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µ′(t)(E)= B(¯µ(t), q) D(¯µ(t), q) µ(t)(dq), − ZE   where µ is a Radon measure on a space of strategies Q andµ ¯(t)= µ(t)(Q). They show a persistence property for such a model and give more precise statements on the asymptotic behavior of the total mass; under the assumption that a unique strategy maximizing the carrying capacity, they show the convergence in the sense of measures to a Dirac mass concentrated on this strategy. An important example of a biological system which allows the experimenter to observe the competitive exclusion principle is the so-called chemostat, a device in which living organisms (like cells, yeasts, bacteria...) are cultivated in a controlled environment where the in- and outflow of nutrients are monitored. Systems of the same type as (1) (as most systems in the literature cited above) are particularly well-adapted to the modeling of a population in a chemostat. We refer to the monograph of Smith and Waltman [44] for a detailed work on mathematical models for the chemostat. Recently, a lot of interest has been given to chemostat models as a mean to study the Darwinian evolution of a population, that is, the changes in frequency of common characteristics due to the processes of selection and mutation. An important difference of chemostat models is that they usually assume Michaelis-Menten functional response, contrary to our system where the functional response if linear. The connections between chemostat and epidemic models has been remarked in [43], where an extensive review of the literature is conducted. A competitive exclusion principle for epidemic models has been established by Bremermann and Thieme [8]. The asymptotic behavior of epidemic models of SIR type has been investigated by Korobeinikov and Wake [30], Korobeinikov and Maini [29], Korobeinikov [26, 27, 28], McCluskey [37, 38], in the context of systems of ordinary differential equations. Our motivation for the study of System (1) comes from a model of Darwinian evolution of spore-producing plant pathogens [16, 9, 10]. The present work represents a first step (without mutation) towards a generalization of this previous model to cases where the mortality rate of the infected population, which is represented by function γ in (1), varies from one strain to another. We add to the existing literature on systems related to (1) by considering the case when the initial condition I0(dx) is a Radon measure. Considering measures as initial data is not a gratuitous generality. Indeed as we will show in Theorem 1.2, initial conditions in the usual space L1(RN ) will converge, in many cases, towards a singular Radon measure. Therefore it is necessary to study the equation in a space of measures in order to investigate the long-time behavior of the solutions to (1). By doing so we were able to include the case of finite systems of ODE into our framework, see Section 1.2 below. Those results are extended to the case of countably many equations under the structural requirement that the coefficients converge to a limit. The system behaves as predicted and the density I converges towards a Dirac measure at some x∗ when the basic reproduction number (fitness) associated to the phenotypic value x defined as (x) := Λ α(x) has a unique maximum on R0 θ

2 the support of I0 and the initial data has a positive mass on the maximizing value x∗, see Section 1.3. The situation is more complex when this assumption is removed. In particular, the asymptotic behaviour of the solution strongly depends on the initial condition I0, which may be related to the absence of mutations in the model. When the density of the initial condition vanished near the maximizing set of 0(I0), we uncovered a subtle dependency on the initial data and on the function γ of (1) (Section 1.3). ItR may happen, if the initial density decreases sufficiently fast near the maximizing set of the basic reproduction number, that the equilibrium reached by the system is different from the one predicted by heuristic arguments on the fitness function. This shows that the dynamics of the system with continuous phenotypic traits x RN is far more rich than the one of discrete models. ∈

1 Main results

In this section we state and discuss the main results we shall prove in this note related to the large time behavior of (1). Before going to our results, we introduce some notations that will be used along this work. For a Borel set K RN , we denote by (K) the set of the signed Radon measures on K of finite mass. Recall that (K) is a Banach⊂ space when endowedM with the absolute variation norm given by: M

µ = µ (K), µ (K). k kAV | | ∀ ∈M We also denote by (K) the set of the finite nonnegative measures on K. Observe that one has (K) M+ M+ ⊂ (K) and +(K) is a closed subset of (K) for the norm topology of AV . An alternate topology on M(K) can beM defined by the so-called Kantorovitch-RubinsteinM norm (see [5,k·k Vol. II, Chap. 8.3 p. 191]), M

µ 0 := sup fdµ : f Lip1(K), sup f(x) 1 , k k ∈ x K | |≤ Z ∈  wherein we have set

Lip (K) := f BC(K): f(x) f(y) x y , (x, y) K2 . 1 ∈ | − |≤k − k ∀ ∈ Here (and below) BC(K) denotes the set of the continuous and bounded functions from K into R. Let us recall (see for instance [5, Theorem 8.3.2]) that the metric generated by 0 on +(K) is equivalent on this set to the weak topology obtained by identifying (K) to the dual spacek·k of BCM(K). Note however that this M equivalence is true only for +(K) and cannot be extended to (K) since the latter space is not (in general) complete for the metric generatedM by . We denote by d thisM metric on (K), that is k·k0 0 M+ d (µ,ν) := µ ν for all µ,ν (K). (2) 0 k − k0 ∈M+ N Now along this note we fix I0 = I0(dx) +(R ). About the parameters arising in (1) our main∈M assumption reads as follows. Assumption 1.1. The constants Λ > 0 and θ > 0 are given. The functions α(x) and γ(x) are continuous from N R into R and there exist positive constants α∞ and γ0 <γ∞ such that

N α(x) α∞, 0 <γ γ(x) γ∞ for all x R . ≤ 0 ≤ ≤ ∈

Finally, define α∗ := supx supp I α(x). We assume that the set ∈ 0

L (I ) := α α∗ ε supp I = x supp I : α(x) α∗ ε ε 0 { ≥ − } ∩ 0 { ∈ 0 ≥ − } is bounded when ε> 0 is sufficiently small.

Let us observe that if S0 0 then (1) equipped with the initial data S(0) = S0 and I(0, dx)= I0(dx) has a unique globally defined solution≥ S(t) 0 and I(t, dx) (RN ) for all t 0. In addition I is given by ≥ ∈M+ ≥ t I(t, dx) = exp γ(x) α(x) S(s)ds t I (dx). − 0   Z0  The above formula ensures that supp I(t, ) = supp I for all t 0. And to describe the large time behavior of · 0 ≥ I, we will use the values of α and γ on the support of I0. Due to the above remark and since I0 is given and fixed, along this paper, for any y R we write ∈ 1 α− (y)= x supp (I ): α(x)= y supp (I ). (3) { ∈ 0 }⊂ 0

3 We also define the two quantities α∗ 0 and (I ) by ≥ R0 0 Λ α∗ =: sup α(x) and 0(I0) := α∗. (4) x supp I R θ ∈ 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 (1) when I0 is an arbitrary Radon measure. We roughly 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 I0(dx) is a finite or countable sum of Dirac masses. We continue our investigations with an absolutely continuous 1 (with respect to Lebesgue measure) initial measure and a finite set α− (α∗). In that setting we are able to fully characterize the points where the measure I(t, dx) concentrates as t . → ∞ 1.1 General results for measure-valued initial data As mentioned above this subsection is concerned with the large time behavior of the solution (S, I) of (1) where the initial measure I0(dx) is an arbitrary Radon measure. Using the above notations our first result reads as follows. Theorem 1.2 (Asymptotic behavior of measure-valued initial data). Let Assumption 1.1 be satisfied. Let S0 0 be given and denote by (S(t), I(t, dx)) the solution of (1) equipped with the initial data S(0) = S0 and I(0≥, dx)= I (dx). Recalling (4), suppose that (I ) > 1. We distinguish two cases depending on the measure 0 R0 0 of this set w.r.t. I0:

1 i) If I0 α− (α∗) > 0, then one has
1 τγ(x) S(t) and I(t, dx) I∗ (dx) := ½α−1(α∗)(x)e I0(dx), t + t + −−−−→→ ∞ α∗ −−−−→→ ∞ ∞ where τ R denotes the unique solution of the equation ∈

τγ(x) θ γ(x)½α−1(α∗)(x)e I0(dx)= 0 1 . RN α R − Z ∗
The convergence of I(t, dx) to I∗ (dx) holds in the absolute variation norm AV . ∞ k·k 1 1 ii) If I α− (α∗) =0, then one has S(t) ∗ and I(t, dx) is uniformly persistent, namely 0 → α
lim inf I(t, dx) > 0. t + RN → ∞ Z 1 Moreover I(t, dx) is asymptotically concentrated as t on the set α− (α∗), in the sense that → ∞ 1 d0 I(t, dx), +(α− (α∗)) 0, t + M −−−−→→ ∞
where d0 is the Kantorovitch-Rubinstein distance.

We continue our general result by showing that under additional properties for the initial measure I0, the 1 function I(t, dx) concentrates in the large times on the set of the points in α− (α∗) that maximize the function γ = γ(x). The additional hypothesis for the initial measure I0(dx) are expressed in term of some properties of its disintegration measure with respect to the function α. We refer to the book of Bourbaki [7, VI, §3, Theorem 1 p. 418] for a proof of the disintegration Theorem B.4 which is recalled in Appendix B. N Let A(dy) be the image of I0(dx) under the continuous mapping α : R R, then there exists a family of nonnegative measures I (y, dx) (the disintegration of I with respect to α)→ such that for almost every y 0 0 ∈ α(supp I0) with respect to A we have:

1 supp I0(y, dx) α− (y), I0(y, dx) = 1 and I0(dx)= I0(y, dx)A(dy) (5) ⊂ −1 Zα (y) Z wherein the last equality means that

N f(x)I0(dx)= f(x)I0(y, dx)A(dy) for all f BC(R ). RN y R α−1(y) ∈ Z Z ∈ Z

Note that, by definition, the measure A is supported on the set α(supp I0).

4 1 n Remark 1.3 (Important example). Suppose that I0 L (R ). 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 RN and that

I (x) 0 L1(RN ). (6) α(x) ∈ |∇ | The coarea formula implies that, for all g L1(RN ), we have ∈

g(x) α(x) dx = g(x) N 1(dx)dy, RN |∇ | R −1 H − Z Z Zα (y) where N 1(dx) is the (N 1)-dimensional Hausdorff measure (see Federer [17, §3.2]). − H −I0(x) Therefore if g(x)= f(x) α(x) we get |∇ |

I0(x) f(x)I0(x)dx = f(x) N 1(dx)dy, (7) RN R −1 α(x) H − Z Z Zα (y) |∇ | and if moreover f(x)= ϕ(α(x)) we get

I0(x) ϕ(y)A(dy)= ϕ(α(x))I0(x)dx = ϕ(y) N 1(dx)dy, R RN R −1 α(x) H − Z Z Z Zα (y) |∇ | where we recall that A(dy) is the image measure of I0(dx) through α. Therefore we have an explicit expression for A(dy): I0(x) A(dy)= N 1(dx)dy (8) −1 α(x) H − Zα (y) |∇ | and (recalling (7)) we deduce the following explicit disintegration of I0:

I0(x) ½x α−1(y) N 1(dx) ∈ α(x) − I0(y, dx)= |∇ | H . (9) I0(z) α−1(y) α(z) N 1(dz) |∇ | H − Equations (8) and (9) give an explicit formula forR the disintegration introduced in (5).

