Benoît Samuel Nordmann
email: samuel.nordmann@ehess.fr

Benoît Perthame
email: benoit.perthame@upmc.fr

Dynamics of concentration in a population structured by age and a phenotypic trait with mutations. Convergence of the corrector

Keywords: Adaptive evolution, Asymptotic behaviour, Dirac concentration, Hamilton-Jacobi equations, Mathematical biology, Renewal equation, Viscosity solutions, correctors AMS Class. No: 35B40, 35F21, 35Q92, 49L25

Dynamics of concentration in a population structured by age and a phenotypic trait with mutations. Convergence of the corrector

Introduction

Main results

We study a mathematical model describing the growth process of a population structured by age and a phenotypic trait, subject to aging, competition between individuals and mutations. Our goal is to describe the asymptotic behavior of the solution, in particular the selection of the fittest traits and the adaptative dynamics of such traits. For ε > 0, we choose m ε (t, x, y) to represent the population density of individuals who, at time t ≥ 0, have age x ≥ 0 and a quantitative phenotipic trait y ∈ R n , solution of a renewal type equation

               ε∂ t m ε + ∂ x [A(x, y)m ε ] + (ρ ε (t) + d(x, y)) m ε = 0,
A(0, y)m ε (t, 0, y) = 1 ε n R n R+ M ( y -y ε )b(x , y )m ε (t, x , y )dx dy , ρ ε (t) = R+ R n m ε (t, x, y)dxdy, m ε (t = 0, x, y) = m 0 ε (x, y) > 0.

(

) 1
The parameter ε > 0 stands for a hyperbolic rescaling (t, y) ↔ (ε -1 t, ε -1 y) in the mutation term. Our main concern is to study the asymptotics of m ε when ε → 0. This work is the continuation of the study begun in [START_REF] Nordmann | Dynamics of Concentration in a Population Model Structured by Age and a Phenotypical Trait[END_REF], where the model without mutations is studied and where we proved that there is a measure µ (typically a Dirac mass) and a bounded profile Q such that m ε (t, x, y) Q(t, x, y) µ(t, y), in other words, the asymptotic singularity is carried in the variable y only. In the present work, the mutation term in the second line of (1) adds a significant difficulty because the profile Q turns out to be strongly related to the limit of the corrector in the spectral problem defining the underlying effective Hamiltonian, i.e., the effective fitness in our context, arising in a Hamilton-Jacobi equation which defines the singular part.

While the classical approach consists in studying the asymptotics of v ε (t, x, y) := ε ln(m ε (t, x, y)), here, and following [START_REF] Nordmann | Dynamics of Concentration in a Population Model Structured by Age and a Phenotypical Trait[END_REF][START_REF] Perthame | Rare Mutations Limit of a Steady State Dispersal Evolution Model[END_REF], we define a priori some kind of variable separation setting m ε (t, x, y) := p ε (t, x, y)e Uε(t,y)-t 0 ρε(•) ε

where the exponential term carries the singular part of the limiting population density and U ε is defined autonomously through a Hamilton-Jacobi equation (see [START_REF] Barles | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF]) involving the effective fitness Λ (defined in [START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adpative dynamics[END_REF]). The main finding of this paper is the proof that, with our choice of U ε , the corrector p ε is uniformly bounded. We also show that p ε converges to a multiple of the principal eigenfunction Q(t, x, y) of the formal limiting operator (defined in [START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adpative dynamics[END_REF]). It justifies that the ansatz U ε is appropriate and carries all the information on the singular behavior of m ε when ε → 0. The formal idea of the method is explained with more details in section 1.3 and consists in keeping as simple as possible the Hamilton-Jacobi equation which defines U ε while all the functional analytic difficulties are carried by the corrector p ε which satisfies a linear equation.

The first step is to define U ε and study its properties.

Theorem 1.1 (Convergence of U ε ) Under the assumptions of section 1.5, when ε → 0, the ansatz U ε (t, y) is well defined through [START_REF] Barles | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF] and converges in W 1,r loc , 1 ≤ r < ∞, to some U ∈ W 1,∞ loc which is semi-convex and is a viscosity solution of the Hamilton-Jacobi equation (1.3).

The second step is to prove that the corrector is uniformly bounded and converges. Theorem 1.2 (Convergence of the corrector) Under the assumptions of section 1.5, for any fixed T > 0, p ε (t, x, y) and x>0 p ε (t, x, y)dx are bounded from above and below, uniformly in ε > 0, t ∈ [0, T ], y ∈ R n .

In addition, for t ∈ [0, T ], x ∈ [0, x], y ∈ R n , up to extraction of a subsequence ε → 0, p ε converges in L ∞ -weak-to γ(t, y)Q(t, x, y), where Q is the eigenfunction defined in [START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adpative dynamics[END_REF] and γ ∈ L ∞ formally satisfies [START_REF] Lorz | Dirac mass dynamics in multidimensional nonlocal parabolic equations[END_REF].

With these informations, it is standard that the concentration points of the population density m ε (t, x, y) are carried by the set S := t ≥ 0, y ∈ R n : U (t, y) = sup y ∈R n U (t, y ) .

(

) 2
Theorem 1.3 (concentration) Under the assumptions of section 1.5, when ε → 0 1. The total population ρ ε converges weakly to some positive ρ ∈ L ∞ and ∀t > 0,

t 0 ρ = sup y∈R n U (t, y).
2. The population m ε vanishes locally uniformly outside the set S.

3. Under further assumptions on the initial conditions, and for small times t ∈ [0, T ], we have

S = {(t, ȳ(t))},
where ȳ(t) follows the Canonical Equation

   d dt ȳ(t) = D 2 y U (t, ȳ(t)) -1 • ∇ y Λ(ȳ(t), 1) + ∂ η Λ(ȳ(t), 1) R n M (z)zdz, ȳ(0) = ȳ0 .
Notice that we establish convergence of the full families U ε and ρ ε , without using the famous uniqueness result of [START_REF] Calvez | Uniqueness of the viscosity solution of a constrained Hamilton-Jacobi equation[END_REF], because of the simple dependency on the unknown ρ ε in our setting. We point out that the main restriction on the coefficients is the assumption that the transport in x outwards the support of b(•, y) occurs in finite time, see [START_REF] Bouin | Thin front limit of an integro-differential Fisher-KPP equation with fat-tailed kernels[END_REF]. The other assumptions, detailed in section 1.5, are formulated directly on the limiting eigenproblem and are quite general.

The model

Some possible biological interpretations of the model ( 1) are as follows. The function A(x, y) is the speed of aging of individuals with age x and phenotypic trait y. The total size of the population at time t is denoted with ρ ε (t). In the mortality term, d(x, y) > 0 represents intrinsic death rate and the nonlocal term ρ ε (t) represents competition. The condition at the boundary x = 0 describes the birth of newborns that happens with rate b(x, y) > 0 and with the probability kernel M for mutations.

The terminology of renewal equation comes from this boundary condition. It is related to the McKendrickvon Foerster equation which is only structured in age (see [START_REF] Perthame | Transport equations in biology[END_REF] for a study of the linear equation). This model has been extended with other structuring variables, as size [START_REF] Metz | The dynamics of physiologically structured volume 68 of Lecture Notes in Biomathematics[END_REF][START_REF] Mischler | Stability in a nonlinear population maturation model[END_REF], DNA content, maturation, etc., in the context of cell divisions [START_REF] Doumic | Eigenelements of a general aggregation-fragmentation model[END_REF], or proliferative and quiescent states of tumor cells [START_REF] Adimy | A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia[END_REF][START_REF] Gyllenberg | A nonlinear structured population model of tumor growth with quiescence[END_REF]. Space structured problems have also been extensively studied [START_REF] Jabin | Selection-Mutation dynamics with spatial dependence[END_REF][START_REF] Mirrahimi | Adaptation and migration of a population between patches[END_REF][START_REF] Mirrahimi | Asymptotic analysis of a selection model with space[END_REF][START_REF] Perthame | Rare Mutations Limit of a Steady State Dispersal Evolution Model[END_REF].

To keep the model (1) quite general, we have allowed the progression speed A to depend on x. Thus, although the variable x is referred to as age, it can represent other biological quantities that evolve throughout the individual lifespan such as, for instance, the size of individuals, a physiological age, a parasite load, etc.

The rescaling parameter ε > 0 comes from a hyperbolic rescaling of t and y. Accordingly, the dynamics are considered in two different time scales. The first one is the individual lifetime scale εt, i.e., the characteristic time for the population to reach the dynamical equilibrium for a fixed y. The second one is the evolutionary time scale t, corresponding to the evolution of the population distribution with respect to the variable y. Formally, at the limit when ε → 0, the time scales are completely separated. This rescaling is a classical way to give a continuous formulation of the adaptive evolution of a phenotypically structured population (see [START_REF] Champagnat | The canonical equation of adaptive dynamics: A mathematical view[END_REF][START_REF] Diekmann | A beginner's guide to adaptive dynamics[END_REF][START_REF] Diekmann | The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach[END_REF][START_REF] Lorz | Dirac mass dynamics in multidimensional nonlocal parabolic equations[END_REF]). Note that the mutation kernel is supposed to be thin-tailed, i.e., it decreases faster than any exponential. A fat-tailed kernel needs a different rescaling, see [START_REF] Bouin | Thin front limit of an integro-differential Fisher-KPP equation with fat-tailed kernels[END_REF].

From the modelization point of view, Theorem 1.3 is a form of mathematical formulation of Natural Selection and Evolution. On an ecological time scale, only the phenotype ȳ(t) which maximizes the ecological fitness U (t, •) can survive. On an evolutionary time scale, we observe the dynamics of ȳ(t). Similar results as Theorem 1.3 have been obtained for various models with parabolic equations [START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result[END_REF][START_REF] Barles | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF][START_REF] Lorz | Dirac mass dynamics in multidimensional nonlocal parabolic equations[END_REF] and integrodifferential equations [START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adpative dynamics[END_REF][START_REF] Desvillettes | On selection dynamics for continuous structured populations[END_REF][START_REF] Lorenzi | Asymptotic Dynamics in Populations Structured by Sensitivity to Global Warming and Habitat Shrinking[END_REF]. More generally, convergence to positive measures in selection-mutation models has been studied by many authors [START_REF] Ackleh | Rate distributions and survival of the fittest: a formulation on the space of measures[END_REF][START_REF] Busse | Mass concentration in a nonlocal model of clonal selection[END_REF][START_REF] Calsina | Steady states of a selection-mutation model for an age structured population[END_REF]. The special case of age-structured populations are also considered in [START_REF] Calvez | Limiting Hamilton-Jacobi equation for the large scale asymptotics of a subdiffusion jump-renewal equation[END_REF][START_REF] Méléard | Trait Substitution Sequence process and Canonical Equation for agestructured populations[END_REF][START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF].

A formal presentation of the method

To analyze the singular perturbation problem at hand, the usual approach relies on the WKB change of unknown ( [START_REF] Barles | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF][START_REF] Diekmann | The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach[END_REF]), which consists in the change of variable m ε (t, x, y) = e vε(t,x,y) ε

. In the context of concentrations, this form is motivated by the heuristics that a Dirac mass is nothing but a narrow Gaussian. Indeed, in a weak sense (πε)

-n 2 e -y-ȳ 2 ε δ ȳ=y as → 0. This approach has been extensively used in works on a similar issue, see for instance [START_REF] Barles | Wavefront propagation for reaction diffusion systems of PDE[END_REF][START_REF] Barles | Front propagation for reaction-diffusion equations arising in combustion theory[END_REF][START_REF] Champagnat | The evolutionary limit for models of populations interacting competitively via several resources[END_REF][START_REF] Evans | A PDE approach to geometric optics for certain semilinear parabolic equations[END_REF]. With this change of unknown, at the limit ε → 0, the function v ε (t, x, y) satisfies a constrained Hamilton-Jacobi equation, on which estimates can be difficult to prove because it carries all the difficulties in the asymptotic analysis, concentration effect and profile defined by the corrector. For that reason, the perturbed test function method has been invented [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF] and widely used, which avoids computing the corrector.

Here, we propose a variant of the method. The principle can be viewed as a Taylor expansion v ε (t, x, y) = v 1 ε (t, y) + εv 2 ε (t, x, y), to choose only some convenient terms to define the Hamilton-Jacobi equation for v 1 ε , and then to prove that the corrector v 2 ε is bounded. With a slight rewriting, we proceed to the change of variable

m ε (t, x, y) = p ε (t, x, y) exp U ε (t, y) - t 0 ρ ε (s)ds ε (3)
where U ε (t, y) is defined ad hoc through a standard Hamilton-Jacobi equation for which classical regularity properties can be proved. Then,the new and difficult step is to prove estimates on the corrector p ε (t, x, y). Note that p ε satisfies a linear equation rather than a constrained Hamilton-Jacobi equation: it makes thus possible to use classical comparison principles and ideas issued from the General Relative Entropy method (see [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]).

