Time fluctuations in a population model of adaptive dynamics

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ScienceDirect

Ann. I. H. Poincaré – AN 32 (2015) 41–58

www.elsevier.com/locate/anihpc

Time fluctuations in a population model of adaptive dynamics

Sepideh Mirrahimi

^a,b,^∗

, Benoît Perthame

^c,d

, Panagiotis E. Souganidis

^e,1

aCNRS, Institut de Mathématiques de Toulouse UMR 5219, 31062 Toulouse, France bUniversité de Toulouse, UPS, INSA, UT1, UTM, IMT, 31062 Toulouse, France

cUPMC Université Paris 06, Laboratoire Jacques-Louis Lions, CNRS UMR7598, 4 pl. Jussieu, 75005 Paris, France dINRIA EPI BANG and Institut Universitaire de France, France

eThe University of Chicago, Department of Mathematics, 5734 S. University Avenue, Chicago, IL 60637, USA Received 29 May 2013; received in revised form 1 October 2013; accepted 5 October 2013

Available online 17 October 2013

Abstract

We study the dynamics of phenotypically structured populations in environments with fluctuations. In particular, using novel arguments from the theories of Hamilton–Jacobi equations with constraints and homogenization, we obtain results about the evolution of populations in environments with time oscillations, the development of concentrations in the form of Dirac masses, the location of the dominant traits and their evolution in time. Such questions have already been studied in time homogeneous environments. More precisely we consider the dynamics of a phenotypically structured population in a changing environment under mutations and competition for a single resource. The mathematical model is a non-local parabolic equation with a periodic in time reaction term. We study the asymptotic behavior of the solutions in the limit of small diffusion and fast reaction. Under concavity assumptions on the reaction term, we prove that the solution converges to a Dirac mass whose evolution in time is driven by a Hamilton–Jacobi equation with constraint and an effective growth/death rate which is derived as a homogenization limit. We also prove that, after long-time, the population concentrates on a trait where the maximum of an effective growth rate is attained. Finally we provide an example showing that the time oscillations may lead to a strict increase of the asymptotic population size.

MSC:35B25; 35K57; 49L25; 92D15

Keywords:Reaction–diffusion equations; Asymptotic analysis; Hamilton–Jacobi equation; Adaptive dynamics; Population biology;

Homogenization

1. Introduction

Phenotypically structured populations can be modeled using non-local Lotka–Volterra equations, which have the property that, in the small mutations limit, the solutions concentrate on one or several evolving in time Dirac masses.

A recently developed mathematical approach, which uses Hamilton–Jacobi equations with constraint, allows us to understand the behavior of the solutions in constant environments[5,12,1,10].

* Corresponding author.

E-mail addresses:sepideh.mirrahimi@math.univ-toulouse.fr(S. Mirrahimi),benoit.perthame@upmc.fr(B. Perthame), souganidis@math.uchicago.edu(P.E. Souganidis).

1 Partially supported by the National Science Foundation grants DMS-0901802 and DMS-1266383.

http://dx.doi.org/10.1016/j.anihpc.2013.10.001

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Since stochastic and periodic modulations are important for the modeling[9,14,15,8,16], a natural and relevant question is whether it is possible to further develop the theory to models with time fluctuating environments.

In this note we consider an environment which varies periodically in time in order, for instance, to take into account the effect of seasonal variations in the dynamics, and we study the asymptotic properties of the initial value problem

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

εnε,t=nεR

x,t ε, Iε(t)

+ε²nε inR^N×(0,∞), n_ε(·,0)=n_0,ε inR^N,

Iε(t):=

R^N

ψ (x)nε(x, t) dx,

(1)

where

R:R^N×R× [0,∞)→R is smooth and 1-periodic in its second argument. (2) The population is structured by phenotypical traits x∈R^N with densityn_ε(x, t)at timet. It is assumed that there exists a single type of resource which is consumed by each individual traitx at a rateψ (x);I_ε(t)is then the total consumption of the population. The mutations and the growth rate are represented respectively by the Laplacian term andR. The novelty is the periodic in time dependence of the growth rateR. The small coefficientεis used to consider only rare mutations and to rescale time in order to study a time scale much larger than the generation one.

To ensure the survival and the boundedness of the population we assume thatR takes positive values for “small enough populations” and negative values for “large enough populations”, i.e., there exists a valueI_M>0 such that

max

0s1, x∈R^NR(x, s, I_M)=0 and X:= x∈R^N, 1 0

R(x, s,0) ds >0

= ∅. (3)

In addition the growth rateRsatisfies, for some positive constantsK_i,i=1, . . . ,7, and all(x, s, I )∈R^N×R×[0, IM] andA >0, the following concavity and decay assumptions:

−2K1D_x²R(x, s, I )−2K2, −K₃−K₁|x|²R(x, s, I )K₄−K₂|x|², (4)

−K₅D_IR(x, s, I )−K₆, (5)

D_x³R∈L^∞

R^N×(0,1)× [0, A]

and D²_x,IRK7. (6)

The “uptake coefficient”ψ:R^N→Rmust be regular and bounded from above and below, i.e., there exist positive constantsψ_m,ψ_M andK8such that