2 Remark 1.4. Suppose that α is a C function with no critical point in supp I0 except for a finite number of regular maxima (in the sense that the bilinear form D2α(x) is non-degenerate at each maximum). This is a N typical situation. Then assumption (6) in Remark 1.3 is automatically satisfied if N 3 and I0 L∞(R ). If N ≥ ∈ N = 2 then a sufficient condition to satisfy (6) with I0 L∞(R ) should involve I0 vanishing sufficiently fast in the neighborhood of each maximum of α. ∈

We shall also make use, for all y A almost everywhere, of the disintegration measure of I0(y, dx) with respect to the function γ, as follows −

α,γ α I0(y, dx)= I0 (y, z, dx)I0 (y, dz), z γ(α−1(y)) Z ∈ that allows to the following reformulation of I0(dx):

α,γ α I0(dx)= I0 (y, z, dx)I0 (y, dz)A(dy). y R z γ(α−1(y)) Z ∈ Z ∈

Now equipped with this disintegration of I0 with respect to α we are now able to state our regularity 1 assumption to derive more refine concentration information in the case where I0 α− (α∗) = 0.

1 Assumption 1.5 (Regularity with respect to α, γ). Recalling (3), assume that α− (α∗) =
and define γ∗ > 0 by 6 ∅ γ∗ := sup γ(x). (10) x α−1(α∗) ∈ We assume that, for each valueγ<γ ¯ ∗ there exist constants δ > 0 and m> 0 such that

m I0(y, dx) for A-almost every y (α∗ δ, α∗]. ≤ γ−1([¯γ,γ∗]) α−1(y) ∈ − Z ∩

5 Remark 1.6. The above assumption means that the initial measure I0(dx) is uniformly positive in a small 1 neighborhood of α− (α∗). For instance, if the assumptions of Remark 1.3 are satisfied and if there exists an open 1 1 set U containing α− (α∗) γ− (γ∗) on which I0(x) is almost everywhere uniformly positive, then Assumption 1.5 is automatically satisfied.∩

The next proposition ensures that, when the initial measure I0 satisfies Assumption 1.5, then the function 1 1 I(t, dx) concentrates on α− (α∗) γ− (γ∗). ∩ 1 Proposition 1.7. Under the same assumptions as in Theorem 1.2, assume that I0 α− (α∗) = 0 and let us furthermore assume that Assumption 1.5 holds, then recalling that γ is defined in (10) one has ∗
1 1 d0 I(t, dx), +(α− (α∗) γ− (γ∗)) 0, t + M ∩ −−−−→→ ∞ as well as the following asymptotic mass
θ I(t, dx) 0(I0) 1 . RN −−−−→t + α γ R − Z → ∞ ∗ ∗ N 1 1
Let U R be a Borel set such that U α− (α∗) γ− (γ∗) = ∅ and ⊂ ∩ ∩ 6 A(dy) Iα(y,dz) 0 α,γ lim inf ess inf ess inf I0 (y, z, dx) > 0, ε 0 α∗ ε y α∗γ∗ ε z γ∗ → − ≤ ≤ − ≤ ≤ ZU then the following persistence occurs lim inf I(t, dx) > 0. t →∞ ZU We now explore some numerical computations of (1) with various configurations.

1 Figure 1: Illustration of Theorem 1.2 in the case i), i.e., when I0 α− (α∗) > 0. Parameters of this simulation

2

½ ½ are: Λ = 2, θ = 1, α(x)=0.5+ Ì[ 0.4, 0.2](x1)+ [0.2,0.8](x1) [ 0.6,0.6](x2) where x = (x1, x2) R and − − −
1 ∈ Ì[ 0.4, 0.2] is the triangular function of height one and support [ 0.4, 0.2], and γ = . Initial condition is

− −
− − 2α ½ given by I0(dx)= I0(x1, x2) dx where I0(x1, x2)= ½[ 0.5,0.5](x1)cos(πx1) [ 0.5,0.5](x2)cos(πx2). In particular, 1 − − we have α∗ =3/2 and α− (α∗) = ( 0.3 [0.2, 0.8]) [ 0.6, 0.6]. {− } ∪ × −

1.2 The case of discrete systems In this subsection we propose an application of the general result, namely Theorem 1.2, to the case of discrete systems. We start with the case of finite systems, i.e., the case when system (1) can be written as follows: d S(t)=Λ θS(t) S(t) α γ I1(t)+ α γ I2(t)+ ... + α γ In(t) dt − − 1 1 2 2 n n  d 1 1
 I (t)= α1S(t) 1 γ1I (t) dt −  d 2
2  I (t)= α2S(t) 1 γ2I (t) (11) dt −  .
.  d  In(t)= α S(t) 1 γ In(t).  n n dt −  
6 1 Figure 2: (continued from Fig. 1) Illustration of Theorem 1.2 in the case i), i.e., when I0 α− (α∗) > 0. Function t S(t) converges towards 1/α∗ = 2/3 and function x I(t, x) at time t = 50 is asymptotically → 1 →
concentrated on α− (α∗) = ( 0.3 [0.2, 0.8]) [ 0.6, 0.6]. {− } ∪ × −

1 Figure 3: Illustration of Theorem 1.2 in the case i), i.e., when I0 α− (α∗) > 0. Function x I(t, x) at time t = 10, 20, 30 and 40. The function I remains bounded in this case. →

7 1 Figure 4: Illustration of Theorem 1.2 in the case ii), i.e., when I0 α− (α∗) = 0. Parameters of this simu-

2

½ Ì lation are: Λ = 2, θ = 1, α(x)=0.5+ Ì[ 0.1,0.8](x1) [ 0.6,0.6](x2) where x = (x1, x2) R and [ 0.1,0.8] − −
∈ − is the triangular function of height one and support [ 0.1, 0.8], γ = 1 . Initial condition is given by

− 2α ½ I0(dx) = I0(x1, x2) dx where I0(x) = ½[ 0.5,0.5](x1) cos(πx1) [ 0.5,0.5](x2) cos(πx2). In particular, α∗ = 3/2 1 − − and α− (α∗)= 0.35 [ 0.6, 0.6]. { }× −

1 Figure 5: (continued from Fig. 4) Illustration of Theorem 1.2 in the case ii), i.e. when I0 α− (α∗) = 0. Function t S(t) converges towards 1/α∗ = 2/3. Function x I(t, x) at time t = 100 is asymptotically → 1 →
concentrated on α− (α∗)= 0.35 [ 0.6, 0.6]. { }× −

8 1 Figure 6: Illustration of Theorem 1.2 in the case ii), i.e., when I0 α− (α∗) = 0. Function x I(t, x) at time t = 20, 40, 60 and 80. The function I asymptotically converges towards a singular measure. →

For the above system, we can completely characterize the asymptotic behavior of the population. To that aim we define the basic reproductive number (in the ecological or epidemiological sense) for species i as follows:

Λ i := α , i =1, . . . , n. R0 θ i

i Then we can show that the only species that do not get extinct are the ones for which 0, i = 1,...,n, is maximal and strictly greater than one. R In the case when several species have the same basic maximal reproductive number, then these species all i survive and the asymptotic distribution can be computed explicitly as a function of the initial data I0 and of the values γi.

Theorem 1.8 (Asymptotic behavior of finite systems). Let n 1 and α1,...,αn and γ1 > 0, ..., γn > 0 be given. Set ≥ α∗ := max α , , α , { 1 ··· n} and assume that Λ 1 n ∗ := α∗ = max , , > 1. R0 θ R0 ··· R0 1 n  Then, for any initial data S0 0, I0 > 0, , I0 > 0, the corresponding solution to (11) converges in the large times to S , I1 , , In ≥given by ··· ∞ ∞ ··· ∞

i 1 i 0 if 0 < 0∗, S = and I = τγ i Ri R for all i =1, . . . , n, ∞ α∗ ∞ e i I if = ∗, ( 0 R0 R0 wherein the constant τ R is defined as the unique solution of the equation: ∈

i τγi θ γiI0 e = ( 0 1). α∗ R − i : i = ∗ { RX0 R0 }

9 Note that in the case when the interaction of the species with the resource is described by the Michaelis- Menten kinetics instead of the mass action law, or when the growth of the resource obeys a logistic law (Hsu [21]), a similar result was already present in Hsu, Hubbell and Waltman [22] and Hsu [21], including the case i when several species have the exact same reproduction number (or fitness) 0. In the case of a countable system we can still provide a complete descriptionR when both α and γ converge to a limit near + . We now investigate the following system ∞ d S(t)=Λ θS(t) S(t) α γ I (t), dt − − i i i i N  X∈ (12) d  Ii(t)= α S(t) 1 γ Ii(t), for i N, dt i − i ∈ 
supplemented with some initial data

S(0) = S > 0 and Ii(0) = Ii > 0, i N with Ii < . (13) 0 0 ∀ ∈ 0 ∞ i N X∈ Since components of (12) starting from a zero initial data will stay equal to zero in positive time, they can be removed from the system without impacting the dynamics and we may without loss of generality assume that i I0 > 0 for all i N (as we did in (13)). ∈ i In the sequel we denote by S(t), I (t) be the corresponding solution to (12) with initial data S(0) = S0 and Ii(0) = Ii for all i N. 0 ∈
Theorem 1.9 (Asymptotic behavior of discrete systems). Let (αi)i N and (γi)i N be bounded sequences with γ > 0 for all i N. Set ∈ ∈ i ∈ α∗ := sup α , i N , { i ∈ } and assume that Λ ∗ := α∗ > 1. R0 θ We distinguish two cases.

i i) If the set i N : αi = α∗ is not empty, then S(t), I (t) converges to the following asymptotic stationary{ state∈ }
i 1 i 0 if 0 < 0∗, S = , and I = τγ i Ri R for all i N . ∞ α∗ ∞ e i I if = ∗, ∈ ∪ {∞} ( 0 R0 R0 where the constant τ R is the unique solution of the equation: ∈

i τγi θ γiI0 e = ( 0∗ 1). α∗ R − i N : i = ∗ { ∈ XR0 R0 }

1 i ii) If the set i N : αi = α∗ is empty, then one has S(t) α∗ and I (t) 0 for all i N as t , while { ∈ } → → ∈ → ∞ lim inf Ii(t) > 0. t + → ∞ i N X∈ If moreover one has αn α∗ and γn γ > 0 as n with n N then the total mass converges to a positive limit → → ∞ → ∞ ∈ i θ lim I (t)= ( 0∗ 1). (14) t + α∗γ R − → ∞ i N ∞ X∈

Note that for more complex countable systems, such as if the ω-limit set of αn or γn contains two or more distinct values, then it is no longer possible in general to state a result independent of the initial data. We will discuss a similar phenomenon for measures with a density with respect to the Lebesgue measure in Section 1.3

1.3 The case when α(x) has a finite number of regular maxima We now go back to our analysis of (1) set on RN . If the function α(x) has a unique global maximum which is accessible to 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 behvaior of the orbit is completely known, independently on the positivity of the initial mass of the fitness maximizing set α(x)= α∗ . { }

10 Theorem 1.10 (The case of a unique global maximum). Let Assumption 1.1 be satisfied. Let S0 0 and N ≥ I (dx) (R ) be a given initial data. Suppose that the function α = α(x) has a unique maximum α∗ on 0 ∈ M+ the support of I attained at x∗ supp I , and that 0 ∈ 0 Λ (I ) := α∗ > 1. R0 0 θ Then it holds that 1 S(t) , d0 (I(t, dx), I∞δx∗ (dx)) 0, t + t + −−−−→→ ∞ α∗ −−−−→→ ∞ where δx∗ (dx) denotes the Dirac measure at x∗ and θ I∞ := ( 0(I0) 1). α∗γ(x∗) R − Next we describe the large time behavior of the solutions when the function α has a finite number of maxima N on the support of I0(dx). We consider an initial data (S0, I0) [0, ) +(R ) with I0 absolutely continuous with respect to the Lebesgue measure dx in RN (in other words∈ and∞ with×M a small abuse of notation, I L1(RN )) 0 ∈ in a neighborhood of the maxima of the fitness function. Recalling the definition of α∗ in (4), throughout this section, we shall make use of the following set of assumptions. 1 N By a small abuse of notation, we will identify in this section the function I0 L (R ) and the associated measure I (x)dx (RN ) when the context is clear. ∈ 0 ∈M+ Assumption 1.11. We assume that:

