Genotype-Environment Interaction and the of Author(s): Sara Via and Russell Lande Reviewed work(s): Source: Evolution, Vol. 39, No. 3 (May, 1985), pp. 505-522 Published by: Society for the Study of Evolution Stable URL: http://www.jstor.org/stable/2408649 . Accessed: 10/11/2011 18:57

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http://www.jstor.org Evolution,39(3), 1985, pp. 505-522

GENOTYPE-ENVIRONMENT INTERACTION AND THE EVOLUTION OF PHENOTYPIC PLASTICITY

SARA VIA' AND RUSSELL LANDE Departmentof , The UniversityofChicago, Chicago, IL 60637

Summary.-Studiesof spatialvariation in theenvironment have primarilyfocused on how geneticvariation can be maintained.Many one-locus genetic models have addressedthis issue,but, for several reasons, these models are not directlyapplicable to quantitative (polygenic)traits. One reasonis thatfor continuously varying characters, the evolution of themean phenotype expressed in differentenvironments (the norm of reaction)is also of interest.Our quantitativegenetic models describe the evolution of phenotypic response to theenvironment, also knownas phenotypicplasticity (Gause, 1947), and illustratehow the normof reaction(Schmalhausen, 1949) can be shapedby selection.These modelsutilize thestatistical relationship which exists between genotype-environment interaction and ge- neticcorrelation to describeevolution of the mean phenotype under soft and hardselection in coarse-grainedenvironments. Just as geneticcorrelations among characters within a single environmentcan constrainthe response to simultaneousselection, so can a geneticcorre- lationbetween states of a characterwhich are expressedin twoenvironments. Unless the geneticcorrelation across environments is ? 1, polygenicvariation is exhausted,or thereis a costto plasticity,panmictic populations under a bivariatefitness function will eventually attainthe optimum mean phenotype for a givencharacter in each environment.However, veryhigh positive or negativecorrelations can substantiallyslow the rate of evolutionand mayproduce temporary maladaptation in one environmentbefore the optimum joint phe- notypeis finallyattained. Evolutionarytrajectories under hard and softselection can differ:in hardselection, the environmentswith the highest initial mean fitness contribute most individuals to the mating pool. In bothhard and softselection, evolution toward the optimum in a rareenvironment is muchslower than it is in a commonone. A subdividedpopulation model revealsthat migration restriction can facilitatelocal adaptation.However, unless there is no migrationor one ofthe special cases discussed for panmicticpopulations holds, no geographicalvariation in the normof reactionwill be maintainedat equilibrium.Implications of theseresults for the interpretationof spatial patternsof phenotypic variation in naturalpopulations are discussed.

