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

We study an equation structured by age and a phenotypic trait describing the growth process of a population subject to aging, competition between individuals, and mutations. This leads to a renewal equation which occurs in many evolutionary biology problems. We aim to describe precisely the asymp-totic behavior of the solution, to infer properties that illustrate the concentration and adaptive dynamics of such a population. This work is a continuation of [38] where the case without mutations is considered. When mutations are taken into account, it is necessary to control the corrector which is the main novelty of the present paper. Our approach consists in defining, by the Hopf transform, a Hamilton-Jacobi equation with an effective Hamiltonian as in homogenization problems. Its solution carries the singular part of the limiting density (typically Dirac masses) and the corrector defines the weights. The main new result of this paper is to prove that the corrector is uniformly bounded, using only the global Lipschitz and semi-convexity estimates for the viscosity solution of the Hamilton-Jacobi equation. We also establish the limiting equation satisfied by the corrector. To the best of our knowledge, this is the first example where such bounds can be proved in such a context.


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 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.
( 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 [38], 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 [38,40], 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 (7)) involving the effective fitness Λ (defined in (6)). 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 (6)). 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 (7) 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 (8).
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 .
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)  Notice that we establish convergence of the full families U ε and ρ ε , without using the famous uniqueness result of [14], 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 (11). 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 [39] for a study of the linear equation). This model has been extended with other structuring variables, as size [32,37], DNA content, maturation, etc., in the context of cell divisions [21], or proliferative and quiescent states of tumor cells [2,26]. Space structured problems have also been extensively studied [27,34,35,40].
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 [16,19,20,30]). 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 [10].
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 phenotypeȳ(t) which maximizes the ecological fitness U (t, ·) can survive. On an evolutionary time scale, we observe the dynamics ofȳ(t). Similar results as Theorem 1.3 have been obtained for various models with parabolic equations [5,7,30] and integrodifferential equations [6,18,29]. More generally, convergence to positive measures in selection-mutation models has been studied by many authors [1,11,12]. The special case of age-structured populations are also considered in [13,31,41].

A formal presentation of the method
To analyze the singular perturbation problem at hand, the usual approach relies on the WKB change of unknown ( [7,20]), 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 ⇀ δȳ =y as ǫ → 0. This approach has been extensively used in works on a similar issue, see for instance [4,8,17,24]. 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 [22] 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 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 [33]).
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).
With the change of variable z = y ′ −y ε , the renewal term becomes When ε is small, we can formally approximate where Then, formally putting ε∂ t p ε = O(ε) in the first line of (4), we end up with the following approximate problem 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 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 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 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 with the Hamiltonian H(y, p) defined by 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 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.
The right hand side of the first line can be controlled with the available a priori bounds on U ε . Indeed, integrating equation (6) to obtain AQ, we compute 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 (2). 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 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 , is a Gaussian probability kernel, for some constants K > 0, r > 0 and where η is determined from the initial conditions, see (18). These assumptions can be substantially generalized. For instance, using the formula (obtained by integrating the first line in (6)) the relation between b ⋆ and d ⋆ is only used to ensure the inequality which is sufficient in the sequel, see (18).

Assumptions on the coefficients
We assume that for some d(·, y), A(·, y) are bounded from above and below in some interval of R + , uniformly in y.
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 [38].
In addition, we assume where x defines the largest support of b(x, y), 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 ∂η Q 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, the eigenvalue Λ(y, η) is defined through the relation We assume there exists Λ < 0 such that 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, and p 0 ε such that, for some γ 0 , γ 0 > 0: where Q is defined through (6) and we define Another condition is also required on η 0 ε , see (18). Note that (15) and (16) ensures that ρ 0 ε := R+×R n m 0 ε is uniformly bounded. Note also that assumption (15) 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 (17), 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 , This assumption implies η ≤ η 0 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 assumption (18) can be written 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 (26), we assume that there exists two constants l, 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 (6) 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.
Differentiating the relation (23), we immediately obtain the derivatives of Λ 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 Proof We use the short notations F := F (y, Λ(y, η)) (where F is defined in (12)) and Λ := Λ(y, η). Differentiationg the first relation in (26) with respect to η, we find Combining it with the expression of ∂ η Λ(y, η) in (26) and (23), we deduce Using the Cauchy-Schwarz inequality and the definition of F in (12), we have and the proof of (27) is completed.

Construction of U ε and a priori estimates
We give a rigourous definition of U ε , formally introduced in (7).
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. (18), and setting we have Proof Simply use Proposition 2.2, Λ ε = −∂ t U ε and assumption (18).

Further estimates
Proposition 2.4 Under the assumptions of section 1.5, with k 0 defined in (13), L, l > 0 in (20) and η in (29), we have 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.
The following corollary is essential when studying the corrector in section 3.

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 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.
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 (11)).
Proof The function p ε (t, x, y) satisfies the following equation, for ε > 0, t > 0, x > 0, y ∈ R n , and Q ε satisfies, for (t, y) as parameters, the equation in the variable x Setting γ ε (t, x, y) := p ε (t, x, y) Q ε (t, x, y) , where 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 .
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 where γ, γ are given by the previous lemma. 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.
Hence, on the one hand, we have On the other hand, 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, Proof The first point is deduced from Lemma 3.2 and Q ≤ γ η A , with A from assumption (9). To prove the second point, we only need to estimate the integrals of Q and Q. Recalling Λ < 0, we compute which proves the bound from above. For the other inequality, we use the non-degeneracy assumption (10) 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 The difficulty in stating rigorously equation (33) 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 (33), 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 [7], 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 (33) we use the dual problem associated with (6); for fixed (y, η) ∈ R n × (0, +∞) consider Φ(x, y, η) the unique solution of We can solve this ordinary differential equation and find Also, multiplying equation (34) by ∂ η Q and integrating by parts, we get and, multiplying equation (34) 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 (31). 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 (31), 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 x, y) := Φ(x, y, η ε (t, y)).