1 (i) the set α− (α∗) (given in (3)) is a finite set, namely there exist x1, ..., xp in the interior of supp (I0) such that x = x for all i = j and i 6 j 6

1 Λα∗ α− (α∗)= x , , x and := > 1. { 1 ··· p} R0 θ

(ii) There exist ε0 > 0, M > 1 and κ1 0,..,κp 0 such that for all i = 1, .., p and for almost all x B(x ,ε ) supp (I ) one has ≥ ≥ ∈ i 0 ⊂ 0 1 κ κ M − x x i I (x) M x x i . | − i| ≤ 0 ≤ | − i| Here and along this note we use to denote the Euclidean norm of RN . | · | (iii) The functions α and γ are of class C2 and there exists ℓ> 0 such that for each i =1, .., p one has

D2α(x )ξ2 l ξ 2, ξ RN . i ≤− | | ∀ ∈

Remark 1.12. Let us observe that since xi belongs to the interior of supp (I0) then Dα(xi) = 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 ≍ → ∞

1 C− g(t) f(t) C g(t) , t T. | | ≤ | |≤ | | ∀ ≥

According to Theorem 1.2 (ii), one has α∗S(t) 1 as t , and as a special case we conclude that → → ∞ 1 t 1 S¯(t)= S(l)dl as t . t → α → ∞ Z0 ∗

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 S0 and I0 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. Theorem 1.13. Let Assumption 1.11 be satisfied. Then the function η = η(t) satisfies the following asymptotic expansion ln t 1 η(t)= ̺ + O , as t . (15) t t → ∞   wherein we have set N + κ ̺ := min i . (16) i=1,...,p 2γ(xi)

11 Moreover there exists ε (0,ε ) such that for all 0 <ε<ε and all i =1, .., p one has 1 ∈ 0 1

γ(x )̺ N+κi I(t, dx) t i − 2 as t . (17) x x ε ≍ → ∞ Z| − i|≤ As a special case, for all ε> 0 small enough and all i =1, .., p one has

1 if i J I(t, dx) ≍ ∈ as t , x xi ε ( 0 if i / J → ∞ Z| − |≤ → ∈ where J is the set defined as N + κ J := i =1, .., p : i = ̺ . (18) 2γ(x )  i  The above theorem states that the function I(t, dx) concentrates on the set of points x , i J (see { i ∈ } Corollary 1.15 below). Here Assumption 1.5 on the uniform positiveness of the measure I0(dx) around the 1 points xi is not satisfied in general, and the measure I concentrates on α− (α∗) as predicted by Theorem 1.2, 1 1 but not necessarily on α− (α∗) γ− (γ∗) as would have been given by Proposition 1.7. In Figure 7 we provide a precise∩ example of this non-standard behavior. The function α(x) is chosen to have two maxima x = 0.5 and x =0.5; the precise definition of α(x) is

1 − 2 È α(x)= È[x δ,x +δ](x)+ [x δ,x +δ](x), (19) 1− 1 2− 2 where (a + b 2x)2

È (x) := max 1 − , 0 [a,b] − (a b)2  −  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 x1 and x2. The function I0(x) is chosen as

8 2 I0(x) = min 1, 1024 (x x1) min 1, 4 (x x2) ½[ 1,1](x), (20) − − −     so that κ1 = 8 and κ2 = 2. Finally we take 1

γ(x)= (21) È 1+ È[x δ,x +δ](x)+3 [x δ,x +δ](x) 1− 1 2− 2 1 1 so that γ(x1)= 2 and γ(x2)= 4 . Summarizing, we have N + κ N + κ 1 =9 > 6= 2 , 2γ(x1) 2γ(x2)

1 1 so that Theorem 1.13 predicts that the mass I(t, dx) will vanish near x = α− (α∗) γ− (γ∗) and concentrate 1 ∩ on x2. 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 (27).

N Corollary 1.14. For each i = 1, ..., p and f Cc(R ), the set of the continuous and compactly supported functions, one has as t : ∈ → ∞

N N+κi γ(xi)̺ κi γ(xi) 2 2 t 2 f (x xi)√t I(t, dx) t − 2 f(x) x exp D α(xi)x dx. (22) RN − ≍ RN | | 2α∗ Z   Z   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 xi given in (17), it readily follows that any limit measures of I(t, dx) belongs to a linear combination of δxi with i J and strictly positive coefficients of each of these Dirac masses. This reads as follows. ∈

Corollary 1.15. Under the same assumptions as in Theorem 1.13, the ω limit set (I0) as defined in Lemma 2.4 satisfies that there exist 0

J (I0) ciδx : (ci)i J [A, B] . O ⊂ i ∈ ∈ (i J ) X∈

12 Figure 7: Illustration of Theorem 1.13 and Corollary 1.14. Parameters of this simulation are: Λ = 2, θ = 1, 1 α(x) is given by (19), I (x) by (20) and γ(x) by (21). In particular, α∗ = 1, α− (α∗)= x , x with x = 0.5, 0 { 1 2} 1 − x2 = 0.5, κ1 = 8, κ2 = 2, γ(x1)=1/2, γ(x2)=1/4, ρ = 6 and J = 2 . The initial condition I0 vanishes more rapidly around x than x so that the solution I(t, x) vanishes around{ } x as t goes to , even though 1 2 1 ∞ γ(x1) >γ(x2), while around x2 it takes the shape given by expression (1.14).

13 1.4 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 1.13), 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 8 which shows such a transient behavior. In this simulation, we took a fitness function presenting one global maximum at x2 = +0.5 and a local maximum at x = 0.5, whose value is close to the global maximum. The precise definition of α(x) is

1 − È α(x)=0.95 È[x δ,x +δ](x)+ [x δ,x +δ](x) with δ =0.2. (23) × 1− 1 2− 2

The function I0(x) is chosen as

2 2 I0(x) = min 1, 4 (x x1) min 1, 4 (x x2) ½[ 1,1](x), (24) − − −     so that κ1 = 2 and κ2 = 2. Finally, 1

γ(x)= (25) È 1+ È[x δ,x +δ](x)+3 [x δ,x +δ](x) 1− 1 2− 2 1 1 so that γ(x1)= 2 and γ(x2)= 4 . Summarizing, we have N + κ N + κ 1 =3 < 6= 2 . 2γ(x1) 2γ(x2)

If α(x) had two global maximum at the same level, Theorem 1.13 would predict that the mass I(t, dx) vanishes near x2 and concentrates on x1. Since the value of α(x2) is slightly higher than the value of α(x1), however, it is clear that the eventual distribution will be concentrated on x2. We observe numerically (see Figure 8) that the distribution first concentrates on x1 on a transient time scale, before the dynamics on x2 takes precedence. We refer to [9] for a related model with mutations where these transient behaviors are analytically characterized.

2 Measure-valued solutions and proof of Theorem 1.2

In this section we derive general properties of the solution of (1) equipped with the given and fixed initial data N S(0) = S [0, ) and I (dx) (R ). Recall that α∗ and (I ) are both defined in (4). Next for ε> 0 0 ∈ ∞ 0 ∈M+ R0 0 let us denote by Lε(I0) the following superlevel set:

1 L (I ) := x supp I : α(x) α∗ ε = α− (y). (26) ε 0 { ∈ 0 ≥ − } α∗ ε y α∗ − [≤ ≤ Then the following lemma holds true.

+ N Lemma 2.1. Let Assumption 1.1 hold and let (S0, I0(dx)) R +(R ) be a given initial condition. Denote S(t), I(t, dx) the corresponding solution of (1). Then S∈(t), I(×Mt, dx) is defined for all t 0 and ≥
min(θ,γ )
Λ 0 < ∗ lim inf S(t) lim sup S(t) < + , θΛ min(θ,γ )+ α∗γ∗ ≤ t + ≤ t + ≤ θ ∞ ∗ → ∞ → ∞ Λ lim sup I(t, dx) < + , t + RN ≤ min(θ,γ ) ∞ → ∞ Z ∗ where γ := infx supp I0 γ(x), γ∗ := supx supp I γ(x) and α∗ := supx supp I α(x). ∗ ∈ ∈ 0 ∈ 0 Proof. We remark that

d S(t)+ I(t, dx) Λ θS(t) γ I(t, dx), dt RN ≤ − − ∗ RN  Z  Z therefore

Λ Λ min(θ,γ0)t S(t)+ I(t, dx) + S0 + I0(dx) e− . RN ≤ min(θ,γ ) RN − min(θ,γ ) Z ∗  Z ∗ 

14 Figure 8: Illustration of a transient behavior for (1). Parameters of this simulation are: Λ = 2, θ = 1, α(x) is 1 given by (23), I (x) by (24) and γ(x) by (25). In particular, α∗ = 1, α− (α∗)= x with x = 0.5, x =0.5. 0 { 2} 1 − 2 Other parameters are κ1 = 2, κ2 = 2, γ(x1)=1/2, γ(x2)=1/4. The value of the local maximum at x1, α(x1)=0.95, being very close to α∗, observe that the distribution I(t, x) first concentrates around x1 before the global maximum x2 becomes dominant (bottom right plot).

15 In particular I(t, dx) is uniformly bounded in (RN ) and therefore we have the global existence of the solution as well as M Λ Λ lim sup I(t, dx) and lim sup S(t) . t + RN ≤ min(θ,γ ) t + ≤ min(θ,γ ) → ∞ Z ∗ → ∞ ∗ Next we return to the S-component of equation (1) and let ε > 0 be given. We have, for t0 sufficiently large and t t , ≥ 0 Λ St =Λ θ + α(x)γ(x)I(t, dx) S(t) Λ θ + α∗γ∗ + ε S(t), − RN ≥ − min(θ,γ )  Z   ∗  therefore

Λα∗γ∗ Λα∗γ∗ θ+ min(θ,γ ) +ε (t t0) Λ min(θ,γ ) θ+ min(θ,γ ) +ε (t t0) S(t) e− ∗  − S(t0)+ ∗ 1 e− ∗  − , ≥ (θ + ε) min(θ,γ )+Λα∗γ∗ − ∗   so that finally by letting t + we get → ∞ min(θ,γ )Λ lim inf S(t) ∗ . t + ≥ (θ + ε) min(θ,γ )+Λα∗γ∗ → ∞ ∗ Since ε> 0 is arbitrary we have shown

min(θ,γ )Λ lim inf S(t) ∗ . t + ≥ θ min(θ,γ )+Λα∗γ∗ → ∞ ∗ The Lemma is proved.