ReceivedApril 9, 1984. AcceptedDecember 11, 1984

Environmental modification of the Schmalhausen, 1949; Bradshaw, 1965). phenotypeis common in thequantitative The models presentedhere concern the (polygenic)characters of organismsthat evolution of quantitative traits in spa- inhabitheterogeneous environments. The tiallyvariable environments.Within this profileof phenotypesproduced by a ge- generalcontext, they explore how an ad- notypeacross environmentsis the "norm vantageouslevel of phenotypicplasticity of reaction" (Schmalhausen, 1949); the mightevolve. extentto which the environmentmodi- Schmalhausen (1949) recognized that fiesthe phenotype is termed"phenotypic mutationsare likelyto be expresseddif- plasticity" (Gause, 1947; Bradshaw, ferentlyin various environments.Mu- 1965). Because phenotypicresponse to tation can thus act to disrupt the most environmentalchange may facilitatethe advantageous norm of reaction by cre- exploitationof some environmentsand ating genetic variation in phenotypic provide protectionfrom others, the level plasticity.Such variation in response to of plasticityin a given traitis thoughtto the environmentis also known as geno- be molded by selection (Gause, 1947; type-environmentinteraction (Falconer, 1981). Schmalhausen's view was that ' Presentaddress: Department of Biology, Uni- molds the norm of re- versityof Iowa, Iowa City,Iowa 52242. action from the variants produced by 505 506 S. VIA AND R. LANDE mutation.In otherwords, genotype-en- titative traits in whichthe influenceof vironmentinteraction is thetype of ge- individualloci cannotbe identified.3) neticvariation required for the evolution Thedynamics of the mean phenotype and of a selectivelyadvantageous level of thenorm of reaction are ofconsiderable phenotypicplasticity. interestin polygeniccharacters; these are AlthoughSchmalhausen argued that not describedby existingmodels. 4) In theoptimal norm of reactionwill even- two-nichemodels with only two alleles, tuallyevolve fromthe variation among theadditive genetic correlation between genotypesin reaction norms generated by theexpression of the genotypes in the two mutation,agriculturalists find that ge- environmentsis always -1, even with notype-environmentinteraction can act dominance,because the breedingvalue as a constraintin the selectionof a ge- of theheterozygote is alwaysintermedi- notypewith broad adaptability (Robert- ate betweenthe two homozygotes. son, 1959; James, 1961; Dickerson, Quantitativegenetic models are used 1962). Genotype-environmentinterac- hereto investigatethe effectsof genetic tion may thushave differentlong- and variationin phenotypicplasticity (ge- short-termeffects on theprocess of evo- notype-environmentinteraction) on both lution. thetrajectories and ratesof evolutionof Many classical population genetic the average phenotypeunder simulta- modelsof spatialvariation in the envi- neous selectionin twodiscrete environ- ronmenthave beenmade sinceLevene's ments.Although these models do not original(1953) "multiple-niche"model considercontinuous environmental vari- (reviewedin Hedricket al., 1976).These ationsuch as temperatureor photoperi- modelsshare a similarintent: to deter- od, theymay be usefulin the interpre- minethe conditionsunder which poly- tationof experimental data in which tests morphismat a singlelocus can be main- areperformed at discretepoints along an tained. The multiple-nichemodel has environmentalcontinuum. substantiallyinfluenced the course of ex- Genotype-EnvironmentInteraction as perimentalwork, motivating many at- GeneticCorrelation.-Falconer (1952) temptsto documentthe effects of envi- firstnoted that a characterexpressed in ronmentalvariation on themaintenance twoenvironments can be viewedas two of geneticvariation in bothelectropho- characterswhich are genetically correlat- reticand polygenictraits (Beardmore and ed. This view is an essentialfeature of Levine, 1963; Powell,1971; McDonald the presentmodels: a separatevariable and Ayala, 1974; Powelland Wistrand, is definedfor the expressionof a given 1978;Mitter et al., 1979;MacKay, 198 1; traitin each environment.We will call Lacy,1982; Jaenike and Grimaldi, 1983). theexpression of a characterin a given However,for severalreasons, existing environmenta characterstate. For ex- one-locusmodels do notprovide an ad- ample,body weight in one environment equate descriptionof evolutionin the and bodyweight in anotherenvironment quantitativetraits often studied with re- can be thoughtof as twogenetically cor- spectto adaptationto differentenviron- relatedcharacter states (Falconer, 1952). ments.1) In thesimplest models of one In this context,the additivegenetic locus withtwo alleles, the mean and the correlationestimates the degree to which geneticvariance are alwaysstatistically the phenotypesexpressed in two envi- dependent,while in polygenictraits, they ronmentshave the same geneticbasis, can oftenevolve independently (Slatkin, attributableeither to pleiotropiceffects 1978). 2) The multiple-nichemodels de- ofgenes or to linkagedisequilibrium be- rivecriteria for the maintenance of poly- tweenalleles at differentloci. A highge- morphismin termsof the mean fitnesses netic correlationacross environments ofalleles in differentenvironments; these impliesthat the same alleles or sets of fitnessescannot be measuredfor quan- alleles influencethe characterstates in G x E AND EVOLUTION 507 thesame way in twoenvironments. Ifthe estimatethe genetic correlation between geneticcorrelation between two charac- characterstates, replicated genotypes or terstates is ? 1, theyshould be consid- familymembers are allowedto develop eredto be exactlythe same character; the in the differentenvironments. Because correlationindicates that they have an measurementsofdifferent character states identicalgenetic basis. In contrast,a cross- mustbe made on separateindividuals, environmentgenetic correlation between the usual statisticalmethods for calcu- + 1 and -1 indicatesthat the phenotypes latingthe geneticcorrelation (e.g., Fal- in eachenvironment are influenced either coner,1981 Ch. 19) are not applicable. bysome different alleles or differentlyby In this case alternativesmay be em- thesame alleles, and thuscan have some ployed,such as thecorrelation of family degreeof independentevolution. Pre- meansor there-expression of genotype- vious polygenicmodels of spatialvaria- environmentinteraction as a genetic tion have assumed that the character covariancebetween character states ex- statesin thetwo environments are iden- pressedin differentenvironments (re- tical,that is, that the additive genetic cor- viewedin Via, 1984). relationacross environments is + 1 (e.g., Both "soft"and "hard" selectionare Bulmer,1971; Slatkin,1978). examined(following the terminologyof Althougha well-definedstatistical re- Christiansen,1975), and, as will be lationshipexists between genotype-en- shown, the evolutionary dynamics of the vironmentinteraction and geneticco- meanphenotype in heterogeneousenvi- variance(Robertson, 1959; Dickerson, ronmentsdiffer for these modes of selec- 1962; Yamada, 1962; Fernandoet al., tion.In all but a fewcircumstances, op- 1984), estimationof the geneticcorre- timallevels of phenotypicplasticity are lationacross environments is the more eventuallyattained at equilibrium.Pop- preciseway of the two to evaluategenetic ulationsubdivision and reducedmigra- variancein responseto theenvironment. tion will be seen to permitmore direct Any geneticcorrelation between char- and rapidlocal adaptationthan is pos- acters expressedin differentenviron- siblewith high levels of migration. How- mentswhich is lessthan + 1 willtranslate ever,the equilibriummean phenotype intoa significantgenotype-environment expressed by thepopulation in each en- interaction(Robertson, 1959). The cor- vironmentgenerally reaches the opti- relationformat is also moreuseful math- mum,where the populationas a whole ematically,because estimatesof geno- is understabilizing, not disruptive selec- type-environmentinteraction cannot be tion.Stabilizing selection depletes addi- incorporatedinto any currenttheory of tivegenetic variance in polygenicchar- evolution.In contrast,genetic correla- acters(Scharloo, 1964; Wright, 1969 Ch. tionsbetween polygenic characters in a 4); observedlevels of additivegenetic singleenvironment are knownto affect varianceat equilibriumare thus pre- boththe direction and rateof joint evo- sumed to be maintainedby mutation- lution (Hazel, 1943; Dickerson,1955; selectionbalance (Lande, 1976, 1980a; Lande, 1979, 1980b,1982). The models Turelli,1984). Spatial variation in selec- presentedhere explore how geneticcor- tion pressureswill persistand augment relationsbetween character states ex- geneticvariance at equilibriumonly when pressedin different environmental niches thereis no migrationor whenthe genetic may similarlyconstrain evolution in correlationacross environments is ? 1. variableenvironments. In particular,the models investigatethe effectsof these GeneticModels of Panmictic Populations correlationson the evolutionof pheno- Assumptionsand Limitationsof the typicplasticity. Models.-Using standardquantitative In thepresent models, any individual geneticstechniques (e.g., Falconer, 1981 experiencesonly one environment.To Ch. 9), variationin thephenotypic value 508 S. VIA AND R. LANDE of thecharacter in the ithenvironment, marily concerned with evolution of phe- zi, can be partitionedinto additivege- notypictraits which have an intermedi- neticand within-or micro-environmen-ate optimum. Many morphological, tal portions,with non-additive genetic physiological,and behavioraltraits, and variancepooled into the within-environ- minor components of fitness like growth mentalvariance. The phenotypicvari- rate,development time, or bodysize are ance of thecharacter state in the ithen- typicallyunder such stabilizing selection vironmentis Pii = Gii + Eii where Eii is (e.g.,Rendel, 1943); thesetraits may be the within-environmentalvariance and selectedin the same or differentdirec- is assumedto followa normal(Gaussian) tionsacross environments.In contrast, distribution.When character states in dif- majorcomponents of fitnesslike fecun- ferentenvironments cannot be measured dityor survivalcan be presumedto be on the same individual,the phenotypic undercontinual directional selection to covariance,Pij withi # j, is undefined increasein all environments.The portion (Falconer,1981 p. 284). It is assumed of the analysisof the presentmodels throughoutthat scales of measurement whichassumes a Gaussianform for the are used suchthat the phenotypic distri- fitnessfunction is notapplicable to such butionis normalin each environment,characters. with the varianceindependent of the In thefirst two models, we assumethat mean.A logarithmictransformation will thepopulation is panmicticand that mat- oftenimprove the fitof the data to the ed adultsdisperse into the available hab- normaldistribution (Wright, 1968 Chs. itatsat randomeach generation.Selec- 10 and 11). tion in each environmentacts on For polygeniccharacters in whichsev- individualsbefore dispersal and mating. eral loci of small effectare assumedto Underweak or moderateselection and contributeto the phenotype,the distri- panmixis,the means of the groupsse- butionof additive genetic effects on each lectedin each environmentwill not di- charactercan also be assumedto be ap- vergeenough in one generationto cause proximatelyGaussian. The matrixof ad- appreciablenon-normality in the phe- ditivegenetic variances and covariances notypicdistribution before selection. of characterstates can be writtenas G, Hard and softselection, as definedby wherethe element Gii is theadditive ge- Christiansen(19 7 5), aredistinguished by neticvariance of the characterstate in theform of population regulation which the ith environmentand Gij is the ad- is presumedto occur.In softselection, ditive geneticcovariance of character populationsin eachenvironment are reg- statesexpressed in the ithand jth envi- ulated independently,so thateach en- ronments. vironmentalniche contributes a constant Selectionis assumedto be weak and fractionof thetotal population. In hard populationsto be large,allowing genetic selection,the contribution of each niche variationdepleted by selectionto be re- to thetotal population is weightedby its plenishedby mutation(Lande, 1976, meanfitness; the population is regulated 1980a). The matrixof geneticvariances globally.Soft selection may apply when and covariancescan thusbe assumedto populationsin differentniches are held remainroughly constant as evolutionof at constantnumbers by resourcelimita- the mean phenotypeproceeds. This as- tion.If populations are limited by evolv- sumptionhas been examinedin itera- ingtraits, such as theability to utilizethe tions of equations approximatingthe availableresources effectively, then hard evolutionof the covariance structure un- selectionmay apply. As evolutionoccurs der selectionin two environments,and underhard selectionto increaseadap- was foundto be robust(Via and Lande, tation in a given environment,that unpubl.). niche'scontribution to the total popu- The models describedhere are pri- lationwill increase. G x E AND EVOLUTION 509

