Jean-Baptiste Burie

Arnaud Ducrot

Quentin Griette
email: quentin.griette@univ-lehavre.fr

Epidemic models in measure spaces: persistence, concentration and oscillations

Keywords: AMS subject classifications (2020). 34D05, 92D25, 37L15, 37N25 Keywords. ordinary differential equations, asymptotic behavior, population dynamics, Radon measure, evolution

We investigate the long-time dynamics of a SIR epidemic model in the case of a population of pathogens infecting a homogeneous host population. The pathogen population is structured by a genotypic variable. When the initial mass of the maximal fitness set is positive, we give a precise description of the convergence of the orbit, including a formula for the asymptotic distribution. We also investigate precisely the case of a finite number of regular global maxima and show that the initial distribution may have an influence on the support of the eventual distribution. In particular, the natural process of competition is not always selecting a unique species, but several species may coexist as long as they maximize the fitness function. In many cases it is possible to compute the eventual distribution of the surviving competitors. In some configurations, species that maximize the fitness may still get extinct depending on the shape of the initial distribution and some other parameter of the model, and we provide a way to characterize when this unexpected extinction happens. Finally, we provide an example of a pathological situation in which the distribution never reaches a stationary distribution but oscillates forever around the set of fitness maxima.

Introduction

In this article we investigate the large time behavior of the SIR epidemic model

:main             
S t (t) = Λ -θS(t) -S(t) X β(x)I(t, dx), I t (t, dx) = β(x)S(t)I(t, dx) -γ(x)I(t, dx), R t (t) = X γ(x)I(t, dx), (1.1a) with the initial data S(0) = S 0 ∈ (0, +∞), I(0, dx) = I 0 (dx) ∈ M + (X), R(0) = R 0 ∈ (0, +∞), (1.1b) where X is a Polish space and M + (X) is the set of nonnegative Borel measures on X. This model describes the evolution of a population of hosts that can be, at any time t > 0, either free of infection (S(t), the susceptible population), infected by a pathogen of type x ∈ X (I(t, dx), the infected population of type x) or removed from the system (R(t), the recovered population), the latter class including two possible outcomes of the infection, complete immunity or death. The parameter Λ > 0 models a constant influx of susceptible hosts, θ > 0 the death rate of the hosts in the absence of infection, β(x) the transmission parameter of the pathogen of type x, and γ(x) the recovery rate of a pathogen of type x. Both β(x) and γ(x) are bounded continuous functions. SIR models are ubiquitous in the literature concerning mathematical epidemiology and have been extensively studied. Without the pretention of reconstructing the entire history of the model, let us cite the works of Kermack and McKendrick [29] that might well be its first occurrence in the literature, and was immediately applied to a plague outbreak in the island of Bombay.

In this article we consider that the phenotype of the pathogen (that is, the values of β(x) and γ(x)) depends on an underlying variable x ∈ X, where X is a set of attainable values that possesses a few mathematical properties. This variable x may be a collection of quantitative phenotypic traits involved the mechanism of transmission, reproduction or replication of the pathogen (expression of surface protein at the cellular or viral level, impact on the host's behavior, ...) or the underlying genes that determine the values of β(x) and γ(x). We do not specify the particular mechanisms that link the underlying variable x ∈ X and the phenotype (β(x), γ(x)) but focus on the dynamics of the population under (1.1) conditionally to the knowledge of these mechanisms. By analogy, because it stands for a hidden process that determines an observable quantity, x will be called the genotypic variable, even if it could well stand for hidden quantitative phenotypic variables. We do not take mutations into account and consider that the pathogen is asexual; therefore, (1.1) can be considered a pure competition model where the pathogens compete for a single resource (the susceptible hosts).

When I 0 (dx) is a finite collection of Dirac masses,

I 0 (dx) = n i=1 I i 0 δ xi (dx),
our problem is reduced to a system of ordinary differential equations,

inite                      d dt S(t) = Λ -θS(t) -S(t) β 1 I 1 (t) + β 2 I 2 (t) + . . . + β n I n (t) d dt I 1 (t) = β 1 S(t)I 1 (t) -γ 1 I 1 (t)
. . .

d dt I n (t) = β n S(t)I n (t) -γ n I n (t), (1.2a)
with the initial data S(0) = S 0 ∈ (0, +∞), I 1 (0) = I 1 0 ∈ (0, +∞), . . . , I n (0) = I n 0 ∈ (0, +∞), (

where β i = β(x i ), γ i = γ(x i ) and I(t, dx) = n i=1 I i (t)δ xi (dx). In this context, Hsu [START_REF] Hsu | Limiting behavior for competing species[END_REF] and Hsu, Hubbell, and Waltman [START_REF] Hsu | A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms[END_REF] showed for closely related systems of ordinary differential equations that the solution eventually converges to an equilibrium which may not be unique but is always concentrated on the equations that maximize the fitness β i /γ i . One of the objectives of the current article is to establish an equivalent result in the context of measure-valued initial conditions that are not necessarily finite sums of Dirac masses (this will be given by Theorem 2.2).

In a recent work [START_REF] Burie | Asymptotic behavior of an epidemic model with infinitely many variants[END_REF] we clarified the asymptotic behavior of an extension of (1.2) in the case of an infinite number of equations. This discrete setting is a particular case of the results that we present here, and we provide a number of illuminating examples that give a glimpse of the diversity of different behaviors that can be expected for solutions to (1.1). In particular, we show that, when the assumptions of Proposition 2.7 are not satisfied, it is possible to construct exotic inital data for which the total mass of pathogen X I(t, dx) does not converge to a limit but oscillates between distinct values. We refer to [START_REF] Burie | Asymptotic behavior of an epidemic model with infinitely many variants[END_REF] for details.

The problem of several species competing for a single resource has received a lot of attention in the literature. In this context, the "Competitive exclusion principle" states that "Complete competitors cannot coexist", which in particular means that given a number of species competing for the same resource in the same place, only one can survive in the long run. This idea was already present to some extent in the book of Darwin, and is sometimes referred to as Gause's law [START_REF] Hardin | The competitive exclusion principle[END_REF]. This problem of survival of competitors has attracted the attention of mathematicians since the '70s and many studies have proved this property in many different contexts -let us mention the seminal works of Hsu [START_REF] Hsu | Limiting behavior for competing species[END_REF] and Hsu, Hubbell, and Waltman [START_REF] Hsu | A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms[END_REF] followed by Armstrong and McGehee [START_REF] Armstrong | Competitive exclusion[END_REF], Butler and Wolkowicz [START_REF] Butler | A mathematical model of the chemostat with a general class of functions describing nutrient uptake[END_REF], Hsu, Smith, and Waltman [START_REF] Hsu | Competitive exclusion and coexistence for competitive systems on ordered Banach spaces[END_REF], Li [START_REF] Li | Global asymptotic behavior of the chemostat: general response functions and different removal rates[END_REF], Rapaport and Veruete [START_REF] Rapaport | A new proof of the competitive exclusion principle in the chemostat[END_REF], Wolkowicz and Lu [START_REF] Wolkowicz | Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates[END_REF], and Wolkowicz and Xia [START_REF] Wolkowicz | Global asymptotic behavior of a chemostat model with discrete delays[END_REF], to cite a few -and also disproved in other contexts, for instance in fluctuating environments, see Cushing [START_REF] Cushing | Two species competition in a periodic environment[END_REF] and Smith [START_REF] Smith | Competitive coexistence in an oscillating chemostat[END_REF]. Ackleh and Allen [START_REF] Ackleh | Competitive exclusion and coexistence for pathogens in an epidemic model with variable population size[END_REF] study the competitive exclusion in an epidemic model with a finite number of strains, and describe how different species can coexist in some cases.

In our model, the fitness of a pathogen with genotype x is given by the formula R 0 (x) = Λβ(x) θγ(x) = Λ θ α(x), where α(x) := β(x) γ(x) ; the competitive exclusion principle implies that the only genotypes that eventually remain are the ones that maximize R 0 (x). When R 0 (x) has a unique maximum then it is clear that I(t, x) converges to a Dirac distribution concentrated at the maximum. But if R 0 (x) attains its maximum at a more complex set -from two isolated maxima to an entire line segment -then the eventual weight of each fitness maximum in the population is less clear. We will give a partial description of this distribution here, that depends on the initial repartition of infected as well as the repartition of the phenotypic value γ(x) in the vicinity of the set of fitness maxima. We will show in particular that, while it is true that the species have to maximize the fitness function in order to survive, the natural process of competition is not selecting a unique genotypic value but several may coexist as long as they maximize the fitness function. In many cases it is possible to compute the eventual repartition of the surviving competitors. In some cases, species that maximize the fitness may still get extinct if the initial population is not sufficient, and we provide a method to characterize when this unexpected extinction happens.

Considering a situation where R 0 has more than one maximum at the same exact level may appear artificial but is not without biological interest. Indeed, the long-time behavior that we observe in these borderline cases can persist in transient time upon perturbing the function R 0 . For example in the epidemiological context of Day and Gandon [START_REF] Day | Applying population-genetic models in theoretical evolutionary epidemiology[END_REF], it has been observed that a strain 1 with a higher value of γ and a slightly lower R 0 value than a strain 2 may nevertheless be dominant for some time (we reproduce such a behavior numerically in Figure 5). These borderline cases shed light on our understanding of the transient dynamics, see also Burie, Djidjou-Demasse, and Ducrot [START_REF] Burie | Asymptotic and transient behaviour for a nonlocal problem arising in population genetics[END_REF] where we explicit transient dynamics for a related evolutionary model depending on the local flatness of the fitness function.

Quantitative traits such as the virulence or the transmission rate of a pathogen, the life expectancy of an individual and more generally any observable feature such as height, weight, muscular mass, speed, size of legs, etc. are naturally represented using continuous variables. Such a description of a population seems highly relevant and has been used mostly in modelling studies involving some kind of evolution [START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result[END_REF][START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics[END_REF][START_REF] Bouin | Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration[END_REF][START_REF] Desvillettes | On selection dynamics for continuous structured populations[END_REF][START_REF] Ducrot | Differential equations and population dynamics I. Introductory approaches[END_REF][START_REF] Griette | Singular measure traveling waves in an epidemiological model with continuous phenotypes[END_REF][START_REF] Jabin | On selection dynamics for competitive interactions[END_REF][START_REF] Lorz | Long-term behaviour of phenotypically structured models[END_REF][START_REF] Lorz | Dirac concentrations in a chemostat model of adaptive evolution[END_REF][START_REF]Mutation and recombination in a model of phenotype evolution[END_REF][START_REF] Magal | Mutation, selection, and recombination in a model of phenotype evolution[END_REF][START_REF] Raoul | Long time evolution of populations under selection and vanishing mutations[END_REF]. In this context, and this has been remarked before [START_REF] Desvillettes | On selection dynamics for continuous structured populations[END_REF][START_REF] Lorenzi | Asymptotic analysis of selection-mutation models in the presence of multiple fitness peaks[END_REF], concentration on the maximum level set of the fitness function R 0 (x) means that the classical mathematical framework of functions is not sufficient to describe accurately the dynamics of the solutions to (1.1). In this article, we will therefore extend our analysis to the case of Radon measures. Note that it is also natural to consider measures as initial data in epidemic models with an age of infection structure to model cohorts of patients, see Demongeot et al. [START_REF] Demongeot | A Kermack-McKendrick model with age of infection starting from a single or multiple cohorts of infected patients[END_REF].

When X = R N , System (1.1) arises naturally as the limit of a mutation-selection model of spore-producing pathogen proposed by [START_REF] Lo Iacono | The evolution of plant pathogens in response to host resistance: factors affecting the gain from deployment of qualitative and quantitative resistance[END_REF] and studied mathematically by Burie, Djidjou-Demasse, and Ducrot [START_REF] Burie | Asymptotic and transient behaviour for a nonlocal problem arising in population genetics[END_REF][START_REF] Burie | Slow convergence to equilibrium for an evolutionary epidemiology integro-differential system[END_REF], Burie et al. [START_REF] Burie | Concentration estimates in a multi-host epidemiological model structured by phenotypic traits[END_REF], Djidjou-Demasse, Ducrot, and Fabre [START_REF] Djidjou-Demasse | Steady state concentration for a phenotypic structured problem modeling the evolutionary epidemiology of spore producing pathogens[END_REF], and Fabre et al. [START_REF] Fabre | An epi-evolutionary model to predict spore-producing pathogens adaptation to quantitative resistance in heterogeneous environments[END_REF] when the dynamics of the spores is very fast. The system (1.1) corresponds to the case of no mutations at all or, equivalently, the case of a fully concentrated kernel (equal to a Dirac mass at 0).

Despite our efforts, we were unable to find a precise description of the behavior of the solutions of (1.1) in the literature when the initial condition I 0 (dx) is a Radon measure. Here we remark that the vector field of (1.1) is locally Lipschitz continuous in the space R × M + (X) × R (when M(X) is equipped with the total variation norm), so the existence of solutions is not the main difficulty. The solution I(t, dx) can be written as

I(t, dx) = e β(x) t 0 S(s)ds-tγ(x) I 0 (dx),
so the solution is always a bounded continuous function multiplied by the initial data I 0 (x) at any finite time t > 0.

But to describe what happens as t → +∞ is not at all trivial. In Theorem 2.2, we distinguish two typical situations. When there is a positive initial mass on the set of maximal fitness ( α(x)=α * I 0 (dx) > 0, where α * = sup x∈supp I0 α(x) and we recall that α(x) = β(x)/γ(x)), then we can show that the distribution of pathogens converges to a stationary distribution that we can compute explicitly. This is done with the help of a Lyapunov function that is essentially the same as the one used by Hsu [START_REF] Hsu | Limiting behavior for competing species[END_REF]. This is point i) of Theorem 2.2. The case when there is no initial mass on the set of maximal fitness ( α(x)=α * I 0 (dx) = 0) is less clear. We compactify the orbits by using the weak-* topology of measures and use this compactness to show the uniform persistence of the population thanks to a general argument from Magal and Zhao [START_REF] Magal | Global attractors and steady states for uniformly persistent dynamical systems[END_REF]. Then we show that the population on the sets of high fitness always grows faster than the one on sets of low fitness, and this allows us to control uniformly the Kantorovitch-Rubinstein distance between the solution I(t, dx) and the space of measures that are concentrated on the set of maximal fitness, M + {α(x) = α * } . Thus in this case also we can prove that the solution eventually concentrates on the set that maximizes the fitness. This is point ii) of Theorem 2.2.