We are now left with the task of finding a good candidate for U ε (t, y) and formally identifying p ε (t, x, y). Injecting (3) in (1), we find

   ε∂ t p ε + ∂ x [A(x, y)p ε ] + d(x, y)p ε + ∂ t U ε (t, y)p ε = 0, A(0, y)p ε (t, 0, y) = 1 ε n R n R+ M ( y -y ε )e Uε(t,y )-Uε(t,y) ε b(x , y )p ε (t, x , y )dx dy . (4)
With the change of variable z = y -y ε , the renewal term becomes

A(0, y)p ε (t, 0, y) = R n R+ M (z)e Uε(t,y+εz)-Uε(t,y) ε b(x , y + εz)p ε (t, x , y + εz)dx dz.
When ε is small, we can formally approximate

A(0, y)p ε (t, 0, y) ≈ η ε (t, y) R+ b(x , y)p ε (t, x , y)dx , (5)
where

η ε (t, y) := R n M (z)e
Uε(t,y+εz)-Uε(t,y) ε dz.

Then, formally putting ε∂ t p ε = O(ε) in the first line of (4), we end up with the following approximate problem

   ∂ x [A(x, y)p ε ] + d(x, y)p ε + ∂ t U ε (t, y)p ε = O(ε),
A(0, y)p ε (t, 0, y) = η ε (t, y) R+ b(x , y)p(t, x , y)dx + O(ε). Considering η ε (t, y) as a parameter, we introduce the following eigenproblem: for fixed (y, η) ∈ R n × (0, +∞), find (Λ(y, η), Q(x, y, η)), solution of

       ∂ x [A(x, y)Q] + d(x, y)Q -Λ(y, η)Q = 0, ∀x > 0, A(0, y)Q(0, y, η) = η, Q > 0, R+ b(x, y)Q(x, y, η)dx = 1. (6)
The third line corresponds to a normalization of the eigenfunction which is convenient for future calculations. Formally, Λ corresponds to the effective fitness, and Q to the age profile at equilibrium in an environment characterized by the parameters (y, η).

In an attempt to indentify p ε with Q, this formal approach suggests to define U ε as a solution of the Hamilton-Jacobi equation

     ∂ t U ε (t, y) = -Λ y, R n M (z)e Uε(t,y+εz)-Uε(t,y) ε dz ∀t ≥ 0, y ∈ R n , U ε (0, y) = U 0 ε (y) ∀y ∈ R n , (7)
for some initial conditions U 0 ε . We stress out that, when ε vanishes, the full term exp Uε(t,y)-t 0 ρε(s)ds ε represents a bounded measure, e.g., a Dirac mass as in the example of Gaussian concentration, and thus

sup y∈R n U ε (t, y) - t 0 ρ ε (s)ds = 0, ∀t ≥ 0,
which explains the first formula in Theorem 1.3. However, because of the non-local term η ε , proving uniform in ε > 0 estimates on U ε is quite technical, a fact that can also be seen on the limiting equation when ε → 0. Taking for granted that U ε converges to a function U locally uniformly, U turns out to be a viscosity solution of the Hamilton-Jacobi equation

∂ t U (t, y) = H(y, ∇ y U ) ∀t ≥ 0, ∀y ∈ R n , U (0, y) = U 0 (y) ∀y ∈ R n . ( 8
)
with the Hamiltonian H(y, p) defined by

H(y, p) := -Λ y, R n M (z)e
∂ 2 t U = -∂ t η ∂ η Λ, we infer that ∂ t η ∈ L 1 loc .
In the sequel, all these estimates are proved on U ε , uniformly in ε > 0. These are the classical (and sharp) general estimates for Hamilton-Jacobi equations with convex Hamiltonians.

With these optimal estimates on U ε in hands, we can bound the corrector p ε . We set

Q ε (t, x, y) := Q(x, y, η ε (t, y)), Λ ε (t, y) := Λ(y, η ε (t, y))
and find that

   ε∂ t Q ε + ∂ x [A(x, y)Q ε ] + d(x, y)Q ε + ∂ t U ε (t, y)Q ε = ε∂ t Q ε , A(0, y)Q ε (t, 0, y) = 1 ε n R n R+ M ( y -y ε )e
Uε(t,y )-Uε(t,y) ε b(x , y )Q ε (t, x , y )dx dy .

Note that the boundary term at x = 0 is obtained using the definition of η ε (t, y) and the normalization R+ b(x, y)Q(x, y, η)dx = 1 for all (y, η).

The right hand side of the first line can be controlled with the available a priori bounds on U ε . Indeed, integrating equation [START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adpative dynamics[END_REF] to obtain AQ, we compute

∂ t Q ε = ∂ t η ε (t, y)∂ η Q(x, y, η ε (t, y)) = ∂ t η ε 1 η ε + ∂ η Λ(y, η ε ) x 0 1 A(•, y) Q ε .
Except from this term, we see that Q ε and p ε satisfy the same equation ( 4), which is linear and admits a comparison principle. Under assumptions on the initial conditions, we deduce that p ε is bounded from above and below by multiples of Q ε . Passing to the limit in the equation, we prove that p ε converges weakly to a multiple of Q. It justifies our approach, especially the approximation (5), and formally proves Theorem 1.2. Now, having in hand that U ε converges (Theorem 1.1) and that p ε is uniformly bounded (Theorem 1.2), Theorem 1.3 can be understood and formally justified as follows. On the one hand, the saturation term "ρ ε (t)" in (1) implies the total population ρ ε to be bounded, uniformly in ε > 0. On the other hand, from (3), the asymptotics of m ε (t, •, •) when ε vanishes are driven by the points y where U (t, •) is maximal. In other words, when ε → 0, m ε vanishes outside the set S defined in [START_REF] Adimy | A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia[END_REF]. Then, we can study the evolutionary dynamics through the dynamics of the (unique) critical point of U (t, •).

Outline of the paper

Section 2 is devoted to the definition of the eigenelements (Q, Λ), the definition of the ansatz U ε , the statement of a priori estimates, and the asymptotics of U ε when ε → 0. The proofs are postponed to section 5. In section 3, we study the corrector p ε and prove Theorem 1.2. Next, we study the asymptotics of the population density, and we prove Theorem 1.3 in section 4. Finally, some longer or more technical proofs are gathered in section 5.

Assumptions

Most of our assumptions are formulated directly on the solution (Λ(y, η), Q(x, y, η)) of the limiting eigenproblem ( 6) and, therefore, may seem abstract to the reader. However, we think that, besides being quite general, this formulation gives a better insight into the nature of our assumptions and the spirit of our approach.

Example

Before stating the general assumptions, we first give for the reader's convenience a concrete set of assumptions on the coefficients A, b, d which are sufficient to fulfil the general assumptions. Note that, besides, we need the initial conditions to be "well prepared," which is not detailed in this example.

To avoid any difficulty when |y| → +∞, we can assume for example that A, b and d have a compact dependence on y. Namely, if ψ is a globally smooth diffeomorphism from R n into the unit ball, we assume A(x, y) := A (x, ψ(y)), b(x, y) := b (x, ψ(y)), d(x, y) := d (x, ψ(y)), where A , b and d are defined on the closed unit ball. This way, the y space R n is compactified. Then, the coefficients can be chosen to fullfill the following conditions, for all x ≥ 0 and uniformly in y ∈ B 1 ,

               A (•, •), b (x, •) ≥ 0, d (x, •) ≥ 0 are C 1 , A (x, y) ≥ A > 0, 1 A (•, y) is integrable on the support of b (•, y), b (x, y) ≤ Ke Kx , ηb (x, y) -d (x, y) ≥ r, M (•) is a Gaussian probability kernel,
for some constants K > 0, r > 0 and where η is determined from the initial conditions, see [START_REF] Champagnat | The canonical equation of adaptive dynamics: A mathematical view[END_REF]. These assumptions can be substantially generalized. For instance, using the formula (obtained by integrating the first line in (6))

Λ(y, η) = R+ (d(x, y) -ηb(x, y)) Q(x, y, η)dx R+ Q(x, y, η)dx ,
the relation between b and d is only used to ensure the inequality

Λ(y, η) ≤ -r < 0, ∀η ≥ η
which is sufficient in the sequel, see [START_REF] Champagnat | The canonical equation of adaptive dynamics: A mathematical view[END_REF].

Assumptions on the coefficients

We assume that for some

A > 0 b ≥ 0, d ≥ 0, A ≥ A > 0 are continuous functions, (9)
A(•, •), b(x, •), d(x, •) are C 1 ,
d(•, y), A(•, y) are bounded from above and below in some interval of R + , uniformly in y. [START_REF] Bouin | Thin front limit of an integro-differential Fisher-KPP equation with fat-tailed kernels[END_REF] which can be viewed as a non-degeneracy condition. Regarding the mutation kernel, we assume that M (•) is a probability kernel and vanishes faster than any exponential.

For instance, M (•) can be a Gaussian distribution or have a compact support. Note that the case without mutations corresponds to M = δ 0 and has been already treated in [START_REF] Nordmann | Dynamics of Concentration in a Population Model Structured by Age and a Phenotypical Trait[END_REF].

In addition, we assume

∃K > 0 such that sup y∈R n x 0 1 A(x , y) dx ≤, K (11)
where x defines the largest support of b(x, y),

x := sup {x ≥ 0 : ∃y ∈ R n , b(x, y) > 0} ∈ [0, +∞].
This assumption means that the transport outwards the support of b occurs in finite time (it can be seen on the characteristics). It is a necessary and sufficient condition for the ratio ∂ηQ Q to be bounded on [0, x], which is used in section 3.1 to prove an estimate on p ε . Our approach would also work if the support of A(•, y) is compact, but we omit this case for simplicity.

We also need the eigenvalue Λ of ( 6) to be well defined and differentiable (w.r.t η). As we see it in Proposition 2.1,defining,

F (y, λ) := R+ b(x, y) A(x, y) exp x 0 λ -d(x , y) A(x , y) dx dx, λ ∈ R, (12)
the eigenvalue Λ(y, η) is defined through the relation

F (y, Λ(y, η)) = 1 η .
We assume there exists Λ < 0 such that

F (y, Λ), ∂ λ F (y, Λ) < +∞ for all y ∈ R n .
This assumption is not very restrictive, it is satisfied if b(x, y) ≤ K e Kx choosing Λ ≤ -K.

Assumptions on the initial conditions.

We need the population m ε (t, x, y) to be "well prepared" for concentration. We write m 0 ε (x, y) = p 0 ε (x, y)e U 0 ε (y) ε according to (3), and we assume that U 0 ε smoothly converges to a function U 0 when ε vanishes,

∃k 0 > 0 such that ∀ε > 0, ∀y ∈ R n , |∇ y U 0 ε (y)| ≤ k 0 , (13)
∃C > 0 such that ∀ε > 0, ∀y ∈ R n , ∂ 2 yi U 0 ε (y) ≥ -C, (14)
J 0 ≤ R n e U 0 ε (y) ε dy ≤ J 0 , for some J 0 , J 0 > 0, (15)
and p 0 ε such that, for some γ 0 , γ 0 > 0:

γ 0 ≤ p 0 ε (x, y) Q(x, y, η 0 ε (y)) ≤ γ 0 , ( 16
)
where Q is defined through (6) and we define

η 0 ε (y) := z∈R n M (z)e U 0 ε (y+εz)-U 0 ε (y) ε dz, Λ 0 ε (y) := Λ(y, η 0 ε (y)). ( 17
)
Another condition is also required on η 0 ε , see [START_REF] Champagnat | The canonical equation of adaptive dynamics: A mathematical view[END_REF]. Note that ( 13) and ( 14) ensures that ρ 0 ε := R+×R n m 0 ε is uniformly bounded. Note also that assumption [START_REF] Calvez | Limiting Hamilton-Jacobi equation for the large scale asymptotics of a subdiffusion jump-renewal equation[END_REF] implies sup R n U 0 = 0.

Assumptions on the distribution of phenotypes

The following assumptions only deal with the dependence of the coefficients on y ∈ R n . In particular, if all the coefficients have a compact dependence on y, then all the following assumptions are automatically satisfied.

First, we need a condition on the initial data, namely that, from (15), η 0 ε (y) is bounded and Λ 0 ε (y) is bounded and negative. More precisely, in accordance to section 1.5.2, we assume that there are two negative constants Λ ≤ Λ < 0 and two positive constants 0 < η ≤ η, such that for all y ∈ R n ,

1 η ≤ F (y, Λ) ≤ F (y, Λ 0 ε (y)) := 1 η 0 ε (y) ≤ F (y, Λ) ≤ 1 η . ( 18
)
This assumption implies η ≤ η 0 ε (y) ≤ η and Λ ≤ Λ 0 ε (y) ≤ Λ (since λ → F (y, λ) is increasing). We will see in Corollary 2.3 that those two inequalities hold for all times, namely η ≤ η ε (t, y) ≤ η and Λ ≤ Λ(y, η ε (t, y)) ≤ Λ. Note that with the notations 1 η(y)

:= F (y, Λ), 1 η(y) := F (y, Λ), (19)
assumption ( 16) can be written η ≤ η(y) ≤ η 0 ε (y) ≤ η(y) ≤ η. We stress out that this assumption implies -Λ(y, η ε (t, y)) > 0. It means that every phenotype has a positive fitness, and is thus able to survive in absence of other phenotypes. This assumption is somehow restrictive, but it is not irrealistic, and allows to avoid some technicalities.