+ N Lemma 2.2. Let Assumption 1.1 hold and let (S0, I0(dx)) R +(R ) be a given nonnegative initial condition. Let S(t), I(t, dx) be the corresponding solution of∈(1).×M Then

1 T 1 lim sup S(t)dt , T + T 0 ≤ α∗ → ∞ Z where α∗ is given in (4). Proof. Let us remark that the second component of (1) can be written as

γ(x)(α(x) t S(s)ds t) I(t, dx)= I (dx)e 0 − , 0 R t t = I0(dx)exp γ(x) S(s)ds α(x) t . (27) 0 − S(s)ds Z " 0 #! Assume by contradiction that the conclusion of the Lemma does not hold,R i.e. there exists ε> 0 and a sequence Tn + such that → ∞ 1 Tn 1 S(t)dt + ε. T ≥ α n Z0 ∗ Then Tn 1 α∗ ε′, Tn ≤ 1 ≤ − 0 S(t)dt + ε α∗ 2 R where ε′ = (α∗) ε + o(ε). Since the mapping x α(x) is continuous, the set L (I )= x supp I : α(x) 7→ ν 0 { ∈ 0 ≥ α∗ ν has positive mass with respect to the measure I0(dx) for all ν > 0, i.e. I0(dx) > 0. This is true, − } Lν (I0) ε′ in particular, for ν = 2 , therefore R

Tn Tn I(Tn, dx)= exp γ(x) S(s)ds α(x) I0(dx) Tn L ′ (I ) L ′ (I ) 0 − Z ε /2 0 Z ε /2 0 Z " 0 S(s)ds#! Tn ε′ R exp γ S(s)ds I0(dx) ≥ ∗ · 2 ZLε′/2(I0) Z0 ! Tn ε′γ = I (dx)exp ∗ S(s)ds , 0 2 ZLε′/2(I0) Z0 !

16 Tn where γ = infx supp I0 γ(x). Since I0(dx) > 0 and S(t)dt + when n + , we have ∗ ∈ Lε′/2(I0) 0 → ∞ → ∞ therefore R R lim sup I(t, dx) lim sup I(Tn, dx)=+ , t + RN ≥ n + L ′ (I ) ∞ → ∞ Z → ∞ Z ε /2 0 which is a contradiction since I(t, dx) is bounded in (RN ) by Lemma 2.1. This completes the proof of the Lemma. M The following kind of weak persistence property holds. Lemma 2.3. Let Assumption 1.1 hold and let (S , I (dx)) R+ (RN ) be a given nonnegative initial 0 0 ∈ ×M+ condition. Let S(t), I(t, dx) be the corresponding solution of (1). Recalling the definition of α∗ in (4), assume that
Λ (I )= α∗ > 1. R0 0 θ Then θ lim sup I(t, dx) 0(I0) 1 > 0, t + RN ≥ α∗γ∗ R − → ∞ Z
where γ∗ := supx supp I γ(x). ∈ 0

Proof. Assume by contradiction that for t0 sufficiently large we have θ I(t, dx) η′ < η =: 0(I0) 1 for all t t0, RN ≤ α γ R − ≥ Z ∗ ∗
with η′ > 0. As a consequence of Lemma 2.2 we have

1 T 1 lim inf S(t) lim sup S(t)dt . (28) t + ≤ T + T 0 ≤ α∗ → ∞ → ∞ Z

Let S := lim inft + S(t). Let (tn)n 0 be a sequence that tends to as n and such that → ∞ ≥ ∞ → ∞ limn + S′(tn) = 0 and limn + S(tn) = S. As RN I(tn, dx) η′ for n large enough we deduce from the equality→ ∞ → ∞ ≤ R S′(tn)=Λ θS(tn) S(tn) α(x)γ(x)I(tn, dx), − − RN Z that 0 Λ θS Sα∗γ∗η′ ≥ − − so that Λ Λ S > ≥ θ + α∗γ∗η′ θ + α∗γ∗η and by definition of η Λ 1 S > = , θ α R0 ∗ which contradicts (28).

N Let us remind that +(R ), equipped with the Kantorovitch-Rubinstein metric d0 defined in (2), is a complete metric space. M

Lemma 2.4 (Compactness of the orbit and uniform persistence). Let Assumption 1.1 hold and let (S0, I0(dx)) + N ∈ R +(R ) be a given nonnegative initial condition. Let S(t), I(t, dx) be the corresponding solution of (1).×M Assume that there exists ε> 0 such that the superlevel set L (I ) (defined in (26)) is bounded. Then, the ε 0
closure of the orbit of I0,

N (I0) := µ +(R ): there exists a sequence tn 0 such that d0(I(tn, dx),µ) 0 , O ∈M ≥ −−−−−→n +  → ∞  is compact for the topology induced by d0 (i.e. the weak topology of measures). If moreover (I ) > 1, then it holds R0 0

lim inf I(t, dx) > 0. t + → ∞ Z

17 Proof. First of all let us remark that

( t S(s)dsα(x) t)γ(x) I(t, dx)= e 0 − I (dx), R 0 and therefore the orbit t I(t, dx) is continuous for the metric d0. By Lemma 2.2 we have7→ 1 T 1 lim sup S(s)ds , T + T 0 ≤ α∗ → ∞ Z where α∗ defined in (4). Let ε > 0 be sufficiently small, so that the set Lε(I0) is bounded and let R := supx L (I ) x . Then there exists T0 = T0(ε) such that ∈ ε 0 k k T ε α∗ for all T T . T ≥ − 2 ≥ 0 0 S(t)dt Therefore if T T , we have R ≥ 0 T T γ(x) 0 S(t)dt α(x) T  − 0 S(t)dt  I(T, dx)= e R R I0(dx) x R x R Zk k≥ Zk k≥ γ(x) T S(t)dt(α(x) α∗+ ε ) e 0 − 2 I (dx) R 0 ≤ x R Zk k≥ ε γ(x) T S(t)dt e− 2 0 I (dx) 0. R 0 ≤ x R −−−−−→T + Zk k≥ → ∞ In particular, the set I(t, dx): t 0 is tight and bounded in the absolute variation norm (see Lemma 2.1), therefore precompact{ for the weak≥ topology} by Prokhorov’s Theorem [5, Theorem 8.6.2, Vol. II p. 202]. d0 Next we show the weak uniform persistence property if 0(I0) > 1. Let tn + be such that I(tn, dx) n + R → ∞ −−−−−→→ ∞ I∞(dx). Then, for ε> 0 sufficiently small we will be fixed in the rest of the proof, we have Λ inf α(x) > 1. (29) x Lε(I0) θ ∈ By Lemma 2.2 we have tn ε α∗ for all T T0, tn ≥ − 2 ≥ 0 S(t)dt for some T0 = T0(ε). Therefore we getR

tn tn γ(x) 0 S(t)dt α(x) tn  − 0 S(t)dt  I(tn, dx)= e R R I0(dx) RN L (I ) RN L (I ) Z \ ε 0 Z \ ε 0 γ(x) tn S(t)dt(α(x) α∗+ ε ) e 0 − 2 I (dx) R 0 ≤ RN L (I ) Z \ ε 0 ε γ(x) tn S(t)dt e− 2 0 I (dx) 0. R 0 ≤ RN L (I ) −−−−−→tn + Z \ ε 0 → ∞

In particular we have N I∞(dx) = 0. Recall that, as a consequence of (29), we have 0(I∞) := R Lε(I0) Λ Λ \ R supx supp(I∞) θ α(x) θ infx Lε(I0) α(x) > 1. By Lemma 2.3 we have the alternative: ∈ ≥ R ∈ θ θ Λ either I(dx) 0 or I∞(dx) ( 0(I∞) 1) inf α(x) 1 > 0. ≡ RN ≥ α γ R − ≥ α γ θ x Lε(I0) − Z ∗ ∗ ∗ ∗  ∈  This is precisely the weak uniform persistence in the metric space (I ), d , which is complete. As a conse- O 0 0 quence of [36, Proposition 3.2] in the complete (and compact) metric space M = (I ) 0 equipped with the
0 metric d , with M = (I ) 0 , ∂M = 0 and O ∪{ } 0 0 O 0 \{ } 0 { }

ρ( )= (dx)= , 1 (RN ),BC(RN ), I RN I hI iM Z the Poincar´emap is uniformly persistent, where the chevron , (RN ),BC(RN ) denotes the canonical bilinear N N N h· ·iM mapping on (R )= BC(R )∗ BC(R ). Hence this yields M × lim inf I(t, dx) > 0, t + RN → ∞ Z and the Lemma is proved.

18 Lemma 2.5. Let Assumption 1.1 hold and let (S , I (dx)) R+ (RN ) be a given nonnegative initial 0 0 ∈ ×M+ condition. Let S(t), I(t, dx) be the corresponding solution of (1). Assume that 0(I0) > 1 and that Lε(I0) (defined in (26)) is bounded for ε> 0 sufficiently small. Then R
1 T 1 lim inf S(t)dt , T + T ≥ α → ∞ Z0 ∗ with α∗ given in (4). Proof. As in the proof of Lemma 2.5, we write

α(x) t S(s)ds t γ(x) I(t, dx)= I (dx)e 0 − , 0 R t
t = I0(dx)exp γ(x) S(s)ds α(x) t . 0 − S(s)ds Z " 0 #! Assume by contradiction that the conclusion of the Lemma does not hold,R i.e. there exists ε> 0 and a sequence Tn + such that → ∞ 1 Tn 1 S(t)dt ε. T ≤ α − n Z0 ∗ Then Tn 1 α∗ + ε′, Tn ≥ 1 ≥ 0 S(t)dt ε α∗ − 2 R where ε′ = (α∗) ε + o(ε), provided ε is sufficiently small. Therefore

Tn Tn I(Tn, dx)dx = exp γ(x) S(s)ds α(x) I0(dx) Tn RN RN 0 − Z Z Z " 0 S(s)ds #! Tn R exp γ0 S(s)ds ε′ I0(dx) ≤ RN − · Z Z0 ! Tn = exp ε′γ0 S(s)ds I0(dx), − RN Z0 ! Z We have therefore

lim inf I(t, dx) lim inf I(Tn, dx)=0, t + RN ≤ n + RN → ∞ Z → ∞ Z which is in contradiction with Lemma 2.4. This completes the proof of the Lemma.

