non-vanishing effects of mutations equilibria in heterogeneous environments and with A Hamilton–Jacobi method to describe the evolutionary C.R. Acad. Sci. Paris, Ser.I

Contents lists available atScienceDirect

C. R. Acad. Sci. Paris, Ser. I

www.sciencedirect.com

Mathematical analysis/Partial differential equations

A Hamilton–Jacobi method to describe the evolutionary equilibria in heterogeneous environments and with non-vanishing effects of mutations

Méthode de Hamilton–Jacobi pour décrire des équilibres évolutifs dans les environnements hétérogènes avec des mutations non évanescentes

Sylvain Gandon

^a

, Sepideh Mirrahimi

^b

aCentred’écologiefonctionnelleetévolutive(CEFE),UMRCNRS5175,34293Montpelliercedex5,France

bCNRS,Institutdemathématiques(UMRCNRS5219),UniversitéPaul-Sabatier,118,routedeNarbonne,31062Toulousecedex,France

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received11October2016

Acceptedafterrevision7December2016 Availableonline15December2016 PresentedbyHaïmBrézis

In this note, we characterize the solution to a system of elliptic integro-differential equationsdescribingaphenotypicallystructuredpopulationsubjecttomutation,selection, and migration. Generalizing an approach based on the Hamilton–Jacobi equations, we identify the dominant terms of the solution when the mutation term is small (but nonzero).Thismethodwasinitiallyused,fordifferentproblemsarisenfromevolutionary biology,toidentifytheasymptoticsolutions,whilethemutationsvanish,asasumofDirac masses.AkeypointisauniquenesspropertyrelatedtotheweakKAMtheory.Thismethod allowsustogofurtherthantheGaussianapproximationcommonlyusedbybiologists,and isanattempttofillthegapbetweenthetheoriesofadaptivedynamicsand quantitative genetics.

©^{2016Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r é s um é

Danscettenote,nousétudions unsystèmed’équationsintégro-différentielleselliptiques, décrivantunepopulationstructuréepartraitphénotypiquesoumiseàdesmutations,àla sélection età desmigrations. Nous généralisons uneapproche baséesur deséquations de Hamilton–Jacobi pour détérminer les termes dominants de la solution lorsque les effetsdesmutationssontpetits(maisnonnuls).Cetteméthodeétaitinitialementutilisée, pour différents problèmes venant de la biologie évolutive, pour identifier les solutions asymptotiques, lorsqueles effetsdes mutationstendent vers0,sous formede sommes de massesdeDirac. Un point-cléest unepropriétéd’unicité en rapportavec lathéorie deKAMfaible.Cetteméthodenouspermetd’allerau-delàdesapproximationsgaussiennes

E-mailaddresses:[email protected](S. Gandon),[email protected](S. Mirrahimi).

http://dx.doi.org/10.1016/j.crma.2016.12.001

1631-073X/©^{2016Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/4.0/).

habituellementutilisées parles biologistes,etcontribueainsi àrelier lesthéories dela dynamiqueadaptativeetdelagénétiquequantitative.

©^{2016Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

During the last decade, an approach basedon Hamilton–Jacobi equationswithconstraints hasbeen developedto de- scribetheasymptoticevolutionarydynamicsofphenotypicallystructured populations,inthelimitofvanishingmutations.

Mathematicalmodelingofsuchphenomenaleadstoparabolic(orellipticforthesteadycase)integro-differentialequations, whose solutions tend, asthe diffusion term vanishes, toward a sum ofDirac masses, corresponding to dominant traits.

TheseasymptoticsolutionscanbedescribedusingtheHamilton–Jacobiapproach.Thereisalargeliteratureonthismethod.

Wereferto[4,17,13]fortheestablishmentofthebasis ofthisapproachforproblemsfromevolutionarybiology.Notethat relatedtoolswerealreadyusedinthecaseoflocalequations(forinstanceKPPtypeequations)todescribethepropagation phenomena(see,forinstance,[8,5]).

In almost all the previous works, the Hamilton–Jacobi approach has been used to describe the limit of the solution, corresponding to thepopulation’s phenotypical distribution, asthe mutations steps vanish.However, from thebiological pointofview,itissometimesmorerelevanttoconsidernon-vanishingmutationsteps.Arecentwork[16]haspointedout that such toolscan alsobeused, forasimplemodel withhomogeneousenvironment, tocharacterizethe solution,while mutationstepsaresmall,butnonzero.Inthisnote,weshowhowsuchresultscanbeobtainedinamorecomplexsituation withaheterogeneousenvironment.

