Proc. Natl. Acad. Sci. USAVol. 84, pp. 7164-7168, October 1987Evolution

Toward a theory for the evolution of cultural communication:Coevolution of signal transmission and reception

(gene-culture coevolution/learning/two-locus theory)

KENICHI AOKIt AND MARCUS W. FELDMANt§tNational Institute of Genetics, Mishima, Shizuoka-ken 411, Japan; and tDepartment of Biological Sciences, Stanford University, Stanford, CA 94305

Communicated by Paul R. Ehrlich, June 11, 1987

ABSTRACT A haploid sexual two-locus model of gene-culture coevolution is examined, in which a dichotomous phe-notype subject to natural selection is transmitted verticallywith probabilities dependent on the chosen parent's genotypeand phenotype and the offspring's genotype. Stability condi-tions for the genetically monomorphic corner equilibria areobtained. In a specialization of this general model, one locuscontrols the transmission and the other controls the receptionof adaptive information. The corner and edge equilibria of thisdoubly coevolutionary model are fully analyzed, and condi-tions for transmission and reception to coevolve are derived interms of the efficiency of vertical transmission, the selectiveadvantage gained from possessing the information, the costs oftransmission and reception, and the recombination fractionbetween the two loci. Possible applications of the model are tothe evolution of semantic alarm calls in vervet monkeys andthe phonetic aspects of human language. In a third model withdiploid genetics, we consider the initial increase of culturaltransmission from a mutation-selection balance in which theadaptive phenotype is the consequence of a dominant gene atone locus. A second gene controls the transmission of the phe-notype in such a way that a new mutant at this second locuspermits learning of the adaptive phenotype from a parent whohas it. This new mutant cannot increase when rare.

Animal communication has been defined in a number ofways (e.g. refs. 1-3), and the definition adopted determinesto a large extent the appropriate model for the evolution ofcommunication. When communication involves the culturaltransmission of information, as, for example, with humanlanguage and perhaps the semantic alarm calls of vervetmonkeys (4, 5), the evolution of communication becomes aspecial case of gene-culture coevolution (6-13). The prob-lem was explicitly addressed by Cavalli-Sforza and Feldman(9), who showed that initial increase of communication isslow unless it occurs primarily among relatives. This wouldseem to imply that a different model is necessary to deal withthe evolution of mating signals in outbreeding species (ref. 2,p. 224).These previous studies have considered the effects of a

single gene on transmission and have not attempted an anal-ysis of the process of transmission itself. In the present studywe examine the role of genetic effects on the transmissionand reception of information, the possession of which con-fers a selective advantage.Communication involves two physiological processes,

transmission and reception of a signal. In our previous stud-ies of cultural inheritance we have referred to these as"teaching" and "learning," respectively (see, e.g., ref. 14).In other contexts their meaning may be more directly phys-iological as in the excretion of a pheromone and its recogni-tion. At whatever level of biological determination the pro-

cesses are viewed, they can be regarded as examples of a"lock-and-key" pairing in which the evolution of each sepa-rate component is likely to have been closely related to thatof the other.A good example of the dual nature of communication is

the alarm call. The alarmist must give the alarm and the in-tended receiver must interpret it as such. In ground squirrelsalarm calls may be directed primarily from mother to off-spring (15). Another interesting example of mother-off-spring communication is provided by the stem-dwelling eu-menine wasps (ref. 16; see ref. 2, p. 186, for summary). Thestructure of the nest contains a "message" from the motherwasp to its larvae. If conditions are experimentally altered sothat a larva misinterprets this message, it dies in a vain at-tempt to burrow out of its cell upon eclosure. In these exam-ples, communication occurs between relatives, and the in-formation transmitted is clearly meaningless without the ca-pacity to receive it. Phonetic aspects of human languagecontain the other element we wish to explicitly model-i.e.,cultural transmission.

It is reasonable to suppose that both components of com-munication are subject to variation and that the set of genesthat affect transmission is, at least to some extent, distinctfrom that affecting reception. We shall model this in the sim-plest way by assuming that each process is under the controlof a single gene and that the two genes are linked, with c, therecombination fraction, between them. These genes affectthe transmission and reception of a dichotomous phenotypethat is subject to natural selection. In our most general mod-el, the phenotypes are acquired either through vertical trans-mission (14) or innately. Within this context we shall consid-er the joint evolution of alleles that allow transmission andreception and show the effect of linkage between them. Inthe process a framework for the more general study of gene-culture coevolution, with two linked genes affecting thetransmission, will be developed.