0< ψ_mψψ_M and ψC²K₈. (7)

We also assume that the initial datum is “asymptotically monomorphic”, i.e., it is close to a Dirac mass in the sense that there existx⁰∈X,ρ⁰>0 and a smoothu⁰_ε:R^N→Rsuch that

n_0,ε=e^u⁰^ε^/ε and, asε→0, (8)

n_ε(·,0)−→

ε→0⁰δ

· −x⁰

weakly in the sense of measures. (9)

In addition there exist constantsL_i>0, i=1, . . . ,4, and a smoothu⁰:R^N→Rsuch that, for allx∈R^N,

−2L1ID_x²u⁰_ε−2L2I, −L₃−L₁|x|²u⁰_ε(x)L₄−L₂|x|², max

x∈R^Nu⁰(x)=0=u⁰ x⁰

(10) and, asε→0,

u⁰_ε−→u⁰ locally uniformly inR^N. (11)

Finally, in order to ensure the same control onD²uas forD²u⁰, it is necessary to impose the following compatibility relation on the parameters in the initial data and in the growth rateR:

4L²₂K₂K₁4L²₁. (12)

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To state our results we first need, as it is the case in homogenization, a cell problem given by the following lemma:

Lemma 1.1.Assume(3),(5)and(7). For allx∈X, there exists a unique1-periodic positive solutionI(x, s): [0,1] → (0, IM)to

d

dsI(x, s)=I(x, s)R

x, s,I(x, s) , I(x,0)=I(x,1).

(13) Moreover, asX x→x₀∈∂X,

0maxs1I(x, s)→0. (14)

In view of(14), forx∈∂X, we define, by continuity,I(x, s)=0. The functionI(x, s)helps us to identify the weak limit ofI_ε(t)(seeLemma 2.1). It also helps to derive, using a homogenization process, an effective growth rate Rwhich will replaceR(seeTheorem 1.2) and is defined, for allx∈R^Nandy∈X (hereX stands for the closure of X), by

R(x, y):=

1 0

R

x, s,I(y, s)

ds. (15)

Notice that, integrating(15)above insand using the periodicity, we always have, forx∈X,

R(x, x)≡0. (16)

Moreover, if y∈∂X, then R(x, y)=₁

0R(x, s,0) ds. Finally, it is immediate from (4) and(15), thatR(x, y)is strictly concave in the first variable.

Our first result is about the behavior of then_ε’s asε→0. It asserts the existence of a fittest traitx(t)and a total population sizeρ(t ) at timet and provides a “canonical equation” for the evolution in time ofx in terms of the

“effective fitness”R(x, y). In the sequel,D₁Rdenotes the derivative ofRwith respect to the first argument.

Theorem 1.2(Limit asε→0). Assume(2)–(12). There exist a fittest traitx∈C¹([0,∞);X)and a total population sizeρ∈C¹([0,∞);(0,∞))such that, along subsequencesε→0,

nε(·, t )− (t)δ

· −x(t)

weakly in the sense of measures, Iε− I:=ψ (x) inL^∞(0,∞)weak-,

and R

x,t

ε, I_ε(t)

− R x, x(t)

weakly in the sense of measures intand strongly inx. Moreover,xsatisfies the canonical equation

x(t)˙ =

−D_x²u

x(t), t₋1

·D₁R

x(t), x(t)

. (17)

We note that, in the language of adaptive dynamics,R(y, x)can be interpreted as the effective fitness of a mutanty in a resident population with a dominant traitx, whileD₁Ris usually called the selection gradient, since it represents the capability of invasion. The extra term(−D_x²u(x(t), t))⁻¹is an indicator of the diversity around the dominant trait in the resident population.

The second issue is the identification of the long time limit of the fittest traitx. We prove that, in the limitt→ ∞, the population converges to a so-called Evolutionary Stable Distribution (ESD) corresponding to a distribution of population which is stable under introduction of small mutations (see [11,6,7]for a more detailed definition). See also[4,13]for recent studies of the local and global stability of stationary solutions of integro-differential population models in constant environments.

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Theorem 1.3(Limit ast→ ∞). In addition to(2)–(12)assume that eitherN=1or, ifN >1,Ris given, for some smoothb, d, B, D:R^N→(0,∞)by

R(x, s, I )=b(x)B(s, I )−d(x)D(s, I ). (18)

Then, ast→ ∞, the population reaches an Evolutionary Stable Distribution_∞δ(· −x_∞), i.e., (t)−→_∞and x(t)−→x_∞, where_∞>0andx_∞are characterized by(Iis defined in(13))

R(x_∞, x_∞)=0=max

x∈R^NR(x, x_∞) and _∞= 1 ψ (x_∞)

1 0

I(x_∞, s) ds. (19)

The proof of the above result for a growth rateRgiven by(18)is based on a Lyapunov functional defined by L(t ):= b(x(t))

d(x(t)).

Notice that we do not claim the uniqueness of the Evolutionary Stable Distribution. Indeed there may exist several (ρ_∞, x_∞)satisfying(19). Here we only prove that there exists(ρ_∞, x_∞)satisfying(19)such that, ast→ ∞, the population converges to_∞δ(· −x_∞).