In general, it is not true that the distribution I(t, dx) eventually reaches a stationary distribution. We construct a counterexample in Section 2.3. By carefully choosing the initial data and the fitness function α(x) = β(x)/γ(x), we construct a solution of (1.1) with I(t, dxdy) that approaches the unit circle of R 2 but never stops turning around it. This fact is illustrated numerically in Section 3.3. We prove in Claim 2.14 that the ω-limit set of the integral of I(t, dxdy) on the upper-half plane, R×R + I(t, dxdy), contains at least two values, therefore I(t, dxdy) does not converge to a stationary distribution. We also refer to [START_REF] Burie | Asymptotic behavior of an epidemic model with infinitely many variants[END_REF] where we construct an exemple in a discrete setting where the total mass X I(t, dx) does not converge to a single value but oscillates between several values.

With some additional assumption we improve the description of the asymptotic behavior of I(t, dx) compared to Theorem 2.2 in case ii). In Assumption 2.5 we impose a condition on the disintegration of I 0 (dx) with respect to α(x) to impose that the distribution of I 0 (dx) is uniformly positive around the maximum of γ(x), γ * := sup α(x)=α * γ(x). Under this assumption, in Proposition 2.7, we refine the localization of the asymptotic concentration set of I(t, dx) and we prove that the total mass of I(t, dx), X I(t, dx), converges to a limit value. We also focus on the special case when the fitness function α(x) attains a finite number of interior regular maxima in the interior of supp I 0 , and when X = R N . In Theorem 2.10, we show that the initial distribution around the maxima of the fitness function plays a crucial role in the asymptotic behavior of the solution. We show that the fitness maxima that keep a nonzero asymptotic population are the ones that maximize an ad-hoc score that involves the value of γ(x) but also the dimension of the Euclidean space and the polynomial decay of the initial data around the fitness maximum.

The structure of the paper is as follows. In section 2 we present our main results. More precisely, we state our results for persistence and concentration in section 2.1, we give precise statements of our results concerning fitness functions with a finite number of regular maxima in section 2.2, and in section 2.3 we provide a counterexample to the convergence of the distribution I(t, dx) when the initial mass of fitness maxima is negligible. In section 3 we illustrate our results with numerical simulations. The corresponding figures are added at the end of the article. In section 4 we prove our results concerning general measure initial data (corresponding to the statements in section 2.1). In section 5 we prove our statements on the systems with a fitness function α(x) having a finite number of regular maxima (corresponding to the statements in section 2.2).

Data availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Main results

:main

Without loss of generality, the system (1.1) can be rewritten as

o-mut    S t (t) = Λ -θS(t) -S(t) X α(x)γ(x)I(t, dx), I t (t, dx) = α(x)S(t) -1 γ(x)I(t, dx), x ∈ X,
(2.1a) eq:SIwith the initial data S(0) = S 0 ∈ (0, +∞),

I(0, dx) ∈ M + (X), (2.1b)
by setting α(x) := β(x)/γ(x), and removing the third equation, which has no impact on the dynamics of the system. In the rest of the article we will study the system (2.1) instead of (1.1). Before going to our results, we introduce some notations that will be used along this work. We work on a Polish space X (i.e. a metrizable space which is separable and complete for at least one metric) equipped with a complete distance d. We denote by M(X) the set of finite signed Radon measures on X. Recall that M(X) is a Banach space when endowed with the total variation norm given by:

µ T V = |µ|(X) = X |µ|(dx), ∀µ ∈ M(X).
This fact is proved for instance in Bogachev [5, Vol. I, Theorem 4.6.1 p. 273]. When X is compact, it is possible to identify M(X) with the dual of the space of continuous functions over X, C(X). This is the Riesz representation theorem [5, Vol. II, Theorem 7.10.4 p.111]. When X is an arbitrary Polish space, while it is true that every measure µ ∈ M(X) yields a continuous linear functional on BC(X) (the space of bounded continuous functions), the converse is no longer true [5, Vol. II, Example 7.10.3 p.111].

We denote by M + (X) the set of the finite nonnegative measures on X. Observe that one has M + (X) ⊂ M(X) and M + (X) is a closed subset of M(X) for the norm topology of • T V . An alternate topology on M(X) can be defined by the Kantorovitch-Rubinstein norm [START_REF] Bogachev | Measure theory[END_REF]Vol. II,Chap. 8.3 p. 191],

µ 0 := sup f dµ : f ∈ Lip 1 (X), sup x∈B |f (x)| ≤ 1 , wherein we have set Lip 1 (X) := f ∈ BC(X) : |f (x) -f (y)| ≤ d(x, y), ∀(x, y) ∈ X 2 .
Let us recall [START_REF] Bogachev | Measure theory[END_REF]Theorem 8.3.2] that the metric generated by • 0 on M + (X) is equivalent to the weak- * topology generated by tests against bounded continuous test functions. Note however that this equivalence is true only for M + (X) and cannot be extended to M(X) since the latter space is not (in general) complete for the metric generated by • 0 . We denote by d 0 this metric on M + (X), that is

d 0 (µ, ν) := µ -ν 0 for all µ, ν ∈ M + (X).
(2.2) eq:kan About the parameters arising in (2.1) our main assumption reads as follows.

nomut Assumption 2.1. The constants Λ > 0 and θ > 0 are given. The functions α(x) and γ(x) are bounded and continuous from X into R and there exist positive constants α ∞ and γ 0 < γ ∞ such that

α(x) ≤ α ∞ , 0 < γ 0 ≤ γ(x) ≤ γ ∞ for all x ∈ X.
We let S 0 > 0 be given and I 0 (dx) ≡ 0 be a finite nonnegative Radon measure and S 0 > 0 be given. We define the two quantities α * ≥ 0 and R 0 (I 0 ) by

α * =: sup x∈supp I0 α(x) and R 0 (I 0 ) := Λ θ α * . (2.3) def-al
We finally assume that the set

L ε (I 0 ) := {x ∈ X : α(x) ≥ α * -ε} ∩ supp I 0 = {x ∈ supp I 0 : α(x) ≥ α * -ε} (2.4
) eq:Lep is compact when ε > 0 is sufficiently small.

Let us observe that if S 0 ≥ 0 then (2.1) equipped with the initial data S(0) = S 0 and I(0, dx) = I 0 (dx) has a unique solution S(t) ≥ 0 and I(t, dx) ∈ M + (X) for all t ≥ 0. This is a direct application of the Cauchy-Lipschitz Theorem in the Banach space R × M(X). It is not difficult to show that S(t) and I(t, dx) are a priori bounded (this will be proved in Lemma 4.1), hence the solution is global. In addition I is given by a quasi-explicit formula:

I(t, dx) = exp γ(x) α(x) t 0 S(s)ds -t I 0 (dx).
The above formula ensures that supp I(t, •) = supp I 0 for all t ≥ 0. We now split our main results into several parts. We first derive very general results about the large time behavior of the solution (S, I) of (2.1) when I 0 is an arbitrary Radon measure. We show that I(t, dx) concentrates on the points that maximize both α and γ. We then apply this result to consider the case where I 0 (dx) is a finite or countable sum of Dirac masses. We continue our investigations with an absolutely continuous initial measure with respect to Lebesgue measure and a finite set {α(x) = α * }. In that setting we are able to fully characterize the points where the measure I(t, dx) concentrates as t → ∞.

Persistence and concentration neral

As mentioned above this subsection is concerned with the large time behavior of the solution (S, I) of (2.1) where the initial measure I 0 (dx) is an arbitrary Radon measure. Using the above notations our first result reads as follows.

sures Theorem 2.2 (Asymptotic behavior of measure-valued initial data). Let Assumption 2.1 be satisfied and suppose that R 0 (I 0 ) > 1. Let (S(t), I(t, dx)) be the solution of (2.1) equipped with the initial data S(0) = S 0 and I(0, dx) = I 0 (dx). We distinguish two cases depending on the measure of the set {α(x) = α * } with respect to I 0 : where τ ∈ R denotes the unique solution of the equation

i) If α(x)=α * I 0 (dx) > 0,
X γ(x)1 α(x)=α * (x)e τ γ(x) I 0 (dx) = θ α * R 0 (I 0 ) -1 .
The convergence of I(t, dx) to I ∞ (dx) holds in the total variation norm • T V .

ii) If α(x)=α * I 0 (dx) = 0, then one has S(t) → 1 α * and I(t, dx) is uniformly persistent, namely lim inf t→+∞ X I(t, dx) > 0.

Moreover I(t, dx) is asymptotically concentrated as t → ∞ on the set {α(x) = α * }, in the sense that

d 0 I(t, dx), M + ({α(x) = α * }) ----→ t→+∞ 0,
where d 0 is the Kantorovitch-Rubinstein distance.

In the statement of the Theorem 2.2 and in the rest of the paper, we stress for clarity that M + {α(x) = α * } is the set of finite positive measures on the closed set {x ∈ X : α(x) = α * }. This set is naturally embedded as a subset of the space M + X , which is closed for the topology induced by the total variation norm • T V and also the one induced by the Kantorovitch-Rubinstein distance d 0 .

Theorem 2.2 can be interpreted as follows. In case i), when the set of pathogens with maximal fitness is "already populated", the behavior of the dynamical system is no different from the case of a finite system: it converges to a equilibrium which is concentrated on the set of maximal fitness. The case ii), when the maximal fitness is not attained for the population but can only be reached asymptotically, is more intricate and we can only prove that the population of pathogens is uniformly persistent and asymptotically concentrated on the set of maximal fitness. We cannot prove the convergence to an equilibrium distribution in general; in fact, it is false, see the example in section 2.3 below and also the counterexamples in [START_REF] Burie | Asymptotic behavior of an epidemic model with infinitely many variants[END_REF]. In fact, as shown in the latter reference, it is not even true in general that the total mass of pathogen X I(t, dx) converges to a limit.

We continue our general result by showing that under additional properties for the initial measure I 0 , the function I(t, dx) concentrates in the large times on the set of the points in {α(x) = α * } that also maximize the function γ.

The additional assumption for the initial measure I 0 (dx) are expressed in term of some properties of its disintegration measure with respect to the function α on L ε (I 0 ) with ε sufficiently small. We refer to the book of Bourbaki [7, VI, §3, Theorem 1 p. 418] for a proof of the disintegration Theorem which is recalled in the Appendix, Theorem C.4.

Let A(dy) be the image of I 0 (dx) under the continuous mapping α : X → R, then there exists a family of nonnegative measures I 0 (y, dx) (the disintegration of I 0 with respect to α) such that for almost every y ∈ α(supp I 0 ) with respect to A we have:

supp I 0 (y, dx) ⊂ {x ∈ X : α(x) = y}, α(x)=y
I 0 (y, dx) = 1 and I 0 (dx) = I 0 (y, dx)A(dy)

(2.5) eq:dis wherein the last equality means that

X f (x)I 0 (dx) = y∈R α(x)=y f (x)I 0 (y, dx)A(dy) for all f ∈ BC(X).
Note that, by definition, the measure A is supported on the set α(supp I 0 ). The measure A is called the pushforward measure of I 0 under the mapping α. Note that the disintegration is unique up to a redefinition on an A-negligible set of fibers, see the disintegration theorem recalled in the Appendix, Theorem C.4. We shall also make use, for all y A-almost everywhere, of the disintegration measure of I 0 (y, dx) with respect to the function γ, as follows

I 0 (y, dx) = R I α,γ 0 (y, z, dx)I α 0 (y, dz),
where I α,γ 0 (y, z, dx) is concentrated on the set {x : α(x) = y and γ(x) = z}. This allows to the following reformulation of I 0 (dx):

I 0 (dx) = y∈R z∈R I α,γ 0 (y, z, dx)I α 0 (y, dz)A(dy). Remark 2.3 (Explicit disintegration in Euclidean spaces). Suppose that X = R N and I 0 ∈ L 1 (R N
). Since we restrict to measures which are absolutely continuous with respect to the Lebesgue measure here, with a small abuse of notation we will omit the element dx when the context is clear. Assume that α is Lipschitz continuous on R N and that

I 0 (x) |∇α(x)| ∈ L 1 (R N ).
(2.6) eq:reg

The coarea formula implies that, for all g ∈ L 1 (R N ), we have

R N g(x)|∇α(x)|dx = R α(x)=y g(x)H N -1 (dx)dy,
where

H N -1 (dx) is the (N -1)-dimensional Hausdorff measure (see Federer [20, §3.2]). Therefore if g(x) = f (x) I0(x) |∇α(x)| we get R N f (x)I 0 (x)dx = R α(x)=y f (x) I 0 (x) |∇α(x)| H N -1 (dx)dy, (2.7) eq:coa and if moreover f (x) = ϕ(α(x)) we get R ϕ(y)A(dy) = R N ϕ(α(x))I 0 (x)dx = R ϕ(y) α(x)=y I 0 (x) |∇α(x)| H N -1 (dx)dy,
where we recall that A(dy) is the image measure of I 0 (dx) through α. Therefore we have an explicit expression for A(dy):

A(dy) = α(x)=y I 0 (x) |∇α(x)| H N -1 (dx)dy (2.8) eq:exp
and (recalling (2.7)) we deduce the following explicit disintegration of I 0 :

I 0 (y, dx) = 1 {x| α(x)=y} I0(x) |∇α(x)| H N -1 (dx) α(x)=y I0(z) |∇α(z)| H N -1 (dz)
.

(2.9) eq:exp Equations (2.8) and (2.9) give an explicit formula for the disintegration introduced in (2.5).

Remark 2.4. In particular, if α is a C 2 function with no critical point in supp I 0 except for a finite number of regular maxima (in the sense that the bilinear form D 2 α(x) is non-degenerate at each maximum; this is a typical situation), then the assumption (2.6) above is automatically satisfied if N ≥ 3 and

I 0 ∈ L ∞ (R N ). If N = 2 then a sufficient condition to satisfy (2.6) with I 0 ∈ L ∞ (R N
) should involve I 0 vanishing sufficiently fast in the neighborhood of each maximum of α.