Ourpurposeinthisnoteistostudythesolutionstothefollowingsystem,forz∈R^,

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

− ε

²n_ε,1

(

z

) =

ⁿε,1

(

z

)

R₁

(

z

,

N_ε,1

) +

^m2n_ε,2

(

z

) −

^m1n_ε,1

(

z

),

− ε

²n_ε,2

(

z

) =

ⁿε,2

(

z

)

R₂

(

z

,

N_ε,2

) +

^m1n_ε,1

(

z

) −

^m2n_ε,2

(

z

),

N_ε,i

=

R

n_ε,i

(

z

)

dz

,

fori

=

1

,

2

,

(1)

with

Ri

(

z

,

Ni

) =

ri

−

gi

(

z

− θ

i

)

²

− κ

iNi

,

with

θ

1

= −θ

and

θ

2

= θ .

(2) Thissystemrepresentstheequilibriumofaphenotypicallystructuredpopulationundermutation,selectionandmigration betweentwohabitats.Formoredetailsonthemodelingandthebiologicalmotivations,seeSection2.

Notethattheasymptoticbehavior,as

ε

→^{0 andalong}subsequences,ofthesolutionstothissystem,undertheassump- tion mi>0,fori=¹,2,andforboundeddomains, was alreadystudiedin[14].Inthe presentwork,we gofurther than the asymptoticlimit along subsequences,andwe obtain uniqueness ofthelimit andidentify thedominant termsofthe solutionwhen

ε

issmallbutnonzero.

Themainelementsofthemethod:Todescribethesolutionsnε,i(z),weuseaWKBansatz n_ε,i

(

z

) = √

¹

2

π ε

^exp

u_ε,i

(

z

) ε

.

Notethat afirstapproximation,whichiscommonlyusedinthetheoryof‘quantitativegenetics’(atheoryinevolutionary biologythatinvestigatestheevolutionofcontinuouslyvaryingtraits, see[18],chapter7),isaGaussiandistributionofthe form:

n_ε,i

(

z

) = √

^Ni 2

π εσ

^exp

− (

z

−

^z∗

)

²

ε σ

²

= √

¹ 2

π ε

^exp

−

_2σ¹2

(

z

−

z^∗

)

²

+ ε

log^N_σⁱ

ε

.

Here,wetrytogofurtherthanthisaprioriGaussianassumptionandtoapproximatedirectlyuε,i.Tothisend,wewritean expansionforuε,iintermsof

ε

:

u_ε,i

=

^ui

+ ε

v_i

+ ε

²w_i

+

^O

( ε

³

).

(3)

We provethatu₁=^u2=^{uistheuniqueviscositysolutiontoa}Hamilton–Jacobiequationwithconstraint.Theuniqueness oftheviscositysolutiontosuchHamilton–Jacobi equationwithconstraintisrelatedtotheuniquenessoftheEvolutionary Stable Strategy(ESS),seeSection 3foradefinitionandfortheresultontheuniquenessoftheESS,andtotheweakKAM theory[7].Insection4,wecomputeexplicitlyu,whichindeedsatisfies

maxR u

(

z

) =

⁰

,

withthemaximumpointsattainedatoneortwopoints corresponding totheESSpoints oftheproblem.We thennotice that,whileu(z)<0,nε,i(z)isexponentiallysmall.Therefore,onlythe valuesof v_i and w_i atthepoints closeto thezero levelsetofu matter,i.e.theESSpoints.Insection5,we providethemainelementstocomputeformally vi andhenceits second-orderTaylorexpansionaroundtheESSpointsandthevalueofwiatthosepoints.Then,weshow,insection6,that theseapproximationstogetherwithafourth-orderTaylorexpansionofuaroundtheESSpointsareenoughtoapproximate themomentsofthepopulation’sdistributionwithanerrorintheorderof

ε

².

Themathematicaldetailsofourresultswillbeprovidedin[12].Thebiologicalapplicationswillbedetailedin[15].