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.
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 In addition, with S defined in (2), we have Remark 4.2 T,he proof of Proposition 4.1 becomes much simpler if we assume that there are r, r > 0 such that Indeed, integrating (1) and using an integration by parts, we obtain 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 (38).
Proof (of Proposition 4.1) We recall that Multiplying by e From 0 < −Λ ≤ ∂ t U ε ≤ −Λ (Proposition 2.2) and the L 1 (dx) estimate on p ε (Proposition 3.3), we have Dividing by εe t 0 ρε ε on both sides we find Then, from assumption (15), we deduce Identically, we infer Now, from the definition of ρ ε (recalled in (39)) and the L 1 estimate on p ε (Proposition 3.3) we find 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 We deduce 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 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.

Adaptive dynamics
We now prove the third statement of Theorem 1.3. We need further assumptions on the initial conditions, y 0 ε converges to someȳ 0 ∈ R n when ε vanishes, , there exists a uniqueȳ(t) ∈ R n on which U (t, ·) reaches its maximum. Moreover, t →ȳ(t) ∈ C 1 and satisfies the Canonical Equation .
Note that (42) 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 [7,14,36].
Proof Since U (defined in Proposition 2.7) satisfies (30) 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 [23]). Consider such a time interval [0, T ], and V ⊂ R n a neighborhood of y 0 . We are interested in the solutions From (40) we know that at initial time there exists a unique solutionȳ 0 of (43). Besides, ∇ 2 y U 0 (ȳ 0 ) is invertible. From the implicit functions theorem, there exists a uniqueȳ(t) ∈ R n satisfying (43), for t in a certain time interval still denoted [0, T ]. We can again choose a smaller T to ensure thatȳ(t) remains in V .
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 (30). 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 (44). Finally, the possible blow-up in finite time of y(t) from the dynamics of the Canonical Equation (42). Regarding the last point, we point out that Λ can sometimes be used as a Lyapunov function. Indeed, we have In particular, if M (·) is even, then d dt [Λ(ȳ(t), 1)] ≤ 0. Thus, ifȳ 0 belongs to a "well" of Λ, thenȳ(t) remains "trapped", which prevents from an blow-up in finite time. It also implies, at least formally, thatȳ(t) converges to a local minimum of Λ(·, 1) when t → +∞.
5 Construction, estimates, and asymptotics of U ε -Proof of Theorem 1.1
For later purpose, we also neeed the following stronger version of (27).

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 [28]. 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 where B y and B y are chosen to be positive, increasing, bounded by 1, and such thatΛ 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 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: For ε > 0, we consider the Cauchy problem for which the classical Cauchy-Lipschitz theorem provides existence and uniqueness of a solutionŨ 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 (17), The full proof of this statement, which is technical, can be found in [38] (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) := ∂ tŨ R ε (t, y). Differentiating (46) with respect to t, we obtain Then, if for some t > 0, V (t, ·) reaches its maximum atȳ ∈ R n , we obtain the inequality Formally, it shows that the maximum value of V is decreasing with time, that is, 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 (18) and ∂ λ F > 0, we have Using Lemma 5.2, we infer Removing the truncation. From (47) and the choice of φ R , for R large enough, we have for all R large enough, ε > 0, t ≥ 0, y ∈ R n . Thus, from the definition ofΛ in (45), we have that is,Ũ R ε is a solution of (7). 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 (20), (21), (26) and (29). In addition, we first need to prove the estimate on a truncated function, then to remove the truncation.
Differentiating (7), we obtain We formally define a truncated problem, for R > 0, and its solution W R ε satisfying where F is defined above and φ R is a truncation function as in (46). We can prove existence and uniqueness of a global solution of (48) by a direct application of the Cauchy-Lipschitz theorem.

Semi-convexity -proof of Proposition 2.5
For convex Hamiltonian, the semi-convexity of the solution is a classical matter, [3,15]. 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 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 Proof Use Jensen's inequality and the definition of η ε in (28).
Next, we prove that Proof Differentiating (7) twice, we find Next, combining Lemma 5.3 with (27), we find Using the above inequality and (52), we find (51).
From inequality (51), 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 atȳ ∈ R n , from (51) 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 (58) in t, we obtain In particular, our assumptions imply inf y∈R n V 0 ε > −∞ uniformly in ε > 0, thus (53) implies that V ε is bounded from below, uniformly in ε.
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 [15]). We obtain that U ε is uniformly bounded in W 2,1 loc .
Using that U ε is uniformly bounded in W 2,1 (Proposition 2.5) and that −∂ η Λ is positively bounded (26), 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.

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 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 ′ |.
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 (30), i.e., U satisfies ∂ t U = H(y, ∇ y U ), U (0, y) = U 0 (y), with H(y, p) := −Λ y, The proof is adapted from classical stability results for viscosity solutions of Hamilton-Jacobi equations (see [3]). 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.
Uniqueness We point out that the Hamiltonian H is Lipschitz continuous in the y variable. We introduce a truncated HamiltonianH