Now equipped with this disintegration of I 0 with respect to α we are now able to state our regularity assumption to derive more refine concentration information in the case where α(x)=α * I 0 (dx) = 0. bound Assumption 2.5 (Regularity with respect to α, γ). Let us define γ * > 0 by

γ * := sup α(x)=α * γ(x).
(2.10) def-ga

We assume that, for each value γ < γ * in a neighborhoof of γ * , there exist constants δ > 0 and m > 0 such that

m ≤ γ(x)∈([γ,γ * ]) and α(x)=y I 0 (y, dx) for A-almost every y ∈ (α * -δ, α * ].
Remark 2.6. The assumption 2.5 means that the initial measure I 0 (dx) is uniformly positive in a small neighborhood of {α(x) = α * } ∩ {γ(x) = γ * }. For instance, if the assumptions (2.6) is satisfied and if there exists an open set U containing {α(x) = α * } ∩ {γ(x) = γ * } on which I 0 (x) is almost everywhere uniformly positive, then Assumption 2.5 is automatically satisfied.

The next proposition ensures that, when the initial measure I 0 satisfies Assumption 2.5, then the function I(t, dx) concentrates on {x ∈ X : α(x) = α * and γ(x) = γ * }. vival Proposition 2.7. Let Assumption 2.1 hold, and suppose that R 0 (I 0 ) > 1. Assume moreover that α(x)=α * I 0 (dx) = 0 and that Assumption 2.5 holds, then:

i) The measure I(t, dx) concentrates on the set {α(x) = α * } ∩ {γ(x) = γ * }: d 0 (I(t, dx), M + ({α(x) = α * } ∩ {γ(x) = γ * })) ----→ t→+∞ 0.
ii) The total mass of I(t, dx) converges to a limit value:

X I(t, dx) ----→ t→+∞ θ α * γ * R 0 (I 0 ) -1 . iii) If there exists a Borel set U ⊂ X such that U ∩ {x : α(x) = α * and γ(x) = γ * } = ∅ and lim inf ε→0 A(dy) ess inf α * -ε≤y≤α * I α 0 (y,dz) ess inf γ * -ε≤z≤γ * U I α,γ 0 (y, z, dx) > 0,
then the following persistence occurs

lim inf t→∞ U I(t, dx) > 0.
Here the notation µ(dy) ess inf K denotes the essential infimum taken on the set K and relatively to the measure µ (i.e. up to redefinition on µ-negligible sets). The condition on the set U in iii) means that the distribution of the population does not vanish in a neighborhood of {x : α(x) = α * and γ(x) = γ * } in U ; the existence of a set U cannot be guaranteed in general, in particular, no such U exists in the case of the counterexample given in subsection 2.3.

Refined concentration estimates: regular fitness maxima in Euclidean spaces

-data

We now deepen our analysis of (2.1) set on X = R N when the fitness function has a finite number of regular maxima. We will not discuss here the case of a unique global maximum, in which the precise asymptotic behavior can be completely determined: see the Appendix for a statement of what we obtain. Rather, we describe the large time behavior of the solutions when the function α has a finite number of maxima on the support of I 0 (dx). We consider an initial data (S 0 , I 0 ) ∈ [0, ∞) × M + (R N ) with I 0 absolutely continuous with respect to the Lebesgue measure dx in R N (in other words and with a small abuse of notation, I 0 ∈ L 1 (R N )) in a neighborhood of the maxima of the fitness function. Recalling the definition of α * in (2.3), throughout this section, we shall make use of the following set of assumptions. By a small abuse of notation, we will identify in this section the function I 0 ∈ L 1 (R N ) and the associated measure

I 0 (x)dx ∈ M + (R N ) when the context is clear.
lculs Assumption 2.8. We assume that: (i) the set {α(x) = α * } is a finite set, namely there exist x 1 , ..., x p in the interior of supp (I 0 ) such that x i = x j for all i = j and

{α(x) = α * } = {x 1 , • • • , x p } and R 0 := Λα * θ > 1.
(ii) There exist ε 0 > 0, M > 1 and κ 1 ≥ 0,..,κ p ≥ 0 such that for all i = 1, .., p and for almost all

x ∈ B(x i , ε 0 ) ⊂ supp (I 0 ) one has M -1 |x -x i | κi ≤ I 0 (x) ≤ M |x -x i | κi .
Here and along this note we use | • | to denote the Euclidean norm of R N .

(iii) The functions α and γ are of class C 2 and there exists > 0 such that for each i = 1, .., p one has

D 2 α(x i )ξ 2 ≤ -|ξ| 2 , ∀ξ ∈ R N .
Remark 2.9. Let us observe that since x i belongs to the interior of supp (I 0 ) then Dα(x i ) = 0.

In order to state our next result, we introduce the following notation: we write f (t) g(t) as t → ∞ if there exists C > 1 and T > 0 such that

C -1 |g(t)| ≤ |f (t)| ≤ C|g(t)|, ∀t ≥ T.
According to Theorem 2.2 (ii), one has α * S(t) → 1 as t → ∞, and as a special case we conclude that

S(t) = 1 t t 0 S(l)dl → 1 α * as t → ∞.
As a consequence the function η(t) := α * S(t) -1 satisfies η(t) = o(1) as t → ∞. To describe the asymptotic behavior of the solution (S(t), I(t, dx)) with initial data S 0 and I 0 as above, we shall derive a precise behavior of η for t 1. This refined analysis will allow us to characterize the points of concentration of I(t, dx). Our result reads as follows.

O-eta

Theorem 2.10. Let Assumption 2.8 be satisfied. Then the function η = η(t) satisfies the following asymptotic expansion Moreover there exists ε 1 ∈ (0, ε 0 ) such that for all 0 < ε < ε 1 and all i = 1, .., p one has |x-xi|≤ε

η(t) = ln t t + O 1 t , as t → ∞. ( 2
I(t, dx) t γ(xi) -N +κ i 2 as t → ∞. (2.13) mass-e
As a special case, for all ε > 0 small enough and all i = 1, .., p one has |x-xi|≤ε

I(t, dx) 1 if i ∈ J → 0 if i / ∈ J as t → ∞,
where J is the set defined as

J := i = 1, .., p : N + κ i 2γ(x i ) = . (2.14) def-J
The above theorem states that the function I(t, dx) concentrates on the set of points {x i , i ∈ J} (see Corollary 2.12 below). Here Assumption 2.5 on the uniform positiveness of the measure I 0 (dx) around the points x i is not satisfied in general, and therefore the measure I concentrates on {α(x) = α * } as predicted by Theorem 2.2, but not necessarily on {α(x) = α * } ∩ {γ(x) = γ * } as would have been given by Proposition 2.7. In Figure 6 we provide a precise example of this non-standard behavior.

In addition, the precise expansion of η = η(t) provided in the above theorem allows us obtain the self-similar behavior of the solution I(t, dx) around the maxima of the fitness function. This asymptotic directly follows from (4.1). milar Corollary 2.11. For each i = 1, ..., p and f ∈ C c (R N ), the set of the continuous and compactly supported functions, one has as t → ∞:

t N 2 R N f (x -x i ) √ t I(t, dx) t γ(xi) -N +κ i 2 R N f (x)|x| κi exp γ(x i ) 2α * D 2 α(x i )x 2 dx. (2.15) eqCORO
Our next corollary relies on some properties of the ω-limit set of the solution I(t, dx). Using the estimates of the mass around x i given in (2.13), it readily follows that any limit measures of I(t, dx) belongs to a linear combination of δ xi with i ∈ J and strictly positive coefficients of each of these Dirac masses. This reads as follows.

-conv Corollary 2.12. Under the same assumptions as in Theorem 2.10, the ω-limit set O(I 0 ) as defined in Lemma 4.6 satisfies that there exist 0 < A < B such that

O(I 0 ) ⊂ i∈J c i δ xi : (c i ) i∈J ∈ [A, B] J .

Oscillations gence

Here we construct a counterexample which shows that, in general, it is hopeless to expect convergence of the genotypic distribution to a stationary measure on X = R 2 . Such a counterexample is new in the case of "continuous" spaces; we provided some counterexamples in the case of discrete spaces in [START_REF] Burie | Asymptotic behavior of an epidemic model with infinitely many variants[END_REF].

Fix L > 0 and let Ω be the parametric curve described as

Ω := ω(τ ) := 1 -e -τ cos π L τ , sin π L τ : τ ∈ R + ,
and define I 0 (dxdy) ∈ M(R 2 ) as the pushforward of the measure e -τ 1 τ ≥0 dτ by ω. That is to say,

R 2 ϕ(x, y)I 0 (dxdy) = R + e -τ ϕ 1 -e -τ cos π L τ , 1 -e -τ sin π L τ dτ, for all ϕ ∈ BC(R 2 ).
Then we select α(x, y)

:= 1 -1 -x 2 + y 2 , γ(x, y) ≡ 1.
In this setting, α attains its global maximum on the unit circle in R 2 , while the support of I 0 (the curve Ω) approaches the unit circle from the inside with an exponentially decreasing mass as the radius converges to 1. We are thus in the situation described in case ii) of Theorem 2.2; in particular, it is true that S(t) = 1 t t 0 S(s)ds → 1 as t → +∞. Yet, explicit computations show that

I(t, dxdy) = e (1-e -τ ) t 0 S(s)ds-t I 0 (τ )ω(dτ ) = e (1-e -τ )tS(t)-t-τ ω(dτ ) =: I(t, τ )ω(dτ ),
where we denote ω(dτ ) the pushforward of the Lebesgue measure on R + onto Ω and, with a small abuse of notation,

I 0 (τ ) = e -τ 1 τ ≥ 0. More precisely, R 2 ϕ(x, y)I(t, dxdy) = R + I(t, τ )ϕ 1 -e -τ cos π L τ , 1 -e -τ sin π L τ dτ, for all ϕ ∈ BC(R 2 ).
Now by using explicit computations, we establish the following claim.

:asym Claim 2.13. The function I(t, τ ) is, up to a multiplicative error of order zero, a solitary wave whose position behaves like τ 0 (t) := ln(t):

I(t, τ ) = e u(τ -τ0(t))+o(1) , (2.16)
where u(τ ) = ln(I ∞ ) -e -τ -τ and I ∞ := θ α * (R 0 -1) is given by Proposition 2.7. We now prove the claim. We note that Assumption 2.5 is clearly satisfied since γ ≡ 1 therefore, by Proposition 2.7, we have +∞ 0

I(t, τ )dτ = I ∞ + o(1) with I ∞ := θ Λ θ α * -1 > 0.
S(t) = 1 + ln(t) t + ln(I ∞ ) t + o 1 t .
We deduce that

I t, τ + τ 0 (t) = exp 1 -e -(τ +τ0(t)) t + ln(t) + ln(I ∞ ) + o(1) -t -(τ + τ 0 (t)) = exp 1 - e -τ t ln(t) + ln(I ∞ ) + o(1) -te -(τ +τ0(t)) -(τ + τ 0 (t)) = exp ln(t) + ln(I ∞ ) + o(1) + e -τ -τ -ln (t) = exp ln(I ∞ ) -e -τ -τ + o(1) ,
which proves the claim.

Note that the above computations and in particular the result of Claim 2.13 are completely independent of the parameter L. We see that I(t, dxdy) is asymptotically equivalent to a rotating mass which becomes concentrated on the unit circle and does not converge to a static distribution. We illustrate this fact in numerical simulations in section 3.3. We can also prove that the distribution does not reach stationarity when the rotation speed of the spiral (which behaves like L -1 ) is very slow. Indeed, the integral in the upper half-space never reaches a stationary value, as we show in the following Claim. m:osc Claim 2.14. There exists a function ε(L) ≥ 0 such that ε(L) → 0 as L → ∞, and two sequences t 1 n := e (2n+1/2)L → +∞ and t 2 n := e (2n+3/2)L → +∞ such that lim inf (2.18) eq:osc Indeed, we have

n→+∞ R×R + I(t 1 n , dxdy) ≥ I ∞ -ε(L) (2.17
R×R + I(t, dxdy) = +∞ k=0 (2k+1)L 2kL I(t, τ )dτ = +∞ k=0 (2k+1)L 2kL e u(τ -τ0(t))+o(1) dτ = 1 + o(1) +∞ k=0 (2k+1)L-τ0(t) 2kL-τ0(t) e u(τ ) dτ = 1 + o(1) +∞ k=0 (2k+1)L-τ0(t) 2kL-τ0(t) I ∞ e -e -τ -τ dτ = 1 + o(1) +∞ k=0 I ∞ exp -e -(2k+1)L+τ0(t) -exp -e -2kL+τ0(t) , (2.19
) eq:221 so when t = t 1 n = e (2n+1/2)L we have

R×R + I(t 1 n , dxdy) = 1 + o(1) +∞ k=0 I ∞ exp -e -(2k+1)L+(2n+1/2)L -exp -e -2kL+(2n+1/2)L = I ∞ 1 + o(1) +∞ k=-n e -e -2kL-L/2 -e -e -2kL+L/2
We note that e -e -(2k±1/2)L ≤ e -(2k-1/2)L as L → ∞ when k ≤ -1, and that for k ≥ 1 we have by a Taylor expansion:

|e -e -(2k-1/2)L -e -e -(2k+1/2)L | ≤ Ce -(2k-1/2)L ,
for some constant C > 0 independent of L and k. Therefore by the dominated convergence theorem we have lim

L→+∞ -1 k=-∞ + +∞ k=1
e -e -2kL-L/2 -e -e -2kL+L/2 = 0. Now for k = 0 we have e -e -2kL-L/2 -e -e -2kL+L/2 = e -e -L/2 -e -e L/2 → 1 as L → +∞. We have shown lim inf

L→+∞ lim inf n→+∞ R×R + I(t 1 n , dxdy) ≥ I ∞ ,
which proves (2.17). Now (2.19) with t = t 2 n = e (2k+3/2)L leads us to

R×R + I(t 2 n , dxdy) = 1 + o(1) +∞ k=0 I ∞ exp -e -(2k+1)L+(2n+3/2)L -exp -e -2kL+(2n+3/2)L = I ∞ 1 + o(1) +∞ k=-n
e -e -2kL+L/2 -e -e -2kL+3L/2 .

Note that we have lim

L→+∞ -1 k=-∞ + +∞ k=1
e -e -2kL+L/2 -e -e -2kL+3L/2 = 0.

by the dominated convergence theorem and, for k = 0, e -e -2kL+L/2 -e -e -2kL+3L/2 = e -e L/2 -e -e 3L/2 → 0 as L → ∞. This shows lim sup

L→+∞ lim sup n→+∞ R×R + I(t 2 n , dxdy) ≤ 0,
which finishes the proof of Claim 2.14.

3 Comments and numerical illustrations erics

Numerical illustrations: the main Theorem

In this section we provide numerical illustrations to some of our results. Note however that the figures are included at the end of the manuscript.

We start with an illustration of the long-time behavior of the solution to (2.1) when the initial mass of the fitness maximum is positive (I 0 ({α(x) = α * }) > 0) in Figure 1. To help with the visual representation, in our example, I 0 is chosen absolutely continuous with respect to the Lebesgue measure, i.e. carried by a L 1 (R 2 ) function. We provide a plot of the fitness function which is proportional to α (top left sub-figure) and the initial data (top right). The fitness function attains its maximum on the union of a rectangle [0.2, 0.5] × [-0.5, 0.5] with a line segment {-0.3} × [-0.5, 0.5] and the support of the initial data intersects this set with non-negligible intersection. We plot the time evolution of the converging function S(t) (bottom left) and a snapshot of the distribution I(t, dx) at t = 50 (bottom right). We observe that the mass that was initially located outside of the fitness maximum has vanished. What remains is a distribution of mass in the initial rectangle of maximal fitness (according to Theorem 2.2, the precise distribution can be computed). The distribution located at {x 1 = -0.3} is still positive, but is negligible with respect to the Lebesgue measure and does not contribute to the mass. In Figure 2 we present four snapshots of the distribution I(t, dx) to monitor the time evolution of this distribution with the same initial distribution.

Next, we consider the case when the initial mass of the fitness maximum is equal to zero (I 0 ({α(x) = α * }) = 0) in Figure 3. Again, we provide a plot of the fitness function (top left) and of the initial data (top right). The fitness function attains its maximum on the line segment {0.35} × [-0.5, 0.5]. The time evolution of S(t) (bottom left) and a snapshot of the distribution I(t, dx) at t = 100 (bottom right) are also displayed. The distribution I(t, dx) at t = 100 is already concentrated on the maximum of the fitness. Figure 4 consists of four snapshots of the example presented in Figure 3.

Transient dynamics on local maxima: a numerical example

In many biologically relevant situations it may be more usual to observe situations involving a fitness function with one global maximum and several (possibly many) local maxima, whose values are not exactly equal to the global maximum but very close. In such a situation, while the long-term distribution will be concentrated on the global maximum, one may observe a transient behavior in which the orbits stay close to the equilibrium of the several global maxima situation (corresponding to Theorem 2.10), before it concentrates on the eventual distribution. We leave the analytical treatment of such a situation open for future studies, however, we present a numerical experiment in Figure 5 which shows such a transient behavior.

In this simulation, we took a fitness function presenting one global maximum at x 2 = +0.5 and a local maximum at x 1 = -0.5, whose value is close to the global maximum. The precise definition of α(x) is

α(x) = 0.95 × P [x1-δ,x1+δ] (x) + P [x2-δ,x2+δ] (x) with δ = 0.2.
(3.1) eq:exa

The function I 0 (x) is chosen as

I 0 (x) = min 1, 4 (x -x 1 ) 2 min 1, 4 (x -x 2 ) 2 1 [-1,1] (x), (3.
2) eq:exa so that κ 1 = 2 and κ 2 = 2. Finally,

γ(x) = 1 1 + P [x1-δ,x1+δ] (x) + 3P [x2-δ,x2+δ] (x) (3.
3) eq:exa so that γ(x 1 ) = 1 2 and γ(x 2 ) = 1 4 . Summarizing, we have

N + κ 1 2γ(x 1 ) = 3 < 6 = N + κ 2 2γ(x 2
) .

If α(x) had two global maximum at the same level, Theorem 2.10 would predict that the mass I(t, dx) vanishes near x 2 and concentrates on x 1 . Since the value of α(x 2 ) is slightly higher than the value of α(x 1 ), however, it is clear that the eventual distribution will be concentrated on x 2 . We observe numerically (see Figure 5) that the distribution first concentrates on x 1 on a transient time scale, before the dynamics on x 2 takes precedence. We refer to Burie, Djidjou-Demasse, and Ducrot [START_REF] Burie | Asymptotic and transient behaviour for a nonlocal problem arising in population genetics[END_REF] for a related model with mutations where these transient behaviors are analytically characterized. Figure 3 illustrates the case when the maximal fitness is negligible for the initial measure (Point ii) of Theorem 2.2). We provide a plot of the fitness function (top left-hand side) and the initial data (top right-hand side). The fitness function attains its maximum on a rectangle with positive Lebesgue measure and the support of the initial data intersects this rectangle with non-negligible intersection. We also plot the time evolution of S(t) (bottom left-hand side) and a snapshot of the distribution I(t, x) at t = 100. We observe that the mass that was initially located outside of the fitness maximum has vanished (bottom right-hand side). What remains is a distribution of mass around the initial line of maximal fitness, which is negligible for the initial data; however the distribution takes very high values.

In Figure 6 we provide a precise example of this non-standard behavior. The function α(x) is chosen to have two maxima x 1 = -0.5 and x 2 = 0.5; the precise definition of α(x) is

α(x) = P [x1-δ,x1+δ] (x) + P [x2-δ,x2+δ] (x),
I 0 (x) = min 1, 1024 (x -x 1 ) 8 min 1, 4 (x -x 2 ) 2 1 [-1,1] (x), (3.5
) eq:exa so that κ 1 = 8 and κ 2 = 2. Finally we take

γ(x) = 1 1 + P [x1-δ,x1+δ] (x) + 3P [x2-δ,x2+δ] (x)
(3.6) eq:exa so that γ(x 1 ) = 1 2 and γ(x 2 ) = 1 4 . Summarizing, we have

N + κ 1 2γ(x 1 ) = 9 > 6 = N + κ 2 2γ(x 2 )
, so that Theorem 2.10 predicts that the mass I(t, dx) will vanish near {x 1 } = {α(x) = α * } ∩ {γ(x) = γ * } and concentrate on x 2 .

Numerical illustration: Oscillations tions

In Figure 7 we present a numerical simulation that illustrates the example provided in section 2.3. The parameter L is set to 1. A supplementary movie is also available (spiraling.avi).

Comments

We have studied the asymptotic behavior of a simple epidemiological model whose originality is that the population is structured by a continuous genotypic variable. Thus the population is divided into a compartment of susceptible and infected by a certain strain of pathogen. The duration of the infectious period (1/γ) and the basic reproduction number (α, up to a multiplicative constant) of the disease depend on the trait considered. In this work, the pathogen population does not mutate and therefore if a trait is absent from the initial pathogen population, it cannot appear in the population afterwards.

Assuming that the basic reproduction number has a maximum strictly greater than one for at least one phenotypic trait of the initial population, we have shown the convergence of the solution of this model towards an endemic equilibrium. Firstly, in the case where the maximum value of the basic reproduction number is reached on a continuum of phenotypic traits of the initial population, we can completely describe the asymptotic number of susceptible as well as the distribution of the infected population with respect to the different variants.

Secondly, in the case where the maximum of R 0 is reached on a set of zero measure, we have shown the persistence of the infectious population and its asymptotic concentration on a subset of the set of the traits maximizing both the R 0 and the additional mortality rate due to the pathogen (or in other words minimizing the infectious period).

To go further in the analysis, we then considered the case of a finite number of traits maximizing the R 0 . In this case, we were able to describe more precisely the set of traits around which the population of infected (and therefore of pathogens) is concentrated, as well as the profile of the asymptotic distribution of infected around these traits.

In particular, we observed that even if there are no infected individuals with a trait maximizing the R 0 initially, the population can concentrate around this trait (but not on this trait since there is no mutation in this model). The selection of traits around which the population concentrates thus depends not only on the value of R 0 and the additional mortality rate, but also on the initial population distribution around the trait. A non-standard behavior may thus appear where the selected strain no longer maximizes γ, the virulence. The question arises whether such a configuration can be observed in vivo.

4 Measure-valued solutions and proof of Theorem 2.2 asure In this section we derive general properties of the solution of (2.1) equipped with the given and fixed initial data S(0) = S 0 ∈ [0, ∞) and I 0 (dx) ∈ M + (X). Recall that α * and R 0 (I 0 ) are both defined in (2.3). Next for ε > 0 we recall that L ε (I 0 ) is the following superlevel set (defined by (2.4) in Assumption 2.1):

L ε (I 0 ) = {x ∈ supp I 0 : α(x) ≥ α * -ε} = α * -ε≤y≤α * {α(x) = y}.
Recall also that the existence and uniqueness of a solution S(t), I(t, dx) ∈ R×M(X) corresponding to (S 0 , I 0 ) in the Banach space R × M(X) (where M(X) is equipped with the norm • T V ) follow directly from the Cauchy-Lipschitz Theorem.

The following lemma holds true. .

Next we return to the S-component of equation (2.1) and let ε > 0 be given. We have, for t 0 sufficiently large and t ≥ t 0 , Assume by contradiction that the conclusion of the Lemma does not hold, i.e. there exists ε > 0 and a sequence

S t = Λ -θ + X α(x)γ(x)I(t, dx) S(t) ≥ Λ -θ + α * γ * Λ min(θ, γ * ) + ε S(t), therefore S(t) ≥ e -θ+ Λα * γ * min(θ,γ * ) +ε (t-t0) S(t 0 ) + Λ min(θ, γ * ) (θ + ε) min(θ, γ * ) + Λα * γ * 1 -e -θ+ Λα * γ * min(θ,γ * ) +ε (t-
T n → +∞ such that 1 T n Tn 0 S(t)dt ≥ 1 α * + ε. Then T n Tn 0 S(t)dt ≤ 1 1 α * + ε ≤ α * -ε , where ε = (α * ) 2 ε + o(ε).
Since the map x → α(x) is continuous, the set L ν (I 0 ) = {x ∈ supp I 0 : α(x) ≥ α * -ν} has positive mass with respect to the measure I 0 (dx) for all ν > 0, i.e. Lν (I0) I 0 (dx) > 0. This is true, in particular, for

ν = ε 2 , therefore L ε /2 (I0) I(T n , dx) = L ε /2 (I0) exp γ(x)
I(T n , dx) = +∞,
which is a contradiction since I(t, dx) is bounded in M(X) by Lemma 4.1. This completes the proof of the Lemma.

An important tool in later proofs is that the mass of I(t, dx) vanishes on any set sufficiently far away from I 0 when the Cesàro mean S(t) = 1 t t 0 S(s)ds of S is sufficiently close to α * , which we prove now. ishes Lemma 4.3. Let Assumption 2.1 hold and S(t), I(t, dx) be the corresponding solution of (2.1). Let {t τ } be a net t τ → ∞ and ε > 0 be such that we have eventually

S(t τ ) = 1 t τ tτ 0 S(s)ds ≤ 1 α * -ε for all τ. (4.2)
Then for any p > 1 we have {α(x)≤α * -pε}

I(t τ , dx) -----→ tτ →+∞ 0. (4.3)
Proof. Indeed, we can write {α(x)≤α * -pε}

I(t τ , dx) = {α(x)≤α * -pε} exp γ(x)t τ S(t τ ) α(x) - 1 S(t τ ) I 0 (dx) ≤ {α(x)≤α * -pε} exp γ(x)t τ S(t τ ) α * -pε - 1 S(t τ ) I 0 (dx) ≤ {α(x)≤α * -pε} exp γ(x)t τ S(t τ )(1 -p)ε I 0 (dx).
Since p > 1 and lim inf S(t τ ) > 0 by Lemma 4.1, the argument of the exponential converges to -∞ as t τ → +∞ therefore {α(x)≤α * -pε}

I(t τ , dx) -----→ tτ →+∞ 0.
The Lemma is proved.

The following weak persistence property holds.

nomut Lemma 4.4. Let Assumption 2.1 hold and suppose that R 0 (I 0 ) > 1. Let S(t), I(t, dx) be the corresponding solution of (2.1). Then

lim sup t→+∞ X I(t, dx) ≥ θ α * γ * R 0 (I 0 ) -1 > 0,
where γ * := sup x∈supp I0 γ(x).

Proof. Assume by contradiction that for t 0 sufficiently large we have

X I(t, dx) ≤ η < η =: θ α * γ * R 0 (I 0 ) -1 for all t ≥ t 0 , with η > 0.
As a consequence of Lemma 4.2 we have lim inf t→+∞ S(t) ≤ lim sup

T →+∞ 1 T T 0 S(t)dt ≤ 1 α * . ( 4

.4) eq:inf

Let S := lim inf t→+∞ S(t). Let (t n ) n≥0 be a sequence that tends to ∞ as n → ∞ and such that lim n→+∞ S (t n ) = 0 and lim n→+∞ S(t n ) = S. As X I(t n , dx) ≤ η for n large enough we deduce from the equality Let us remind that M + (X), equipped with the Kantorovitch-Rubinstein metric d 0 defined in (2.2), is a complete metric space. nomut Lemma 4.5 (Compactness of the orbit and concentration). Let Assumption 2.1 hold and Let S(t), I(t, dx) be the corresponding solution of (2.1). Then, the closure of the orbit of (S 0 , I 0 ),

S (t n ) = Λ -θS(t n ) -S(t n ) X α(x)γ(x)I(t n , dx), that 0 ≥ Λ -θS -Sα * γ * η so that S ≥ Λ θ + α * γ * η > Λ θ + α *
O(S 0 , I 0 ) := (s, µ) ∈ R × M + (X) : there exists (t n ) such that d 0 (I(t n , dx), µ) -----→ n→+∞ 0 and |S(t n ) -s| -----→ n→+∞ 0 , is compact.
Moreover R 0 (I 0 ) > 1 and t n → +∞ is an arbitrary sequence along which

lim inf n→+∞ X I(t, dx) > 0, ( 4
I(t, dx) = 0.
Thus given any threshold ν > 0, there is T ν such that I(t, X\L ε (I 0 )) ≤ ν for all t ≥ T ν , and since I 0 is Radon there exists a compact set

K ν ⊂ X such that I 0 (X\K) ≤ νe -γ ∞ Tν α ∞ sup 0≤s≤Tν S(s) so that X\K I(t, dx) ≤ X\K I 0 (dx)e γ ∞ Tν (α ∞ sup 0≤s≤Tν S(s)-1) ≤ ν.
Thus for all t ≥ 0 we have I t, X\ K ν ∪ L ε (I 0 ) . Next let (t n ) be an arbitrary sequence along which (4.5) holds. Thanks to the compactness of the orbit, we extract from (S(t n ), I(t n , dx)) a subsequence still denoted t n , such that S(t n ) → S ∞ and I(t n , dx) → I ∞ (dx) weakly. Clearly S (t + t n ) is bounded independently on n; differentiating the first line in (2.1a) we see that S (t + t n ) is also bounded independently on n when t is in an arbitrary compact set. Thus, up to a diagonal extraction, we may assume that both S(t + t n ) and S (t + t n ) converge locally uniformly in t to a limit S ∞ (t) and (S ∞ ) (t). This proves the final statement of the Lemma. We can now take the weak- * limit in the formula

I(t + t n , dx) = I(t n , dx) exp γ(x)t α(x) 1 t t 0 S(t n + s)ds -1
which shows that, for fixed but arbitrary t ∈ R, we have that I(t n + t, dx) converges weakly to

I ∞ (t, dx) := I ∞ (dx) exp γ(x)t α(x) 1 t t 0 S ∞ (s)ds -1 .
In particular (S ∞ (t), I ∞ (t, dx)) is a complete orbit of the equation (2.1). Since 1 ∈ BC(X) we have that

X I ∞ (dx) = X 1I ∞ (dx) =
α(x)≤α * -1 k I ∞ (dx) = lim n→+∞ α(x)≤α * -1 k I(t n , dx) = 0, for all k ∈ N.
Thus by Fatou's Lemma

α(x)<α * I ∞ (dx) ≤ lim k→+∞ α(x)≤α * -1 k I ∞ (dx) = 0.
This proves (4.7) and completes the proof of Lemma 4.5.