2. Modelandmotivation

Thesolutionto(1)correspondstothesteadysolutiontothefollowingsystem,for(t,z)∈R⁺×R^,

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

∂

tn_i

(

t

,

z

) − ε

²

^∂

2

∂

z²n_i

(

t

,

z

) =

ⁿi

(

t

,

z

)

R_i

(

z

,

N_i

(

t

)) +

^mjn_j

(

t

,

z

) −

^min_i

(

t

,

z

),

i

=

¹

,

2

,

j

=

²

,

1

,

N_i

(

t

) =

R

n_i

(

t

,

z

)

dz

,

fori

=

¹

,

2

.

⁽⁴⁾

Thissystemrepresentsthedynamicsofapopulationthatisstructuredbyaphenotypicaltrait z,andlivesintwohabitats.

Wedenotebyn_i(t,z)thedensityofthephenotypicaldistributioninhabitati,andbyN_ithetotalpopulation’ssizeinhabitat i.ThegrowthrateRi(z,Ni)isgivenby(2),whererirepresentsthemaximumintrinsicgrowthrate,giisthestrengthofthe selection,θi istheoptimaltrait inhabitati,and

κ

i representstheintensityofthecompetition.The constantsmi arethe migrationratesbetweenthehabitats.Inthisnoteweassumethatthereispositivemigrationrateinbothdirections,i.e.

mi

>

0

,

i

=

1, 2

.

(5)

However,thesourceandsinkcase,whereforinstancem2=^{0,canalsobeanalyzedusingsimilartools.Wereferto[15]for} theanalysisofthiscase.Weadditionallyassumethat

max

(

r1

−

m1

,

r2

−

m2

) >

0

.

(6)

Thisguaranteesthatthepopulationdoesnotgetextinct.

Such phenomena havealready beenstudied by severalapproaches. A first class of resultsare basedon the adaptive dynamics approach,where one considers that the mutations are very rare,so that the population hastime to attain its equilibrium betweentwo mutations and hencethe population’s distribution has discrete support (one or two points in a two-habitat model)[11,2,6]. Asecond class ofresults isbased on an approachknown as‘quantitative genetics’,which allowsmorefrequentmutationsanddoesnotseparatetheevolutionaryandtheecologicaltimescales.Amainassumption inthisclassofworksisthat oneconsiders thatthepopulation’s distributionisa Gaussian[9,19] or,totake intoaccount thepossibilityofdimorphicpopulations,asumofoneortwoGaussiandistributions[20,3].

Inourwork,asinthequantitativegeneticsframework,wealsoconsidercontinuousphenotypicaldistributions.However, we do not assume anya prioriGaussian assumption. We compute directlythe population’s distribution andinthisway wecorrectthepreviousapproximations.Tothisend,wealsoprovidesomeresultsintheframeworkofadaptivedynamics and, in particular, we generalize previous results on the identification of the ESS to the caseof nonsymmetric habitats.

Furthermore,ourworkmakesaconnectionbetweenthetwoapproachesofadaptivedynamicsandquantitativegenetics.

3. Theadaptivedynamicsframework

In thissection, we introduce some notions fromthe theory ofadaptive dynamics that we will be using in the next sections[11].Wealsoprovideourmainresultinthisframework.

Effectivefitness.Theeffectivefitness W(z;^N1,N2)isthelargesteigenvalueofthefollowingmatrix:

A

(

z

;

^N1

,

N2

) =

R₁

(

z

;

^N1

) −

^m1 m₂ m₁ R₂

(

z

;

^N2

) −

^m2

.

(7)

Thisindeed corresponds tothe effective growthrateassociated withtrait z inthe whole metapopulationwhen thetotal populationsizesaregivenby(N1,N2).

Demographicequilibrium.Considerasetofpoints = {^z1,· · ·^zm}^.Thedemographicequilibriumcorrespondingtothis setisgivenby(n1(z),n2(z)),withthetotalpopulationsizes (N1,N2),suchthat

n_i

(

z

) =

m

j=¹

α

i,j

δ(

z

−

^zj

),

N_i

=

m

j=¹

α

i,j

,

W

(

z_j

,

N₁

,

N₂

) =

⁰

,

andsuchthat(

α

1,j,

α

2,j)^TistherighteigenvectorassociatedwiththelargesteigenvalueW(zj,N1,N2)=^{0 of}A(zj;^N1,N2).