Finally, the next assumptions deal with the derivatives of Λ. We need ∂ η Λ, ∇ y Λ and ∇ y ∂ η Λ to be bounded, and Λ to be semi-convex. According to [START_REF] Evans | Partial differential equations[END_REF], we assume that there exists two constants l, L > 0 such that for all

y ∈ R n , λ ∈ [Λ, Λ], l ≤ ∂ λ F (y, λ) ≤ L, (20)
|∇ y F (y, λ)|, |∇ y ∂ λ F (y, λ)| ≤ L, (21) ∀i
∈ {1, . . . , n}, ∂ 2 yi F (y, λ) ≥ -L. ( 22
)
2 Definition and properties of U ε

To make sense to the above heuristic, we first give a rigorous definition of the eigenelements (Λ, Q), which only uses classical arguments. Then, we define U ε , formally introduced in section 1.3, and state some a priori estimates. We use those results to derive estimates on η ε and Λ ε . Note that the study of U ε is autonomous and can be done separately from the analysis of the corrector. Finally, we pass to the limit, as ε → 0, in the quantities U ε , η ε and Λ ε to recover a viscosity solution of the Hamilton-Jacobi equation. The longer or more technical proofs are postponed to section 5.

The eigenproblem and effective Hamiltonian

We consider the limiting problem [START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adpative dynamics[END_REF] and prove the existence of the eigenelements (Λ, Q), along with some properties. The proof only uses elementary arguments and is postponned to section 5.1.

Proposition 2.1 Under the assumptions of section 1.5, for fixed y ∈ R n and η ∈ (η(y), η(y)) (from (1.5.4)), there exists a unique couple (Λ(y, η), Q(x, y, η)) which satisfies (6). Moreover, with F defined in [START_REF] Busse | Mass concentration in a nonlocal model of clonal selection[END_REF], Λ and Λ defined in [START_REF] Champagnat | The canonical equation of adaptive dynamics: A mathematical view[END_REF], the eigenvalue Λ(y, η) is continuously differentiable and it holds, for all y ∈ R n , η ∈ (η(y), η(y)),

F (y, Λ(y, η)) = 1 η , (23)
Λ ≤ Λ(y, η) ≤ Λ < 0, (24)
Q(x, y, η) = η 1 A(x, y) exp x 0 Λ(y, η) -d(x , y) A(x , y) dx . ( 25
)
Proof See section 5.1.

Differentiating the relation [START_REF] Diekmann | The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach[END_REF], we immediately obtain the derivatives of Λ

∂ η Λ(y, η) = -1 η 2 ∂ λ F (y, Λ(y, η)) ≤ -1 η 2 L , ∇ y Λ(y, η) = -∇ y F (y, Λ(y, η)) ∂ λ F (y, Λ(y, η)) . (26)
In particular, the property ∂ η Λ < 0, turns out to be fundamental in the following section. We also have some kind of concavity property for Λ. For all y ∈ R n , η ∈ (η(y), η(y)), we have

∂ 2 η Λ(y, η) + ∂ η Λ(y, η) η ≤ 0. ( 27
)
Proof We use the short notations F := F (y, Λ(y, η)) (where F is defined in [START_REF] Busse | Mass concentration in a nonlocal model of clonal selection[END_REF]) and Λ := Λ(y, η).

Differentiationg the first relation in [START_REF] Evans | Partial differential equations[END_REF] with respect to η, we find

∂ 2 λ F (∂ η Λ) 2 + ∂ λ F ∂ 2 η Λ = 2 η 3 , ∂ 2 η Λ = 2 η 3 ∂ λ F - 1 η 4 (∂ λ F ) 3 ∂ 2 λ F.
Combining it with the expression of ∂ η Λ(y, η) in ( 23) and ( 20), we deduce

∂ 2 η Λ + ∂ η Λ η = 1 (∂ λ F ) 3 η 3 (∂ λ F ) 2 -F ∂ 2 λ F .
Using the Cauchy-Schwarz inequality and the definition of F in [START_REF] Busse | Mass concentration in a nonlocal model of clonal selection[END_REF], we have

(∂ λ F ) 2 -F ∂ 2 λ F ≤ 0
and the proof of ( 24) is completed.

Construction of U ε and a priori estimates

We give a rigourous definition of U ε , formally introduced in (7).

Proposition 2.2 Under the assumptions of section 1.5, for all ε > 0 there exists a solution U ε (t, y) of [START_REF] Barles | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF].

In addition, it satisfies, with Λ, Λ < 0 defined in [START_REF] Champagnat | The canonical equation of adaptive dynamics: A mathematical view[END_REF],

-Λ ≤ ∂ t U ε (t, y) ≤ -Λ, ∀ε > 0, t ≥ 0 y ∈ R n .
Proof See section 5.2.

In fact, U ε is the unique solution of (7) with a locally bounded time derivative. As a direct consequence of Proposition 2.2, we deduce the following useful corollary.

Corollary 2.3 With Λ, Λ, η, η defined in [START_REF] Champagnat | The canonical equation of adaptive dynamics: A mathematical view[END_REF], and setting

η ε (t, y) := R n M (z)e Uε(t,y+εz)-Uε(t,y) ε dz, Λ ε (t, y) := Λ(y, η ε (t, y)), (28)
we have

η ≤ η ε (t, y) ≤ η, Λ ≤ Λ ε (t, y) ≤ Λ < 0. ( 29
)
Proof Simply use Proposition 2.2, Λ ε = -∂ t U ε and assumption (16).

Further estimates

Proposition 2.4 Under the assumptions of section 1.5, with k 0 defined in (1.5.3), L, l > 0 in [START_REF] Champagnat | The evolutionary limit for models of populations interacting competitively via several resources[END_REF] and η in (26), we have

|∇ y U ε (t, y)| ≤ k 0 + L lη 2 t, ∀ε > 0, t ≥ 0, y ∈ R n .
Proof See section 5.3.

Note that the coefficient L lη 2 comes from a bound on |∇ y Λ(y, η ε )|. We will see that, at the limit when ε → 0, we can prove Lipschitz continuity globally in time.

We also need the following control of second order derivatives.

Proposition 2.5 (Semi-convexity) The function U ε is semi-convex in (t, y), that is, for all the ν ∈ S := {(t, y) ∈ R n+1 : t 2 + |y| 2 = 1}, ∂ 2 νν U ε are bounded from below, uniformly in ε > 0, y ∈ R n , locally uniformly in t ≥ 0. Therefore U ε belongs to W 2,1 loc in (t, y), uniformly in ε > 0.
Proof The idea is to use that the Hamiltonian has properties closely related to convexity, namely [START_REF] Evans | A PDE approach to geometric optics for certain semilinear parabolic equations[END_REF] and to use the Lipschitz bounds. See section 5.4.

The following corollary is essential when studying the corrector in section 3.

Corollary 2.6

We have η ε ∈ W 1,1 loc , uniformly in ε > 0. In addition, denoting η(t) := sup

y∈R n η ε (t, y), we have T 0 |∂ t η ε (t)| dt is bounded uniformly in ε > 0, ∀T ≥ 0.
Proof See section 5.5.

Asymptotics

With these regularity properties, we are ready to establish the asymptotics of U ε when ε vanishes.

Proposition 2.7 Under the assumptions of section 1.5, when ε vanishes, U ε converges locally uniformly (and in

W 1,r loc , 1 ≤ r < ∞) to a function U (t, y) ∈ W 1,∞ loc which is a semi-convex viscosity solution of      ∂ t U (t, y) = -Λ y, R n M (z)e ∇yU (t,y)•z dz , ∀t > 0, ∀y ∈ R n , U (0, y) = U 0 (y), ∀y ∈ R n . (30)
Moreover, under the assumption that M is not degenerate (i.e., M (•) > 0 in a neighborhood of 0), the function U (t, y) is globally Lipschitzian.

Proof See section 5.6 and section 5.7.

We also point out that, from Proposition 4.3 in [START_REF] Nordmann | Dynamics of Concentration in a Population Model Structured by Age and a Phenotypical Trait[END_REF],

p → -Λ y, R n M (z)e p•z dz is a convex mapping, ∀y ∈ R n .
This class of Hamiltonian has been widely studied, and numerous results on regularity as well as representation formula are available [START_REF] Bianchini | SBV regularity for Hamilton-Jacobi equations with Hamiltonian depending on (t,x)[END_REF][START_REF] Fleming | Deterministic and Stochastic Optimal Control[END_REF].

As a direct consequence of the L r loc convergence of ∇U ε to ∇U , we have the following corollary.

Corollary 2.8 When ε → 0, η ε converges in L r loc , 1 ≤ r < ∞, to η(t, y) := R n M (z)e ∇yU •z dz.
Consequently, Λ ε (t, y) := Λ(t, η ε (t, y)) converges to Λ(y, η(t, y)) and

Q ε (t, x, y) := Q(x, y, η ε (t, y)) to Q(x, y, η(t, y)) in L r loc , 1 ≤ r < ∞.
3 Asymptotics of the corrector -Proof of Theorem 1.2

We now turn to the main new results of the paper. These are boundedness from above and below and the asymptotics of the corrector p ε (t, x, y) defined through the factorisation (3), according to the definition of U ε in (7).

Estimates on p ε

This section is devoted to the proof of the first statement of Theorem 1.2, that is, p ε (t, x, y) and x>0 p ε (t, x, y) are bounded uniformly in ε > 0, x ≥ 0, y ∈ R n , locally uniformly in t ≥ 0.

Our first result states a control of p ε , from above and below, for x ∈ [0, x] (with x from (10)).

Lemma 3.1 Under the assumptions of section 1.5, for any fixed T > 0, there exists two constants γ, γ > 0

such that γ Q ε (t, x, y) ≤ p ε (t, x, y) ≤ γ Q ε (t, x, y), for all ε > 0, t ∈ [0, T ], y ∈ R n , x ∈ [0, x],
where

Q ε (t, x, y) := Q(x, y, η ε (t, y)).
Proof The function p ε (t, x, y) satisfies the following equation, for

ε > 0, t > 0, x > 0, y ∈ R n ,        ε∂ t p ε + ∂ x [A(x, y)p ε ] + (d(x, y) -Λ ε ) p ε = 0, A(0, y)p ε (t, 0, y) = x>0, z∈R n M (z)e Uε(t,y+εz)-Uε(t,y) b(x, y + εz)p ε (t, x, y + εz)dxdz.
and Q ε satisfies, for (t, y) as parameters, the equation in the variable x

∂ x [A(x, y)Q ε ] -(d(x, y) -Λ ε (t, y))Q ε = 0, A(0, y)Q ε (t, 0, y) = η ε (t, y).
Setting

γ ε (t, x, y) := p ε (t, x, y) Q ε (t, x, y) , we have          ∂ t γ ε + A(x, y) ε ∂ x γ ε = - ∂ t Q ε Q ε γ ε , γ ε (t, 0, y) = x>0, z∈R n J ε (t, x, y, z)γ ε (t, x, y + εz)dz, (31)
where

J ε (t, x, y, z) := 1 η ε M (z)e Uε(t,y+εz)-Uε(t,y) b(x, y + εz)Q ε (t, x, y + εz). ( 32
)
Our goal is to infer some bounds on γ ε . First, from the definition of η ε and the normalization x>0 b(x, y)Q(x, y, η)dx = 1, we see that J ε is a probability kernel,

x>0, z∈R n J ε (t, x, y, z)dxdz = 1, ∀t ≥ 0, y ∈ R n .
We need to estimate ∂tQε Qε . We compute, using the representation formula [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF],

∂ t Q ε Q ε (t, x, y) = ∂ t η ε (t, y) ∂ η Q Q (x, y, η ε (t, y)) = ∂ t η ε (t, y) 1 η ε (t, y) + ∂ η Λ(y, η ε (t, y)) x 0 1 A(x , y) dx .
Because η ε is bounded from below, see [START_REF] Gyllenberg | A nonlinear structured population model of tumor growth with quiescence[END_REF], and ∂ η Λ is bounded, see [START_REF] Evans | Partial differential equations[END_REF] and assumption [START_REF] Champagnat | The evolutionary limit for models of populations interacting competitively via several resources[END_REF], we have

∂ t Q ε Q ε (t, x, y) ≤ K|∂ t η ε (t, y)| 1 + x 0 1 A(x , y) dx ,
for some constant K > 0. Then, using assumption (10), we have, for x ∈ [0, x],

∂ t Q ε Q ε (t, x, y) ≤ K|∂ t η ε (t, y)|
for some constant still denoted by K. Setting η ε (t) := sup y∈R n η ε (t, y), we define

γ ε (t, x, y) := γ ε (t, x, y) exp -K t 0 |∂ t η ε (t )|dt ,
so that γ ε is a subsolution to [START_REF] Kim | Well-posedness for constrained Hamilton-Jacobi equations[END_REF], namely

         ∂ t γ ε + A(x, y) ε ∂ x γ ε ≤ 0 γ ε (t, 0, y) ≤ x>0, z∈R n J ε (t, x, y, z)γ ε (t, x, y + εz)dzdx
(in fact, equality holds in the second line). From the comparison principle, we deduce

γ ε (t, x, y) ≤ sup x∈[0,x] y∈R n γ ε (0, x, y) ≤ γ 0 ,
where γ 0 comes from assumption [START_REF] Calvez | Uniqueness of the viscosity solution of a constrained Hamilton-Jacobi equation[END_REF]. This gives a control from above on γ ε (t, x, y) which implies

p ε (t, x, y) ≤ γ 0 Q ε (t, x, y) exp K t 0 |∂ t η ε (t )|dt .
By Corollary 2.6, t 0 |∂ t η ε | is bounded uniformly in ε > 0, therefore, for some constant γ, p ε (t, x, y) ≤ γQ ε (t, x, y).

