Michel Philippe
email: philippe.michel@ec-lyon.fr

Optimal proliferation rate in a cell division model

Keywords: cell division, long time asymptotic, eigenvalue, min-max, variation, asymmetry AMS class. No: 35P05, 92B05, 93B60

We consider a size structured cell population model where a mother cell gives birth to two daughter cells. We know that the asymptotic behavior of the density of cells is given by the solution to an eigenproblem. The eigenvector gives the asymptotic shape and the eigenvalue gives the exponential growth rate and so the Maltusian parameter. The Maltusian parameter depends on the division rule for the mother cell, i.e., symmetric (the two daughter cells have the same size) or asymmetric. We use a min-max principle and a differentiation principle to find the variation of the first eigenvalue with respect to a parameter of asymmetry of the cell division. We prove that the symmetrical division is not always the best fitted division, i.e., the Maltusian parameter may be not optimal.

Introduction

General models of cell division are known for a long time. Although the most classical case is division into two equal new cells, it is now well established that this is not always the case (see [START_REF] Stewart | Aging and death in an organism that reproduces by morphologically symmetric division[END_REF][START_REF] Kirkwood | The evolution of ageing and longevity[END_REF][START_REF] Partridge | Optimality, mutation and the evolution of ageing[END_REF][START_REF] Ackermann | Senescence in a bacterium with asymmetric division[END_REF][START_REF] Nystrom | Aging in bacteria[END_REF][START_REF] Sinclair | Paradigms and pitfalls of yeast longevity research[END_REF]). In particular in hematopoiesis (see [START_REF] Brummendorf | Asymmetric cell divisions sustain long-term hematopoiesis from single-sorted human fetal liver cells[END_REF][START_REF] Huang | Symmetry of Initial Cell Divisions Among Primitive Hematopoietic Progenitors Is Independent of Ontogenic Age and Regulatory Molecules[END_REF] or [START_REF] Sugimoto | A Cell Kinetic and Cytological Study on the Asymmetric Cell Division of Thymic Lymphoblasts of the Embryonic Rat, Development[END_REF] for thymic lymphoblasts), in the large class of budding yeasts (see [START_REF] Jazwinski | Growing old: Metabolic control and yeast aging[END_REF][START_REF] Breitenbach | Yeast as a model for ageing and apoptosis research[END_REF][START_REF] Barker | Replicative ageing in the fission yeast Schizosaccharomyces pombe[END_REF][START_REF] Qu | Coordination of cell growth and cell division: a mathematical modeling study[END_REF]), E.Coli or for some other bacteria [START_REF] Dens | Cell division theory and individual-based modeling of microbial lag: part I. The theory of cell division[END_REF] like Physcomitrella protoplast, division is not always symmetric and a mother cell can give birth to a bigger and a smaller cells. The goal of the present paper is to propose a possible explaination for the existence of different way of cell division (symmetric or asymmetric) based on adaptive dynamic. In some cases, the symmetric division is not the best fitted. The natural model to study it is a cell division model (see [START_REF] Diekmann | On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory[END_REF][START_REF] Metz | The dynamics of physiologically structured populations[END_REF]) in which the density of cells n(t, y) is structured by their size y and the evolution is described by the master equation

∂ ∂t n(t, y) + ∂ ∂y n(t, y) + B(y)n(t, y) = 1 σ B( y σ )n(t, y σ ) + 1 1 -σ B( y 1 -σ )n(t, y 1 -σ ), (1.1)
where a cell of size y gives birth to a cell of size yσ and another one of size (1 -σ)y, with σ ∈]0, 1[. A similar model also arises to describe fragmentation in physics [START_REF] Fournier | Exponential trend to equilibrium for discrete coagulation equations with strong fragmentation and without a balance condition[END_REF][START_REF] Laurençot | Steady states for a fragmentation equation with size diffusion[END_REF] and the growth term ∂ y n arises after rescalling [START_REF] Bertoin | Asymptotic laws for nonconservative self-similar fragmentations[END_REF][START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF]. The division part of this equation can be understood as follows. The cells of size y are produced from the cells of size y/σ (resp. y/(1 -σ)). The number of cells which size belongs to [y, y + δy] that appear after the division are δy σ B( y σ )n(t, y σ ) (resp. δy 1-σ B( y 1-σ )n(t, y 1-σ )). We have then the population balance equation (1.1). We know from the general theory (see [START_REF] Michel | Existence of a Solution to the Cell Division Eigenproblem[END_REF][START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]) that the asymptotic behavior of such an evolution equation is given by the rate λ, the eigenvalue of the following eigenproblem

                     ∂ ∂y N (y) + [λ + B(y)]N (y) = 1 σ B( y σ )N ( y σ ) + 1 1 -σ B( y 1 -σ )N ( y 1 -σ ),
-∂ ∂y φ(y) + λφ(y) = B(y) φ(σy) + φ((1 -σ)y) -φ(y) , N (y = 0) = 0, N, φ ≥ 0, ∞ 0 N (y)φ(y)dy = 1, φ(0) = 1.

(1.2)

More precisely, we know from general relative entropy method [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF][START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF], that there exists a constant C only depending on the initial condition n(0, .) such that : -CN (.)e λt is a solution to (1.1), -φe -λt is a solution to the backward equation of (1.1), or similarly, φ is a solution to the dual eigenproblem, -finally, we have the convergence in the weighted space L 1 ([0, ∞[, φdy) with the norm f := | f (y) | φ(y)dy, lim t→∞ n(t, .)e -λt = C N (.) in L 1 ([0, ∞[, φdy).

Consequently, the dynamics of the density n(t, y) is strongly related to the eigenvalue in (1.2) that gives an invasive parameter of the population (and thus the fitness of the population). We use a min-max method [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to non linear propagation phenomena[END_REF][START_REF] Horn | MatrixAnalysis[END_REF] and a differentiation method to study the variation of the first eigenvalue λ with respect to the asymmetry parameter σ. We prove that when a cell divides early then the asymmetric division gives a better growth, i.e., a larger first eigenvalue. On the contrary, for late division, asymmetric division is better.

This paper is organized as follows. Firstly, in Section 2, we give the main results on the variation of the invasive parameter (first eigenvalue in (1.2)) with respect to the asymmetry of the division. We show that the convexity of the dual eigenfunction φ is related to this problem of variation. Then, in Section 3, we show that a min-max method and a differentiation method can be used to study the variation of the invasive parameter (first eigenvalue) with respect to the parameter of cell division asymmetry σ in (1.1). In Section 4, we consider birth rate B such that we obtain directly the convexity of the dual eigenfunction. In Section 5, we extend the results found in Section 2 to more general cell division models. Finally, in the last section, we give some simple applications of the main methods.

Main results

The purpose of this section is to show that the symmetric division is not necessarily the best fitted division. We study two different cases, when the birth rate B has a compact support separated from

y = 0, Supp B ⊂ [a, b], a > b/2, (2.1)
or Supp B contains y = 0. In this case we suppose that

Supp B = [0, b], λ(σ) λ(σ) -B(0) B(0) < B (y) B(y) ≤ 0, ∀y ∈]0, bσ 0 [, ∀σ ∈ [1/2, σ 0 [, (2.2)
with σ 0 ∈ [1/2, 1[, λ(σ) the first eigenvalue λ in (1.
2) where σ is the asymmetric parameter. We notice that B(y) = 1 [0,b] satisfies assumption (2.2). This condition means that the cells begin to divide early (at size y = 0) and the birth rate decreases. Theorem 2.1 Assume that Supp B is compact and (2.1) holds. Then

d dσ λ(σ) ≤ 0, ∀σ ∈ [1/2, a/b[. (2.3)
Moreover, we have

sup σ∈]1-a/b,a/b[ λ(σ) = λ(1 -a/b). Theorem 2.2 Assume that b > 0, σ 0 ∈ [1/2,
λ(σ) = λ(1/2).
Proof of Theorems 2.1-2.2 First, we will recall some results on the existence of a solution to the eigenproblem. Then, we differentiate the first eigenvalue λ(σ) with respect to the parameter σ to study the variation of λ(σ). We notice that the variation of the first eigenvalue is directly linked to the properties of the dual eigenfunction φ σ to be concave or convex. Finally, we prove that under the assumptions of Theorems 2.1-2.2, we can directly prove the convexity or concavity of the dual eigenfunction.

Step 1. Existence and uniqueness of the solution to eigenproblem (1.

2) The eigenproblem (1.2) associated to the cell division model can be written with more compact notations

L * σ (φ σ ) = λ(σ)φ σ , L σ (N σ ) = λ(σ)N σ , ∞ 0 N σ (y)φ σ (y)dy = 1, (2.5)
where

L 1 w ([0, ∞[) := L 1 ([0, ∞[, (1 + B(y))(1 + y)dy), L σ (f ) := - ∂f ∂y (.) -B(.)f (t, .) + 1 σ B( . σ )f ( . σ ) + 1 1 -σ B( . 1 -σ )f ( . 1 -σ ), f ∈ L 1 w ([0, ∞[), (2.6)
and the dual operator,

L * σ (g) := ∂g ∂y (.) -B(.)g(t, .) + B(.) g(.σ) + g .(1 -σ) , g ∈ L 1 w ([0, ∞[) * . (2.7)
Remark. We notice that it is enough to consider σ ∈ [1/2, 1[, since, using the symmetry of (1.2), we have λ(σ) = λ(1 -σ).

Here, we cannot expect to find an exact solution as in the McKendrick-VonFoerster model for instance. Nevertheless, we have the following lemma. Lemma 2.3 Under the assumption (2.1) (resp. (2.2)), there exists a solution (N σ , λ(σ), φ σ ) to (2.5)-(2.7). Moreover, we have

∀σ ∈]0, 1[, ∃C σ < ∞ :            N σ (y) ≤ C σ e -λ(σ)y , φ σ (y) ≤ C σ (1 + y), | ∂ ∂y φ σ (y) |≤ C σ (B(y) + 1)(1 + y). (2.8)
We refer to [START_REF] Michel | Existence of a Solution to the Cell Division Eigenproblem[END_REF] for the proof of this result.

Step 2. The differentiation method. Lemma 2.4 The function σ → λ(σ) is well defined on ]0, 1[ and is differentiable. Moreover, we have

∂ ∂σ λ(σ) = ∞ 0 B(y) ∂ ∂y φ σ (σy) - ∂ ∂y φ σ ((1 -σ)y) N σ dy.
(2.9)

We prove this lemma in Section 3.1 (as an application of Lemma 3.2) and using the same method, we find :

Lemma 2.5 The following equality holds :

( ∂ 2 ∂σ 2 λ(σ)) σ=1/2 = 2 ∞ 0 B(y)y 2 ∂ 2 ∂y 2 φ 1/2 (y/2)N 1/2 (y)dy.
Proof. We use Lemma 2.4 and the same method to prove that ∂ 2 ∂σ 2 λ(σ) exists and satisfies the above equality.

2

Now, we can give a direct corollary of Lemma 2.4. We suppose that

σ 0 ∈ [1/2, 1[, (2.10)
K = Supp B ⊂ [0, ∞[. (2.11)
Corollary 2.6 Assume that φ σ0 is a convex (resp. concave) function on [1 -σ 0 , σ 0 ].ConvHull(K).

Then we find that

( ∂ ∂σ λ σ ) σ=σ0 ≥ 0 (resp.( ∂ ∂σ λ σ ) σ=σ0 ≤ 0).
Proof. Indeed, using Lemma 2.4 and that φ σ0 is a convex function, we find that ∂ ∂y φ σ0 is increasing. Since σ 0 ≥ (1 -σ 0 ), we have ( ∂ ∂σ λ σ ) σ=σ0 ≥ 0. 2 Remark. The strict convexity (resp. strict concavity) of φ σ0 on K implies ∂ ∂σ λ > 0 (resp. ∂ ∂σ λ < 0).

Thus, we reduce the problem of the variation of λ(σ) with respect to σ to a problem on φ σ . In the next section we prove that under some assumptions on B, the dual eigenfunction φ σ is a convex or concave function.

Step 3. Convexity of the dual eigenfunction φ σ When the birth rate has a compact support, the following result holds.