Model 1: Two-Locus Haploid Genetics and VerticalTransmission

This is a haploid two-locus version of that developed in ref.6. Consider two genes with alleles A and a at the first andalleles B and b at the second. Each genotype may appear inone of two phenotypes so that the eight phenogenotypes areAB, Ab, alB, ab, AB, Ab, aB, ab, and these appear with fre-quencies ul, u2, U3, U4, v1, v2, V3, V4 in the population. Thetwo phenotypes are referred to as "bar" and "not bar" andthe relative (Darwinian) fitness of bar to not bar is 1 + s: 1.Mating occurs randomly and from each mating the offspringgenotypes result with the standard probabilities, using therecombination fraction c. The phenotype of each offspring isdetermined by choosing one of its parents at random andassigning to the offspring the bar phenotype with a probabili-

§To whom reprint requests should be addressed.

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ty that is governed by the chosen parent's genotype and phe-notype and the offspring's genotype. Thus if the chosen par-ent's genotype is i(i = 1, 2, 3, 4), the child's genotype isj(j =1, 2, 3, 4), and the chosen parent's phenotype is 1 (for bar) or2 (for not bar), then the probability that the child is bar is bjjor bij 2, respectively. Thus we have transmission matrices B1= ||bij,1i and B2 = Ijbij,2I1 for parents of phenotype bar and notbar, respectively. If lj(b1j,l - bij2) > Ej(b2j,1 - b2j,2), forexample, then AB transmits the trait better than Ab. Fromthe matings AB x ab and ab x AB, for example, which eachoccur with frequency u1v4, the offspring are AR, Ab, aB, ab,AB, Ab, aB, ab with probabilities (1 - c)[(b1j,1/2) +(b4l,2/2)]/2, c[(b12,1/2) + (b42,2/2)]/2, c[(b13,1/2) +(b43,2/2)]/2, (1 - c)[(b14,j/2) + (b44,2/2)]/2, (1 - c)[l -

(bll,1/2) - (b4l,2/2)]/2, c[l - (bl2,j/2) - (b42,2/2)]/2, c[l -

(bl3,1/2) - (b43,2/2)]/2, (1 - c)[1 - (bi4,1/2) - (b44,2/2)]/2,respectively. If selection takes place after transmission, thenthe recursions for AB and AB, for example, are

TU' = (1 + s){(biijui + bl1,2V1)[1 + U1 + V1 - C(U4 + V4)]/2+ (b2101U2 + b21,2v2)[1U + V1 + C(U3 + V3)]/2+ (b31,1u3 + b31,2v3)[u1 + V1 + C(U2 + V2)]/2+ (1 - c)(b41,0u4 + b41,2v4)(u1 + vj)/2}, [l.la]

TV' = [(1 - bjj,1)uj + (1 - bll,2)V1][1 + U1 + V1 - C(U4 + v4)]/2+ [(1 - b21,1)u2 + (1 - b2l,2)vA[ul + V1 + C(U3 + V3)]/2+ [(1 - b31,1)u3 + (1- b3l,2)v3][u1 + V1 + C(U2 + V2)]/2+ (1 - c)[(1 - b41,l)u4 + (1- b41,2)v4](u1 + v1)/2, [1.lb]

where T is the sum of the right sides of the eight recursions inU1, V1, U2, V2, U3, V3, U4, V4.Remark: If the transmission is uniparental-i.e., from, for

example, the mother-then the same recursions obtain.We proceed to an analysis of the chromosomal fixation

states. When genotype AB is fixed, for example, the equilib-rium values of the frequency of the phenogenotype AB arethe roots of the quadratic [see Cavalli-Sforza and Feldman(14)]

u2s(b11 - b11,2) + u1[1 + sbjj,2- (1 + s)(b11,1 - b11,2)] - b1j,2(1 + s) = 0. [1.2]