Identically, we infer the bound from below, and the proof of Lemma 3.1 is completed.

With the previous lemma in hand, we now estimate p ε for all x ≥ 0 (which is useless if x = +∞). We set

Q(x, y) := γ η η(y) Q(x, y, η(y)) = γ η A(x, y) exp x 0 Λ -d(x , y) A(x , y) dx , Q(x, y) := γ η η(y) Q(x, y, η(y)) = γ η A(x, y) exp x 0 Λ -d(x , y) A(x , y) dx ,
where γ, γ are given by the previous lemma.

Lemma 3.2 Under the same condition as in the previous lemma, we have

Q(x, y) ≤ p ε (t, x, y) ≤ Q(x, y), ∀ε > 0, t ∈ [0, T ], x ≥ 0, y ∈ R n .
Note that, when restricted to [0, x] these bounds are weaker than in Lemma 3.1 since

1 γ Q ≤ Q ε ≤ 1 γ Q.
Proof From Lemma 3.1, we deduce γ η ≤ A(0, y)p ε (t, 0, y) ≤ γ η.

Hence, on the one hand, we have

ε∂ t p ε + ∂ x [A(x, y)p ε ] + d(x, y) -Λ p ε ≤ 0, A(0, y)p ε (t, 0, y) ≤ γ η, for ε > 0, t ∈ [0, T ],
x > 0, y ∈ R n . On the other hand,

ε∂ t Q + ∂ x A(x, y)Q + d(x, y) -Λ Q = 0, A(0, y)Q(0, y) = γ η.
From the comparison principle, we deduce p ε ≤ Q. The lower bound can be proved similarily.

We are now ready to prove our first main result Proposition 3.3 (Uniform estimates on p ε in L 1 ∩ L ∞ ) Under the assumptions of section 1.5 and for any fixed T > 0, p ε (t, x, y) and x>0 p ε (t, x, y)dx are bounded from above and below, uniformly in ε > 0,

t ∈ [0, T ], x ≥ 0, y ∈ R n .
Proof The first point is deduced from Lemma 3.2 and Q ≤ γ η A , with A from assumption (8). To prove the second point, we only need to estimate the integrals of Q and Q. Recalling Λ < 0, we compute

+∞ 0 Q(x, y)dx ≤ +∞ 0 γ η A(x, y) exp x 0 Λ A(x , y) dx = γ η Λ exp x 0 Λ A(x , y) dx +∞ x=0 = γ η -Λ 1 -exp +∞ 0 Λ A(x , y) dx ≤ γ η -Λ ,
which proves the bound from above.

For the other inequality, we use the non-degeneracy assumption (9) which implies R+ Q(x, y, η)dx > α, for some α > 0.

Asymptotics of p ε

We complete the proof of Theorem 1.2 by showing that, when ε → 0, the mutations affect the equilibrium distribution Q(x, y, η) only through a multiplicative factor γ(t, y), Proposition 3.4 (Convergence of the corrector) For t ∈ [0, T ] (with T > 0 fixed), x ∈ [0, x], y ∈ R n , and up to extraction of a subsequence when ε → 0, p ε converges in L ∞ weak-to γ(t, y)Q(x, y, η(t, y)) with γ(t, y) ∈ L ∞ which formally satisfies the equation

     ∂ t γ + ∂ η Λ z∈R n M (z)e ∇U •z zdz • ∇ y γ + ∂ t η ∂ 2 η Λ 2∂ η Λ γ = 0, γ(t = 0) = γ 0 . (33)
The difficulty in stating rigorously equation [START_REF] Lorz | Dirac mass dynamics in multidimensional nonlocal parabolic equations[END_REF] is that ∂ t η and z∈R n M (z)e ∇U •z zdz are nothing more than bounded measures which is not smooth enough since γ is just L ∞ . Nevertheless, we can prove establish the convergence of the full sequence p ε to γQ if T is small enough: we use the regularity of the initial condition to rigourously pass to the limiting equation [START_REF] Lorz | Dirac mass dynamics in multidimensional nonlocal parabolic equations[END_REF], and then use a standard uniqueness result on this equation.

Note also that, according to Theorem 1.3, the population concentrates on S where U (t, y) achieved its maximum. From [START_REF] Barles | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF], U is differentiable at points of S and one has ∂ t U = ∇ y U = 0 and η = 1. Thus p ε converges to a multiple of Q(x, y, 1) on S. In addition, if M (•) is even, then the drift term vanishes and the equation can be written

∂ t [∂ η Λ(y, η(t, y)) 1/2 γ] = 0.
To prove Proposition 3.4, and establish (30) we use the dual problem associated with [START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adpative dynamics[END_REF]; for fixed (y, η) ∈ R n × (0, +∞) consider Φ(x, y, η) the unique solution of

   A(x, y)∂ x Φ + [Λ(y, η) -d(x, y)] Φ = -ηb(x, y)Φ(0, y, η), ∀x > 0, x>0 Q(x, y, η)Φ(x, y, η)dx = 1. ( 34
)
We can solve this ordinary differential equation and find

Φ(x, y, η) = -∂ η Λ(y, η)η +∞ x b(x , y) A(x , y) exp x x Λ(y, η) -d(x , y) A(x , y) dx dx .
Also, multiplying equation ( 31) by ∂ η Q and integrating by parts, we get

Φ(0, y, η) = -∂ η Λ(y, η), (35)
and, multiplying equation ( 31) by ∂ 2 ηη Q, we end up with

+∞ 0 Q(x, y, η)∂ η Φ(x, y, η)dx = - ∂ 2 η Λ 2∂ η Λ . ( 36
)
Proof We fix T > 0 and, throughout the proof, choose t ∈ [0, T ], x ∈ [0, x], y ∈ R n . We define γ ε := pε Qε and recall equation [START_REF] Kim | Well-posedness for constrained Hamilton-Jacobi equations[END_REF]. From Lemma 3.1, we know that γ ε is bounded and thus converges (up to extraction of a subsequence) to some γ in L ∞ weak-. Passing to the limit in [START_REF] Kim | Well-posedness for constrained Hamilton-Jacobi equations[END_REF], we deduce ∂ x γ ≡ 0 (in the sense of distributions), and thus γ only depends on t and y. Since Q ε (strongly) converges to some Q (see Corollary 2.8), we deduce that p ε (t, x, y) converges (up to extraction of a subsequence) to γ(t, y)Q(t, x, y) in L ∞ weak-.

We are now left with the task of identifying γ. To do so, we set

E ε (t, y) := +∞ 0 γ ε (t, x, y)Q ε (t, x, y)Φ ε (t, x, y)dx, Φ ε (t, x, y) := Φ(x, y, η ε (t, y)).
Note that Φ ε (t, x, y) ≡ 0 for x > x and thus E ε (t, y) := 6), ( 28) and ( 31), we can write

x 0 γ ε Q ε Φ ε . From (
ε∂ t E ε = x>0 ε∂ t γ ε Q ε Φ ε + ε x>0 γ ε ∂ t [Q ε Φ ε ] = - x>0 A∂ x γ ε Q ε Φ ε -ε x>0 γ ε ∂ t Q ε Φ ε + ε x>0 γ ε ∂ t [Q ε Φ ε ] = [Aγ ε Q ε Φ ε ](x = 0) + x>0 γ ε ∂ x [AQ ε Φ ε ] + ε x>0 γ ε Q ε ∂ t Φ ε (from ∂ x [AQ ε Φ ε ] = -η ε bQ ε Φ ε (x = 0)
, and with the probability kernel J ε defined in ( 29))

= η ε Φ ε (x = 0) x>0, z∈R n J ε (t, x, y, z) (γ ε (t, x, y + εz) -γ ε (t, x, y)) dzdx + ε x>0 γ ε Q ε ∂ t Φ ε
Recalling Corollary 2.8, we know that Φ ε converges to some Φ when ε → 0. Dividing by ε and passing to the limit ε → 0 (after extracting a subsequence, in the sense of distributions), we deduce

∂ t γ x>0 QΦ = ∂ t γ = Φ(t, 0, y) z∈R n M (z)e ∇U •z z • ∇ y γ + γ x>0 Q∂ t Φ. ( 37
)
Injecting ∂ t Φ ε (t, x, y) = ∂ t η ε ∂ η Φ(x, y, η ε
) and ( 32)-( 33) in [START_REF] Mirrahimi | Adaptation and migration of a population between patches[END_REF], we end up with equation [START_REF] Lorz | Dirac mass dynamics in multidimensional nonlocal parabolic equations[END_REF]. From classical uniqueness results, we deduce that γ ε converges weakly to γ for the whole sequence ε → 0 (and not for an extracted subsequence).

Concentration of the population density -proof of Theorem 1.3

We now conclude on the consequences of our study on U ε and p ε with the concentration effect for the population density m ε itself.

Selection of the fittest phenotypes

The following result states that the total population ρ ε is uniformly bounded and converges when ε → 0. Recalling (3), the two first statements of Theorem 1.3 are direct consequences of the following proposition and the uniform L 1 estimate on p ε (Proposition 3.3). Proposition 4.1 Under the assumptions of section 1.5, 1. There exist two positive constants ρ, ρ > 0 such that

ρ ≤ ρ ε (t) ≤ ρ, ∀ε > 0, t ≥ 0.
In addition, ρ ε converges to some ρ in the L ∞ -weak * topology.

The integral

R n e Uε(t,y)-t 0 ρε ε dy is bounded away from 0, uniformly in ε > 0, t ≥ 0. Consequently, when ε vanishes, we have

t 0 ρ = sup y∈R n U (t, y), ∀t ≥ 0. (38)
In addition, with S defined in (2), we have

S := t ≥ 0, y ∈ R n : U (t, y) = sup R n U (t, •) = t ≥ 0, y ∈ R n : U (t, y) = t 0 ρ .
Remark 4.2 T,he proof of Proposition 4.1 becomes much simpler if we assume that there are r, r > 0 such that r ≤ b(x, y) -d(x, y) ≤ r, ∀x ≥ 0, y ∈ R n , Indeed, integrating (1) and using an integration by parts, we obtain

ε d dt ρ ε (t) = R n ×R+ 1 ε n R n M ( y -y ε )dy b(x, y ) -d(x, y ) m ε (t, x, y )dxdy -ρ 2 ε (t) ≤ rρ ε (t) -ρ 2 ε (t).
which implies 0 ≤ ρ ε (t) ≤ max r, ρ 0 ε and provides us with an a priori upper bound on ρ ε . With the same method, we also infer a positive lower bound on ρ ε .

Then, using the uniform L 1 estimate on p ε from Proposition 3.3, we directly deduce [START_REF] Mirrahimi | Asymptotic analysis of a selection model with space[END_REF].

Proof (of Proposition 4.1) We recall that

ρ ε (t) =
x,y p ε (t, x, y)e Uε(t,y)-t 0 ρε ε dxdy.

(

) 39
Multiplying by e t 0 ρε ε we have

ρ ε (t)e t 0 ρε ε = R n e Uε(t,y) ε R+ p ε (t, x, y)dxdy,
and integrating over (0, t), we deduce

ε e t 0 ρε ε -1 = t 0 R n e Uε(t,y) ε R+ p ε (t, x, y)dxdydt. From 0 < -Λ ≤ ∂ t U ε ≤ -Λ (Proposition 2.
2) and the L 1 (dx) estimate on p ε (Proposition 3.3), we have

ε e t 0 ρε ε -1 ≥ I t 0 R n ε -Λ ∂ t U ε (t, y) ε e Uε(t,y) ε dy dt ≥ εI -Λ R n e Uε(t,y) ε -e U 0 ε (y) ε dy.
Dividing by εe t 0 ρε ε on both sides we find

1 -e - t 0 ρε ε ≥ I -Λ R n e Uε(t,y) ε -e U 0 ε (y) ε e - t 0 ρε ε dy, that we rewrite, with u ε (t, y) = U ε (t, y) - t 0 ρ ε R n e uε(t,y) ε dy ≤ -Λ I 1 -e - t 0 ρε ε + e - t 0 ρε ε R n e U 0 ε (y) ε dy.
Then, from assumption (13), we deduce

R n e uε(t,y) ε dy ≤ -Λ I 1 -e - t 0 ρε ε + J 0 e - t 0 ρε ε , and R n e uε(t,y) ε dy
≤ K := max -Λ I , J 0 .
Identically, we infer Now, from the definition of ρ ε (recalled in [START_REF] Mirrahimi | A class of Hamilton-Jacobi equations with constraint: uniqueness and constructive approach[END_REF]) and the L 1 estimate on p ε (Proposition 3.3) we find

I K ≤ ρ ε (t) ≤ I K.
Since ρ ε (t) is uniformly bounded, there exists a sequence ε k → 0 such that ρ ε k converges to some ρ in the L ∞ -weak * topology when k → +∞. Since, in addition, U ε converges locally uniformly to U (Proposition 2.7), we deduce that u ε k converges locally uniformly to some u.

Now, from K ≤ R n e uε(t,•) ε ≤ K, at the limit k → +∞, we have ∀t > 0, sup y∈R n u(t, y) = 0.
We deduce

t 0 ρ = sup y∈R n U (t, y).
Therefore, t 0 ρ does not depend on the extracted subsequence, and the convergence occurs for the whole sequence ε → 0, which achieves the proof.