Next we show the weak uniform persistence property if R 0 (I 0 ) > 1.

nomut Lemma 4.6 (Uniform persistence). Let Assumption 2.1 hold and suppose that R 0 > 1. Let S(t), I(t, dx) be the corresponding solution of (2.1). Then lim inf t→+∞ X I(t, dx) > 0.

Proof. We adapt here the argument of Magal and Zhao [START_REF] Magal | Global attractors and steady states for uniformly persistent dynamical systems[END_REF]Proposition 3.2] to our context. Suppose by contradiction that there exists a sequence t n → +∞ such that lim n→+∞ X I(t n , dx) = 0.

By Lemma 4.1 we know that S(t n ) ≥ S > 0 for all n ∈ N. By Lemma 4.4, for each n ∈ N sufficiently large, there exists s < t n such that

X I(s n , dx) = θ 2α * γ * R 0 (I 0 ) -1 and X I(t, dx) ≤ θ 2α * γ * R 0 (I 0 ) -1 for all s n ≤ t ≤ t n .
(4.8) eq:230

Up to replacing t n by a subsequence, we will assume without loss of generality that t n-1 < s n < t n for all n ∈ N.

Thanks to Lemma 4.5 and up to a further extraction, the shifted orbits S(t+s n ), I(t+s n , dx) converge to a complete orbit S ∞ (t), I ∞ (t, dx) which satisfies 0 < S ≤ S ∞ (t) ≤ Λ θ and X I ∞ (t, dx) > 0 for all t ∈ R. Moreover I ∞ (t, dx) is concentrated on the set {x : α(x) = α * }, and in particular R 0 (I ∞ (0, dx)) = R 0 (I 0 ) > 1. By Lemma 4.4 we have therefore lim sup

t→+∞ X I ∞ (t, dx) ≥ θ α * γ * R 0 (I 0 ) -1 .
(4.9) eq:230

Next we investigate the time T n := t n -s n . Up to extracting a subsequence, there are two options.

T n → T as n → +∞. In that case we have X I ∞ (T, dx) = 0 so I ∞ (T, dx) ≡ 0, and by the uniqueness of the solution to (2.1) we have I ∞ (t, dx) ≡ 0 for all t ≥ T . This contradicts (4.9).

T n → +∞ as n → +∞. In that case, recalling (4.8), we have X I ∞ (t, dx) ≤ θ 2α * γ * R 0 (I 0 ) -1 for all t ≥ 0, and this also contradicts (4.9). This completes the proof of Lemma 4.6.

o-mut

Lemma 4.7. Let Assumption 2.1 hold. Let S(t), I(t, dx) be the corresponding solution of (2.1). Assume that R 0 (I 0 ) > 1. Then

lim inf T →+∞ 1 T T 0 S(t)dt ≥ 1 α * ,
with α * given in (2.3).

Proof. Assume by contradiction that the conclusion of the Lemma does not hold, i.e. there exists ε > 0 and a sequence

T n → +∞ such that 1 T n Tn 0 S(t)dt ≤ 1 α * -ε. Then T n Tn 0 S(t)dt ≥ 1 1 α * -ε ≥ α * + ε , where ε = (α * ) 2 ε + o(ε)
, provided ε is sufficiently small. Therefore by Lemma 4.3 we have lim inf

t→+∞ X I(t, dx) ≤ lim inf n→+∞ X I(T n , dx) = 0,
which is in contradiction with Lemma 4.6. This proves the Lemma. Proof. Let ε > 0 be as in the statement of Lemma 4.9. By Lemma 4.2, there exists T ≥ 0 such that for all t ≥ T we have t t 0 S(s)ds

≥ α * - ε 2 .
Hence Lemma 4.3 implies that X\Lε(I0)

I(t, dx) ----→ t→+∞ 0.
In particular, if I(t, dx)| Lε(I0) denotes the restriction of I(t, dx) to L ε (I 0 ), we have I(t, dx)-I(t, dx)| Lε(I0) T V ----→ t→+∞ 0 and hence

d 0 I(t, dx), I(t, dx)| Lε(I0) ≤ d T V I(t, dx), I(t, dx)| Lε(I0) ----→ t→+∞ 0.
Here ε > 0 can be chosen arbitrarily small. By Lemma 4.1 we know moreover that lim sup

t→+∞ X I(t, dx) ≤ Λ min(θ, γ * )
, so that for t sufficiently large, we have

X I(t, dx) ≤ 2 Λ min(θ, γ * )
.

Finally by using Proposition B.5 (proved in the Appendix), we have

d 0 I(t, dx), M + ({α(x) = α * }) ≤ d 0 I(t, dx), I(t, dx)| Lε(I0) + d 0 I(t, dx)| Lε(I0) , M + ({α(x) = α * }) ≤ d 0 I(t, dx), I(t, dx)| Lε(I0) + 2 Λ min(θ, γ * ) sup x∈Lε(I0) d x, {α(x) = α * } . Since sup x∈Lε(I0) d x, {α(x) = α * } ---→ ε→0 0,
the Kantorovitch-Rubinstein distance between I(t, dx) and M {α(x) = α * } can indeed be made arbitrarily small as t → +∞. This proves the Lemma.

const Lemma 4.10. Let Assumption 2.1 hold and suppose moreover that α(x) ≡ α * for all x ∈ supp I 0 and that R 0 (I 0 ) > 1.

Let S(t), I(t, dx) be the corresponding solution of (2.1) and let S * := 1 α * and i * (x) := e τ γ(x) where τ ∈ R is the unique solution of the equation X γ(x)e τ γ(x) I 0 (dx) = θ α * (R 0 -1) . (4.10) eq:id-

Define g(x) := x -ln(x) and let

D(V ) := S, i(x) ∈ R × L ∞ i * (I 0 ) : S > 0 and I0(dx) ess inf i(x) i * (x) > 0 ,
where

L ∞ i * (I 0 ) is the weighted L ∞ space equipped with the norm i L ∞ i * (I0) := i i * L ∞ (I0) . Then D(V ) is open in R × L ∞ i * (I 0 ) and the functional V (S, i) := S * g S S * + X i * (x)g i(x) i * (x) I 0 (dx)
is well-defined and continuous on D(V ). Moreover if i(t, x) is the Radon-Nikodym derivative of I(t, dx) with respect to I 0 (in other words, .11) eq:Lya

I(t, dx) = i(t, x)I 0 (dx)), then t → V (S(t), i(t, x)) is of class C 1 and we have d dt V S(t), i(t, x) ≤ -θ (S(t) -S * (t)) 2 S(t) for all t ∈ R. ( 4
Proof. The well-definition of τ is clear since the left-hand side of (4.10) is strictly increasing and connects 0 when τ → -∞, to +∞ when τ → +∞. The openness of D(V ), the well-definition and continuity of V (S, i) are also clear. Let us check (4.11). We first remark that i(t, x) = e tγ(x)(α(x)S(t)-1) , so it is clear that i(t, x) ∈ D(V ) and that t → V (S(t), i(t, x)) ∈ C 1 (R).

We show (4.11). Let us write V 1 (S) = S * g S(t) S * and V 2 (t) = X i * (x)g i(t,x) i * (x) I 0 (dx), then we have

V 1 (t) = S * S (t) S * g S(t) S * = Λ -θS(t) -S(t) X α * γ(x)i(t, x)I 0 (dx) 1 - S * S(t) = Λ -θS(t) -S(t) X α * γ(x)i(t, x)I 0 (dx) -Λ + θS * + S * X α * γ(x)i * (x)I 0 (dx) 1 - S * S(t) = -θ (S(t) -S * ) 2 S(t) + S * X α * γ(x)i * (x)I 0 (dx) -S(t) X α * γ(x)i(t, x)I 0 (dx) 1 - S * S(t) , = -θ (S(t) -S * ) 2 S(t) + S * X α * γ(x)i * (x)I 0 (dx) - (S * ) 2 S(t) X α * γ(x)i * (x)I 0 (dx) -S(t) X α * γ(x)i(t, x)I 0 (dx) + S * X α * γ(x)i(t, x)I 0 (dx),
and

V 2 (t) = X i * (x) i t (t, x) i * (x) g i(t, x) i * (x) I 0 (dx) = X γ(x) (α * S(t) -1) i(t, x) 1 - i * (x) i(t, x) I 0 (dx) = X γ(x) (α * S(t) -1) (i(t, x) -i * (x)) I 0 (dx) = X γ(x)α * S(t)i(t, x)I 0 (dx) - X γ(x)i(t, x)I 0 (dx) - X γ(x)α * S(t)i * (x)I 0 (dx) + X γ(x)i * (x)I 0 (dx).
Recalling S * = 1 α * , we have therefore

d dt V (S(t), i(t, •)) = d dt V 1 (t) + d dt V 2 (t) = -θ (S(t) -S * ) 2 S(t) + 2 X γ(x)i * (x)dx - (S * ) 2 S(t) X α * γ(x)i * (x)I 0 (dx) - X α * γ(x)S(t)i * (t)I 0 (dx). Since X α * γ(x)i * (x) S(t) + (S * ) 2 S(t) I 0 (dx) ≥ X α * γ(x)i * (x) × 2S * I 0 (dx),
which stems from the inequality a + b ≥ 2 √ ab, we have indeed proved that (4.11) holds.

Next we can determine the long-time behavior when the initial measure I 0 puts a positive mass on the set of maximal fitness. unded Lemma 4.11. Let Assumption 2.1 hold. Assume that R 0 (I 0 ) > 1 and suppose that I 0 ({α(x) = α * }) > 0, or in other words,

α(x)=α * I 0 (dx) > 0. Let η(t) := α * S(t) -1, then there exists a constant η such that -∞ < -η ≤ tη(t) ≤ η < +∞, (4.12
) eq:230 for all t ≥ 0. If moreover lim inf t→-∞ α(x)=α * I(t, dx) > 0, then up to changing the constant η, (4.12) holds for all t ∈ R.

Proof. We first remark that I(t, dx) can be written as

I(t, dx) = exp γ(x)t η(t) + (α(x) -α * ) 1 t t 0 S(s)ds I 0 (dx).
By Jensen's inequality we have

exp α(x)=α * γ(x)tη(t) I 0 (dx) α(x)=α * I 0 ≤ α(x)=α * e γ(x)tη(t) I 0 (dx) α(x)=α * I 0 (dz) , so that tη(t) ≤ α(x)=α * I 0 α(x)=α * γ(x)I 0 (dx) ln X e γ(x)tη(t) I 0 (dx) X I 0 = α(x)=α * I 0 α(x)=α * γ(x)I 0 (dx) ln 1 α(x)=α * I 0 α(x)=α * I(t, dx) .
Applying Lemma 4.1, I(t, dx) is bounded and we have indeed an upper bound for tη(t). Next, writing

I(t, dx) = exp γ(x)tη(t) + (α(x) -α * ) t 0 S(s)ds I 0 (dx)
and recalling that t 0 S(s)ds → +∞ as t → +∞, the function exp γ(x)tη(t) + (α(x) -α * ) t 0 S(s)ds converges almost everywhere (with respect to I 0 ) to 0 on X\{α(x) = α * }, so that by Lebesgue's dominated convergence theorem, we have

lim t→+∞ X\{α(x)=α * } I(t, dx) = X\{α(x)=α * } lim t→+∞ exp γ(x)tη(t) + (α(x) -α * ) t 0 S(s)ds I 0 (dx) = 0.
Next it follows from Lemma 4.7 that lim inf t→+∞ I(t, dx) > 0, so that lim inf

t→+∞ α(x)=α * I(t, dx) = lim inf t→+∞ X I(t, dx) > 0.
Assume by contradiction that there is a sequence

(t n ) such that t n η(t n ) → -∞, then α(x)=α * I(t, dx) = α(x)=α * e γ(x)tnη(tn) I 0 (dx) ≤ α(x)=α * e γ * tnη(tn) I 0 (dx) = e γ * tnη(tn) α(x)=α * I 0 (dx) ----→ t→+∞ 0,
where γ * := inf x∈supp I0 γ(x) > 0. This is a contradiction. Therefore there is a constant η > 0 such that

tη(t) ≥ -η > -∞.
In particular, the function t → tη(t) is bounded by two constants,

-∞ < -η ≤ tη(t) ≤ η < +∞.
This completes the proof of Lemma 4.11.