1 T Remark 2.6. By combining Lemma 2.2 and Lemma 2.5 we obtain that T 0 S(t)dt admits a limit when T + and → ∞ 1 T 1 R lim S(t)dt = . T + T α → ∞ Z0 ∗ Lemma 2.7. Let Assumption 1.1 hold and let (S , I (dx)) R+ (RN ) be a given nonnegative initial 0 0 ∈ ×M+ condition. Let S(t), I(t, dx) be the corresponding solution of (1). Assume that Lε(I0) (defined in (26)) is bounded for ε> 0 sufficiently small. Then one has
1 d0 I(t, dx), +(α− (α∗)) 0. t + M −−−−→→ ∞ Proof. Let ε > 0 be as in the statement of Lemma 2.7. By Lemma
2.2, there exists T 0 such that for all t T we have ≥ ≥ t ε α∗ , t ≥ − 2 0 S(s)ds where α := sup α(x). Hence for t T we have ∗ x supp I0 R ∈ ≥ 1 t t I(t, dx)= exp γ(x) S(s)ds α(x) t I0(dx) RN L (I ) RN L (I ) · t 0 − S(s)ds Z \ ε 0 Z \ ε 0 Z 0 !! 1 t R t exp γ(x) S(s)ds α∗ ε t I0(dx) ≤ RN L (I ) · t 0 − − S(s)ds Z \ ε 0 Z 0 !! ε γ(x) t S(s)ds e− 2 0 I (dx) 0. R R 0 ≤ RN L (I ) −−−−→t + Z \ ε 0 → ∞

19 In particular, if I(t, dx) L (I ) denotes the restriction of I(t, dx) to Lε(I0), we have (I(t, dx) I0) L (I ) AV | ε 0 k − | ε 0 k −−−−→t + 0 and hence → ∞ d0 I0, +(Lε(I0)) dAV I0, +(Lε(I0)) 0. t + M ≤ M −−−−→→ ∞ Here ε> 0 can be chosen arbitrarily small. By
Lemma 2.1 we know moreover
that Λ lim sup I(t, dx) , t + RN ≤ min(θ,γ ) → ∞ Z ∗ so that for t sufficiently large, we have Λ I(t, dx) 2 . RN ≤ min(θ,γ ) Z ∗ Finally by using Proposition A.5, we have

1 1 d I(t, dx), α− (α∗) d I(t, dx), I(t, dx) + d I(t, dx) , α− (α∗) 0 M+ ≤ 0 |Lε (I0) 0 |Lε(I0) M+


Λ 1

d0 I(t, dx), I(t, dx) Lε (I0) +2 sup d x, α− (α∗) . ≤ | min(θ,γ ) x L (I ) ∗ ε 0
∈
Since 1 sup d x, α− (α∗) 0, x L (I ) −−−→ε 0 ∈ ε 0 →
1 the Kantorovitch-Rubinstein distance between I(t, dx) and α− (α∗) can indeed be made arbitrarily small as t + . This proves the Lemma. M → ∞
Lemma 2.8. Let Assumption 1.1 hold and let (S , I (dx)) R+ (RN ) be a given nonnegative initial 0 0 ∈ ×M+ condition. Let S(t), I(t, dx) be the corresponding solution of (1). Assume that α(x) α∗ is a constant + ≡ 1 function on supp I such that (I ) > 1. There exists a stationary solution (S∗,i∗) R L (I ) such that 0
R0 0 ∈ × 0 1 S(t) S∗ = , t + −−−−→→ ∞ α∗ AV I(t, dx) i∗(x)I0(dx). t + −−−−→→ ∞ N i∗ is a Borel-measurable function on R , which is unique up to a negligible set with respect to I0(dx). Moreover τγ(x) it satisfies i∗(x)= e , where τ is the unique solution to the equation

τγ(x) θ γ(x)e I0(dx)= ( 0 1) . (30) RN α R − Z ∗ Proof. First we check that the proposed stationary solution is indeed unique and a stationary solution. By 1 Lemma 2.2–2.5, S∗ = α∗ is the only possible choice for S∗. Next, we remark that the map

τγ(x) τ γ(x)e I0(dx) 7→ RN Z is strictly increasing and maps R onto (0, + ), therefore (30) has a unique solution τ and the corresponding τγ(x) ∞ function i∗(x)I0(dx) := e I0(dx) satisfies θ γ(x)i∗(x)I0(dx)= ( 0 1) . RN α R − Z ∗

Therefore it is not difficult to check that (S∗,i∗(x)I0(dx)) is a stationary solution to the system of differential equations

S′(t)=Λ θS(t) α∗γ(x)I(t, dx) − − RN  Z I′(t, dx)= γ(x) (α∗S(t) 1) I(t, dx),  − which is equivalent to (1) on supp I0. Next we show the convergence of an initial condition to (S∗, I∗) where I∗(dx) = i∗(x)I0(dx). To that aim we introduce the Lyapunov functional

S I(x) V (S, I) := S∗g + i∗(x)g I0(dx), S RN i (x)  ∗  Z  ∗ 

20 1 where g(s)= s ln(s). V (S, I) is well-defined when ln(I(x)) L (I0). Let us denote I(t, dx)= i(t, x)I0(dx) and − ∈ α(x)S(t) γ(x)t remark that V (S(t),i(t, x)) is always well-defined since i(t, x)= e − . We claim that V ′(S(t),i(t, )) S(t) i(t,x) · ≤ 0. Indeed, writing V1(S)= S∗g S∗ and V2(t)= RN i∗(x)g i∗(x) I0(dx), we have   R   S′(t) S(t) S∗ V1′(t)= S∗ g′ = Λ θS(t) S(t) α∗γ(x)i(t, x)I0(dx) 1 S S − − RN − S(t) ∗  ∗   Z    S∗ = Λ θS(t) S(t) α∗γ(x)i(t, x)I0(dx) Λ+ θS∗ + S∗ α∗γ(x)i∗(x)I0(dx) 1 − − RN − RN − S(t)  Z Z    2 (S(t) S∗) S∗ = θ − + S∗ α∗γ(x)i∗(x)I0(dx) S(t) α∗γ(x)i(t, x)I0(dx) 1 , − S(t) RN − RN − S(t)  Z Z    2 2 (S(t) S∗) (S∗) = θ − + S∗ α∗γ(x)i∗(x)I0(dx) α∗γ(x)i∗(x)I0(dx) − S(t) RN − S(t) RN Z Z

S(t) α∗γ(x)i(t, x)I0(dx)+ S∗ α∗γ(x)i(t, x)I0(dx), − RN RN Z Z and

it(t, x) i(t, x) i∗(x) V2′(t)= i∗(x) g′ I0(dx)= γ(x) (α∗S(t) 1) i(t, x) 1 I0(dx) RN i (x) i (x) RN − − i(t, x) Z ∗  ∗  Z  

= γ(x) (α∗S(t) 1) (i(t, x) i∗(x)) I0(dx) RN − − Z = γ(x)α∗S(t)i(t, x)I0(dx) γ(x)i(t, x)I0(dx) γ(x)α∗S(t)i∗(x)I0(dx) RN − RN − RN Z Z Z + γ(x)i∗(x)I0(dx). RN Z 1 Recalling S∗ = α∗ , we have therefore d d d V (S(t),i(t, )) = V (t)+ V (t) dt · dt 1 dt 2 2 2 (S(t) S∗) (S∗) = θ − +2 γ(x)i∗(x)dx α∗γ(x)i∗(x)I0(dx) − S(t) RN − S(t) RN Z Z

α∗γ(x)S(t)i∗(t)I0(dx). − RN Z Since 2 (S∗) α∗γ(x)i∗(x) S(t)+ I0(dx) α∗γ(x)i∗(x) 2S∗I0(dx), RN S(t) ≥ RN × Z   Z which stems from the inequality a + b 2√ab, we have proved that ≥ d V (S(t),i(t, )) 0. dt · ≤

It then follows from classical arguments that S(t) S∗ and i(t, ) i∗( ) as t + (where the last limit 1 → · → · → ∞ holds in L (I0)). The Lemma is proved.

Next we can determine the long-time behavior when the initial measure I0 puts a positive mass on the set 1 of maximal fitness. Recall that α− (α∗) (see (3) and (4)) is the set of points in the support of I0 that have maximal fitness, i.e.

1 α− (α∗)= L (I )= x supp I : α(x) α(y) for all y supp I . ε 0 { ∈ 0 ≥ ∈ 0} ε>0 \ Lemma 2.9. Assume that Lε(I0) is bounded for ε > 0 sufficiently small and that 0(I0) > 1. Suppose that 1 R I0(α− (α∗)) > 0, or in other words,

I0(dx) > 0. −1 ∗ Zα (α ) 1 Then the limit of I(t, dx) is completely determined by the part of I0 in α− (α∗), that is to say,

d0 I(t, dx), I∗ (dx) 0, t + ∞ −−−−→→ ∞
where I∗ (dx) is the stationary measure given by Lemma 2.8 associated with the initial condition I0∗(dx) := ∞ 1 I −1 ∗ (dx), the restriction of I (dx) to the set α− (α∗). 0|α (α ) 0

21 1 Proof. First, let us define α∗ := supx supp I α(x) (so that α(x) is a constant equal to α∗ on α− (α∗)) and ∈ 0 1 t η(t) := α∗S(t) 1, with S(t)= S(s)ds. − t Z0 Then I(t, dx) can be written as

1 t I(t, dx) = exp γ(x)t η(t) + (α(x) α∗) S(s)ds I (dx). − t 0   Z0  We remark that the function t tη(t) is bounded. Indeed, by Jensen’s inequality we have 7→ I (dx) I (dx) exp γ(x)tη(t) 0 eγ(x)tη(t) 0 , −1 ∗ I ≤ −1 ∗ I (dz) Zα (α ) α−1(α∗) 0 ! Zα (α ) α−1(α∗) 0 so that R R

−1 ∗ I0 I (dx) −1 ∗ I0 1 tη(t) α (α ) ln eγ(x)tη(t) 0 = α (α ) ln I(t, dx) . ≤ γ(x)I (dx) RN I γ(x)I (dx) I −1 ∗ α−1R(α∗) 0 Z RN 0  α−1R(α∗) 0 α−1(α∗) 0 Zα (α ) ! ApplyingR Lemma 2.1, I(t, dx) is bounded andR we have indeedR an upper bound forR tη(t). Next, writing

t I(t, dx) = exp γ(x)tη(t) + (α(x) α∗) S(s)ds I (dx) − 0  Z0  t t and recalling that S(s)ds + as t + , the function exp γ(x)tη(t) + (α(x) α∗) S(s)ds converges 0 → ∞ → ∞ − 0 N 1 almost everywhereR (with respect to I0) to 0 on R α− (α∗), so that by Lebesgue’s dominatedR convergence theorem, we have \

t lim I(t, dx)= lim exp γ(x)tη(t) + (α(x) α∗) S(s)ds I0(dx)=0. t + RN α−1(α∗) RN α−1(α∗) t + − 0 → ∞ Z \ Z \ → ∞  Z 

Next it follows from Lemma 2.5 that lim inft + I(t, dx) > 0, so that → ∞

lim inf I(t, dx) = lim inf I(t, dx) > 0. t + −1 ∗ t + RN → ∞ Zα (α ) → ∞ Z Assume by contradiction that there is a sequence (t ) such that t η(t ) , then n n n → −∞

γ(x)tnη(tn) γ∗tnη(tn) γ∗tnη(tn) I(t, dx)= e I0(dx) e I0(dx)= e I0(dx) 0, −1 ∗ −1 ∗ ≤ −1 ∗ −1 ∗ −−−−→t + Zα (α ) Zα (α ) Zα (α ) Zα (α ) → ∞ where γ := infx supp I γ(x) > 0. This is a contradiction. Therefore there is a constant η > 0 such that ∗ ∈ 0 tη(t) η > . ≥− −∞ In particular, the function t tη(t) is bounded by two constants, 7→ < η tη(t) η< + . −∞ − ≤ ≤ ∞

Suppose that there exists a sequence t + and η∗ [ η, η] such that n → ∞ ∈ −

lim tnη(tn)= η∗. n + → ∞

Upon replacing tn by a subsequence, the function S(tn) converges to a limit S0∗ and therefore the shifted orbits satisfy (S(t + tn), I(t + tn, dx)) (S∗(t), I∗(t, dx)) n + −−−−−→→ ∞ locally uniformly in time. The resulting orbit (S∗(t), I∗(t, dx)) is a solution to (1), defined for all times t R, and satisfying ∈

S∗(0) = S0, γ(x)η∗ I∗(0, dx)= e I0∗(dx),

22 1 1 where we recall that I0∗(dx) is the restriction of I0(dx) to α− (α∗). By Lemma 2.8, this implies that S0∗ = α∗ and τγ(x) I∗(0, dx)= e I0∗(dx), where τ is uniquely defined by (30) (and independent of the sequence tn). Therefore η∗ = τ. We conclude that lim tη(t)= τ, t + → ∞ where τ is the constant uniquely defined by (30) with the initial measure I0∗(dx). This ends the proof of the Lemma.