Notice that the bound on ρ(s) can be made more precise in the Cesaro sense

∀t > 0, -Λ ≤ 1 t t 0 ρ(s)ds ≤ -Λ
since, from Proposition 2.2 we know that

t 0 ρ = sup R n U (t, •) ≤ sup R n U 0 (•) -Λt = -Λt
and the bound from below is similar. Corollary 4.4 Under the same assumptions, t → ρ(t) ∈ C 1 ([0, T ]) and for all t ∈ [0, T ],

ρ(t) = -Λ(ȳ(t), 1) and d dt ρ(t) = -∇ y Λ • ∇ 2 y U • ∇ y Λ -∂ η Λ R n M (z)zdz • ∇ 2 y U 2 • ∇ y Λ.
where ∇ 2 y U is evaluated in (t, ȳ(t)), and the derivatives of Λ in (ȳ(t), 1).

In particular, if M (•) is even, then d dt ρ(t) ≥ 0 and -Λ ≤ ρ ≤ -Λ. Proof Follows from ∂ t U (t, ȳ(t)) = ρ(t) and d dt [∂ t u(t, ȳ(t))] = d dt ρ(t).
Remark 4.5 The limitation of Proposition 4.3 to a short time interval is merely due to three independant phenomena. First, the possible loss of concavity, or apparition of singularities for U , coming from the Hamilton-Jacobi equation [START_REF] Jabin | Selection-Mutation dynamics with spatial dependence[END_REF]. Secondly, the possible "jump" of the point where U reaches its maximum, contradicting max y∈V U (t, y) = max y∈R n U (t, y) in [START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF]. Finally, the possible blow-up in finite time of ȳ(t) from the dynamics of the Canonical Equation [START_REF] Perthame | Transport equations in biology[END_REF]. Regarding the last point, we point out that Λ can sometimes be used as a Lyapunov function. Indeed, we have

d dt [Λ(ȳ(t), 1)] = ∇ y Λ(ȳ(t), 1) • ẏ(t) = ∇ y Λ(ȳ(t), 1) • ∇ 2 y U -1 • ∇ y Λ(ȳ(t), 1) + ∂ η Λ(ȳ(t), 1)∇ y Λ(ȳ(t), 1) • R n M (z)zdz ≤ ∂ η Λ(ȳ(t), 1)∇ y Λ(ȳ(t), 1) • R n M (z)zdz.
In particular, if M (•) is even, then d dt [Λ(ȳ(t), 1)] ≤ 0. Thus, if ȳ0 belongs to a "well" of Λ, then ȳ(t) remains "trapped", which prevents from an blow-up in finite time. It also implies, at least formally, that ȳ(t) converges to a local minimum of Λ(•, 1) when t → +∞.

5 Construction, estimates, and asymptotics of U ε -Proof of Theorem 1.1

The eigenproblem -proof of Proposition 2.1

An immediate calculation on [START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adpative dynamics[END_REF] gives the explicit solution [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF] for Q in terms of Λ. Multiplying by b(x, y) and integrating in x, we obtain formula [START_REF] Diekmann | The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach[END_REF]. Next, from [START_REF] Champagnat | The canonical equation of adaptive dynamics: A mathematical view[END_REF] we have, for y ∈ R n , ∀η ∈ (η(y), η(y))

F (y, Λ) ≤ 1 η ≤ F (y, Λ).
As ∂ λ F > 0, we conclude the existence and uniqueness of Λ(y, η) as the unique solution of (20). Now, using [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF], we obtain existence and uniqueness of Q.

Finally, the bounds (21) follow from ∂ λ F > 0 and

F (y, Λ) ≤ F (y, Λ(y, η)) ≤ F (y, Λ).
For later purpose, we also neeed the following stronger version of (24).

Lemma 5.1 There exists δ > 0 such that, for all y ∈ R n , η ∈ (η(y), η(y)), we have

∂ 2 η Λ(y, η) + ∂ η Λ(y, η) η ≤ -δ.
Proof From the proof of ( 24), we have

∂ 2 η Λ + ∂ η Λ η = 1 (∂ λ F ) 3 η 3 (∂ λ F ) 2 -F ∂ 2 λ F . Since 1 (∂ λ F ) 3 η 3 ≥ 1 L 3 η 3 , our goal is to show that (∂ λ F ) 2 -F ∂ 2 λ F ≤ -δ,
for all y ∈ R n , η ∈ (η(y), η(y)).

We set, for all x ≥ 0,

y ∈ R n , λ ∈ R, f (x, y, λ) := b(x, y) A(x, y) exp x 0 λ -d(x , y) A(x , y) dx .
According to [START_REF] Busse | Mass concentration in a nonlocal model of clonal selection[END_REF], we have F (y, λ) = R+ f (x, y, λ)dx. We also define the probability measure f (x, y, λ) := f (x,y,λ)

F (y,λ) . Now, setting A(x, y) :=

x 0 1 A(x ,y) dx , we have

∂ λ F (y, λ) = R+ A(x, y)f (x, y, λ)dx, ∂ 2 λ F (y, λ) = R+ A(x, y) 2 f (x, y, λ)dx.
It gives,

(∂ λ F ) 2 -F ∂ 2 λ F = -F 2 R+ A(x, y) - R+ A f dx 2 f dx
(the function are evaluated on y and λ = Λ(y, η)). We have F 2 ≤ 1 η 2 , and assumption (9) implies that the above term is negative uniformly in y ∈ R n , η ∈ (η(y), η(y)).

Construction of U ε -proof of Proposition 2.2

We present a proof of existence based on a regularization argument. For the constrained Hamilton-Jacobi equation, a fixed point method has also been proposed in [START_REF] Kim | Well-posedness for constrained Hamilton-Jacobi equations[END_REF]. Our proof is divided into three parts. First, we construct U ε on a truncated problem. Then, we prove a uniform a priori estimate on ∂ t U ε , which allows finally to remove the truncation.

Extending Λ. Proposition 2.1 defines Λ(y, η) only for y ∈ R n and η ∈ η(y), η(y) . We first need to artificially extend Λ(y, η) for η ∈ (0, +∞). For y ∈ R n , we set

Λ(y, η) =      Λ -B y (η) if η < η(y), Λ(y, η) if η(y) ≤ η ≤ η(y), Λ + B y (η) if η > η(y),
where B y and B y are chosen to be positive, increasing, bounded by 1, and such that Λ is smooth. Note that the extension of Λ is completely arbitrary, but we will show, a posteriori, that η ε ∈ η(y), η(y) . We consider the following problem

     ∂ t Ũε (t, y) = -Λ y, R n M (z)e Ũε(t,y+εz)-Ũε(t,y) ε dz , ∀t ≥ 0, ∀y ∈ R n , Ũε (0, y) = u 0 ε (y), ∀y ∈ R n . ( 45
)
Solution for the truncated problem. For a fixed R > 0, we consider a truncation function φ R : R → R which is smooth, increasing and satisfies the following conditions:

• φ R (r) = r for r ∈ [-R 2 , R 2 ], • φ R (r) = R for r ≥ 2R, • φ R (r) = -R for r ≤ -2R, • φ R ≥ 0 is uniformly bounded.
For ε > 0, we consider the Cauchy problem

     ∂ t Ũ R ε (t, y) = φ R -Λ y, R n M (z)e Ũ R ε (t,y+εz)-Ũ R ε (t,y) ε dz , Ũ R ε (0, •) = U 0 ε . (46)
for which the classical Cauchy-Lipschitz theorem provides existence and uniqueness of a solution Ũ R ε , defined globally in time.

Estimate on the time derivative. Lemma 5.2 We have, with ∂ t U 0 ε := -Λ y, η 0 ε (y) and η 0 ε is defined in [START_REF] Cannarsa | Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control[END_REF],

inf y∈R n ∂ t U 0 ε (y) ≤ ∂ t Ũ R ε (t, y) ≤ sup y∈R n ∂ t U 0 ε (y), ∀ε > 0, t > 0, y ∈ R n .
The full proof of this statement, which is technical, can be found in [START_REF] Nordmann | Dynamics of Concentration in a Population Model Structured by Age and a Phenotypical Trait[END_REF] (proof of Proposition 4.7, Appendix D). We give the formal idea of the method. Let us fixe ε > 0, R > 0 and set V (t, y) := ∂ t Ũ R ε (t, y). Differentiating (43) with respect to t, we obtain

∂ t V (t, y) = R n K(t, y, z) V (t, y + εz) -V (t, y) ε dz,
where K(t, y, z)

:= -φ R ∂ η Λ M (z)e Ũ R ε (t,y+εz)-Ũ R ε (t,y) ε
. Since ∂ η Λ < 0, we have K ≥ 0. Then, if for some t > 0, V (t, •) reaches its maximum at ȳ ∈ R n , we obtain the inequality

∂ t V (t, ȳ) = R n K(t, ȳ, z) V (t, ȳ + εz) -V (t, ȳ) ε dz ≤ 0.
Formally, it shows that the maximum value of V is decreasing with time, that is,

sup y V (t, y) ≤ sup y V (0, y) = sup y ∂ t U 0 ε .
With the same method we show inf y V ≥ inf y ∂ t U 0 ε , which conclude the proof of Lemma 5.2.

Hereafter, from assumption [START_REF] Champagnat | The canonical equation of adaptive dynamics: A mathematical view[END_REF] and ∂ λ F > 0, we have

-Λ ≤ ∂ t U 0 ε (y) ≤ -Λ.
Using Lemma 5.2, we infer

-Λ ≤ φ R -Λ y, R n M (z)e Ũ R ε (t,y+εz)-Ũ R ε (t,y) ≤ -Λ. (47)
Removing the truncation. From (44) and the choice of φ R , for R large enough, we have

-φ R Λ y, R n M (z)e Ũ R ε (t,y+εz)-Ũ R ε (t,y) = Λ y, R n M (z)e Ũ R ε (t,y+εz)-Ũ R ε (t,y)
.

Besides, since ∂ η Λ < 0, we have

η(y) ≤ R n M (z)e Ũ R ε (t,y+εz)-Ũ R ε (t,y) ≤ η(y),
for all R large enough, ε > 0, t ≥ 0, y ∈ R n . Thus, from the definition of Λ in (42), we have

-Λ y, R n M (z)e Ũ R ε (t,y+εz)-Ũ R ε (t,y) ε dz = -Λ y, R n M (z)e Ũ R ε (t,y+εz)-Ũ R ε (t,y) ε dz , that is, Ũ R
ε is a solution of [START_REF] Barles | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF]. The proof is thereby achieved.

A priori Lipschitz estimate -proof of Proposition 2.4

We follow the same idea as for Lemma 5.2. However, there are some technical difficulties. First, we have to deal with a "source term" ∇ y Λ (y, η ε (t, y)) which is bounded by a constant L lη 2 , using ( 17), ( 18), ( 23) and [START_REF] Gyllenberg | A nonlinear structured population model of tumor growth with quiescence[END_REF]. In addition, we first need to prove the estimate on a truncated function, then to remove the truncation.

We fix i ∈ {1, . . . , n}, T > 0, and we set

W ε (t, y) := ∂ yi U ε (t, y).
Differentiating [START_REF] Barles | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF], we obtain

∂ t W ε (t, y) = -∂ yi Λ(y, η ε ) -∂ η Λ(y, η ε ) M (z)e
Uε(t,y+εz)-Uε(t,y) ε W ε (t, y + εz) -W ε (t, y) ε dz

:= F(t, y, W ε (t, •)).
We formally define a truncated problem, for R > 0, and its solution W R ε satisfying

W R ε (t, y) = φ R ∂ yi U 0 ε (y) + t 0 F(s, y, W R ε (s, •))ds , (48)
where F is defined above and φ R is a truncation function as in (43). We can prove existence and uniqueness of a global solution of (45) by a direct application of the Cauchy-Lipschitz theorem.

We

set W R ε := W R ε -Ct with C := L lη 2 . Our goal is to show ∀t ∈ [0, T ], ∀y ∈ R n , W R ε (t, y) ≤ sup ∂ yi U 0 ε . (49)
By contradiction, assume (46) does not hold, i.e., there exist y

0 ∈ R n , t 0 ∈ [0, T ] such that W R ε (t 0 , y 0 ) -sup ∂ yi U 0 ε > 0. ( 50
) For β > 0, α > 0 small enough, t ∈ [0, T ], y ∈ R n we introduce ϕ α,β (t, y) := W R ε (t, y) -αt -β|y -y 0 |.
As W R ε is bounded, ϕ α,β reaches its maximum on [0, T ] × R n at a point ( t, ȳ). We have

∀z ∈ R n , ϕ α,β ( t, ȳ + εz) ≤ ϕ α,β ( t, ȳ).
Then, we obtain the inequality

∀z ∈ R n , W R ε ( t, ȳ + εz) -W R ε ( t, ȳ) ε ≤ β |ȳ + z -y 0 | -|ȳ -y 0 | ε ≤ β|z|.
We choose α small enough so that ϕ α,β (t 0 , y 0 ) > ϕ α,β (0, y 0 ) = ∂ yi U 0 ε (y 0 ), which is possible thanks to (47). It implies t > 0. Hence ∂ t ϕ α,β ( t, ȳ) ≥ 0, i.e. ∂ t W R ( t, ȳ) ≥ α (if t = T , then ∂ t stands for the left derivative). Differentiating (45) at ( t, ȳ), we have

α ≤ ∂ t W R ε ( t, ȳ) ≤ -sup φ R × ∂ yi Λ(y, η ε (t, y)) -C + sup φ R × (-∂ η Λ(y, η)) M (z)e Uε(t, ȳ+εz)-U (t, ȳ) ε W R ε ( t, ȳ + εz) -W R ε ( t, ȳ) ε dz. Now, from |∂ yi Λ(y, η ε (t, y))| ≤ L lη 2 = C and 0 ≤ -∂ η Λ(y, η) ≤ L η 2 , we have α ≤ L η 2 M (z)e Uε(t, ȳ+εz)-U (t, ȳ) ε |z|dz × β ≤ L η 2 M (z)e -2ΛT ε +k0|z| |z|dz × β.
Then, passing to the limit β → 0 we obtain α ≤ 0: contradiction. Thus, we have

W R ε ≤ sup |∂ yi U 0 ε | = k 0 .
We proceed similarily to obtain the reverse inequality

W R ε ≥ k 0 . We have, for all R > 0, ε > 0, t ∈ [0, T ], y ∈ R n |W R ε (t, y)| ≤ k 0 + Ct.
Finally, the bound on W R ε is uniform in R so we can remove the truncation, as detailed in section 5.2. Thus, W R ε = W ε for R large enough and

|∂ yi U ε (t, y)| ≤ k 0 + Ct.