SoftSelection. -With qi definedas the si = [f zipi(z1) Wi (zi)dzi]/ WJ - f,. (3) proportionof individuals entering the ith habitat(Z qi = 1), 2i as the mean value Expressingthe changein mean fitness ofthe character state expressed in theith withan incrementalchange in 2iand us- environment,Pii-I as thereciprocal of the ing(3), theobserved selection differential phenotypicvariance in the ithenviron- can be rewrittenin termsof the gradient ment,and si as thedifference between the of the logarithmof mean fitnessin the mean phenotypebefore and afterselec- ithenvironment (Lande, 1979): tionin theith environment, the dynam- ical equationfor soft selection in twoen- (4) vironmentsis: Pii-Isi=VilnW where the gradient operator Vi = 3/62i actingon lnW represents the direct effect Az2/ kG21 G22,Iq2P22-'2f (1) ofselection in theith environment on the Thus, characterstate expressed there, with the statesexpressed in otherenvironments A = qG11P1I-'sI held constant. This selective force, + (1 -q)G2P22-IS2 VilnWi, is also equal to thepartial regres- sioncoefficient of individualrelative fit- AZ2 = qG2IPII IsI ness in the ithenvironment (Wi/W) on + (1 - q)G22P22-Is2, (2) zi (Lande and Arnold,1983). By substi- tuting(4) into(1), thedynamic equations whereq = q,. Eqs. 2 illustratethat the evolutionof each of thecharacter states forsoft selection in two environments includesboth a directresponse to selec- can be rewrittenas tion in the environmentin whichthe characterstate of interest is expressedand (A2), _ (G1 2 a correlatedresponse to selectionon the AZ2J VG21 G22J stateexpressed in the other environment. Thus,if two characterstates are simul- A22/~~~ I I - taneouslyselected to increasein equally V )n[ 1 W2(1 q].(5) frequentenvironments (s, > 0 and52 > 0, q = 0.5), a low or negativegenetic co- variancein the phenotypeacross envi- Thefunction W = 2(1 - q) givesthe ronments(G12) will slow the rate of jointmean fitness under soft selection as evolution. Similarly,evolution under theweighted geometric mean ofthe mean disruptiveselection (s, > Oand 52 < Oor fitnessesin thetwo environments, defin- conversely)will be retardedby positive ingan adaptivetopography for evolution geneticcovariance. If multipletraits in intwo environments. Assuming a Gauss- each environmentare included,then se- ian fitnessfunction in eachenvironment, lectionon othercharacters in the same thistopography is plottedin FigureIA. environmentcan also cause correlated The heightof this adaptive landscape for changesin themean phenotype. All the phenotypesis thejoint mean fitnessat modelsdescribed here can be generalized each value of the vectorof mean phe- to includemultiple environments with notypesin the two environments;the manycharacters expressed in each en- peakdefines an equilibriumat whichthe vironmentby expandingthe dimen- phenotypeis at thejoint optimumand sionalityof thematrices in (1). thegeometric mean fitness is maximized The fitnessof an individualof phe- (Wright,1969 Ch. 4; Lande, 1979). To- notypez in environmenti is definedas getherwith the geneticcovariance ma- Wi(zi). With a normal distributionof trix,the adaptive topography determines phenotypes,p1(zj), and mean fitnessWi, the rate and directionof multivariate theselection differential in the ithenvi- evolution.The formof thejoint fitness ronmentis functionis thusan importantelement in 510 S. VIA AND R. LANDE thedynamics of phenotypic evolution in temproduces a setof axes (theeigenvec- variableenvironments. tors)with the rateof evolutionin each The evolutionarydynamics of mean directiondetermined by the correspond- fitnesscan be determinedby expanding ing eigenvalues.If the eigenvaluefor a Win a Taylorseries around an arbitrary givendirection is small,X < 1, thenthe point.Assuming weak selection,higher rateof approach to theequilibrium from orderterms can be ignoredto yield thatdirection is (1 + X)t e ex', wheret is the numberof generations.This rate A ln[ W1 W2(I-q)] = (A21, Az2) correspondsto a timescale of -1 /Xgen- *'Vln[WqW2(-] erations,where the time scale is thenum- l/e = (VlnW)TGVln W ber of generationsrequired to move fromthe initial position > (6) of thedistance 0, in phenotypespace to theoptimum in a directionwhich is parallelto the corre- whereVT = (V1,V2), and T indicatesma- spondingeigenvector. trixtransposition. The quadraticform in The effectof the genetic correlation on (6) ensuresthat the evolution of the mean theeigenvalues, and thuson theevolu- phenotypesselected in differentenviron- tionaryrate, can be most simplyillus- mentsalways occurs in a directionwhich tratedfor the symmetrical case in which increasesthe joint mean fitnessin the q = 0.5, GI, = G22= G, G12= G21 = yG, population. However, correlatedre- Pll = P22 = P, and w1=2 =w. The dy- sponsesto selectioncaused by genetic co- namicalequation in (5) can thenbe re- varianceof characterstates in differentwritten as environmentscan cause meanfitness to decreasetemporarily in one of the en- vironmentsduring the courseof evolu- tion.Even so, unlessthe geneticcorre- A22 2(W2 + P) ( (22 - 02) (8) lation across environmentsis ? 1, an optimallevel of phenotypic plasticity will The eigenvaluesof thissystem are X,= eventuallybe attainedat equilibrium. The -G(1 + y)/2(W2+ P)andX2=-G(1 -y)/ mean fitnessin each environmentwill 2(W2 + P). These are associatedrespec- thenbe at a local maximum.For ex- tivelywith the eigenvectors correspond- ample,when the fitnessfunction in the ingto evolutionof the character states in ithenvironment is of a Gaussian form the same direction [xIT = (1 /V2)(1, 1)] with width wiand optimum Oi, WV(z1) = and evolutionof the characterstates in exp { -(zi - O)2/2wi2}.The mean fitness oppositedirections [x2T = (I/VF)(I, - 1)]. of phenotypezi is then High or low values of -ylead to in- creasingdiscrepancy of the time scales forevolution in thedirections of theei- Wi= WJ(z)pi(z) dz, genvectors.Negative genetic correlations acrossenvironments will slow evolution = \1Wi2/(Wi2 + PJ along the eigenvectorassociated with parallelchange, xl. In contrast,positive exp{-(2, - OI)2/2(w12+ PJJ)}. (7) geneticcorrelations across environments will retardmovement along x2, which At equilibrium,unless G is singular, correspondsto the evolutionof geo- VilnWi= -(2, - O13/(Wi2+ P1,) = 0, so graphicalvariation under selection in dif- that i = 0i. ferentdirections in the two environ- Rates of Evolutionunder Soft Selec- ments.Any evolutionary trajectory can tion.-The dynamicsof phenotypic evo- be consideredas a combinationof move- lutionunder soft selection are described mentin thedirections of thetwo eigen- by (5). WithGaussian fitnessfunctions vectors,and evolutionwill proceed at a as in (7), thedynamical system is linear. ratedetermined by the appropriate com- Analysisof the eigenstructure ofthis sys- binationof theeigenvalues. G x E AND EVOLUTION 511

SOFT HARD

60-

50-

40

IN030

20-

A B 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60

ZI Z1

FIG. 1. Adaptivetopographies for a characterselected in twoenvironments under either soft selection (A) or hardselection (B). The phenotypesof the trait in thetwo environments are consideredto be two separate,but genetically correlated, character states with mean phenotypes z, and Z2. Contoursrepresent levelsof joint mean fitnessat differentcombinations of mean phenotypesin the two environments. Contoursare 0.1 unitsapart. Under soft selection, joint meanfitness is W1qW2(0- q), whileunder hard selection,the joint mean fitness is qW1+ (1 - q)W2,where W, is themean fitness in theith environment, givenby Equation (7). Parametersfor both plots are G =G22 = 10, P11 = P22= 20, W12= W2 = 200, q = 0.5, and 01 = 02 = 50.

