Alexandre Sedoglavic

Alexey V Smirnov

The tensor rank of 5 × 5 matrices multiplication is bounded by 98 and its border rank by 89

Keywords:

We present a non-commutative algorithm for the product of 3 × 5 by 5 × 5 matrices using 58 multiplications. This algorithm allows to construct a non-commutative algorithm for multiplying 5 × 5 (resp. 10 × 10, 15 × 15) matrices using 98 (resp. 686, 2088) multiplications. Furthermore, we describe an approximate algorithm that requires 89 multiplications and computes this product with an arbitrary small error.

INTRODUCTION

Even if matrix multiplication is one of the most fundamental tool in scientific computing, it is still not completely understood. Strassen's algorithm [START_REF] Strassen | Gaussian elimination is not optimal[END_REF], with 7 recursive multiplications and 18 additions, was the first sub-cubic time algorithm for matrix product (with a cost of 𝑂 (𝑛 log 2 7 )) and finding explicit algorithms for small size matrix product remains today a challenge. In order to describe the contributions presented in this paper, let us recall some well-known terminologies for the sake of clarity. Definitions 1.1. We denote by M 𝑚,𝑛,𝑝 the bilinear map representing the matrix product of a 𝑚 × 𝑛-matrix by a 𝑛 × 𝑝 matrix. In particular, given any field F, there exist 𝑟 𝑚 × 𝑝 matrices (𝐶 𝑖 ) 1≤𝑖 ≤𝑟 , 𝑟 linear forms (ℓ 𝑖 ) 1≤𝑖 ≤𝑟 from F 𝑚×𝑛 into F and 𝑟 linear forms (ℓ ′ 𝑖 ) 1≤𝑖 ≤𝑟 from F 𝑛×𝑝 into F such that the product of the 𝑚 × 𝑛 matrix 𝐴 by the 𝑛 × 𝑝 matrix 𝐵 is computed by the following computational scheme:

M 𝑚,𝑛,𝑝 (𝐴, 𝐵) = 𝐴 • 𝐵 = 𝑟 ∑ 𝑖=1 ℓ 𝑖 (𝐴)ℓ ′ 𝑖 (𝐵)𝐶 𝑖 . (1)
This non-commutative scheme is classically interpreted as a tensor (see precise encoding in Section 2) and we recall that the number 𝑟 of its summands is the rank of that tensor. In this work, the notations ⟨𝑚 × 𝑛 × 𝑝 : 𝑟 ⟩ stands for a tensor of rank 𝑟 encoding the product M 𝑚,𝑛,𝑝 . We denotes by ⟨𝑚 × 𝑛 × 𝑝⟩ the whole family of such schemes independently of their rank. The tensor rank R⟨𝑚 × 𝑛 × 𝑝⟩ of the considered matrix product is the smallest integer 𝑟 such that there is a tensor ⟨𝑚 × 𝑛 × 𝑝 : 𝑟 ⟩ in ⟨𝑚 × 𝑛 × 𝑝⟩. Similarly, {𝑚 × 𝑛 × 𝑝 : 𝑟 } denotes a computational scheme of rank 𝑟 involving a parameter 𝜖 whose limit computes the matrix product M 𝑚,𝑛,𝑝 exactly as 𝜖 tends to zero. The border rank of M 𝑚,𝑛,𝑝 is the smallest integer 𝑟 such that there exits an approximate scheme {𝑚 × 𝑛 × 𝑝 : 𝑟 }.

The tensor rank R⟨𝑛 × 𝑛 × 𝑛⟩ is related to the number of multiplications needed to compute the product of two 𝑛 × 𝑛 and gives a measures of its complexity such as the exponent 𝜔 of matrix multiplication equal to lim 𝑛→∞ R⟨𝑛 × 𝑛 × 𝑛⟩. There is no result relative to asymptotic complexity in the present work and we refer to [START_REF] Alman | A refined laser method and faster matrix multiplication[END_REF] for a presentation of these researches, bibliographic references and the last best asymptotic bound 𝑂 (𝑛 2.3728596 ) known to date.

Alongside these galactic algorithms, a substantial amount of work was devoted to the practical design of computational schemes for small matrix products (see [START_REF] Drevet | Optimization techniques for small matrix multiplication[END_REF] for bibliographic references and [START_REF] Sedoglavic | Fast matrix multiplication database[END_REF] for last known results). Our contribution follows this path and improves the complexity of the product of 5 × 5 matrices and several other algorithms.

The structure of ⟨2 × 2 × 2⟩ is broadly understood: there is basically just one rank 7 tensor ⟨2 × 2 × 2 : 7⟩ introduced in [START_REF] Strassen | Gaussian elimination is not optimal[END_REF] up to symmetries as shown in [10, Thm 0.1]; furthermore, in that case the tensor rank is known to be 7 (see [START_REF] Winograd | On multiplication of 2 × 2 matrices[END_REF]Thm 3.1]). But already the structure of ⟨3 × 3 × 3⟩ remains unresolved and very little is known about the actual complexity of small matrix products.

Pursuing the works done the last 53 years in [START_REF] Makarov | A noncommutative algorithm for multiplying 5 × 5 matrices using 102 multiplications[END_REF][START_REF] Makarov | A non-commutative algorithm for multiplying 5 × 5 matrices using one hundred multiplications[END_REF][START_REF] Schachtel | A noncommutative algorithm for multiplying 5 × 5 matrices using 103 multiplications[END_REF][START_REF] Sedoglavic | A non-commutative algorithm for multiplying 5 × 5 matrices using 99 multiplications[END_REF], we propose a new non-commutative algorithm for multiplying 5 × 5 matrices using 98 multiplications. More precisely, by presenting explicit algorithms, we establish the following result: Theorem 1.1. The tensor rank of matrix multiplication tensor encoding the product of a 3 × 5 by a 5 × 5 matrix is bounded by 58.

This implies that the tensor rank of matrix multiplication tensor encoding the product of 5 × 5 matrices is bounded by 98.

Furthermore, the border rank of matrix multiplication tensor encoding the product of 5 × 5 matrices is bounded by 89.

This theorem results mainly from the work initiated by the second author in [START_REF] Smirnov | The bilinear complexity and practical algorithms for matrix multiplication[END_REF]. This work induces many algorithms (e.g. see [START_REF] Sedoglavic | Fast matrix multiplication database[END_REF] for a complete list) and especially an algorithm ⟨3 × 3 × 6 : 40⟩ that leads to tensors ⟨6 × 6 × 6 : 160⟩ and ⟨12 × 12 × 12 : 1040⟩ (this last one defines a generic algorithm with cost 𝑂 (𝑛 2.796 ) for suitable square matrices).

TENSOR REPRESENTATION OF MATRIX MULTIPLICATION

In the forthcoming sections, we present several matrix product algorithms as trilinear forms. To do so, let us review the needed notions through an well-known example. The matrix product 𝐶 = 𝐴 • 𝐵 could be computed using Strassen algorithm by performing the following computations (see [START_REF] Strassen | Gaussian elimination is not optimal[END_REF]):

𝜌 1 ← 𝑎
𝑐 21 𝑐 22 = 𝜌 5 +𝜌 4 -𝜌 2 +𝜌 6 𝜌 6 +𝜌 3 𝜌 2 +𝜌 1 𝜌 5 +𝜌 7 +𝜌 1 -𝜌 3 .
(2)

In order to consider this algorithm under a geometric standpoint, we present it as a tensor. Matrix multiplication is a bilinear map:

F 𝑚×𝑛 × F 𝑛×𝑝 → F 𝑚×𝑝 , (𝐴, 𝐵) → 𝐴 • 𝐵, (3)
where the spaces F •×• are finite vector spaces over a field F that can be endowed with the Frobenius inner product:

⟨𝑀, 𝑁 ⟩ = Trace(𝑀 ⊺ • 𝑁 ). (4)
Hence, this inner product establishes an isomorphism between F •×• and its dual space F •×• ★ allowing for example to associate matrix multiplication and the trilinear form Trace(𝐶 ⊺ • 𝐴 • 𝐵):

F 𝑚×𝑛 × F 𝑛×𝑝 × (F 𝑚×𝑝 ) ★ → F, (𝐴, 𝐵, 𝐶 ⊺ ) → ⟨𝐶, 𝐴 • 𝐵⟩. (5)
As by construction, the space of trilinear forms is the canonical dual space of order three tensor product, we could associate the Strassen multiplication algorithm (2) with the tensor S defined by:

7 𝑖=1 𝑃 𝑖 ⊗𝑄 𝑖 ⊗𝑆 𝑖 = 1 0 0 0 ⊗ 0 1 0 -1 ⊗ 0 0 1 1 + 1 1 0 0 ⊗ 0 0 0 1 ⊗ -1 0 1 0 + 0 0 1 1 ⊗ 1 0 0 0 ⊗ 0 1 0 -1 + 0 1 0 -1 ⊗ 0 0 1 1 ⊗ 1 0 0 0 + 1 0 0 1 ⊗ 1 0 0 1 ⊗ 1 0 0 1 + 0 0 0 1 ⊗ -1 0 1 0 ⊗ 1 1 0 0 + -1 0 1 0 ⊗ 1 1 0 0 ⊗ 0 0 0 1 (6) in (F 𝑚×𝑛 ) ★ ⊗ (F 𝑛×𝑝 ) ★ ⊗ F 𝑚×𝑝 with 𝑚 = 𝑛 = 𝑝 = 2.
As stated in Definitions 1.1, this tensor is a particular description of the matrix product bilinear map. Indeed, there is an infinite set of equivalent tensors shown by the following theorem:

Theorem 2.1 ([9, § 2.8]). The isotropy group of any matrix multiplication tensor in ⟨𝑚 × 𝑛 × 𝑝⟩ is (psl(𝑚) × psl(𝑛) × psl(𝑝)) ⋊ 𝔖 3 , where psl stands for the group of matrices of determinant ±1 and 𝔖 3 for the symmetric group on 3 elements.

The following proposition gives another description from an invariant perspective. Proposition 2.2 ([16, § 2.5.1]). The tensor defining the product of a 𝑚 × 𝑛-matrix by a 𝑛 × 𝑝-matrix is isomorphic to the tensor Id 𝑚×𝑚 ⊗ Id 𝑛×𝑛 ⊗ Id 𝑝×𝑝 . This isomorphism gives the well-known expression of the classical matrix multiplication tensor:

𝑚 ∑ 𝑖=1 𝑛 ∑ 𝑗=1 𝑝 ∑ 𝑘=1 𝐸 𝑗 𝑖 ⊗ 𝐸 𝑘 𝑗 ⊗ 𝐸 𝑖 𝑘 , (7)
where 𝐸 𝑗 𝑖 denotes the matrix with its coefficient at the intersection of line 𝑖 and column 𝑗 equal to 1 and all its other coefficients equal to 0.

Let us introduce now an invariant for the action described in Theorem 2.1 that will be useful in our presentation. Definition 2.1. Given a tensor P decomposable as sum of rankone elementary tensors:

P = 𝑞 ∑ 𝑖=1 𝑃 𝑖 ⊗ 𝑄 𝑖 ⊗ 𝑆 𝑖 (8)
where 𝑃 𝑖 , 𝑄 𝑖 and 𝑆 𝑖 are matrices of suitable sizes for 𝑖 in {1, . . . , 𝑞}.

The type of a tensor P is the list [(rank 𝑃 𝑖 , rank 𝑄 𝑖 , rank 𝑆 𝑖 )] 𝑖=1...𝑞 .

Inspired by [13, § 4], we encode the type of a tensor P as the following polynomial:

𝑇 P (𝑋, 𝑌, 𝑍 ) = 𝑞 ∑ 𝑖=1 𝑋 rank 𝑃 𝑖 𝑌 rank 𝑄 𝑖 𝑍 rank 𝑆 𝑖 . (9)
Hence, the type of Strassen's tensor is 𝑋 2 𝑌 2 𝑍 2 + 6 𝑋𝑌 𝑍 . Now we precise the trilinear form of tensor that is used in the sequel of this work. Given any triple (𝐴, 𝐵, 𝐶) of suitable size matrices, one can explicitly express from tensor S the Strassen matrix multiplication algorithm computing 𝐴 • 𝐵 by the complete contraction {S, 𝐴 ⊗ 𝐵 ⊗ 𝐶}:

(F 𝑚×𝑛 ) ★ ⊗ (F 𝑛×𝑝 ) ★ ⊗F 𝑚×𝑝 ⊗ F 𝑚×𝑛 ⊗F 𝑛×𝑝 ⊗ (F 𝑚×𝑝 ) ★ → F, S ⊗ (𝐴 ⊗ 𝐵 ⊗ 𝐶) → 7 𝑖=1 ⟨𝑃 𝑖 , 𝐴⟩⟨𝑄 𝑖 , 𝐵⟩⟨𝑆 𝑖 , 𝐶⟩ (10)
that is equal to Trace(𝐴 • 𝐵 • 𝐶). Hence, for inputs 𝐴 = (𝑎 𝑖 𝑗 ), 𝐵 = (𝑏 𝑖 𝑗 ) and 𝐶 = (𝑐 𝑖 𝑗 ) of suitable sizes (𝑖 = 1, 2 and 𝑗 = 1, 2), the representation of Strassen Algorithm (2) as a trilinear form is:

𝑎 11 (
Before ending this section let us recall that, as stated in introduction, the ⟨2 × 2 × 2⟩ is pretty well-understood even in its rectangular counterpart as shown by the following proposition: Proposition 2.3 ([14, Thm 1]). The 𝑚 × 2 by 2 × 𝑛 matrix product can be encoded by a ⟨𝑚 × 2 × 𝑛 : ⌈(3𝑚𝑛 + max(𝑚, 𝑛))/2⌉⟩ tensor.

The next section is devoted to gather classical results used in the sequel of this paper to construct new algorithm for small matrix product.

Constructing composite tensors from small "atomic" ones

Introducing tensor to represents matrix product and their relationship with the trace operator induces naturally several interesting results on matrix product algorithms. First, let us remark thatgiven three matrices 𝐴, 𝐵 and 𝐶 of suitable sizes-the following properties of the trace operator:

Trace (𝐴 • 𝐵 • 𝐶) = Trace (𝐶 • 𝐴 • 𝐵), = Trace (𝐵 • 𝐶 • 𝐴), = Trace (𝐴 • 𝐵 • 𝐶) ⊺ , = Trace 𝐶 ⊺ • 𝐵 ⊺ • 𝐴 ⊺ , (12)
show that the following relations hold:

⟨𝑚 × 𝑛 × 𝑝 : 𝑟 ⟩ = ⟨𝑝 × 𝑚 × 𝑛 : 𝑟 ⟩ = ⟨𝑛 × 𝑝 × 𝑚 : 𝑟 ⟩, = ⟨𝑝 × 𝑛 × 𝑚 : 𝑟 ⟩ = ⟨𝑚 × 𝑝 × 𝑛 : 𝑟 ⟩, = ⟨𝑛 × 𝑚 × 𝑝 : 𝑟 ⟩. (13)
Furthermore, for all ℓ such that 1 ≤ ℓ ≤ 𝑚 -1 there is a natural isomorphism between F 𝑚×𝑛 and F ℓ×𝑛 ⊕ F (𝑚-ℓ)×𝑛 ; The same remark shows that Id 𝑚×𝑚 is equal to Id ℓ×ℓ ⊕ Id (𝑚-ℓ)×(𝑚-ℓ) . This relation and the tensor product's properties imply that the tensor Id 𝑚×𝑚 ⊗ Id 𝑛×𝑛 ⊗ Id 𝑝×𝑝 is equal to

Id ℓ×ℓ ⊗ Id 𝑛×𝑛 ⊗ Id 𝑝×𝑝 + Id (𝑚-ℓ)×(𝑚-ℓ) ⊗ Id 𝑛×𝑛 ⊗ Id 𝑝×𝑝 . ( 14
)
These simple remarks and Proposition 2.2 recall that small sizes matrix product algorithms allow by their direct sum to construct an algorithm for the product of matrices of bigger sizes as shown by the following well-known lemma:

Lemma 2.1. Given ⟨ℓ × 𝑛 × 𝑝 : 𝑟 ⟩ and ⟨(𝑚ℓ) × 𝑛 × 𝑝 : 𝑠⟩, one can construct ⟨𝑚 × 𝑛 × 𝑝 : 𝑟 + 𝑠⟩ as follow:

⟨𝑚 × 𝑛 × 𝑝 : 𝑟 + 𝑠⟩ = ⟨ℓ × 𝑛 × 𝑝 : 𝑟 ⟩ ⊕ ⟨(𝑚 -ℓ) × 𝑛 × 𝑝 : 𝑠⟩. (15)
There is a similar construction using the tensor product:

⟨𝑚𝑢 × 𝑛𝑣 × 𝑝𝑤 : 𝑟𝑠⟩ = ⟨𝑚 × 𝑛 × 𝑝 : 𝑟 ⟩ ⊗ ⟨𝑢 × 𝑣 × 𝑤 : 𝑠⟩. (16)
We say that a tensor is "atomic" if it is not constructed using the above constructions, Proposition 2.3 or if it is induced by Strassen's algorithm (see [8, § 2] and [START_REF] Sedoglavic | A non-commutative algorithm for multiplying 7 × 7 matrices using 250 multiplications[END_REF] for such constructions).

The previous sections were devoted to the notions and notations necessary to describe concisely the new results on which the next sections focus. We could now present the main result of this paper.

NEW EXACT ALGORITHMS

The rank of the classical tensor in ⟨3 × 5 × 5⟩ is 75. Proposition 2.3 and Lemma 2.1 improves the resulting bound to 63 as shown by the following relations:

⟨3 × 3 × 5 : 38⟩ = ⟨3 × 3 × 3 : 23⟩ ⊕ ⟨3 × 3 × 2 : 15⟩, (17)
⟨3 × 5 × 5 : 63⟩ = ⟨3 × 3 × 5 : 38⟩ ⊕ ⟨3 × 2 × 5 : 25⟩. (18)
This bound was superseded using ⟨3 × 3 × 5 : 36⟩ introduced in [START_REF] Smirnov | The bilinear complexity and practical algorithms for matrix multiplication[END_REF] and above standard constructions.

Our results are also rooted in the same kind of experimental mathematics combining computer power and human efforts. Thus, let us give a short account of a method allowing to find new atomic tensors in the next section.

How to construct small rank matrix multiplication tensors

As any tensor in ⟨𝑚 × 𝑛 × 𝑝⟩ encodes the same bilinear map, Proposition 7 implies that the following relation always holds:

⟨𝑚 × 𝑛 × 𝑝 : 𝑟 ⟩ - 𝑚 ∑ 𝑖=1 𝑛 ∑ 𝑗=1 𝑝 ∑ 𝑘=1 𝐸 𝑗 𝑖 ⊗ 𝐸 𝑘 𝑗 ⊗ 𝐸 𝑖 𝑘 = 0. ( 19
)
Using an ansatz with undetermined coefficients for ⟨𝑚 × 𝑛 × 𝑝 : 𝑟 ⟩, this relation defines the Brent over-determined system of (𝑚𝑛𝑝) 2 cubic polynomial equations in (𝑚𝑛 + 𝑛𝑝 + 𝑝𝑚)𝑟 unknowns (see [5, § 5, eq 5.03]). Theoretically, a Gröbner basis computation allows to describe all solutions of this system and thus close the topic.

But such a resolution is not possible in practice for any matrix size of interest. Nevertheless several original matrix multiplication algorithms where found by hand (e.g. ⟨3 × 3 × 3 : 23⟩ in [START_REF] Laderman | A noncommutative algorithm for multiplying 3 × 3 matrices using 23 multiplications[END_REF] and probably Strassen's algorithm [START_REF] Strassen | Gaussian elimination is not optimal[END_REF]) and almost every method for solving were tried (e.g. sat solver in [13, § 2]) with-up to our knowledge-few complexity improvements.

For now, the almost only productive approach remains numerical optimization method using least-squares methods and heuristics. In fact, while objective functions derived from Equation ( 19) are non-convex and nonlinear, they can be splitted in three linear subsystems (by helding two components of the unknown tensor fixed for example) and their resolution boils down to linear algebra. Nevertheless, in order to obtain new results, a regularization term needs to be added and the following is chosen in this paper:

Arg min 𝑃 𝑖 , 𝑄 𝑖 , 𝑆 𝑖 𝑞 𝑖=0 𝑃 𝑖 ⊗𝑄 𝑖 ⊗𝑆 𝑖 -𝑚 𝑖=0 𝑛 𝑗=0 𝑝 𝑘=0 𝐸 𝑗 𝑖 ⊗𝐸 𝑘 𝑗 ⊗𝐸 𝑖 𝑘 +𝜆 𝑞 𝑖=0 𝑃 𝑖 -𝑃 𝑖 + 𝑄 𝑖 -𝑄 𝑖 + 𝑆 𝑖 -𝑆 𝑖 . (20)
with model matrices 𝑃 𝑖 , 𝑄 𝑖 , 𝑆 𝑖 defining the regularization and a scalar parameter 𝜆 that determines the weight of the regularization term. The models are designed to drive the solution to match a desired structure and are choosed carefully for each iteration (see [START_REF] Smirnov | The bilinear complexity and practical algorithms for matrix multiplication[END_REF] for a detailed presentation). This approach gives a uniform method for deriving exact algorithm but also approximate one when they are a order 1 polynomial approximation w.r.t their parameter 𝜖.

A better precision-an approximation order greater then 1requires several other heuristics. If all dimensions of the problem are greater than 3, it is not yet possible to obtain acceptably short exact algorithms. Hence, the success of the resolution presented here relies on heuristical expertise and tyazhelaya rabota.However, the works ot that topic done since [START_REF] Smirnov | The bilinear complexity and practical algorithms for matrix multiplication[END_REF] show that the objective function of the found approximate algorithms allows to presumably estimate the exact rank of large problems. This point will appear in a future work.

Let us now describe the latest exact matrix multiplication tensor found with this method.

⟨3 × 5 × 5 : 58⟩ description

Before the detailed description done in the next section, let us first present the type introduced in Definition 2.1 of this tensor:

17 𝑋 2 𝑌 2 𝑍 2 + 2 𝑋𝑌 4 𝑍 + 𝑋 3 𝑌 2 𝑍 + 𝑋𝑌 2 𝑍 3 +5 𝑋 3 𝑌𝑍 + 5 𝑋𝑌𝑍 3 + 2 𝑋 2 𝑌 2 𝑍 + 2 𝑋𝑌 2 𝑍 2 + 𝑋𝑌 3 𝑍 +𝑋 2 𝑌𝑍 + 𝑋𝑌𝑍 2 + 13 𝑋𝑌 2 𝑍 +7 𝑋𝑌𝑍 = 𝑇 (𝑋, 𝑌, 𝑍 ). (21)
Remark 3.1. Considering that the indeterminates commute, the relation 𝑇 (𝑋, 𝑌, 𝑍 ) = 𝑇 (𝑍, 𝑌, 𝑋 ) holds. Unfortunately, even if this property suggests the existence of a symmetry (see Equation [START_REF] Hedtke | Search and test algorithms for triple product property triples[END_REF] and [9, 𝜋 13 in Thm 3.4] for a detailed description), this tensor does not have a non-trivial stabilizer. Such stabilizer are not uncommon for tensor found using the method sketched in Section 3.1: for example the stabilizer ((𝐶 2 × 𝐶 2 ) ⋊ 𝑆 4 ) ⋊ 𝐶 2 of ⟨3 × 3 × 6 : 40⟩ is of order 192 (see [START_REF] Sedoglavic | Fast matrix multiplication database[END_REF]). So, even if Comon's conjecture was disproved in full generality (see [START_REF] Shitov | A counterexample to comon's conjecture[END_REF]), there might be a tensor in ⟨3 × 5 × 5⟩ of rank 58 with stabilizer 𝐶 2 × 𝐶 2 × 𝑆 3 .

As two matrix multiplication tensor of same rank could share the same type, this invariant is not a faithful descriptions and we give its explicit expression in the next section.

Trilinear form of ⟨3 × 5 × 5 : 58⟩. We split the forthcoming description in 13 expressions whose components have the type corresponding to their indices as follow: (

⟨3 × 5
) 35
Induced geometry. As shown by Theorem 2.1, the action of the group (psl(3) × psl(5) × psl( 5)) ⋊ 𝐶 2 on the tensor introduced in the previous section defines classically a manifold of dimension 52 of tensors ⟨3 × 5 × 5 : 58⟩. Furthermore, this tensor have 8 serendipitous equalities that is couple of summands that shares the same factor (e.g. the two first summands of Equation 35 share the factor 𝑏 12 ). Up to our knowledge, this property was first introduced in [27, § 9.3] but does not seem to receive the attention it deserves. For example, this property allows to define new transformation of a matrix multiplication tensor into another as shown by the following lemma: Lemma 3.1. Given any invertible 𝑞 × 𝑞-matrix 𝑀, the tensor with serendipitous equalities 𝑞 𝑖=0 𝑈 𝑖 ⊗ 𝑉 𝑖 ⊗ 𝑊 involving the component𝑊 is equal to the tensor 𝑞 𝑖=0 𝛼 𝑖 ⊗ 𝛽 𝑖 ⊗ 𝑊 defined by:

𝛼 1 . . . 𝛼 𝑞 = Transpose(𝑀) 𝑈 1 . . . 𝑈 𝑞 , 𝛽 1 . . . 𝛽 𝑞 = 𝑀 -1 𝑉 1 . . . 𝑉 𝑞 . ( 36
)
The proof of this lemma reduces to the trivial computation of the expression

𝑞 𝑖=0 𝑈 𝑖 ⊗ 𝑉 𝑖 ⊗ 𝑊 -𝛼 𝑖 ⊗ 𝛽 𝑖 ⊗ 𝑊 .
This lemma shows that the dimension of the manifold induced by the tensor introduced in this section is greater then that could be expected.

The following sections are devoted to present other interesting consequences.

Upper bound 98 on tensor rank

of ⟨5 × 5 × 5⟩ Lemma 2.1 and the tensor presented in Section 3.2 allows to construct the following tensor:

⟨5 × 5 × 5 : 98⟩ = ⟨2 × 5 × 5 : 40⟩ ⊕ ⟨3 × 5 × 5 : 58⟩, (37)
with the construction of tensor ⟨2 × 5 × 5 : 40⟩ taken from [START_REF] Hopcroft | On minimizing the number of multiplication necessary for matrix multiplication[END_REF] (its explicit expression is given in [START_REF] Sedoglavic | Fast matrix multiplication database[END_REF]). Remark that the best theoretical lower bound on the corresponding tensor rank is 48 (see [START_REF] Bläser | A 5 2 𝑛 2 -lower bound for the rank of 𝑛 × 𝑛-matrix multiplication over arbitrary fields[END_REF]Theorem 2]). Furthermore, this new atomic tensor also improves the construction of the following algorithms:

⟨10 × 10 × 10 : 686⟩ = ⟨2 × 2 × 2 : 7⟩ ⊗ ⟨5 × 5 × 5 : 98⟩, (38)
⟨15 × 15 × 15 : 2088⟩ = ⟨3 × 5 × 5 : 58⟩ ⊗ ⟨5 × 3 × 3 : 36⟩. (39)
All these tensors are explicitly presented via [START_REF] Sedoglavic | Fast matrix multiplication database[END_REF].

Consequence of this new upper bound. A group-theoretic approach to the conception of matrix multiplication algorithm related to Fourier transform on finite groups was introduced by Cohn and Umans in [START_REF] Cohn | A group-theoretic approach to fast matrix multiplication[END_REF]. The new tensor constructed in Equation (37) allows to exhibit a limitation of this approach as shown by the following remark. Remark 3.2. It is shown in [START_REF] Hart | A fast search algorithm for (m,m,m) triple product property triples and application for 5 × 5 matrix multiplication[END_REF] that no group can realize 5 × 5 matrix multiplication better then Makarov's algorithm ⟨5 × 5 × 5 : 100⟩ using the group-theoretic approach of Cohn and Umans [START_REF] Cohn | A group-theoretic approach to fast matrix multiplication[END_REF]. Hence, the tensor presented in this note shows that this approach does not produce better algorithms then ⟨5 × 5 × 5 : 98⟩. The same assertion holds for ⟨3 × 3 × 3 : 23⟩ and ⟨4 × 4 × 4 : 49⟩ (see [START_REF] Hedtke | Search and test algorithms for triple product property triples[END_REF]Theorem 7.3]).

The next section is devoted to describe a new approximate algorithm {5 × 5 × 5 : 89}.