Semi-convexity -proof of Proposition 2.5

For convex Hamiltonian, the semi-convexity of the solution is a classical matter, [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF][START_REF] Cannarsa | Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control[END_REF]. Here, we have to deal with a nonlocal operator which features a difference rather than a gradient Semi-convexity in t. For shorter formulas, we need some notations

V ε := ∂ 2 t U ε , J ε (t, y, z) := M (z)e
Uε(t,y+εz)-Uε(t,y) ε

.

We begin with two results that are used later and express some properties usually connected to the convexity of the Hamiltonian. Firstly we observe that Lemma 5.3 For all ε > 0, t > 0, y ∈ R n , we have

(∂ t η ε ) 2 ≤ η ε R n ∂ t U ε (t, y + εz) -∂ t U ε (t, y) ε 2 J ε dz.
Proof Use Jensen's inequality and the definition of η ε in [START_REF] Fleming | Deterministic and Stochastic Optimal Control[END_REF].

Next, we prove that

∂ t V ε ≥ -∂ η Λ(y, η ε ) R n V ε (t, y + εz) -V ε (t, y) ε J ε dz. (51)
Semi-convexity in y. Let us show how the method can be adapted to prove that ∂ 2 i U ε (the second derivative w.r.t. y i ) is bounded from below. We set W ε := ∂ 2 i U ε . Differentiating (7) twice, we find

∂ t W ε = -∂ η Λ R n W ε (t, y + εz) -W ε (t, y) ε J ε dz -∂ 2 η Λ (∂ i η ε ) 2 -∂ η Λ R n ∂ i U ε (t, y + εz) -∂ i U ε (t, y) ε 2 J ε dz -∂ 2 i Λ -2∂ 2 i,η Λ∂ i η ε . (57)
In contrast with the equation (49) on ∂ 2 t U ε , we need to deal with a source term and a linear term in the last line.

For any constant K > 0, Young's inequality implies

-2∂ 2 i,η Λ∂ i η ε ≥ -K 2 |∂ 2 i,η Λ| 2 - 1 K 2 (∂ i η ε ) 2 .
Applying this inequality and Lemma 5.3 (replacing ∂ t by ∂ i ) in (54), we obtain

∂ t W ε ≥ -∂ η Λ R n W ε (t, y + εz) -W ε (t, y) ε J ε dz -∂ 2 η Λ + ∂ η Λ η ε + 1 K 2 (∂ i η ε ) 2 -∂ 2 i Λ -K 2 |∂ 2 i,η Λ| 2 . Using lemma Lemma 5.1, and choosing K ≥ 1 √ δ , we have ∂ t W ε ≥ -∂ η Λ R n W ε (t, y + εz) -W ε (t, y) ε J ε dz -∂ 2 i Λ -K 2 |∂ 2 i,η Λ| 2 .
From assumptions ( 18)-( 19), the source term -∂ 2 i Λ -K 2 |∂ 2 i,η Λ| 2 is bounded from below by some constant -K < 0. We end up with

∂ t W ε ≥ -∂ η Λ R n W ε (t, y + εz) -W ε (t, y) ε J ε dz -K .
Then, applying the same method as in the proof of Lemma 5.4 (see also the proof of Proposition 2.2), we show W ε (t, y) ≥ inf y∈R n W ε (t = 0, y) -K t, ∀ε > 0, t ≥ 0, y ∈ R n . Finally, we conclude the lower bound since W ε (t = 0, y) ≥ -C (from assumption ( 12)), we have

W ε (t, y) ≥ -C -K t, ∀ε > 0, t ≥ 0, y ∈ R n .
Other directional derivatives and conclusion. Lower bounds on other second order derivatives in directions ν ∈ S := {(t, y) ∈ R n+1 : t 2 + |y| 2 = 1} can be obtained by a slight adaptation of the previous steps show. We deduce that U ε is semi-convex and that ∇U ε is uniformly in BV loc (see Proposition 1.1.3 and Theorem 2.3.1 in [START_REF] Cannarsa | Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control[END_REF]). We obtain that U ε is uniformly bounded in W 2,1 loc .

Proof of Corollary 2.6

We recall the definition of η ε in [START_REF] Fleming | Deterministic and Stochastic Optimal Control[END_REF], and note that differentiating [START_REF] Barles | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF], we obtain

∂ 2 t U ε = -∂ t η ε (t,
Using that U ε is uniformly bounded in W 2,1 (Proposition 2.5) and that -∂ η Λ is positively bounded [START_REF] Evans | Partial differential equations[END_REF], we deduce that η ε is bounded in W 1,1 , which proves the first part of Corollary 2.6.

Let us now fix T > 0 and prove the second part. We recall that U ε is semi-concave (Proposition 2.5). Thus, there exists a constant K > 0 such that ∂ t η ε (t, y) ≥ -K for all ε > 0, (t, y) ∈ [0, T ] × R n . Denoting η(t) := sup y∈R n η(t, y), we deduce ∂ t η ε (t) ≥ -K (indeed, the mapping θ y : t → η ε (t, y)+Kt is nondecreasing, and so is sup y∈R n θ y ). The last inequality should be understood in the sense of the distributions.

We deduce

T 0 |∂ t η ε (t)|dt = T 0 ∂ t η ε (t)dt + 2 T 0 (∂ t η ε (t)) -dt ≤ η ε (t) + 2Kt ≤ η + 2Kt,
where -denotes the negative part, and the last inequality comes from [START_REF] Gyllenberg | A nonlinear structured population model of tumor growth with quiescence[END_REF]. The proof is complete.

5.6 Asymptotics of U ε -proof of Proposition 2.7

Extraction of a subsequence From the a priori estimate of Proposition 2.2 and Ascoli's theorem, we know that U ε converges locally uniformly to some U , up to extraction of a subsequence. Incidentally, this convergence also occurs in W Thus, the convergence occurs in W 1,r loc , 1 ≤ r < ∞. Also notice that Proposition 2.5 implies that U is semi-convex, uniformly in y ∈ R n , locally uniformly in t.

Viscosity solution We are going to show that U is a viscosity solution of [START_REF] Jabin | Selection-Mutation dynamics with spatial dependence[END_REF], i.e., U satisfies The proof is adapted from classical stability results for viscosity solutions of Hamilton-Jacobi equations (see [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF]). However, this case is not completely standard because of the nonlocal term R n M (z)e Uε(t,y+εz)-Uε(t,y) ε dz.

∂ t U = H(y, ∇ y U ), U (0, y) = U 0 (y), (59) with
Lemma 5.5 The function U is a viscosity solution of (56) in (0, ∞) × R n . Also, for all T > 0, the viscosity inequalities stand for t ∈ (0, T ].

Proof We are going to prove that U is a subsolution of (56). Let us consider a test function ϕ and a point (t 0 , y 0 ) such that U -ϕ reaches a global maximum at (t 0 , y 0 ). From classical results, there exists (t ε , y ε ) such that    (t ε , y ε ) -→ ε→0 (t 0 , y 0 ), max t,y

U ε -ϕ = (U ε -ϕ)(t ε , y ε ).
For all z ∈ R n , ϕ(t ε , y ε + εz) -U ε (t ε , y ε + εz) ≥ ϕ(t ε , y ε ) -U ε (t ε , y ε ), thus we have

ϕ(t ε , y ε + εz) -ϕ(t ε , y ε ) ε ≥ U ε (t ε , y ε + εz) -U ε (t ε , y ε ) ε .
Since ∂ η Λ < 0, equation (56) gives As ε goes to 0, ∂ t ϕ(t 0 , y 0 ) ≤ -Λ y 0 , R n M (z)e ∇yϕ(t0,y0)•z = H(y 0 , ∇ y ϕ(t 0 , y 0 )), then U is a viscosity subsolution of (56). With the same method, we prove that U is also a viscosity supersolution. It completes the first part of the proof. The second part of the statement is a well-known result, and proof can be found in [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF].

Uniqueness We point out that the Hamiltonian H is Lipschitz continuous in the y variable. We introduce a truncated Hamiltonian Since η ≤ η ε (t, y) ≤ η (from ( 26)), we have

∂ t U = H(y, ∇ y U ).
For this equation, a classical uniqueness result is in order (see e.g [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF][START_REF] Nordmann | Dynamics of Concentration in a Population Model Structured by Age and a Phenotypical Trait[END_REF]). We deduce that U ε converges to U for the whole sequence ε → 0 (and not for an extracted subsequence). From the assumption that M (•) is not degenerate, we deduce that, for some r 0 > 0, and for all z ∈ R n such that |z| = r 0 , then U ε (t, y + εz) -U ε (t, y) ε ≤ C

A posteriori

for some constant C (independant of ε, t, y and z). Then, chosing z and ε such that y -y = εz, we have U ε (t, y) -U ε (t, y ) ≤ C|y -y |. As ε → 0, we prove the goal (57).

H

  (y, p) := -Λ y, R n M (z)e p•z dz .

∂

  t ϕ(t ε , y ε ) = -Λ y ε , R n M (z)e Uε(tε,yε+εz)-Uε(tε,yε) ε dz ≤ -Λ y ε , R n M (z)eϕ(tε,yε+εz)-ϕ(tε,yε) ε dz .

M

  (z)e p•z dz ∈ [η, η], 0 otherwise.

  p•z dz .

	From this equation, it is classical to prove uniform a priori bounds on ∂ t U (indeed, ∂ t U satisfies a transport
	equation). We deduce that H(y, ∇ y U ) is bounded, then that
	η(t, y) :=	M (z)e ∇yU (t,y)•z dz
	R n
	is bounded. Besides, since p → H(y, p) is convex, we deduce that U is semi-convex, namely in space-
	time, the Hessian D 2 U is bounded from below. Since ∂ t U is bounded, we have ∂ 2 t U ∈ L 1 loc . Then, using

  1,1 , from the W 2,1 estimate in Proposition 2.4 and a classical compact embedding. In addition, we know from Proposition 2.2 and Proposition 2.4 that U ε is locally Lipschitz continuous, uniformly in ε > 0: for all t ≥ t ≥ 0 and y, y ∈ R

n |U (t, y) -U (t , y )| ≤ -Λ(t -t ) + k 0 + L lη 2 t |y -y |.

  Lipschitz estimate -proof of the global Lipschitz regularity in Proposition 2.7From Proposition 2.2 and Proposition 2.4, we know that U is Lipschitz, globally in t and locally in y. Our goal is to show that U is globally Lipschitz continuous, i.e., that there exists a constant C > 0 such that∀t ≥ 0, ∀(y, y ) ∈ (R n ) 2 , U (t, y) -U (t, y ) ≤ C|y -y |.(60)Let us fix t ≥ 0 and (y,y ) ∈ (R n ) 2 .With η ε defined in[START_REF] Fleming | Deterministic and Stochastic Optimal Control[END_REF] and the bound η ε ≤ η from (26), we have, for all ε > 0, z ∈ R n

	M (z)e
	R n

Uε(t,y+εz)-U (t,y) ε dz ≤ η ε (t, y) ≤ η.

‡ Acknowledgment. BP has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 740623)

Adaptive dynamics

We now prove the third statement of Theorem 1.3. We need further assumptions on the initial conditions,

ȳ0 ε converges to some ȳ0 ∈ R n when ε vanishes,

Proposition 4.3 Under the assumptions of section 1.5 and (37)- [START_REF] Nordmann | Dynamics of Concentration in a Population Model Structured by Age and a Phenotypical Trait[END_REF], for a short time interval [0, T ], there exists a unique ȳ(t) ∈ R n on which U (t, •) reaches its maximum. Moreover, t → ȳ(t) ∈ C 1 and satisfies the Canonical Equation

.

Note that (39) features a drift term ∂ η Λ R n M (z)zdz. If the mutation kernel M (•) is even, this term vanishes and we recover the classical Canonical Equation. Let us also recall that uniqueness and regularity of a unique concentration point (monomorphism) is a hard questions with few progresses, see [START_REF] Barles | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF][START_REF] Calvez | Uniqueness of the viscosity solution of a constrained Hamilton-Jacobi equation[END_REF][START_REF] Mirrahimi | A class of Hamilton-Jacobi equations with constraint: uniqueness and constructive approach[END_REF].

