END_REF] showed that the average size of fish was longer for all size-at-age with warming, but also that this is related to a gradual higher growth at early life stage. This size-dependent warming effect on fish growth are known to change regulation and dynamic of populations and communities by affecting the species interactions [START_REF] Lindmark | Temperaturedependent Body Size Effects Determine Population Responses to Climate Warming[END_REF]2019).

The inter-annual variations of YOY size at the end of the growing season observed in the Rhône River, as well as other rivers [START_REF] Bolland | Lifetime Consequences of Variable 0 Year Group Length in Riverine Populations of Chub Leuciscus cephalus (L.)[END_REF], could be explained both by an important variability in hatching time (earlier in the warmest environment or years), in the length of their growing period and by differential growth rates among individuals and/or populations. Moreover, our study results evidenced the major effect of thermal conditions (expressed in GDD) experienced by YOY during their life and are in accordance with the TSR. However, our results also showed that other factors might mitigate the influence of temperature on the early life stage of fish.

FIGURE 1 |

 1 FIGURE 1 | Map of the Caderousse reach on the lower Rhône River. Chub were sampled in the bypassed section (BPS), in the reservoir (RES) of the Caderousse dam and power plant, and in the tailrace (TAIL) of the Bollène power plant. The gray areas represent the sampling sites. The arrows locate the four gauging stations and the stars locate the hobo thermometers.

FIGURE 2 |

 2 FIGURE 2 |The logistic growth curve with the four parameters: asymptotic size (L ∞ ), minimal asymptotic size (α), time at the inflexion point (T), and maximum growth rate (K), derived from the scale parameter δ and computed as K (L ∞ -α)/(4δ).

FIGURE 3 |

 3 FIGURE 3 | Hatching date distributions (A,B), daily mean discharges (m 3 s -1 ) (C,D), and daily mean water temperature ( °C) patterns (E,F) during the two studied years (2013, left panels and 2015, right panels).

FIGURE 4 |

 4 FIGURE 4 | Relationship between growing degree-days (GDD > 12 °C) and chub size at capture (L cpt ), between years and sections. Curves are the estimated exponential regressions.

FIGURE 5 |

 5 FIGURE 5 | Average values (dots) and their associated credible intervals (95%; vertical lines) of the posterior distributions of the four parameters of the logistic growth curves, per sections and year, with the maximum asymptotic size (L ∞ ), the minimal asymptotic size (α), the time at the inflexion point (T), and the maximum growth rate (K).

FIGURE 6 |

 6 FIGURE 6 | The average growth curves (lines) and their associated credible intervals (95%; bands) for each section and each year during the maximum growth periods observed.

TABLE 1 |

 1 Root mean square errors (mm) of the relationships between growing degree-days (GDD) and size at capture (L cpt ) for the different sections and years for chub collected in the reach of Caderousse (Rhône River, France).

	Section	RMSE (mm)
		2013	2015
	BPS	6.02	6.32
	RES	8.18	10.90
	TAIL	5.10	7.25
	BPS, bypassed section; RES, reservoir of Caderousse; TAIL, tailrace of the Bollène
	power plant.

TABLE 2 |

 2 Average root mean square errors (RMSE; mm) of individual growth curves and their associated credible interval CI (95%).

	Section	Year	RMSE (mm)
	BPS	2013	0.433 (0.128-0.911)
		2015	0.298 (0.147-0.474)
	TAIL	2013	0.556 (0.178-0.856)
		2015	0.428 (0.204-0.946)
	RES	2013	0.383 (0.136-0.899)
		2015	0.376 (0.154-0.636)
	BPS, bypassed section; RES, reservoir of Caderousse; TAIL, tailrace of the Bollène
	power plant.

TABLE 3 |

 3 Summary of the posterior distribution of the dispersion parameters (τ k ).

	Section	Year	α	L '	T	δ
	BPS	2013	1.75 (1.49-2.10)	11.19 (9.63-13.21)	10.49 (9.04-12.37)	4.60 (3.95-5.45)
		2015	2.45 (1.98-3.12)	12.01 (9.83-15.13)	10.35 (8.49-13.03)	5.49 (4.46-6.97)
	TAIL	2013	3.38 (2.38-5.15)	12.93 (9.38-19.29)	8.65 (6.29-12.90)	5.14 (3.68-7.75)
		2015	2.86 (2.26-3.75)	12.32 (9.86-15.99)	7.93 (6.36-10.30)	5.02 (3.99-6.57)
	RES	2013	2.12 (1.71-2.71)	14.52 (11.79-18.55)	10.35 (8.40-13.16)	5.58 (4.51-7.13)
		2015

Frontiers in Environmental Science | www.frontiersin.org December 2021 | Volume 9 | Article 784751

ACKNOWLEDGMENTS

The authors would like to thank Baptiste Testi and Pierre Favriou for their help with fish collection, Jean-Jacques Boreux of the University of Liége for his precious help concerning Bayesian hierarchical models, Christine Argillier for her help in the preparation of this paper and Coralie Garron for her precious help with the English.

FUNDING

This study was funded by the French Office for Biodiversity (OFB), by Electricity of France (EDF), and by the national research institute in science and technology for the environment and agriculture (INRAE).

DATA AVAILABILITY STATEMENT

The original contributions presented in the study are publicly available. This data can be found here. https://doi.org/10.6084/ m9.figshare.16691323 [START_REF] Logez | Fish_growth_data_set[END_REF].

ETHICS STATEMENT

Ethical review and approval was not required for the animal study because All relevant international, national, and/or institutional guidelines for the care and use of animals were followed. Fish were sacrificed using anaesthetic overdose (eugenol) in accordance with the European Directive 2010/63/EU. Fish were collected in accordance with the authorisation numbers: 2013-203-0005 and 2015-229-DDTSE01 delivered by the prefecture.

AUTHOR CONTRIBUTIONS

This study was designed by GC and FM. Field collections were made GC, FM, XC, PG, and VD. Otolith analyses were conducted by BB, PG, XC, and FM. Statistical analyses were conducted by ML. The manuscript was written by ML, FM, GC, and BB.

SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fenvs.2021.784751/ full#supplementary-material Conflict of Interest: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's Note: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.