Ifthe environments are not equally fre- adapted.It can be seen that,when only quent(q # 0.5), theeigenvalues are X = a slightasymmetry exists in the initial -G[l ?V1- 4q(l - q)(l - y2)]/2(W2+ conditions(Fig. 2A), thegenetic covari- P). Whenone environmentis veryrare anceacross environments has littleeffect (q<< 1), thenthe smallest eigenvalue can on theevolutionary trajectories under soft be approximated by X = -qG(1 - y2)/ selectionregardless of whetherselection (W2 + P). Thus, the rate of approach of on thetwo character states is in thesame the mean phenotypeto the joint opti- or differentdirections (compare Fig. 2A, mumwhen one environmentis rarewill C). However,the genetic correlation be- be extremelyslow. tweencharacter states in the two envi- To aid in interpretationof the equa- ronmentswill affect the rate of evolution tions,several numerical examples of evo- as discussedabove: therate of approach lutionof one traitunder soft selection in to equilibriumunder soft selectionis two environmentsare presentedin Fig- slowestfor extremely negative correla- ures2 and 3. All examplesconcern pop- tions. Under disruptiveselection (Fig. ulationswhich have been perturbed from 2C), theslowest evolution will occur with thejoint optimumby variousamounts. a highpositive correlation between states. The effectof thegenetic correlation be- Whenone ofthe habitats comprises a tweenthe character states on thecourse largerfraction of the totalenvironment of evolutionback towardthe joint opti- thanthe other and bothcharacter states mumis thenexamined. areselected to increase,the effects of neg- In the firstexamples, individuals are ativegenetic covariance across environ- initiallypoorly adapted to bothhabitats. mentson thejoint trajectoryare more An exampleof such a case mightbe an striking.In Figure3A, 70% of theindi- herbivorewhich invades a geographical viduals are selectedin environment1: areacontaining a new constellation of host evolutionthus occurs more rapidly there, plantsto whichit is uniformlypoorly and muchof the changein the average 512 S. VIA AND R. LANDE SOFT HARD

CD Ln) +1 - -75 -.75 CD

C\JD /A

CD

CD

7 ~~~~~~~~~~~~+75

CD oI I II I I I I I I

CD 0 -10 0 3 0 0 0 10 20 310 5

GD C CZl D Zl 0 10 20 30 40 50 0 10 20 30 40 50

FIG.2. Effectof the genetic correlation in theexpression of a characterin twoenvironments on the evolutionarytrajectories of themean phenotype under soft selection (A, C) and hardselection (B, D). Thevalues of the genetic correlations across environments are: + 1(E),0.75(0), 0.375(A), 0(+), -0.375( x ), -0.75(K0),- 1(V).Selected trajectories are labelled with the corresponding value of the genetic correlation. Markerson theplots are at 50 generationintervals. Parameters for all plotsare as in Figure1 exceptthat in A and B, theinitial values are z1 = 25, z2 = 27, and in C and D, theinitial conditions are 2, = 30, Z2 = 27 and theoptimum is at (5, 50). Arrowsindicate the direction of evolution.

phenotypein therare environment is due thus produce temporarymaladaptation to a correlatedresponse (see Eqs. 2). When to the rare environmentand cause the the phenotypesin the two environments approach to the joint optimum to be are negativelygenetically correlated, ad- greatlyslowed. aptation to the rare environmentcannot As a finalexample, consider the case proceed untilthe population approaches in whichthe population is perturbedfrom the local optimumin the more common the joint optimum much more in one environmentand the intensityof selec- environmentthan in the other. For ex- tion diminishes there. When the envi- ample, if a new host plant were intro- ronmentsoccur in unequal frequencies, duced into the range of an herbivore an unfavorablecorrelation structure can species which was already well adapted G x E AND EVOLUTION 513 SOFT HARD

CD +1 +

IN -X75 CD

CD]