Finally we prove that any complete orbit that is already concentrated on {α(x) = α * } is constant, provided the mass can be bounded when t → -∞. This is a kind of LaSalle principle, since we have a partial Lyapunov functional by Lemma 4.10. stant Lemma 4.12. Let Assumption 2.1 hold and assume that α(x) ≡ α * for all x ∈ supp I 0 . Suppose that S(t), I(t, dx) is a complete orbit of (2.1) with initial data S 0 , I 0 (dx) , and assume that lim inf

t→-∞ X I(t, dx) > 0. Then S(t) ≡ 1 α * , for all t ∈ R.
Proof. Thanks to our assumption and the results of Lemma 4.11, we know that tη(t) is bounded for t ∈ R; moreover by Lemma 4.10, the functional V (S(t), i(t, x)) is well-defined and decreasing along the orbit S(t), i(t, x) , t ∈ R. Since V (S(t), i(t, x)) is bounded and decreasing there exists

V ∞ ∈ R such that V (S(t), i(t, x)) ----→ t→-∞ V ∞ . Let t n → -∞ be a sequence with S(t n ) → S -∞ and t n η(t n ) → η -∞ when t n → -∞, so that I(t n , dx) = e tnη(tn)γ(x) I 0 (dx) converges when t → -∞ to I -∞ 0 (dx) := e η -∞ I 0 (dx).
Then the shifted orbits S(t+t n ), I(t+t n , dx) converge, as t n → -∞, to a complete orbit S -∞ (t), I -∞ (t, dx) with I -∞ (0, dx) = I -∞ 0 (dx) = e η -∞ I 0 (dx). By the continuity of V , along the new orbit S -∞ (t), I -∞ (t, dx) , we have that Then it follows from the first line in (2.1a) that 0 = Λ -θS * -S * X α * γ(x)I -∞ (t, dx) for all t ∈ R, therefore in particular

V S -∞ (t), I -∞ (t, dx) ≡ V -∞ is a constant. Thus d dt V S -∞ (t), I -∞ (t,
X γ(x)e η ∞ γ(x) I 0 (dx) = θ α * R 0 -1 .
Thus η -∞ is a solution of (4.10) and, by the uniqueness of the solution, we have η -∞ = τ and I -∞

0 (dx) = i * (x)I 0 (dx). Thus V -∞ = V S -∞ , e η -∞ γ(x) = V S * , i * (x) = 0.
Thus V -∞ is the smallest possible value of V (S, i(x)). Since t → V (S(t), i(t, x)) is nonincreasing, we have therefore V S(t), i(t, x) ≡ 0, d dt V S(t), i(t, x) ≡ 0, for all t ∈ R.

By (4.11), we have therefore

S(t) ≡ S * = 1
α * for all t ∈ R, which completes the proof of Lemma 4.12.

itive Lemma 4.13. Let Assumption 2.1 hold. Assume that R 0 (I 0 ) > 1 and suppose that I 0 ({α(x) = α * }) > 0, or in other words,

α(x)=α * I 0 (dx) > 0. Then S(t) ----→ t→+∞ 1 α * and I(t, dx) -I ∞ (dx) T V ----→ t→+∞ 0,
where we have the following formula for I ∞ (dx), with τ being the unique solution of (4.10),

I ∞ (dx) = e τ γ(x) 1 α(x)=α * I 0 (dx).
Proof. Suppose that there exists a sequence t n → +∞ and η I(s, dx) > 0 for all t ∈ R.

Thus we can apply Lemma 4.12 which shows that

S ∞ (t) ≡ 1 α * for all t ∈ R.
Thus (S ∞ ) (t) ≡ 0 and by using the first line in (2.1a) we find that

X γ(x)e η * γ(x) 1 α(x)=α * I 0 (dx) = θ α * R 0 (I 0 ) -1 .
This is (4.10) which has a unique solution η * = τ . Since we can extract from any sequence t n → +∞ a subsequence with t n η(t n ) → τ , we conclude that lim t→+∞ tη(t) = τ therefore I(t, dx)

• T V ----→ t→+∞ e τ γ(x) 1 α(x)=α * I 0 (dx).
We show similarly that S(t) → 1 α * as t → +∞. When the set of maximal fitness {α(x) = α * } is negligible for I 0 , it is more difficult to obtain a general result for the long-time behavior of I(t, dx). We start with a short but useful estimate on the rate η(t) Proof. Assume by contradiction that there exists a sequence t n → +∞ such that t n η(t n ) has a uniform upper bound as t n → +∞, then observe that the quantity e γ(x)(α(x)S(tn)-1)tn = e γ(x)(α(x)-α * )S(tn)tn+γ(x)η(tn)tn is uniformly bounded in t n and vanishes as t n → +∞ almost everywhere with respect to I 0 (dx). By a direct application of Lebesgue's dominated convergence Theorem, we have therefore

X I(t n , dx) = X e γ(x)(α(x)S(tn)-1)tn I 0 (dx) -----→ tn→+∞ 0,
which is in contradiction with Lemma 4.6. We conclude that tη(t) → +∞ as t → +∞.

We are now in the position to prove Theorem 2.2.

Proof of Theorem 2.2. The convergence of S(t) and I(t, dx) in case i) was proved in Lemma 4.13.

Let us focus on case ii), that is to say, we assume

α(x)=α * I 0 (dx) = 0.
The uniform persistence of I(t, dx) is a consequence of 4.6. The concentration on the maximal fitness was proved in Lemma 4.9. Let us show that S(t) → 1 α * . Suppose by contradiction that it is not the case, then there exists ε > 0 and a sequence t n → +∞ with S(t n ) -1 α * ≥ ε. By Lemma 4.5 we can extract t n a subsequence such that the shifted orbits S(t + t n ), I(t + t n , dx) converge to S ∞ (t), I ∞ (t, dx) . We have α(x)<α * I ∞ (t, dx) = 0, X I ∞ (t, dx) > 0, and lim inf t→-∞ X I ∞ (t, dx) ≥ lim inf t→+∞ X I(t, dx) > 0, so by Lemma 4.12 we have lim n→+∞ S(t n ) = S ∞ (0) = 1 α * . This is obviously a contradiction. Theorem 2.2 is proved.

We now turn to the proof of Proposition 2.7 and we first prove that I(t, dx) concentrates on the set of points maximizing both α and γ. This property is summarized in the next lemma. Proof. We decompose the proof in several steps.

Step 1: We show that I(t, dx) and 1 α(x)S(t)≥1 I(t, dx) are asymptotically close in • T V . That is to say,

I(t, dx) -1 α(•)S(t)≥1 I(t, dx) T V ----→ t→+∞ 0.
Indeed we have I(t, dx) -1 α(x)S(t)≥1 I(t, dx) = 1 α(x)S(t)<1 I(t, dx) = 1 α(x)S(t)<1 e γ(x)(α(x)S(t)-1)t I 0 (dx).

First note that the function 1 α(x)S(t)<1 I(t, dx) = 1 α(x)S(t)<1 e γ(x)(α(x)S(t)-1)t I 0 (dx) is uniformly bounded. On the other hand, since I 0 ({α(x) = α * }) = 0 recall that S(t) → 1 α * for t → ∞, so that 1 α(x)S(t)<1 → 0 as t → ∞ almost everywhere with respect to I 0 . It follows from Lebesgue's dominated convergence Theorem that X 1 α(x)S(t)<1 e γ(x)(α(x)S(t)-1)t I 0 (dx) ----→ t→+∞ 0.

Step 2: We show that the measure 1 S(t)y≥1 e γ(yS(t)-1)t A(dy) is bounded when t → ∞ for all γ < γ * . Recall that A is the pushforward measure of I 0 by the continuous map α. Note that I 0 ({α(x) = α * }) = 0 implies that A({α * }) = 0 and remark that one has X 1 α(x)S(t)≥1 I(t, dx) = α * min(α * ,1/S(t)) α(x)=y e γ(x)(yS(t)-1)t I 0 (y, dx)A(dy), so, according to Step 1, for t sufficiently large one has e γ(yS(t)- e γ(yS(t)-1)t A(dy) < ∞.

Note that, if the constant m is independent of γ, then the above estimate does not depend on γ either.

Step 3: We show that 1 γ(x)<γ 1 S(t)α(x)≥1 I(t, dx) vanishes whenever γ < γ * .

Fix γ < γ * and let 0 < ε≤ γ * -γ 2 . Then we have I 0 (y, dx)e -ε(yS(t)-1)t e γ(yS(t)-1)t A(dy).

Reducing ε if necessary we may assume that γ ε > 1. Therefore it follows from Hölder's inequality that γ(x)≤γ and α(x)≥1/S(t)

I(t, dx) ≤ α * min(α * ,1/S(t))
e -ε(yS(t)-1)t γ ε e γ(yS(t)-1)t A(dy)

ε γ ×   α * min(α * ,1/S(t)) α(x)=y I 0 (y, dx) γ γ-ε e γ(yS(t)-1)t A(dy)   1-ε γ ≤ α * min(α * ,1/S(t)) A(dy) ε γ α * min(α * ,1/S(t))
e γ(yS(t)-1)t A(dy)

1-ε γ = I 0 (L α * -min(α * ,1/S(t)) (I 0 )) ε γ α * min(α * ,1/S(t))
e γ(yS(t)-1)t A(dy) Next we prove the asymptotic mass. Pick a sentence t n → +∞. By the compactness of the orbit (proved in Lemma 4.6) we can extract from t n a subsequence t n such that there exists a Radon measure I ∞ (dx) with d 0 (I(t, dx), I ∞ (dx)) ----→ t→+∞ 0, and since S(t) → 1 α * and upon further extraction, S (t n ) → 0. Therefore,

X α(x)γ(x)I(t n , dx) = Λ -S (t n ) S(t n ) -θ -----→ n→+∞ α * Λ -θ = θ (R 0 (I 0 ) -1)
.

By the concentration result in Lemma 4.15,

I ∞ is concentrated on {α(x) = α * } ∩ {γ(x) = γ * }. Therefore α * γ * I ∞ (dx) = α(x)γ(x)I ∞ (dx) = lim n→+∞ X α(x)γ(x)I(t n , dx) = θ (R 0 (I 0 ) -1) , so that lim n→+∞ I(t n , dx) = I ∞ (dx) = θ α * γ * (R 0 (I 0 ) -1) .
Since the limit is independent of the sequence t n , we have indeed shown that

I(t, dx) + f (t) ≤ lim inf t→+∞ f (t) Remark that 1 U (x)1 γ≥γ 1 S(t)y≥1 I(t, dx) = α * min(α * ,1/S(t)) γ * γ {γ(x)=z} 1 U (x)e z(yS(t)-1)t I α,γ 0 (y, z, dx)I α 0 (y, dz)A(dy) = α * min(α * ,1/S(t)) γ * γ {γ(x)=z} 1 U (x)I α,γ 0 (y, z, dx)e z(yS(t)-1)t I α 0 (y, dz)A(dy) ≥ α * min(α * ,1/S(t)) γ * γ m 2 e z(yS(t)-1)t I α 0 (y, dz)A(dy) ≥ f (t) m 2 ,
provided t is sufficiently large and γ is sufficiently close to γ * , where

m := lim inf ε→0 A(dy) ess inf α * -ε≤y≤α * I α 0 (y,dz) ess inf γ * -ε≤z≤γ * 1 x∈U I α,γ 0 (y, z, dx) > 0. Therefore lim inf t→+∞ U I(t, dx) ≥ m 2 lim inf t→+∞ f (t) > 0.
This completes proof of Proposition 2.7.

5 The case of a finite number of regular maxima axima

In this section we prove Theorem 2.10. To that aim, we shall make use of the following formula

I(t, dx) = exp γ(x) α(x) t 0 S(s)ds -t I 0 (dx). ( 5

.1) formul

Recall also the definition of η(t):

η(t) = α * 1 t t 0 S(s)ds -1 = α * S(t) -1.
Proof of Theorem 2.10. We split the proof of this result into three parts. We first derive a suitable upper bound. We then derive a lower bound in a second step and we conclude the proof of the theorem by estimating the large time asymptotic of the mass of I around each point of {α(x) = α * }.

Upper bound: Let i = 1, .., p be given. Recall that ∇α(x i ) = 0. Now due to (iii) in Assumption 2.8 there exist m > 0 and T > ε -2 0 large enough such that for all t ≥ T and for all y ∈ B 0, t -1 2 we have

α(x i + y) -α * ≤ -α * m y 2 .
As a consequence, setting Γ(x) = γ(x) α(x) α * , we infer from (5.1) and the lower estimate of I 0 around x i given in Assumption 2.8 (ii), that for all t > T

xi-x ≤t -1 2 I(t, dx) ≥ M -1 |y|≤t -1 2 |y| κi exp tη(t)Γ(x i + y) -tγ(x i + y)m|y| 2 dy.
Next since the function I = I(t, dx) has a bounded mass, there exists some constant C > 0 such that

R N I(t, dx) ≤ C, ∀t ≥ 0.
Coupling the two above estimates yields for all t > T |y|≤t -1 2 |y| κi exp tη(t)Γ(x i + y) -tγ(x i + y)m|y| 2 dy ≤ M C.

Hence setting z = y √ t into the above integral rewrites as

|z|≤1 t -κ i 2 |z| κi exp tη(t)Γ(x i + t -1 2 z) -γ(x i + t -1 2 z)m|z| 2 dz t N/2 ≤ M C, ∀t > T.
Now, since γ and α are both smooth functions, we have uniformly for |z| ≤ 1 and t 1:

Γ(x i + t -1 2 z) = γ(x i ) + O t -1 2 , γ(x i + t -1 2 z) = γ(x i ) + O t -1 2 .
This yields for all t

1 |z|≤1 t -κ i 2 |z| κi exp tη(t) γ(x i ) + O t -1 2 -γ(x i )m|z| 2 dz t N/2 ≤ CM, t -κ i 2 -N 2 e tη(t) γ(xi)+O t -1 2 |z|≤1
|z| κi e -γ(xi)m|z| 2 dz ≤ CM, that also ensures the existence of some constant c 1 ∈ R such that

tη(t) γ(x i ) + O t -1 2 - N + κ i 2 ln t ≤ c 1 , ∀t 1, or equivalently η(t) ≤ N + κ i 2γ(x i ) ln t t + O 1 t as t → ∞.
Since the above upper-bound holds for all i = 1, .., p, we obtain the following upper-bound

η(t) ≤ ln t t + O 1 t as t → ∞, (5.2) upper
where is defined in (2.12).