Evolutionarystablestrategy.Asetofpoints^∗= {^z∗1,· · ·,z_m^∗}^iscalledanevolutionarystablestrategy(ESS)if

W

(

z

,

N^∗₁

,

N₂^∗

) =

⁰

,

forz

∈

Aand

,

W

(

z

,

N₁^∗

,

N^∗₂

) ≤

⁰

,

forz

∈ /

A,

where(N₁^∗,N^∗₂)arethetotalpopulationsizescorrespondingtothedemographicequilibriumassociatedwiththeset^∗. Since,thereareonlytwohabitatsforwhichweexpectthatatmosttwodistincttraitscoexistattheevolutionarystable equilibrium.Weproveindeedthefollowing.

Theorem3.1.Assume(5)–(6).Thereexistsauniquesetofpoints^∗whichisanevolutionarystablestrategy.Suchsethasatmosttwo elements.

We call anevolutionary stablestrategy whichhasone (respectively two)element(s), amonomorphic (respectively di- morphic)ESS. Wecanindeedgiveacriterion tohavemonomorphicordimorphicESS,andwe canidentifythedimorphic ESSinthegeneralcase(see[12]formoredetails).

4. Howtocomputethezero-ordertermsui

The identification of thezero-order terms u_i is based onthe following result. Note that the part(ii) of the theorem belowisavariantofTheorem1.1in[14].

Theorem4.1.Assume(5)–(6).

(i)As

ε

_→0,(nε,1,nε,2)convergesto(n^∗₁,n^∗₂),thedemographicequilibriumoftheuniqueESSofthemodel.Moreover,as

ε

_→0,Nε,i convergestoN_i^∗,thetotalpopulationsizeinpatchi correspondingtothisdemographicequilibrium.

(ii)As

ε

→^{0,bothsequences}(uε,i)_ε,fori=¹,2,convergealongsubsequencesandlocallyuniformlyinR^{toacontinuousfunction} u∈^C(R),suchthatu isaviscositysolutiontothefollowingequation

−|

^u

|

²

=

^W

(

z

,

N₁^∗

,

N^∗₂

),

in

R,

maxz∈Ru

(

z

) =

⁰

.

⁽⁸⁾

Moreover,wehavethefollowingconditiononthezero-levelsetofu:

suppn^∗₁

=

^suppn∗2

⊂ {

^z

|

^u

(

z

) =

⁰

} ⊂ {

^z

|

^W

(

z

,

N₁^∗

,

N^∗₂

) =

⁰

} .

(iii)Thereexistsconstants(λi,

ν

i),fori=¹,2,whichcanbedeterminedexplicitlyfromm₁,m₂,g₁,g₂,

κ

1,

κ

2andθ,suchthat,under thecondition

r₂

= λ

1r₁

+ ν

1

,

r₁

= λ

2r₂

+ ν

2

,

(9)

wehave

suppn^∗₁

=

^suppn∗₂

= {

^z

|

^u

(

z

) =

⁰

} = {

^z

|

^W

(

z

,

N₁^∗

,

N^∗₂

) =

⁰

}.

⁽¹⁰⁾

Thesolutionto(8)–(10)isunique,andhencethewholesequence(uε,i)_εconvergeslocallyuniformlyinR^tou.

Note thata Hamilton–Jacobiequation oftype (8)ingeneralmightadmit severalviscositysolutions.Here,the unique- ness isobtainedthanks to (10)andaproperty fromtheweak KAMtheory,whichis thefact that viscositysolutionsare completelydeterminedby onevaluetakenoneachstaticclassoftheAubryset([10],Chapter5and[1]).Inwhatfollows, weassumethat(9)andhence(10)alwayshold.Wethengiveanexplicitformulaforu consideringtwocases.

(i) MonomorphicESS. We considerthe casewhere thereexists a unique monomorphic ESS z^∗ andwhere thecorre- spondingdemographicequilibriumisgivenby(N^∗₁δ(z^∗),N₂^∗δ(z^∗)).Then uisgivenby

u

(

z

) = −

^z

z^∗

−

^W

(

x

;

^N∗₁

,

N₂^∗

)

dx

.

(11)

(ii)DimorphicESS.WenextconsiderthecasewherethereexistsauniquedimorphicESS(z^∗_a,z^∗_b)withthedemographic equilibrium:ni=

ν

a,iδ(z−^z∗a)+

ν

b,iδ(z−^z_b^∗),and

ν

a,i+

ν

b,i=^N∗_i^{.Thenuisgivenby}

u

(

z

) =

^max

− |

z z^∗_a

−

^W

(

x

;

^N∗₁

,

N₂^∗

)

dx

|, −|

z z^∗_b

−

^W

(

x

;

^N∗₁

,

N₂^∗

)

dx

|

.