The difference between our conclusions and the results for time homogeneous environments in [10]is that, in the canonical equation(17), the growth rate R is replaced by an effective growth rate Rwhich is derived after a homogenization process. Moreover, we are only able to prove that theIε’s converge inL^∞weak-∗and not a.e., which is the case for constant environments in[10]. This adds a difficulty inTheorem 1.3and it is the reason why we are not able to describe, without additional assumptions, the long-time limit behavior of the fittest traitxfor general growth rateRwhenN >1. This remains an open question.

In Section3.3, we give an example ofRnot satisfying the assumptions of Theorem 1.3for whichx(t)exhibits a periodic behavior. This example fits the structure (22)below with general concavity properties onRbut it is not necessarily derived from a homogenization limit.

The proofs use in a fundamental way the classical Hopf–Cole transformation

uε=εlnnε, (20)

which yields the following Hamilton–Jacobi equation foru_ε:

⎧⎨

⎩

u_ε,t=R

x,t ε, I_ε(t)

+ |D_xu_ε|²+εu_ε inR^N×(0,∞),

uε(·,0)=u⁰_ε inR^N.

(21)

The next theorem describes the behavior of theu_ε’s, asε→0 (recall thatRis defined in Section2).

Theorem 1.4.Assume(2)–(12). Along subsequences ε→0,u_ε→ulocally uniformly inR^N× [0,∞), where u∈ C(R^N× [0,∞))is a solution of

⎧⎪

⎪⎨

⎪⎪

⎩

u_t=R x, x(t)

+ |D_xu|² inR^N×(0,∞), max

x∈R^Nu(x, t )=0=u x(t), t

in(0,∞),

u(·,0)=u⁰ inR^N.

(22)

Note that the convergence of theu_ε’s inTheorem 1.4and, thus, the convergence of then_ε’s inTheorem 1.2are established only along subsequences. To prove convergence for allε, we need that(22)has a unique solution. This is, however, not known even for non-oscillatory environments except for some particular form of growth rateR (see [2,12]).

When the environment is time homogeneous, it is possible to derive such Hamilton–Jacobi equations without any concavity assumption on R andu⁰ (see [12,1]). However, here we need assumptions(4),(10) and(12)to ensure a priori thatn_ε goes to a single Dirac mass at the pointx(t). This information is needed in order to write(22). We

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also point out that, forx∈R,Ris monotonic with respect tox and without the concavity assumption, it is possible to show that then_ε’s still converge to a single Dirac mass. This suggests that it may be possible to relax the concavity assumptions in this latter case or in similar situations where such information is known a priori. Here, however, we concentrate on the difficulty coming from the time fluctuations and do not try to state a more general result.

In Section4we consider a non-concave rate of the form R(x, s, I )=b(x)B(s, I )−D(s, I ),

and show that the effective growth rateRcan be computed independently ofx(t)and has the form R(x, F )=

1 0

I(F, s) ds b(x)

F −1

,

with

F (t ):=lim

ε→0

ψ (x)b(x)n_ε(x, t) dx

I_ε(t) ,

in the Hamilton–Jacobi limit. Note that, if the(nε)ε’s converge to a single Dirac massρ(t )δ(x−x(t)), then we can computeF (t )=b(x(t)). This example is even more interesting, since it is possible to compute explicitly the effective growth rateR, while, in general, not much is known about the structure ofR.

In Section5, we study the particular case R(x, I )=b(x)−D₁(s)D₂(I ).

We compute the population size at the evolutionary stable state of the oscillatory model and compare it to the one of a non-oscillatory averaged environment. We observe that the time oscillations lead to a strict increase in the population size. This example emphasizes the importance of including the time oscillations in population models.

The paper is organized as follows. In Section2we study the asymptotic behavior of the solution under rare mutations (limitε→0) and we provide the proofs ofTheorems 1.2 and 1.4. In Section3 we consider the long time behavior of the dynamics (limitt→ ∞) and we give the proof ofTheorem 1.3. In Section4 we study a particular form of growth rateR, for which the results can be proved without any concavity assumptions onRand the effective growth rateRhas a natural structure. Finally in Section5we present an example of an oscillatory environment which yields an asymptotic effective population density that is strictly larger than the averaged one.

2. The behavior asε→0 and the proofs ofTheorems 1.2 and 1.4

We present the proofs of Theorems 1.2 and 1.4which are closely related. Since the argument is long, we next summarize briefly the several steps. First we obtain some a priori bounds onu_εandI_ε. Then we identify the equation for the fittest traitx. The limit of theI_ε’s is studied inLemma 1.1. The last three steps are the identification (and properties) of the effective growth rateR, the effective Hamilton–Jacobi equation and the canonical equation.