Lower bound:

Let ε 1 ∈ (0, ε 0 ) small enough be given such that for all i = 1, .., p and |y| ≤ ε 1 one has

α(x i + y) ≤ α * - 2 |y| 2 .
Herein > 0 is defined in Assumption 2. Recall that Γ(x) = α(x)γ(x) α *

and ∇Γ(x) = 1 α * (α(x)∇γ(x) + γ(x)∇α(x)). Consider M > 0 such that for all k = 1, .., p and all |x -

x k | ≤ ε 1 one has |Γ(x) -γ(x k ) -∇γ(x k ) • (x -x k )| ≤ M |x -x k | 2 . ( 5

.3) Gamma

Next fix i = 1, .., p and ε ∈ (0, ε 1 ). Then one has for all t > 0 |x-xi|≤ε

I(t, dx) ≤ |x-xi|≤ε exp tη(t)Γ(x) -tm|x -x i | 2 I 0 (dx) ≤ e tη(t)γ(xi) |x-xi|≤ε exp t η(t)∇γ(x i ) • (x -x i ) -(m + O(η(t))|x -x i | 2 I 0 (dx).
Now observe that for all t 1 one has

η(t)∇γ(x k ) • (x -x i ) -(m + O(η(t)))|x -x i | 2 = -(m + O(η(t))) x -x i - η(t)∇γ(x i ) 2(m + O(η(t))) 2 + η(t) 2 ∇γ(x i ) 2 4(m + O(η(t))
) , so that we get, using Assumption 2.8 (ii), that |x-xi|≤ε

I(t, dx) ≤ e tη(t)γ(xi)+ tη(t) 2 ∇γ(x i ) 2 4(m+O(η(t)) |x-xi|≤ε exp -(m + O(η(t))t x -x i - η(t)∇γ(x i ) 2(m + O(η(t)) 2 I 0 (dx) ≤ M e tη(t)γ(xi)+ tη(t) 2 ∇γ(x i ) 2 4(m+O(η(t)) × |x-xi|≤ε |x -x i | κi exp -(m + O(η(t))t x -x i - η(t)∇γ(x i ) 2(m + O(η(t)) 2 dx.
We now make use of the following change of variables in the above integral

z = √ t x -x i - η(t)∇γ(x i ) 2(m + O(η(t))
, so that we end up with

C(t) → C ∞ := M R N |z| κi e -m 2 |z| 2 dz ∈ (0, ∞) as t → ∞.
As a conclusion of the above analysis, we have obtained that there exists some constant C such that for all ε ∈ (0, ε 1 ) and all i = 1, .., p one has Now recalling the definition of and J in (2.12) and (2. Estimate of the masses: In this last step we turn to the proof of (2.13). Observe first that the upper estimate directly follows from the asymptotic expansion of η(t) in (2.11) together with (5.4). Next, the proof for the lower estimate follows from similar inequalities as the one derived in the second step above. Thus x ∈ B 2 . The equality is proved. We conclude that P -1 K (B H (C, R)) is a Borel set for all C ∈ K(K) and R > 0, and since those sets form a basis of the Borel σ-algebra, P K is indeed Borel measurable. The Lemma is proved. Proof. Indeed, let us choose a Borel measurable metric projection P K on K as in Theorem B.4. Let µ K be the image measure defined on B(K) by µ K (B) := µ P -1 K (B) , for all B ∈ B(K). Then in particular for all f ∈ BC(M ) we have

R N f (P K (x))dµ(x) = K f (x)dµ K (x).
Let f ∈ Lip 1 (M ), then we have Therefore d 0 (µ, µ K ) ≤ µ T V sup x∈supp K d(x, K) and, since µ K ∈ M + (K),

R N f (x)d(µ -µ K )(x) = R N f (x)dµ(x) - R N f (x)dµ K (x) = R N f (x)dµ(x) -
d 0 µ, M + (K) ≤ d 0 (µ, µ K ) ≤ µ T V sup x∈supp µ d(x, K).
The Proposition is proved.

C Disintegration of measures ation

C.1 Bourbaki's disintegration theorem

We recall the disintegration theorem as stated in [7, VI, §3, Theorem 1 p. 418]. We use Bourbaki's version, which is proved by functional analytic arguments, for convenience, although other approaches exist which are based on measure-theoretic arguments and may be deemed more intuitive. We refer to Ionescu Tulcea and Ionescu Tulcea for a disintegration theorem resulting from the theory of (strong) liftings [START_REF] Ionescu Tulcea | On the lifting property. IV. Disintegration of measures[END_REF][START_REF] Ionescu Tulcea | Topics in the theory of lifting[END_REF].

Let us first we recall some background on adequate families. This is adapted from [7, V.16 §3] to the context of finite measures of R N . We let T and X be locally compact topological spaces and µ ∈ M + (T ) be a fixed Borel measure. The initial condition I 0 vanishes more rapidly around x 1 than x 2 so that the solution I(t, x) vanishes around x 1 as t goes to ∞, even though γ(x 1 ) > γ(x 2 ), while around x 2 it takes the shape given by expression (2.11).

  then one has S(t) ----→ t→+∞ 1 α * and I(t, dx) ----→ t→+∞ I ∞ (dx) := 1 α(x)=α * (x)e τ γ(x) I 0 (dx),

  ) eq:osc and lim supn→+∞ R×R + I(t 2 n , dxdy) ≤ ε(L).

( 3 . 4 )

 34 eq:exa where P [a,b] (x) := max 1 -(a + b -2x) 2 (a -b) 2 , 0 is the downward parabolic function of height one and support [a, b] and δ = 0.2. The function α(x) has the exact same local behavior in the neighborhood of x 1 and x 2 . The function I 0 (x) is chosen as

  where γ * = inf x∈supp I0 γ(x). Since L ε /2 (I0) I 0 (dx) > 0 and Tn 0 S(t)dt → +∞ when n → +∞, we have therefore lim sup t→+∞ X I(t, dx) ≥ lim sup n→+∞ L ε /2 (I0)

Remark 4 . 8 . 9 .

 489 By combining Lemma 4.2 and Lemma 4.7 we obtain that 1 T T 0 S(t)dt admits a limit when T → +∞ and lim Let Assumption 2.1 hold. Let S(t), I(t, dx) be the corresponding solution of (2.1). Then one has d 0 (I(t, dx), M + ({α(x) = α * })) ----→ t→+∞ 0.

  dx) = 0 and, by(4.11),S -∞ (t) ≡ S * = 1 α * and (S -∞ ) (t) ≡ 0 for all t ∈ R.

ta(t) Lemma 4 . 14 .t t 0 S

 4140 Let Assumption 2.1 hold. Assume that R 0 (I 0 ) > 1. Suppose that I 0 ({α(x) = α * }) = 0 and set η(t) := α * S(t) -1, with S(t) = 1 (s)ds, where α * := sup x∈supp I0 α(x). Then it holds tη(t) ---→ t→∞ +∞.

vival Lemma 4 . 15 .

 415 Let Assumptions 2.5 hold. Assume that R 0 (I 0 ) > 1 and that I 0 ({α(x) = α * }) = 0. Recalling the definition of α * in (2.3) and γ * in Assumption 2.5, set Γ 0 (I 0 ) be the set of maximal points of γ on {α(x) = α * }, defined by Γ 0 (I 0 ) := {x : γ(x) = γ * and α(x) = α * }. Then one has d 0 (I(t, dx), M + (Γ 0 (I 0 ))) ----→ t→+∞ 0.

γ

  (x)∈[γ,γ * ] I(t, dx) = α * min(α * ,1/S(t)) α(x)=y and γ(x)∈[γ,γ * ] e γ(x)(yS(t)-1)t I 0 (y, dx)A(dy) + o(1) ≥ α * min(α * ,1/S(t)) α(x)=y and γ(x)∈[γ,γ * ] e γ(yS(t)-1)t I 0 (y, dx)A(dy) + o(1) = α * min(α * ,1/S(t)) α(x)=y and γ(x)∈[γ,γ * ] I 0 (y, dx)e γ(yS(t)-1)t A(dy) + o(1) ≥ m α * min(α * ,1/S(t))

γ

  (x)≤γ and α(x)≥1/S(t)I(t, dx) = α *min(α * ,1/S(t)) γ(x)≤γ and α(x)≥1/S(t) e γ(x)(yS(t)-1)t I 0 (y, dx)A(dy) ≤ α * min(α * ,1/S(t)) γ(x)≤γ and α(x)≥1/S(t) I 0 (y, dx)e (γ * -2ε)(yS(t)-1)t A(dy) ≤ α * min(α * ,1/S(t)) γ(x)≤γ and α(x)≥1/S(t)

13 )

 13 eq:upp Since S(t) → 1/α * as t → ∞, I 0 (L (I 0 )) ---→ →0 0 and by the boundedness of α * min(α * ,1/S(t)) e γ(yS(t)-1)t A(dy) shown in Step 2, we have indeed γ(x)≤γ and α(x)≥1/S(t) I(t, dx) ----→ t→+∞ 0, and this completes proof of Lemma 4.15. Proof of Proposition 2.7. The concentration of the distribution to M + ({α(x) = α * } ∩ {γ(x) = γ * }) was shown in Lemma 4.15.

  lim t→+∞ X I(t, dx) = θ α * γ * (R 0 (I 0 ) -1) .To prove the last statement, setf (t) := α * min(α * ,1/S(t)) γ * γ e z(yS(t)-1)t I α 0 (y, dz)A(dy),where γ < γ * . It follows from (4.13) that γ(x)≤γ and α(x)≥1/S(t) I(t, dx) ----→ e z(yS(t)-1)t I α,γ 0 (y, z, dx)I α 0 (z, dy)A(dy) = γ≤γ(x)≤γ * and α(x)≥1/S(t) I(t, dx) satisfies 0 < lim inf t→+∞ I(t, dx) = lim inf t→+∞ γ(x)≤γ and α(x)≥1/S(t)

  8 (iii). Next define m > 0 by i + y) > 0.

2 e

 2 dx) ≤ C t -N +κ i tη(t)γ(xi) , ∀t 1.(5.4) esti1Since I(t, dx) concentrates on {α(x) = α * }, then for all ε ∈ (0, ε 1 ) one hasR N I(t, dx) = p i=1 |x-xi|≤ε I(t, dx)dx + o(1) as t → ∞.Using the persistence of I stated in Theorem 2.2 (see Lemma 4.1), we end-up with 0 < lim inf t→∞ R N I(t, dx) ≤ lim inf t→∞ p i=1 |x-xi|≤ε I(t, dx), so that (5.4) ensures that there exists c > 0 and T > 0 such that 0 < c ≤ p i=1 e γ(xi) tη(t)-N +κ i 2γ(x i ) ln t , ∀t ≥ T. (5.5) esti2

n ≥ 1 and k ≥ 1 .

 1 If z ∈ C then there is a sequence z ϕ(k) such that z = lim z ϕ(k) , and (by assumption) we haveP K (x) ∩ B(z ϕ(k) , R + 1/k) = ∅. Therefore d z, P K (x) = lim k→+∞ d z ϕ(k) , P K (x) ≤ lim k→+∞

ction Theorem B. 4 (

 4 Existence of a regular metric projection). Let K ⊂ M be compact. There exists a Borel measurable mapP K : M → K such that d x, P K (x) = d(x, K).Proof. The proof is immediate by combining Proposition B.3 Proposition B.2. -supp Proposition B.5 (Metric projection on measure spaces). Let K ∈ K(M ) be a given compact set. Let µ ∈ M + (M ) be a given nonnegative Borel measure on M . Then the Kantorovitch-Rubinstein distance between µ and M + (K) can be bounded by the distance between K and the furthest point in supp µ: d 0 (µ, M + (K)) ≤ µ T V sup x∈supp µ d(x, K).

f

  (x) -f (P K (x)) dµ(x) ≤ supp µ |x -P K (x)|dµ(x) ≤ sup y∈supp µ d(y, K) supp µ 1dµ = µ T V sup x∈supp K d(x, K).

Figure 1 :

 1 Figure 1: Illustration of Theorem 2.2 in the case i), i.e., when I 0 {α(x) = α * } > 0. Parameters of this simulation are: Λ = 2, θ = 1, α(x) = 0.5 + T [-0.4,-0.2] (x 1 ) + 1 [0.2,0.8] (x 1 ) 1 [-0.6,0.6] (x 2 ) where x = (x 1 , x 2 ) ∈ R 2 and T [-0.4,-0.2] is the triangular function of height one and support [-0.4, -0.2], and γ = 1 2α . Initial condition is given by I 0 (dx) = I 0 (x 1 , x 2 ) dx where I 0 (x 1 , x 2 ) = 1 [-0.5,0.5] (x 1 ) cos(πx 1 )1 [-0.5,0.5] (x 2 ) cos(πx 2 ). In particular, we have α * = 3/2 and {α(x) = α * } = ({-0.3} ∪ [0.2, 0.5]) × [-0.5, 0.5]. Function t → S(t) converges towards 1/α * = 2/3 and function x → I(t, x) at time t = 50 is asymptotically concentrated on {α(x) = α * } = ({-0.3} ∪ [0.2, 0.5]) × [-0.5, 0.5].

Fig1

  Fig1

Figure 2 :

 2 Figure 2: Illustration of Theorem 2.2 in the case i), i.e., when I 0 {α(x) = α * } > 0. Function x → I(t, x) at time t = 10, 20, 30 and 40. The function I remains bounded in this case.

Fig3

  Fig3

Figure 3 :

 3 Figure 3: Illustration of Theorem 2.2 in the case ii), i.e., when α(x)=α * I 0 (dx) = 0. Parameters of this simulation are: Λ = 2, θ = 1, α(x) = 0.5 + T [-0.1,0.8] (x 1 )1 [-0.5,0.5] (x 2 ) where x = (x 1 , x 2 ) ∈ R 2 and T [-0.1,0.5] is the triangular function of height one and support [-0.1, 0.5], γ = 1 2α . Initial condition is given by I 0 (dx) = I 0 (x 1 , x 2 ) dx where I 0 (x) = 1 [-0.5,0.5] (x 1 ) cos(πx 1 )1 [-0.5,0.5] (x 2 ) cos(πx 2 ). In particular, α * = 3/2 and {α(x) = α * } = {0.35} × [-0.5, 0.5]. The function t → S(t) converges towards 1/α * = 2/3. The function x → I(t, x) at time t = 100 is asymptotically concentrated on {α(x) = α * } = {0.35} × [-0.5, 0.5].

Fig4

  Fig4

Figure 4 :

 4 Figure 4: Illustration of Theorem 2.2 in the case ii), i.e., when I 0 {α(x) = α * } = 0. Function x → I(t, x) at time t = 20, 40, 60 and 80. The function I asymptotically converges towards a singular measure.