Only the positive root u1 of Eq. 1.2 is of interest (and whenb11,2 > 0 this is the only valid root of Eq. 1.2) and is stable inthe interior of the u1 - v, boundary of the frequency space.The initial increase of the six phenogenotypes Ab, Ab, aB,aB, ab, ab in the neighborhood of u'2, vl = (1 - t21) is gov-erned by the 6 x 6 local stability matrix L:

£' = Le, [1.3]

where T. = (,eu2,v EV3 EU EV4) is the vector of smallfrequencies of die six phenogenotypes and the structure ofLis such that the six eigenvalues emerge as three pairs, onepair for each of the rare chromosomes. Each pair comprisesthe roots of a quadratic. The roots of the Ab and aB quadrat-ics are less than unity provided

sju'jbjjj - (bj,1/2) - (bjj/2)]+ f1[bll,2 - (blj,2/2) - (bfi,2/2)]} > 0, [1.4]

with j = 2 for Ab, j = 3 for aB. Clearly the initial increase of

ab near fixation of genotype AB depends on c. In fact, thetwo stability eigenvalues for ab are less than unity if

sludl[blil - (b14,1/2) - (b44,1/2)] + Vl[bll,2 - (b14,2/2)- (b44,2/2)]} + c[l + (b44,1/2)(1 + s) - (b44,2/2)+ (s/2)(b14,1 1 + b14,p2 ) - (1 + s i,)(b44,1 - b44,2)]+ c2(1 + si)1)(b44,1 - b44,2)/2 > 0. [1.5]

The dependence of the inequality (Eq. 1.5) on c is illustrat-ed by setting b11,2 = bl4,2 = b44,2 = 0. Then if

(b14,1/2) + (b44,1/2) > bl1il > (b14,1/4) + (b44,1/4), [1.6]

the (a1, vi1) equilibrium is unstable at c = 0 and is stable for c= 1/2. It may also be shown that if this chromosomal fixa-tion is stable for some c0 it is stable for c > c0.The most natural specialization of model 1 takes bij,2 = 0.

This is natural in the context of communication because un-less the transmitter has the bar trait no transmission is ex-pected. Of course this ignores other modes of transmission,such as horizontal and oblique that have been discussed ex-tensively elsewhere (14). With b11,2 = 0 the polymorphicroot of Eq. 1.2 is valid and stable in the u1-vl boundary if (1+ s)b1l,l > 1. When (1 + s)bll,l < 1 the valid root of Eq. 1.2is al = 0, so that the population is entirely not bar at equilib-rium. In this case the bar phenotype can enter the populationonly if one of the other three chromosomal types invades.From the three local stability quadratics we see that the con-ditions for Ab, aB, and ab to increase when rare are, respec-tively,

(1 + s)b22,1/2 > 1, (1 + s)b33,1/2 > 1,

(1 - c)(1 + s)b44,1/2 > 1. [1.7]

These stability analyses can be repeated for each of the chro-mosome fixation states and the same results obtain with theappropriate subscript changes among the transmission pa-rameters b11,l and bUj,2.

In addition to equilibria where a single chromosome isfixed, there is a class of boundary equilibria where an alleleis fixed. For example, it is possible to have just AB, AB, Ab,Ab present at equilibrium. These are called (genotypic) edgeequilibria in two-locus population genetic theory, but in thepresent phenogenotypic context they are boundary points ofan eight-dimensional simplex. When bij,2 = 0 for all i and j,and (1 + s)bll,l > 1, (1 + s)b22,1> 1, for example, there arestable phenotypic polymorphisms when each of genotypesAB and Ab are fixed. If, in addition, the conditions corre-sponding to Eq. 1.4 hold-namely,

b11,1 - (b12,1/2) - (b22,1/2) < 0,b22,1 - (b2l,1/2) - (bil,1/2) < 0, [1.8]

then neither chromosomal fixation in AB nor in Ab is stable.Thus we would expect some genetic polymorphism of ABand Ab and phenotypic dimorphism to be stable.From Eqs. 1.6 and 1.8 we see that when the two genes are

tightly linked, the condition for a rare chromosome to in-crease is that it increase the average level of vertical trans-mission of the adaptive trait. We also see that linkage mayplay a role in this increase. Together with our earlier results(8), these results suggest that this haploid model does notproduce a polymorphism with both phenotypes and all fourchromosomes present.