Theorem 2.7 Assume that (2.1) (resp. (2.2) with σ 0 ∈ [1/2, 1[) holds. Then for all σ ∈]1 -a/b, a/b[ (resp. σ ∈ [1/2, σ 0 ]), φ σ is a strictly convex (resp. strictly concave) function.
We prove Theorem 2.7 in section 4.

Step 4. Conclusion We can now prove the main Theorems 2.1-2.2 on the variation of the eigenvalue λ(σ) with respect to the asymmetry parameter σ. Indeed, using Corollary 2.6 and the results proved in Section 2, we get directly Theorems 2.1-2.2. 2

Main methods

Formally, we have a family of linear bounded operators (and their dual operators) which depend on a parameter p ∈ I, where

I ⊂] -∞, ∞[ is an interval, L p : B → C, L * p : C * → B * , (3.1)
B and C are Banach spaces of real functions (in duality ., . with B * and C * ) such that

∀p ∈ I ∃(N p , λ(p), φ p ) ∈ B + ×] -∞, ∞[×C * + : L p (N p ) = λ(p)N p , L * p (φ p ) = λ(p)φ p , (3.2)
where

B + (resp C * + ) is the positive cone of B (resp C * ) and ∀p ∈ I φ p , N p = 1. (3.3)
We will study the variation of λ(p) with respect to p. We notice that the variation of N p , φ p in Banach spaces of real functions makes the problem rather complex.

Our purpose is to find the variation of the first eigenvalue in a general eigenvalue problem (3.1)-(3.3). To study the variation of the first eigenvalue, we develop two points of view, one which favors the min-max principle and the other which uses the differentiation of the eigenvalue (and eigenproblem). Even if the second one needs the operators in the eigenproblem to be more regular, it gives better results in some problems as the cell division one.

Min-max method: We have the following principle Lemma 3.1 Assume that there exists a solution (N p , λ(p), φ p ) to (3.1)-(3.3). Then we have for all p ∈ I,

λ(p) = max f ∈Λ min g∈Λ * g, L p f g, f , (3.4)
where Λ ⊂ B + which satisfies N p ∈ Λ, φ p ∈ Λ * ⊂ C * + and g, f > 0 for all f ∈ Λ and g ∈ Λ * . Moreover, if L p is monotone with respect to p ∈ I then so is λ(p). More precisely, we have

∀p ≤ q ∈ J, ∀(f, g) ∈ Λ × Λ * g, L p (f ) ≤ g, L q (f ) =⇒ λ(p) ≤ λ(q). (3.5)
Finally, if L p is continuous with respect to p,

∀(f, g) ∈ Λ × Λ * lim p→q | g, L p (f ) -L q (f ) |= 0, (3.6)
then λ(p) is continuous with respect to p.

Differentiation method :

We have the following principle Lemma 3.2 Assume, there exists (N p , λ(p), φ p ) solution to (3.1)-(3.3) and ∀q ∈ J, lim

p→q∈ J N p = N q , strongly in B, (3.7
)

∀q ∈ J ∃ ∂ p L * p | p=q ∈ L(C * , B * ) : ∀f ∈ Λ, lim p→q∈ J L * p -L * q p -q φ q , f = ∂ p L * q | p=q φ q , f . (3.8)
Then p → λ(p) is differentiable and we have for all q ∈ J ⊂ I,

d dp λ (q) = ∂ p L * q | p=q φ q , N q . (3.9)
Moreover, under the assumptions of Lemma 2.4, we have (2.9).

Proof of Lemma 3.1

Since N p ∈ Λ and L p N p = λ(p)N p then we have

∀g ∈ Λ * , λ(p) = g, L p N p g, N p .
Hence we obtain the inequality

λ(p) ≤ max f ∈Λ min g∈Λ * g, L p f g, f .
Now, assume there exists f 0 ∈ Λ such that

λ(p) < min g∈Λ * g, L p f 0 g, f 0 .
In particular, this inequality holds for g = φ p and so we have

λ(p) < φ p , L p f 0 φ p , f 0 = L * p φ p , f 0 φ p , f 0 = λ(p),
which is impossible. Thus (3.4) is satisfied. Now, assume that L p is increasing in the sense of (3.5).

Then we have the inequalities g, L p (f ) ≤ g, L q (f ) , and (using (3.4)),

min g∈Λ * g, L p f g, f ≤ min g∈Λ * g, L q f g, f ≤ λ(q).
In particular, for f = N p , we find :

λ(p) = min g∈Λ * g, L p N p g, N p ≤ min g∈Λ * g, L q N p g, N p ≤ λ(q).
Finally, if L p is continuous with respect to p, then using the min-max property, we have,

g, L p f g, f ≤ g, L q f g, f + (p, q),
where (p, q) → p→q 0. Thus, we obtain that

min g∈Λ * g, L p f g, f ≤ min g∈Λ * g, L q f g, f + (p, q) ≤ λ(q) + (p, q),
and so λ(p) ≤ λ(q) + (p, q).

Similarly, λ(q) ≤ λ(p) + 1 (p, q), where 1 (p, q) → p→q 0. Therefore, we prove that λ(p) is continuous with respect to p. 2

Proof of Lemma 3.2

For all q, p ∈ J, we have : λ q = L * q φ q , N q . Thus, we find

λ q -λ p = L * q φ q , N q -L * p φ p , N p . (3.10)
We obtain, using the normalization (3.3), i.e. φ q , N q = φ p , N p = 1, and

(φ s , N s ) ∈ C * × (B C), λ q -λ p = L * q φ q -L * p φ q , N p + L * q φ q , N q -N p -L * p φ p -L * p φ q , N p ,
that gives, using (1.1), (3.2), and the duality between L q and L * q :

λ q -λ p = L * q φ q -L * p φ q , N p + λ q φ q , N q -N p -λ p φ p -φ q , N p .

Therefore, using the normalization (3.3), we find

λ q -λ p = L * q φ q -L * p φ q , N p + λ q φ q , N q -N p -λ p φ q , N q -N p , So, we obtain (λ q -λ p ) 1 -φ q , N p -N q = L * q φ q -L * p φ q , N p . ( 3
1 -φ q , N p -N q = 1, lim p→q L * q φ q -L * p φ q , N p q -p = ∂ p L * q | p=q φ q , N p ,
and so λ(p) is differentiable and by passing to the limit, we find (3.9).

We prove Lemma 2.4. We have (see the equation (3.11)),

(λ(σ)-λ(σ-)) 1- ∞ 0 φ σ-(y)-φ σ (y) N σ-(y)dy = ∞ 0 L * σ (φ σ )(y)-L * σ-(φ σ )(y) N σ-(y)dy.
Using (2.7), we have

(L * σ -L * σ-)(g) = B(y) g(yσ) -g(yσ -y ) + g y(1 -σ) -g(y(1 -σ) + y ) .
Thus we find

(λ(σ) -λ(σ -)) 1 - ∞ 0 φ σ-(y) -φ σ (y) N σ-(y)dy = ∞ 0 B(y) φ σ (yσ) -φ σ (yσ -y ) + φ σ y(1 -σ) -φ σ (y(1 -σ) + y ) N σ-dy,
which can be written also as

(λ(σ) -λ(σ -)) 1 - ∞ 0 φ σ-(y) -φ σ (y) N σ-(y)dy = ∞ 0 B(y) yσ y(σ-) ∂ ∂y φ σ (s)ds + y(1-σ) y(1-σ+ ) ∂ ∂y φ σ (s)ds N σ-dy.
Finally, dividing by , we have

λ(σ) -λ(σ -) = ∞ 0 B(y) yσ y(σ-) ∂ ∂y φ σ (s) ds + y(1-σ) y(1-σ+ ) ∂ ∂y φ σ (s) ds N σ-dy 1 - ∞ 0 φ σ-(y) -φ σ (y) N σ-(y)dy . ( 3
.12) Thus, using [START_REF] Michel | Existence of a Solution to the Cell Division Eigenproblem[END_REF], Lemma 2.3 and the Lebesgue dominated convergence theorem, we pass to the limit in (3.12) as → 0. Therefore the function λ(σ) is differentiable. Moreover, passing to the limit, we obtain (2.9). 2

Proof of the convexity of the dual eigenfunction

In this section, we are interested in global conditions on the birth rate B such that the dual function φ σ (.) is convex (resp. concave) on the real line. We prove here Theorem 2.7.

I -Assume that (2.1) holds. Using (1.2), we have,

φ σ (y) = Cte e λ(σ)y , ∀y ∈ [0, a],
where φ σ is convex on [0, σb] ⊂ [0, a] for all σ such that σb < a.

II -Assume that (2.2) holds. Then we first prove the concavity of the dual eigenfunction in a neighborhood of y = 0 and we extend the property of concavity to y ∈ [0, σ 0 b].

We notice that φ σ (0) > 0, ∂φσ ∂y (0) = (λ σ -B(0))φ σ (0) < 0 since B is decreasing (and non constant) and λ σ < B(0) = sup B. Moreover, assumption (2.2) implies

∂ 2 φ σ ∂y 2 (0) < 0. Using - ∂φ σ ∂y + (B(y) + λ σ )φ σ (y) = B(y) φ σ (σy) + φ σ ((1 -σ)y) , y ≥ 0,
we find,

∂ 2 φ σ ∂y 2 = λ σ ∂ ∂y φ σ (y) - B (y) B(y) - ∂ ∂y φ σ + λ σ φ σ (y) -B(y) σ ∂ ∂y φ σ (σy) + (1 -σ) ∂ ∂y φ σ ((1 -σ)y) - ∂ ∂y φ σ (y) , y ≥ 0.
Now, assume that there exists 0 < α < bσ 0 such that

∂ 2 φ σ ∂y 2 (y) < 0, y ∈ [0, α[, ∂ 2 φ σ ∂y 2 (α) = 0, (4.1)
and

∂φ σ ∂y (y) < 0, y ∈ [0, α[. (4.2)
Then we have

-B(y) σ ∂ ∂y φ σ (σy) + (1 -σ) ∂ ∂y φ σ ((1 -σ)y) - ∂ ∂y φ σ (y) < 0, y ∈ [0, α],
and

∂φ σ ∂y (y) < ∂φ σ ∂y (0) = λ σ -B(0) φ σ (0) < 0, y ∈ [0, α], 0 < φ σ (y) < φ σ (0), y ∈ [0, α].
Assumption (2.2) implies

λ σ + B (y) B(y) ≥ λ σ + λ σ (λ σ -B(0)) B(0) = λ 2 σ B(0) ≥ 0, ∀y ∈ [0, bσ 0 ].
Thus we obtain

∂ 2 φ σ ∂y 2 (y) < λ σ + B (y) B(y) (λ σ -B(0)) - B (y) B(y) λ σ φ σ (0), y ∈ [0, α], ∂ 2 φ σ ∂y 2 (y) < λ σ λ σ -B(0) B(0) - B (y) B(y) φ σ (0)B(0), y ∈ [0, α],
and finally, we have

∂ 2 φ σ ∂y 2 (y) < 0, y ∈ [0, α].
Therefore, the strict inequalities (4.1) and (4.2) are satisfied on [0, bσ 0 ] and φ σ is a concave function on [0, σ 0 b] for all σ ∈ [1/2, σ 0 ]. 2

Extension to a more general cell division model

In this section we extend the main results to a more general model of cell division. More precisely, in Section 5.1, we generalize the results to the homogeneous cell division (see [START_REF] Michel | Existence of a Solution to the Cell Division Eigenproblem[END_REF]) and in Section 5.2, we allow cells to have non constant rate of division.

Homogeneous cell division

A cell of size y may give birth to a cell of size ys, s ∈ [0, 1], with "probability" Θ(s) where

Θ(s) = Θ(1 -s), (5.1)
1 0 sΘ(s)ds = 1, 1 0 Θ(s)ds = 2, (5.2) 1 η Θ(s)ds ≤ 1, for η ∈]0, 1[. ( 5
Again, we prove, using General Relative Entropy method that the asymptotic behavior of such an evolution equation is given by the following eigenproblem (5.6)

                       ∂ ∂y N (
Then we have λ Θ ≥ λ sym , (resp. λ Θ ≤ λ sym ).

Thus, the less (resp. the 'best') fitted division is the symmetric division.

Proof of Theorem 5.2 Under assumption (2.1) (resp. (2.2)), we directly obtain that the dual eigenfunctions are convex (resp. concave) on

ConvHull(Supp Θ).Supp B := {ry, r ∈ ConvHull(Supp Θ), y ∈ Supp B}, for all Θ that satisfies (5.1)-( 5.3) and (5.6). We use a min-max principle to conclude the proof of the theorem.

Step 1. Min-max method: Let (5.8)

We have the min-max lemma (see Section 3). Lemma 5.3 Assume that there exists (N, λ, φ) solution to (5.5). Then we have for all Θ,

λ Θ = max f ∈Λ min g∈Λ * g, Lf g, f , (5.9)
where φ ∈ Λ * ⊂ D om (L * ) and N ∈ Λ ⊂ D om (L).

Step 2. Application to the homogeneous cell division: Let (5.12)

L sym (f ) := - ∂f ∂y (.) -B(.)f (t, .) + 4B(2.)f (2.), ( 5
Proof. We have shown that for all Θ satisfying (5.1)-( 5.3) and (5.6), and B satisfying (2.1) (resp. 2.2)), the dual functions φ (and in particular φ sym ) are convex (resp. concave) functions on ConvHull(Supp Θ).Supp B. Thus, we can choose Λ * as the set of positive convex (resp. concave) functions on ConvHull(Supp Θ).Supp B. We have

L * (f ) -L * sym (f ) = B(.) 1 0 Θ(s) f (s.) -f (./2) ds, ∀f ∈ Λ * ,
which can be rewritten as

L * (f ) -L * sym (f ) = 2B(.) 1 Θ(s) f (s.) + f ((1 -s).) 2 -f (./2) ds, ∀f ∈ Λ * .
Since f ∈ Λ * is a convex (resp. concave) function on ConvHull(Supp Θ).Supp B, then we have

f (s.)+f ((1-s).) 2
-f (./2) ≥ 0 (resp. ≤ 0), and

L * (f ) -L * sym (f ) ≥ 0, ∀f ∈ Λ * , (resp. ≤ 0
). Thus, using the min-max Lemma 5.3, we have λ(Θ) ≥ λ sym (resp. λ(Θ) ≤ λ sym ). 2

Cell division model with non constant speed rate

In this section we consider the cell division equation

∂ ∂t n(t, y) + ∂ ∂y V (y)n(t, y) + B(y)n(t, y) = 1 0 Θ(s)B(y/s)n(t, y/s)ds/s, (5.13)
where the birth rate satisfies Supp B(y) ⊂ [a, b] and the speed rate V (y) = y µ . As above, we prove, using General Relative Entropy method that the asymptotic behavior of such an evolution equation is given by the following eigenproblem Here, we only have to prove the convexity (resp. the concavity) of the dual eigenfunction in (5.14). First, we recall results on the existence of a solution to the eigenproblem. Then, we use the differentiation method in order to find the variation of the invasive parameter with respect to Θ by proving that the solution φ Θ to the dual problem is convex (or concave).

                       ∂ ∂y (V (
Step 1. Existence and uniqueness of the solution to the eigenproblem (5.14). We have the following lemma. Lemma 5.6 Under the assumption µ > -1, there exists a solution (N σ , λ(σ), φ σ ) to (2.5). We refer to [START_REF] Michel | Existence of a Solution to the Cell Division Eigenproblem[END_REF] for the proof of this result.

Step 2. Corollary of the min-max method Using the proof of Theorem 5.2 and Lemmas 5.3 and 5.4, we have the following lemma.

Lemma 5.7 If φ σ is a convex (resp. concave) function on [1/2 -η, 1/2 + η].Supp B = σSupp B with Θ = δ σ + δ 1-σ for σ ∈ [1/2 -η, 1/2 + η], then ∀ 1/2 -η < σ 0 < σ 1 < 1/2 + η, λ σ0 < λ σ1 (resp. > λ σ1 ).
Moreover, if we assume that the probabilities Θ satisfy (5.1)-( 5.3) and (5.6), and the dual eigenfunction φ is convex (resp. concave) on ConvHull(Supp Θ).Supp B, then we have

inf Θ λ Θ = λ sym (resp. sup Θ λ Θ = λ sym ).
Proof. Indeed, we have,

L * σ0 (f ) -L * σ1 (f ) = B(.) f (σ 0 .) + f ((1 -σ 0 ).) -f (σ 1 .) -f ((1 -σ 1 ).) ds, ∀f ∈ Λ * ,
where Λ * is the set of positive convex (resp. concave) functions on ConvHull(Supp Θ).Supp B. Since 1/2 -η < σ 0 < σ 1 < 1/2 + η, then we find that the constant t given by the equality

t = σ 0 + σ 1 -1 2σ 1 -1 ∈]0, 1[, satisfies σ 0 = tσ 1 + (1 -t)σ 1 and 1 -σ 0 = tσ 1 + (1 -t)σ 1 . We have f (σ 0 y) + f ((1 -σ 0 )y) -f (σ 1 y) -f ((1 -σ 1 )y) = f (σ 0 y) -tf (σ 1 y) -(1 -t)f ((1 -σ 1 )y) + f ((1 -σ 0 )y) -tf ((1 -σ 1 )y) -(1 -t)f (σ 1 y) ,
with f positive convex (resp. concave) function and t ∈]0, 1[. Thus, we find that

L * σ0 (f ) -L * σ1 (f ) ≤ 0 (resp. ≥ 0), ∀f ∈ Λ * .
We conclude using the extension of the min-max Lemma 5.3. Next, we assume that for all Θ satisfying (5.1)-(5.3) and (5.6), the dual functions φ (and in particular φ sym ) are convex (resp. concave) functions on ConvHull(Supp Θ).Supp B = [1 -σ, σ].Supp B. Thus, we can choose, in Lemma 5.3, Λ * as the set of positive convex (resp. concave) functions on [1 -σ, σ].Supp B. We have

L * (f ) -L * sym (f ) = B(.) 1 0 Θ(s) f (s.) -f (./2) ds, ∀f ∈ Λ * ,
which can be rewritten as

L * (f ) -L * sym (f ) = 2B(.) 1 1/2 Θ(s) f (s.) + f ((1 -s).) 2 -f (./2) ds, ∀f ∈ Λ * .
Since f ∈ Λ is a positive convex (resp. concave) function, then we have f (s.)+f ((1-s).)

2 -f (./2) ≥ 0 (resp. ≤ 0), L * (f ) -L * sym (f ) ≥ 0 (resp. ≤ 0), ∀f ∈ Λ * .
Thus, using the min-max Lemma 5.3, we have λ(Θ) ≥ λ sym (resp. λ(Θ) ≤ λ sym ) .

2

Step 3. Concavity of the dual eigenfunction We have

φ (y) = λ 2 V (y) 2 1 -V /λ , y ∈ Supp B,
with V (y) = y µ and Supp B = [a, b]. Therefore, we have clearly that assumptions (5.16) or (5.17) lead to φ (y) > 0 on ConvHull(Supp Θ).Supp B and condition (5.15) leads to φ (y) < 0 on ConvHull(Supp Θ).Supp B.

Step 4. Conclusion Thus, using Lemma 5.7 and step 3, we prove the theorem. 2

6 Simple application of the differentiation and the min-max methods

In this section, in order to apply both methods, we focus our attention on a simple model of McKendrick-VonFoerster in which cells are structured by their age (see for instance [START_REF] Cushing | An Introduction to Structured Population Dynamics[END_REF][START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF][START_REF] Iannelli | Mathematical Theory of Age-Structured Population Dynamics[END_REF][START_REF] Mischler | Stability in a Nonlinear Population Maturation Model[END_REF][START_REF] Webb | Theory of non-linear age-dependent population dynamics[END_REF]). The density of the population of age a at time t is denoted by n(t, a) and follows the renewal equation

         ∂ ∂t n(t, a) + ∂ ∂a n(t, a) + d(a)n(t, a) = 0, n(t, 0) = ∞ 0 B(a)n(t, a)da, (6.1)
where B is the growth rate and d the death rate. As above, the GRE gives the asymptotic behavior of n(t, .) ∼ cst N (.)e λt , where (N, λ, φ) is a solution to the eigenproblem The relevance of this model is its simplicity, contrary to the cell division model (1.1). We can easily compute (N, λ, φ). Indeed, if there exists λ such that Thus we have the following lemma. 6.1 Evolution of the invasive parameter λ with respect to the growth of the death rate d or the birth rate B.

                     ∂ ∂a N (
The aim of this part is to study the evolution of the first eigenvalue λ when the death rate (resp. the birth rate) increases.

Growth of death rate. Let n 1 (resp. n 2 ) be the density of the population satisfying (6.1) with the death rate d = d 1 (resp. d = d 2 ) that verifies the assumptions of Lemma 6.1.

Let λ 1 (resp. λ 2 ) be the invasive parameter, that is the eigenvalue, associated to the first population n 1 (resp. the second population n 2 ) Lemma 6.2 Assume that 0 ≤ d 1 ≤ d 2 . Then λ 2 ≤ λ 1 . Thus, the population n 2 is less fitted than the population n 1 .

Proof. Here we can use both methods. Indeed, let

L 1 (f ) = -[ ∂ ∂a f (a) + d 1 (a)f (a)] + δ a=0 ∞ 0 B(a )f (a )da , L 2 (f ) = -[ ∂ ∂a f (a) + d 2 (a)f (a)] + δ a=0 ∞ 0 B(a )f (a )da , then L 1 (f ) = L 2 (f ) + (d 2 -d 1 )(a)f (a) ≥ L 2 (
f ) for all f ≥ 0 and thus, using Lemma 3.2, we obtain that λ 1 ≥ λ 2 . 2 Remark. The differentiation method could be used to prove the lemma. Indeed, let

L * p (f ) = -[- ∂ ∂a f (a) + d p (a)f (a)] + B(a)f (0), with d p = pd 2 + (1 -p)d 1 and p ∈ [0, 1]. Then L * p (f ) -L * q (f ) = d q (a) -d p (a) f (a) = (p -q)(d 1 -d 2 ),
and so (3.8) is satisfied. Moreover, the implicit form of λ(p) and the explicit form of N p imply directly condition (3.7). Therefore, we have,

d dp λ(p) = ∞ 0 (d 1 -d 2 )(a)φ p (a)N p (a)da ≤ 0,
and we find λ

2 = λ(1) ≤ λ(0) = λ 1 .
Growth of birth rate. Let n 1 (resp. n 2 ) be the density of the population satisfying (6.1) with the birth rate B = B 1 (resp. B = B 2 ) that verifies the assumptions of Lemma 6.1. Let λ 1 (resp. λ 2 ) be the invasive parameter, i.e., eigenvalue, associated to the first population n 1 (resp. the second population n 2 ) Lemma 6.3 Assume that 0 ≤ B 1 ≤ B 2 . Then λ 1 ≤ λ 2 . Thus, the population n 2 is better fitted than the population n 1 .

Proof. Here we can use both methods. Indeed, let

L 1 (f ) = -[ ∂ ∂a f (a) + d(a)f (a)] + δ a=0 ∞ 0 B 1 (a )f (a )da , L 2 (f ) = -[ ∂ ∂a f (a) + d(a)f (a)] + δ a=0 ∞ 0 B 2 (a )f (a )da . Then L 1 (f ) = L 2 (f ) + δ a=0 ∞ 0 (B 1 (a ) -B 2 (a ))f (a )da ≤ L 2 (
f ) for all f ≥ 0 and thus, using Lemma 3.2, we obtain that λ 1 ≤ λ 2 .

2 Remark. The differentiation method could be used to prove the lemma. Indeed, let

L * p (f ) = -[- ∂ ∂a f (a) + d(a)f (a)] + B p (a)f (0), with B p = pB 2 + (1 -p)B 1 and p ∈ [0, 1]. Then (as above) d dp λ(p) = ∞ 0 (B 2 -B 1 )(a)φ p (0)N p (a)da ≥ 0,
and we find λ 2 λ(1) ≥ λ(0) = λ 1 .

We notice that both examples give the same results using the min-max method or the differentiation method. Here, we give an example which is more difficult to study with the min-max method.

6.2 Evolution of the invasive parameter λ with respect to a complex variation of the death rate d.

Child Vs Oldness death. Let n p be the density of the population satisfying (6.1) with the death rate d p (y) = 1 y∈[p,p+1] d(y) that verifies the assumptions of Lemma 6.1. More precisely, the death rate d p has its support on the interval [p, p + 1], which means that for small p, only childs die and for large p, only old individuals die. Let λ(p) be the invasive parameter, that is the eigenvalue, associated to the population n p . In this part, we assume that

Supp B = [α, β] ⊂ [0, ∞[, (6.5
)

and d ∈ C 1 ([0, ∞[) decreases on [0, β -1]. (6.6)
Then we have the following result.

Lemma 6.4 Assume that (6.5) and (6.6) are satisfied. Then λ(p) is increasing, that is the juvenile death rate is less fitted than 'oldness' death rate.

Proof. Here we can use only the differentiation method. Indeed, let Therefore, we obtain that λ(p) is increasing. 2

L * p (f ) = -[- ∂ ∂a f (
We notice that the min-max method is not well adapted when the growth of the operators L p with respect to p is hard to prove. As for instance in the McKendrick-Von Foerster model when the death rates cannot be compared. The question of knowing the variation of the invasive parameter with respect to the asymmetry parameter leads to the same remark and therefore, we first use the differentiation method.

Discussion

In this paper we have studied the variation of the Malthusian growth rate of a cell population due to the variability of the symmetry of the cell division during mitosis [START_REF] Huang | Symmetry of Initial Cell Divisions Among Primitive Hematopoietic Progenitors Is Independent of Ontogenic Age and Regulatory Molecules[END_REF]. We show that under the assumption of a constant speed rate, equal mitosis is not necessarily the division giving the better growth rate. More precisely, when the cells divide lately, that is for large size relatively to the average size of a dividing cell, then equal mitosis is not the best fitted division. When cells can divide early, then equal mitosis is the best fitted division.

Obviously, this model is a simple one and the conclusion depends on the assumptions such as : -there is no interaction between cells, no competition between species with different asymmetry parameters, -the speed rate is constant (or a power of the size) which cannot be the case in general, -there is no apoptosis in the model.

Nevertheless, the tools we have used to study this problem, that is the differentiation method and the min-max method, seem to be well fitted to study the variation of the Malthusian growth rate with respect to a parameter for various problems (see Section 6).

An interesting question to study is the competition between two species of cells (with two different cell divisions) as for instance for normal and tumoral (due to mutation) cells.

0 N( 5 . 5 ) 5 . 1

 05551 y) + [λ + B(y)]N (y) = φ(ys)ds -φ(y) , N (y = 0) = 0, N, φ ≥ 0, ∞ (y)φ(y)dy = 1, φ(0) = 1. Definition We call λ Θ the first eigenvalue associated to eigenproblem (5.5) and λ sym the eigenvalue associated to Θ(s) = δ s=1/2 . Then we have the following theorem. Theorem 5.2 Assume that (2.1) (resp. (2.2)) holds and ConvHull(Supp Θ) := Convex Hull(Supp Θ) ⊂ [1 -a/b, a/b] (resp.[1 -σ 0 , σ 0 ]).

1 0Θ

 1 operator, L * (f ) := ∂f ∂y (.) -B(.)f (t, .) + B(.) (s)f (.s)ds -f (.) .

. 10 )Lemma 5 . 4

 1054 and the dual operator,L * sym (f ) := ∂f ∂y (.) -B(.)f (t, .) + B(.) 2f (./2) -f (.) . (5.11) Assuming (2.1) (resp. 2.2)), we have for all Θ satisfying (5.1)-(5.3) and (5.6), λ Θ ≥ λ sym (resp ≤ λ sym ).

1 0Θ 1 0Θ 5

 115 y)N (y)) + [λ + B(y)]N (y) = (s)B(y/s)N (y/s)ds/s, y ≥ 0, -V (y) ∂ ∂y φ(y) + [λ + B(y)]φ(y) = B(y) (s)φ(ys)ds, y ≥ 0,N (0) = 0, N ≥ 0, φ ≥ 0, N(y)φ(y)dy = 1.(5.14)If λ sym = 0 and one of these conditions holds 0 < µ < 1 and 0 < λ sym < µa µ-1 , (5.15)µ > 1 and λ sym > µa µ-1 , Assume that Θ satisfies (5.1)-(5.3) and (5.6), and µ satisfies (5.15) (resp. (5.16) or (5.17)). Then we have sup Θ λ Θ ≤ λ sym (resp. ≥ λ sym ).(5.18)

  a) + d(a)N (a) + λN (a) = 0, N (0) = ) + d(a)φ(a) + λφ(a) = B(a)φ(0),

∞ 0 B 0 B

 00 (a )e -a 0 (d(s)+λ)ds da = 1) + d(a)f (a)] + δ a=0 ∞ (a )f (a )da , L * (g) = -[-∂ ∂a g(a) + d(a)g(a)] + g(0)B(a).

Lemma 6 . 1 0 B

 610 Assume that B, d ∈ L ∞ ([0, ∞[, [0, ∞[) and Supp B ⊂ [0, A] is compact.Then we have the existence of a solution (N, λ, φ) to eigenproblem (6.2). Moreover, we have,L(f ) : L 2 ([0, A]) → M([0, A]), L * (f ) : C 0 0 ([0, ∞[) → L 2 ([0, ∞[),where M([0, A]) is the set of a bounded measure on [0, A] andN ∈ C 0 b ([0, A]), N ≤ 1, Supp φ ⊂ [0, A], φ ≤ A(s)dse λA . Therefore (3.1)-(3.3) are satisfied for B = L 2 and C = C 0 0 ([0, ∞[).

  a) + d p (a)f (a)] + B(a)f (0). Then for d p (y) = 1 y∈[p,p+1] d(y) = 1 p∈[y-1,y] d(y) we have -∂ ∂p d p (y) = -δ p=y-1 d(p + 1) + δ p=y d(p), and so, we find d dp λ(p) = d(p)φ p (p)N p (p) -d(p + 1)φ p (p + 1)N p (p + 1). Therefore, using (6.4),we obtain, d dp λ(p) = -d(p + 1) ∞ p+1 B(a )e -a 0 (d(s)+λ)ds da + d(p) ∞ p B(a )e -a 0 (d(s)+λ)ds da , and so, we have, d dp λ(p) = d(p) -d(p + 1) ∞ p+1 B(a )e -a 0 (d(s)+λ)ds da + d(p) p+1 p B(a )e -a 0 (d(s)+λ)ds da . Since d increases (see (6.6)), then d(p) -d(p + 1) ∞ p+1 B(a )e -a 0 (d(s)+λ)ds da ≥ 0 and d dp λ(p) ≥ 0.


P Michel
email: philippe.michel@ec-lyon.fr

General Relative Entropy in a nonlinear McKendrick model

Keywords: Mathematics Subject Classification. 35Bxx General Relative Entropy, Nonlinear McKendrick model, Global convergence

Introduction

In the study of populations of cells, animals or humans, one of the most used model is the McKendrick-Von Foerster model ( [START_REF] Metz | The dynamics of physiologically structured populations[END_REF][START_REF] Tuljapurkar | Structured population models in marine, terrestrial and freshwater ecosystems[END_REF]) where the density of population n(t, y) at time t and age y, is described by the master equation where B ≥ 0 is the birth rate, d ≥ 0 is the death rate. It is well known that the long time asymptotic is described by the first eigenvalue λ and positive eigenvector N of the eigenproblem (1.2). Indeed, for a large time, n ∼ Cst N e λt , this can be proved by using Laplace transform ( [START_REF] Metz | The dynamics of physiologically structured populations[END_REF][START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF]) and more recently by using the General Relative Entropy (GRE) inequalities ( [START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF][START_REF] Mischler | Stability in a Nonlinear Population Maturation Model[END_REF][START_REF] Michel | General entropy equations for structured population models and scattering[END_REF]10]). We recall the GRE results, if there exist (N, λ, φ) solution to the following eigenproblem

(1.2)                ∂ ∂y N (y) + (d(y) + λ)N (y) = 0,
with the dissipation of entropy -D H given by (1.4)

-D H (ne -λt /N ) = N (0)φ(0) H( ∞ 0 ne -λt N dµ) - ∞ 0 H( ne -λt N )dµ ,
where dµ(y) = B(y)N (y)dy/ ∞ 0 B(y )N (y )dy is a probability measure. In particular, for H(x) = x, we obtain the conservation of ∞ 0 n(t, y)e -λt φ(y)dy. The convergence of ne -λt to Cst N implies the exponential growth or extinction. This of course does not take in account the use of resources for instance. In order to take in account the consumption of nutrient, we can for instance change the linear birth term in (1.1) by a nonlinear birth term ([2, 9, 17]) where P is related to the population number and the non linearities to the consumption of nutrients. Here we study the following nonlinear problem where f describes the nonlinearity and so the change of birth term with the environment alteration, for example we can choose f (x) = x ω with 0 < ω < 1. We notice that (1.6) is a particular case of the problem (1.5), B(y , P (t)) = B(y )f (P (t))/P (t), d(y , P (t)) = d(y ), P (t) = ∞ 0 B(y )n(t, y )dy .

(1.5)                    ∂ ∂t n(t,
Even though this is not the most biologically relevant situation at least it is simple enough to classify the type of results we can obtain. There are several some theorems giving the local asymptotic behavior, i.e., in a neighbourhood of a steady state [START_REF] Cushing | An Introduction to Structured Population Dynamics[END_REF][START_REF] Farkas | Stability Conditions for the Non-Linear McKendrick Equations[END_REF][START_REF] Gurtin | Non-linear age-dependent populations dynamics[END_REF][START_REF] Iannelli | Mathematical Theory of Age-Structured Population Dynamics[END_REF][START_REF] Webb | Theory of nonlinear age-dependent population dynamics[END_REF], in which, one uses the linearisation of (1.6), then checks for the eigenvector and eigenvalue of the linear system and according to the sign of the eigenvalue concludes to the local asymptotic convergence or not. Here we analyze how the GRE method can be used for this class of nonlinear problem and to the more general case where y is the age and x the space parameter. As, we will see, in several situations we can arrive at global behaviors. This paper is organized as follows : in section 2 we extend the definition of (N, λ, φ) to this nonlinear problem and prove the longtime convergence under some assumptions on f , especially we arrive to global pictures for extinction or attraction to a steady state. Then in section 4 we compare then results obtained by local asymptotic method and the GRE results for periodic or oscillating solutions. In section 6 we extend the result to (1.7) and (1.8).

Nonlinear McKendrick

First we define the eigenproblem, and in particular, the adjoint eigenproblem in the nonlinear equation (1.6), then we use the GRE introduce in (1.3) (see [START_REF] Mischler | Stability in a Nonlinear Population Maturation Model[END_REF][START_REF] Michel | General entropy equations for structured population models and scattering[END_REF]10]). Thus, under global assumptions on f we prove the decay of specific entropy and the global convergence in long time in (1.6). We notice that (1.6) can be written in a simpler way. We introduce the following notations that will be used throughout the paper Notice that dµ(.) is a probability measure. Then we set (2.5) m(t, y) := βn(t, y)e D(y) , and m(t, .) satisfies the master equation in

L p ([0, ∞[) L 1 ([0, ∞[, φ(y)dy) (2.6)                ∂ ∂t m(t, y) + ∂ ∂y m(t, y) = 0, m(t, 0) = g( ∞ 0 m(t, y )dµ(y )), m(0, .) ∈ L ∞ + ([0, ∞[) L 1 + ([0, ∞[, φ (
y)dy), where p > 1. From now, we study the behavior of m and we give assumptions on g in order to simplify the notations. We prove the existence of a decreasing GRE and the longtime convergence of m under some assumptions on g. At this stage we cover two simple cases : global extinction, global attraction to a non zero steady state.

Eigenproblem in nonlinear case

3.1. Extinction. As for the linear case, the GRE principle is based on the existence of an adjoint eigenfunction problem, here we focus on the case where the eigenvalue is zero, i.e., the adjoint steady state. We notice there always exists φ uniquely defined up to multiplicative constant, solution to

(3.1) - ∂ ∂y φ = φ(0)B(y)e -D(y) /β, φ ≥ 0, φ ≡ 0, φ(∞) = 0.
Indeed, the function Proof. We prove the convergence (3.6), using the equation (2.6) and (3.2), we have

(3.7) d dt ∞ 0 m(t, y)φ(y)dy = φ(0)m(t, 0) -φ(0) ∞ 0 m(t, y)dµ(y), (3.8) d dt ∞ 0 m(t, y)φ(y)dy = φ(0) g ∞ 0 m(t, y)dµ(y) - ∞ 0 m(t, y)dµ(y) , using (3.
3), we have,

(3.9) d dt ∞ 0 m(t, y)φ(y)dy ≤ 0.
We prove the convergence of ∞ 0 m(t, y)dµ(y) to 0 using a method developped in [START_REF] Mischler | Stability in a Nonlinear Population Maturation Model[END_REF].

Step1-L ∞ bound. For all C > 0, we have (as a particular case of a GRE)

d dt ∞ 0 (m(t, y) -C) + φ(y)dy = φ(0)(m(t, 0) -C) + -φ(0) ∞ 0
(m(t, y) -C) + dµ(y), (3.10) using (3.3) and Jensen inequality, we have, (2.6)), we have (3.12) m(t, y) ≤ m(0, y), ∀t, y ≥ 0, and m(t, .) is uniformly bounded in L ∞ ([0, ∞[). Therefore, we may choose a sequence of times t k → ∞ so that the sequence of functions

d dt ∞ 0 (m(t, y) -C) + φ(y)dy = φ(0) g ∞ 0 m(t, y) -dµ(y) -C + - ∞ 0 (m(t, y) -C) + dµ(y) ≤ 0. (3.11) Since m(0, .) ∈ L ∞ ([0, ∞[) (see
m k (t, y) = m(t + t k , y) converges weak- * in L ∞ ([0, T ] × [0, ∞[),
for any T > 0, to a function denoted m(t, y).

Step2-L 1 bound. From (3.9), we have the decay of ∞ 0 m(t, y)φ(y)dy and thus its convergence to a positive number. Now, using (3.8), we obtain

(3.13) ∞ 0 ∞ 0 m(t, y)dµ(y) -g ∞ 0 m(t, y)dµ(y) dt ≤ ∞ 0 m(0, y)φ(y)dy,
and thus

(3.14) T 0 ∞ 0 m k (t, y)dµ(y) -g ∞ 0 m k (t, y)dµ(y) dt → 0 as k → ∞.
Step3-Compactness. For any smooth function ψ(y), and using equation (2.6), we have that

(3.15) d dt ∞ 0 m k (t, y)ψ(y)dy is bounded in L ∞ ([0, ∞[).
Therefore, after approximating Be -D by smooth functions, we deduce that ∞ 0 m k (t, y)dµ(y) converges in C([0, T ]) to ∞ 0 m(t, y)dµ(y). Now using that g is u.s.c. and satisfies (3.3), we have

0 ≤ T 0 ∞ 0 m(t, y)dµ(y) -g ∞ 0 m(t, y)dµ(y) dt ≤ lim inf k→∞ T 0 ∞ 0 m k (t, y)dµ(y) -g ∞ 0 m k (t, y)dµ(y) dt = 0, (3.16) thus T 0 ∞ 0 m(t, y)dµ(y) -g ∞ 0 m(t, y)dµ(y) dt = 0, but using (3.3), we have lim t k →∞ ∞ 0 m(t + t k , y)dµ(y) = ∞ 0 m(t, y)dµ(y) = 0.
Using transport equation (2.6) with boundary condition that converges to 0, we have the global convergence of m(t, .) to 0 in L ∞ ([0, ∞[).

Theorem 3.2. Assume there exists λ 0 < 0 such that (3.17 indeed φ is well defined (using ∞ 0 dµ(y) = 1) and satisfies (3.18). Now we prove the exponential convergence under assumption (3.17), we have, using φ defined by the equation (3.18) ∃x 0 > 0, g(x 0 ) = x 0 , then there exists N ≥ 0 (N = 0) such that N (x 0 ) = x 0 and

d dt ∞ 0 m(t, y)e -λ0t φ(y)dy = φ(0)e -λ0t m(t, 0) - β ∞ 0 B(y )e -D(y )-
(3.26)        ∂ ∂y N (y) = 0, N (0) = g ∞ 0 N (y )dµ(y ) .
And there exists φ ≥ 0, φ ≡ 0 that satisfies the following adjoint eigenproblem

(3.27) - ∂ ∂y φ = φ(0)B(y)e -D(y) , ∞ 0 N (y)φ(y)dy = 1.
Proof. Indeed, let (3.28) N (y) := x 0 ,

(3.29) φ(y) := φ(0) ∞ y dµ(y ),
then N satisfies (3.26) and φ satisfies (3.27).

GRE and first results.

Here we consider the relative entropy introduce in (1.3) and give assumptions on g that allow us to use it. From now we assume there exist a strictly positive steady state and adjoint steady state solution (N, φ) to (3.26), (3.27). First we prove the existence of a GRE entropy structure in the nonlinear problem (2.6). Under some assumptions on g, we prove, the decay of entropy, some bounds on m(t, .) and the convergence of m(t, .) to N (.). Theorem 3.5. Under the assumption (2.4), with the notations of section 2, and these of lemma 3.4, let the entropy defined by

H(t) := ∞ 0 H(m/N )N φdy,
with N and φ solution to (3.26), (3.27) and H a convex function. Then we have

∂ ∂t H(t) = N (0)φ(0) H g ∞ 0 m(t, y )dµ(y ) g(N (0)) - ∞ 0 H(m/N )N (y)dµ(y) N (0) , (3.30) and define equivalently (N (y) = x 0 ) -D H (m/N )(t) := ∂ ∂t H(t) = N (0)φ(0) H g x 0 ∞ 0 m(t, y ) N (y ) dµ(y ) g(x 0 ) - ∞ 0 H m N dµ(y) . (3.31) Proof. Indeed we have ∂ ∂t H(t) = ∞ 0 H (m/N )φ ∂ ∂t m(t, y)dy = - ∞ 0 H (m/N )φ ∂ ∂y m(t, y)dy, (3.32) Figure 1. Condition of convergence. ∂ ∂t H(t) = - ∞ 0 ∂ ∂y H(m/N )φN dy + ∞ 0 H(m/N ) ∂ ∂y φN dy - ∞ 0 H (m/N )φ m N 2 ∂ ∂y N dy, (3.33)
and finally we obtain

∂ ∂t H(t) = H(m/N )φN (t, y = 0) - ∞ 0 H(m/N )N φ(0)Be -D(y) dy/β, (3.34) ∂ ∂t H(t) = N (0)φ(0) H g ∞ 0 m(t, y )dµ(y ) g(N (0)) - ∞ 0 H(m/N )N (y)dµ(y) N (0) , (3.35)
where N (y) = N (0) = x 0 and the conclusion follows.

Under some assumptions on g (and thus on f , see (2.2)) we prove the decay of some entropy and the convergence of m to N . More precisely, (3.36)

           ∃x 0 > 0 : g(x 0 ) = x 0 , x < g(x) < x 0 , ∀x < x 0 , x 0 < g(x) < x, ∀x > x 0 .
This implies directly the uniqueness of the fixed point of g and thus the uniqueness of a strictly positive steady state of (1.6).

Remark 3.6. For instance (3.36) is satisfied for g(x) = x ω with 0 < ω < 1 and more generally if g and so f is an increasing concave function. See Figure 1.

Here we prove some L ∞ a priori bounds, using theorem 3.5 and assumption (3.36).

Lemma 3.7. Assume (2.4), (3.36) and 0 ≤ m(0, .) ≤ KN (.) for some K ≥ 1 then

(3.37) 0 ≤ m(t, y) ≤ KN (y), ∀t, y ≥ 0.
Proof. We have for

H(z) = (z -K) + , using (3.31) with K ≥ 1, -D H (m/N )(t) = N (0)φ(0) [ g ∞ 0 m(t, y ) N (y ) dµ(y )x 0 g(x 0 ) -K + - ∞ 0 m N -K + dµ(y)], (3.38) but using (3.36), we have -D H (m/N )(t) = N (0)φ(0) [ ∞ 0 m(t, y ) N (y ) dµ(y ) -K + - ∞ 0 m N -K + dµ(y)] ≤ 0. (3.39)
Thus, m(0, .) ≤ KN (.), implies ∞ 0 (m(t, y)/N (y) -K) + N (y)φ(y)dy = 0 for t = 0, decreases and is positive thus

∞ 0 (m(t, y)/N (y) -K) + N (y)φ(y)dy = 0, ∀t ≥ 0,
and the conclusion follows.

Lemma 3.8. Assume (2.4), (3.36) and m(0, .) ≥ kN (.) for some 1 ≥ k > 0 then

(3.40) m(t, y) ≥ kN (y), ∀t, y ≥ 0.
Proof. We have for

H(z) = (m -z) + , using (3.31) with 0 < m ≤ 1, -D H (m/N )(t) = N (0)φ(0) [ k - g ∞ 0 m(t, y ) N (y ) dµ(y )x 0 g(x 0 ) + - ∞ 0 k - m N + dµ(y)], (3.41) but using (3.36) we have -D H (m/N )(t) = N (0)φ(0) [ k - ∞ 0 m(t, y ) N (y ) dµ(y ) + - ∞ 0 k - m N + dµ(y)] ≤ 0. (3.42)
Thus, m(0, .) ≥ kN (.), implies ∞ 0 (k -m(t, y)/N (y)) + N (y)φ(y)dy = 0 for t = 0, decreases and is positive thus Moreover, we have

(3.46) lim t→∞ m(t, .)/N (.) = 1, L 1 ([0, ∞[, N (y )φ(y )dy ).
Theorem 3.10. We make the assumptions of theorem 3.9, for some α > 0, (3.47) dµ(y) dy = B(y)e -D(y) /β ≥ αφ(y)/φ(0), ∀y ≥ 0, (3.48)

J := min 1 - sup 1≤z≤max(1,J) g(zx0) g(x0) -1 z -1 , inf min(1,J)≤z≤1 g(zx0) g(x0) -1 z -1 -1 > 0,
where

(3.49) J := inf t≥0 ∞ 0 m(t, y )dµ(y ) ≥ 0, J := sup t≥0 ∞ 0 m(t, y )dµ(y ) ≤ ∞. Then ∞ 0 | m(t, y ) -N (y ) | φ(y )dy ≤ e -JαN (0)φ(0)t ∞ 0 | m(0, y ) -N (y ) | φ(y )dy . (3.50) Remark 3.11. We notice that when g is C 1 g(zx 0 ) g(x 0 ) -1 z-1
gives the slope of g, and J > 0 means that the slope of g is strictly less than 1 at the right of the fixed point x 0 and strictly greater than 1 at the left of x 0 . Moreover (3.49) can be replaced by (3.51)

J := inf t≥t0 ∞ 0 m(t, y ) B(y )e -D(y ) dy β ≥ 0,
for some t 0 > 0, the same for (3.49) and then appears a constant C(t 0 ) in the right hand side of (3.50).

Proof. of theorem 3.9 First we prove (3.45), (3.44) then we prove the convergence result (3.46).

Step1. Bound (3.45). Under the assumptions (3.36) and lemmas 3.7 and 3.8, the dissipation of entropy associated to

H(z) =| 1 -z | satisfies, -D H = -D |1-z| = -D (1-z)+ -D (1-z)-≤ 0, and (3.52) ∞ 0 | m(t, y) -N (y) | φ(y)dy ≤ ∞ 0 | m(0, y) -N (y) | φ(y)dy < ∞, thus using m(0, y) ∈ L 1 ([0, ∞[, φ(y)dy) and (3.23), we obtain, ∞ 0 m(t, y)φ(y)dy ≤ ∞ 0 N (y)φ(y)dy + ∞ 0 | m(0, y) -N (y) | φ(y)dy < ∞, ∀t ≥ 0. (3.53)
Step2. Bound L p . Under the assumptions (3.36) and adapting lemmas 3.7 and 3.8, the dissipation of entropy associated to Step3. Bound (3.44). Since Supp B ⊂ Supp φ and using the Hölder inequality with 1/p + 1/q = 1, 1 < p ≤ p and (3.24), we find (B(y)e -D(y) ) q β q φ(y) q/p dy

H(z) =| 1 -z | p satisfies, -D H = -D |1-z| p = -D ((1-z)+) p -D ((1-z)-) p ≤ 0,
1/q ≤ K(p) ∞ 0 m(t, y) p φ(y)dy 1/p , (3.58) with K(p) := supp φ (B(y)e -D(y) ) p/(p-1) β p/(p-1) φ(y) 1/(p-1) dy (p-1)/p < ∞.
Now, we use the Hölder inequality for 1/p + 1/q = 1, 1 < p ≤ p and we obtain

∞ 0 m(t, y)φ(y)dy ≤ ∞ 0 m(t, y) p φ(y)dy 1/p ∞ 0 φ(y)dy 1/q = ∞ 0 ydµ(y) 1/q ∞ 0 m(t, y) p φ(y)dy 1/p . (3.59)
Similarly, the Hölder inequality with 1/p + 1/q = 1, q :

= p-1 p-1 > 1 implies ∞ 0 m p (t, y)φ(y)dy = ∞ 0 m 1/p m p-1/p (t, y)φ(y)dy ≤ ∞ 0 m(t, y)φ(y)dy 1/p ∞ 0 m (p-1/p )q (t, y)φ(y)dy 1/q ≤ ∞ 0 m(t, y)φ(y)dy 1/p ∞ 0 m (p-1+1/q )q (t, y)φ(y)dy 1/q ≤ ∞ 0 m(t, y)φ(y)dy
-D |1-z| p (m/N )(t) ≤ N (0)φ(0) | 1 - ∞ 0 m(t, y) N (y) dµ(y) | p - ∞ 0 | 1 - m N | p dµ(y) , (3.63)
thus m/N = ρ independent of y. Moreover if ρ(t) N satisfies (1.6) and (3.44), with assumption (3.36) that implies uniqueness of the strictly positive steady state then ρ(t) = 1. Using the transport equation with bounds given by lemma 3.8 and 3.7, we have for all t k ∞, m(t + t k , .)/N (.) has a weak L ∞ ([0, ∞[, dµ) limit, by passing to the limit in the dissipation of entropy and in the equation (1.6) (we refer to [START_REF] Mischler | Stability in a Nonlinear Population Maturation Model[END_REF] of to the proof of lemma 3.1 to the complete proof), we have lim k→∞ m(t + t k , .)/N (.) = 1.

Proof. of theorem 3.10 Now, we prove the exponential decay. We recall that for all H positive convex function we have

∂ ∂t ∞ 0 H(m/N )N φdy = N (0)φ(0) H g ∞ 0 m(t, y ) N (y ) dµ(y )x 0 g(x 0 ) - ∞ 0 H m N dµ(y) ,
thus using (3.49) we have

∂ ∂t ∞ 0 (m/N -1) + N φdy = N (0)φ(0) g ∞ 0 m(t, y ) N (y ) dµ(y )x 0 g(x 0 ) -1 + - ∞ 0 n N -1 + dµ(y) ≤ N (0)φ(0) γ + ∞ 0 m(t, y ) N (y ) dµ(y ) -1 + - ∞ 0 m N -1 + dµ(y) , with γ + = sup 1≤x≤max(1,J) g(xx0)/g(x0)-1 x-1
and (3.47) implies

∂ ∂t ∞ 0 (m/N -1) + N φdy ≤ α(γ + -1) ∞ 0 (m/N -1) + N φdy.
In the same way, using (3.49) we have

∂ ∂t ∞ 0 (m/N -1) -N φdy = N (0)φ(0) g ∞ 0 m(t, y ) N (y ) dµ(y )x 0 g(x 0 ) -1 - - ∞ 0 m N -1 - dµ(y) ≤ N (0)φ(0) γ - ∞ 0 m(t, y ) N (y ) dµ(y ) -1 - - ∞ 0 m N -1 - dµ(y) ,
with γ -= inf min(J,1)≤x≤1 g(xx0)/g(x0)-1

x-1 and (3.47) implies

∂ ∂t ∞ 0 (m/N -1) -N φdy ≤ α(1 -γ -) ∞ 0 (m/N -1) -N φdy.
Therefore (3.50) follows from Gronwall lemma.

This theorem gives results if we have some informations such as the conservation law given by the linear case for instance or the uniqueness of the steady state in the nonlinear case. Remark 3.12. We notice that the condition

(3.64) 1 < g(zx 0 )/g(x 0 ) < 2 -z, ∀z < 1, 1 > g(zx 0 )/g(x 0 ) > 2 -z, ∀z > 1,
is satisfied for g(x) = min(x ω , a) with ln(2-a) ln(a) < ω < 0, 0 < a < 1, and implies the convergence to N as in the theorem 3.9.

Blow-up.

Here, we give some conditions on g that imply the blow-up of m solution to (2.6). Theorem 3.13. Assume (2.4) and there exist γ > 1, α > 0 such that Now, it is interesting to compare this result to the linearisation method in order to prove the local asymptotic convergence.

Local asymptotic convergence

In this section we compare the linearisation and GRE results in the case where there exists only one x 0 > 0 such that g(x 0 ) = x 0 , i.e, there exists one and only one steady state different from 0. Now we give some assumptions on g such that the strictly positive steady state N of the equation (1.6) be locally asymptotically stable. We begin with conditions for the stability (instability) of the linearized problem then we prove a sharp result on the nonlinear problem. ∞ 0 e -ry dµ(y ) < ∞, ∀r ∈] -∞, ∞[, then the steady state associated to x 0 , N (y) = x 0 is a locally asymptotically stable steady state when

(4.2) 0 <| g (x 0 ) |< 1,
and N (y) = x 0 is a locally asymptotically unstable steady state when

(4.3) g (x 0 ) > 1.
Proof. In a neighbourhood of the steady state we set m = N + ˜ and (4.4)

       ∂ ∂t ˜ (t, y) + ∂ ∂y ˜ (t, y) = 0, ˜ (t, 0) = g (x 0 ) ∞ 0 ˜ (t, y )dµ(y ) + O(sup y ˜ 2 (t, y)).
Therefore m solution to (1.6) converges locally asymptotically to the steady state N if the eigenvalue associated to the system is strictly negative (see [START_REF] Cushing | An Introduction to Structured Population Dynamics[END_REF][START_REF] Farkas | Stability Conditions for the Non-Linear McKendrick Equations[END_REF][START_REF] Gurtin | Non-linear age-dependent populations dynamics[END_REF][START_REF] Iannelli | Mathematical Theory of Age-Structured Population Dynamics[END_REF][START_REF] Webb | Theory of nonlinear age-dependent population dynamics[END_REF]).

Using the General Relative Entropy in the linear case, and the approximation ˜ ∼ where satisfies the linear problem (4.5)

       ∂ ∂t (t, y) + ∂ ∂y (t, y) = 0, (t, 0) = g (x 0 ) ∞ 0 (t, y )dµ(y ),
we deduce the asymptotic behavior of .

Step1. 0 <| g (x 0 ) |< 1. In this case, there exists (λ, φ) solution to (and thus converges to 0) when (4.2) is satisfied.

P. MICHEL

Step2. g (x 0 ) > 1. In this case, there exists ( , λ, φ) solution to (4.7) Indeed, using (4.1), we have,

                           ∂ ∂y (y) + λ (y) = 0, (0) = g (x 0 ) ∞ 0 (
∞ 0 e -λy dµ(y ) = 1/g (x 0 ), φ(y) = φ(0)g (x 0 )
∞ y e -λ(y -y) dµ(y).

Then, using the GRE, we obtain, for all ρ ∈] -∞, ∞[,

d dt ∞ 0 | (t, y)e -λt -ρ (y) | φ(y)dy := -D H (t),
where

-D H (t) = φ(0)g (x 0 ) | ∞ 0 (t, y)e -λt -ρ (y) dµ(y) | - ∞ 0 | (t, y)e -λt -ρ (y) | dµ(y) ≤ 0.
Therefore λ > 0 and using the decay of GRE when g > 1, we obtain the convergence result lim t→∞ (t, .)e -λt -ρ (.

) L 1 ([0,∞[,dµ) = 0,
where ρ is given by the conservation law ∞ 0 (t, y)e -λt φ(y)dy = ∞ 0 (0, y)φ(y)dy, and thus does not converges to 0. Theorem 4.2. Assume there exist η > 0 and an unique x 0 > 0 such that g(x) = x 0 for all x ∈]x 0 -η, x 0 +η[ then the steady state associated to x 0 , N (y) = x 0 is locally asymptotically stable.

Proof. Indeed, in this case m(t, 0) = g( ∞ 0 m(t, y)dµ(y)) = x 0 when m(t, .) is in a neigborhood of N (.) = x 0 , therefore N (.) = x 0 is locally asymptotically stable.

Remark 4.3. Notice that (3.36) or the inequalities given in remark 3.12 give weaker locally conditions in a neighborhood of the steady state than the condition (4.2). In other words theorem 4.1 gives a weaker convergence than the theorems 3.9, 3.10 but needs weaker assumptions on g than the theorems 3.9, 3.10. Now we prove that under some assumptions on g the conditions (3.36) are equivalent to the condition (4.2). Lemma 4.4. Assume g belongs to C 1 ([0, ∞[) is increasing and strictly concave, and there exists only one strictly positive solution to g(x 0 ) = x 0 then condition (3.36) is equivalent to 0 < g (x 0 ) < 1.

Using the proof of theorem 3.9 we prove there exists some bounds on n(t = 0, .) that gives the local exponential convergence. As we see it in section 5 the following result is sharp because for 'large variations' of the initial data, there can be oscillations.

Theorem 4.5. Assume (3.66) for α > 0, g belongs to C 1 ([0, ∞[), and there exists an x 0 > 0 such that g(x 0 ) = x 0 and

(4.8) 0 < g (x 0 ) < 1,
then there exists η + , η -> 0 such that

(4.9) x < g(x) < x 0 , ∀x ∈]x 0 (1 -η -), x 0 [, (4.10) x 0 < g(x) < x, ∀x ∈]x 0 , x 0 (1 + η + )[.

and

(4.11)

I := min(1 -sup 1≤z≤1+η+ g (zx 0 ), inf 1-η-≤z≤1 g (zx 0 ) -1) > 0.
Also for all m(0, .) that satisfies

(4.12) | m(0, .)/N (.) -1 | ∞ < η := min(η + , η -),
we have

(4.13) | m(t, .)/N (.) -1 | ∞ < η, ∀t ≥ 0,
and the local exponential convergence holds,

(4.14) ∞ 0 | m(t, y)-N (y) | φ(y)dy ≤ e -IαN (0)φ(0)t ∞ 0 | m(0, y)-N (y) | φ(y)dy.
Proof. The inequalities (4.9), (4.10) are just taylor formula. Now, we prove the L ∞ bound then we use the proof of theorem 3.9 to conclude.

Step1. Bound (4.13). Let r(t, y) = m(t, y)/N (y) then using the transport equation, we have,

r(t, y) = m(0, y)/N (y)1 t<y + g x 0 ∞ 0 r(t -y, y )dµ(y ) g(x 0 ) -1 1 y≤t ,
and r(t, y) := L(r)(t, y). But m(0, y)/N (y)

∈ {f :| f -1 | ∞ < η} and L : {f :| f -1 | ∞ < η} -→ {f :| f -1 | ∞ < η}, implies that r ∈ {f :| f -1 | ∞ < η} for all t ≥ 0.
Step2. Convergence. Then we adapt the proof of theorem 3.9, noticing that in this case J ≤ 1 + η + , J ≥ 1 -η -and J defined in (3.48) is equal to

(4.15) J = min(1 -sup 1≤z≤1+η+ z 1 g (sx 0 )ds z -1 , inf 1-η-≤z≤1 z 1 g (sx 0 )ds z -1 -1) > 0,
thus I ≤ J and (4.14) follows.

Remark 4.6. Notice that when g belongs to C 1 ([0, ∞[), is convex and decreasing, and there exists only one strictly positive solution to g(x 0 ) = x 0 then the inequalities given in remark 3.12 are equivalent to -1 < g (x 0 ) < 0.

Examples of oscillating solutions

In this part we give some examples where oscillations appear, i.e., there is no convergence of the solution to a steady state.

5.1. Only unstable steady states. When all the steady states are unstable and then a bounded solution to (1.6) is bounded then the solution to (1.6) can oscillate. We see in the proof of theorem 4.1 and theorem 4.2 that, in the neighborhood of the steady state N , the solution behaves as N + with (t, .) ∼ (.)e λt with λ given by ∞ 0 e -λy dµ(y ) =| 1/g (x 0 ) |, when | g (x 0 ) |< 1 or g (x 0 ) > 1. Now, we are interested in the case where g (x 0 ) < -1. The existence of oscillating solution needs more information on dµ, (5.1)

   g(x 0 ) = x 0 , g (x 0 ) < -1, p 0 = sup Re(p) {p ∈ C : B(p) = 1/g (x 0 )}, Re(p 0 ) ≥ 0.
t→∞ Ω(t) = 0, and x 0 is an unstable steady state.

Proof. We refer to [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF][START_REF] Iannelli | Mathematical Theory of Age-Structured Population Dynamics[END_REF] for the proof of this lemma which is based on the Laplace Transform.

We give an example of g such that all its steady states are 'unstable' and an example of dµ satisfying (5.1).

Example 5.2. Let g(z) = κz(1-z) + then the only steady states are {0, 1-1/κ} and g (z) = κ(1 -z) + -κz sgn + (1 -z), and g (0) = κ, g (1 -1/κ) = 2 -κ and all the steady states are 'unstable' when κ > 3.

The existence of a solution to (5.1) is not automatic. For instance, let dµ(y) = 1 [0,1] (y)dy and g (x 0 ) = -2 then (5.1) is not satisfied and converges to 0. , and

e -a = 1 cos(b) 1 - 1 κ ln - b κ sin(b)
.

For all κ > 1, there exists b ∈]3π/2, 2π[ such that - b κ sin(b) = 1 cos(b) 1 - 1 κ ln - b κ sin(b) ,
and for κ large enough 0 < -b κ sin(b) < 1 and a = -ln(-b κ sin(b) ) > 0.

Theorem 5.4. Assume g is bounded and all its fixed points satisfy

| g (x 0 ) |> 1,
and dµ satisfies (5.1) then if m(0, .) ∈ {fixed points of g} the solution cannot converge to a steady state.

Proof. Indeed, the solution of (2.6) exists and is bounded, more precisely we have m(t, y) ≤ sup g(z), ∀t < y, and m(t, y) = m(y-t), ∀t ≥ y. Now, assume that m(t, .) converges in L 1 ([0, ∞[, dµ) to m(.) then m is a fixed point of g. But in a neighborhood of m, we have | m-m |∼ (.)e λmt with λ m > 0 and thus m -m L 1 ([0,∞[,dµ) cannot converge to 0.

5.2. At most one unstable steady state. We focus on the cases where there exist almost one unstable steady state, for instance, let d = 0, B(y) = (1/2)1 y∈ [START_REF] Diekmann | On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory[END_REF][START_REF] Farkas | Stability Conditions for the Non-Linear McKendrick Equations[END_REF] and g

(x) = 2 if x ≥ 1, g(x) = 0 if x < 1 then m(t, y) = 2 ∞ i=-∞ 1 y-t∈[3i,3i+1] ,
= ∞ i=-∞ 4 2 1 t-y∈[3i,3i+1] dy = ∞ i=-∞ t-2 t-4 1 y∈[3i,3i+1] dy, but t-2 t-4 1 y∈[3i,3i+1] dy ≥ 1 iff [t -4, t -2] ⊃ [3i, 3i + 1],
1 y∈[3i,3i+1] dy ≥ 1 iff ∃i 0 s.t. [t -4, t -2] ⊃ [3i 0 , 3i 0 + 1].
Thus we have

g ∞ i=-∞ t-2 t-4 1 y∈[3i,3i+1] dy = 2 iff ∃i 0 , s.t. [t -4, t -2] ⊃ [3i 0 , 3i 0 + 1], g ∞ i=-∞ t-2 t-4 1 y∈[3i,3i+1] dy = 0 iff ∀i 0 [t -4, t -2] not contain [3i 0 , 3i 0 + 1],
finally we obtain that

(5.5) g ∞ i=-∞ t-2 t-4 1 y∈[3i,3i+1] dy = 2 ∞ i0=-∞ 1 [t-4,t-2]⊃[3i0,3i0+1] , (5.6) g
∞ i=-∞ t-2 t-4 1 y∈[3i,3i+1] dy = 2 ∞ i0=-∞ 1 t∈[3i0,3i0+1] , and (5.7) g
∞ 0 m(t, y)B(y)dy = m(t, 0), is a periodic solution that oscillates between the stable steady states 0 and 2. See Figure 2.

Here we prove that under the assumption of linearity of g around the unstable steady state and a condition over the Fourier transform of Be -D ,

F(Be -D ) = ∞ -∞
Be -D (y)1 y≥0 e -iξy dy, the unstable steady state generates a periodic solution to (1.6) (see [START_REF] Mischler | Stability in a Nonlinear Population Maturation Model[END_REF]).

Lemma 5.5. Assume g is linear in a neighbourhood of the unstable steady state, i.e., if n = N + with small enough then

F(˜ )(ξ) = R(ξ)F(˜ )(ξ), and
F(˜ )(ξ)(1 -R(ξ)
), thus using (5.8), we obtain that F(˜ )(ξ) = ηδ ξ0 (ξ), η small enough, is a periodic solution.

5.3. Only one locally stable steady state. The following example gives a local asymptotic convergence, i.e., a stable steady state in the case when g satisfies (4.2) and does not satisfy (3.36), and which has oscillating solutions. This proves that the smallness assumptions, in theorem 4.2 and so, in the Theorem 4.5 is necessary. We notice that the limit case where B(y) = δ y=1 gives the simplest oscillating solutions, indeed, for all function J such that J(t) = g(J(t -1)), then m(t, y) = J(t -y), is a solution to (2.6). Moreover, for all g that satisfies (4.2), g(1) = 1 = x 0 , g(1 + ) = 1 -and g(1 -) = 1 + , with 0 < < 1, then one can take for instance

J(t) := 1 + ∞ k=-∞ (-1) k 1 t∈]k,k+1[ ,
that gives a periodic solution. The following theorem gives an example of periodic solution for a g such that the only strictly positive steady state 1 is locally asymptotically stable and nevertheless there exists some oscillating solution. Let g satisfies (5.10)

g(x) = g > (1 + η)/(1 -η), ∀x < 1 -η, (5.11) g(x) = 0, ∀x > 1 + η, (5.12) g(x) = 1, 1 -η ≤ x ≤ 1 + η,
Theorem 5.6. Assume (5.10)-(5.12) for 0 < η < 1/2 and (5.13)

Supp B = K ⊂ [α, β],
with ∞ > α, β > 0 then for (see Figure 3) Proof. The idea of the proof is simple, we use the transport equation and its boundary condition to prove that after some times t 1 , t 2 , t 3 , t 4 , T the solution comes back to the initial state on y ∈ [0, β]. Indeed, consider the initial data (5.15).

For t 1 = 1 , such that, (5.17)

0 + g 1 = 1 -η, we have (5.18) n(t 1 , y) = g 1 [0,α+ 1] (y) + 1 [α+ 1,α+ 1+ 0] (y), y ≤ β. For t 2 = t 1 + 2 , such that, (5.19) 0 + g 1 + g 2 = 1 + η, we have (5.20) n(t 2 , y) = 1 [0, 2] + g 1 [ 2,α+ 1+ 2] (y) + 1 [α+ 1+ 2,α+ 0+ 1+ 2] (y), y ≤ β.
For t 3 = t 2 + β -3 , such that, (5.21)

2 + g 3 = 1 + η, we have (5.22) n(t 3 , y) = 1 [β-3-2,β-3] + g 1 [β-3,β] (y), y ≤ β.
for t 4 = t 3 + 4 , such that, (5.23)

2 + g 3 -g 4 = 1 -η, we have (5.24) n(t 4 , y) = 1 [0, 4] + 1 [β-4-2,β-4] + g 1 [β-4,β] (y), y ≤ β.
Finally, for T = t 4 + β -4 , we have

(5.25) n(T, y) = g 1 [0,α] (y) + 1 [α,α+ 4] (y), y ≤ β.
Using (5.21), (5.23), we have 4 = 2η/g and thus using (5.14), we have 0 = 4 and n(T, y) = n(0, y) for all y ≤ β.

Generalization

In this section we extend the GRE to the problem (1.7) and (1.8), using the same idea developed in section 4. with

               ∂ ∂y N (y) + d(y)N (t, y) = 0, N (0) = f ∞ 0 B(y)N (y)
-D H (n/N ) = N (0)φ(0) H( f β 0 ∞ 0 n N dµ B (y), β 1 ∞ 0 n N dµ θ (y ) f (β 0 , β 1 ) ) - ∞ 0 H( n N ) χ(y)N (y) N (0) dy .
Proof. Indeed, we have,

∂ ∂t

             ∃(α 0 , α 1 ) : 1 = f (xx0,yy0) f (x0,y0) , ∀(x, y) : α0x+α1y α0+α1 = 1, 1 < f (xx0,yy0) f (x0,y0) < α0x+α1y α0+α1 , ∀(x, y) : α0x+α1y α0+α1 > 1, α0x+α1y α0+α1 < f (xx0,yy0) f (x0,y0) < 1, ∀(x, y) : α0x+α1y α0+α1 < 1,
and we have the following theorem.

Theorem 6.2. Assume (6.7) and let The proof follows that of the theorem 3.9 and we do not reproduce it here.

Remark 6.3. Assume there exist α 0 , β 0 such that (6.9) Then we have the following property, here we only use the dissipation term associated to the age. Theorem 6.7. If additionally, we assume that for some µ > 0, we have B(y, x) ≥ αφ(y, x)/φ(0, x), ∀y, x ≥ 0, and

             ∃(α 0 , α 1 ) 1 = f (xx0,yy0) f (x0,
I = min 1 - sup x∈]-∞,∞[,z≥1 f (zm0(x)) f (m0(x)) -1 z -1 , inf x∈]-∞,∞[,z≤1 f (zm0(x)) f (m0(x)) -1 z -1 -1 > 0.
Moreover, if r = Iα inf x≥0 N (0, x) then we obtain The proof of this theorem follows that of theorems 3.9 and 3.10 and we leave it to the reader. The existence proof of (N, λ, φ) solution to (6.12) can be performed using Schauder's fixed point theorem (see [START_REF] Laurençot | Steady states for a fragmentation equation with size diffusion[END_REF][START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF][START_REF] Michel | Existence of a solution to the cell division eigenproblem[END_REF]).

0 B

 0 t, y) + ∂ ∂y n(t, y) + d(y)n(t, y) = 0, n(t, 0) = ∞ (y )n(t, y )dy ,

0 B

 0 t, y) + ∂ ∂y n(t, y) + d(y)n(t, y) = 0, n(t, 0) = f ( ∞ (y )n(t, y )dy ),

( 1

 1 t, y) + ∂ ∂y n(t, y) + d(y)n(t, y) = 0,n(t, 0) = f ( ∞ 0 B(y )n(t, y )dy , ∞ 0 θ(y )n(t, y )dy ),or for the following space-age model, t, y, x) + ∂ ∂y (V (y, x)n(t, y, x)) + d(y, x)n(t, y, x)-∂ ∂x ν(y, x) ∂ ∂x n(t, y, x) = 0, n(t, 0, x) = f ( ∞0b(y , x)n(t, y , x)dy ),

  dµ(y) := B(y)e -D(y) dy/β, where, and this is a first assumption, )e -D(y ) dy < ∞.

  )e -D(y ) dy /β = φ(0) ∞ y dµ(y), is well defined because we have ∞ 0 dµ(y) = 1 and satisfies (3.1). Theorem 3.1. Assume (2.4), m(0, .) ∈ L ∞ L 1 ([0, ∞[, φ(y)dy) for eq. (2.6) and (3.3) g(x) < x, g u.s.c., then N = 0 is a globally attractive steady state. We have ∞ 0 m(t, y)φ(y)dy decreases, (3.4) 0 ≤ m(t, .) ≤ m(0, .) ∞ ,

0 B 0 B

 00 ) g(y) ≤ γy, with γ := β ∞ 0 B(y )e -D(y )-λ0y dy > 0, then N = 0 is a stable steady state and there exists φ, uniquely defined up to a multiplicative constant, a solution to (3.18) -∂ ∂y φ + λ 0 φ(y) = φ(0)B(y)e -D(y) / ∞ (y )e -D(y )-λ0y dy , and for this φ we have (3.19) ∞ 0 m(t, y )φ(y )dy ≤ e λ0t ∞ 0 m(0, y )φ(y )dy . Proof. The φ are given by the following formula : (3.20) φ(y) := φ(0) ∞ y B(y )e -D(y )-λ0(y -y) dy / ∞ (y )e -D(y )-λ0y dy ,

  y)e -D(y) ) p/(p-1) ∞ y dµ(y ) 1/(p-1) dy < ∞. Remark 3.3. The condition (3.24) is satisfied for a large class of B and d. a) If Supp B is compact, max Supp B = y B and B(y) ∼ y→y B C r (y B -y) r where 0 < C r < ∞ and -1 < r < +∞ then (3.24) is satisfied for p > max(1, (r + 2)/(r + 1)). b) If B(y)e -D(y) ≤ C r e -ry where 0 < C r < ∞ and r > 0 then (3.24) is satisfied for all p > 1. More precisely we have Lemma 3.4. Assume (3.23) and (3.25)

∞ 0 (

 0 k -m(t, y)/N (y)) + N (y)φ(y)dy = 0, ∀t ≥ 0, and the conclusion follows. Now we prove the following convergence result, Theorem 3.9. Assume (2.4), (3.23), (3.24), (3.36), (3.43) m(0, .) ∈ L 1 ([0, ∞[, φ(y)dy) L p ([0, ∞[, φ(y)dy), p > p, with p defined in (3.24) and ∞ 0 m(0, y )φ(y )dy > 0, then (3.44) I := inf t≥0 ∞ 0 m(t, y )φ(y )dy > 0, (3.45) I := sup t≥0 ∞ 0 m(t, y )φ(y )dy < ∞.

  , y) -N (y) p φ(y)dy ≤ ∞ 0 m(0, y) -N (y) p φ(y)dy < ∞, thus using (3.43), (3.23) and (3.45), we obtain, ∞ 0 m(t, y) p φ(y)dy ≤ ∞ 0 N p (y)φ(y)dy y) -N (y) p φ(y)dy 1/p p < ∞, ∀t ≥ 0. (3.56) Moreover we have the uniform L p ([0, ∞[, φ(y)dy) bound (3.57) M p := sup t≥0 ∞ 0 m(t, y) p φ(y)dy < ∞.

∞ 0 m

 0 (t, y)dµ(y) = ∞ 0 m(t, y)B(y)e -D(y) /βdy = supp φ m(t, y)φ(y) 1/p B(y)e -D(y) βφ(y) 1/p dy ≤ ∞ 0 m(t, y) p φ(y)dy 1/p supp φ

1 /p ∞ 0 m 3

 103 p (t, y)φ(y)dy 1/q . (3.60) Therefore, using (3.57) and (3.23), for all 1 < p < p, we have t, y)φ(y)dy ≤ C p Thus, using (3.36) and (3.31) for H(z) = z, we have, d dt ∞ 0 m(t, y)φ(y)dy > 0, when ∞ 0 m(t, y)dµ(y) < 1, , and (3.44) follows with I ≥ min(C, ∞ 0 m(0, y)φ(y)dy) > 0. Step4. Convergence. Moreover, for H(z) =| 1 -z | p , D H = 0 if and only if m/N is a constant ρ(t), indeed using (3.58) and (3.61), we have (, y)dµ(y) < ∞, and

  y)e -D(y) /β ≥ αφ(y)/φ(0), ∀y ≥ 0, y)φ(y)dy.Proof. We prove the convergence (3.67), using the equation (2.6) and (3.2) introduced in section 3.1, we have (3.68) d dt ∞ 0 m(t, y)φ(y)dy = φ(0)m(t, 0) -φ(0) , y)φ(y)dy = ∞.

Theorem 4 . 1 .

 41 (Linearisation method) Assume there exists an unique x 0 > 0 such that g(x 0 ) = x 0 and (4.1)

0 e 0 | 0 | 0 | 0 |

 00000 φ(y) + λφ(y) = φ(0) | g (x 0 ) | B(y)e -D(y)/β, φ(y) ≥ 0, ∀y ≥ 0. Indeed, using (4.1), there exists λ < 0 s.t.∞ -λy dµ(y ) =| 1/g (x 0 ) |, and thus, for all φ(0) > 0, φ is uniquely defined andφ(y) = φ(0) | g (x 0 ) | ∞ y e -λ(y -y) dµ(y ).Then, we use the GRE and we find d dt∞ (t, y)e -λt | φ(y)dy := -D H (t),where-D H (t) = φ(0) | g (x 0 ) y)e -λt dµ(y) | -∞ (t,y)e -λt | dµ(y) ≤ 0, Therefore λ < 0 and using the decay of GRE we have ∞ (t, y) | φ(y)dy ≤ e λt ∞ (0, y) | φ(y)dy

  y )dµ(y ), -∂ ∂y φ(y) + λφ(y) = φ(0)g (x 0 )B(y)e -D(y)/β, (y), φ(y) ≥ 0, ∀y ≥ 0, ∞ 0 (y )φ(y )dy = 1.

Lemma 5 . 1 .

 51 Assume (5.1) then (5.2) (t, 0) = Ce p0t (1 + Ω(t)),where

Example 5 . 3 .

 53 Let dµ(y) = 1 [0,1] (y)dy and g (x 0 ) = -κ where κ is large enough then (5.1) is satisfied and oscillates. Indeed, it suffices to prove that (5.4) (1 -e -p )/p = -1/κ, have almost a solution with a strictly positive real part. Let p = a + ib, with a, b ∈] -∞, ∞[, (a, b) = (0, 0), then (5.4) becomes e -a = -b κ sin(b)

Figure 2 .

 2 Figure 2. transport of a periodic m(t, y)

  dµ(y ), and the Fourier transform of g ( ∞ 0 B(y )N (y )dy )B(y)e -D(y) /β given by R(ξ) := F(g ∞ 0 B(y )N (y )dy B(y)e -D(y) /β)(ξ), satisfies (5.8) ∃ξ 0 : R(ξ 0 ) = 1,then there exists an oscillating solution in a neighbourhood of the unstable steady state.

Figure 3 .( 5

 35 Figure 3. Example of g

( 5 .

 5 14) 0 = 2η/g , (5.15) n(0, y) = g 1 [0,α] (y) + 1 [α,α+ 0] (y), y ≤ β, there exists T > 0 such that (5.16) n(T, y) = g 1 [0,α] (y) + 1 [α,α+ 0] (y), y ≤ β.

6. 1 . 0 BTheorem 6 . 1 .

 1061 Generalized age model (1.7). dµ B (y) := B(y)N (y)dy/ ∞ (y )N (y )dy , and (6.4) dµ θ (y ) := θ(y)N (y)dy/ ∞ 0 θ(y )N (y )dy . Assume there exist a positive steady state to the problem (1.7) and its dual, (6.5)

  )N (y)dy , -∂ ∂y φ(y) + d(y)φ(y) = φ(0)χ(y), φ ≥ 0, for some function χ ≥ 0. Then for all positive convex function H, we have, (t, y)/N (y))N (y)φ(y)dy := -D H (n/N ),

∞ 0 H 0 H 0 H 0 H 0 H 0 H

 000000 (n(t, y)/N (y))N (y)φ(y)dy = ∞ (n(t, y)/N (y)) ∂ ∂t n(t, y)φ(y)dy, ∂ ∂t ∞ (n(t, y)/N (y))N (y)φ(y)dy = -∞ (n(t, y)/N (y))d(y)n(t, y)φ(y)dy -(t, y)/N (y))N (y)φ(y)dy = -∞ (n(t, y)/N (y))d(y)n(t, y)φ(y)dy +H(n(t, 0)/N (0))N (0)φ(0) -∞ 0 H(n(t, y)/N (y)) N (y)φ(0)χ(y) dy + ∞ (n(t, y)/N (y))φ(y)d(y)n(t, y)dy.Thus we have-D H (n/N ) = N (0)φ(0) using (6.1)-(6.2) we conclude the proof of the theorem. Now, we return to the longtime asymptotic. Assumption (3.36) is modified as (6.7)

(6. 8 ) 1 +

 81 χN (y) N (0) dy = (α 0 dµ B (y) + α 1 dµ θ (y))/(α 0 + α 1 ) ≥ 0, then the entropy associated toH(z) =| z -1 | is decreasing and n(t, .) → N (.) in L 1 ([0, ∞[×[0, ∞[, N (x, y)φ(x, y)dxdy).Proof. Using the theorem 6.1 and the equation (6.7), with H + (z) = (z -1) + and H -(z) = (z -1) -, we obtain-D H+ (n/N ) = N (0)φ(0) find -D H+ (n/N ) ≤ N (0)φ(0) χ(y)N (y) N (0)dy , but (6.8) implies that-D H+ (n/N ) ≤ N (0)φ(0)Proceeding analogously, we find that -D H-(n/N ) ≤ N (0)φ(0)

  y0) , ∀(x, y) : α0x+α1y α0+α1 = 1,1 < f (xx0,yy0) f (x0,y0) < 2 -α0x+α1y α0+α1 , ∀(x, y) : α0x+α1y α0+α1 < 1, 2 -α0x+α1y α0+α1 < f (xx0,yy0) f (x0,y0) < 1, ∀(x, y) : α0x+α1y α0+α1 > 1, with χN (y) N (0) dy = (α 0 dµ B (y) + α 1 dµ θ (y))/(α 0 + α 1 ) ≥ 0,then the entropy associated to H(z) =| 1 -z | decreases and we have the global convergence of n to N .Remark 6.4. The function f (x, y) = (x + y) ω with 0 < ω < 1 satisfies (6.7) with α = x 0 /(x 0 + y 0 ) and β = y 0 /(x 0 + y 0 ).Proof. Indeed, proceeding analogously to the theorem 6.1, we have d dt (y, x)N (y, x)φ(y, x)dydx, then using (6.10), (6.11), we easily complete the proof.Assumption (3.36) becomes (6.13)   ∀x ≥ 0, ∀z > 1, 1 < f (zm 0 (x))/f (m 0 (x)) < z,∀x ≥ 0, ∀0 < z < 1, 1 > f (zm 0 (x))/f (m 0 (x)) > z.

Theorem 6 . 6 .

 66 Assume y , x )φ(y , x )dy dx > 0, if there exists a solution to (6.12), then , y , x )φ(y , x )dy dx ≥ c. Moreover, if n(0, ., .)/N (., .) ∈ L p ([0, ∞[×[-∞, ∞[, N (y , x )φ(y , x )dy dx ) for some p ≥ 1 then we have (6.15) lim t→∞ n(t, .)/N (.) = 1, L p ([0, ∞[×[-∞, ∞[, N (y , x )φ(y , x )dy dx ).

  t, y , x ) -N (y , x ) | φ(y , x )dy dx ≤ y , x ) -N (y , x ) | φ(y , x )dy dx .

Then for all positive convex function H, we define,

and we have

) 2 ν(y, x)N (y, x)φ(y, x)dydx, where U (0, x) = N (0, x)V (0, x)φ(0, x).