NEW APPROXIMATE ALGORITHMS

Approximate matrix multiplication tensors were first introduced by Bini et ali in [START_REF] Bini | 𝑂 (𝑛 2.7799 ) complexity for 𝑛 × 𝑛 approximate matrix multiplication[END_REF] in order to improve asymptotic bounds. From a practical point of view, these approximate algorithms could be used efficiently when the coefficients are in Z/𝑝Z (see [START_REF] Boyer | Matrix multiplication over word-size modular rings using approximate formulae[END_REF]). Furthermore, from a theoretical point of view, these tensors allow to work with Euclidean closure of the Brent algebraic variety defined by Equation [START_REF] Makarov | A non-commutative algorithm for multiplying 5 × 5 matrices using one hundred multiplications[END_REF] and not the Zariski closure induced by dealing with exact tensors. As the Zariski closure is often much larger then the Euclidean closure, this shift of standpoint brings usually lower bounds. Hence, the exact tensor ⟨2 × 3 × 3 : 15⟩ presented in [START_REF] Hopcroft | On minimizing the number of multiplication necessary for matrix multiplication[END_REF] is optimal; Smirnov describes {2 × 3 × 3 : 14} in [START_REF] Smirnov | The bilinear complexity and practical algorithms for matrix multiplication[END_REF] and that bound on the corresponding border rank was proved to be optimal in [START_REF] Conner | New lower bounds for matrix multiplication and the 3x3 determinant[END_REF]Theorem 1.4]. Similarly, while the best upper bound for tensor rank of ⟨3 × 3 × 3⟩ is 23 ( [START_REF] Laderman | A noncommutative algorithm for multiplying 3 × 3 matrices using 23 multiplications[END_REF]), {3 × 3 × 3 : 20} could be found in [START_REF] Smirnov | The bilinear complexity and practical algorithms for matrix multiplication[END_REF] ([7, Theorem 1.1] reports that the lower bound is 17 for the corresponding border rank).

Lemma 2.1 shows that these atomic approximate matrix multiplication tensors lead to the following tensor used in the sequel:

{5 × 3 × 3 : 34} = {3 × 3 × 3 : 20} ⊕ {2 × 3 × 3 : 14}. (40)
The results in [START_REF] Bini | 𝑂 (𝑛 2.7799 ) complexity for 𝑛 × 𝑛 approximate matrix multiplication[END_REF] are based on a partial matrix multiplication algorithm that computes approximately the product of a 2 × 2-matrix 𝐴 with one element vanishing (e.g. 𝑎 22 = 0) and a 2 × 2-matrix 𝐵 a full matrix. This kind of tensor were used to improve the bound on the exponent of matrix multiplication (see [21, § 3]).

Let us now show how to complete a tensor constructed using Equation 40 in order to define {5 × 5 × 5 : 89}.

A partial approximate matrix multiplication

This section presents a partial approximate tensor T 𝜖 that defines an algorithm computing the product of a 5 × 5-matrix 𝐴 with 9 vanishing elements:

𝑎 𝑖 𝑗 = 0, ∀(𝑖, 𝑗) such that 1 ≤ 𝑖 ≤ 3, 3 ≤ 𝑗 ≤ 5. ( 41
)
and a full 5 × 5-matrix 𝐵.

The type of T 𝜖 . Remarks that the isotropy introduced in Theorem 2.1 and the associated the invariant presented in Definition 2.1 for exact matrix multiplication tensors remain obviously valid for approximates ones. The type of T 𝜖 is:

20 𝑋 2 𝑌 2 𝑍 2 +3 𝑋 2 𝑌 2 𝑍 + 2 𝑋 2 𝑌𝑍 2 + 4 𝑋𝑌 2 𝑍 2 +7 𝑋 2 𝑌𝑍 + 6 𝑋𝑌 2 𝑍 + 8 𝑋𝑌𝑍 2 +5 𝑋𝑌𝑍 . (42)
Again, we are going to split its description by trilinear form into several summands whose subscript indicate the type of their components as follow:

T 𝜖 =
Remark that there is 4 serendipitous equalities in this tensor. We conclude the construction of an approximate tensor {5 × 5 × 5 : 89} in the next section.

PERSPECTIVES

We have presented there the upper bound 98 (resp. 89) for the tensor (resp. border) rank of the 5 × 5-matrix product while the best theoretical lower bound is 48 (resp. 45) (see [START_REF] Bläser | A 5 2 𝑛 2 -lower bound for the rank of 𝑛 × 𝑛-matrix multiplication over arbitrary fields[END_REF]Theorem 2] (resp. [17, Corollary 1.2])). Furthermore, the new ⟨3 × 5 × 5 : 59⟩ improves also

• ⟨10 × 10 × 10 : 686⟩ = ⟨2 × 2 × 2 : 7⟩ ⊗ ⟨5 × 5 × 5 : 98⟩ • ⟨15 × 15 × 15 : 2088⟩ = ⟨3 × 5 × 5 : 58⟩ ⊗ ⟨5 × 3 × 3 : 36⟩ and we have a quite clear idea of all exact matrix multiplication tensors up to size 32 × 32 × 32 induced by the best known-to-date "atomic" such tensors. But as ⟨15 × 15 × 15 : 2088⟩'s tensor rank is lesser then the first decomposition ⟨3 × 3 × 3 : 23⟩ ⊗ ⟨5 × 5 × 5 : 98⟩ that comes to mind, the most obvious decomposition does not always lead to the current most efficient one. Hence, there is a-simple but requiring expensive calculation-computer based search to do in order to construct a database for approximate tensor similar to [START_REF] Sedoglavic | Fast matrix multiplication database[END_REF].

4. 2

 2 New upper bound 89 on border rank of ⟨5 × 5 × 5⟩Using the approximate matrix multiplication tensor defined in the previous section and the construction made in Equation (40), one can construct easily:{5 × 5 × 5 : 89} = T 𝜖 + {3 × 3 × 3 : 20} + {2 × 3 × 3 : 14}. (52)Remark that the best theoretical lower bound on the corresponding border rank is 45 (see[START_REF] Landsberg | New lower bounds for the border rank of matrix multiplication[END_REF] Corollary 1.2])).

  11 (𝑏 12 -𝑏 22 ), 𝜌 2 ← (𝑎 11 + 𝑎 12 )𝑏 22 , 𝜌 4 ← (𝑎 12 -𝑎 22 )(𝑏 21 + 𝑏 22 ), 𝜌 3 ← (𝑎 21 + 𝑎 22 )𝑏 11 , 𝜌 5 ← (𝑎 11 + 𝑎 22 )(𝑏 11 + 𝑏 22 ), 𝜌 6 ← 𝑎 22 (𝑏 21 -𝑏 11 ), 𝜌 7 ← (𝑎 21 -𝑎 11 )(𝑏 11 + 𝑏 12 ),

	𝑐 11 𝑐 12

  𝑏 12 -𝑏 22 ) (𝑐 21 + 𝑐 22 ) + (𝑎 11 + 𝑎 12 ) 𝑏 22 (-𝑐 11 + 𝑐 21 ) + (𝑎 21 + 𝑎 22 ) 𝑏 11 (𝑐 12 -𝑐 22 ) + (𝑎 12 -𝑎 22 ) (𝑏 21 + 𝑏 22 ) 𝑐 11 + (𝑎 11 + 𝑎 22 ) (𝑏 11 + 𝑏 22 ) (𝑐 11 + 𝑐 22 ) +𝑎

22 (𝑏 21 -𝑏 11 ) (𝑐 11 + 𝑐 12 ) + (𝑎 21 -𝑎 11 ) (𝑏 11 + 𝑏 12 ) 𝑐 22 .

  × 5 : 58⟩ = 𝜏 222 + 𝜏 141 + 𝜏 321 + 𝜏 123 + 𝜏 311 + 𝜏 113 + 𝜏 221 + 𝜏 122 + 𝜏 131 + 𝜏 211 + 𝜏 112 + 𝜏 121 + 𝜏 111 . 𝜏 141 =𝑎 24 𝑏 13 + 𝑏 22 + 𝑏 23 -𝑏 24 + 𝑏 33 -𝑏 35 -𝑏 43 + 𝑏 44 + 𝑏 45 (𝑐 41 + 𝑐 42 ) -(𝑎 14 -𝑎 24 ) 𝑏 14 -𝑏 32 + 𝑏 34 + 𝑏 41 + 𝑏 42 -𝑏 44 + 𝑏 51 -𝑏 54 -𝑏 55 𝑐 41 . 𝑎 14 -𝑎 21 -𝑎 23 -𝑎 24 (𝑏 12 + 𝑏 35 ) (𝑐 51 -𝑐 21 -𝑐 22 ) + (𝑎 24 -𝑎 14 + 𝑎 15 ) (-𝑏 41 -𝑏 51 + 𝑏 54 + 𝑏 55 ) (𝑐 41 -𝑐 12 ) + (𝑎 34 + 𝑎 15 ) (𝑏 41 + 𝑏 53 ) (𝑐 31 + 𝑐 13 ) + (𝑎 32 + 𝑎 25 ) (𝑏 22 -𝑏 24 + 𝑏 53 ) (𝑐 32 -𝑐 43 ) + (𝑎 33 -𝑎 25 + 𝑎 35 ) (𝑏 12 -𝑏 32 + 𝑏 52 + 𝑏 34 ) (𝑐 43 -𝑐 22 ) + (𝑎 13 + 𝑎 14 -𝑎 24 ) (𝑏 34 + 𝑏 42 -𝑏 32 ) (𝑐 41 -𝑐 22 ) . 𝑎 13 -𝑎 23 + 𝑎 33 + 𝑎 14 -𝑎 24 -𝑎 25 + 𝑎 35 (𝑏 12 -𝑏 32 + 𝑏 34 ) 𝑐 22 . 𝜏 311 = 𝑎 13 + 𝑎 15 + 𝑎 22 + 𝑎 33 + 𝑎 35 -𝑎 12 -𝑎 23 𝑏 31 (𝑐 21 + 𝑐 11 + 𝑐 41 ) + 𝑎 23 + 𝑎 32 -𝑎 13 -𝑎 22 -𝑎 14 -𝑎 33 -𝑎 34 𝑏 33 (𝑐 23 -𝑐 33 + 𝑐 43 ) + 𝑎 11 -𝑎 21 + 𝑎 31 + 𝑎 14 -𝑎 24 -𝑎 25 + 𝑎 35 𝑏 14 (𝑐 52 -𝑐 42 ) + 𝑎 11 -𝑎 21 + 𝑎 31 -𝑎 12 + 𝑎 22 + 𝑎 15 + 𝑎 35 𝑏 11 (𝑐 11 + 𝑐 51 ) + 𝑎 21 + 𝑎 32 -𝑎 11 -𝑎 14 -𝑎 22 -𝑎 31 -𝑎 34 𝑏 13 (𝑐 53 -𝑐 33 ) . 𝜏 113 = (𝑎 22 -𝑎 23 ) 𝑏 22 𝑐 11 + 𝑐 12 + 𝑐 21 + 𝑐 22 + 𝑐 23 -𝑐 32 -𝑐 33 + 𝑐 41 + 𝑐 42 + 𝑐 43 -(𝑎 31 + 𝑎 35 ) 𝑏 55 𝑐 11 -𝑐 13 + 𝑐 42 + 𝑐 43 + 𝑐 51 -𝑐 52 -𝑐 53 + (𝑎 11 + 𝑎 14 ) (𝑏 35 -𝑏 45 ) 𝑐 31 -𝑐 33 + 𝑐 41 + 𝑐 42 -𝑐 51 -𝑐 52 + 𝑐 53 + (𝑎 22 -𝑎 21 ) 𝑏 25 𝑐 11 + 𝑐 12 -𝑐 32 -𝑐 33 + 𝑐 51 + 𝑐 52 + 𝑐 53 -(𝑎 33 + 𝑎 35 ) 𝑏 52 (𝑐 11 + 𝑐 21 + 𝑐 41 -𝑐 22 -𝑐 13 -𝑐 23 ) . 𝜏 221 = (𝑎 22 -𝑎 32 + 𝑎 24 ) (𝑏 22 + 𝑏 13 + 𝑏 23 + 𝑏 33 -𝑏 24 ) 𝑐 32 -(𝑎 12 + 𝑎 14 + 𝑎 34 ) (𝑏 41 + 𝑏 13 + 𝑏 33 -𝑏 43 ) 𝑐 31 . (31) 𝜏 122 = 𝑎 15 (𝑏 41 + 𝑏 51 -𝑏 52 -𝑏 55 ) (𝑐 11 + 𝑐 41 -𝑐 13 ) +𝑎 25 (𝑏 22 -𝑏 24 + 𝑏 54 + 𝑏 55 ) (𝑐 12 + 𝑐 42 + 𝑐 43 ) . 𝜏 131 = (𝑎 25 -𝑎 35 ) 𝑏 12 + 𝑏 14 -𝑏 32 + 𝑏 34 + 𝑏 52 + 𝑏 53 -𝑏 54 𝑐 43 . 𝜏 121 =𝑎 32 (𝑏 21 -𝑏 53 ) 𝑐 13 -𝑎 12 (-𝑏 21 + 𝑏 41 + 𝑏 22 + 𝑏 25 ) (𝑐 11 + 𝑐 12 ) +𝑎 32 (𝑏 22 + 𝑏 23 -𝑏 53 + 𝑏 25 ) (𝑐 32 + 𝑐 33 ) +𝑎 34 (𝑏 42 + 𝑏 43 + 𝑏 53 + 𝑏 45 ) (𝑐 33 -𝑐 31 ) +𝑎 13 (𝑏 32 -𝑏 31 -𝑏 42 ) (𝑐 21 + 𝑐 41 ) +𝑎 33 (𝑏 32 + 𝑏 33 + 𝑏 35 -𝑏 12 -𝑏 52 ) (𝑐 23 + 𝑐 43 ) +𝑎 23 (𝑏 22 + 𝑏 34 ) (𝑐 22 + 𝑐 42 ) +𝑎 31 (𝑏 13 + 𝑏 15 -𝑏 55 ) 𝑐 53 +𝑎 21 (𝑏 14 + 𝑏 15 + 𝑏 25 -𝑏 35 ) 𝑐 52 +𝑎 11 (𝑏 12 + 𝑏 15 + 𝑏 35 -𝑏 11 -𝑏 45 ) 𝑐 51 +𝑎 34 (𝑏 44 + 𝑏 53 -𝑏 42 ) 𝑐 43 -(𝑎 15 + 𝑎 35 ) (𝑏 11 + 𝑏 31 -𝑏 51 + 𝑏 53 ) 𝑐 13 + (𝑎 22 -𝑎 12 ) (𝑏 11 + 𝑏 21 + 𝑏 31 + 𝑏 22 -𝑏 24 ) 𝑐 12 . 𝜏 111 = (𝑎 31 + 𝑎 33 ) 𝑏 12 𝑐 23 + (𝑎 21 + 𝑎 23 + 𝑎 24 -𝑎 11 -𝑎 13 -𝑎 14 ) 𝑏 12 (𝑐 51 -𝑐 21 ) + (𝑎 21 + 𝑎 23 + 𝑎 24 ) 𝑏 35 (𝑐 51 + 𝑐 52 -𝑐 21 -𝑐 22 ) + (𝑎 33 + 𝑎 34 ) 𝑏 35 (𝑐 53 -𝑐 23 -𝑐 43 )

	(29)
	(22)
	Let us start with 𝜏 141 :
	(23)
	(30)
	(32)
	(33)
	(34)
	(26)
	𝜏 321 =

𝜏 211 = (𝑎 12 -𝑎 22 -𝑎 25 ) (𝑏 22 -𝑏 24 ) (𝑐 12 + 𝑐 32 + 𝑐 42 ) (24) 𝜏 112 = (𝑎 12 + 𝑎 14 -𝑎 15 ) 𝑏 41 (𝑐 11 + 𝑐 31 + 𝑐 12 ) . (25) 𝜏 222 = (𝑎 12 -𝑎 22 + 𝑎 23 ) (𝑏 22 + 𝑏 31 ) (𝑐 11 + 𝑐 12 + 𝑐 21 + 𝑐 41 ) + (𝑎 32 -𝑎 22 + 𝑎 23 ) (𝑏 22 -𝑏 33 ) (𝑐 23 + 𝑐 43 -𝑐 32 -𝑐 33 ) + (𝑎 33 + 𝑎 15 + 𝑎 35 ) (𝑏 52 -𝑏 31 ) (𝑐 11 + 𝑐 21 + 𝑐 41 -𝑐 13 ) + (𝑎 13 + 𝑎 14 + 𝑎 34 ) (𝑏 42 + 𝑏 33 + 𝑏 35 ) 𝑐 31 + 𝑐 23 -𝑐 33 + 𝑐 43 + (𝑎 25 -𝑎 31 -𝑎 35 ) (𝑏 14 + 𝑏 55 ) (𝑐 52 -𝑐 42 -𝑐 43 ) + (𝑎 24 -𝑎 11 -𝑎 14 ) (𝑏 14 -𝑏 35 + 𝑏 45 ) (𝑐 52 -𝑐 41 -𝑐 42 ) + (𝑎 21 + 𝑎 12 -𝑎 22 ) (𝑏 11 + 𝑏 25 ) (𝑐 11 + 𝑐 51 + 𝑐 12 ) + (𝑎 21 -𝑎 22 + 𝑎 32 ) (𝑏 13 -𝑏 25 ) (𝑐 32 + 𝑐 33 -𝑐 53 ) + (𝑎 31 + 𝑎 15 + 𝑎 35 ) (𝑏 55 -𝑏 11 ) (𝑐 11 + 𝑐 51 -𝑐 13 ) + (𝑎 12 + 𝑎 24 ) 𝑏 43 + 𝑏 24 -𝑏 22 -𝑏 13 -𝑏 23 -𝑏 33 (𝑐 41 + 𝑐 32 + 𝑐 42 ) + (𝑎 11 + 𝑎 14 + 𝑎 34 ) (𝑏 13 -𝑏 35 + 𝑏 45 ) (𝑐 31 -𝑐 33 + 𝑐 53 ) + 𝑎 13 + (27) 𝜏 123 = -(𝑎 13 + 𝑎 14 ) (𝑏 42 + 𝑏 35 ) 𝑐 31 -𝑐 21 -𝑐 22 + 𝑐 23 -𝑐 33 + 𝑐 43 . (28) +𝑎 12 (𝑏 13 + 𝑏 23 + 𝑏 33 -𝑏 43 ) (𝑐 31 + 𝑐 41 + 𝑐 32 + 𝑐 42 ) + (𝑎 24 + 𝑎 15 -𝑎 14 -𝑎 25 ) (𝑏 54 + 𝑏 55 -𝑏 51 ) 𝑐 12 + (𝑎 32 -𝑎 34 + 𝑎 35 ) 𝑏 53 (𝑐 13 + 𝑐 33 + 𝑐 43 ) .

  𝜌 222 + 𝜌 221 + 𝜌 212 + 𝜌 122 + 𝜌 211 + 𝜌 121 + 𝜌 112 + 𝜌 111 . (43) 𝜌 222 = 𝑎 11 -𝑎 22 𝜖 3 𝑏 12 -𝑏 21 -𝑏 11 𝜖 3 𝑐 21 + 𝑐 12 𝜖 3 + 𝑎 11 + 𝑎 52 𝜖 3 𝑏 41 + 𝑏 51 -𝑏 31 𝑐 15 + 𝑐 51 𝜖 2 + 𝑎 11 + 𝑎 53 + 𝑎 52 𝜖 3 𝑏 13 + 𝑏 31 𝜖 3 𝑐 31 + 𝑎 41 -𝑎 43 + 𝑎 55 𝜖 3 𝑏 34 + 𝑏 53 𝜖 3 𝑐 35 -𝑐 44 -𝑐 45 𝜖 3 + 𝑎 11 -𝑎 54 + 𝑎 51 𝜖 + 𝑎 52 𝜖 3 𝑏 14 -𝑏 41 𝜖 3 𝑐 41 + 𝑐 25 + 𝑐 42 𝜖 3 + (𝑎 21 -𝑎 53 ) 𝑏 13 + 𝑏 35 𝜖 2 -𝑏 32 𝜖 3 𝑐 32 + + (𝑎 22 + 𝑎 45 ) 𝑏 55 + 𝑏 23 𝜖 3 𝑐 54 + 𝑐 32 𝜖 3 + (𝑎 32 + 𝑎 44 ) 𝑏 41 + 𝑏 24 𝜖 3 𝑐 14 + 𝑐 43 𝜖 3 + (𝑎 12 + 𝑎 43 ) 𝑏 32 + 𝑏 25 𝜖 3 𝑐 24 + 𝑐 51 𝜖 3 + (𝑎 22 + 𝑎 41 ) 𝑏 15 + 𝑏 22 𝜖 3 𝑐 54 + 𝑐 22 𝜖 3 + (𝑎 43 -𝑎 52 ) 𝑏 33 + 𝑏 25 𝜖 3 𝑐 34 -𝑐 55 𝜖 3 + (𝑎 45 -𝑎 32 ) 𝑏 51 + 𝑏 23 𝜖 3 𝑐 14 -𝑐 33 𝜖 3 + (𝑎 41 -𝑎 32 ) 𝑏 11 + 𝑏 22 𝜖 3 𝑐 14 -𝑐 23 𝜖 3 + (𝑎 22 + 𝑎 44 ) 𝑏 45 -𝑏 24 𝜖 3 𝑐 54 -𝑐 42 𝜖 3 + (𝑎 41 + 𝑎 52 ) 𝑏 13 -𝑏 22 𝜖 3 𝑐 34 -𝑐 25 𝜖 3 + (𝑎 12 + 𝑎 44 ) 𝑏 42 -𝑏 24 𝜖 3 𝑐 24 -𝑐 41 𝜖 3 + (𝑎 32 + 𝑎 43 ) 𝑏 31 -𝑏 25 𝜖 3 𝑐 14 -𝑐 53 𝜖 3 + (𝑎 44 -𝑎 52 ) 𝑏 43 -𝑏 24 𝜖 3 𝑐 34 + 𝑐 45 𝜖 3 + (𝑎 45 -𝑎 12 ) 𝑏 52 -𝑏 23 𝜖 3 𝑐 24 + 𝑐 31 𝜖 3 . 𝑎 31 + 𝑎 53 𝜖 3 𝑏 13 𝑐 15 + 𝑐 55 𝜖 -𝑐 53 𝜖 2 -𝑐 33 𝜖 3 + (𝑎 31 -𝑎 54 ) 𝑏 14 𝑐 43 -𝑎 52 𝜖 𝑏 14 + 𝑏 15 𝜖 + 𝑏 51 𝜖 3 𝑐 15 + 𝑎 41 + 𝑎 55 𝜖 3 𝑏 14 + 𝑏 34 -𝑏 54 + 𝑏 15 𝜖 + 𝑏 55 𝜖 2 𝑐 44 -𝑎 21 + 𝑎 55 𝜖 3 𝑏 14 + 𝑏 15 𝜖 + 𝑏 55 𝜖 2 -𝑏 52 𝜖 3 𝑐 25 . 𝜌 121 =𝑎 32 𝑏 21 + 𝑏 22 + 𝑏 23 -𝑏 24 + 𝑏 25 𝜖 3 -𝑏 41 𝑐 14 + 𝑎 51 + 𝑎 54 𝜖 3 𝑏 14 -𝑏 45 𝜖 2 + 𝑏 44 𝜖 3 𝑐 45 + 𝑎 51 + 𝑎 53 𝜖 3 𝑏 13 + 𝑏 34 + 𝑏 35 𝜖 2 + 𝑏 33 𝜖 3 𝑐 35 +𝑎 52 𝑏 33 + 𝑏 43 -𝑏 13 -𝑏 53 + 𝑏 22 + 𝑏 23 -𝑏 24 + 𝑏 25 𝜖 3 𝑐 34 -𝑎 12 𝑏 32 + 𝑏 42 -𝑏 12 -𝑏 52 + 𝑏 22 + 𝑏 23 -𝑏 24 + 𝑏 25 𝜖 3 𝑐 24 -𝑎 22 𝑏 15 + 𝑏 45 + 𝑏 55 -𝑏 35 + 𝑏 22 + 𝑏 23 -𝑏 24 + 𝑏 25 𝜖 3 𝑐 54 . (47) 𝜌 122 =𝑎 53 𝑏 13 + 𝑏 35 𝜖 2 𝑐 32 -𝑐 33 -𝑐 31 + 𝑐 52 -𝑐 53 -𝑐 51 𝜌 211 = 𝑎 42 + 𝑎 22 -𝑎 43 𝜖 3 𝑏 25 -𝑏 35 𝜖 3 𝑐 54 + 𝑐 52 𝜖 3 + 𝑎 45 + 𝑎 52 -𝑎 42 𝜖 3 𝑏 23 -𝑏 53 𝜖 3 𝑐 35 -𝑐 34 𝜖 3 + 𝑎 43 -𝑎 41 -𝑎 51 -𝑎 53 + 𝑎 55 𝜖 3 𝑏 34 𝑐 35 -𝑐 45 𝜖 3 + 𝑎 41 + 𝑎 45 + 𝑎 55 𝜖 3 𝑏 54 + 𝑏 53 𝜖 3 𝑐 44 + 𝑐 45 𝜖 3 + 𝑎 42 + 𝑎 12 -𝑎 41 𝜖 3 𝑏 22 -𝑏 12 𝜖 3 𝑐 24 + 𝑐 21 𝜖 3 𝑎 51 + 𝑎 22 -𝑎 11 + 𝑎 21 𝜖 3 𝑏 12 -𝑏 11 𝜖 3 𝑐 12 . 𝜌 112 =𝑎 43 𝑏 25 𝑐 52 + 𝑐 55 -𝑐 51 -𝑐 53 + 𝑐 14 + 𝑐 54 -𝑐 24 -𝑐 34 𝜖 3 -𝑎 41 𝑏 22 𝑐 22 + 𝑐 25 -𝑐 21 -𝑐 23 + 𝑐 14 + 𝑐 54 -𝑐 24 -𝑐 34 𝜖 3 -𝑎 45 𝑏 23 𝑐 32 + 𝑐 35 -𝑐 31 -𝑐 33 + 𝑐 14 + 𝑐 54 -𝑐 24 -𝑐 34 𝜖 3 -𝑎 44 𝑏 24 𝑐 43 + 𝑐 14 -𝑐 24 -𝑐 34 -𝑐 44 -𝑐 54 𝜖 3 -𝑎 11 𝑏 21 𝑐 51 𝜖 + 𝑐 11 + 𝑐 15 -𝑐 12 𝜖 3 +𝑎 31 𝑏 11 𝑐 13 + 𝑐 55 -𝑐 53 𝜖 + 𝑐 15 𝜖 + 𝑎 21 -𝑎 51 𝜖 3 𝑏 12 𝑐 22 -𝜌 111 = 𝑎 42 -𝑎 44 𝜖 3 𝑏 24 -𝑏 44 𝜖 3 (𝑐 41 + 𝑐 42 -𝑐 45 )𝜖 3 + 𝑐 44 + 𝑎 12 𝜖 3 + 𝑎 11 𝑏 11 𝜖 3 + 𝑏 21 𝜖 𝑐 11 𝜖 3 + 𝑐 21 + 𝑎 32 -𝑎 42 𝜖 3 𝑏 11 -𝑏 31 + 𝑏 51 -𝑏 21 𝜖 3 𝑐 14 -𝑐 13 𝜖 3 +𝑎 54 𝑏 43 𝑐 35 + 𝑎 31 𝑏 12 𝑐 23 .

							𝜌 221 =	𝑎 55 -𝑎 11 𝜖 3	-	𝑎 51 𝜖 2 -(46)
		+	𝑏 15 𝜖 2 +	𝑏 21 -𝑏 13 𝜖 3
							𝑐 52 𝜖	-	𝑐 25 -𝑐 12 𝜖 3	𝑐 51 𝜖	+	𝑐 15 𝜖 3
	+ 𝑎 51 -+ 𝑐 15 𝑎 31 𝜖 𝑏 15 + 𝑏 11 𝜖 -𝑏 13 𝜖 𝑐 55 -𝑐 53 𝜖 + 𝜖 3	𝑐 15 𝜖	. (50)
	+ (𝑎 21 + 𝑎 54 ) 𝑏 42 -	𝑏 45 𝜖	+	𝑏 14 𝜖 3 𝑐 52 𝜖	+	𝑐 25 𝜖 3
							𝜖
							+	𝑐 55 𝜖 2 +	𝑐 25 -𝑐 35 𝜖 3
							+𝑎 21 𝜖 2	𝑏 32 -𝑏 42 -𝑏 52 + + 𝑏 55 + 𝑏 45 -𝑏 35 𝜖	𝑏 15 𝜖 2 + 𝑏 12 -𝑏 13 𝜖 3	𝑐 52 +	𝑐 25 𝜖 2
							+	𝑎 54 𝜖 2 𝑏 45 -	𝑏 14 𝜖 2	𝑐 42 -𝑐 41 -𝑐 43 + 𝑐 55 𝑐 25 + 𝑐 45 𝜖 2 + 𝜖 3
							+	𝑎 55 𝜖	𝑏 14 𝜖	+ 𝑏 15 + 𝑏 55 𝜖	𝑐 25 -𝑐 44 𝜖	+ 𝑐 55 .
							(48)
	𝜌 212 =	𝑎 51 𝜖 2 -			𝑐 15 𝜖 3 -	𝑐 55 𝜖 2 .	(44) (45)	+ + (49) 𝑎 31 -𝑎 55 𝜖 𝑏 14 𝜖 + 𝑏 15 𝑐 15 𝜖 + 𝑐 55


Lecomte Jean-Baptiste

Le Pape Olivier

Baillif Hélène

Nevoux Marie

Vermard Youen

Savina-Rolland Marie

Veron Matthieu

Lehuta Sigrid

Hunter Ewan

Rivot Etienne

Olivier Le Pape
email: olivier.le.pape@agrocampus-ouest.fr

State-space modeling of multi-decadal mark-recapture data reveals low adult dispersal in a nursery-dependent fish metapopulation

Keywords: Population structure, capture-mark-recovery, adult connectivity, multi-event modeling, Solea solea L., Eastern

Quantifying connectivity within fish metapopulations is an important component in understanding population dynamics and providing an evidence base for assessment and management. We investigate meta-population connectivity of the common sole (Solea solea L.) in the Eastern English Channel (EEC). The EEC common sole stock is currently assessed as a single and spatially homogeneous population but connectivity induced through adult movements within this stock and with nearby stocks remains unknown. To fill this knowledge gap, we developed a state-space mark-recovery model, designed to estimate adult connectivity, using mark-recapture data from multiple release experiments from 1970 to 2018 across the EEC and adjacent management areas. The model estimates seasonal fish movements between five predefined areas, Western English Channel, Eastern English Channel (split into three discrete sub-areas) and North Sea. Over 32000 fish were tagged, 4036 of which were recovered via fisheries. Our results suggest minimal large-scale adult movements between these areas: movements among spatial units within the EEC were very low with even lower levels of immigration from areas adjoining the EEC. Our results support the hypothesis of segregated populations within the EEC. The importance of accommodating population substructure in the fisheries management is considered.

INTRODUCTION

Movements of individuals determine connectivity between habitats, control metapopulation dynamics [START_REF] Hanski | Metapopulation dynamics[END_REF][START_REF] Benhamou | Of scales and stationarity in animal movements[END_REF]) and resilience of populations to natural and anthropic stressors. Understanding the movement and dispersal patterns of wild animals at every stage in their life cycle is therefore critical for a full understanding of population dynamics and the subsequent provision of an evidence base for population assessment and management. In fisheries, an accurate definition of the spatial structure of fish populations is necessary for fish stock assessment and for setting appropriate fisheries management strategies [START_REF] Kutkuhn | Stock definition as a necessary basis for cooperative management of Great Lakes fish resources[END_REF][START_REF] Smith | Electrophoretic studies and the stock concept in marine teleosts[END_REF][START_REF] Begg | An holistic approach to fish stock identification[END_REF][START_REF] Fogarty | Population connectivity and spatial management of marine fisheries[END_REF]. A misspecification of the spatial limits of a stock can lead to biased estimates of population vital rates [START_REF] Cadrin | Stock Identification Methods: An Overview[END_REF][START_REF] Kerr | Lessons learned from practical approaches to reconcile mismatches between biological population structure and stock units of marine fish[END_REF]. Dispersal process must be addressed at all life-history stages to accurately assess the spatial and temporal delineation of populations and to accurately specify spatial management measures that could potentially target specific areas and life stages (e.g. protecting nursery areas and spawning aggregations, [START_REF] Fogarty | Population connectivity and spatial management of marine fisheries[END_REF]. Recent studies have employed a wide variety of tools and methods to assess connectivity within populations (Le [START_REF] Bris | Migration Patterns and Putative Spawning Habitats of Atlantic Halibut (Hippoglossus Hippoglossus) in the Gulf of St. Lawrence Revealed by Geolocation of Pop-up Satellite Archival Tags[END_REF][START_REF] Rogers | Fine-Scale Population Dynamics in a Marine Fish Species Inferred from Dynamic State-Space Models[END_REF][START_REF] Moreira | Population Structure of the Blue Jack Mackerel (Trachurus Picturatus) in the NE Atlantic Inferred from Otolith Microchemistry[END_REF]. Genetic studies using genetic markers such as microsatellites [START_REF] Cuveliers | Multi-Marker Estimate of Genetic Connectivity of Sole (Solea Solea) in the North-East Atlantic Ocean[END_REF][START_REF] Jasonowicz | Love the One You're with: Genomic Evidence of Panmixia in the Sablefish (Anoplopoma Fimbria)[END_REF][START_REF] Martinez | Population genomics reveals high gene flow in grass rockfish (Sebastes rastrelliger)[END_REF] or single nucleotide polymorphism [START_REF] Milano | Outlier SNP Markers Reveal Fine-Scale Genetic Structuring across European Hake Populations (Merluccius merluccius)[END_REF][START_REF] Laconcha | New Nuclear SNP Markers Unravel the Genetic Structure and Effective Population Size of Albacore Tuna (Thunnus alalunga)[END_REF][START_REF] Bekkevold | Gene-associated markers can assign origin in a weakly structured fish, Atlantic herring[END_REF] are often used to assess the spatial structure of a population and reproductive isolation within populations Page 4 of 47 [START_REF] Östman | Inferring spatial structure from population genetics and spatial synchrony in demography of Baltic Sea fishes: Implications for management[END_REF][START_REF] Marandel | Insights from genetic and demographic connectivity for the management of rays and skates[END_REF]. Otoliths are also extensively used in connectivity studies, either through otolith morphology [START_REF] Bacha | Insights on stock structure of round sardinella Sardinella aurita off north-west Africa based on otolith shape analysis[END_REF][START_REF] Hüssy | Evaluation of Otolith Shape as a Tool for Stock Discrimination in Marine Fishes Using Baltic Sea Cod as a Case Study[END_REF][START_REF] Mahe | Identifying Blue Whiting (Micromesistius Poutassou) Stock Structure in the Northeast Atlantic by Otolith Shape Analysis[END_REF] or otolith chemistry [START_REF] Tanner | Otolith Chemistry in Stock Delineation: A Brief Overview, Current Challenges and Future Prospects[END_REF][START_REF] Callicó Fortunato | Environmental migratory patterns and stock identification of Mugil cephalus in the Spanish Mediterranean Sea, by means of otolith microchemistry[END_REF][START_REF] Régnier | Otolith chemistry reveals seamount fidelity in a deepwater fish[END_REF][START_REF] Moreira | Population Structure of the Blue Jack Mackerel (Trachurus Picturatus) in the NE Atlantic Inferred from Otolith Microchemistry[END_REF]. Morphometry and meristics [START_REF] Allaya | Identification of Atlantic Chub Mackerel Scomber Colias Population through the Analysis of Body Shape in Tunisian Waters[END_REF][START_REF] Sley | Morphometric and Meristic Characters of Blue Runner Caranx Crysos and False Scad Caranx Rhonchus (Pisces: Carangidae) from the Gulf of Gabes, Tunisia, Eastern Mediterranean[END_REF], parasites [START_REF] Catalano | Parasites as Biological Tags to Assess Host Population Structure: Guidelines, Recent Genetic Advances and Comments on a Holistic Approach[END_REF][START_REF] Mackenzie | Parasites as Biological Tags. Stock Identification Methods[END_REF] and life history traits (Begg et al. 1999;[START_REF] Erlandsson | Spatial structure of body size of European flounder (Platichthys flesus L.) In the Baltic Sea[END_REF][START_REF] Du Pontavice | Investigating spatial heterogeneity of von Bertalanffy growth parameters to inform the stock structuration of common sole, Solea solea, in the Eastern English Channel[END_REF][START_REF] Randon | Could we consider a single stock when spatial sub-units present lasting patterns in growth and asynchrony in cohort densities? A flatfish case study[END_REF] have also been used to assess stock structure. In the past decade, stock delineation studies have resulted in revised stock boundaries for numerous stocks (e.g. blue whiting [START_REF] Mahe | Synthèse bibliographique des principales espèces de Manche orientale et du golfe de Gascogne[END_REF], Atlantic cod Zemeckis et al. 2014, and horse mackerel Abaunza et al. 2008).

Although the information derived from mark-recapture experiments is limited (e.g. they cannot be used to quantify gene flow, [START_REF] Cadrin | Stock Identification Methods: An Overview[END_REF], they have nevertheless proven useful for the investigation of fish movements and the spatial structure of populations [START_REF] Howe | Winter flounder movements, growth, and mortality off Massachusetts[END_REF][START_REF] Hilborn | Determination of fish movement patterns from tag recoveries using maximum likelihood estimators[END_REF][START_REF] Gillanders | Estimates of movement and lifehistory parameters of yellowtail kingfish (Seriola lalandi): How useful are data from a cooperative tagging programme?[END_REF][START_REF] Patterson | Movement of tagged red snapper in the northern Gulf of Mexico[END_REF][START_REF] Mcgarvey | Estimating rates of fish movement from tag recoveries: Conditioning by recapture[END_REF][START_REF] Adlerstein | Seasonal movements of Chinook salmon in Lake Michigan based on tag recoveries from recreational fisheries and catch rates in gill-net assessments[END_REF][START_REF] Cadrin | Stock Identification Methods: An Overview[END_REF][START_REF] Hanselman | Move it or lose it: Movement and mortality of sablefish tagged in Alaska[END_REF][START_REF] Bris | Migration Patterns and Putative Spawning Habitats of Atlantic Halibut (Hippoglossus Hippoglossus) in the Gulf of St. Lawrence Revealed by Geolocation of Pop-up Satellite Archival Tags[END_REF][START_REF] Liljestrand | Multi-state dead recovery markrecovery model performance for estimating movement and mortality rates[END_REF]. When considering commercial fish stocks, tagging data generally consist of mark-release experiments conducted on scientific surveys, with tag-recovery facilitated via the fisheries, the latter largely dependent on volunteer reporting by the harvesters. A weakness of such data is that the detection probability is usually unknown, and non-reporting can be substantial [START_REF] Henny | A reward band study of mallards to estimate band reporting rates[END_REF][START_REF] Frusher | Strategies for improving the precision of fishing and natural mortality estimates from multiyear tagging models: A case study[END_REF][START_REF] Pollock | A general model for tagging on multiple component fisheries: An integration of age-dependent reporting rates and mortality estimation[END_REF][START_REF] Cadigan | Semiparametric estimation of tag loss and reporting rates for tag-recovery experiments using exact time-at-liberty data[END_REF][START_REF] Cadigan | Reporting and shedding rate estimates from tagrecovery experiments on Atlantic cod (Gadus morhua) in coastal Newfoundland[END_REF]). However, ancillary information is often available (McGarvey & Feenstra Page 5 of 47 Analysis) typically provide estimates of total and fishing mortality, and thereby total abundance. Such information can be used as input in the analysis of tagging data of harvested fish populations [START_REF] Mcgarvey | Estimating rates of fish movement from tag recoveries: Conditioning by recapture[END_REF].

The common sole (Solea solea, L.) is a flatfish substantively harvested in the Western English Channel to the North Sea (ICES 2017). Today, stock assessment is conducted separately for three ICES stock divisions (VIIe, VIId, IV; ICES 2017). The Eastern English Channel (EEC, ICES division VIId) stock, managed as one homogeneous population, has been overexploited over the last ten years (ICES 2017). Dispersal of the subadult components has been described: the larvae drift passively over short distances to local coastal nurseries [START_REF] Rochette | Effet des perturbations anthropiques sur la survie des juvéniles de poissons marins dans les nourriceries et conséquences sur les renouvellement des populations[END_REF], where the juveniles remain prior to migration offshore at maturity [START_REF] Riou | Contribution respective de differentes nourriceries cotieres aux populations adultes de sole et de plie: Etude par couplage de modeles lineaires generalises avec un systeme d information geographique[END_REF], where the fish reproduce and enter the fishery.

Observed contrast in key life history traits (density-at-age and length-at-age data) have suggested potential spatial structuring within the adult component of the EEC common sole stock within three discrete spatial areas [START_REF] Du Pontavice | Investigating spatial heterogeneity of von Bertalanffy growth parameters to inform the stock structuration of common sole, Solea solea, in the Eastern English Channel[END_REF][START_REF] Randon | Could we consider a single stock when spatial sub-units present lasting patterns in growth and asynchrony in cohort densities? A flatfish case study[END_REF]. However, the movements of adult sole between these areas remain largely unknown. Here we have drawn on an extensive mark-recapture database collected over 49 years to analyze fish movements and assess connectivity of the adult common sole population within the EEC and adjacent management areas. We tested the hypothesis of metapopulation structure within the EEC and estimated the level of mixing between sub-populations. For that purpose, we developped a capture-recapture model [START_REF] Royle | Spatial capture-recapture[END_REF][START_REF] Mccrea | Analysis of capture-recapture data[END_REF] built in a state-space model framework (recently reviewed in [START_REF] Aeberhard | Review of State-Space Models for Fisheries Science[END_REF] in the context of fisheries) to estimate adult and subadult movement probabilities. Finally, based on our observations, we consider the implications for management of the common sole stock in the EEC (and more widely) of Page 6 of 47 acknowledging and incorporating population substructure in management and assessment.

MATERIALS AND METHODS

Spatial structure

The spatial structure of the common sole population along the English Channel is analyzed using five areas (Figure 1): the Western Channel (WC), the North Sea (NS) and three areas for the EEC: English coast of the Eastern English Channel (UK), Southern French coast (FrW) and Northern French coast of the Eastern English Channel (FrE). The WC area is based on an existing spatial management unit (VIIe ICES 2017). The spatial management unit IV currently used for stock assessment in the North Sea (ICES 2017) is divided into three spatial subunits. Our study is restricted to the spatial subunit IVc (ICES 2017) which contains the major part of the common sole commercial catches in the North Sea. The division of the EEC into three areas (UK, FrW, FrE) is based on the occurrence of natural barriers of habitat unsuitable for common sole [START_REF] Rochette | Coupling hydrodynamic and individual-based models to simulate long-term larval supply to coastal nursery areas[END_REF]. These natural barriers consist of deep gravel grounds, which occur centrally in the EEC (separating UK from FrW & FrE), and wide rocky reefs from the coasts out into deeper waters (separating FrW from FrE). These natural delineations could potentially define the spatial structure of the common sole population within the EEC and have recently been used to study its connectivity in the EEC [START_REF] Du Pontavice | Investigating spatial heterogeneity of von Bertalanffy growth parameters to inform the stock structuration of common sole, Solea solea, in the Eastern English Channel[END_REF][START_REF] Randon | Could we consider a single stock when spatial sub-units present lasting patterns in growth and asynchrony in cohort densities? A flatfish case study[END_REF].

Data Collection

The dataset collates four mark-recapture experiments carried out by (i) the Center for Environment Fisheries and Aquaculture Science (CEFAS, UK) from 1955to 1985and 2002to 2007[START_REF] Burt | Movement of sole in the southern North Sea and eastern English Channel from tagging studies 1955-2004[END_REF] and (ii) the French Research Institute for Exploitation of the Sea (IFREMER, France) from 1976 to 1980 and 2016 to 2018 (Table 1, Figure 2). For each release experiment, Petersen discs were used [START_REF] Burt | Movement of sole in the southern North Sea and eastern English Channel from tagging studies 1955-2004[END_REF]. Discs are uniquely numbered and are securely attached to the dorsal flank of the fish. Adults and subadults were released close to the capture position as soon as was possible after capture. Recaptured marked fish were recovered by fishermen, who provided dates and spatial coordinates of the recaptures. 1 and2). Data collected before 1970 were discarded, as information was not sufficient to estimate fish movements from mark-recovery data (i.e. no reliable estimate of fishing mortality, see section 2.4.2). Recaptured fish with no release or recovery position or date of recovery data were discarded. The mark-recapture dataset consists of 32739 released and 7010 recovered fish with known dates and areas of release and recovery respectively.

Time at liberty and time step

A discrete 4 months time step was used in our modeling approach, based on the life cycle of the adult common sole [START_REF] Rochette | Effet des perturbations anthropiques sur la survie des juvéniles de poissons marins dans les nourriceries et conséquences sur les renouvellement des populations[END_REF]: spawning (February to May), foraging (June to September) and overwintering (October to January). For each individual, the time at liberty (time elapsed between the date of release and the date of recapture) was recorded. The time at liberty was computed in a number of time steps (Table 1 and Figure 3). Individuals recovered within the time step of release were removed from the analysis (2974 fish, 7% of released fish).

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Modeling approach

Core modeling structure

The final dataset used in the modeling approach consists of 32739 released and 4036 recovered fish with the following associated information (individual based): dates and areas of release and recovery and release experiment. The dataset was analyzed through a multi-event mark-recapture modeling approach [START_REF] Pradel | Principles and interest of GOF tests for multistate captureRecapture models[END_REF] 1). At any time step 𝑡, a fish can be in one (and only one) of those 15 states. The transition between time step 𝑡 and 𝑡 + 1 consists of a three-step series of transitions representing ecological and observation processes (Figure 4): 1. Movement process. Any alive fish at time 𝑡 can remain in its current area 𝑖 (𝑖 = 1,…,5

) or move to another area 𝑗 (𝑗 = 1,…,5) between 𝑡 and 𝑡 + 1. Page 9 of 47 3. Observation process. The observation process (that ultimately defines the likelihood) links observation to the true biological state of each individual at time 𝑡.

Only fish caught by the fishery can potentially be observed. Alive fish or fish dead from natural causes are unobservable. Once a marked fish is caught by the fishery, it can eventually be declared. The declaration of a marked fish caught is modeled through a Bernoulli distribution with declaration rate 𝑇 (see details hereafter).

Parameterization

We tested several versions of the model accounting for different levels of spatial and temporal variations in the parameters. Below, we detail the parameterization used in the full model that accounts for seasonal movements, spatial heterogeneity and inter-annual variation of survival, and survey-specific declaration rate (model 1, Table 3). Other challenging models are detailed in section 2.4.3.

Movement process

Probabilities of movement 𝜙(𝑖,𝑗,𝑠) are estimated separately for each pair of departure ( 𝑖 = 1,…,5) and arrival areas (𝑗 = 1,…,5) and for each season 𝑠 (𝑠 = 1,…,4). No variability between years is considered. Movement parameters from the WC to the NS, and from the NS to the WC are set to 0 and are not estimated (Figure 1). No fish were observed making these two migrations patterns in a single time step (i.e. 4 months). Therefore, it was assumed that the large distance separating the two areas does not allow for such movement in one time step.

Survival process

Survival probabilities for each time step are fixed and assumed known without errors, defined as:

𝑓 𝑓 + 𝑚 (1 -𝑒𝑥𝑝( -(𝑓 + 𝑚)) 𝑚 𝑓 + 𝑚 (1 -𝑒𝑥𝑝( -(𝑓 + 𝑚)) 𝑟 𝑠 = 1 -(𝑟 𝑓 + 𝑟 𝑚 ) Can.
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Sci. 𝑟 𝑓 = 𝑟 𝑚 = Page 10 of 47 where 𝑟 𝑓 is the probability of death by fishing, 𝑟 𝑚 is the probability of natural fish death and 𝑟 𝑠 is the probability of survival. 𝑚 and 𝑓 respectively are natural and fishing mortality (per time step of 4 months) as directly derived from published stock assessment evaluations (ICES 2017). For the three ICES areas (IVc, VIId and VIIe) the spatially and temporally constant natural mortality was fixed to 𝑚 = 0.1 × 𝑡𝑖𝑚𝑒𝑠𝑡𝑒𝑝 -1 (ICES 2017). Fishing mortality rate 𝑓 was also considered known but with years and spatial variability. Available information from stock assessment differ among areas (ICES 2017): estimates of fishing mortality are available since 1955 for the NS area (ICES division IVc), from 1969 for the WC area (ICES division VIIe) and from 1982 for the UK, FrW, FrE areas jointly (ICES division VIId) (Figure 5).The data collected before 1970 were discarded because no reliable estimate of 𝑓 in the EEC could be found before 1970. In order to use the longest time series of release data, we assumed that fishing mortality was equal in areas VIIe and VIId between 1970 to 1981. Temporal variations were smoothed to limit the influence of uncertainty that surrounds short term (year-to-year) variations in estimates of fishing mortality (Figure 5). Preliminary analysis demonstrated that smoothed time series of 𝑓 improve numerical stability of dispersal estimates. The mortality rates were allocated to each time step within the same year by considering that 𝑚 and 𝑓 were homogeneous among seasons within a year.

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Declaration

The probabilities that marked fish caught by the fishery are declared (and then observed) are estimated, and considered to be variable between release experiments (but spatially homogeneous and constant among seasons).

Hypotheses testing and sensitivity analysis

The full model (model 1, Table 3) accounts for (i) a movement process with parameters (estimated) that vary between seasons, (ii) yearly and spatially varying survival (fixed parameters), and (iii) declaration rates (estimated parameters) considered to be variable between release experiments. In all models, to avoid confusion between parameters and ensure stable numerical results in the maximum likelihood procedure, estimated parameters are the movement probabilities and the declaration rates, and all mortality rates (fishing and natural) are considered known. The existence of dispersal patterns that differ among seasons is tested by fitting models with no variability of movement probabilities between seasons (models 2, 5 and 7, Table 3) and comparing the results obtained with the full model (model 1, Table 3). The variation of the declaration rates between release experiments is tested by fitting a model parameterized with a unique (but still estimated) declaration rate over the full time series (model 3, Table 3). The existence of inter-annual differences in movements during the time series is assessed by fitting separate models to the full time series of data ; model 1, Table 3) and to the most recent (2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017)(2018); model 8, Table 3) and historical time series only (1970[START_REF] Hanski | Metapopulation dynamics[END_REF]model 9, Table 3). Hypotheses made on the fishing mortality (considered known in our model) are critical. We therefore also assessed the robustness of our results to the quantity of expertize on the spatiotemporal variability of fishing mortality brought into the model. Models with less refined hypotheses were fitted using: (i) fishing mortality constant over time but different between areas, computed as the mean of fishing mortality over time series per area (models 4 and 5, Table 3), and (ii) fishing mortality constant over time and Page 12 of 47 homogeneous in space, calculated as the mean over areas and times series (models 6 and 7, Table 3).

Model fit and selection

All models were built and fitted using the E-SURGE program (version 1.9.0, Choquet and Nogue 2010). Details about the implementation of the model in E-SURGE are given in Appendix A. Parameters (movement and declaration rates) were estimated in the maximum likelihood framework (MLE). The data consist in the sequence of observation events for all individuals, starting with 𝑖 (the area of release, 𝑖 = 1,...,5) at the tagging event, and ending with 𝑗 (the area of recovery, 𝑗 = 1,...,5) when the fish is recaptured and declared (with a series of 0 between the tagging and the recapture event as the fish can only be recaptured once) or ending with a series of 0 for fish that are never recaptured. The likelihood writes down as the product of Bernoulli distributions (declaration process) over all time steps and all fish (considered independent), marginalized over the probability distribution of all possible hidden states defined from the product of Multinomial distributions for the movement and mortality processes. The maximum likelihood is estimated using the Maximization-Expectation algorithm [START_REF] Choquet | E-SURGE 1.7 user's manual[END_REF]. A goodness-of-fit (GOF) test was conducted prior to model selection to check if our release and recapture data met the assumption of a general model (the Arnason-Schwarz Model, [START_REF] Pradel | A proposal for a goodness-of-fit test to the Arnason-Schwarz multisite capture-recapture model[END_REF]. We performed a GOF test using the U-CARE software (version 2.2, [START_REF] Choquet | U-CARE: Utilities for performing goodness of fit tests and manipulating CAptureREcapture data[END_REF]. GOF tests a 'trap-dependence' effect by comparing future recapture histories between individuals released on the current occasion versus individuals released on a previous occasion, for all individuals that are seen again (for more details on GOF test, see Appendix A). Model selection was used to assess which model formulation in Table 3 was the best supported by the data. Model selection was based on the Akaike's information criterion corrected for overdispersion (QAIC). The model with the lowest score of QAIC is considered as the best model. Only models with the same dataset (same length of time series of data) were compared together.

Sensitivity to area boundaries

A sensitivity analysis to the delineation of the areas was performed through a buffering method consisting in removing fish released and recovered at or near a border. Such individuals were potentially crossing the border using migration paths shorter than those for centrally-based fish, creating a potential edge effect and introducing bias (i.e. overestimation of movements among spatial areas) in estimates of movement probabilities. We computed a 30 km buffer around each boundary to remove a substantial amount (i.e. 10%) of released and recovered fish inside the buffer (3364 fish were removed).

RESULTS

The GOF test performed on the full dataset (from 1970 to 2018) revealed a lack of fit (𝜒 2 2 8 8 = 57.476, P = 0.000 and df = 24) when over-dispersion is not accounted for. An overdispersion coefficient (𝑐) of 2.39 was applied to all models to account for overdispersion (𝑐 superior to one, Burnham andAnderson 2003, Choquet et al. 2009).

Overall, the full model (model 1, Table 3) best explains the tagging-recovery data including seasonal movements, specific declaration rate for each release experiment, and fishing mortality varying across space and time. Page 14 of 47 The data also support the hypotheses of different detection rates per release experiments (QAIC model 1 < QAIC model 3, Table 3) with the lowest declaration rates estimated for the CEFAS2002, comparable detection rate between IFREMER1976 and IFREMER2016 experiments and a relatively high detection rate for the CEFAS1970 release experiment. Declaration rate estimates presented in Figure 6 for model 1 were similar among models 1, 2 and 4-7 (Supplementary Files; Table S1). Even if the estimated movement's probabilities between areas remain overall very low, the results support the existence of seasonal movements (Figure 7 and Figure S2 in Supplementary Files for a focus on movement between areas). Furthermore, the conclusion that the data support the existence of seasonal movements patterns is robust to changes in the time series of fishing mortality used as input to the model. Indeed, models with seasonal movements in general presented better QAIC scores when considering pairs of models with the same survival input (QAIC model 1 < QAIC model 2; QAIC model 4 < QAIC model 5; Table 3). Overall, the results support the hypothesis of very low connectivity between the different areas (model 1, Figure 7). For most areas and seasons, estimates of movement probability are very low, if not null (i.e. probabilities of staying in the areas of origin are close to one). Virtually no emigration is estimated from the NS and WC to the three areas of the EEC. This result highlights the primarily residential nature of the adult component of the common sole population, with only a very small proportion of adult fish migrating among areas. Low probabilities of movement are estimated (𝛷𝑖,𝑗 < 0.25) for fish in the UK and FrE areas which move to WC and NS. Thus low to moderate exports outside of the EEC occur between the overwintering and spawning season. Low probabilities of emigration from the FrW to the WC is also estimated between spawning Page 15 of 47 and foraging seasons. Movement probabilities estimated with model 2 (no seasonal variability of movements) also support the hypothesis of very low dispersion of common sole between areas (Supplementary Files; Figure S2). Results are also robust to the delineation of the five areas. The analysis performed on a dataset without fish released and recovered near a border provided results (Supplementary Files; Figure S3) comparable to those obtained with no buffering (Supplementary Files; Tables S2, S3

and S4).

Finally, the conclusion that very low connectivity exists between areas is robust to the time series of data considered. No strong differences exist between movement probability estimates obtained with the whole dataset (1970-2018), and with the shorter time series 1970-1998, corresponding to historical release experiments (CEFAS1970 and

IFREMER1976; Tables S2, S3
and S4 in Supplementary Files). Movement probability between the FrE and NS areas is an exception. Between these areas, higher movements are estimated when considering historical release experiments only.

Between foraging and overwintering seasons, all fish from NS move to FrE (Table S3), whereas from spawning to foraging and overwintering to spawning seasons all fish from the FrE move to NS (Supplementary Files; Table S2). However, these estimated probabilities of movements can be explained by the small amount of fish released and recovered in the FrE area: 82 over the 1970-1998 period in comparison with the 1307 fish released and recovered in the NS in the most recent period. Movement probabilities inferred from recent tagging data collected from 2002 to 2018 are similar to the ones estimated with the entire dataset (Supplementary Files; Tables S2, S3 and S4), with only small movements among areas. There are however small differences in estimates of movement probabilities between the recent tagging data and the full-time series, which Page 16 of 47 mostly concern the three areas of the EEC. Between the foraging and overwintering seasons, higher than average movement probability are estimated from the UK to the FrE areas, as well as from the FrW to the FrE areas (Supplementary Files; Table S2). Limited movements are also estimated from the FrE to the WC and FrW areas from spawning to foraging seasons (Supplementary Files; Table S2). Confidence intervals are greater for the more recent period than for 1970-2018 and 1970-1998 time series, because of the smaller amount of fish released during CEFAS2002 and IFREMER2016 in comparison with the CEFAS1970 and IFREMER1976 experiments (Table 3a).