1 When the set of maximal fitness α− (α∗) is negligible for I0, 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)

Lemma 2.10. Assume that Lε(I0) is bounded for ε > 0 sufficiently small and that 0(I0) > 1. Suppose that 1 R I0(α− (α∗))=0 and set 1 t η(t) := α∗S(t) 1, with S(t)= S(s)ds, − t Z0 where α∗ := supx supp I α(x). Then it holds ∈ 0 tη(t) + . t −−−→→∞ ∞

Proof. Assume by contradiction that there exists a sequence tn + such that tnη(tn) has a uniform upper bound as t + , then observe that the quantity → ∞ n → ∞ γ(x)(α(x)S(t ) 1)t γ(x)(α(x) α∗)S(t )t +γ(x)η(t )t e n − n = e − n n n n is uniformly bounded in tn and vanishes as tn + almost everywhere with respect to I0(dx). By a direct application of Lebesgue’s dominated convergence→ Theorem,∞ we have therefore

γ(x)(α(x)S(tn) 1)tn I(tn, dx)= e − I0(dx) 0, RN RN −−−−−→tn + Z Z → ∞ which is in contradiction with Lemma 2.4. We conclude that tη(t) + as t + . → ∞ → ∞ 1 We can now state our convergence result for measures which vanish on α− (α∗), provided the behavior of I0 at the boundary is not too pathological. Basically, it says that the selection filters the low values of γ(x) near 1 boundary points x α− (α∗). ∈ We are now in the position to prove Theorem 1.2.

1 Proof of Theorem 1.2. To show the convergence of S(t) to α∗ (which is present in both i) and ii)), we first remark that 1 t 1 S(t)= S(s)ds , (31) t −−−−→t + α Z0 → ∞ ∗ as a consequence of Lemma 2.2 and 2.5. Next, let t + be an arbitrary sequence, then by the compactness n → ∞ of the orbit proved in Lemma 2.4 we can extract from S(tn) a subsequence which converges to a number S∗. It 1 follows from (31) that S∗ = α∗ . The convergence of I(t, dx) in case i) was proved in Lemma 2.9. The uniform persistence of I(t, dx) in case ii) is a consequence of 2.4. The concentration on the maximal fitness was proved in Lemma 2.7. The Theorem is proved. We now turn to the proof of Proposition 1.7 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.

Lemma 2.11. Assume that Lε(I0) is bounded for ε > 0 sufficiently small and that 0(I0) > 1. Suppose that 1 R I0(α− (α∗)) = 0 and that Assumption 1.5 holds. Recalling the definition of α∗ in (4) and γ∗ in Assumption 1 1.5, set Γ0(I0) be the set of maximal points of γ on α− (α∗), defined by

1 1 1 1 Γ (I ) := x α− (α∗): γ(x) γ(y) for all y α− (α∗) = γ− ( γ∗ ) α− (α∗). 0 0 ∈ ≥ ∈ { } ∩ Then one has  d0 (I(t, dx), +(Γ0(I0))) 0. t + M −−−−→→ ∞

23 Proof. We decompose the proof in several steps.

Step 1: We show that I(t, dx) and ½α(x)S(t) 1I(t, dx) are asymptotically close in AV . That is to say, ≥ k·k

I(t, dx) ½ I(t, dx) AV 0. α( )S(t) 1 t + k − · ≥ k −−−−→→ ∞ Indeed we have

γ(x)(α(x)S(t) 1)t

½ ½ I(t, dx) ½α(x)S(t) 1I(t, dx)= α(x)S(t)<1I(t, dx)= α(x)S(t)<1e − I0(dx). − ≥

γ(x)(α(x)S(t) 1)t ½ First note that the function ½α(x)S(t)<1I(t, dx) = α(x)S(t)<1e − is uniformly bounded. On the 1 1 other hand, since I (α− (α∗)) = 0 recall that S(t) ∗ for t , so that ½ 0 as t almost 0 → α → ∞ α(x)S(t)<1 → → ∞ everywhere with respect to I0. It follows from Lebesgue’s dominated convergence Theorem that

γ(x)(α(x)S(t) 1)t ½α(x)S(t)<1e − I0(dx) 0. RN −−−−→t + Z → ∞

γ¯(yS(t) 1)t Step 2: We show that the measure ½S(t)y 1e − A(dy) is bounded when t for all γ<γ¯ ∗. 1 ≥ → ∞ Note that I (α− (α∗)) = 0 implies that A( α∗ ) = 0 and remark that one has 0 { } α∗ γ(x)(yS(t) 1)t ½α(x)S(t) 1I(t, dx)= e − I0(y, dx)A(dy), RN ≥ min(α∗,1/S(t)) x α−1(y) Z Z Z{ ∈ } so, according to Step 1, for t sufficiently large one has

α∗ γ(x)(yS(t) 1)t I(t, dx)= e − I0(y, dx)A(dy)+ o(1) γ−1([¯γ,γ∗]) min(α∗,1/S(t)) x α−1(y) γ−1([¯γ,γ∗]) Z Z Z ∈ ∩ α∗ γ¯(yS(t) 1)t e − I0(y, dx)A(dy)+ o(1) ≥ min(α∗,1/S(t)) α(x)=y γ−1([¯γ,γ∗]) Z Z{ }∩ α∗ γ¯(yS(t) 1)t = I0(y, dx)e − A(dy)+ o(1) min(α∗,1/S(t)) α(x)=y γ−1([¯γ,γ∗]) Z Z{ }∩ α∗ γ¯(yS(t) 1)t m e − A(dy)+ o(1), ≥ ∗ Zmin(α ,1/S(t)) wherein m > 0 is the constant associated withγ ¯ in Assumption 1.5. Recalling the upper bound for I(t, dx) from Lemma 2.1, we have

α∗ γ¯(yS(t) 1)t 1 Λ lim sup e − A(dy) lim sup I(t, dx) < + . t + min(α∗,1/S(t)) ≤ t + m RN ≤ m min(θ,γ0) ∞ → ∞ Z → ∞ Z This implies that γ¯(yS(t) 1)t lim sup e − A(dy) < . t + α(supp(I )) ∞ → ∞ Z 0

Note that, if the constant m is independent ofγ ¯, then the above estimate does not depend onγ ¯ either. ½ Step 3: We show that ½γ(x)<γ¯ S(t)α(x) 1I(t, dx) vanishes whenever γ<γ¯ ∗. γ∗ γ¯ ≥ Fixγ<γ ¯ ∗ and let ε := R2− . Then we have

α∗ γ(x)(yS(t) 1)t I(t, dx)= e − I0(y, dx)A(dy) γ−1(( ,γ¯]) α−1([1/S(t), )) min(α∗,1/S(t)) x γ−1(( ,γ¯]) α−1(y) Z −∞ ∩ ∞ Z Z ∈ −∞ ∩ α∗ (γ∗ 2ε)(yS(t) 1)t I0(y, dx)e − − A(dy) ≤ min(α∗,1/S(t)) x γ−1(( ,γ¯]) α−1(y) Z Z ∈ −∞ ∩ α∗ ε(yS(t) 1)t γ¯(yS(t) 1)t I0(y, dx)e− − e − A(dy). ≤ min(α∗,1/S(t)) x α−1(y) Z Z ∈ }

24 γ¯ Reducing ε if necessary we may assume that ε > 1. Therefore it follows from H¨older’s inequality that

ε α∗ γ¯ γ¯ ε(yS(t) 1)t ε γ¯(yS(t) 1)t I(t, dx) e− − e − A(dy) γ−1(( ,γ¯]) α−1([1/S(t), )) ≤ min(α∗,1/S(t)) Z −∞ ∩ ∞ Z !   ε γ¯ 1 γ¯ α∗ γ¯−ε − γ¯(yS(t) 1)t I0(y, dx) e − A(dy) ×  ∗  Zmin(α ,1/S(t)) Zα(x)=y ! ε ε  ∗ ∗ 1 α γ¯ α − γ¯ γ¯(yS(t) 1)t A(dy) e − A(dy) ≤ ∗ ∗ Zmin(α ,1/S(t)) ! Zmin(α ,1/S(t)) ! 1 ε α∗ − γ¯ ε γ¯ γ¯(yS(t) 1)t = I0(Lα∗ min(α∗,1/S(t))(I0)) e − A(dy) . ∗ − Zmin(α ,1/S(t)) ! (32)

∗ α γ¯(yS(t) 1)t Since S(t) 1/α∗ as t , I0(Lǫ(I0)) 0 and by the boundedness of ∗ e − A(dy) ǫ 0 min(α ,1/S(t)) → → ∞ −−−→→ shown in Step 2, we have indeed R

I(t, dx) 0, γ−1(( ,γ¯]) α−1([1/S(t), )) −−−−→t + Z −∞ ∩ ∞ → ∞ and this completes proof of Lemma 2.11.

1 1 Proof of Proposition 1.7. The concentration of the distribution to +(α− (α∗) γ− (γ∗)) was shown in Lemma 2.11. M ∩ Next we prove the asymptotic mass. Pick a sentence t + . By the compactness of the orbit (proved in n → ∞ Lemma 2.4) we can extract from tn a subsequence tn′ such that there exists a Radon measure I∞(dx) with

d0(I(t, dx), I∞(dx)) 0, t + −−−−→→ ∞ 1 and since S(t) ∗ and upon further extraction, S′(t′ ) 0. Therefore, → α n →

Λ St(tn′ ) α(x)γ(x)I(tn′ , dx)= − θ α∗Λ θ = θ ( 0(I0) 1) . RN S(t ) − −−−−−→n + − R − Z n′ → ∞ 1 1 By the concentration result in Lemma 2.11, I∞ is concentrated on α− (α∗) γ− (γ∗). Therefore ∩

α∗γ∗ I∞(dx)= α(x)γ(x)I∞(dx) = lim I(t′ , dx)= θ ( 0(I0) 1) , n + n R − Z Z → ∞ so that θ lim I(t′ , dx)= I∞(dx)= ( 0(I0) 1) . n + n α γ R − → ∞ Z Z ∗ ∗ Since the limit is independent of the sequence tn, we have indeed shown that θ lim I(t, dx)= ( 0(I0) 1) . t + RN α γ R − → ∞ Z ∗ ∗

To prove the last statement, set

α∗ γ∗ z(yS(t) 1)t α f(t) := e − I0 (y, dz)A(dy), ∗ Zmin(α ,1/S(t)) Zγ¯ whereγ<γ ¯ ∗. It follows from (32) that

I(t, dx) 0, γ−1(( ,γ¯]) α−1([1/S(t), )) −−−−→t + Z −∞ ∩ ∞ → ∞ therefore

α∗ γ∗ z(yS(t) 1)t α,γ α f(t)= e − I0 (y, z, dx)I0 (z, dy)A(dy)= I(t, dx) min(α∗,1/S(t)) γ¯ γ(x)=z γ−1([¯γ,γ∗]) α−1([1/S(t), )) Z Z Z{ } Z ∩ ∞

25 satisfies 0 < lim inf I(t, dx) = lim inf I(t, dx) lim inf f(t) t + t + γ−1(( ,γ¯]) α−1([1/S(t), )) ≤ t + → ∞ Z → ∞ Z −∞ ∩ ∞ → ∞ Remark that

α∗ γ∗

z(yS(t) 1)t α,γ α

½ ½ ½ ½U (x) γ γ¯ S(t)y 1I(t, dx)= U (x)e − I0 (y, z, dx)I0 (y, dz)A(dy) ≥ ≥ min(α∗,1/S(t)) γ¯ γ(x)=z Z Z Z Z{ } α∗ γ∗ α,γ z(yS(t) 1)t α = ½U (x)I0 (y, z, dx)e − I0 (y, dz)A(dy) min(α∗,1/S(t)) γ¯ γ(x)=z Z Z Z{ } α∗ γ∗ m z(yS(t) 1)t α e − I0 (y, dz)A(dy) ≥ ∗ 2 Zmin(α ,1/S(t)) Zγ¯ m f(t) , ≥ 2 provided t is sufficiently large andγ ¯ is sufficiently close to γ∗, where

A(dy) Iα(y,dz) 0 α,γ m := lim inf ess inf ess inf ½x U I0 (y, z, dx) > 0. ε 0 α∗ ε y α∗γ∗ ε z γ∗ ∈ → − ≤ ≤ − ≤ ≤ Z Therefore m lim inf I(t, dx) lim inf f(t) > 0. t + ≥ 2 t + → ∞ ZU → ∞ This completes proof of Proposition 1.7.

3 The case of discrete systems. Proof of Theorem 1.8 and 1.9

In this section we show how the theory for discrete systems can be included in the theory for measure-valued solutions to (1) in RN . Rather than doing a direct proof of the results, we show how the general results from Section 2 can be applied to prove Theorem 1.8 and 1.9. In particular, we rely heavily on Theorem 1.2, which has been proven in Section 2 (independently of Theorem 1.8 and 1.9).

Proof of Theorem 1.8. Let us choose n distinct real numbers x1,...,xn. Then there exist continuous functions α(x) and γ(x) such that α(xi)= αi and γ(xi)= γi for all i =1, . . . , n. There are many ways to construct α(x) and γ(x); for instance one can work with Lagrange polynomials and interpolate with a constant value outside of a ball and when the values of γ(x) become close to 0. In particular one can impose that α(x) and γ(x) are bounded and that γ(x) > 0 for all x RN , thus meeting Assumption 1.1. Now define the initial data ∈ n i I0(dx) := I0δxi . i=1 X Clearly, the solution S(t), I(t, dx) to (1) can be identified with the solution (S(t), Ii(t)) to (11) by the formula

n i I(t, dx)= I (t)δxi . i=1 X

Since the set x : α(x) = α∗ has non-zero measure for I0, we are in the situation i) of Theorem 1.2 and Theorem 1.8 can{ therefore be deduced} from Theorem 1.2. Proof of Theorem 1.9. As in the proof of Theorem 1.8, we identify the solutions to (12) with the solutions I(t, dx) to (1) through the formula

+ + ∞ ∞ i i I0(dx) := I0δxi , I(t, dx)= I (t)δxi , i=1 i=1 X X 2 only this time we choose xi = (αi,γi) R . Because of this particular choice, it is fairly easy to construct α(x) and γ(x) by the formula ∈

x1 x2 γ0 α(x , x ) := α∞f , γ(x , x ) := γ∞f − + γ , 1 2 α 1 2 γ 0 ∞  ∞   

26 where α∞ := sup α , γ∞ := sup γ , γ = inf γ and f(x) := min(max(x, 1), 1). Then the conclusions of | i| | i| 0 i − Theorem 1.9 in case i) are given by a direct application of Theorem 1.2. Suppose that the set i : αi = α∗ is empty, then we are in case ii) of Theorem 1.2, and we can readily conclude that { }

+ 1 ∞ S(t) and lim inf Ii(t) = lim inf I(t, dx) > 0. −−−−→t + α t + t + R2 → ∞ ∗ → ∞ i=1 → ∞ X Z If we assume moreover that α α∗ and γ γ∗, then the maximum of α(x) on supp I is attained at a single n → n → 0 point (α∗,γ∗) and we can apply Theorem 1.10 to find that

I(t, dx) ⇀ I∞δ(α∗,γ∗), t + −−−−→ ∞ θ where I∞ = ∗ ( ∗ 1). This finishes the proof of Theorem 1.9. γ R0 − 4 The case of a finite number of regular maxima

In this section we prove Theorem 1.13. To that aim, we shall make use of the following formula

t I(t, dx) = exp γ(x) α(x) S(s)ds t I (dx). (33) − 0   Z0  Recall also the definition of η(t):

1 t η(t)= α∗ S(s)ds 1= α∗S(t) 1. t − − Z0

Proof of Theorem 1.13. 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 1 large time asymptotic of the mass of I around each point of α− (α∗). Upper bound: Let i = 1, .., p be given. Recall that α(xi) = 0. Now due to (iii) in Assumption 1.11 there exist m > 0 and 2 ∇ 1 T >ε− large enough such that for all t T and for all y B 0,t− 2 we have 0 ≥ ∈  2  α(x + y) α∗ α∗m y . i − ≤− k k As a consequence, setting α(x) Γ(x)= γ(x) , α∗ we infer from (33) and the lower estimate of I0 around xi given in Assumption 1.11 (ii), that for all t>T

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

I(t, dx) C, t 0. RN ≤ ∀ ≥ Z Coupling the two above estimates yields for all t>T

κi 2 y exp tη(t)Γ(xi + y) tγ(xi + y)m y dy MC. − 1 y t 2 | | − | | ≤ Z| |≤   Hence setting z = y√t into the above integral rewrites as

κi 1 1 dz 2 κi 2 2 2 t− z exp tη(t)Γ(xi + t− z) γ(xi + t− z)m z N/2 MC, t > T. z 1 | | − | | t ≤ ∀ Z| |≤ h i Now, since γ and α are both smooth functions, we have uniformly for z 1 and t 1: | |≤ ≫ 1 1 Γ(xi + t− 2 z)= γ(xi)+ O t− 2 ,

1  1  γ(xi + t− 2 z)= γ(xi)+ O t− 2 .  

27 This yields for all t 1 ≫

κi 1 dz 2 κi 2 2 t− z exp tη(t) γ(xi)+ O t− γ(xi)m z N/2 CM, z 1 | | − | | t ≤ Z| |≤ h    i − 1 κi N 2 2 tη(t) γ(xi)+O t κi γ(xi)m z t− 2 − 2 e    z e− | | dz CM, z 1 | | ≤ Z| |≤ that also ensures the existence of some constant c R such that 1 ∈ 1 N + κi tη(t) γ(x )+ O t− 2 ln t c , t 1, i − 2 ≤ 1 ∀ ≫    or equivalently N + κ ln t 1 η(t) i + O as t . ≤ 2γ(x ) t t → ∞ i   Since the above upper-bound holds for all i =1, .., p, we obtain the following upper-bound

ln t 1 η(t) ̺ + O as t , (34) ≤ t t → ∞   where ̺ is defined in (16). Lower bound: Let ε (0,ε ) small enough be given such that for all i =1, .., p and y ε one has 1 ∈ 0 | |≤ 1

ℓ 2 α(x + y) α∗ y . i ≤ − 2| | Herein ℓ> 0 is defined in Assumption 1.11 (iii). Next define m> 0 by

ℓ m = min min γ(xi + y) > 0. 2 i=1,..,p y ε1 | |≤

α(x)γ(x) 1 Recall that Γ(x) = α∗ and Γ(x) = α∗ (α(x) γ(x)+ γ(x) α(x)). Consider M > 0 such that for all k =1, .., p and all x x ε one∇ has ∇ ∇ | − k|≤ 1 Γ(x) γ(x ) γ(x ) (x x ) M x x 2. (35) | − k − ∇ k · − k |≤ | − k| Next fix i =1, .., p and ε (0,ε ). Then one has for all t> 0 ∈ 1

2 I(t, dx) exp tη(t)Γ(x) tm x xi I0(dx) x x ε ≤ x x ε − | − | Z| − i|≤ Z| − i|≤   tη(t)γ(xi) 2 e exp t η(t) γ(xk) (x xi) (m + O(η(t)) x xi I0(dx). ≤ x x ε ∇ · − − | − | Z| − i|≤ 
 Now observe that for all t 1 one has ≫ η(t) γ(x ) 2 η(t)2 γ(x ) 2 η(t) γ(x ) (x x ) (m + O(η(t))) x x 2 = (m + O(η(t))) x x ∇ i + k∇ i k , ∇ k · − i − | − i| − − i − 2(m + O(η(t))) 4(m + O(η(t)))

so that we get, using Assumption 1.11 (ii), that

2 2 2 tη(t) k∇γ(xi)k tη(t)γ(xi)+ η(t) γ(xi) I(t, dx) e 4(m+O(η(t)) exp (m + O(η(t))t x xi ∇ I0(dx) x x ε ≤ x x ε − − − 2(m + O(η(t)) Z| − i|≤ Z| − i|≤ " # 2 2 tη(t)γ(x )+ tη(t) k∇γ(xi)k Me i 4(m+O(η(t)) ≤ 2 κi η(t) γ(xi) x xi exp (m + O(η(t))t x xi ∇ dx. × x x ε | − | − − − 2(m + O(η(t)) Z| − i|≤ " #

We now make use of the following change of variables in the above integral

η(t) γ(x ) z = √t x x ∇ i , − i − 2(m + O(η(t))  

28 so that we end up with

2 2 N+κi tη(t) k∇γ(xi)k tη(t)γ(xi)+ I(t, dx) t− 2 e 4(m+O(η(t)) C(t), x x ε ≤ Z| − i|≤ with C(t) given by κ m+O(η(t)) z 2 C(t) := M z + √tO(η(t)) i e− 2 | | dz. z √t(ε+O(η(t))) | | Z| |≤ Now let us recall that Lemma 2.10 ensures that

lim tη(t)= . t →∞ ∞ Hence one already knows that η(t) 0 for all t 1. Moreover (34) ensures that ≥ ≫ lim √tη(t)=0, t →∞ so that Lebesgue convergence theorem ensures that

κ m z 2 C(t) C := M z i e− 2 | | dz (0, ) as t . → ∞ RN | | ∈ ∞ → ∞ Z As a conclusion of the above analysis, we have obtained that there exists some constant C′ such that for all ε (0,ε ) and all i =1, .., p one has ∈ 1 N+κi tη(t)γ(x ) I(t, dx) C′t− 2 e i , t 1. (36) x x ε ≤ ∀ ≫ Z| − i|≤ 1 Since I(t, dx) concentrates on α− (α∗), then for all ε (0,ε ) one has ∈ 1 p I(t, dx)= I(t, dx)dx + o(1) as t . RN → ∞ i=1 x xi ε Z X Z| − |≤ Using the persistence of I stated in Theorem 1.2 (see Lemma 2.1), we end-up with

p 0 < lim inf I(t, dx) lim inf I(t, dx), t RN ≤ t →∞ →∞ i=1 x xi ε Z X Z| − |≤ so that (36) ensures that there exists c> 0 and T > 0 such that

p N+κi γ(xi) tη(t) 2γ(x ) ln t 0

N+κi γ(xi) tη(t) ln t e − 2γ(xi) 0 as t ,   → → ∞ i/J X∈ and (37) rewrites as c γ(x )(tη(t) ̺ ln t) 0 < e i − , t 1. 2 ≤ ∀ ≫ i J X∈ This yields lim inf (tη(t) ̺ ln t) > , t →∞ − −∞ that is ln t 1 η(t) ̺ + O as t . (38) ≥ t t → ∞   Then (15) follows coupling (34) and (38). Estimate of the masses: In this last step we turn to the proof of (17). Observe first that the upper estimate directly follows from the asymptotic expansion of η(t) in (15) together with (36). Next, the proof for the lower estimate follows from similar inequalities as the one derived in the second step above.