Proof Since U (defined in Proposition 2.7) satisfies [START_REF] Jabin | Selection-Mutation dynamics with spatial dependence[END_REF] with smooth initial datum, it is uniformly C 2 in the y variable for short times t ∈ [0, T ], T > 0 (this can be proved with the method of the characteristics, see Chapter 3.2 in [START_REF] Evans | Partial differential equations[END_REF]).

Consider such a time interval [0, T ], and V ⊂ R n a neighborhood of y 0 . We are interested in the solutions

From [START_REF] Mischler | Stability in a nonlinear population maturation model[END_REF] we know that at initial time there exists a unique solution ȳ0 of [START_REF] Perthame | Rare Mutations Limit of a Steady State Dispersal Evolution Model[END_REF]. Besides, ∇ 2 y U 0 (ȳ 0 ) is invertible. From the implicit functions theorem, there exists a unique ȳ(t) ∈ R n satisfying [START_REF] Perthame | Rare Mutations Limit of a Steady State Dispersal Evolution Model[END_REF], for t in a certain time interval still denoted [0, T ]. We can again choose a smaller T to ensure that ȳ(t) remains in V .

From [START_REF] Nordmann | Dynamics of Concentration in a Population Model Structured by Age and a Phenotypical Trait[END_REF], we can also choose T and V small enough to guarantee, ∀t ∈ [0, T ],

Hence, for all t ∈ [0, T ], the solution ȳ(t) of ( 40) must satisfy

which proves the first part of the proposition and that t → ȳ(t) is C 1 .

For the Canonical Equation, we differentiate [START_REF] Perthame | Rare Mutations Limit of a Steady State Dispersal Evolution Model[END_REF] with respect to t, and noting that

we obtain

and (39) follows, for t ∈ [0, T ].

Proof Differentiating (7) twice, we find

Next, combining Lemma 5.3 with (24), we find

Using the above inequality and (49), we find (48).

From inequality (48), we deduce Lemma 5.4 V ε is uniformly bounded from below and more precisely, with V 0 ε (y) := V ε (t = 0, y), we have

The proof follows closely the method of section 5.3. The formal idea is the following. If, for some t > 0, V ε (t, •) reaches its minimum at ȳ ∈ R n , from (48) we obtain ∂ t V ε (t, ȳ) ≥ 0. Formally, it shows that the minimum value of V ε is increasing with time, that is, inf y V ε (t, y) ≥ inf y V ε (0, y). Then, we conclude with the fact that inf y V ε (0, y) is bounded, uniformly in ε > 0.

Proof Differentiating (55) in t, we obtain

In particular, our assumptions imply inf y∈R n V 0 ε > -∞ uniformly in ε > 0, thus (50) implies that V ε is bounded from below, uniformly in ε.

We prove (50) by contradiction. We assume that there exists (T, y 0 ) ∈ (0, +∞) × R n such that

For β > 0, α > 0 small and for t ∈ [0, T ], y ∈ R n , we also introduce ϕ α,β (t, y) := V ε (t, y) + αt + β|y -y 0 |.

From (51) and for a fixed ε > 0, we have V ε (t, y) is bounded from below uniformly in t ∈ [0, T ], y ∈ R n . Therefore, ϕ α,β goes to +∞ as |y| → +∞ and reaches its minimum on [0, T ] × R n at a point ( t, ȳ). We have

We choose α small enough to ensure ϕ α,β (T, y 0 ) < ϕ α,β (0, y 0 ), which is possible thanks to assumption (52). It implies t > 0.

) stands for the left-derivative). From (48) at ( t, ȳ), using (53), Lη . As β goes to 0, we obtain α ≤ 0, which is absurd. The proof is thereby achieved.


Merzoug Soltane
email: merzoug-soltane@univ-eloued.dz

Yudith Cardinale
email: ycardinale@usb.ve

Rafael Angarita
email: rafael.angarita@isep.fr

Philippe Rosse
email: philippe.roose@univ-pau.fr

Marta Rukoz
email: mrukoz@dauphine.fr

Derdour Makhlouf

Kazar Okba
email: kazarokba@gmail.com

A Self-adaptive Agent-based System for Cloud Platforms

Cloud computing is a model for enabling on-demand network access to a shared pool of computing resources, that can be dynamically allocated and released with minimal effort. However, this task can be complex in highly dynamic environments with various resources to allocate for an increasing number of different users requirements. In this work, we propose a Cloud architecture based on a multi-agent system exhibiting a self-adaptive behavior to address the dynamic resource allocation. This self-adaptive system follows a MAPE-K approach to reason and act, according to QoS, Cloud service information, and propagated run-time information, to detect QoS degradation and make better resource allocation decisions. We validate our proposed Cloud architecture by simulation. Results show that it can properly allocate resources to reduce energy consumption, while satisfying the users demanded QoS.

I. INTRODUCTION

Cloud computing is the most flexible computing model to provide on-demand platforms for accessing custom resources and applications in the form of several types of services. Most common services in the Cloud are Platform as a Service (PaaS), Infrastructure as a Service (IaaS), Software as a Service (SaaS), Data as a Service (DaaS). The constantly increasing number of such services on the Cloud arises new challenges for ensuring their correct use, while guarantying self-adaptivity property, especially to manage user requirements (e.g., ensure QoS) [START_REF] Gutierrez-Garcia | Agentbased cloud service composition[END_REF] and optimize the use of resources (e.g., by minimizing energy consumption [START_REF] Srikantaiah | Energy aware consolidation for cloud computing[END_REF]). Taking into account these aspects, resource allocation becomes a complex task in highly dynamic environments, because of the increasing number of managed resources, as well as the increasing number of users with different requirements.

Existing resource allocation strategies lead to two unwanted situations: (i) the waste of resources when the demand is more than the capacity and demands cannot be attended, and (ii) the waste of resources when the demand is less than the capacity. In the first case, there are users not satisfied; in both cases, there are idle resources that are misused. A self-adaptive approach can direct the system to an ideal situation in which capacity is dynamically adapted to the demand of resources.

The main aim of this work is to address the resource allocation in Cloud platforms, towards a better utilization of resources and satisfaction of users requirements, by offering a self-adaptive approach. We propose a Cloud computing architecture that benefits from the combination of the Cloud and multi-agent technologies. The self-adaptive multiagent system integrated into the proposed Cloud architecture is inspired on our previous work presented in [START_REF] Angarita | A knowledge-based approach for selfhealing service-oriented applications[END_REF]. This self-adaptive system is a MAPE-K-based (monitoring, analyzing, planning, executing, and knowledge) approach [START_REF] Didac | Mape-k formal templates to rigorously design behaviors for self-adaptive systems[END_REF] to reason and act, according to QoS (e.g., time, price), Cloud service information, and propagated run-time information, to detect QoS degradation, failures, or unavailability and make better resource allocation decisions for all users requests. The agents application knowledge for decision making comprises offline precomputed global and local information, user QoS preferences, and propagated actual resource state information. We validate our proposed multiagent based Cloud architecture by prototyping and simulation. Results show that it can properly allocate resources to reduce energy consumption, while satisfying the QoS demanded by users.

II. RELATED WORK

Self-adaptive and autonomous management of computing resources have been topics of research interest and development for many years. In this section, we present recent literature concerning self-adaptive systems and agent-based approaches for cloud computing. A. Self-adaptive system in Cloud computing

In [START_REF] Kertesz | An interoperable and self-adaptive approach for sla-based service virtualization in heterogeneous cloud environments[END_REF], a self-adaptive approach for Service-Level Agreement (SLA) based service virtualization in Cloud environments is proposed. The pro-posed architecture allows the interoperability of service executions in heterogeneous, distributed, and virtualized environments. Inter Cloud is a selfmanagement and SLA handling approach proposed in [START_REF] Buyya | Intercloud: Utility-oriented federation of cloud computing environments for scaling of application services[END_REF]. It is a Cloud federation oriented provisioning environment, that offers just in-time, opportunistic, and scalable application services. The idea of Inter Cloud is to envision utility-oriented federated IaaS systems that are able to predict application service behavior for intelligent down-and up-scaling infrastructures. Though it addresses self-management and SLA handling, the unified utilization of other services like PaaS and SaaS are not studied. These SLA-based works demonstrate that the combination of negotiation, brokering, and deployment using SLA-aware extensions and autonomic computing principles are suitable for achieving reliable and efficient service operation in distributed environments. However, its application in resource allocation has not been proved.

An approach for self-adaptive and selfconfigurable CPU resource provisioning for virtualized servers using Kalman filter 1 is presented in [START_REF] Kalyvianaki | Self-adaptive and selfconfigured CPU resource provisioning for virtualized servers using kalman filters[END_REF]. That work deals with the integration of a Kalman filter into feedback controllers to dynamically allocate CPU resources to virtual machines hosting server applications, creating a new resource management scheme. The novelty of this approach is the use of the Kalman filter to track the CPU usage and update the allocations accordingly. Another approach based on monitored data is described in [START_REF] Wang | Self-adaptive cloud monitoring with online anomaly detection[END_REF], which consists on a self-adaptive Cloud monitoring approach with on-line anomaly detection. The main contribution of this work is the design of a self-adaptive monitoring framework, that can efficiently collect monitoring data from various distributed systems deployed in a Cloud computing environment. The authors focused on a correlation-based method to select key metrics representing other metrics. The problem with these works relies on the limitation of their applications. They are not generic and applicable for all kind of resource allocation.

Other works focus on QoS compliant [START_REF] Evans | Dynamically reconfigurable workflows for time-critical applications[END_REF], [START_REF] Spezzano | Using service clustering and self-adaptive mopso-cd for qos-aware cloud service selection[END_REF]. In [START_REF] Evans | Dynamically reconfigurable workflows for time-critical applications[END_REF], it is proposed a distributed self-adaptive architecture based on the Edge Computing concept with container-based technologies, such as Docker and Kubernetes, to ensure QoS for time-critical applications. For each container, features of resources required for the host can be allocated upon monitoring data and operational strategies defined by end-users, application developer, and administrator. In [START_REF] Spezzano | Using service clustering and self-adaptive mopso-cd for qos-aware cloud service selection[END_REF], it is proposed a QoS-aware Cloud Service Selection approach using service clustering 1 Kalman filtering is an algorithm that uses a series of measurements observed over time and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone. and self-adaptation to efficiently support service selection in which runtime changes in the services QoS are taken into account to adapt to changes. As our work, these studies focus on QoS satisfaction; however, they do not consider local and remote information for more accurate decisions.

Based on the MAPE model, concepts of autonomic computing for adaptive management of Grid Computing are leveraged in [START_REF] Lee | Adaptive workflow processing and execution in pegasus[END_REF]. However, this approach does not consider the deployment and virtualization mechanism in Cloud datacenters.

B. Agent-based Clouds

Agents are networked software entities that can do specific tasks on behalf of a user and have a degree of intelligence that allows them to perform parts of their tasks autonomously and to interact with their environment in a successfully way. Agents are characterized by important features such as autonomy, sociality, rationality, responsiveness, proactiveness, and mobility [START_REF] Rodriguez-Aguilar | A survey on sensor networks from a multiagent perspective[END_REF]. Some studies have been proposed to design Cloud platforms based on multi-agent systems [START_REF] Al-Ayyoub | Multi-agent based dynamic resource provisioning and monitoring for cloud computing systems infrastructure[END_REF], [START_REF] Fortino | Integration of agent-based and cloud computing for the smart objects-oriented iot[END_REF], [START_REF] Singh | Autonomous agent based load balancing algorithm in cloud computing[END_REF]. In [START_REF] Fortino | Integration of agent-based and cloud computing for the smart objects-oriented iot[END_REF], it is proposed the integration of agent-based system and Cloud computing for smart objects. This approach is suitable to effectively model Cooperative Smart Objects (CSO). In particular, a CSO is a smart object able to sense, store, and interpret information. In [START_REF] Singh | Autonomous agent based load balancing algorithm in cloud computing[END_REF], it is presented a mechanism to provide dynamic load balancing for Clouds based on autonomous agents. The proposed mechanism is based on Ant mobile agents and whenever the load of a Virtual Machine (VM) reaches a threshold value, it initiates a search for a candidate VM from other datacenters, reducing in this way the allocation time. This work only considers the workload of VMs and does not consider the physical machines. In [START_REF] Al-Ayyoub | Multi-agent based dynamic resource provisioning and monitoring for cloud computing systems infrastructure[END_REF], a multi-agent system is proposed to manage the Cloud resources, while taking into account the customers QoS requirements. In this work a VM migration occurs when its hosting physical machine is facing an overloading or under loading problem. This approach is the most similar to ours, however, our approach is autonomic self-adaptive, allowing controlling the user requirements depending on the system and agents states.

To overcome the limitation of existing works in the field of self-adaptive model for Cloud computing, we propose a novel approach for the management of resources in Cloud platforms. Our approach is based on a multi-agent system implementing the MAPE-K model to enable self-adaptation. Particularly, we enhance the Cloud computing architecture with agent technologies to tackle the energy of consumption problem.