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Model 2: A Simple Coevolutionary Model with SelectiveCosts to Transmission and Reception

When the allele A produces in parents a greater ability totransmit the signal than a, and B produces a greater ability toreceive the signal than b, we may reduce the number oftransmission parameters. Thus we may set bU,2 = 0, and

b11,1 = b21,1 = b13,1 = b23,1 = bTR,

b31,1 = b41,1 = b33,1 = b43,1 = btRb12,1 = b22,1 = b14,1 = b24,1 = bTr,

b32,1 = b42,1 = b34,1 = b44,1 = btr [2.1]

reducing the set to a total of four parameters correspondingto all combinations of parents (A-, a-) and offspring (-B,-b). In the simplest case, alleles a and b do not permit trans-mission and reception, respectively, so that bR = bT, = btr =O and we use the notation 8 for bTR.The selective advantage to possessors of the transmitted

information-i.e., the bar phenotype-remains, as before,s. We now assume that the fitness of allele A, causing trans-mission, is reduced below that of a by yT and that ofB, caus-ing reception, is reduced by YR below that of b. These fitnessdecrements are additive, and 0< yT, YR < yT + YR < 1. Theycan be viewed as the cost of supporting the necessary biolog-ical machinery for communication. A transmitter may, forexample, incur an additional cost-e.g., warning calls wouldtend to attract predators. (Note that a transmitter of the notbar phenotype transmits nonadaptive noise.)

In this model the phenogenotypes Ab and _ii do not existbecause offspring carrying b do not acquire the trait in thisspecial case. There are six possible phenogenotypes AB,AB, Ab, aB, aB, ab, and their frequencies among adults justbefore random mating are ul, vi, v2, U3, V3, V4, respectively.The life cycle is random mating, reproduction, communica-tion, and natural selection, in that order. The recursions inthe six variables, where the prime indicates the next genera-tion, are

Tu' = (1 - YT - 11R) 1(1 + s)3u1l(l + u1 + v1 - cv4) [2.2a]

Tv1 = (1 - yT - VR)[u1 + v1 - cD

- 1 ul(l + Ul + vl -CV4)]

Tv2 = (1 - YT)(V2 + cD)

Tu3 = (1 - YR)1(1 + S)fBU1(U3 + V3 + CV4)

Tv3 = (1 - YR)[U3 + V3 + cD

- 1 fui(U3 + v3 + CV4)]

Tv4 = V4- cD,

D = (u1 + V)V4 - V2(U3 + v3),

T = 1 - YT(Ul + V1 + V2) - YR(Ul + V1 + U3 + v3)1

± 2SfPUl[(l - YR)(1 + Ul + Vl + U3 + V3)

VT(++ Ul + V1 - CV4)]. [2.2h]

The corner and edge equilibria of the system of Eqs. 2.2a-

Table 1. Existence and stability of the corner and edge equilibriaof model 2

Phenogenotypes present Can equilibrium ShorthandAR AB Ab aB aB Ab be stable? nomenclature0 1 0 0 0 0 No Noneo 0 1 0 0 0 No Noneo 0 0 0 1 0 No Noneo 0 0 0 0 1 Yes Eab* * O 0 0 0 Yes EAB* * 0 * * No EB-* * 0 * * Yes EB+* * * 0 0 0 No EA

An asterisk indicates positive frequency.

2.2h can be obtained by setting various appropriate combina-tions of the six phenogenotype frequencies simultaneouslyequal to 0. They are summarized in Table 1. There are fourchromosomal fixation states in which only the phenotypenot bar is present. Four boundary equilibria are geneticallyor phenotypically polymorphic and exist for appropriate pa-rameter values. Under our assumptions on the parametervalues, three of the four monomorphic equilibria are unsta-ble, and two of the four polymorphic equilibria (denoted EB-and EA) are unstable whenever they exist. We have not beenable to determine whether any completely polymorphic equi-libria, with all chromosomes and both phenotypes represent-ed, are possible. The boundary equilibria of most evolution-ary interest here are the stable ones and we restrict our at-tention in what follows to the three denoted Eab, EAB, andEB+. Detailed derivations of existence and stability proper-ties ofEB- and EB+ are omitted but are available on request.The equilibrium Eab is monomorphic for ab. It represents

the primitive state from which transmission and receptionare assumed to have coevolved. Eab is unstable if

(1 - YT - YR)(1 + s)f3(l - c) > 2, [2.3]and it is stable if the inequality is reversed. As with model 1,Eq. 2.3 is more likely to be satisfied if c is small-i.e., inva-sion of transmitters and receivers of an adaptive trait is morelikely with tight linkage between the two loci (17).The equilibrium EAB is genetically monom2phic but phe-

notypically polymorphic. Two phenotypes AB and AB arepresent, where the frequency of AB is (14)

[2.2b]U, = [(1 + s)A - 1]/(sB). [2.4]

[2.2c] It is clear from Eq. 2.4 that EAB exists only if (1 + s),B > 1.Note that if ,3 = 1-i.e., communication is perfect-then ul

[2.2d] = 1. Straightforward local stability analysis shows that EABis stable if

YT + YR/[1l - 1/(l + S)13] < 1,

2yT/[l - 1/(l + s)18] + YR < 1,

(VT + YR)/[1 - (1 - C)/(1 + S)V] < 1.[2.2e]

[2.5a][2.5b][2.5c]

[2.2f] The equilibrium EB+ is one of a pair of polymorphic equi-libria obtained by solving a quadratic equation. Its conjugate

[2.2g] denoted EB- is unstable. Four phenogenotypes AB, AR, aB,aB are present with both A and a segregating at the transmit-ter locus but B fixed at the receiver locus. EB+ exists if

2yT/[l - 1/(1 + s)13] + yR > 1, [2.6a](opposite of Eq. 2.5b) and

8yT/(l+ s)P+ yR ' 1 for 2 < (1 + s)f3 4, [2.6b]

yT/[l - 2/(1 + s)13] + YR < 1 for 4 < (1 + s),3. [2.6c]

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The explicit formulae are

dl = M+k+, vi = m+(1 - k+), U3 = (1 - m+)l+, V13

= (1 - m+)(1 - 1+), [2.7a]

where

2yTn+ = 1 - 2yT - yR + (1 - yR)[I - 8yT/(1 - YR)(l + S)/]11/2, [2.7b]

2sk+ = (1 + s){1 + [1 - 8yT/(1 - YR)(1 + S)p]]21}, [2.7c]

1+ = (1 + s)f3m+k+/(2 + sf3m+k+). [2.7d]

The equilibrium EB+ is stable to perturbations that leave Bfixed-i.e., on the edge with V2 = V4 = 0. It can also bestable in the whole space. A sufficient condition for stabilityis

tB+ = (1 - YT - YR)(1 + s)f3(m+ + 1)/2 > 1, [2.8]

where TB+ is the mean fitness at equilibrium and m+ is givenby Eq. 2.7b. In particular, if Eq. 2.3 holds, then ?TB+ > (m++ 1)/(1 - c). And since existence of EB+ implies 0 < m+ <1, EB+ is stable if it exists and if Eq. 2.3 holds (see below).Let us now summarize the existence and stability of the

three equilibria Eab, EAB, and EB+ with reference to Fig. 1.The inequalities 2.3, 2.5a-2.5c define the boundaries of theregions in Fig. 1. RI is the horizontally hatched region, RI, isvertically hatched, and R.,, has crosses. Note that the isos-celes triangular region defined by Eq. 2.3-i.e., RI, U R.,, asdrawn-exists only if

c < 1 - 2/(1 + s), [2.9]

in which case Eq. 2.3 can be rewritten as yT + YR < 1 - 2/(1+ s)/3(1 - c). Note also that RI,, exists only if Eqs. 2.3 and2.6a are simultaneously satisfied. This requires that

c < 1 - 4/[1 + (1 + s)f3]. [2.10]

(If Eq. 2.9 is satisfied but Eq. 2.10 is not, then RI, is an isos-

1

1(1+s)B

1- 2(1+s)B(1-c)

0 1-1c 1(1+s)83YrT

FIG. 1. Existence and stability of the three biologically interest-ing equilibria of model 2. Regions are defined in the YT - yR parame-ter space. The figure is drawn for the case c < 1 - 4/[1 + (1 + s),B];in fact, (1 + s),B = 5 and c = 1/5. Region RI is hatched horizontally,RI, is hatched vertically, and RI,, is marked with crosses. Equilibri-um Eab is stable outside RI, U RI11, EAB is stable in R1 U RII, and EB+is stable in RI,,.

celes triangular region bounded above and to the right byRI.)

If (1 + s)p3 < 1, then from 2.4, 2.6a-2.6c, EAB and EB+ donot exist. In this case, from Eq. 2.3, Eab is the unique stablecorner or edge equilibrium.

If 1 < (1 + s)f3 . 2, then from Eq. 2.6a-2.6c, EB+ does notexist, but Eab and EAB exist. From Eq. 2.3 Eab is stable.From Eq. 2.5a-2.5c EAB is stable or unstable depending onthe parameter values (see next paragraph).

If 2 < (1 + s)P3, then EB+ can also exist. (Eab and EAB bothexist.) From Eq. 2.6a, EB+ exists only in the region aboveand to the right of the line 2yT/[l - 1/(1 + s)P] + YR = 1. Inparticular, EB+ exists in RII, if Eq. 2.10 holds. Referring toFig. 1 and keeping in mind conditions 2.9 and 2.10, Eab isstable outside RI, U RI,,, EAB is stable in RI U RI,, and EB+ isstable in RI,,. In fact, EAB is the unique stable corner or edgeequilibrium when the parameter values lie in RIH, and thesame is true of EB+ in RI,,. Note that EB+ may be stableoutside RII, since Eq. 2.8 constitutes only a sufficient condi-tion for stability.A useful way to view Fig. 1 is that RI, RI,, and RII, place

bounds on YT and YR beyond which the evolution of commu-nication is precluded. If s and P3 are large enough to over-come these, then the receptor allele can proceed to fixationwhile the transmitter allele increases either to polymorphismor to fixation. As might be expected, increasing (1 + s)f3enlarges the region RI U RI, U RH!, that permits the evolutionof communication.

Model 3: A Diploid Model with Genetic Control of Learning

Again there are two loci each with two alleles, A, a and B, b,but now the genotypes are diploid. There are two pheno-types, bar and not bar, such that with genotype BB at the B-locus, the genotypes AA and Aa acquire bar with probabilityone. Again the relative advantage of bar to not bar is 1 + s to1. Genotype aa is maintained by mutation at rate ,u from A toa, and aaBB is always not bar. In other words, there is amutation-selection balance between A and a when B is fixed.The recombination fraction between the loci is c.Now at this mutation-selection equilibrium the mutation b

arises at the B- locus, and Bb heterozygote offspring of par-ents with the bar phenotype also have that phenotype withprobability f(0 < 13 < 1). In matings between parents of dif-ferent phenotypes the transmitting parental phenotype ischosen at random from the parental pair. When neither par-ent is bar the offspring is always not bar. Homozygous BBoffspring have their phenotype determined by their A- lo-cus genotype whatever their parents' phenogenotypes. Thusthe function of b is to convert the bar-not bar dichotomyfrom one that is entirely determined by the genotype at theA- locus to one that is copied at rate 3 from the transmittingparent.The initial increase of b is studied in terms of the eight rare

phenogenotypes AB/Ab, AB/Ab, AB/ab, AB/ab, Ab/aB,Ab/aB, aB/ab, aB/ab. b cannot increase when rare. Thisresult that is clearly negative for the evolution of communi-cation is a consequence of the fact that b reduces the trans-mission of an adaptive trait in association with the commongenotypes (AA and Aa) and increases its transmission only inassociation with aa, the rare genotype. Thus it appears thatthe failure of the new modifier to invade primarily reflects itsdepression of the rate of transmission of an adaptive trait.

Discussion

We have addressed the problem of the evolution of commu-nication using a coevolutionary model for the cultural trans-mission of adaptive information. The model is coevolution-

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ary in two ways: (i) the genetic capacity for signal transmis-sion coevolves with that for signal reception; (ii) the com-bined genetic capacity coevolves with the culturally trans-mitted information. In constructing our model, we have hadthe evolution of human language capacity in mind. For ex-ample, with respect to its phonetic aspects, Lieberman (18)has suggested that supralaryngeal morphology and the neu-rons sensitive to certain sounds may have coevolved. Such acommunication system probably developed because thetransmitted information was adaptive.However, our evolutionary model most probably has wid-

er applicability, even to communication in nonhuman spe-cies. Semantic alarm calls in vervet monkeys are a case inpoint. Vervet monkeys innately produce several differentalarm calls, each triggered by a class of predator, or even anirrelevant stimulus such as a falling leaf. Each such genericalarm call is transformed through social learning into a highlyspecific call, specific to a certain predator in a certain local-ity (5). Alarm calls have usually been modeled as an altruis-tic trait. It is interesting to note that in previous analyses theinitial increase of communication has required that similarconditions be satisfied to those required for the evolution ofaltruism (9).The model predicts the existence of a genetically and phe-

notypically polymorphic, stable equilibrium-i.e., EB+-ifefficiency of vertical transmission is high and selection isstrong [large (1 + s),f] and if the cost of transmission is highrelative to that of reception (i.e., yT large and yR small as inRI,, of Fig. 1). The alarm calls just discussed may satisfythese requirements, in which case the model would predict agenetic polymorphism for callers and noncallers. It is possi-ble to construct a two-locus genetic model for the coevo-lution of transmission and reception but without culturaltransmission. In such a model, all transmitters are innatelycapable of transmitting the adaptive information. This is thetwo-locus analog of Cavalli-Sforza and Feldman's one-locusmodel (9). All equilibria in this model are monomorphic. Ittherefore appears that the genetically polymorphic equilibri-um EB+ in our model 2 is generated by cultural transmission.The effect of linkage in model 2 can be qualitatively stated

as follows. Tight linkage facilitates the initial increase oftransmission and reception (i.e., instability of Eab), but con-ditions for the maintenance of cofixation (i.e., stability ofEAB) are not so stringent. Tandemly duplicated genes are, ofcourse, very tightly linked. If similar physiological processesare involved in both transmission and reception, then the re-sponsible genes may be homologous and may even haveoriginated as tandem duplications.Model 2 is a special case of model 1 except that costs have

been introduced for transmission and reception. In the con-text of this specific model these costs are realistic, and theyalso simplify local stability analysis by removing unit eigen-values. Model 1 is a general model, and within its frameworkit should be possible to study many problems of gene-cul-ture coevolution. A possible application is to alcohol drink-ing and the alcohol dehydrogenase, aldehyde dehydrogenasepolymorphisms in human populations.

In a sense, the result of model 2 that communication canevolve for certain parameter values is built into the model.One might realistically introduce the additional assumptionthat individuals with allele b (i.e., genotypes Ab and ab) in-nately possess the bar phenotype. Thus, in our example ofthe eumenine wasps, larvae or the eclosed adults might con-

ceivably have been able to properly orient themselves usingsome external cue rather than the maternal message. Model2 can be modified to take this assumption into account bysetting bi4,1 = bi4,2 = Pf*, say, where (3* is the probability ofinnate expression of the bar phenotype. The stability of thegenetically monomorphic equilibrium in this model corre-sponding to Eab in the old can be tested by applying Eqs. 1.2,1.4, and 1.5 (after appropriate permutation of the sub-scripts). We ignore the costs of transmission and reception.If 1 + s(l* > (1 + s)f3/2, then the equilibrium is stable for allc. In particular, this is true if (* = 1.Model 3 is similar in spirit to genetic modifier models.

With B fixed the adaptive trait is entirely innate and A/apolymorphism is the consequence only of mutation and se-lection. Allele b converts the model to a coevolutionary onewith cultural transmission. The finding that cultural trans-mission cannot succeed here is analogous to the findings ofCavalli-Sforza and Feldman (9, 10) that some structure isneeded in the population before the state of genetic determi-nation can be invaded. Presumably, this will usually involvegenetic relationships between transmitters and receivers.A structural connection between models 2 and 3 is highly

desirable and may be possible in terms of a modification ofthe vertical transmission rate (3 in the neighborhood of thepolymorphic point EB+. This poses some interesting and dif-ficult analytical questions.

This research was supported in part by National Institutes ofHealth Grants GM 28016 and GM 10452, a grant to the Institute ofPopulation and Resources at Stanford University from the Alfred P.Sloan Foundation, and by a Grant-in-Aid for Special Project Re-search from the Japan Ministry of Education, Science and Culture.This is contribution no. 1724 from the National Institute of Genetics.

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Proc. NatL Acad Sci. USA 84 (1987)

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