DISCUSSION

In the present study, we have analyzed for the first time a substantive database of markrecapture data, with 32739 subadult and adult common sole being released over a period of 49 years. Our results suggest minimal movements of sole among areas in the EEC, and even lower immigration from adjacent areas, but low to moderate emigration to the NS and to the WC, mainly between the overwinter and spawning seasons. Our results provide evidence of low connectivity among common sole subpopulations within the EEC, and no inputs from the adjacent spatial management areas. This evidence of local segregation is consistent with recent studies using patterns in key life history traits [START_REF] Du Pontavice | Investigating spatial heterogeneity of von Bertalanffy growth parameters to inform the stock structuration of common sole, Solea solea, in the Eastern English Channel[END_REF][START_REF] Randon | Could we consider a single stock when spatial sub-units present lasting patterns in growth and asynchrony in cohort densities? A flatfish case study[END_REF] for the common sole in the EEC.

Reliability of the estimates of movement

Commercially exploited fish populations are often widely monitored, which offers the opportunity to use survey-generated estimates of population vital rates, expert knowledge and/or stock assessment [START_REF] Mcgarvey | Estimating rates of fish movement from tag recoveries: Conditioning by recapture[END_REF]. In our study, we used estimates of fishing and natural mortality from published stock assessments (ICES Page 17 of 47 2017) as input fixed a priori in our state-space capture-mark-recapture modeling approach. Models incorporating all data available on the survival process (e.g. yearly and spatially varying fishing mortality) best explain the data. Seasonal movement probabilities estimated in our study revealed low movements, with small variations when considering different periods within the time series. However, the historical release experiments (CEFAS1970 and IFREMER1976) were not designed specifically to study movement patterns at the scale of the English Channel, and relatively little tagging effort was expended in the two areas along the French coast (FrW and FrE). This deficit of data in the oldest release experiments can explain the difference in movement parameters estimated for the 1970 to 1998 time series with regards to the full time series or to the recent release experiment. Only the recent release experiments (CEFAS2002 and IFREMER2016) sampled all three EEC areas more evenly. Jointly analyzing all four tagging experiments allowed for a more balanced dataset, and reduced singularities and potential bias incorporated by the spatial locations of fish releases between release experiments. We recognize, however, some weaknesses in our approach, some of which open up exciting opportunities for future research. Our model is structured on a seasonal time step of 4-months that reflect biological seasons of the life cycle of the sole in the Eastern Channel. Although this is justified from a biological point of view, this choice excludes data from fish that were released and recovered within the same seasonal time step. The percentage of fish removed because of the chosen time step remains low (7% of released fish). For those removed fish, 94% were recovered in their zone of release, in accordance with the results obtained on a 4-monthly basis. In all models, mortality rates (fishing and natural) were considered known and estimated parameters are the Page 18 of 47 movement and declaration probabilities. Estimating movement, mortality and detectability is possible in certain configuration of mark-recapture models [START_REF] Royle | Spatial capture-recapture[END_REF] but separating temporal and spatial variations in those parameters is still difficult overall. In our model, attempts to estimate both natural mortality and declaration rates from our data have shown unstable numerical results and estimates that were nonrobust to small changes in hypotheses, revealing some statistical confusion between mortality and declaration rates. In other terms, different combinations of mortality and declaration rates provide the same likelihood for the data. Because the total mortality estimated from statistical catch-at-age stock assessment models is robust [START_REF] Quinn | Quantitative fish dynamics[END_REF]ICES 2017), and because no expertize exists on the declaration rate, our choice here was to consider mortality as known, and to estimate declaration rates from the data. Although stock assessment results provide some measure of uncertainty about 𝑓 (confidence intervals), only point estimates were considered in our approach. This simplification answers practical considerations as no methods exist to account for uncertainty in 𝑓 using the E-SURGE software. However, we carefully assessed the sensitivity of our results to the hypothesis made on the temporal and spatial variation of the fishing mortality. Because temporal and spatial variation in 𝑓 are much higher than estimation uncertainty (ICES 2017), the robustness of our results to those changes in 𝑓 demonstrates the robustness of our inferences on movements to uncertainty about 𝑓. However, estimates from stock assessment models (and especially fishing mortality) can be sensitive to hypotheses made on the spatial structure of the population (see for instance results by [START_REF] Archambault | Adult-mediated connectivity affects inferences on population dynamics and stock assessment of nursery-dependent fish populations[END_REF] on the same case study), and make inferences on dispersion by using fixed 𝑓 from previous assessments that do not consider dispersion to be an issue. This paper is an important step toward the construction of an integrated model to simultaneously estimate fishing mortality and dispersion by integrating tagging data within the stock assessment model. Tagging data, however, can have a complex structure, which when coupled with a poorly balanced sampling design, can provide further challenge to obtaining robust statistical inferences from tagging-integrated models [START_REF] Maunder | A review of integrated analysis in fisheries stock assessment[END_REF]. Consequently, we attempted here to develop a first analysis of the tagging data before integrating them in a more complex structure.

Our approach also considers natural mortality to be constant in space and in time. In stock assessment evaluations, natural mortality is usually fixed and the fishing mortality is estimated relative to the fixed value of natural mortality (ICES 2017). Preliminary exploration of the data (not shown) suggested that consideration of a higher value of 𝑚 (but still considering 𝑚 to be constant in time and space) has a simple and intuitive impact on estimates of declaration rates. Considering a higher value of 𝑚 leads to lower estimates of declaration rates, as the same number of recorded marked fish must be explained by a lower actual number of marked fish in the sea due to higher 𝑚. On the opposite, considering a smaller value of 𝑚 leads to higher estimates of declaration rates. The effect of introducing spatial heterogeneity in 𝑚 which is operationally challenging was not tested here. However, there is no evidence of a spatially varying natural mortality for the common sole in the English Channel. Consequently, our assumption of constant natural mortality among areas was considered to be a realistic and parsimonious modeling approach for the estimation of fish movement.

Although Du Pontavice et al. ( 2018) and [START_REF] Archambault | Adult-mediated connectivity affects inferences on population dynamics and stock assessment of nursery-dependent fish populations[END_REF] suggested heterogeneity of fishing mortality within the EEC, with the fishing pressure in FrE being higher than in the UK and FrW areas, no spatial variability of fishing mortality within the Page 20 of 47 three EEC areas was considered in our approach. However, because the EEC common sole stock is considered homogeneous, no stock assessment results were available for the individual 'sub-stock' areas (UK, FrW, FrE), and no spatialized catch data were available in the time series. Furthermore, although the fishery is strongly seasonal (Vermard and Savina, com. pers.), available times series of within-year fishing effort distribution were too short to be considered in our analysis (2000( -2016( for French and 2004( -2016 for foreign fleets), precluding the application of varying fishing mortality between the 4-month 'seasons'. However, given (i) the very low level of dispersal estimated among the different areas in the EEC and (ii) the high robustness of these results to hypotheses on spatial and temporal variations in the fishing mortality that were tested, it is unlikely that using more refined data would have dramatically altered the conclusions. Furthermore, recapture probability in our approach depends only on 𝑚 and 𝑓. However because the fleet and associated gear is not routinely recorded for mark recovery, heterogeneity due to the use of different fishing gears, which could have introduced further realism to our model, was unable to be considered. Finally, we found strong statistical evidence in favor of different declaration rates between the tagging periods. Unfortunately, we were not able to find definitive explanations for those differences. One potential explanation might be haphazard levels of communication with the fishermen to publicize the tagging programs and associated rewards at different periods in time. We considered the declaration rate to be homogeneous in time and space within a period, which might have been a source of bias, in that different fleets operating in different zones may not have had equivalent information, interest or motivation to participate in scientific tagging programs. As a result, our estimated movement probabilities may have been biased by reducing the number of declarations of marked fish caught in a particular area. We have assumed that the aggregation of several tagging surveys from both UK and French research institutes in the present modeling approach should limit the overall related biases.

Combining different approaches to improve estimates of connectivity

Several authors have recently highlighted the importance of combining multiple approaches to study stock delineation [START_REF] Pita | Conceptual and practical advances in fish stock delineation[END_REF]. Indeed, individual methods each have their own strengths and weaknesses, and combining results of different methods generally improves the overall reliability of the results [START_REF] Cadrin | Stock Identification Methods: An Overview[END_REF][START_REF] Izzo | Integrated approach to determining stock structure: Implications for fisheries management of sardine, Sardinops sagax, in Australian waters[END_REF]. For marine fish in particular, high mortality in early life stages (eggs, larvae and juveniles, Le Pape and Bonhommeau 2015) and the lack of barriers in the ocean often rend the genetic signal of segregation relatively weak [START_REF] Selkoe | Seascape genetics and the spatial ecology of marine populations[END_REF]. Only a very small proportion of fish dispersing at each generation will erase genetics differences between populations [START_REF] Palmer | Inter-individual variability in dispersal behaviours impacts connectivity estimates[END_REF], which can result in a mismatch between ecological and genetic connectivity [START_REF] Hawkins | Fisheries stocks from an ecological perspective: Disentangling ecological connectivity from genetic interchange[END_REF].

The common sole stocks in the English Channel and North Sea are recognized as genetically distinct populations [START_REF] Diopere | Seascape genetics of a flatfish reveals local selection under high levels of gene flow[END_REF] and have been managed separately for decades. While common sole in the EEC has been managed as a single homogeneous stock unit thus far, evidence from different approaches is now accumulating in support of considering spatial structure at a finer scale. Here we demonstrate very low exchanges between the three delineated EEC areas, and virtually no immigration from adjacent stocks. This demographic isolation is consistent with the marked discrepancies in growth between the areas [START_REF] Du Pontavice | Investigating spatial heterogeneity of von Bertalanffy growth parameters to inform the stock structuration of common sole, Solea solea, in the Eastern English Channel[END_REF][START_REF] Randon | Could we consider a single stock when spatial sub-units present lasting patterns in growth and asynchrony in cohort densities? A flatfish case study[END_REF]) and the lasting synchrony among density-at-age time series inside each area [START_REF] Randon | Could we consider a single stock when spatial sub-units present lasting patterns in growth and asynchrony in cohort densities? A flatfish case study[END_REF]).

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Sci. Page 22 of 47 From low adult-mediated connectivity to segregation in EEC areas In the EEC, larval connectivity of the common sole is low, since spawning areas directly feed adjacent coastal nursery grounds [START_REF] Rochette | Coupling hydrodynamic and individual-based models to simulate long-term larval supply to coastal nursery areas[END_REF]. After metamorphosis, juveniles grow on shallow nursery grounds [START_REF] Riou | Contribution respective de differentes nourriceries cotieres aux populations adultes de sole et de plie: Etude par couplage de modeles lineaires generalises avec un systeme d information geographique[END_REF][START_REF] Rochette | Effect of nursery habitat degradation on flatfish population: Application to Solea solea in the Eastern Channel (Western Europe)[END_REF]. Limited movements of juvenile flatfish (Le Pape & Cognez 2016), and the dependence of the juvenile common sole upon shallow nursery habitats [START_REF] Riou | Contribution respective de differentes nourriceries cotieres aux populations adultes de sole et de plie: Etude par couplage de modeles lineaires generalises avec un systeme d information geographique[END_REF] result in low juvenile connectivity [START_REF] Coggan | Movements of juvenile Dover sole, Solea solea (L.), In the Tamar Estuary, South-western England[END_REF]. After about two years on nursery grounds, the common sole move to deeper offshore adult foraging grounds. In addition to premature segregation, the lack of connectivity during the adult phase is a potentially important driver of segregation [START_REF] Frisk | Moving beyond the current paradigm in marine population connectivity: Are adults the missing link?[END_REF]. However, estimated levels of dispersal demonstrate that at subadult and adult stages, EEC export a low fraction of individuals to the NS and a very low fraction to the WC. The EEC may act as a limited source of fish for these adjacent stocks. These movements occur mainly during the overwintering period of movements of subadults [START_REF] Dorel | Seasonal distribution of young sole (Solea solea (L.)) In the nursery ground of the Bay of Vilaine (Northern Bay of Biscay)[END_REF] and adults [START_REF] Horwood | The Bristol Channel sole (Solea solea (L.)): A fisheries case study[END_REF][START_REF] Burt | Movement of sole in the southern North Sea and eastern English Channel from tagging studies 1955-2004[END_REF] when common sole move out towards deeper sea areas.

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Management implications

Ignoring the spatial structure of exploited fish population dynamics may induce mismatches between the stock units considered for stock assessment and management and the biological population structure [START_REF] Carson | Evaluating the importance of demographic connectivity in a marine metapopulation[END_REF][START_REF] Frisk | Moving beyond the current paradigm in marine population connectivity: Are adults the missing link?[END_REF][START_REF] Kerr | Lessons learned from practical approaches to reconcile mismatches between biological population structure and stock units of marine fish[END_REF]. A better consideration of the spatial structure of populations and of metapopulation dynamics is critical for sound evidence-based decision-making [START_REF] Heino | Synchronous dynamics and rates of extinction in spatially structured populations[END_REF][START_REF] Porch | The efficacy of VPA models in the presence of complicated movement patterns[END_REF]Ulrich et al. 2017). Inconsistencies between the spatial structure of populations and delineation of stock units could impede effectiveness and appropriateness of management measures [START_REF] Kerr | Lessons learned from practical approaches to reconcile mismatches between biological population structure and stock units of marine fish[END_REF], resulting in the overexploitation of the less productive subunits [START_REF] Cadrin | Accounting for spatial population structure in stock assessment: Past, present, and future[END_REF]Ying et al. 2011;[START_REF] Goethel | Accounting for Spatial Complexities in the Calculation of Biological Reference Points: Effects of Misdiagnosing Population Structure for Stock Status Indicators[END_REF]. Adults' movements play a critical role in population connectivity [START_REF] Frisk | Moving beyond the current paradigm in marine population connectivity: Are adults the missing link?[END_REF], especially for species with a low level of connectivity at previous life stages. Failure to appropriately consider the role of adults' movements (or local segregation) can lead to bias in assessment of stock status and the proposal of management non-adapted to the true dynamics and productivity of stocks [START_REF] Punt | Some effects of ignoring mixing when managing fish populations subject to limited mixing[END_REF][START_REF] Porch | The efficacy of VPA models in the presence of complicated movement patterns[END_REF]. [START_REF] Rochette | A hierarchical Bayesian model for embedding larval drift and habitat models in integrated life cycles for exploited fish[END_REF] and [START_REF] Archambault | Adult-mediated connectivity affects inferences on population dynamics and stock assessment of nursery-dependent fish populations[END_REF] have developed an integrated life cycle Bayesian model of the EEC common sole stock. The life cycle sequentially considers the production of eggs by adults, larval survival and drift, survival and settlements in coastal nurseries, habitat-dependent survival of juveniles on nurseries and finally natural and fishing mortality of the adult population. [START_REF] Archambault | Adult-mediated connectivity affects inferences on population dynamics and stock assessment of nursery-dependent fish populations[END_REF] consider two contrasted hypotheses: (i) one single homogeneous adult population supplemented by all coastal nurseries; (ii) three isolated subpopulation supplemented by separated pools of coastal nurseries. Results show that accounting for spatial segregation markedly influences stock assessment results [START_REF] Archambault | Adult-mediated connectivity affects inferences on population dynamics and stock assessment of nursery-dependent fish populations[END_REF].