III. AGENTS FOR THE CLOUD

One of the foundations of autonomic computing is the MAPE-K model for building autonomous self-adaptive agents [START_REF] Didac | Mape-k formal templates to rigorously design behaviors for self-adaptive systems[END_REF]. MAPE-K-based agents can sense diverse properties of the environment and make rational decisions based on them. Such an agent can monitor changes on the environment where it exists, analyze the situation based on the sensed information, plan what it is going to do next based on this situation and its own capabilities and options, and execute the plan. All of these steps are supported by the knowledge the agent has about the environment and itself. This model is usually implemented as a loop to permanently sense the environment and adapt to changes.

Figure 1 shows our proposed general multi-agent Cloud Architecture, based on three layers: user interface, Cloud instances, and datacenter infrastructure. The user layer provides an interface to access the Cloud services and through which users specify their resource allocation needs. Analyzer Agents identify the resources and services demanded by users and build specific queries. Cloud instances layer represents an inter-layer between upper layer (i.e., users) and lower layer (i.e., datacenters). Scheduler Agents work at this layer. Datacenter layer provides resources as a service. It consists of a physical layer (e.g., servers, host, physical manage) and a virtual layer (i.e., instances of VMs). There exists a Controller Agent per each physical machine in datacenters. Additionally, the Coordinator Agent supervises the whole process. Figure 2 shows our proposed agent architecture blueprint. Internal agent components rely on a modular architecture, favoring the understanding and reuse of the main components of each agent. Each component focuses on a specific function required for accomplishing the purpose of the agent. Each agent can communicate with other agents through its communication interfaces. In this section we detail the analyzer agent, the scheduling agent, and the controller agent. The coordinator agent supervises the whole process.

A. The Analyzer Agent

The analyzer agent is the first agent launched by the system. We illustrate its detailed architecture in -Individual knowledge: it reflects the agent selfknowledge, including the following:

• name: Agent analyzer (AA);

• address: the location where it is deployed; e.g., web Cloud service interface; • individual objectives: its objective is analyze the resources requested by a user; • state: it can have the following execution states: receiving user request, sending resource allocation request, and getting resource allocation response to/from the Scheduling Agent. -Analytical knowledge: it concerns rules stored in its knowledge base dictating its behavior and actions.

Its detailed behavior is illustrated in the algorithm presented in Alg. 1. It receives the user requirement, which is analyzed by the Request Analyser component (lines 2 to 4 in Alg. 1), by identifying the required resources (line 5 in Alg. 1). This information is received by the Resources Requester module (line 9 in Alg. 1), which in turn build a request with the required resources to send it to the Scheduling Agent (lines 10 and 11 in Alg. 1).

B. The Scheduling Agent

The Scheduling Agent is the central agent in the system. Its goal is to allocate resources needed by users. We illustrate its architecture in Figure 4. The Resource Allocator module receives requests from the Analyzer Agent and makes allocation decisions respecting the requested resources and QoS, and according the available resources. The Resource Availability Monitor is in charge of updating the resources state.

The knowledge base of the scheduling agent is composed by two parts: -Individual knowledge: reflects the agent selfknowledge, including:

• name: Agent Scheduling (SA);

• address: the location where it is deployed; e.g., the resource management system in the data center; • individual objectives: its objective is the resource allocation; • state: this agent can have the following execution states: receiving request from the Analyzer Agent, sending a request of available resources, and waiting for a response to that request. -Allocation knowledge:

• Available resources: the available resources on all physical machines; • Self-adaptive options: for example, retry, resource replacement or reallocation. • Inference engine and rules: the inference engine applies rules to the knowledge base to deduce new information. Rules can specify the actions to take given certain conditions. Its detailed behavior is illustrated in Figure 5 and in the Alg. 2. If it receives a resource request, it checks the availability of those resources and sends an allocation request to the Controller Agent (lines 3 to 5 in Alg. 2). When it receives from the Controller Agent the availability of resources, it updates the information in its knowledge base and does the reserves the resources (allocation process) for the corresponding query (lines 8 to 16 in Alg. 2). For the allocation process, the Scheduling Agent verifies the system state and the SLA (see Alg. 3). If the system state is normal and the SLA = acceptable, it allocates the resources (lines 1 to 7 in Alg. 3); if the system state is degraded, but the SLA = acceptable, it allocates the resources (lines 8 to 12 in Alg. 3); otherwise it applies its rules and acts according the available options ( lines 13 to 17 Alg. 3).

C. The Controller Agent

The goal of the controller agent is to manage datacenter resources, by tracking their status; through the Resource Status Updater component. Figure 6 illustrates its detailed architecture. The knowledge base of this agent contains the following information: -Individual knowledge: reflects the agent selfknowledge, including:

• name: Controller Agent (CA);

• address: the location where it is deployed; e.g., physical servers in the data center; • individual objectives: the objective of this agent is to track resource status; • state: this agent can have the following execution states: receiving the request from the Scheduling Agent and sending the available resources to Scheduling Agent. -Controller knowledge: it refers to the information this agent has about the status of the different resources in the datacenter.

Finally, its detailed behavior is illustrated in the algorithm presented in Alg. 4. It permanently monitors the resources' states and updates this information in its knowledge base (lines 7 to 15 in Alg. 4) . When a request from the Scheduling Agent arrives, it responds with the resources' statuses (line 18 in Alg. 4).

D. Resource allocation and inter-agent Interaction

The interaction among agents is illustrated in Figure 7. The interaction process starts when the Analyzer Agent receives a user request, which is analyzed, interpreted, sent it to the Scheduling Agent. The Scheduling Agent verifies the system state (state == normal) and sends a request to the Controller Agent asking for information about the free resources. As a reminder, the Controller Agent tracks the status of resources in the datacenter. When the Scheduling Agent gets the response from the Controller Agent, it verifies if the proposed QoS and available resources match the user requirements before making a final decision about resource allocation. Finally, the Scheduling Agent sends a detailed report about the process and decisions made to the Coordinator Agent that supervises the whole process. The possible agent states are normal, degraded, and broken. An agent is in the normal state if the execution goes as expected concerning QoS allocation and system failures. An agent is in the degraded state if there is a degraded QoS allocation or failures in the system, but the QoS allocation still falls within the allowed range. Finally, an agent is in the broken state if the QoS allocation is not acceptable or there are irreparable failures in the system.

IV. ILLUSTRATIVE EXAMPLE

To illustrate our approach, we propose a scenario of resource allocation, shown in Figure 8. The steps of the allocation process are the following:

1) In the first step, a user requests the desired services and resources through a web interface, by expressing the services/resources needed and the expected QoS (e.g., desired resource capacity, acceptable range of capacity, acceptable time and price). The Analyzer Agent gets this request and studies the existing services/resources to extract the requirements concerning the needed resources and the expected QoS. Some examples of resources are CPU, RAM, and disk space. At the end of this step, the Analyzer Agent builds a query, called Q, specifying the needed resources and expected QoS, and forwards it to the Scheduling Agent. We represent the combination of resources and their corresponding requested capacities and QoS to build the suitable V M R as:

V M R =      r 1 c 1 qos 1 r 2 c 2 qos 2 . . . . . . r n c n qos n     
3) After the identification of requested resources, the Scheduling Agent sends a request to the Controller Agent to demand the corresponding free resources.

4) The Controller Agent receives the request from the Scheduling Agent and responds with the current state of the physical machines.

There exists one Controller Agent, CA i , for each physical machine, P M i , in the Cloud. CA i manages the resources and information of its corresponding P M i , including:

• The set of virtual machines that run in P M i , denoted as

V i = (V M 1 , V M 2 , ..., V M m );
• The set of capacities corresponding to each virtual machine, denoted as

C i = (c.V M 1 , c.V M 2 , ..., c.V M m ); • The
T.P Mi =<      V M1 c.V M1 V M2 c.V M2 . . . . . . V Mn c.V Mn      ,      ra1 ca1 ra2 ca2 . . . . . . rap cap      > 5)
In the next step, the Scheduling Agent stores the information received from the Controller Agent in its knowledge base; then, it builds a global matrix of available resources at time t, denoted as A(t):

A(t) =      P M 1 RA 1 CA 1 P M 2 RA 2 CA 2 . . . P M n RA n CA n     
where P M i are physical machines and RA i and CA i are the set of available resources with their corresponding capacities. 6) Finally, after building the global resource matrix, the Scheduling Agent applies its selfadaptive allocation algorithm to build from A(t) a V M that satisfies V M Q .

V. EXPERIMENTS

We measured the performance of our multi-agent system in terms of energy consumption, by simulation. To do so, we used OMNeT [START_REF] Varga | An overview of the omnet++ simulation environment[END_REF] and ICan Cloud [START_REF] Núñez | icancloud: A flexible and scalable cloud infrastructure simulator[END_REF] to simulate its functionalities. OMNeT is an extensible, modular, component-based C++ simulation library and framework, primarily for building network simulators. ICan Cloud is a simulation platform aimed to model and simulate Cloud computing systems. This simulation framework has been developed on the top of OMNeT++ and INET frameworks. The multi-agents system and the scheduling algorithm have been developed in C++, by using sockets to simulate their communication.

We simulated a scenario for allocating resources as mentioned in Section IV. Subsequently, we implemented a Cloud computing service provider to offers Cloud services to users. The data center implemented for this Cloud runs five heterogeneous servers. We randomly generated queries (Q = (R, C, QoS)) to submit to the Cloud in randomly time intervals (in seconds) and measured the aggregated energy consumption of the five servers.

Figure 9 shows an extract of the simulation running. Note that queries are submitted at different time intervals: the first one at time 0, the second one at 500ms, the third one at 5sec, and so on until the end of the simulation at 102min. All queries were satisfied in terms of needed resources and required QoS. Figures 10 and11 show the energy consumption rates in Server 1 and Server 2, respectively. When we apply our self-adaptive strategy for resource allocation, we observe a remarkable change in the level of energy consumption, showing the efficiency of the scheduling algorithm. For Server 1 (Figure 10), the decrease of energy consumption was from 2500 megawatts to 480 megawatts, while for Server 2 (Figure 11), it was from 3500 megawatts to 600 megawatts. These decreases of energy consumption happen at times ranging from 0 minutes to 102 minutes.

These results put in evidence the performance of our algorithm of QoS allocation, confirming the choice of using a multi-agent system implementing a self-adaptive mechanism for the smart management of resources in Cloud data centers.

VI. CONCLUSIONS

In this work, we tackled the problem of resource allocation in Cloud platforms. To that end, we have proposed a multi-agent architecture based on the MAPE-K model. For our future work, we plan to present a formal model of our approach defining the query for resource allocation, user preferences, agent rules, and state transitions. We also plan to describe in detail the scheduling algorithm and present additional experiments to evaluate other performance aspects, besides energy consumption.

Figure 1 .

 1 Figure 1. Agent-based Cloud Architecture.

Figure 2 .

 2 Figure 2. Agent Architecture Blueprint.

Figure 3 .

 3 Figure 3. Analyzer Agent Architecture.

Figure 3 .

 3 Figure 3. Its goal is to study a user requirement (done by the Request Analyzer Component) and to build a request specifying the resources needed to satisfy it (through the Resources Requester component). This agent relies on a knowledge base containing individual knowledge and analytical knowledge, as follows:-Individual knowledge: it reflects the agent selfknowledge, including the following:

Input: 2 case input interface do 3 if service description then 4 analyzerProcess 1 :

 2341 (i), service description; (ii), SAL; (iii), id. user Output : (i), needed resources 1 switch communication interface do (service description); 5 return resources = { CPU, RAM, disk,... }; Analyzer Agent (AA)

Figure 4 .

 4 Figure 4. Scheduling Agent Architecture.

Figure 5 .

 5 Figure 5. Scheduling Agent Behavior (General Description).

2 case input interface do 3 if resources request then 4 check available resources; 5 send 8 case input interface do 9 if 13 return TAB.resources; 14 store resources' data in the knowledge base; 15 //allocation process 16 if available resources ≡ needed resources then 17 Alg 2 :

 23458913141516172 Input : (i), resources request; (ii), available resources Output : (i), allocated resources 1 switch communication interface do request to the Controller Agent; analyzerProcess returns resources then 10 TAB.resources = [null]; 11 while receiving response from the controller agent do 12 TAB.resources = [CA1.resources, CA2.resources,...]; Scheduling Agent (SA)

3 :Figure 6 .

 36 Figure 6. Controller Agent Architecture.

2 case input interface do 3 if 15 store 4 :

 23154 Input: (i), Scheduling Agent's request Output : (i), available resources response 1 switch communication interface do Scheduling resources.stat's data in the knowledge base; Controller Agent (CA)

Figure 7 .

 7 Figure 7. Inter-agent Interaction.

Figure 8 .

 8 Figure 8. Allocation Scenario.

Figure 9 .

 9 Figure 9. Extract of the Simulation Output.

Figure 10 .

 10 Figure 10. Energy consumption Server 1.

Figure 11 .

 11 Figure 11. Energy consumption Server 2.

  = (ra 1 , ra 2 , ..., ra p ); • The set of capacities corresponding to each available resource, denotes as CA i = (ca 1 , ca 2 , ..., ca p ).

		set of available resources
	on	in	P M i ,	denoted	as
	RA i Then, the Controller Agent sends this infor-
	mation to the Scheduling Agent. We model
	the response of a Controller Agent CA i as
	two matrices: