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Long time evolutionary dynamics of phenotypically structured populations in time periodic environments
Susely Figueroa Iglesias, Sepideh Mirrahimi
To cite this version:
Susely Figueroa Iglesias, Sepideh Mirrahimi. Long time evolutionary dynamics of phenotypically
structured populations in time periodic environments. SIAM Journal on Mathematical Analysis,
Society for Industrial and Applied Mathematics, 2018, �10.1137/18M1175185�. �hal-01727549v2�
Long time evolutionary dynamics of phenotypically structured populations in time-periodic environments
Susely Figueroa Iglesias
∗, Sepideh Mirrahimi
†January 4, 2019
Abstract
We study the long time behavior of a parabolic Lotka-Volterra type equation considering a time-periodic growth rate with non-local competition. Such equation describes the dynamics of a phenotypically struc- tured population under the effect of mutations and selection in a fluctuating environment. We first prove that, in long time, the solution converges to the unique periodic solution of the problem. Next, we des- cribe this periodic solution asymptotically as the effect of the mutations vanish. Using a theory based on Hamilton-Jacobi equations with constraint, we prove that, as the effect of the mutations vanishes, the solu- tion concentrates on a single Dirac mass, while the size of the population varies periodically in time. When the effect of the mutations is small but nonzero, we provide some formal approximations of the moments of the population’s distribution. We then show, via some examples, how such results could be compared to biological experiments.
Keywords: Parabolic integro-differential equations; Time-periodic coefficients; Hamilton-Jacobi equation with constraint; Dirac concentrations; Adaptive evolution.
AMS subject classifications: 35B10; 35B27; 35K57; 92D15
1 Introduction
1.1 Model and motivations
The purpose of this article is to study the evolutionary dynamics of a phenotypically structured population in a time-periodic environment. While the evolutionary dynamics of populations in constant environments are widely studied (see for instance [10, 6, 7, 31, 24]), the theoretical results on varying environments remain limited (see however [22, 28]). The variation of the environment may for instance come from the seasonal effects or a time varying administration of medications to kill cancer cells or bacteria. Several questions arise related to the time fluctuations. Could a population survive under the fluctuating change? How the population size will be affected? Which phenotypical trait will be selected? What will be the impact of the variations of the environment on the population’s phenotypical distribution?
Several frameworks have been used to study the dynamics of populations under selection and mutations.
Game theory has been one of the first approaches to study evolutionary dynamics [15, 32]. Adaptive dynamics which is a theory based on the stability of dynamical systems allows to study evolution under rare mutations [8, 9]. Integro-differential models are used to study evolutionary dynamics of large populations (see for instance [4, 7, 10, 23]). Probabilistic tools allow to study populations of small size [5] and also to derive the above models
∗Institut de Math´ematiques de Toulouse; UMR 5219, Universit´e de Toulouse; CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France; E-mail: Susely.Figueroa@math.univ-toulouse.fr
†Institut de Math´ematiques de Toulouse; UMR 5219, Universit´e de Toulouse; CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France; E-mail: Sepideh.Mirrahimi@math.univ-toulouse.fr
in the limit of large populations [6].
Here, we are interested in the integro-differential approach. We study in particular the following Lotka-Volterra type model
∂tn(t, x)−σ∆n(t, x) =n(t, x)[a(t, x)−ρ(t)], (t, x)∈[0,+∞)×Rd, ρ(t) =
Z
Rd
n(t, x)dx, n(t= 0, x) =n0(x).
(1)
Here, n(t, x) represents the density of individuals with traitxat time t. The mutations are represented by a Laplace term with rateσ. The terma(t, x) is a time-periodic function, corresponding to the net growth rate of individuals with traitxat timet. We also consider a death term due to competition between the individuals, whatever their traits, proportional to the total population sizeρ(t).
A main part of our work is based on an approach using Hamilton-Jacobi equations with constraint. This approach has been developed during the last decade to study asymptotically the dynamics of populations under selection and small mutations. There is a large literature on this approach. We refer for instance to [24, 29]
where the basis of this approach for problems coming from evolutionary biology were established. Note that re- lated tools were already used to study the propagation phenomena for local reaction-diffusion equations [11, 12].
Our work follows an earlier article on the analysis of phenotype-structured populations in time-varying en- vironments [28]. In [28], the authors study a similar equation to (1) using also the Hamilton-Jacobi approach, but with a different scaling than our paper. They indeed obtain a homogenization result by simultaneously accelerating time and letting the size of the mutations vanish. In this paper, we study first a long time limit of this equation and next we describe asymptotically such long time solutions as the effect of the mutations vanishes. Our scaling, being motivated by biological applications (see Section 6), leads to a different qualitative behavior of solutions and requires a totally different mathematical analysis.
1.2 Assumptions
To introduce our assumptions, we first define a(x) = 1
T Z T
0
a(t, x)dt.
We then assume thata(t, x) is a time-periodic function with periodT, andC3with respect to x, such that a(t, x) =a(t+T, x), ∀(t, x)∈R×Rd, and∃d0>0 :ka(t,·)kL∞(Rd)≤d0 ∀ t∈R. (H1) Moreover, we suppose that there exists a uniquexm which satisfies for some constantam,
0< am≤max
x∈Rd
a(x) =a(xm). (H2)
In order to guarantee that the initial condition do not explode, we make the following assumption
0≤n0(x)≤eC1−C2|x|, ∀x∈Rd, (H3) for some positive constantsC1,C2.
Furthermore, just for Section 2, that is the case withσ= 0, we assume additionally that H =
∂2a
∂xi∂xj
(xm)
i,j
is negative definite, (H4)
i.e., its eigenvalues are all negative. Also, let us suppose that there exist some positive constantsδandR0such that
a(t, x)≤ −δ, for allt≥0, and|x| ≥R0. (H5) Finally, letM and d1 be positive constants, it is assumed again for the case of no mutations, that
kn0kW3,∞ ≤M, kakW3,∞ ≤d1. (H6)
1.3 Main results
We begin the qualitative study, with a simpler case, whereσ= 0, which means there is no mutation. The model reads as follows
∂tn(t, x) =n(t, x)[a(t, x)−ρ(t)], (t, x)∈[0,+∞)×Rd, ρ(t) =
Z
Rd
n(t, x)dx, n(t= 0, x) =n0(x).
(2)
Our first result is the following.
Proposition 1 (caseσ= 0)
Assume (H1)-(H6). Letn be the solution of (2). Then,
(i) ast→+∞,kρ(t)−%(t)ke L∞ →0, where%(t)e is the unique positive periodic solution of equation ( d%e
dt =%(t) (a(t, xe m)−%(t))e , t∈(0, T),
%(0) =e %(Te ),
(3) given by
%(t) =e
1−exp
"
− Z T
0
a(s, xm)ds
#
exp
"
− Z T
0
a(s, xm)ds
#Z t+T t
exp Z s
t
a(θ, xm)dθ
ds
. (4)
(ii) Moreover, n(t, x)
ρ(t) converges weakly in the sense of measures toδ(x−xm)ast→+∞. As a consequence, n(t, x)−%(t)δ(xe −xm)*0 ast→+∞,
in the sense of measures.
This result implies that the trait with the highest time average of the net growth rate over the time interval [0, T], will be selected in long time, while the size of the population oscillates with environmental fluctuations.
To present our results for problem (1), we first introduce the following parabolic eigenvalue problems ∂tp−σ∆p−a(t, x)p=λp, in [0,+∞)×Rd,
0< p: T−periodic, (5)
∂tpR−σ∆pR−a(t, x)pR=λRpR, in [0,+∞)×BR, pR= 0, on [0,+∞)×∂BR, 0< pR: T−periodic,
(6) where BR is the ball in Rd centered at the origin with radius R > 0. It is known that (see [14]) if a ∈ L∞([0,+∞)×BR), then there exists a unique principal eigenpair (λR, pR) for (6) with kpR(0,·)kL∞(BR) = 1.
Moreover, as R→+∞,λR&λand pR converges along subsequences top, with (λ, p) solution of (5) (see for instance [17]).
We next assume a variant of hypothesis (H5), that is, there exist positive constants,δandR0such that a(t, x) +λ≤ −δ, for all 0≤t, andR0≤ |x|. (H5σ) Under the above additional assumption, which means that “a” takes small values at infinity, the eigenpair (λ, p) is also unique, (see Lemma 6).
We next define theT−periodic functionsQ(t) andP(t, x) as follows Q(t) =
R
Rda(t, x)p(t, x)dx R
Rdp(t, x)dx , P(t, x) = p(t, x) R
Rdp(t, x)dx. (7)
We deduce from previous Proposition that if and only ifRT
0 Q(t)>0, then there exists a unique positive periodic solutioneρ(t) for the problem
( dρe
dt =ρe[Q(t)−ρ]e, t∈(0, T), ρ(0) =e ρ(Te ).
We can then describe the long time behavior of the solution of (1) Proposition 2 (caseσ >0, long time behavior)
Assume (H1),(H2),(H3)and (H5σ). Letnbe the solution of (1), then (i) if λ≥0then the population will go extinct, i.e. ρ(t)→0,ast→ ∞, (ii) ifλ <0then |ρ(t)−ρ(t)| →e 0, ast→ ∞.
(iii) Moreover
n(t, x)
ρ(t) −P(t, x) L∞
−→0, ast→ ∞. Consequently we have, ast→ ∞
kn(t,·)−ρ(t)Pe (t,·)kL∞ →0, ifλ <0 and knkL∞→0, ifλ≥0. (8) Remark 3 Assuming (H2) implies that λ <0, providedσis small enough.
We prove this remark in Lemma 9.
Proposition 2 guarantees, whenλ <0, the convergence inL∞−norm of the solutionn(t, x) of the equation (1) to the periodic functionen(t, x) =ρ(t)P(t, x) and it is not difficult to verify thate enis in fact a solution of (1).
We next describe the periodic solutionen, asymptotically as the effect of mutations is small. To this end, with a change of notation, we takeσ=ε2 and study (nε, ρε), the unique periodic solution of the following equation
∂tnε−ε2∆nε=nε[a(t, x)−ρε(t)], (t, x)∈[0,+∞)×Rd, ρε(t) =
Z
Rd
nε(t, x)dx, nε(0, x) =nε(T, x).
(9)
We expect thatnε(t, x) concentrates as a Dirac mass asε→0.
In order to study the limit ofnε, as ε→0, we make the Hopf-Cole transformation nε= 1
(2πε)d/2expuε
ε
, (10)
which allows us to prove
Theorem 4 (caseσ=ε2, asymptotic behavior)
Let nε solve (9)and assume (H1),(H2)and (H5σ). Then (i) As ε→0, we have
kρε(t)−%(t)ke L∞ →0, and nε(t, x)−%(t)δ(xe −xm)*0, (11) point wise in time, weakly inxin the sense of measures, with %(t)e given by (4).
(ii) Moreover asε→0,uεconverges locally uniformly to a functionu(x)∈C(R), the unique viscosity solution to the following equation
−|∇u|2= 1 T
Z T 0
(a(t, x)−%(t))dt,e x∈Rd, max
x∈Rd
u(x) =u(xm) = 0.
(12)
In the case x∈R,uis indeed a classical solution and is given by u(x) =−
Z x xm
p−a(x0) +% dx0
(13) where %= T1 RT
0 %(t)dt.e
To prove Theorem 4, we first prove some regularity estimates onuεand then pass to the limit in the viscosity sense using the method of perturbed test functions. We finally show that (12) has a unique solution, and hence all the sequence converges. Note that in order to prove regularity estimates onuε, a difficulty comes from the fact thatuε is time-periodic and one cannot use, similarly to previous related works [3, 27], the bounds on the initial condition to obtain such bounds for all time and further work is required.
1.4 Some heuristics and the plan of the paper
We next provide some heuristic computations which allow to better understand Theorem 4, but also suggest an approximation of the population’s distributionnε, when εis small but nonzero.
Replacing (10) in (9), we first notice thatuε solves 1
ε∂tuε−ε∆uε = |∇uε|2+a(t, x)−ρε(t), (t, x)∈[0,+∞)×Rd,
uε(t= 0, x) = u0ε(x) =εlnn0ε(x). (14)
We then write formally an asymptotic expansion foruε andρεin powers ofε
uε(t, x) =u(t, x) +εv(t, x) +ε2w(t, x) +o(ε2), ρε(t) =ρ(t) +εκ(t) +o(ε), (15) where the coefficients of the developments are time-periodic.
We substitute in (14) and organize by powers ofε, that is 1
ε(∂tu(t, x)) +ε0
∂tv(t, x)− |∇u|2−a(t, x) +ρ(t)
+ε[∂tw−∆u−2∇u· ∇v+κ(t)] +o(ε2) = 0.
From here we obtain
∂tu(t, x) = 0⇔u(x, t) =u(x), and
∂tv(t, x) =|∇u|2+a(t, x)−ρ(t).
Integrating this latter equation int∈[0, T], we obtain that 0 =
Z T 0
|∇u|2dt+ Z T
0
a(t, x)dt− Z T
0
ρ(t)dt,
because of theT−periodicity ofv. This implies that
−|∇u|2= 1 T
Z T 0
(a(t, x)−ρ(t))dt,
which is the first equation in (12). Keeping next the terms of orderεwe obtain that
∂tw−∆u= 2∇u· ∇v−κ(t), and again integrating in [0, T] we find
−∆u= 2 T∇u
Z T 0
∇vdt−κ, with κ= 1 T
Z T 0
κ(t)dt.
Evaluating the above equation atxm we obtain that
∆u(xm) =κ.
Then, using the averaged coefficients a(x) = T1RT
0 a(t, x)dt and ρ = T1RT
0 ρ(t)dt, we deduce, combining the above computations, thatv(t, x) satisfies
∂tv = a(t, x)−a(x)−ρ(t) +ρ,
−∆u = 2 T
Z T 0
∇u· ∇vdt−κ, (16)
which allows to determinev.
We will use these formal expansions in Section 5, to estimate the moments of the population’s distribution using the Laplace’s method of integration. Note that such approximations were already used to study the phenotypical distribution of a population in a spatially heterogeneous environment [25, 13] (see also [26] where such type of approximation was first suggested). We next show, via two examples, how such results could be interpreted biologically. In particular, our work being motivated by a biological experiment in [18], we suggest a possible explanation for a phenomenon observed in this experiment.
The paper is organized as follows. In Section 2 we deal with problem (2) and prove Proposition 1. In Section 3 we study the long time behavior of (1) and provide the proof of Proposition 2. Next, in Section 4 we study the asymptotic behavior ofnεasε→0, and prove Theorem 4. In Section 5, we use the above formal arguments to estimate the moments of the population’s phenotypical distribution. Finally we use these results in Section 6 to study two biological examples considering two different growth rates.
2 The case with no mutations
In this section we study the qualitative behavior of (2), whereσ= 0, and provide the proof of Proposition 1.
To this end, we defineN(t, x) =n(t, x)eR0tρ(s)dswhich solves
∂tN =a(t, x)N(t, x).
From the periodicity ofaand the Floquet theory we obtain thatN has the following form N(t, x) =eµ(x)tp0(t, x), with p0(0, x) =p0(T, x), and µ(x) =a(x) = 1
T Z T
0
a(s, x)ds.
2.1 Long time behavior of ρ
In this subsection we prove Proposition 1 (i).
Integrating equation (2) with respect toxwe obtain d
dtρ(t) = Z
Rd
n(t, x)a(t, x)dx−ρ(t)2=ρ(t) Z
Rd
n(t, x)a(t, x)
ρ(t) dx−ρ(t)
. (17)
Then we claim the following Lemma that we prove at the end of this subsection.
Lemma 5 Assume (H1)-(H3)and (H6)then
Z
Rd
n(t, x)a(t, x)
ρ(t) dx−a(t, xm)
−→0, ast→+∞.
Proof. (Proposition 1)(i) From Lemma 5,ρ(t) satisfies
d
dtρ(t) =ρ(t)[a(t, xm) + Σ(t)−ρ(t)],
where Σ(t) → 0 as t → ∞. In order to prove the convergence to a periodic function, we adapt a method introduced in [21].
After a standard substitutionκ(t) = 1/ρ(t) in order to linearize the latter equation, and integration with the help of an integrating factor, the solutionρcan be written as follows
1
ρ(t) = exp
− Z t
0
(a(s, xm) + Σ(s))ds 1 ρ0 +
Z t 0
exp Z s
0
(a(θ, xm) + Σ(θ))dθ
ds
.
We then write 1
ρ((k+ 1)T) as function of 1
ρ(kT), that is 1
ρ((k+ 1)T)= exp −
Z (k+1)T kT
(a(s, xm) + Σ(s))ds
! 1 ρ(kT)+
Z (k+1)T kT
eRkTs (a(θ,xm)+Σ(θ))dθds
! ,
and we obtain a recurrent sequence forρk =ρ(kT) as follows 1
ρk+1
=ξk+ηk ρk
, where
ηk = exp −
Z (k+1)T kT
(a(s, xm) + Σ(s))ds
!
, ξk=ηk
Z (k+1)T kT
exp Z s
kT
(a(θ, xm) + Σ(θ))dθ
ds.
From the T−periodicity of aand the fact that Σ(t)→0 we obtain easily thatηk →η and ξk →ξas k→ ∞, where
η= exp − Z T
0
a(t, xm)dt
!
, ξ=η Z T
0
exp Z t
0
a(θ, xm)dθ
dt.
From these convergences we have that for all >0, there existsK such that ξ−≤ξk≤ξ+, η−≤ηk≤η+, ∀k≥K, which implies
ξ−+η− ρk ≤ 1
ρk+1 ≤ξ++η+ ρk . Noteκk= 1
ρk
then
ξ−+ (η−)κk≤κk+1≤ξ++ (η+)κk. (18) From the inequality at the right hand side of (18), denotingκ∗= lim sup
k→+∞
κk, we obtain κ∗ ≤ξ++ (η+)κ∗, ∀ >0.
Then thanks to assumption (H2), which impliesη <1, we have κ∗≤ ξ
1−η.
Analogously, from the left hand side inequality in (18), and denotingκ∗= lim inf
k→+∞κk, we deduce that κ∗≥ ξ
1−η. Sinceκ∗≤κ∗, we obtain
κ∗=κ∗= lim
k→+∞κk= ξ 1−η. Going back to variableρk, it implies
k→∞lim ρk =1−η ξ . Finally we can make a translation fromρ0 to obtain
e%(t) = lim
k→∞ρ(kT+t), with%(t) the unique periodic solution of equation (3) given by (4).e
Finally we prove Lemma 5.
Proof. LetK={x∈Rd:|x| ≤R0}forR0as in assumption (H5) then Z
Rd
n(t, x)a(t, x) ρ(t) dx=
Z
Kc
n(t, x)a(t, x)dx+ Z
K
p0(t, x)etµ(x)a(t, x)dx Z
Kc
n(t, x)dx+ Z
K
p0(t, x)etµ(x)dx .
Thanks to (2) and assumptions (H1), (H3) and (H5) we can control the integral terms taken outside the compact setK as follows
Z
Kc
n(t, x)a(t, x)dx≤ kakL∞e−δt Z
Kc
n0(x)dx≤Ce−δt Z
Kc
e−C2|x|dx−→0, ast→ ∞, (19) and an analogous inequality holds forR
Kcn(t, x)dx. Next for the remaining terms, we use Taylor expansions around the pointx=xm until third order terms, forxmgiven by (H2), that is,
I(t) = Z
K
p0(t, x)etµ(x)a(t, x)dx
= Z
K
a(t, xm) +∇a(t, xm)(x−xm) +1 2
t(x−xm)D2a(t, xm)(x−xm) +O(|x−xm|3)
·
p0(t, xm) +∇p0(t, xm)(x−xm) +1 2
t(x−xm)D2a(t, xm)(x−xm) +O(|x−xm|3)
·exp t
2
t(x−xm)D2µ(xm)(x−xm) +tO(|x−xm|3)
dx, wheretxindicates the transpose vector ofx.
We organizeI(t) by powers of|x−xm|as below I0(t) = a(t, xm)p0(t, xm)
Z
K
et2t(x−xm)D2µ(xm)(x−xm)dx, I1(t) =
Z
K
[a(t, xm)∇p0(t, xm) +p0(t, xm)∇a(t, xm)] (x−xm)e2tt(x−xm)D2µ(xm)(x−xm)dx= 0, I2(t) =
Z
K
t(x−xm) 1
2a(t, xm)D2p0(t, xm) +t∇a(t, xm)∇p0(t, xm) +1
2p0(t, xm)D2a(t, xm)
(x−xm)
·et2t(x−xm)D2µ(xm)(x−xm)o dx, I3(t) =
Z
K
(1 +t)O(|x−xm|3)et2t(x−xm)D2µ(xm)(x−xm)dx.
By performing a change of variables asy=√
t(x−xm) we obtain for the non null integrals I0(t) = 1
td/2a(t, xm)p0(t, xm) Z
Ket
e12tyD2µ(xm)ydy,
I2(t) = 1 td/2+1
Z
Ket
ty 1
2a(t, xm)D2p0(t, xm) + t∇a(t, xm)∇p0(t, xm) +1
2p0(t, xm)D2a(t, xm)
ye12tyD2µ(xm)ydy,
I3(t) = 1 +t td+32
Z
Ket
O(|y|3)e12tyD2µ(xm)ydy≈O 1
td+12
,
withKet={x∈Rd:|x| ≤√ tR0}.
Note thatI1(t) = 0 because it is the integral of an odd function in a symmetric interval. Moreover we obtain the approximation for I3(t) thanks to assumption (H6), which implies that the derivatives of µ and a, and consequentlyp0, up to order 3, are globally bounded.
Moreover, if we denoteA(t) = αij(t)
i,j the periodic matrix inside the crochets inI2(t), i.e A(t) =1
2a(t, xm)D2p0(t, xm) + t∇a(t, xm)∇p0(t, xm) +1
2p0(t, xm)D2a(t, xm),
we obtain, thanks to the periodicity of a and p0, that all the coefficients of A(t) are bounded as t → ∞.
Moreover,
tyA(t)y =
d
X
i,j=1
αij(t)yiyj
≤
d
X
i,j=1
|αij(t)||yi||yj| ≤C|y|2, for someC >0.
Then forI2 we have, by using (H4)
|I2(t)| ≤ C td/2+1
Z
Ket
|y|2e12tyD2µ(xm)ydy≈O 1
td/2+1
as t→ ∞.
By arguing in the same way we obtain for the denominator term Z
K
p0(t, x)etµ(x)dx= 1
td/2p0(t, xm) Z
Ket
e12tyD2µ(xm)ydy+O 1
td+12
,
and we conclude multiplying bytd/2and using again (H4)
Z
Rd
n(t, x)a(t, x) ρ(t) dx=
a(t, xm)p0(t, xm) Z
Ket
ey
2
2D2µ(xm)dy+O 1
√t
p0(t, xm) Z
Ket
ey
2
2D2µ(xm)dy+O 1
√t
=a(t, xm) +O 1
√t
,
fortlarge enough.
2.2 Convergence to a Dirac mass
In this subsection we prove Proposition 1 (ii).
Proof. (ii)
We begin by defining
f(t, x) = n(t, x)
ρ(t) = p0(t, x)eµ(x)t Z
Rd
p0(t, x)eµ(x)tdx .
Therefore, since Z
Rd
f(t, x)dx= 1, there exists a sub-sequence (ftk) that converges weakly to a measureν, i.e.
Z
Rd
ftkϕdx→ Z
Rd
νϕdx ∀ϕ∈Cc(Rd).
We first prove Z
Ωcζ
n(tk, x)
ρ(tk) ϕ(x)dx−→0 as tk→ ∞ ∀ϕ: suppϕ⊂Ωcζ, where Ωζ ={x∈Rd:|x−xm|< ζ}. (20) We can rewrite the above integral as below
Z
Ωcζ
f(t, x)ϕ(x)dx= 1 I(t)
Z
Ωcζ
p0(t, x)eµ(x)tϕ(x)dx, where
I(t) = Z
Rd
p0(t, y)eµ(y)tdy.
We estimateI(t) using the Laplace’s method for integration and the assumption (H4). It follows I(t)∼ etµ(xm)p0(t, xm)
p|detH|
2π t
d/2
as t→ ∞,
withµ(xm) the strict maximum that is attained at a single point thanks to assumption (H2), andH given by (H4). Sincep0(t, x) is positive and periodic with respect tot, there exist positive constantsK1,K2such that
K1
etµ(xm)
td/2 ≤ I(t)≤K2
etµ(xm) td/2 . Next we note that
td/2e−tµ(xm) Z
Ωcζ
p0(t, x)eµ(x)tϕ(x)dx−→0, as t→+∞,
sinceµ(x)−µ(xm)≤ −β for some β >0, andϕhas compact support, which immediately implies (20).
We deduce from (20) by lettingζ→0, that as t→+∞along subsequences n(t, x)
ρ(t) * ωδ(x−xm).
We then prove thatω= 1, and hence all the sequence converges to the same limit.
LetKR={x∈Rd:|x| ≤R}, forR >0.
We can write using (2) that
n(t, x) =n0(x)eR0t(a(s,x)−ρ(s))ds. (21) Thanks to assumption (H3) and (H5), forR0≤R, by making an analogous analysis to (19) we obtain
Z
KRc
n(t, x)dx→0 as t→+∞.
Moreover, thanks to Section 2.1, we know thatρconverges to%, a periodic and positive function. Therefore, ine long time,ρis bounded from below and above by positive constants. We deduce that
Z
KcR
f(t, x)dx= Z
KRc
n(t, x)
ρ(t) dx→0 as t→ ∞.
Thanks to the above convergence and the fact that Z
Rd
f(t, x)dx = 1, we deduce that ∀ζ > 0 there exists a compact setK andt0>0 such that, for allt≥t0
1−ζ≤ Z
K
f(t, x)dx.
Moreover, we know that f converges weakly to a measure ωδ(x−xm), thus choosing a smooth compactly supported functionϕ such thatϕ(x) = 1 if x∈ K, ϕ(x) = 0 if x∈(K0)c for another compactK0 such that K K0 and 0< ϕ(x)<1 forx∈K0\K, we obtain
Z
Rd
f(t, x)ϕ(x)dx= Z
K
f(t, x)dx+ Z
Kc
f(t, x)ϕ(x)dx −→ω,
where the first term in the RHS is bigger than 1−ζ and the second one is positive. It follows that 1−ζ≤ω, for all 0< ζ <1 and henceω= 1. We conclude that
n(t, x)
ρ(t) * δ(x−xm) as t→+∞, which implies, using the convergence result forρ,
n(t, x)−%(t)δ(xe −xm)*0 as t→+∞, weakly in the sense of measures.
3 The case with mutations: long time behavior
In this section we study (1) withσ >0 and provide the proof of Proposition 2.
To this end, we first introduce a linearized problem. Letnsolve (1), we definem(t, x) =n(t, x)eR0tρ(s)ds which solves
∂tm(t, x)−σ∆m(t, x) = m(t, x)a(t, x), (t, x)∈[0,+∞)×Rd,
m(t= 0, x) = n0(x), (22)
and associate to (22) the parabolic eigenvalue problem (5). In subsection 3.1, we provide a convergence result for (22). Next, using this property, we prove Proposition 2 in subsection 3.2.
3.1 A convergence result for the linearized problem
In this section we provide a convergence result for the linearized problem.
Lemma 6 Assume (H1),(H3) and (H5σ). Then,
(i) there exists a unique principal eigenpair(λ, p)for the problem (5), with p∈L∞(R×Rd), up to normali- zation of p. Moreover, the eigenfunction p(t, x)is exponentially stable, i.e. there exist a constant α >0 such that the solutionm(t, x)to problem (22)satisfies
km(t, x)eλt−αp(t, x)kL∞(Rd)→0 ast→ ∞, (23) exponentially fast.
(ii) Moreover, letδandR0 given by (H5σ), then we have p(t, x)≤ kpkL∞e−
√δ
σ(|x|−R0), ∀(t, x)∈[0,+∞)×Rd. (24)
Proof. The proof of (i).
We will apply a result from [17] to equation
∂tme −σ∆me =m[a(t, x) +e λ+δ], (t, x)∈R×Rd, (25) withδgiven in assumption (H5σ). This result allows to show that there exists a unique principal eigenpair for the equation (25), with an eigenfunction which is exponentially stable.
Consider the problem
( ∂tφeR−σ∆φeR = φeR[a+λ+δ], inR×BR,
φeR = 0, onR×∂BR. (26)
Thanks to (H1) and (H5σ) we can chooseRandδ >0 such that there existsdδ >0 ka(t, x) +λ+δkL∞([0,+∞)×BR)< dδ, a(t, x) +λ+δ <0, ∀ |x| ≥R0.
Note thatφeR=pRet(δ−λR+λ)is a positive entire solution to (26). Moreover, it satisfies the hypothesis (H1) of [17], that is
kφeR(t,·)kL∞(BR) kφeR(s,·)kL∞(BR)
= kpR(t,·)kL∞(BR) kpR(s,·)kL∞(BR)
e(δ−λR+λ)(t−s)≥Ce(δ−λR+λ)(t−s), t≥s,
withδ−λR+λ >0 forR large enough.
Therefore Theorem 2.1 (and its generalization Theorem 9.1) in [17] implies that there exists a unique positive entire solutionφefor problem (25), which is given by
φ(t, x) = lime
R→∞φeR(t, x).
Moreover, forp=φee −δt we obtain
p(t, x) = lim
R→∞pR(t, x),
and sincepR is the solution of (6), thenpis a positive periodic eigenfunction to (5).
Furthermore, Theorem 2.2 in [17] implies also that
km(t, x)e −αeφ(t, x)kL∞(Rd)
kφ(t,e ·)kL∞(Rd)
−→0, exponentially fast ast→ ∞.
Noting that every solutionmof problem (22) can be written asm=mee −λt−δt, we obtain km(t, x)eλt−αp(t, x)kL∞(Rd)−→0 as t→+∞,
and this convergence is also exponentially fast.
The proof of (ii).
Next we prove (24) following similar arguments as in the proof of Lemma 2.4 in [30]. Letea(t, x) =a(t, x) +λ thenpis a positive bounded solution of the following equation
∂tp−σ∆p=pea(t, x), inR×Rd. (27)
Note that we have definedpin (−∞,0] by periodic prolongation. LetkpkL∞(R×Rd)=M. We define ζ(t, x) =M e−δ(t−t0)+M e−ν(|x|−R0),
whereν = qδ
σ andR0is given by (H5σ). One can verify that
M ≤ζ(t, x) if|x|=R0ort=t0. Furthermore if|x|> R0 ort > t0 evaluating in (27) shows
∂tζ−σ∆ζ−ζea(t, x) =M e−δ(t−t0)(−δ−ea(t, x)) +M e−ν(|x|−R0)
−σν2−ea(t, x) +σνd−1
|x|
≥0, sinceea(t, x)≤ −δthanks to assumption (H5σ). Thus ζis a supersolution of (27) on
Q0={(t, x)∈(t0,∞)×Rd;|x|> R0},
which dominatespon the parabolic boundary ofQ0. Applying the maximum principle toζ−p, we obtain p(t, x)≤M e−δ(t−t0)+M e−ν(|x|−R0), |x| ≥R0, t∈(t0,∞).
Taking the limitt0→ −∞yields
p(t, x)≤M e−ν(|x|−R0), |x| ≥R0, t≤+∞, forν=
qδ
σ. We conclude thatpsatisfies (24).
3.2 The proof of Proposition 2
To prove Proposition 2 we first prove the following Lemmas.
Lemma 7 Assume (H1) and (H3) and letC3=σC22+d0 then the solutionn(t, x)to equation (1)satisfies n(t, x)≤exp (C1−C2|x|+C3t), ∀(t, x)∈(0,+∞)×Rd.
Proof. Define the functionen(t, x) = exp (C1−C2|x|+C3t).
We prove that n≤en. To this end we proceed by a comparison argument. One can easily verify that forC3 defined above, we have the following inequality
∂tne−σ∆en−[a(t, x) +ρ(t)]en=e(C1−C2|x|+C3t)
C3−σC22+σC2(d−1)
|x| −a(t, x) +ρ(t)
≥0, a.e inR×Rd. Moreover, we have for t = 0, n(0, x) ≤ en(0, x) thanks to assumption (H3). We can then apply a Maximum Principle, in the class ofL2 functions, and we conclude that
n(t, x)≤en(t, x), ∀(t, x)∈(0,+∞)×Rd.
Lemma 8 Assume (H1),(H3) and (H5σ)then
Z
Rd
n(t, x)a(t, x)
ρ(t) dx−Q(t)
−→0, ast→+∞, withQ(t) given by (7).
Proof. From (23), we obtain that
n(t, x)eR0tρ(s)ds+λt=αp(t, x) + Σ(t, x),
withkΣ(t, x)kL∞ →0 exponentially fast, ast→ ∞.
We define the compact setKt={x∈Rd:|x| ≤At}, for someA >>1 large enough and compute
1 ρ(t)
Z
Rd
n(t, x)a(t, x)dx= Z
Kt
αp(t, x)a(t, x)dx+ Z
Kt
Σ(t, x)a(t, x)dx+ Z
Ktc
(αp(t, x) + Σ(t, x))a(t, x)dx Z
Kt
αp(t, x)dx+ Z
Kt
Σ(t, x)dx+ Z
Ktc
(αp(t, x) + Σ(t, x)) dx
.
We then notice that Z
Kt
Σ(t, x)a(t, x)dx
≤ kakL∞kΣ(t,·)kL∞|Kt| →0 ast→ ∞,
sincekΣ(t,·)kL∞ converges exponentially fast to zero and the measure of Kt is at most algebraic int. Making the same analysis for
R
KtΣ(t, x)dx
it will just remain to prove that the integral terms taken outside the compact setKtvanish ast→+∞.
We have, trivially Z
Ktc
(αp(t, x) + Σ(t, x))a(t, x)dx
≤ kakL∞
Z
Kct
n(t, x)eR0t(ρ(s)+λ)dsdx .
Then we use Lemma 7 to obtain Z
Kct
n(t, x)eR0t(ρ(s)+λ)dsdx≤ Z
Ktc
eC1−C2|x|+M tdx≤eC1+M t Z
Ktc
e−C2|x|dx→0, ast→+∞.
forA > M large enough, where M ≥ρM +λ+C3.
Combining the last two inequalities we obtain that the integral terms taken outside the compact, vanish as t→+∞. This concludes the proof.
Proof of Proposition 2
Convergence ofρ.
By integrating equation (1) inx, we obtain that Z
Rd
∂tn(t, x)dx= Z
Rd
n(t, x)[a(t, x)−ρ(t)]dx, and using Lemma 8 we deduce that
dρ dt =ρ(t)
Z
Rd
n(t, x)a(t, x)
ρ(t) dx−ρ(t)
=ρ(t) [Q(t) + Σ0(t)−ρ(t)], where Σ0(t)→0 exponentially ast→ ∞, andQ(t) is given by (7).
Following similar arguments as in the proof of Proposition 1 we obtain|ρ(t)−eρ(t)| →0 ast→ ∞, withρethe unique solution of
( dρe
dt =ρ(t) [Q(t)e −eρ(t)], ρ(0) =e ρ(Te ),
provided Z T
0
Q(t)dt >0. Moreover if Z T
0
Q(t)dt≤0, thenρ(t)→0 ast→ ∞. Note also that
λ=−1 T
Z T 0
Q(t)dt. (28)
We consider, indeed, the eigenvalue problem (5) and integrate it inx∈Rd
∂t Z
Rd
p(t, x)dx= Z
Rd
a(t, x)p(t, x)dx+λ Z
Rd
p(t, x)dx.
We divide by Z
Rd
p(t, x)dxand integrate now in t∈[0, T], to obtain Z T
0
∂tR
Rdp(t, x)dx R
Rdp(t, x)dx dt= Z T
0
Q(t)dt+λT, which implies that
0 = ln Z
Rd
p(T, x)dx
−ln Z
Rd
p(0, x)dx
= Z T
0
Q(t)dt+λT, and hence (28).
This ends the proof of statements (i)−(ii) of Proposition 2.
Convergence of n ρ.
LetKt={x∈Rd:|x|< At}, forA > R0, as in the proof of Lemma 8, we can write n(t, x)
ρ(t) = αp(t, x) + Σ(t, x)
Z
Kt
(αp(t, x) + Σ(t, x))dx+ Z
Ktc
(αp(t, x) + Σ(t, x))dx .
Following similar arguments as in Lemma 8 we obtain that
n(t, x)
ρ(t) −P(t, x) L∞
−→0, withP(t, x) as in (7).
Consequently, whenλ <0 we obtain that
kn(t,·)−ρ(t)Pe (t,·)kL∞ −→0 ast→ ∞, and this concludes the proof of (iii).
4 Case σ << 1. Small mutations
In this section we chooseσ=ε2 and we prove that forεsmall enough, the principal eigenvalue λgiven in (5) is negative. As a consequence, thanks to Proposition 2, any solution of (9) converges to the unique periodic solution (nε, ρε). Next, we prove Theorem 4, which allows to characterizenε, as ε→0.
Consider now the problem (9) and let (λε, pε) be the eigenelements of problem (5) forσ=ε2, then we have the following result.
Lemma 9 Under assumption (H2)there exists λm>0andε0>0 such that for allε < ε0 we haveλε≤ −λm. Proof. We follow the proof for the case of bounded domains, given in [14].
ForR >0 defineBR:=B(xm, R) andaR(t) = min
x∈BR
a(t, x). Then we chooseR1 small enough such that
Z T 0
aR1(t)dt >0. (29)
This is possible thanks to (H2) and the continuity ofawith respect tox.
We first consider the periodic-parabolic Dirichlet eigenvalue problem on [0,+∞)×BR1,
∂tw−ε2∆w−aR1(t)w=µεw, in [0,+∞)×BR1, w= 0, on [0,+∞)×∂BR1, w: T−periodic int.
(30)
We calculateµεby the Ansatzw(t, x) =α(t)ϕ1(x) whereϕ1>0 is the principal eigenfunction of −∆ϕ1 = γ1ϕ1, inBR1,
ϕ1 = 0, on∂BR1, with principal eigenvalueγ1>0. By substituting in (30) we deduce that
α0(t)ϕ1+γ1ε2α(t)ϕ1−aR1(t)α(t)ϕ1=µεα(t)ϕ1, and consequently
α(t) =α(0) exp Z t
0
aR1(τ)dτ−(γ1ε2−µε)t
. Forwto beT−periodic we must have
Z T 0
aR1(τ)dτ−γ1ε2T+T µε= 0.
We deduce indeed that choosing
µε=γ1ε2− 1 T
Z T 0
aR1(t)dt,
we obtain the principal eigen-pair (w, µε) for (30). Next we consider the periodic-parabolic eigenvalue problem
∂tw−ε2∆w−a(t, x)w=λR1w, in [0,+∞)×BR1, w= 0, on [0,+∞)×∂BR1, w: T−periodic int.
SinceaR1(t)≤a(t, x) on [0,+∞)×BR1 we haveλR1≤µε(Lemma 15.5 [14]). By monotony of eigenvalues with respect to the domain we obtainλε≤λR1 ≤µε. Finally, thanks to (29) we conclude that there existλm>0, ε0>0 such that for allε≤ε0, we haveλε≤ −λm.
In the following subsections we provide the proof of Theorem 4. In Subsection 4.1, we give some global bounds for ρε. Next, in Subsection 4.2, we prove that (uε) is locally uniformly bounded, Lipschitz with respect to x and locally equicontinuous in time. In the last subsection we conclude the proof of Theorem 4 lettingεgoes to zero and describing the limits ofuε,nεandρε.
4.1 Uniform bounds for ρ
εIn this section we provide uniform bounds forρε.
Lemma 10 Assume (H1),(H5σ). Then for everyε >0, there exist positive constantsρm andρM such that
0< ρm≤ρε(t)≤ρM ∀t≥0. (31)
Proof. From equation (9) integrating inx∈Rd and using assumption (H1) we get dρε
dt = Z
Rd
nε(t, x)[a(t, x)−ρε(t)dx]≤ρε(t)[d0−ρε(t)]. (32) This implies that
ρε(t)≤ρM := max(ρ0ε, d0).
To obtain the lower bound we recall thatnε(t, x) =ρε(t)Pε(t, x), withρε(t) the unique periodic solution of dρε
dt =ρε(t)[Qε(t)−ρε(t)], and withQε(t) andPε(t, x) given by (7). From (4) we know that
ρε(t) =
1−exp
"
− Z T
0
Qε(s)ds
#
exp
"
− Z T
0
Qε(s)ds
#Z t+T t
exp Z s
t
Qε(θ)dθ
ds
. (33)
From Lemma 9, we note that,λε=−1 T
Z T 0
Qε(t)dt≤ −λm, thus
exp
"
− Z T
0
Qε(θ)dθ
#
≤e−T λm.
Also from (H1) and (7) we get Z t+T
t
exp Z s
t
Qε(θ)dθ
ds≤ Z t+T
t
ed0Tds=T ed0T. Combining the above inequalities with (33) we obtain
0< ρm:= 1
Te−d0T eλmT −1
≤ρε(t), ∀t≥0.
4.2 Regularity results for u
εIn this section we study the regularity properties of uε = εln (2πε)d/2nε
, where nε is the unique periodic solution of equation (9).
Theorem 11 Assume (H1),(H2)and (H5σ). Thenuεis locally uniformly bounded and locally equicontinuous in time in[0, T]×Rd. Moreover, for someD >0,ωε=√
2D−uε, is Lipschitz continuous with respect toxin (0,∞)×Rd and there exists a positive constant C such that we have the following
|∇ωε| ≤C, in[0,+∞)×Rd, (34)
∀R >0, sup
t∈[0,T], x∈BR
|uε(t, x)−uε(s, x)| →0 asε→0. (35) We prove this theorem in several steps.
4.2.1 An upper bound foruε
We recall from (8) thatnε(t, x) =ρε(t) pε(t, x) R
Rdpε(t, x)dx, where
∂tpε−ε2∆pε−a(t, x)pε=λεpε, inR×Rd, 0< pε: T −periodic,
kpε(0, x)kL∞(Rd)= 1.
(36)
Defineqε(t, x) =pε(t, xε), which satisfies
∂tqε−∆qε=aε(t, x)qε, inR×Rd,
qε: T−periodic, (37)
foraε(t, x) =a(t, xε) +λε. Note that aε is uniformly bounded thanks to the L∞−norm of a, which together with Lemma 9 implies that−d0≤λε≤ −λm.
Sincekpε(0, x)kL∞(Rd)= 1 we can choose x0 such that pε(0, x0) = 1. Moreoverqε is a nonnegative solution of (37) in (0,2T)×B(xε0,1). Let δ0, be such that 0< δ0< T, then we apply the Theorem 2.5 [16] which is an elliptic-type Harnack inequality for positive solutions of (37) in a bounded domain, and we have∀t∈[δ0,2T]
sup
x∈B(xε0,1)
qε(t, x)≤C inf
x∈B(xε0,1)
qε(t, x), whereC=C(δ0, d0). Returning topεthis implies
pε(t0, x0)≤ sup
y∈B(x0,ε)
pε(t0, y)≤Cpε(t0, x), ∀(t0, x)∈[δ0,2T]×B(x0, ε). (38) Since pε is T−periodic we conclude that the last inequality is satisfied ∀ t ∈ [0, T]. From (31), (38) and the upper bound (24) forpεwithσ=ε2, we obtain
nε(0, x)≤ρM
pε(0, x) R
Rdpε(0, x)dx ≤ CρMpε(0, x) R
B(x0,ε)pε(0, x0)dx ≤ρM
Cpε(0, x)
|B(x0, ε)| ≤C0ε−dexp
C0 1−C0
2|x|
ε ,
for allε≤ε0, withε0 small enough, where the constantC0 depends on ρM,kpkL∞(Rd)and the constantC in (38) andC10 andC20 depend on the constants of hypothesis (H5σ). Next we proceed with a Maximum Principle argument as in Lemma 7 to obtain for every (t, x)∈[0,+∞)×Rd andC3= (C20)2+d0,
nε(t, x)≤C0expC
0 1−C0
2|x|
ε +C3t.
From here and the periodicity ofuε, with an abuse of notation for the constants, we can write, for allε≤ε0 uε(t, x)≤C10 −C20|x|, ∀(t, x)∈[0,+∞)×Rd. (39) 4.2.2 A lower bound foruε
Using the bounds forain (H1) and forρεin (31) we obtain for Ce=d0+ρM
∂tnε−ε2∆nε≥ −Cne ε. Letn∗ε be the solution of the following heat equation
∂tn∗ε−ε2∆n∗ε+Cne ∗ε= 0, n∗ε(0, x) =n0ε,