Ao- _

CD GD 75 +75 -7 7

CD

o-GDI I I I I I I CD INC\ CD

GD C D GD - I 0 10 20 30 40 50 0 10 20 30 40 50

z z I I

FIG. 3. Effectsof geneticcorrelation across environments on thetrajectory of themean phenotype whenenvironments are represented in unequalfrequencies or when the population is initiallywell adapted to onlyone environment.Except as noted,parameters are the same as Figure1; symbolsare as in Figure 2. Optimaare O,= 40 and 02 = 50. In A and B, theinitial values are 2, = 15,22 = 27, q = 0.7; in C and D, theinitial values are 2, = 37, 22 = 27 and q = 0.5. to anotherplant species, the population ronmentwhere the initial mean pheno- mightbe expected to be close to the op- typewas welladapted, while negative ge- timum phenotypeon the old plant, but netic correlations lead to similar poorlyadapted to thenew hostplant. The maladaptiveevolution in the opposite numerical example shown in Figure 3C direction.As expectedfrom the eigen- illustratesthat the most directevolution values,high genetic correlations of either in this situation occurs when the char- signalso substantiallydecrease the rate acterof interestis geneticallyuncorrelat- ofapproach to thejoint optimum; in both ed across environments.High positive cases, some evolutionmust occur in a genetic correlations between character directionwith a verysmall eigenvalue. statesin differentenvironments cause an In theseexamples, the population may overshoot of the optimum in the envi- experienceselection in differentdirec- 514 S. VIA AND R. LANDE tions in the two environmentswhen it is The gradientvector in (1 1) reveals that farfrom the joint optimum.This can in- thejoint mean fitness( V) in hard selec- crease the genetic variation. However, tionis thearithmetic average ofthe mean numericalexamples in which (1) is cou- fitnessesin the separate environments. pled with an equation for the evolution Thus, the adaptive topographiesfor soft of genetic variance under soft selection and hard selectiondiffer (Fig. 1); in hard reveal that the qualitative patternspre- selection,the joint fitnessfunction is no sented in the figuresare not greatlyaf- longerbivariate normal. It is of interest fectedby evolutionarychange in the ge- to note that both of the adaptive topog- netic covariance matrix, G (Via and raphiesin thesepolygenic models are the Lande, unpubl.). same as derivedin thecorresponding sin- Hard Selection. -The dynamical gle locus treatments (e.g., Li, 1955; equation forphenotypic evolution under Dempster, 1955). hard selectioncan be expressedin a form The.differencein the adaptive land- whichdiffers from (1) onlyby theweight- scapes forthe two modes of selection is ing function:the contributionof each en- reflectedin the evolutionarytrajectories vironmentto the total population is now which are expected followinga pertur- the product of its frequency,q, and the bation.Under hardselection, even a small relative mean fitness(WK/W) of individ- asymmetryin the initial location of the uals selected there,where, as shown be- phenotypein the two niches relative to low, W= qW1 + (1 - q)W2. Forthe two the joint optimum (Fig. 2B, D) causes environmentcase, evolutionof the character state expressed in theenvironment with the lowest initial mean fitness(here, 2,) to be dominated A522J by the correlatedresponse to selectionon the characterstate in the otherenviron- (G1I, G12 ( [qWi/ W]PIjls) ment. When that characterstate (22) ap- VG21G22J \[1- q) W2/RI1P^2- 2'S proaches its optimum, the intensityof {GI, G128 [WI/q RI V7tln WI8 selectionon it will diminishand thechar- - q) W2/W] V21n acters in other environmentswill begin kG21 G22! [(1 V2/ to proceed more directlytoward their op- (9) tima. Only ifno asymmetriesexist in the variances,the selection parameters, or the Noting that W2 is not a functionof 2I, initial distance fromthe phenotypicop- so that a = 0, W2/bfl timumin the two environments,will the (q W1/W)VIln WI evolutionary trajectories of the mean phenotypesproceed directlytoward the - (q/ W)(b W1/b21) optimumregardless of the cross-environ- - (1/I )[q W1/321 ment geneticcorrelation. This degree of - symmetryis unlikelyin natural popula- + (1 q)b W2/321] tions. Note that maladaptive correlated W Vlln (10) responses can occur whetherthe char- and similarlyfor the second termin the acters are selected in the same or in dif- selectionvector on the rightside of (10). ferentdirections (compare Fig. 2B, D). Hence, As in softselection, adaptation will oc- cur firstin the most abundant environ- -A (GGI G12\ ment, and only later in rare ones (Fig. A2 G21 G22/ 3B). When individuals are initiallymuch betteradapted to one environmentthan *.(V)ln[q W, + (1 - q) W2]. the other,Figure 3D illustratesthat high genetic correlations between character (11) statescan cause evolution away fromthe G x E AND EVOLUTION 515 optimum for the characterin the envi- mentbecause the genes which determine ronmentto which individuals were ini- them will be carried by migrantsto the tiallybetter adapted. alternateenvironment where they will be Because the equations for hard selec- expressed.For thetwo environmentcase, tion (9, 1 1) are non-linear,eigenvalues of four variables are definedfor each trait a linearized version describe the rates of of interest:ziJ is the characterstate which evolutiononly when thejoint mean phe- is expressed in the ith environmentbut notypeis near the optimum.In thatcase, carried by individuals located in the jth the weightingsare roughlyconstant and environment.When i =#j, the character the eigenvalues can be approximatedby stateis not expressedand will evolve only thosealready presented for soft selection. by correlatedresponses to selection on Numerical iterationsof the hard selec- the expressedstate. Thus, z11is the char- tion models show that the rates are in- acterstate which is both expressedin en- deed close to those under softselection. vironment 1 and carried by individuals The evolutionaryrates in the examples located there,while z21is the value of the can also be roughlycompared by using characterwhich is expressedin environ- the generationmarkers on the figures. ment 2, but which is carriedby individ- Using a proof analogous to (6), it can uals located in environment1. Therefore, be shown thatthe joint mean fitnessun- only z11 and z22 are exposed to direct der hard selectionwill also increaseuntil selection, while z12 and z21 are unex- an equilibrium is reached in each envi- pressed. Note that in a subdivided pop- ronmentwith the mean phenotypeat the ulation, unexpressed characters may optimum. However, as in softselection, temporarilydiverge from the versions mean fitnessin one of the environments expressedin the otherenvironment (i.e., can temporarilydecrease under hard se- Z21 =# z22 and 212 =# 2wI). As before,the lection during the course of evolution. gradientsof mean fitness,V1ln W and This decrease can be pronounced,tend- V21nW2, are functionsonly of 2 1I and 222 ing to occur in the rare environmentor respectively. the one in which mean fitnessis initially Individuals migrate before selection lowest. with equal propensityand enterthe two environmentsrandomly in proportionto GeneticModels of Subdivided theirrepresentation in the migrantpool. Populations Mating,reproduction and selectionthen occur in each environment.Under soft Most theoreticaltreatments of subdi- selection, niches contributeto the pool vided populations have noted that pop- in constantproportions. Under hard se- ulation divergenceis increasinglylikely lection,the contributionof each niche to withreduced migration (Maynard Smith, the migrantpool varies through time, 1966; Dickinson and Antonovics, 1973; weightedboth by niche frequency(q) and Felsenstein, 1981). Subdivided popula- by local relative mean fitness(WilW). tion models of evolutionin two environ- Thus, the hard selectionmodel is equiv- ments were made to determinewhether alent to a form of group selection me- restricted migration can decrease the diated by differentialmigration. If selec- constrainingeffects of high genetic cor- tionis assumed to be weak and themeans relations across environmentsand thus in the two populations remain similar, facilitatelocal adaptation. then the matrixof geneticvariances and When thepopulation is subdivided,the covariances can be assumed to remain distributionsof genotypesmay differin nearlyconstant throughout the course of the two environments.As before,each evolution. If the means of the subpopu- characterstate is expressed in only one lations do become appreciablydifferent, environment.However, the unexpressed theseapproximations will not be veryac- statesmust be followedin each environ- curate because they are based on a nor- 516 S. VIA AND R. LANDE

mal distributionof phenotypes before se- pressed in each environmentmight dif- lection. fer. From the derivationsin the Appendix, If any of threewell definedconditions the dynamic equation for phenotypic pertain, however, the vectors of mean change under soft selection in a subdi- phenotypesin thesubpopulations will not vided population is be equal at equilibrium: 1) if thereis no migrationamong environments;2) if ge- /G11G12 O O netic variance in one of the character (A211 is exhausted; or 3) if the genetic AZ21 = G21 G22 0 0 states is ? 1. A212 0 0 G11 G12 correlationacross environments Moreover,if there is a cost to phenotypic 0 0 G21 G22 A5222 plasticity,the population may also equil- ibrate away from the optimum. These [1 -m(l - q)]V1lnWV cases are discussedbelow. Finally,if there m(l - q)V21nW2 is very low migration,low geneticvari- mqV IlnW11 ation, or a veryhigh geneticcorrelation, (1 - mq)V2nWj it may take so long to reach equilibrium thatthe joint optimummay move before m(1 q)(21, -212) it is attained. Thus, geographicalvaria- tion in the vector of mean breedingval- Iq(212 - 21l1) ues could be visible fora verylong period ifthe correlationstructure -22 1) oftime genetic \q(222 is unfavorable. (12) To determinethe rate of approach to thejoint optimumunder population sub- where m is the proportionof each pop- division, a stabilityanalysis of a sym- ulation which is composed of migrants metrizedversion of this model with q = each generationand selectionis assumed 0. 5 and the same patternof variationand to act independentlyin each environ- selection in each niche was performed ment. This independence allows the (see theAppendix). From theeigenvalues combination of Equations A6 into this given in the Appendix, it can be shown single equation, with zeros in the off-di- thatif migrationis a much strongerforce agonal submatricesof G. With panmixis, than selection [m > G/(W2+ P)], geo- m = 1 and A21 = qA211+ (1 - q)A212. graphicalvariation in the vectorof mean Then (12) reduces to (5), the basic equa- breedingvalues is reduced on the time tion forsoft selection in a singlepanmic- scale of 1/m generations;as migration tic population. increases,the time required to homoge- At equilibrium,the populations under nize the populations geneticallyis re- selection in differentenvironments will duced. In this case, as in the panmictic generallyconverge, not necessarilyto the model, the time scale forthe approach of same optimum, but to the same vector the mean phenotype to the joint opti- of mean breeding values, so that indi- mum is roughlythe largerof 2(W2 + P)/ rectlyselected characters attain the same G( 1 y) generations.In contrast,if se- value as theirdirectly selected counter- lection is the strongerforce (G/(W2+ parts(2,1 = 2l2 and 221 = 222, see Appen- P) > m), then geographicalvariation in dix fordetails). Thus, a reciprocaltrans- the vector of mean breedingvalues de- plant or its laboratoryanalog in which creases on a time scale which is approx- samples from several populations at imatelythe largerof 2/(1 ? y)m gener- equilibriumwere raised in each environ- ations, and the mean phenotypein each mentwould reveal no geographicalvari- environment evolves toward its opti- ation in the vector of mean phenotypes mumon a timescale of (W2 + P)/Ggen- (i), even thoughthe mean phenotypeex- erations. G x E AND EVOLUTION 517

The dynamical equations forhard se- phenotypicoptimum is nearly attained. lection in a subdivided population are During this phase, genetic correlations derived in the Appendix. betweenthe phenotypesexpressed in two differentniches can cause the evolution- DISCUSSION ary trajectoryof the mean bivariate phe- Using the observation that a traitex- notypeto deviate farfrom a directcourse pressedin two environmentscan be con- toward the joint optimum (e.g., Fig. 2B, sidered to be two geneticallycorrelated D). With an unfavorable correlation characters (Falconer, 1952; Robertson, structure,populations under either soft 1959; Yamada, 1962; Fernando et al., or hard selection will evolve to the op- 1984), the models derived here link evo- timum most rapidly in common or fa- lutionin heterogeneousenvironments to vorable environments;adaptation to rare existingtheory describing the evolution- or marginalhabitats will occur later(Fig. aryeffects of geneticcorrelations on evo- 3C, D). Unless the geneticcorrelation is lution in single environments (Hazel, + 1, however, the maladaptation which 1943; Dickerson, 1955; Lande, 1979). If, may occur will be temporary.Because as we assume, thecovariance between the conditions in natural populations will characterstates expressed in differenten- rarelybe perfectlysymmetrical, it is pos- vironmentsis primarilydue to pleiotro- sible that the sort of asymmetricalevo- py (Lande, 1980a), then the geneticcor- lution illustratedin the examples may relation across environmentsestimates oftenoccur. The addition of more char- the extentto whichthe phenotypein dif- acters or environmentsmay furtherre- ferenthabitats is determinedby the same tard the course of evolution. alleles actingin the same way (Falconer, The differencesbetween the evolution- 1981; Pirchner,1983). A significantge- arypatterns shown in the figuresare pri- notype-environmentinteraction reveals marilydetermined by the relative posi- only that the genetic correlationacross tions of the initial mean phenotypeand environmentsis less than + 1; the reso- the joint optimum, not by whetherthe lution of genotype-environmentinter- optima in the two environmentsare the action is thus too poor fora quantitative same or different.If the phenotypicop- understandingof its evolutionaryeffects tima differacross environments,then (Robertson, 1959). these models describe the evolution of Two typesof selectionhave been con- phenotypicplasticity. The case in which sidered in coarse-grainedenvironments the optima in the two environmentsare where individuals experience only one the same is a special case of this more selectionregime. Under softselection, the generalevolutionary problem. fractionof the total population contrib- These models illustratethat when con- uted fromeach environmentis constant, ditions are such that the joint optimum while under hard selection, the contri- is attained by the population at equilib- butionof each nichedepends on themean rium,genetic variation will not be main- fitnessof individuals in it relativeto that tained in quantitativecharacters by en- in other environments.Under hard se- vironmental heterogeneity. At the lection,even slightasymmetry in the dis- optimum, the population will be under tance ofthe characters from their optima, bivariatestabilizing selection, which can- in the genetic parameters,or in the in- not maintaingenetic variation. Variance tensityof selection in the two environ- around the equilibriumnorm of reaction mentsleads to an asymmetryin the con- will,however, be generatedby mutation. tribution of the environments to the The genotype-environmentinteraction mating pool (9): the environmentwith in fitnesswhich this mutation produces the highestinitial mean fitnesswill con- will equilibrate in a mutation-selection tributeincreasing numbers of individuals balance (Lande, 1976, 1980a; Turelli, to thetotal population until its individual 1984). 518 S. VIA AND R. LANDE

Severalcircumstances exist which can ment,migration among subpopulations preventthe joint optimumfrom being maybe greatlydecreased, speeding local attainedat equilibrium,precluding the adaptationand populationdivergence. evolutionof a selectivelyadvantageous In extremecases, this could lead to such level of phenotypicplasticity. 1) If the low migrationthat the subpopulations geneticcorrelation of characterstates in couldbecome reproductively isolated be- two environmentsis + 1, thenG is sin- forethe joint optimumis attained.5) If gularand no geneticvariation exists for thereis a "cost" to plasticity,then selec- evolutionin one of the principledirec- tionwill act to makethe character states tions in phenotypespace (one of the expressedin differentenvironments more eigenvalueswill be zero). Almost any similar.Eventually, this force is expected perturbationfrom the joint optimum will to opposethe force of selectionfor plas- thenresult in a permanentdisplacement ticity,resulting in an equilibriumwith ofthe population from the optimum. In themean phenotype located somewhere thiscase, therewill be spatialvariation betweenthe joint optimumand a point in the directionof selectionand a cor- at whichthe mean phenotypes expressed respondingincrease in the geneticvari- in each environmentare the same (Via, ance (Slatkin,1978; Via and Lande,un- unpubl.).The frequencyof any of these publ.). In an analogous way, if n cases relativeto the situationin which charactersare simultaneouslyselected to the optimumis reachedat equilibrium increasein a rangeof environments, then remainsas an empiricalquestion. an averagecorrelation of - 1 /(n - 1) can The possibilitythat many field popu- haltthe response to selection(Dickerson, lationsmay not yet be at equilibriumdue 1955). Thus,even smallnegative corre- to an unfavorablecorrelation structure lationsamong a largesuite of characters complicatesthe interpretation of exper- simultaneouslyselected in severalenvi- imentaldata: differencesobserved be- ronmentscan virtuallypreclude evolu- tweenthe mean phenotypesin different tionarychange. Genetic correlations may environmentsmay reflect a deflectionof thusplace an upperlimit on thenumber theevolutionary trajectory due to genetic of environmentsto whicha population correlationsamong homologoustraits can simultaneouslyadapt. 2) If genetic ratherthan indicating variation in phe- variationin eithercharacter state is ex- notypicoptima in differenthabitats. This hausted,evolution toward the joint op- problemcan be particularlyimportant timumwill cease. Thus, some optima whenthe environments under consider- may neverbe attainedsimply because ationare theproduct of relatively recent thereis no geneticvariation in thatdi- man-madedisturbance (e.g., Snaydon, rection.This is anotherway of describing 1970; Antonovics,1971). In such situa- a "developmentalconstraint." 3) If no tions,non-equilibrium phenotypic states gene flow occurs betweensubdivided maybe morethe rule than the exception. populations(m = 0 in Eq. 12), popula- A theoryof evolution in heterogeneous tionsin differentenvironments will nev- environmentsbased on geneticcorrela- erconverge to the joint optimum because tionmay allow some long-standingeco- the stateswhich are not expressedin a logicalhypotheses to be tested.For ex- givenenvironment are not subjectedto ample, host plantsoften form discrete selectionthere. In thiscase, if individuals patcheswhich act as differentselective fromeach environmentwere allowed to environmentsfor herbivores (e.g., Gil- developboth in thehome and in theal- bert, 1978). Geneticallybased "trade- ternateenvironment, geographical vari- offs"in the expression of traits like growth ation in the vectorsof breedingvalues or feedingbehavior on differenthost wouldbe expectedat equilibrium.4) If plants(estimable as geneticcorrelations individualscan selecttheir habitat, or acrosshosts) have been thought to be im- tend to returnto theirhome environ- portantin the evolution of herbivore spe- G x E AND EVOLUTION 519 cialization (Rausher, 1984; Futuyma, at a selectiveoptimum in each environ- 1985).By formalizing ecological hypoth- ment.The expectedequivalence of the eses like the "tradeoff"concept, these phenotypicmean vector between subdi- models suggestthe experimentaldata vided populationsprovides an experi- whichshould be takento testtheir role mentalcriterion for equilibrium: if sam- in theevolution of populationsin vari- plesfrom several environments which are able environments. testedin all environmentshave signifi- Estimatesof the geneticcovariance cantlydifferent mean vectors, then it can structurewill, of course,be mostuseful be hypothesizedthat further evolution in the interpretationof fielddata when shouldbe expectedunless there is no gene accompaniedby estimatesof the shape flowor thegenetic covariance matrix is ofthe selective surface and theintensity singular. ofselection (cf. Lande and Arnold, 1983), as well as by informationon the extent ACKNOWLEDGMENTS of migrationand therelative niche pro- We thank J. Cheverud, D. Futuyma, portions.For example,if one observes M. Kirkpatrick,T. Price,M. Slatkin,and phenotypicdifferentiation among popu- M. Wade for comments on the manu- lations inhabitingdifferent environ- scriptand discussionsof this topic. Sup- ments,the estimationof a directional portwas providedby National Research componentto selectionin thevarious en- ServiceAward GM-09597 to S. Via and vironmentswill revealthat the pheno- U.S. Public Health Service grant GM- typesobserved are notat theindividual 27120 to R. Lande. optimaand can also providean estimate of the location of the joint optimum. LITERATURE CITED Then,estimates of non-zero genetic vari- ANTONOVICS, J. 1971. The effectsof a heteroge- ation,a geneticcorrelation between -1 neous environmenton the of natural and + 1, and non-zeromigration can be populations. Amer. Scientist59:593-599. BEARDMORE, J.A., AND L. LEVINE. 1963. Fitness used to eliminatethe possibility that the and environmentalvariation. I. A studyof some populationis atan equilibriumaway from polymorphicpopulations of Drosophila pseu- thejoint optimum. In thiscase, one could doobscura. Evolution 17:121-129. inferthat evolution of the normof re- BRADSHAW, A. D. 1965. Evolutionarysignificance of phenotypicplasticity in plants. Adv. Genet. actionis stillin progress. 13:115-155. The subdividedpopulation model sug- BULMER, M. G. 1971. Stable equilibria under the geststhat the vectors of breedingvalues two island model. Heredity27:321-330. of expressedand unexpressedtraits es- CHRISTIANSEN, F. B. 1975. Hardand soft selection in a subdivided population. Amer. Natur. 109: timatedfrom samples of populationsin 11-16. differentenvironments are generallyex- DEMPSTER, E. 1955. Maintenanceof genetic het- pectedto be equal at equilibrium,even erogeneity.Cold Spr. Harb. Symp. Quant. Biol. thoughthe mean characterstates ex- 20:25-32. pressedmay differ across environments. DICKERSON, G. E. 1955. Geneticslippage in re- sponse to selectionfor multiple objectives. Cold Thus, geneticvariation among popula- Spr. Harb. Symp. Quant. Biol. 20:213-224. tions,manifest in variationin thevector 1962. Implications of genetic-environ- ofmean breeding values, cannot be used mental interactionin animal breeding. Anim. as a criterionfor adaptation to special Prod. 4:47-64. environmentalcircumstances. However, DICKINSON, H., AND J. ANTONOVICS.1973. The- oreticalconsiderations of sympatricdivergence. becausesome phenotypic variance about Amer. Natur. 107:256-274. the mean will usuallyexist in each en- FALCONER, D. S. 1952. The problemof environ- vironmentat equilibrium,techniques for ment and selection.Amer. Natur. 86:293-298. estimatingthe existence and intensityof 1981. Introductionto Quantitative Ge- netics,2nd ed. 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FERNANDO, R. L., S. A. KNIGHTS, AND D. GIANOLA. response to environmentalheterogeneity. Na- 1984. On a method of estimatingthe genetic ture250:572-574. correlationbetween characters measured in dif- MITTER, C., D. J.FUTUYMA, J. C. SCHNEIDER, AND ferentexperimental units. Theoret. Appl. Genet. J. D. HARE. 1979. Genetic variationand host 67:175-178. plant relationsin a parthenogeneticmoth. Evo- FUTUYMA, D. J. 1985. The role of behaviorin lution 33:777-790. host-associated divergence in herbivorous in- PIRCHNER, F. 1983. PopulationGenetics in Ani- sects. In M. D. Huettel (ed.), EvolutionaryGe- mal Breeding,2nd ed. Plenum, N.Y. netics of InvertebrateBehavior. Plenum, N.Y. POWELL, J. R. 1971. Genetic polymorphismsin In press. varied environments.Science 174:1035-1036. GAUSE, G. F. 1947. Problemsof evolution. Trans. POWELL, J.R., AND H. WISTRAND. 1978. The ef- Conn. Acad. Sci. 37:17-68. fectof heterogeneousenvironments and a com- GILBERT, L. G. 1978. Development of theoryin petitoron geneticvariation in Drosophila.Amer. the analysis of insect-plantinteractions. Anal. Natur. 112:939-947. Ecol. Syst. 8:117-154. RAUSHER, M. D. 1984. Tradeoffsin performance HAZEL, L. N. 1943. The genetic basis for con- on differenthosts: Evidence from within and structingselection indices. Genetics 28:476-490. between site variation in the beetle, Deloyala HEDRICK, P. W.,M. E. GINEVAN, AND E. P. EWING. guttata.Evolution 38:582-595. 1976. Geneticpolymorphism in heterogeneous RENDEL, J. M. 1943. Variations in the weightsof environments.Ann. Rev. Ecol. Syst. 7:1-32. hatchedand unhatchedduck's eggs.Biometrika JAMES, J. W. 1961. Selection in two environ- 33:48-58. ments. Heredity 16:145-152. ROBERTSON, A. 1959. The samplingvariance of JAENIKE, J., AND D. GRIMALDI. 1983. Genetic thegenetic correlation coefficient. Biometrics 15: variation forhost preferencewithin and among 469-485. populations of Drosophila tripunctata.Evolu- SCHARLOO, W. 1964. The effectof disruptive and tion 37:1023-1033. stabilizingselection on theexpression of cubitus LACY,R. C. 1982. Niche breadthand abundance interruptusin Drosophila. Genetics 50:553-562. as determinantsof genetic variation in popu- SCHMALHAUSEN,I. I. 1949. Factorsof Evolution: lations of mycophagousdrosophilid flies(Dip- The Theory of StabilizingSelection. Blakiston, tera: Drosophilidae). Evolution 36:1265-1275. Philadelphia, PA. LANDE, R. 1976. The maintenanceof genetic vari- SLATKIN, M. 1978. Spatialpattems in thedistri- abilityby mutationin a polygeniccharacter with butionsof polygenic characters. J. Theoret. Biol. linked loci. Genet. Res. 26:221-235. 70:213-228. . 1979. Quantitative genetic analysis of SNAYDON, R. W. 1970. Rapid population differ- multivariateevolution, applied to brain: body entiationin a mosaic environment.I. Response size allometry.Evolution 33:402-416. ofAnthoxanthum odoratum populations to soils. . 1980a. The genetic covariance between Evolution 24:257-269. charactersmaintained by pleiotropic mutations. TURELLI, M. 1984. Heritablegenetic variation via Genetics 94:203-215. mutation-selectionbalance: Lerch's zeta meets 1980b. Sexual dimorphism,sexual selec- the abdominal bristle.Theoret. Popul. Biol. 25: tion and adaptation in polygenic characters. 138-193. Evolution 34:292-305. VIA,S. 1984. The quantitativegenetics of polyph- . 1982. A quantitativegenetic theory of life agy in an insect herbivore.II. Genetic correla- historyevolution. 63:607-615. tions in larval performancewithin and across LANDE, R., AND S. J. ARNOLD. 1983. The mea- host plants. Evolution 38:896-905. surementof selection on correlatedcharacters. WRIGHT, S. 1968. Evolutionand theGenetics of Evolution 37:1210-1226. Natural Populations, Vol. 1. Genetic and Bio- LEVENE, H. 1953. Geneticequilibrium when more metric Foundations. Univ. of Chicago Press, than one ecological niche is available. Amer. Chicago, IL. Natur. 87:311-313. 1969. Evolution and the Genetics of Nat- LI, C. C. 1955. The stabilityof an equilibrium ural Populations. Vol. 2. The Theory of Gene and the average fitnessof a population. Amer. Frequencies. Univ. of Chicago Press, Chicago, Natur. 89:281-295. IL. MACKAY, T. F. C. 1981. Genetic variation in YAMADA, Y. 1962. Genotypex environmentin- varyingenvironments. Genet. Res. 37:79-93. teractionand genetic correlationof the same MAYNARD SMITH, J. 1966. Sympatricspeciation. traitunder differentenvironments. Jap. J. Ge- Amer. Natur. 100:637-650. net. 37:498-509. McDONALD, J.F., AND F. J.AYALA. 1974. Genetic CorrespondingEditor: C. Strobeck G x E AND EVOLUTION 521

