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

A Hamilton–Jacobi method to describe the evolutionary equilibria in heterogeneous environments and with non vanishing effects of mutations JID CRASS1 AID 5846 /FLA Doctopic Mathematical analysis [m3G;[.]

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Mathematical analysis/Partial differential equations

Méthode de Hamilton–Jacobi pour décrire des équilibres évolutifs dans les

environnements hétérogènes avec des mutations non évanescentes

Sylvain Gandona, Sepideh Mirrahimib

aCentre d’écologie fonctionnelle et évolutive (CEFE), UMR CNRS 5175, 34293 Montpellier cedex 5, France

bCNRS, Institut de mathématiques (UMR CNRS 5219), Université Paul-Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France

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

Article history:

Received 11 October 2016

Accepted after revision 7 December 2016

Available online xxxx

Presented by Hạm Brézis

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 u m é

Danscettenote,nousétudions unsystèmed’équationsintégro-différentielleselliptiques, décrivantunepopulationstructuréepartraitphénotypiquesoumisềdesmutations,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-mail addresses:sylvain.gandon@cefe.cnrs.fr (S Gandon), sepideh.mirrahimi@math.univ-toulouse.fr (S Mirrahimi).

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

1631-073X/©2016 Académie des sciences Published by Elsevier Masson SAS This is an open access article under the CC BY-NC-ND license

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,

⎧

⎪

⎨

⎪

⎩

− ε2n

ε ,1(z) =n ε ,1(z)R1(z,N ε ,1) +m2n ε ,2(z) −m1n ε ,1(z),

− ε2n

ε ,2(z) =n ε ,2(z)R2(z,N ε ,2) +m1n ε ,1(z) −m2n ε ,2(z),

N ε , i=



R

n ε , i(z)dz, for i=1,2,

(1)

with

R i(z,N i) =r i−g i(z− θi)2− κi N i, withθ1= −θandθ2= θ. (2)

Thissystemrepresentstheequilibriumofaphenotypicallystructuredpopulationundermutation,selectionandmigration betweentwohabitats.Formoredetailsonthemodelingandthebiologicalmotivations,seeSection2

Notethattheasymptoticbehavior,as ε →0 andalongsubsequences,ofthesolutionstothissystem,underthe assump-tion m i>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 εissmallbutnonzero

The main elements of the method: Todescribethesolutionsnε ,( ),weuseaWKBansatz

n ε , i(z) = √1

2π εexp



u ε , i(z)

ε



.

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

n ε , i(z) = N i

√

2π εσ exp



−(z−z∗)2

ε σ2



2π εexp

2σ2(z−z∗)2+ εlogN i

σ

Here,wetrytogofurtherthanthisaprioriGaussianassumptionandtoapproximatedirectlyuε ,.Tothisend,wewritean expansionforuε , intermsof ε:

We provethatu1=u2=u istheuniqueviscositysolutiontoaHamilton–Jacobiequationwithconstraint.Theuniqueness

oftheviscositysolutiontosuchHamilton–Jacobi equationwithconstraintisrelatedtotheuniquenessoftheEvolutionary Stable Strategy(ESS),seeSection 3foradefinitionandfortheresultontheuniquenessoftheESS,andtotheweakKAM theory[7].Insection4,wecomputeexplicitlyu,whichindeedsatisfies

max

R u(z) =0,

withthemaximumpointsattainedatoneortwopoints corresponding totheESSpoints oftheproblem.We thennotice that,whileu( ) <0,nε ,( )isexponentiallysmall.Therefore,onlythe valuesof v i and w i atthepoints closeto thezero levelsetofu matter,i.e.theESSpoints.Insection5,we providethemainelementstocomputeformally v i andhenceits second-orderTaylorexpansionaroundtheESSpointsandthevalueofw iatthosepoints.Then,weshow,insection6,that theseapproximationstogetherwithafourth-orderTaylorexpansionofu aroundtheESSpointsareenoughtoapproximate themomentsofthepopulation’sdistributionwithanerrorintheorderof ε2