Results, from both this and the previous study, show that estimates of fishing mortality and management reference points such as MSY of the common sole in the EEC strongly depend upon the underpinning hypothesis of adult connectivity. Further extension of the integrated life cycle modeling framework of [START_REF] Archambault | Adult-mediated connectivity affects inferences on population dynamics and stock assessment of nursery-dependent fish populations[END_REF] to integrate the analysis of tagging data within the integrated life cycle model would allow the simultaneous estimation of fishing mortality in the different areas of the EEC and connectivity between those areas. We suggest that such an exercise would contribute to Page 24 of 47 improving scientific advice for the spatial management of the fishery (Methot 2009;[START_REF] Goethel | Incorporating Spatial Structure in Stock Assessment: Movement Modeling in Marine Fish Population Dynamics[END_REF][START_REF] Griffiths | Scaling marine fish movement behavior from individuals to populations[END_REF].

ACKNOWLEDGEMENT

We would like to thank Coline Lazard, Sophie Parrad, Nicolas Goascoz, Ronan Lebras, David Osmond and Yves Verin for their extensive involvement in the tagging program. We also thank Elodie Réveillac for fruitful discussions about ecological concepts and the Associate Editor and two anonymous referees for their useful comments in the review process.

FUNDING

This work is part of the SMAC (Sole de Manche Est Amélioration des Connaissances pour une meilleure gestion du stock) research program. This work is supported by the fisheries organization "France Filière Pêche" and the French Ministry in charge of fisheries "Direction des Pêches Maritimes et de l'Aquaculture". [START_REF] Burt | Movement of sole in the southern North Sea and eastern English Channel from tagging studies 1955-2004[END_REF]IFREMER 2019).

GOODNESS-OF-FIT TEST

Goodness-of-fit (GOF) tests are an important component of capture-recapture modeling: they are used to assess the accuracy of a model in capturing the variance of the data [START_REF] Pradel | Principles and interest of GOF tests for multistate captureRecapture models[END_REF]. The GOF test is divided into two components the 3G test, which compares the capture histories of newly-tagged and previously-tagged individuals released at the same time, and the M component testing a trap-dependence effect by comparing future capture histories between individuals released on the current occasion versus individuals released on a previous occasion, for all individuals that are seen again. In our case study, the 3G test does not exist, because our datasets only includes dead recoveries [START_REF] Gauthier | Analysis of band-recovery data in a multistate capture-recapture framework[END_REF]. The GOF test then relies on the M component only.

GOF is performed on a reduced model that considers parameters to be state and timedependent with only six states 'dead' or 'alive' in each area [START_REF] Pradel | A proposal for a goodness-of-fit test to the Arnason-Schwarz multisite capture-recapture model[END_REF][START_REF] Duriez | Estimating survival and movements using both live and dead recoveries: A case study of oystercatchers confronted with habitat change[END_REF][START_REF] Fernández-Chacón | Causes of mortality in depleted populations of Atlantic cod estimated from multi-event modelling of markRecapture and recovery data[END_REF]. GOF cannot handle multiple unobservable states and models with more than 10 different events. We then pooled together subareas of the EEC and summarized observations in seven types of events (not encountered = 0, encountered alive in WC = 1, recovered dead in WC = 4, encountered alive in EEC = 2, recovered dead in ECC = 5, encountered alive in NS = 3, recovered dead in NS = 6). The GOF test was conducted prior to model selection using the U-CARE software (version 2.2, [START_REF] Choquet | U-CARE: Utilities for performing goodness of fit tests and manipulating CAptureREcapture data[END_REF]. The GOF test performed on our data was significant relvealing a lack-of-fit (𝜒 2 = 57.476, P = 0.000 and df = 24) to the general Arnason-Schwarz model [START_REF] Pradel | A proposal for a goodness-of-fit test to the Arnason-Schwarz multisite capture-recapture model[END_REF]. To address this 𝜒 2 of the GOF test divided by the total number of degrees of freedom, to correct for potential lack of fit [START_REF] Burnham | Model selection and multimodel inference: A practical information-theoretic approach[END_REF][START_REF] Choquet | U-CARE: Utilities for performing goodness of fit tests and manipulating CAptureREcapture data[END_REF][START_REF] Fernández-Chacón | Causes of mortality in depleted populations of Atlantic cod estimated from multi-event modelling of markRecapture and recovery data[END_REF].

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  Figure 1: Map of the model geographical definition with the five areas potentially structuring common sole stocks (ICES. 2017; IFREMER 2019): Western Channel (WC, ICES division VIIe), English coast of the Eastern English Channel (UK, ICES division VIId), Southern French coast (FrW, ICES division VIId), Northern French coast of the Eastern English Channel (FrE, ICES division VIId) and the southern part of the North Sea (NS, ICES division IVc).

Figure 2 :

 2 Figure2: Release locations (in blue) and recapture locations (in red) of tagged common sole in the English Channel, east of the Western Channel and west of North Sea[START_REF] Burt | Movement of sole in the southern North Sea and eastern English Channel from tagging studies 1955-2004[END_REF] IFREMER 2019).

Figure 3 :Figure 4 :Figure 5 :Figure 7 :

 3457 Figure 3: Time at liberty in number of seasons (4-month periods) for the 4036 marked

  To assess temporal variations in key model parameters (see below), the entire dataset was divided into four release experiments

denoted (1) CEFAS1970 from 1970 to 1985, (2) CEFAS2002 from 2002 to 2007, (3) IFREMER1976 from 1976 to 1980 and (4) IFREMER2016 from 2016 to 2018 (Tables

  {𝐴 𝑊𝐶 , 𝐴 𝑈𝐾 , 𝐴 𝐹𝑟𝑊 , 𝐴 𝐹𝑟𝐸 , 𝐴 𝑁𝑆 , 𝐹 𝑊𝐶 , 𝐹 𝑈𝐾 , 𝐹 𝐹𝑟𝑊 , 𝐹 𝐹𝑟𝐸 , 𝐹 𝑁𝑆 ,𝑀 𝑊𝐶 , 𝑀 𝑈𝐾 , 𝑀 𝐹𝑟𝑊 , 𝑀 𝐹𝑟𝐸 , 𝑀 𝑁𝑆 } State 𝐴 relates to live fish, state 𝐹 relates to fish dying by fishing and state 𝑀 relates to fish dying of natural causes, all located respectively in areas WC, UK, FrW, FrE, NS, Figure

	built in a
	state-space modeling framework that accounts for both processes and observation
	errors. The process model tracks the history of each individual released fish on a
	'seasonal' (4-month discrete time step) basis. The trajectory of each individual starts
	with the release event, for which the date and areas are assumed to be recorded without
	errors. In our multi-event model, we consider 15 potential states for the true fate of
	individuals at each time step (Figure 4):

  For each individual, this transition is modeled as a Multinomial distribution with survival rate, mortality rate by fishing or mortality rate by natural causes as parameters (denoted 𝑟 𝑠 , 𝑟 𝑓 and 𝑟 𝑚 , respectively; see details hereafter).

			The dispersion of all fish
	present in the area 𝑖 at time 𝑡 is modeled through a Multinomial distribution with
	movement probabilities 𝜙(𝑖,𝑗) and ∑	5	𝜙(𝑖,𝑗) = 1 (see details hereafter).
	2. Survival. The model assumes that mortality occurs after movements. After
	movements, a fish can survive or die from either natural mortality or fishing mortality
	between 𝑡 and 𝑡 + 1.

545 546 Table 1 :

 5461 Number of marked and recovered fish, used in the analysis, per release experiment and associated average time at liberty in number of 4-month periods and recapture rates.

							Page 26 of 47
		record.
		version of
	on 09/24/19	final official	Experiment CEFAS1970	Marked Recovered Average time at liberty Recapture rate 3627 3627 6.26 1
	Can. J. Fish. Aquat. Sci. Downloaded from www.nrcresearchpress.com by IFREMER BIBLIOTHEQUE LA PEROUSE	use only. This Just-IN manuscript is the accepted manuscript prior to copy editing and page composition. It may differ from the	IFREMER1976 305 CEFAS2002 29 IFREMER2016 85 Total 4046	305 29 85 4046	4.07 5.59 3.07 6.02	1 1 1 1
		personal
		For

Table 2 :

 2 Number of fish released per area and time series and in parentheses the number of recovered fish per area and time series.

	Areas 1970-1998	2002-2018 1970-2018
	1 WC	5025 (949)	250 (10)	5275 (959)
	2 UK	6897 (1298)	1861 (13)	8758 (1311)
	3 FrW	1448 (90)	1048 (7)	2496 (97)
	4 FrE	1143 (93)	2148 (55)	3291 (148)
	5 NS	12509 (1505) 410 (16)	12919 (1521)
	6 Total 27022 (3935) 5717 (101) 32739 (4036)

Table 3 :

 3 Model definition. Models 1 to model 7 are fitted to the full time series. Model 1 is the most complex model. Model selection is performed on the full dataset(1970 to 2018) from model 1 to less complex alternatives in the dispersal, fishing mortality inputs and detection rates. 𝛥QAIC represents the difference in QAIC in comparison to model 1. Models 8 and 9 have the same structure as model 1, but are fitted to truncated time series and hence are not considered in the model selection process.

	Model Dispersal Fishing mortality Detection	Time Series QAIC	𝛥QAIC
	1	Seasonal Area x Years	Release experiments 1970-2018	144966 -
	2	Constant Area x Years	Release experiments 1970-2018	144976 10
	3	Seasonal Area x Years	Unique	1970-2018	145882 916
	4	Seasonal Area	Release experiments 1970-2018	145156 190
	5	Constant Area	Release experiments 1970-2018	145187 221
	6	Seasonal Homogeneous	Release experiments 1970-2018	145148 182
	7	Constant Homogeneous	Release experiments 1970-2018	145132 166
	8	Seasonal Area x Years	Release experiments 1970-1998	-	-
	9	Seasonal Area x Years	Release experiments 2002-2018	-	-

Table A1 :

 A1 Matrix 𝐼: initial state probabilities (𝜋) are assigned to alive state only, because tagging is performed on alive fish only. 𝐴 𝑊𝐶 𝐴 𝑈𝐾 𝐴 𝐹𝑟𝑊 𝐴 𝐹𝑟𝐸 𝐴 𝑁𝑆 𝐹 𝑊𝐶 𝐹 𝑈𝐾 𝐹 𝐹𝑟𝑊 𝐹 𝐹𝑟𝐸 𝐹 𝑁𝑆 𝑀 𝑊𝐶 𝑀 𝑈𝐾 𝑀 𝐹𝑟𝑊 𝑀 𝐹𝑟𝐸 𝑀 𝑁𝑆 †

	* -Fish. 𝜋 𝜋 𝜋 𝜋 from www.nrcresearchpress.com by IFREMER BIBLIOTHEQUE LA PEROUSE on 09/24/19 This Aquat. Sci. Downloaded Just-IN manuscript
	only.
	use
	For personal

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APPENDIX A MODEL STRUCTURE AND E-SURGE SPECIFICATION

E-SURGE [START_REF] Choquet | E-SURGE 1.7 user's manual[END_REF] is used to implement multi-event models which are defined by steps representing biological and observation process. Each process is associated to a row-stochastic matrix for which each row corresponds to a multinomial distribution. All cells probability in the same row must sum to one and so one parameter is defined as the complement of the others (i.e. 1 -∑others) and is denoted by the symbol ⋆ in the following matrices. Cells with probabilities fixed at 0 are denoted with a dash (-). Matrices rows correspond to the starting state and columns correspond to the arrival state. In our multi-event model, we consider 16 biological states to inform on the true fate of individuals at each occasion:

State 𝐴 relates to live fish, state 𝐹 relates to fish killed by fishing and state 𝑀 relates to fish dying of natural causes, all located respectively in areas WC, UK, FrW, FrE, NS (Figure 1). An extra state † is introduced to represent an unobservable dead state which discriminates between the "newly dead" fish from the unobservable "long-timedead" fish [START_REF] Lebreton | Competing events, mixtures of information and multistratum recapture models[END_REF][START_REF] Fernández-Chacón | Causes of mortality in depleted populations of Atlantic cod estimated from multi-event modelling of markRecapture and recovery data[END_REF]. This state is required to estimate the detection rate of fish caught by the fisheries and to distinguish the different causes of mortality. In practice, a fish, that died from fishing or natural causes at time step 𝑡, is assigned to the state "long time dead" at the time step 𝑡 + 1 and can not change of state afterwards.

Initialization

Can.

----

Dispersal

The first five rows of matrix 𝐷 corresponds to fish that are alive at time 𝑡 and move (or not) during the transition to time 𝑡 + 1. For instance, 𝜙𝑊𝐶,𝑈𝐾 is the probability for a fish in state 𝐴𝑊𝐶 at time 𝑡 to move from the 𝑊𝐶 area to the 𝑈𝐾 area and be in state 𝐴𝑈𝐾 at time step 𝑡 + 1. The last 10 rows correspond to fish that are dead and by definition cannot move to another area and hence cannot change state at time 𝑡 + 1.

Survival

Can.

After movement, fish survive (or not) following the survival transition matrix 𝑆. 𝑟 𝑓 is the probability of dying from fishing and 𝑟 𝑚 fom natural causes. The first five rows of matrix 𝑆 corresponds to fish that are alive at time 𝑡 and can survive or not during the transition to time 𝑡 + 1. The last 10 rows correspond to fish that are 'long-time-dead'. 𝑟 𝑓 and 𝑟 𝑚 are input from external knowledge (see section 2.4.2., ICES 2017) and are not estimated from the tagging data in E-SURGE.

Can.

First encounter event

This matrix is defined differently for the first encounter event (tag-release, 𝑘 = 1) than for the second encounter event (tag-recovery 𝑘 = 2).

J.

Second encounter event

The second elementary matrix presents the probability of each state given the observation for the second and solely encounter (𝑘 = 2). In the second encounter event, only fish caught by the fishery can be seen. The matrix contains the detection rates 𝜓:

Can.