5. Howtocomputethenext-orderterms

Inthissection, wegive themain elements tocompute formally vi andthevalue of wi atthe ESSpoint, withvi and w_i thecorrectorsintroducedby(3),inthecaseofamonomorphicpopulation.Forthedetailsofthecomputationsforboth monomorphicanddimorphicpopulations,werefertheinterestedreaderto[12].

Weconsiderthecaseofmonomorphicpopulationwherethedemographicequilibriumcorresponding tothemonomor- phicESSisgivenby(N^∗₁δ(z−^z∗),N^∗₂δ(z−^z∗)).Onecancompute,using(11),aTaylorexpansionoforder4 aroundtheESS pointz^∗:

u

(

z

) = −

^A

2

(

z

−

^z∗

)

²

+

^B

(

z

−

^z∗

)

³

+

^C

(

z

−

^z∗

)

⁴

+

^O

(

z

−

^z∗

)

⁵

.

Toprovideanapproximationofthemomentsofthepopulation’sdistribution,wehavetocompute constants Di,Eiand Fi suchthat

v_i

(

z

) =

^vi

(

z^∗

) +

^Di

(

z

−

^z∗

) +

^Ei

(

z

−

^z∗

)

²

+

^O

(

z

−

^z∗

)

³

,

w_i

(

z^∗

) =

^Fi

.

Afirst elementof thecomputationsis obtainedbyreplacing thefunctionsu, vi andwi bythe aboveapproximationsto compute Nε,i=

Rnε,i(z)dz.Thisleadsto v_i

(

z^∗

) =

^log

N^∗_i

√

A

,

N_ε,i

=

^N_i^∗

+ ε

K^∗_i

+

^O

( ε

²

),

with K^∗_i

=

^N∗_i

3C A²

+

^Ei

A

+

^Fi

.

Notealsothatwriting(1)intermsofuε,i weobtain

⎧ ⎪

⎨

⎪ ⎩

− ε

u_ε,1

(

z

) = |

^u_ε,1

|

²

+

^R1

(

z

,

N_ε,1

) +

^m2exp

u_ε,2

−

^uε,1

ε

−

^m1

,

− ε

u_ε,2

(

z

) = |

^u_ε,2

|

²

+

^R2

(

z

,

N_ε,2

) +

^m1exp

u_ε,1

−

^uε,2

ε

−

^m2

.

(12)

Asecondelementisobtainedbykeepingthezero-ordertermsinthefirstlineof(12)andusing(8)toobtain v₂

(

z

) −

^v1

(

z

) =

^log

₁

m₂

W

(

z

,

N₁^∗

,

N^∗₂

) −

^R1

(

z

,

N^∗₁

) +

^m1

.

(13)

Thelastelementisderivedfromkeepingthetermsoforder

ε

in(12),whichleadsto

−

^u

=

^2uv_i

− κ

iK_i^∗

+

^mjexp

(

v_j

−

^vi

)(

w_j

−

^wi

),

for

{

ⁱ

,

j

} = {

¹

,

2

}.

⁽¹⁴⁾

ThefunctionsviandthecoefficientsDi,EiandFicanbecomputedbycombiningtheaboveelements.

6. Approximationofthemoments

Theaboveapproximationsofu, viand wi aroundtheESSpoints allowustoestimate themomentsofthepopulation’s distribution.Inthemonomorphiccasetheseapproximationsaregivenbelow:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

N_ε,i

=

n_ε,i

(

z

)

dz

=

^N∗_i

(

1

+ ε (

F_i

+

^Ei A

+

^3C

A²

)) +

^O

( ε

²

), μ

_ε,i

₌

¹

N_ε,i

zn_ε,i

(

z

)

dz

=

^z∗

+ ε (

^3B A²

+

^Di

A

) +

^O

( ε

²

), σ

_ε²_,i

=

¹

N_ε,i

(

z

− μ

ε,i

)

²n_ε,i

(

z

)

dz

= ε

A

+

^O

( ε

²

),

s_ε,i

=

¹

σ

_ε³_,iN_ε,i

(

z

− μ

ε,i

)

³n_ε,i

(

z

)

dz

=

^6B A³²

√ ε +

^O

( ε

³²

).