Themathematicaldetailsofourresultswillbeprovidedin[12].Thebiologicalapplicationswillbedetailedin[15]

2 Model and motivation

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

⎧

⎪

⎨

⎪

⎩

∂t n i(t,z) − ε2∂2

∂ 2n i(t,z) =n i(t,z)R i(z,N i(t)) +m j n j(t,z) −m i n i(t,z), i=1,2, j=2,1,

N i(t) =



R

Thissystemrepresentsthedynamicsofapopulationthatisstructuredbyaphenotypicaltrait z,andlivesintwohabitats

Wedenotebyn i(t z thedensityofthephenotypicaldistributioninhabitati,andbyN ithetotalpopulation’ssizeinhabitat

i.ThegrowthrateR i( ,N i)isgivenby(2),wherer irepresentsthemaximumintrinsicgrowthrate,g iisthestrengthofthe selection,θi istheoptimaltrait inhabitati,and κi representstheintensityofthecompetition.The constantsm i arethe migrationratesbetweenthehabitats.Inthisnoteweassumethatthereispositivemigrationrateinbothdirections,i.e

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

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 The adaptive dynamics framework

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

Effective fitness Theeffectivefitness W( ;N1,N2)isthelargesteigenvalueofthefollowingmatrix:

A (z;N1,N2) =



R1(z;N1) −m1 m2

m1 R2(z;N2) −m2



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

Demographic equilibrium Considerasetofpoints  = {z1, · · ·z m}.Thedemographicequilibriumcorrespondingtothis setisgivenby(n1( ),n2( )),withthetotalpopulationsizes (N1,N2),suchthat

n i(z) =

m

j=1

αi , jδ(z−z j), N i=

m

j=1

αi , j, W(z j,N1,N2) =0,

andsuchthat( α , α )TistherighteigenvectorassociatedwiththelargesteigenvalueW( ,N ,N ) =0 ofA(z ;N ,N )

Evolutionary stable strategy Asetofpoints∗= {z∗

1, · · · ,z∗

m}iscalledanevolutionarystablestrategy(ESS)if

W(z,N∗

1,N∗

2) =0, for z∈ Aand, W(z,N∗

1,N∗

2) ≤0, for z∈ / A,

where(N∗

1,N∗

2)arethetotalpopulationsizescorrespondingtothedemographicequilibriumassociatedwiththeset∗.

Since,thereareonlytwohabitatsforwhichweexpectthatatmosttwodistincttraitscoexistattheevolutionarystable equilibrium.Weproveindeedthefollowing

Theorem 3.1.Assume(5)–(6) There exists a unique set of points∗which is an evolutionary stable strategy. Such set has at most two

elements.

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

4 How to compute the zero-order termsu i

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

Theorem 4.1.Assume(5)–(6).

(i) As ε →0,(nε ,1,nε ,2)converges to(n∗

1,n∗

2), the demographic equilibrium of the unique ESS of the model Moreover, as ε →0, Nε ,

converges to N∗

i , the total population size in patch i corresponding to this demographic equilibrium.

(ii) As ε →0, both sequences(uε ,)ε , for i=1,2, converge along subsequences and locally uniformly inRto a continuous function

u∈C( R), such that u is a viscosity solution to the following equation

−|u|2=W(z,N∗

1,N∗

2), inR,

max

Moreover, we have the following condition on the zero-level set of u:

supp n∗

1=supp n∗

2⊂ {z|u(z) =0} ⊂ {z|W(z,N∗

1,N∗

2) =0}.

(iii) There exists constants(λi, νi), for i=1,2, which can be determined explicitly from m1, m2, g1, g2, κ1, κ2andθ, such that, under the condition

we have

supp n∗

1=supp n∗

2= {z|u(z) =0} = {z|W(z,N∗

1,N∗

The solution to(8)–(10)is unique, and hence the whole sequence(uε ,)ε converges locally uniformly inRto u.