APPENDIX

SubdividedPopulation Models: Soft Selection. -As describedin thetext, a singlecharacter which is expressedin twoenvironments is considered. Assume that mating occurs within environments and that productionof offspring is followed by selection which proceeds separately in eachenvironment. Then in thefirst environment, the change in themean phenotype after selection is

(Al =(GI, G12) (VIlnWV) (A 1) so thatthe mean breeding values, x*,,, after selection are *1 GI, G12 lnw l G 2 z, ) (A2) kx!l1 21- G21G22 (V1lnJV0 ,I\+ Z2 Similarly,the breeding values after selection in environment2 are 12=_I G12 0 f1 22 G21G22,\V21nVW2 /(2\ (A3) In softselection, each environment contributes migrants to a migrantpool according to its representation in thetotal environment. Thus, the mean breeding values in themigrant pool are

q Xm) (x 1 ) (I q) 1X22) (A4) Aftermigration, migrants compose the same fraction, m, of the population in eachenvironment, so that aftermigration, the breeding values in environment1 are

(x21) m) x 1 ) M m)

- [1 - m(l - q)](X*i) + m(lI q) (X*2) (A5) and similarlyfor X',2 and XZ22.Using (A2) and (A3), the changesin the mean phenotypesin thetwo environmentsare

- 2 [1 - m(l - q)] GIIVllnW-) (I q) G12V2lnW2_ m(l - q) (fl f2) (A6) and

AzI2) = (1 mq)(G12V2lnW2) + m (G11VIlnJW'Vm\ (z12- -fl (Az22/ - ~\G22V2lnW2 mq\GVlnV,J- mq Rearrangementleads to (12) in thetext. Equilibria.-At an equilibrium,there is no changein themean breeding values. Hence,

- GI,G12 ~[I1- m(1 q)]VIIVln m(1 q) fl -1 G21 G22,\ m(1 - q)V2lnW2 m( 2 - 22 and

(GI,G12\ mqVIlnn m (12 -2ll G21 G22, (I - mq)V21n W2 2 - (2A Thus,if 0 < q < 1, IGI # 0 and m #60, at equilibrium:

(VIlnW,)=0 and I (=212 (A8) VnJV21nW Iz1, \222 Thisresult means that the mean phenotype is at a local maximumof W in eachenvironment and there is no geographicvariation in thevector of mean breeding values (z) whensamples of individuals derived fromdifferent environments are raisedin each ofthe environmental alternatives. Stability.-With Gaussianfitness functions, stability properties can be determinedfor the symmetrical 522 S. VIA AND R. LANDE case withG1 = G= G, GI2 = G, = yG,wI = W2 = w,PI = P22 = P, q = 0.5, and arbitrarilysetting 01 = 02 = 0. Then,(12) is

(M1A \/, - (1 + )r)m/2 m/2 -yom/2 0 \/?ll\ AZ12 = (1 + O)rm/2 -m/2 -yo(I - m/2) 0 112 (A9) A222 -yOm/2 0 d - (1 + O)m/2 m/2 Z22I A21 -y(l - m/2) 0 (1 + O)rm/2 -rm/2 \21 where, - G/(W2+ P). The foureigenvalues of the matrix are always real and negative, with a magnitude less thanor equal to unity: X= 1/2{[I - (1 - y)m/2]0 - m ? \([1 - (1 - y)rm/21(3- M)2 + 2(1 + y)rmn} X= 1/2{[l- (1 + y)m/2],B- m ? \/([1 - (1 + y)m/2]0- M)2 + 2(1 - y)mf}. (A10) Thus,the equilibrium with the mean phenotypes at theoptimum in eachenvironment is asymptotically stable,and no geographicalvariation is maintainedin thevector of mean breeding values unless y ?1 or m= 0. Hard Selection.-The hardselection model is constructedsimilarly to thesoft selection one, except thatthe contribution ofeach niche to themigrant pool is weightedboth by its proportional representation and byits mean fitness:

(X,m)= [qW1/ + [(1 -q) W2/W12(x2) (Al 1) whereW = qJW1+ (1 - q) W2.The selectionin eachenvironment before migration is thesame as in soft selection(Eqs. A2 and A3), and therest of the model proceeds in thesame way as in Eqs. A5 and A6 to producethe dynamic equations for hard selection.