Fig5

  Fig5

Figure 5 :

 5 Figure 5: Illustration of a transient behavior for (2.1). Parameters of this simulation are: Λ = 2, θ = 1, α(x) is given by (3.1), I 0 (x) by (3.2) and γ(x) by (3.3). In particular, α * = 1, {α(x) = α * } = {x 2 } with x 1 = -0.5, x 2 = 0.5. Other parameters are κ 1 = 2, κ 2 = 2, γ(x 1 ) = 1/2, γ(x 2 ) = 1/4. The value of the local maximum at x 1 , α(x 1 ) = 0.95, being very close to α * , observe that the distribution I(t, x) first concentrates around x 1 before the global maximum x 2 becomes dominant (bottom right plot).

Fig8

  Fig8

Figure 6 :

 6 Figure 6: Illustration of Theorem 2.10 and Corollary 2.11. Parameters of this simulation are: Λ = 2, θ = 1, α(x) is given by (3.4), I 0 (x) by (3.5) and γ(x) by (3.6). In particular, α* = 1, {α(x) = α * } = {x 1 , x 2 } with x 1 = -0.5, x 2 = 0.5, κ 1 = 8, κ 2 = 2, γ(x 1 ) = 1/2, γ(x 2 ) = 1/4,ρ = 6 and J = {2}. The initial condition I 0 vanishes more rapidly around x 1 than x 2 so that the solution I(t, x) vanishes around x 1 as t goes to ∞, even though γ(x 1 ) > γ(x 2 ), while around x 2 it takes the shape given by expression(2.11).

Fig7

  Fig7

Figure 7 :

 7 Figure 7: Numerical solution of the example provided in Section 2.3 of the main text with L = 1. The x, y axes correspond to the underlying physical space and we plot I(t, τ ) on the z axis at every x = (1 -e -τ ) cos(τ ) and y = (1 -e -τ ) sin(τ ). The observation times are spaced exponentially from one another (t = 0, 10, 10 2 , 10 3 , 10 4 , 10 5 ) to observe constant shifts of the fixed asymptotic profile. These plots are snapshots of the supplementary movie spiraling.avi.g:osc

  .11) expans

	wherein we have set	:= min i=1,...,p	N + κ i 2γ(x i )	.	(2.12) def-va

  By computing the integral of I(t, τ ), we will now obtain a precise description of the behavior of S(t) as t → +∞. Indeed,

	0	+∞	I(t, τ )dτ =	e tS(t)-t tS(t)	0	+∞	tS(t)e -τ e -tS(t)e -τ	dτ =	e tS(t)-t tS(t)	e -tS(t)e -τ +∞ 0
			=	e tS(t)-t tS(t)	1 -e -tS(t) =	e tS(t)-t tS(t)	-	e -t tS(t)	=	e tS(t)-t tS(t)	+ o(1),

therefore, recalling S(t) = 1 + o(1), we have e tS(t)-t = tI ∞ + o(t), and finally

  Lemma 4.2. Let Assumption 2.1 hold. Let S(t), I(t, dx) be the corresponding solution of (2.1). Then

	lim sup T →+∞ α Proof. Let us remark that the second component of (2.1) can be written as 1 T T 0 1 S(t)dt ≤
	I(t, dx) = I 0 (dx)e γ(x)(α(x) t 0 S(s)ds-t) ,
	= I 0 (dx) exp γ(x)	0	t	S(s)ds α(x) -	t 0 S(s)ds t	.	(4.1) eq:I-n
						t0) ,
	so that finally by letting t → +∞ we get
	lim inf t→+∞ (θ + ε) min(θ, γ Since ε > 0 is arbitrary we have shown min(θ, γ * )Λ S(t) ≥
	lim inf

* ) + Λα * γ * . t→+∞ S(t) ≥ min(θ, γ * )Λ θ min(θ, γ * ) + Λα * γ * .

The Lemma is proved.

o-mut * ,

where α * is given in (2.3).

  .5) eq:170 then one can extract from (t n ) a subsequence (t n k ) such that the shifted orbitst → S(t + t n k ), I(t + t n k , dx)converge weak- * pointwise to a complete orbit S ∞ , I ∞ (t, dx) satisfying the following properties: defined in (2.3). We prove that the family {I(t, dx), t ≥ 0} is uniformly tight. Let ε > 0 be sufficiently small, so that the set L ε (I 0 ) is compact. By Lemma 4.3 and Lemma 4.2 we have

	Proof. First of all let us remark that
	I(t, dx) = e ( t 0 S(s)dsα(x)-t)γ(x) I 0 (dx),
	and therefore the orbit t → I(t, dx) is continuous for the metric d 0 .
	By Lemma 4.2 we have
	lim sup T →+∞ α lim sup 1 T T 0 1 S(s)ds ≤
	t→+∞ X\Lε(I0)

X I ∞ (t, dx) > 0 for all t ∈ R, (4.6) eq:170 and α(x)<α * I ∞ (t, dx) = 0. (4.7) eq:170

Finally the convergence S(t

+ t n k ) → S ∞ (t) holds locally uniformly in C 1 (R). * ,

where α *

  The set {I(t, dx) : t ≥ 0} is uniformly tight. Moreover it is bounded in the total variation norm (see Lemma 4.1) in the complete separable metric space X, therefore precompact for the weak topology by Prokhorov's Theorem [5, Theorem 8.6.2, Vol. II p. 202].

  lim

			n→+∞ X	1I(t + t n , dx) > 0,
	which shows (4.6).
	Next it follows from Lemma 4.2 that lim sup n→+∞	1 tn	tn 0 S(s)ds ≤ 1 α

* and thus by

Lemma 4.3 that

  1 α(x)=α * I 0 (dx) = I ∞ (t, dx). We know that supp I ∞ (t, dx) ⊂ {α(x) = α * } and byLemma 4.4

	By Lemma 4.5, the shifted orbits S(t + t n ), I(t + t n , dx) converge to a complete orbit S ∞ (t), I ∞ (t, dx) with
	I(t n , dx) ----→
	we have
	X	I ∞ (t, dx) ≥ lim inf s→+∞ X

* ∈ [-η, η] such that lim n→+∞ t n η(t n ) = η * . t→+∞ e γ(x)η *

  1)t A(dy) + o(1), wherein m > 0 is the constant associated with γ in Assumption 2.5 and we used the Landau notation o(1) to collect terms that converges to 0 as t → +∞. Recalling the upper bound for I(t, dx) from Lemma 4.1, we have

	lim sup t→+∞	α * min(α * ,1/S(t))	e γ(yS(t)-1)t A(dy) ≤ lim sup t→+∞	1 m X	I(t, dx) ≤	Λ m min(θ, γ 0 )	< +∞.
	This implies that
			lim sup
			t→+∞ α(supp(I0))

  [START_REF] Day | Applying population-genetic models in theoretical evolutionary epidemiology[END_REF], the upper bound for η(t) provided in (5.2) implies

		e	γ(xi) tη(t)-
		i / ∈J
	and (5.5) rewrites as	0 <	c 2	≤
	This yields
				lim inf
	that is	η(t) ≥	ln t t	+ O	1 t	as t → ∞.	(5.6)

N +κ i 2γ(x i ) ln t → 0 as t → ∞, i∈J e γ(xi)(tη(t)-ln t) , ∀t 1. t→∞ (tη(t) -ln t) > -∞, lower

Then (2.11) follows coupling (5.2) and (5.6).

J.-B. Burie and A. Ducrot are supported by the ANR project ArchiV ANR-18-CE32-0004. Q. Griette was partially supported by ANR grant "Indyana" number ANR-21-CE40-0008. A.D. and Q.G. acknowledge the support of the Math AmSud program for project 22-MATH-09.

Appendix A The case of a unique fitness maximum

If the function α(x) has a unique global maximum in the support of the initial data, then our analysis leads to a complete description of the asymptotic state of the population. This may be the unique case when the behavior of the orbit is completely known, independently on the positivity of the initial mass of the fitness maximizing set {α(x) = α * }.

e-max

Theorem A.1 (The case of a unique global maximum). Let Assumption 2.1 be satisfied. Suppose that the function α = α(x) has a unique maximum α * on the support of I 0 attained at x * ∈ supp I 0 , and that R 0 (I 0 ) := Λ θ α * > 1.

Then it holds that S(t) ----→ t→+∞ 1 α * , d 0 (I(t, dx),

where δ x * (dx) denotes the Dirac measure at x * and

B Existence of a regular metric projection

In this Section we let (M, d) be a complete metric space. Let K(M ) be the set of compact subsets in M and let K ∈ K(M ). We first recall that we can define a kind of frame of reference, internal to K, which allows to identify each point in K.

Let us denote K(M ) the set formed by all compact subsets of M . Recall that (K(M ), d H ) is a complete metric space, where d H is the Hausdorff distance

(Metric coordinates).

There exists a finite number of points x 1 , . . . , x n ∈ K with the property that each y ∈ K can be identified uniquely by the distance between y and x 1 , . . . , x n . In other words the map

. . .

is one-to-one. Moreover c K is continuous and its reciprocal function c -1 K : c K (K) → K is also continuous. Proof. Let us choose x 1 ∈ K and x 2 ∈ K such that x 1 = x 2 . We recursively construct a sequence x n and a compact set K n such that

the choice of x n+1 being arbitrary. Clearly K n is a compact set and K n+1 K n . Suppose by contradiction that K n = ∅ for all n ∈ N, then (because K is compact) one can construct a sequence x ϕ(n) , extracted from x n , and which converges to a point

In particular K ∞ is not empty. However we see that, by definition of K ∞ , we have d(x, x n ) = d(x, x 1 ) > 0 for all n ∈ N, which contradicts the fact that lim

Hence we have shown by contradiction that there exists n 0 ∈ N such that K n0 = ∅ and K n0-1 = ∅. This is precisely the injectivity of the map c K : K → R n0 .

To show the continuity, we remark that c K is continuous, and therefore for each closed set F ⊂ K, F is compact so that c K (F ) is compact and therefore closed. Therefore (c

The proposition is proved.

Recall that the Borel σ-algebra B(M ) is the closure of the set of all open sets in P(M ) = 2 M under the operations of complement and countable union. A function ϕ : M → N is Borel measurable if the reciprocal image of any Borel set is Borel, i.e. ϕ -1 (B) ∈ B(M ) for all B ∈ B(N ). hoice Proposition B.2 (Borel function of choice). There exists a Borel measurable map c :

where the minimum is taken with respect to the lexicographical order in R n0 (which is a total order and therefore identifies a unique minimum for each K ∈ K(K)). Since the map K ⊂ R n0 → min y∈K y is Borel for the topology on K(R n0 ) induced by the Hausdorff metric, so is c. The proposition is proved. ction Proposition B.3 (Borel measurability of the metric projection). Let K ⊂ M be compact. The map P K : M → K(K) defined by

is Borel measurable.

Proof. First we remark that the map

is well-defined for each x ∈ M , and therefore forms a mapping from M into K(K) ⊂ K(M ). Indeed P K (x) is clearly closed in the compact space K, therefore is compact.

To show the Borel measurability of P K , we first remark that, given a compact space K ⊂ K, the set

is closed. Indeed let x n → x be a sequence in P -1 K (K ), then by definition there exists y n ∈ K such that d(x n , y n ) = d(x n , K). By the compactness of K , there exists y ∈ K and a subsequence y ϕ(n) extracted from y n such that y ϕ(n) → y. Because of the continuity of z → d(z, K), we have

therefore y ∈ P K (x) ∩ K , which shows that x ∈ P -1 K (K ). Hence P -1 K (K ) is closed. We are now in a position to show the Borel regularity of P K . Let C ∈ K(K) and R > 0 be given. We define B H (C, R) the ball of center C and radius R in the Hausdorff metric:

, where B 1 := {x ∈ M : d(y, C) ≤ R for all y ∈ P K (x)}, and

It can be readily seen that B 1 is a Borel set by writing

where V R (C) := {y ∈ K : d(y, C) ≤ R}. To see that B 2 is a Borel set, we choose a sequence z n which is dense in C and write

Indeed if x ∈ B 2 then P K (x) intersects every ball of radius R and center z ∈ C; in particular P K (x) intersects every ball of radius R + 1/k and center z n . Conversely suppose that P K (x) intersects every ball B(z n , R + 1/k) for Definition C.1 (Scalarly essentially integrable family). Let Λ : t → λ t be a mapping from T into M + (X). Λ is scalarly essentially integrable for the measure µ if for every compactly supported continuous function f ∈ C c (X), the function t → X f (x)λ t (dx) is in L 1 (µ). Setting ν(f ) = T X f (x)λ t (dx)µ(dt) defines a linear form on C c (X), hence a measure ν, which is the integral of the family Λ, and we denote

Recall that every positive Borel measure µ on a locally compact space X defines a positive bounded linear functional on C c (X) equipped with the inductive limit of the topologies on C c (K) when K runs over the compact subsets of X. Conversely if µ is a positive bounded linear functional on C c (X), there are two canonical ways to define a measure on the Borel σ-algebra.

1. Outer-regular construction. Let U ⊂ X be a open, then one can define It is always true that µ • ≤ µ * , however it may happen that µ * = µ • when µ * is not finite, see e.

Definition C.2 (Pre-adequate and adequate families). We follow [7, Definition 1, V.17 §3]. Let Λ : t → λ t be a scalarly essentially µ-integrable mapping from T into M + (X), ν the integral of Λ. We say that Λ is µ-pre-adequate if, for every lower semi-continuous function f ≥ 0 defined on X, the function t

We say that Λ is µ-adequate if Λ is µ -pre-adequate for every positive Borel measure µ ≤ µ.

The last notion we need to define is the one of µ-proper function.

Definition C.3 (µ-proper function). We say that a function p : T → X is µ-proper if it is µ-measurable and, for every compact set K ⊂ X, the set p

If µ is Radon, in particular, then every µ-measurable mapping p : T → X (X being equipped with the Borel σ-algebra) is µ-proper. The following Theorem is taken from [7, Theorem 1, VI.41 No.1, §3]. ation Theorem C.4 (Disintegration of measures). Let T and X be two locally compact spaces having countable bases, µ be a positive measure on T , p be a µ-proper mapping of T into X, and ν = p(µ) the image of µ under p. There exists a ν-adequate family x → λ x (x ∈ X) of positive measures on T , having the following properties: a) λ x = 1 for all x ∈ p(T ); b) λ x is concentrated on the set p -1 ({x}) for all x ∈ p(T ), and λ x = 0 for x ∈ p(T ); c) µ = λ x ν(dx).

Moreover, if x → λ x (x ∈ X) is a second ν-adequate family of positive measures on T having the properties b) and c), then λ x = λ x almost everywhere in B with respect to the measure ν.