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) Monomorphic ESS We considerthe casewhere thereexists a unique monomorphic ESS z∗ andwhere the

corre-spondingdemographicequilibriumisgivenby(N∗

1δ(z∗),N∗

2δ(z∗)).Then u isgivenby

u(z) = −

z



z∗



−W(x;N∗

1,N∗

(ii) Dimorphic ESS WenextconsiderthecasewherethereexistsauniquedimorphicESS( ∗

a,z∗

b)withthedemographic equilibrium:n i= νa ,δ(z−z∗

a) + νb ,δ(z−z∗

b),and νa , + νb , =N∗

i.Thenu isgivenby

u(z) =max

− |

z



z∗



−W(x;N∗

1,N∗

2)dx|, −|

z



z∗



−W(x;N∗

1,N∗

2)dx|  .

5 How to compute the next-order terms

Inthissection, wegive themain elements tocompute formally v i andthevalue of w i atthe ESSpoint, withv i and

w i thecorrectorsintroducedby(3),inthecaseofamonomorphicpopulation.Forthedetailsofthecomputationsforboth monomorphicanddimorphicpopulations,werefertheinterestedreaderto[12]

Weconsiderthecaseofmonomorphicpopulationwherethedemographicequilibriumcorresponding tothe monomor-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 D i,E iand F i

suchthat

v i(z) =v i(z∗) +D i(z−z∗) +E i(z−z∗)2+O(z−z∗)3, w i(z∗) =F i.

Afirst elementof thecomputationsis obtainedbyreplacing thefunctionsu, v i andw i bythe aboveapproximationsto compute Nε , = Rnε ,( )dz.Thisleadsto

v i(z∗) =log

N∗

i

√

A

, N ε , i=N∗

i + εK∗

i +O( ε2), with K∗

i =N∗

i

3C

A2+E i

A +F i

.

Notealsothatwriting(1)intermsofuε , weobtain

⎧

⎪

⎪

− εu

ε ,1(z) = |u

ε ,1|2+R1(z,N ε ,1) +m2expu ε ,2−u ε ,1

ε



−m1,

− εu

ε ,2(z) = |u

ε ,2|2+R2(z,N ε ,2) +m1expu ε ,1−u ε ,2

ε



−m2.

(12)

Asecondelementisobtainedbykeepingthezero-ordertermsinthefirstlineof(12)andusing(8)toobtain

v2(z) −v1(z) =log

m2



W(z,N∗

1,N∗

2) −R1(z,N∗

1) +m1

Thelastelementisderivedfromkeepingthetermsoforder εin(12),whichleadsto

−u=2uv

i− κi K∗

Thefunctionsv iandthecoefficientsD i,E iandF icanbecomputedbycombiningtheaboveelements

6 Approximation of the moments

Theaboveapproximationsofu, v iand w i aroundtheESSpoints allowustoestimate themomentsofthepopulation’s distribution.Inthemonomorphiccasetheseapproximationsaregivenbelow:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

N ε , i=



n ε , i(z)dz=N∗

i(1 + ε (F i+E i

A +3C

A2)) +O( ε2),

με , i= 1

N ε , i



zn ε , i(z)dz=z∗+ ε (3B

A2+ D i

A) +O( ε2),

σε2, i= 1

N ε , i



(z− με , i)2n ε , i(z)dz= ε

A+O( ε2),

s ε , i= 1

σ3

ε , i N ε , i



(z− με , i)3n ε , i(z)dz=6B

A3

√

ε +O( ε3).

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

considered asoforder√

εinthe integration.Thisis why,toobtain afirst-order approximationof theintegralsinterms

of ε,itisenough tohave afourth-order approximationof u( ),a second-orderapproximation of v i( ),anda zero-order approximationofw i( ),intermsofz around 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

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weassumethat(9)andhence(10)alwayshold.Wethengiveanexplicitformulaforu