APPENDIX

29 A Measure theory on metric spaces

In this Section we let (M, d) be a complete metric space. Let (M) be the set of compact subsets in M and let K (M). We first recall that we can define a kind of frameK of reference, internal to K, which allows to identify∈ eachK point in K. Let us denote (M) the set formed by all compact subsets of M. Recall that ( (M), d ) is a complete K K H metric space, where dH is the Hausdorff distance

dH (K1,K2) = max sup d(x, K2), sup d(x, K1) . x K x K  ∈ 1 ∈ 2 

Proposition A.1 (Metric coordinates). There exists a finite number of points x1,...,xn K with the property that each y K can be identified uniquely by the distance between y and x ,...,x . In other∈ words the map ∈ 1 n d(y, x1) c y K . Rn , 7−→  .  ∈ + d(y, x )  n    1 is one-to-one. Moreover c is continuous and its reciprocal function c− : c (K) K is also continuous. K K K → Proof. Let us choose x K and x K such that x = x . We recursively construct a sequence x and a 1 ∈ 2 ∈ 1 6 2 n compact set Kn such that K = y K : d(y, x )= d(y, x ) for all 1 i n , n { ∈ i 1 ≤ ≤ } x K , n+1 ∈ n the choice of xn+1 being arbitrary. Clearly Kn is a compact set and Kn+1 Kn. Suppose by contradiction that K = ∅ for all n N, then (because K is compact) one can construct a sequence x , extracted from n 6 ∈ ϕ(n) xn, and which converges to a point

x = lim xϕ(n) Kn =: K . n + ∈ ∞ → ∞ n N \∈ In particular K is not empty. However we see that, by definition of K , we have d(x, xn)= d(x, x1) > 0 for all n N, which∞ contradicts the fact that ∞ ∈

lim d(x, xϕ(n))=0. n + → ∞

Hence we have shown by contradiction that there exists n0 N such that Kn = ∅ and Kn 1 = ∅. This is ∈ 0 0− 6 precisely the injectivity of the map c : K Rn0 . K → To show the continuity, we remark that cK is continuous, and therefore for each closed set F K, F is 1 1 ⊂ compact so that cK (F ) is compact and therefore closed. Therefore (cK− )− (F ) = cK (F ) is closed in cK (K). The proposition is proved. Recall that the Borel σ-algebra (M) is the closure of the set of all open sets in P(M)=2M under the operations of complement and countableB union. A function ϕ : M N is Borel measurable if the reciprocal 1 → image of any Borel set is Borel, i.e. ϕ− (B) (M) for all B (N). ∈B ∈B Proposition A.2 (Borel function of choice). There exists a Borel measurable map c : (K), dH (K, d) such that K →
c(K′) K′ for all K′ (K). ∈ ∈ K n Proof. Let c : K R 0 be the map constructed in Proposition A.1. For a compact K′ K we define K → ⊂ 1 c(K′) := cK− min y , y c (K′)  ∈ K  where the minimum is taken with respect to the lexicographical order in Rn0 (which is a total order and n0 therefore identifies a unique minimum for each K′ (K)). Since the map K R miny K y is Borel for ∈ K ⊂ → ∈ the topology on (Rn0 ) induced by the Hausdorff metric, so is c. The proposition is proved. K e Proposition A.3 (Borel measurability of the metric projection). Let K M be compact. The map PK : M (K) defined by ⊂ → K P (x)= y K : d(x, y)= d(x, K) , K { ∈ } is Borel measurable.

30 Proof. First we remark that the map

P (x) := y K : d(x, y)= d(x, K) (M), K { ∈ } ∈ K is well-defined for each x M, and therefore forms a mapping from M info (K) (M). Indeed PK (x) is clearly closed in the compact∈ space K, therefore is compact. K ⊂ K To show the Borel measurability of P , we first remark that, given a compact space K′ K, the set K ⊂ 1 P − (K′) := x M : P (x) K′ = ∅ K { ∈ K ∩ 6 } g 1 is closed. Indeed let x x be a sequence in P − (K′), then by definition there exists y K′ such that n → K n ∈ d(xn,yn) = d(xn,K). By the compactness of K′, there exists y K′ and a subsequence yϕ(n) extracted from y such that y y. Because of the continuityg of z d(z,K),∈ we have n ϕ(n) → 7→

d(x, y) = lim d(xϕ(n),yϕ(n)) = lim d(xϕ(n),K)= d(x, K), n + n + → ∞ → ∞

1 1 therefore y T (x) K′, which shows that x PK− (K′). Hence PK− (K′) is closed. We are now∈ in a∩ position to show the Borel∈ regularity of P . Let C (K) and R> 0 be given. We defined K ∈ K BH (C, R) the ball of center C and radius R in theg Hausdorff metric:g

B (C, R)= C′ (K): d (C, C′) R . H { ∈ K H ≤ } Then 1 P − (B (C, R)) = x M : d (P (x), C) R = B B , K H { ∈ H K ≤ } 1 ∩ 2 where

B := x M : d(y, C) R for all y P (x) , and 1 { ∈ ≤ ∈ K } B := x M : d(z, P (x)) R for all z C . 2 { ∈ K ≤ ∈ }

It can be readily seen that B1 is a Borel set by writing

1 1 B1 = P − (VR(C)) M P − (K VR+ 1 (C)) , K \ K \ n n 1 \≥    g g where V (C) := y K : d(y, C) R . To see that B is a Borel set, we choose a finite cover of C by N balls R { ∈ ≤ } 2 B1,...,BN (in M) of radius at most R and write

N 1 B2 = PK− (Bi). i=1 \ g 1 We conclude that P − (B (C, R)) is a Borel set for all C (K) and R> 0, and since those sets form a basis K H ∈ K of the Borel σ-algebra, PK is indeed Borel measurable. The Lemma is proved. Theorem A.4 (Existence of a regular metric projection). Let K M be compact. There exists a Borel measurable map P : M K such that ⊂ K →

d x, PK (x) = d(x, K).

Proof. The proof is immediate by combining Proposition
A.3 Proposition A.2. Proposition A.5 (Metric projection on measure spaces). Let K (RN ) be a given compact set. Let µ ∈ K ∈ +(K) be a given nonnegative measure on K. 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: M+

d0(µ, +(K)) µ AV sup d(x, K). M ≤k k x supp µ ∈ K Proof. Indeed, let us choose a Borel measurable metric projection PK on K as in Theorem A.4. Let µ be the image measure defined on (K) by B K 1 µ (B) := µ P − (B) , for all B (K). K ∈B

31 Then in particular for all f BC(RN ) we have ∈ K f(PK (x))dµ(x)= f(x)dµ (x). RN Z ZK Let f Lip (RN ), then we have ∈ 1 f(x)d(µ µK )(x)= f(x)dµ(x) f(x)dµK (x) RN − RN − RN Z Z Z = f(x)dµ(x) f(PK (x))dµ(x) RN − RN Z Z = f(x) f(PK (x))dµ(x) RN − Z x P (x) dµ(x) ≤ | − K | Zsupp µ

sup d(y,K) 1dµ = µ AV sup d(x, K). ≤ y supp µ supp µ k k x supp K ∈ Z ∈ K K Therefore d0(µ,µ ) µ AV supx supp K d(x, K) and, since µ +(K), ≤k k ∈ ∈M K d0 µ, +(K) d0(µ,µ ) µ AV sup d(x, K). M ≤ ≤k k x supp µ ∈
The Proposition is proved.

B Disintegration of measures

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 [23, 24]. Let us first we recall some background on adequate families. This is adapted from [7, V.16 §3] to the context N of finite measures of R . We let T and X be locally compact topological spaces and µ +(T ) be a fixed Borel measure. ∈ M

Definition B.1 (Scalarly essentially integrable family). Let Λ : t λt be a mapping from T into +(X). Λ is scalarly essentially integrable for the measure µ if for every7→ compactly supported continuousM function 1 f Cc(X), the function t X f(x)λt(dx) is in L (µ). Setting ν(f)= T X f(x)λt(dx)µ(dt) defines a linear form∈ on C (X), hence a measure7→ ν, which is the integral of the family Λ, and we denote c R R R

λt µ(dt) := ν. ZT Recall that every positive Borel measure µ on a locally compact space X defines a positive bounded linear functional on Cc(X) equipped with the inductive limit of the topologies on Cc(K) when K runs over the compact subsets of X. Conversely if µ is a positive bounded linear functional on Cc(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 ⊂

µ∗(U) := sup µ(f): f C (X), 0 f(x) ½ (x) , { ∈ c ≤ ≤ U } then for an arbitrary Borel set B,

µ∗(B) := inf µ∗(U): U open, B U . { ⊂ } This notion corresponds to that of the upper integral discussed in [7, IV.1 §1].

2. Inner-regular construction. If U X is open, we define µ•(U) := µ∗(U) and similarly if K X is ⊂ ⊂ compact, then µ•(K) := µ∗(K). Then for an arbitrary Borel set B which is contained in an open set of finite measure: B U with µ•(U) < + , we define ⊂ ∞ µ•(B) := sup µ•(K): K compact, K B . { ⊂ }

Else µ•(B)=+ . This corresponds to the essential upper integral discussed in [7, V.1, §1]. ∞

32 It is always true that µ• µ∗, however it may happen that µ∗ = µ• when µ∗ is not finite, see e.g. [5, II§7.11 ≤ 6 p.113] or [7, V.1, §1]. If µ is a Borel measure, then we define the corresponding notions of µ• and µ∗ associated with the linear functional f f(x)µ(dx). Note that if µ is Radon, then µ∗ = µ = µ•. 7→ X Definition B.2 (Pre-adequateR and adequate families). We follow [7, Definition 1, V.17§3]. Let Λ : t λ be 7→ t a scalarly essentially µ-integrable mapping from T into +(X), ν the integral of Λ. We say that Λ is µ-pre-adequate if, for every lower semi-continuousM function f 0 defined on X, the function ≥ t f(x)λ•(dx) is µ-measurable on T and 7→ t R f(x)ν•(dx)= f(x)λt•(dx)µ•(dt). ZX ZT ZX

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 B.3 (µ-proper function). We say that a function p : T X is µ-proper if it is µ-measurable and, 1 → 1 for every compact set K X, the set p− (K) is µ•-measurable and µ•(p− (K)) < + . ⊂ ∞ 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]. Theorem B.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) of positive measures on T , having the following properties: 7→ x ∈ a) λ =1 for all x p(T ); k xk ∈ 1 b) λ is concentrated on the set p− ( x ) for all x p(T ), and λ =0 for x p(T ); x { } ∈ x 6∈ c) µ = λx ν(dx).

Moreover,R if x λ′ (x X) is a second ν-adequate family of positive measures on T having the properties b) 7→ x ∈ and c), then λx′ = λx almost everywhere in B with respect to the measure ν.

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