Proofs of Theorems 1.2 and 1.4. Step 1. A priori bounds.Multiplying(1)byψ (x), integrating with respect tox, it and arguing as in[10], we find, that for allt0,

0< I_ε(t)I_M+O(ε). (23)

Next we differentiate(21)twice with respect toxand use(4),(10),(12)and the maximum principle to get, for some C₁, C₂>0,ε1 and all(x, t)∈R^N× [0,∞),

−2L1D²_xu_ε−2L2 and −L3−L1|x|²−C1tu_ε(x, t)L4−L2|x|²+C2t. (24) Note that to obtain the second inequality, we do not need to differentiate with respect to x, but use directly the maximum principle for(21)and the fact that the lower and upper bounds are respectively sub- and super-solutions.

For a detailed proof of(24)seeAppendix A.

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It follows from the uniform bounds on u_ε andD_xx² u_ε that D_xu_ε is locally uniformly bounded and, hence, we obtain, from(21), that, for all ballsB_R centered at the origin and of radiusR, there existsC₃=C₃(R) >0 such that

uε,t(x, t)

L^∞(BR×(0,∞))C3. (25)

Finally the regularity properties of the “viscous” Hamilton–Jacobi equations and(6)yield that, for allT >0, there existsC₄=C₄(R, T ) >0 such that

D_x³u_ε

L^∞(B_R×[0,T])C₄. (26)

Hence, after differentiating(21)inx, the previous estimates yield aC₅=C₅(R, T ) >0 such that D_t,x² u_ε

L^∞(B_R×[0,T])C₅. (27)

All the above bounds allow us to pass to the limit, along subsequencesε→0, and to obtainu:R^N× [0,∞)→R such that, asε→0,

uε−→u inCloc

R^N× [0,∞)

and, for allT >0 and(x, t)∈R^N× [0, T],

−2L1D_x²u(x, t )−2L2 and u, D²_t,xu, D_x³u∈L^∞

R^N× [0, T] . For more details on the above arguments see[10].

Step 2. The fittest trait.In view of the strict concavity ofu_ε, for eachε >0, there exists a uniquex_ε(t)such that u_ε

x_ε(t), t

=max

x∈R^Nu_ε(x, t) and D_xu_ε x_ε(t), t

=0.

Differentiating the latter equality with respect totand using(21)we find x˙_ε(t)·D²_xu_ε

x_ε(t), t

= −D_xu_ε,t x_ε(t), t

= −DxR

xε(t),t ε, Iε(t)

−2D²_xuε

xε(t), t

·Dxuε

xε(t), t

−εDxuε

xε(t), t

= −D_xR

x_ε(t),t ε, I_ε(t)

−εD_xu_ε x_ε(t), t

.

SinceD_x²uε(xε(t), t)is invertible andD_x³uL^∞(B_R×[0,T])C4, it follows thatx˙ε(t)is bounded in(0, T ), and, hence, along subsequencesε→0,x_ε→xinC_loc((0,∞)), for somex∈C^0,1((0,∞))such that

⎧⎪

⎪⎨

⎪⎪

⎩ u

x(t ), t

=max

x∈R^Nu(x, t ), D_xu x(t), t

=0, and

x(t)˙ =

−D²_xu

x(t), t₋1

·D_x

R

x(t),t ε, I_ε(t)

,

(28)

where the bracket denotes the weak limit ofR(x(t ),^t_ε, I_ε(t))which exists, sinceRis locally bounded.

Step 3. The weak limit of I_ε.To identify the weak limit of theI_ε’s, we consider the first exit timeT^∗>0 of x fromX, i.e., the smallest timeT^∗>0 such thatx(t )∈X for all 0t < T^∗andx(T^∗)∈∂X ifT^∗<∞. Note that T^∗is well defined sincex(0)=x⁰∈X. The last step of the ongoing proof is to show thatT^∗= ∞.

The weak limit of theI_ε’s fort∈(0, T^∗)follows fromLemma 1.1and the lemma below. Their proofs are given after the end of the ongoing one.

Lemma 2.1.Assume(3),(5)and(7). LetT_ε^∗be the smallest timeT_ε^∗>0such thatx_ε(t)∈X for all0t < T_ε^∗and x(T_ε^∗)∈∂XifT_ε^∗<∞. Then, for all0< t < T_ε^∗,

lnI_ε(t)−lnI

x_ε(t),t ε

lnI_ε(0)−lnI

x_ε(0),0e⁻^K^ε⁶^t +C√ ε,

whereConly depends on the constantsK_i. Moreover, asε→0,T_ε^∗→T^∗. Consequently, ifx(t)∈Xfor0t < T^∗ andx(T^∗)∈∂X, then, asε→0andt→T^∗,I_ε(t)−→0.

(7)

It follows that, asε→0, I_ε(·)− I (·)=

1 0

I x(·), s

ds >0 inL^∞ 0, T^∗

weak- . (29)

OnceI is known, it is possible to compute the weightof the Dirac mass. Indeed, we show in the next steps that, as ε→0,

I_ε(·)=

R^N

ψ (x)n_ε(x,·) dx− I (·)=(·)ψ x(·)

inL^∞ 0, T^∗

weak- .

Step 4. The effective growth rate.We can now explain the average used to determine the effective growth rate.