One can obtain similar approximations inthe case ofdimorphic ESS. Tocompute theabove integrals,replacing the ap- proximation(3)intheintegrals,a naturalchangeofvariableistotake z−z^∗=√

ε

y.Therefore,eachterm z−z^∗ canbe considered asoforder√

ε

inthe integration.Thisis why,toobtain afirst-order approximationof theintegralsinterms of

ε

,itisenough tohave afourth-order approximationof u(z),a second-orderapproximation of vi(z),anda zero-order approximationofwi(z),intermsofzaround z^∗.

Acknowledgements

S. Mirrahimi has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 639638), and from the French ANR projects KIBORD ANR-13-BS01-0004andMODEVOLANR-13-JS01-0009.

References

[1]G.Contreras,ActionpotentialandweakKAMsolutions,Calc.Var.PartialDiffer.Equ.13 (4)(2001)427–458.

[2]T.Day,Competitionandtheeffectofspatialresourceheterogeneityonevolutionarydiversification,Amer.Nat.155 (6)(2000)790–803.

[3]F.Débarre,O.Ronce,S.Gandon,Quantifyingtheeffectsofmigrationandmutationonadaptationanddemographyinspatiallyheterogeneousenviron- ments,J.Evol.Biol.26(2013)1185–1202.

[4]O.Diekmann,P.-E.Jabin,S.Mischler,B.Perthame,Thedynamicsofadaptation:anilluminatingexampleandaHamilton–Jacobiapproach,Theor.Popul.

Biol.67 (4)(2005)257–271.

[5]L.C.Evans,P.E.Souganidis,APDEapproachtogeometricopticsforcertainsemilinearparabolicequations,IndianaUniv.Math.J.38 (1)(1989)141–172.

[6] C.Fabre,S.Méléard,E.Porcher,C.Teplitsky,A.Robert,Evolutionofastructuredpopulationinaheterogeneousenvironment,preprint.

[7]A.Fathi,WeakKamTheoreminLagrangianDynamics,CambridgeStudiesinAdvancedMathematics,vol. 88,CambridgeUniversityPress,Cambridge, UK,2016.

[8]M.Freidlin,Geometricopticsapproachtoreaction–diffusionequations,SIAMJ.Appl.Math.46(1986)222–232.

[9]A.Hendry,T.Day,E.B.Taylor,Populationmixingandtheadaptivedivergenceofquantitativetraitsindiscretepopulations:atheoreticalframeworkfor empiricaltests,Evolution55 (3)(2001)459–466.

[10]P.-L.Lions,GeneralizedSolutionsofHamilton–JacobiEquations,ResearchNotesinMathematics,vol. 69,PitmanAdvancedPublishingProgram,Boston, MA,USA,1982.

[11]G.Meszéna,I.Czibula,S.Geritz,Adaptivedynamicsina2-patchenvironment:atoymodelforallopatricandparapatricspeciation,J.Biol.Syst.5 (02) (1997)265–284.

[12] S.Mirrahimi,AHamilton–Jacobiapproachtocharacterizetheevolutionaryequilibriainheterogeneousenvironments,forthcoming.

[13]S.Mirrahimi,ConcentrationPhenomenainPDEsfromBiology,PhDthesis,UniversitéPierre-et-Marie-Curie,Paris-6,2011.

[14]S.Mirrahimi,Migrationandadaptationofapopulationbetweenpatches,DiscreteContin.Dyn.Syst.,Ser.B18 (3)(2013)753–768.

[15] S.Mirrahimi,S.Gandon,Theequilibriumbetweenselection,mutationandmigrationinspatiallyheterogeneousenvironments,forthcoming.

[16]S.Mirrahimi,J.-M.Roquejoffre,UniquenessinaclassofHamilton–Jacobiequationswithconstraints,C.R.Acad.Sci.Paris,Ser.I353(2015)489–494.

[17]B.Perthame,G.Barles,DiracconcentrationsinLotka–VolterraparabolicPDEs,IndianaUniv.Math.J.57 (7)(2008)3275–3301.

[18]S.H.Rice,EvolutionaryTheory:MathematicalandConceptualFoundations,SinauerAssociates,Inc.,2004.

[19]O.Ronce,M.Kirkpatrick,Whensourcesbecomesinks:migrationmeltdowninheterogeneoushabitats,Evolution55 (8)(2001)1520–1531.

[20]S.Yeaman,F.Guillaume,Predictingadaptationundermigrationload:theroleofgeneticskew,Evolution63 (11)(2009)2926–2938.