Again(5)andLemma 2.1yield that, asε→0,

T^∗

0

R

x,t ε, I_ε(t)

−R

x,t ε,I

x(t),t

ε

dtK5 T^∗

0

Iε(t)−I

x(t),t ε

dt−→0.

Therefore the weak limit in(28)is computed as the weak limit (in time) ofR(x,_ε^t,I(x(t ),_ε^t)). It follows that, for 0tT^∗,

R

x, t

ε, Iε(t)

=R x, x(t)

, (30)

whereRis defined by(15).

Step 5. The limiting Hamilton–Jacobi equation.It is now possible to pass to the limitε→0 in(21)for(x, t)∈ R^N× [0, T^∗). To this end, observe that

ϕ_ε(x, t):=u_ε(x, t)− t 0

R

x,τ ε, I_ε(τ )

dτ solves

ϕ_ε,t−εϕ_ε=ε t 0

R

x,τ ε, I_ε(τ )

dτ+

D_xϕ_ε+ t 0

D_xR

x,τ ε, I_ε(τ )

dτ

2

.

Since theu_ε’s converges locally uniformly from Step 1 andR(x,_ε^t, I_ε(t))converges weakly int and strongly inx to R(x, x(t))from Step 2, we find that, asε→0,

ϕ_ε(x, t)−→ϕ(x, t )=u(x, t )− t 0

R x, x(τ )

dτ inC_loc R^N×

0, T^∗ .

Moreover, in view of(4),(6)andLemma 1.1, for anyT ∈(0, T^∗)andR >0, there existsC=C(R, T ) >0 such that, for(x, t)∈BR× [0, T]

t 0

R

x,τ ε, Iε(τ )

dτ

C, and, asε→0,

t 0

D_xR

x,τ ε, I_ε(τ )

dτ−D_xR

x,τ ε,I

x(τ ),τ

ε

dτK7

T 0

I_ε(t)−I

x(t),t ε

dt−→0.

(8)

Thus, asε→0 and for all(x, t)∈B_R× [0, T], t

0

DxR

x,τ ε, Iε(τ )

dτ−→

t 0

D1R x, x(τ )

dτ.

It follows from the stability of viscosity solutions thatϕis a viscosity solution to ϕ_t=

D_xϕ+ t 0

D₁R x, x(τ )

dτ

2

in R^N× 0, T^∗

, which, written in terms ofu, reads

u_t=R x, x(t)

+ |D_xu|² inR^N× 0, T^∗

.

The constraint max_x_∈RNu(x, t )=0 follows from(20)and(23)(see[12,1]). We then conclude following[12]that, for allt∈(0, T^∗),

n_ε(·, t )− (t)δ

· −x(t)

weakly in the sense of measures.

Step 6. The canonical equation.The canonical equation(17)now follows from(28)and(30).

Step 7. The global timeT^∗= ∞.AssumeT^∗<∞. Thenx(T^∗)∈∂X. It follows from the canonical equation(17) that, for allt∈(0, T^∗),

d dt

1 0

R

x(t), s,0

ds= ˙x(t) 1 0

D_xR

x(t), s,0 ds

=D₁R

x(t), x(t)

−D²u

x(t), t₋1

1 0

D_xR

x(t), s,0 ds, while, whent=T^∗,Lemma 2.1yields thatD₁R(x(t), x(t))=₁

0D_xR(x(t ), s,0) ds. Hence d

dt 1 0

R x

T^∗ , s,0

ds >0,

which is a contradiction because, by the definition of the open set X, ₁

0R(x(t), s,0) ds >0 for t∈ [0, T^∗)and ₁

0 R(x(T^∗), s,0) ds=0. 2

Proof of Lemma 1.1. First we prove that, for a fixedx∈X, there exists a solutionIof(13). To this end observe that J:=lnIsolves

d

dsJ(x, s)=R

x, s,exp

J(x, s)

ins∈ [0,1], J(x,0)=α.

(31) It turns out that it is possible to chooseαlnI_M so thatJ(x,0)=J(x,1). Indeed, the definition ofI_M in(3)yields that, ifα=lnI_M, thenJ(x,1) < α. On the other hand, forαvery small we claim thatJ(x,1) > α, which is enough to conclude, sinceJ(x, s)been a continuous increasing function ofα, it has a fixed pointα^∗. Choosingα=α^∗yields a periodic solution.

To prove the claim, we set μ=1

0R(x, s,0)ds >0 sincex ∈X. BecauseR is locally bounded, there exists a constantC >0, which is independent ofα, such thatJ(x, s)α+Cand, forαsmall enough,

J(x,1)=J(x,0)+ 1

0

R

x, s,exp

J(x, s)

dsJ(x,0)+ 1 0

R(x, s,0) ds+O e^α

J(x,0)+μ 2. This proves the claim and the existence of a periodic solution.

(9)

The uniqueness follows from a contraction argument. Indeed letJ1andJ2be two periodic solutions to(31). Then d

ds(J1−J2)=R

x, s,exp

J1(x, s)

−R

x, s,exp

J2(x, s) .

Multiplying the above equation by sgn(J1−J2)and using the monotonicity inI according to(5), we find d

ds|J1−J2|−C|J1−J2|, and, after integration,

C 1 0

J1(s)−J2(s)ds−J1(1)−J2(1)+J1(0)−J2(0)=0, and, hence,J1=J2.

Finally we prove(14). It follows from(31)that, forx∈X, 0=

1 0

R

x, s, e^J^(x,s)

ds

1 0

R(x, s,0) ds−K₆e^min⁰^s¹^J^(x,s).

Ifx→x₀∈∂X, then₁

0R(x, s,0) ds→0 and, since the variations ofJ(x, s) are bounded, becauseR is locally bounded, the result follows. 2

Proof of Lemma 2.1. We identify the weak limit ofIε and prove (29). We begin with the observation that in the

“Gaussian”-type concentration,x−x_ε(t)scales as√ ε.

Indeed multiplying(1)byψand integrating with respect toxwe find (recall that withJ_ε:=lnI_ε), εd

dtJε(t)=

R^Nψ (x)n_ε(x, t)R(x,_ε^t, I_ε(t)) dx

R^Nψ (x)n_ε(x, t) dx +ε²

R^Nψ (x)n_ε(x, t) dx

R^Nψ (x)n_ε(x, t) dx .

Note that in order to justify the integration by parts above, we first replaceψbyψ_L=χ_Lψwhereχ_Lis a compactly supported smooth function such thatχ_L≡1 in B(0, L)andχ_L≡0 inR^N\B(0,2L). Then we integrate by parts and finally letL→ +∞.

Returning to the above equation we find εd

dtJ_ε(t)=

R^Nψ (x)euε (x,t)−uε (xε (t),t)

ε R(x,_ε^t, I_ε(t)) dx

ε dx

+O ε²

=

ε [R(x,^t_ε, I_ε(t))−R(x_ε(t),_ε^t, I_ε(t))]dx

ε dx

+R

x_ε(t),t ε, I_ε(t)

+O

ε² . Using Laplace’s method for approximation of integrals,(24)and(26), we find that the first term is of order√

εand, hence,

εd

dtJ_ε(t)=R

x_ε(t),t ε, I_ε(t)

+O(√

ε ).

Next we compute εd

dt

J

x_ε(t),t ε

−J_ε(t)

=R

x_ε(t),t ε,exp

J

x_ε(t),t

ε

−R

x_ε(t),t ε,exp

J_ε(t) +O(√

ε )+εD_xJ

x_ε(t),t ε

x˙_ε(t).

Multiplying the above equality by sgn(J(x_ε(t),^t_ε)−J_ε(t)), using our previous estimates and employing the mono- tonicity property in(5), we get

(10)

εd dt

J

x_ε(t),t ε

−J_ε(t) = −

R

x_ε(t),t ε,exp

J

x_ε(t),t

ε

−R

x_ε(t),t ε,exp

J_ε(t)+O(√ ε ) −K₆

J

x_ε(t),t ε

−J_ε(t) +O(√

ε ).

The first claim ofLemma 2.1is now immediate. Moreover, sincex_ε(t)→x(t), locally uniformly asε→0, we obtain that, asε→0,

T_ε^∗→T^∗.

The last claim is a consequence of the previous steps andLemma 1.1. 2 3. The long time behavior

3.1. Convergence ast→ ∞whenN=1(the proof ofTheorem 1.3(i))

Throughout this subsection we assume thatN=1. The goal is to prove the existence of somex_∞∈R^N such that, ast→ ∞,x(t)→x_∞and

R(x_∞, x_∞)=0=max

x∈RR(x, x_∞). (32)

To this end, we consider the mapA:R−→Rdefined byA(x)=y, whereyis the unique maximum point ofR(·, x).

We obviously have DxR

A(x), x

=0.

We consider the following three cases depending on the comparison between x(·)andA(x(·)). Ifx(t) < A(x(t)), thenDxR(x(t ), x(t)) >0 and, ifx(t) > A(x(t)), thenDxR(x(t), x(t)) <0. It then follows using(17)and the concavity of u that, ifx(t) < A(x(t)) (resp.x(t) > A(x(t))), thenx(t) >˙ 0 (resp.x(t) <˙ 0). Ifx(t)=A(x(t)), then D_xR(x(t), x(t))=0 and hence again(17)yieldsx(t)˙ =0. We also notice that, ifA(x(0))=x(0)=x_∞, then from the above argument we have that for allt0,x(t)=x_∞, withx_∞satisfying(32).

Now we assume thatA(x(0)) > x(0)(the caseA(x(0)) < x(0)can be treated similarly) and set t0:=inf

t∈R:A x(t)

x(t) .

Ift₀<∞, thenA(x(t₀))=x(t₀)and, hence, for alltt₀,x(t)=x(t₀)=x_∞. Ift₀= ∞, thenx(t) >˙ 0 for allt0, and thus, since the setB= {x(t): t∈ [0,∞)}is compact (see below), there existsx₀∈Rsuch that

tlim→∞x(t)=x₀.

The compactness ofBfollows from the observation that, in view of(15),(4)and(5), R

x, x(t)

K₄−K₂|x|², and, sinceR(x(t), x(t))=0,

x(t)(K₄/K₂)^1/2. (33) We now claim thatx₀satisfies(32). Indeed, if there existsz∈R^N such thatR(z, x₀) >0, then using(22), we have limt→∞u(z, t )= +∞, a contradiction to the constraint maxx∈Ru(x, t )=0.

3.2. Convergence for a particular case withN >1(the proof ofTheorem 1.3(ii))

We proveTheorem 1.3in the multi-d case with a growth rateRas in(18). In this case we find b

x(t)

B(s, I (t)

−d x(t)

D s, I (t)

=0, (34)

and thus

(11)

−D²_xu

x(t), tx(t)˙ =b x(t)

B s, I (t)

−d x(t)

D s, I (t)

=

b(x(t ))

b(x(t)) −d(x(t)) d(x(t))

d x(t)

D s, I (t)

.

Therefore, after taking inner product withx(t), dividing by˙ d(x(t))D(s, I (t ))and using the strict concavity ofu, we obtain

x(t)˙ ²C d dtln

b(x(t)) d(x(t))

.

This proves thatt→L(t )=b(x(t ))/d(x(t ))is a Lyapunov functional which is increasing and thus converges, as t→ ∞, to some constantl. For this we need to show that{x(t): t∈ [0,∞)}is bounded, a fact which follows exactly as in the proof of(33). Finally, in view of(34), we also have

tlim→∞

D(s, I (t )) B(s, I (t)) =l.

We prove next that l=max

x

b(x) d(x).

Arguing by contradiction we assume that l <maxx(b(x)/d(x)). Then there must exist x ∈ R^N such that l <

(b(x )/d(x )), in which case 0<lim inf

t→∞ b(x ) B

s, I (t)

−d(x ) D

s, I (t) . Finally, sinceusolves

∂tu= |Dxu|²+b(x) B

s, I (t)

−d(x) D

s, I (t) , we find

tlim→∞u(x, t )= ∞,

a contradiction to the constraint max_x_∈R^Nu(x, t )=0.

3.3. A counterexample in the multi-dimension case

In this subsection we present an example showing that, whenN >1, the x’s may not converge, ast → ∞, at least for the Hamilton–Jacobi problem(22). Indeed we find a strictly concave with respect to the first variableR: R^N×R^N→R, a 1-periodic mapt→x(t)and a functionu:R^N× [0,∞] →Rwhich satisfies

⎧⎪

⎪⎨

⎪⎪

⎩

∂_tu− |D_xu|²=R x, x(t)

inR^N×(0,∞), max

x∈R^Nu(x, t )=u x(t), t

=0, u(·,0)=u0 inR^N.

(35)

We choose G:R^N→R^N so that the ODE x˙=G(x)has a periodic solution; note that such function exists only forN >1. A simple example forN =2 isG(x₁, x2)=(−x2, x1), which admits(x1(t), x2(t))=(rcost, rsint )as periodic solutions.

LetF:R^N→Rbe an arbitrary smooth function and defineR:R^N×R^N→Rby R(x, y)= −

DF (y)G(y)+4F (y)²

|x−y|²+2F (y)G(y)(x−y). (36)

It is immediate thatRis a concave function with respect toxand satisfiesR(x, x)=0. It is also easily verified that u(x, t )= −Fx(t)x−x(t)² withx(t)˙ =G

x(t)

andx(0)=x₀,

(12)

Fig. 1. Dynamics of the total populationIε(t)forR(x, s, I )=(2+sin(2π s))²_I⁻_+.5^x²−.5,ψ (x)=1 andε=0.01. TheIε’s oscillate with period of orderεaround a monotone curveI.

is a viscosity solution of(35)forRas in(36)andu0(x)= −F (x0)|x−x0|². Moreover the canonical equation(28)is written as

x(t)˙ =

−D²_xu

x(t), t₋1

D_xR

x(t), x(t)

= 2F

x(t)₋1 2F

x(t) G

x(t)

=G x(t)

.

Finally we choosex₀∈R^N such thatt →x(t)withx(t₀)=x₀is 1-periodic. Then the limit limt→∞x(t)does not exist.

Note that the counterexample presented above is for the Hamilton–Jacobi problem(22). We do not know if such periodic oscillation can arise in theε→0 limit of the viscous Hamilton–Jacobi equation(21). When the growth rate independent of time, a result similar toTheorem 1.3was proved in[10]for generalR. In that problem, the key point leading to the convergence, as t→ ∞, of the x(t)’s is thatI (t ), which is the strong limit of theI_ε’s asε→0, is increasing in time. In the case at hand, we can only prove that theI_ε’s converge weakly toI. We know nothing about the monotonicity ofI. We remark that numerical computations suggest (seeFig. 1) that monotonicity holds, if at all, in the average.

4. A particular case with a natural structure forR

The concavity assumption (4) is very strong. Here we study, using a different method based on BV estimates, a class of growth ratesRwhich do not satisfy(4). Throughout this section, for several arguments, we follow[1]which studies a similar problem but without time oscillations.

We consider growth rates of the form

R(x, s, I )=b(x)B(s, I )−D(s, I ), (37)

with

B, D:R× [0,∞)→R 1-periodic with respect to the first argument (38) and we assume that, for all(s, I )∈R× [^I₂^m,2I_M]andx∈R^N,

0< B(s, I ), 0< D(s, I ) and 0< b_mb(x)b_M, (39)

whereI_M>I_m>0 are such that max

0s1, x∈R^NR(x, s,I_M)=0 and min

0s1, x∈R^NR(x, s,I_m)=0, (40)

(13)

and there exist constantsa₁>0 anda₂>0 such that, for all(s, I )∈R× [0,∞),

DIB(s, I ) <−a1 and a2< DID(s, I ). (41)

As far asn_ε(·,0)is concerned, we replace(9)–(11)by Im

ψ (x)nε(x,0)dxIM, and n_ε(x,0)exp

−A|x| +B ε

for someA, B >0 and allx∈R^N. (42)

Even though(37)seems close to(18), no concavity assumption is made and the analysis of Section3does not apply here.

Theorem 4.1.Assume (7)and (41)–(42). Along subsequences ε→0, theu_ε’s converge locally uniformly to u∈ C(R^N×R)satisfying the constrained Hamilton–Jacobi equation

⎧⎨

⎩

u_t=R x, F (t)

+ |D_xu|² inR^N×(0,∞), max_x_∈RNu(x, t )=0,

u(·,0)=u⁰ inR^N,

(43)

with

R(x, F )= 1 0

I(F, s) ds b(x)

F −1

and F (t )=lim

ε→0

ψ (x)b(x)nε(x, t) dx

I_ε(t) ,

andI defined in(49)below. In particular, along subsequencesε→0and in the sense of measures,nε− nwith suppn⊂ {(x, t): u(x, t )=0} ⊂ {(x, t): R(x, F (t))=0}.

As inTheorem 1.3, we can deduce the long time convergence to the Evolutionary Stable Distribution. To this end we assume that

there exists a uniquex_∗∈R^N such thatb(x_∗)=max

x∈R^Nb(x). (44)

Theorem 4.2. Assume(7), (41)–(42) and (44). Then, as t→ ∞, the population reaches the Evolutionary Stable Distributionρ_∗δ(· −x_∗), i.e.,

n(·, t )−

t→∞ρ_∗δ(· −x_∗) in the sense of measures, (45)

with

ρ_∗= 1 ψ (x_∗)

1 0

I

b(x_∗), s ds.

Proof of Theorem 4.1. It follows easily from(40),(42)and the arguments in[1]that

Im+O(ε)Iε(t)IM+O(ε). (46)

Define next Fε(t):=

b(x)ψ (x)n_ε(x, t) dx

I_ε(t) ,

and note that

b_mF_ε(t)b_M. (47)

(14)

We next prove thatF_ε∈BVloc(0,∞)uniformly inε. Indeed, using(7),(39)and(46), we find d

dtF_ε(t)=I_ε⁻²

I_ε

n_ε,tbψ dx−

n_ε,tψ dx

n_εbψ dx

=I_ε⁻²

Iε εnε+ε⁻¹nε

bB

t ε, Iε

−D t

ε, Iε

bψ dx

−

n_εbψ dx εn_ε+ε⁻¹n_ε

bB t

ε, I_ε

−D t

ε, I_ε

ψ dx

=O(ε)+ εI_ε²₋1

B t

ε, I_ε

n_εψ dx·

n_εb²ψ dx−

n_εbψ dx 2

O(ε). (48)

Then(47)and(48)yield that, for eachT >0, there existsC=C(T ) >0 such that T

0

d dtFε

dtC.

It follows that, along subsequencesε→0, theF_ε’s converge a.e. and inL¹to someF. To conclude we need a result similar to the one ofLemma 1.1

Lemma 4.3. Assume (7) and (41)–(42). For all t ∈ R, there exists a unique, 1-periodic solution I(t,·)∈ C¹(R,[I_m,I_M])to

⎧⎨

⎩ d dsI

F (t), s

=I

F (t), s F (t )B

s,I

F (t), s

−D s,I

F (t), s , I

F (t ),0

=I F (t ),1

.

(49)

Moreover, for allT >0and asε→0,_T

0 |I_ε(t)−I(F (t),_ε^t)|dt→0.

The first claim is proved as inLemma 1.1. We postpone the proof of the second assertion to the end of this section.

Using(7),(41)–(42)and following[1], we show that theu_ε’s are bounded and locally Lipschitz continuous uniformly inεand, hence, converge along subsequencesε→0 to a solutionuof

u_t= |D_xu|²+ 1 0

B s,I

F (t), s

ds b(x)− 1 0

D s,I

F (t), s ds.

SinceI(F,·)is a periodic solution to(49), we have 1

0

B s,I

F (t), s

ds F (t )− 1 0

D s,I

F (t), s ds=0.

It follows that

u_t= |D_xu|²+R x, F (t) with

R x, F (t)

= 1 0

D s,I

F (t), s ds

b(x) F (t )−1

.

The last claim ofTheorem 4.1can be proved using(20),(43)and following[12]. 2