Fundamental theorem of natural selection in biocultural populations

THEORETICAL

POPULATION

BIOLOGY

Fundamental

38, 367-384 (1990)

Theorem Biocultural

of Natural Selection Populations

in

SCOTT FINDLAY Ottawa-Carleton Institute of Biology, Department of Biology, University of Ottawa, 30 George Glinski, Ottawa, Ontario, Canada KIN 6N5

Received May 8, 1989

Fisher’s (1930) fundamental theorem of natural selection states that the rate of change in the mean biological fitness of a population is equal to the additive genetic variance in fitness. Since variances are always positive, this implies that mean fitness never decreases,an elegant result which neatly captures the flavor of Darwin’s assertion that natural selection leads inevitably to the improvement of the species. Subsequent investigations have shown that the theorem is effectively true only when departures from Hardy-Weinberg proportions are negligible, when genotypic fitnesses are constant over time, and when the genesaffecting fitness are in approximate linkage equilibrium (Wright, 1955;Kimura, 1958;Nagylaki and Crow, 1974; Crow and Nagylaki, 1976; Ewens, 1979). Violation of these assumptions may increase or decreasethe rate of change in population fitness depending on the circumstances. However, numerical experiments indicate that relatively large deviations from Fisher’s original assumptions are required if mean fitness is to decrease (Ewens, 1979). For example, with random mating, the theorem is a good approximation unless linkage is very tight or epistasis very strong (Crow and Kimura, 1970). Moreover, under weak selection, deviations from Hardy-Weinberg proportions decay rapidly (Nagylaki, 1974), so that with constant titnesses the relative error in Fisher’s theorem after several generations is of order O(s), where s is the intensity of natural selection (Crow and Nagylaki, 1976). That mean biological fitness is nondecreasing underlies most sociobiological theories of human behavior. These theories postulate that variation for certain behaviors (e.g., male-female differences in sexual behavior (Symons, 1979), nepotism (Daly and Wilson, 1983), and homicide (Chagnon, 1988)) at one time existed, that this variation was in part attributable to additive gene effects, and that it correlated with biological fitness. The fundamental theorem-or more precisely, its extension to 367 OO4O-58O9/90 $3.00 Copyright 0 1990 by Academic Press, Inc. All rights 01 reproduction in any form reserved.

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characters correlated with fitness (e.g., Crow and Nagylaki, 1976; Nagylaki, 1989)-then implies that those behavioral variants conferring high fitness will come to enjoy greater and greater representation as the population adapts to its environment. Consequently, at equilibrium one expects the most prevalent behaviors to have high fitness, and populations to be generally well-adapted. Because this conclusion has been construed as a testable prediction of sociobiological theory, the observation that populations often do not appear to be well-adapted is used as ammunition against sociobiological theories of human behavior (Vining, 1986). There are at least two possible responses to this criticism. The first (see, for example, Alexander, 1979; Dawkins, 1982; Fox, 1983) is that human populations now occupy environments markedly different from those in which they evolved. If-as many suspect-there has been accelerated anthropogenic modification of the environment during historical times, it is not surprising that natural selection has lagged behind in producing well-adapted populations. A second possibility is that for specieswith the capacity to transmit and receive behavioral information through both biological (e.g., Mendelian inheritance) and cultural (e.g., social learning) means, Fisher’s fundamental theorem may not apply. A general result emerging from recent formal investigations of behavioral evolution in biocultural systems (Feldman and Cavalli-Sforza, 1976, 1977, 1984, 1988; Cavalli-Sforza and Feldman, 1978, 1983; Aoki, 1987; Lumsden and Wilson, 1981; Boyd and Richerson, 1985; Findlay ef al., 1989a,b) is that the results from standard (that is, purely biological) evolutionary theory often cannot be directly applied to the biocultural case.For example, when phenotypes have different litnesses but genotypic effects are restricted to differences in the efficiency of cultural transmission, the resulting “induced” genotypic selection is not equivalent to standard Darwinian selection insofar as heterozygote superiority in the transmission of an advantageous phenotype is not sufficient for the establishment of a genetic polymorphism (Feldman and Cavalli-Sforza, 1976). In a similar vein, Findlay et al. (1989a, b) have shown that the classical ESS conditions are neither necessary nor sufficient for evolutionary stability in biocultural games where strategies are culturally transmitted. Evidently, then, there is a problem. On the one hand, sociobiological arguments often rest on assumptions taken directly from standard evolutionary theory about how the mean fitness of a population evolves. On the other hand-as suggested by the studies cited above-these assumptions may not be valid in the biocultural case. The solution is to develop from first principles a fundamental theorem of natural selection that is actually appropriate to biocultural systems, not one that we only assume is appropriate. That, in a nutshell, is what the following paper attempts to do.

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EVOLUTION IN BIOCULTURAL POPULATIONS

For modeling evolution in purely biological systems, it is often sufficient to derive equations of motion for the distribution of genotypes in a population. This is because one can usually assume that the probability of a particular genotype developing a given phenotype is constant over time. Of course, this need not be true in the case of a changing environment, but even here one can usually assume that any modifications in the correspondence between genotypes and phenotypes depend on exogenous conditions, and are independent of the genotypic or phenotypic composition of the population itself. In the biocultural case, things are not quite as simple. Here one must keep track of the joint distribution of genotypes and phenotypes (CavalliSforza and Feldman, 1976, 1978; Feldman and Cavalli-Sforza, 1976, 1977, 1984; Boyad and Richerson, 1985), referred to as the phenogenotype distribution. This complication arises because the probability of an individual transmitting or adopting a particular phenotype may depend on both its genotype and the social environment. For example, with oblique cultural transmission (i.e., from members of the parental generation to offspring), the probability of an individual adopting a particular phenotype may depend on its exposure to the trait during socialization: the greater the exposure, the more likely it is to be adopted (Cavalli-Sforza and Feldman, 1981; Lumsden and Wilson, 1981). The degree of exposure will in turn depend on the frequency of the trait in the social environment. So the correspondence between genotype and phenotype is influenced by endogenous factors such as the phenotype distribution itself. Consequently, knowledge of the genotype distribution alone is not sufficient to predict future evolutionary changes. THE MODEL In this section, I develop a fundamental theorem for biocultural systems. In doing so, I consider a scenario similar to that envisioned by Fisher (1930) in his original statement of the theorem with the added complication of biocultural (versus purely biological) transmission of the phenotype. In particular, I restrict my attention to the single-locus case in an (effectively) infinite population with random mating at both the genotypic and phenotypic levels. While these conditions are perhaps unlikely to apply to actual biocultural populations, they allow direct comparison with Fisher’s original result, and provide a basis for extensions to the generalized mating, multiple loci case. Consider a diploid population of size N, with N assumed sufficiently

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large that stochastic effectsarising from finite population size are negligible. Suppose that the fitness of an individual is influenced by a single diallelic autosomal locus A. Let 4ij(m’) be the fitness density of individuals with ordered genotype AiAj such that #ii dm’ is the proportion of individuals in the population with genotype AiAj and fitness lying in [m’, m’ + dm’], and assume random mating at both the genotypic and phenotypic levels. Following Fisher (1930) and Kimura (1958), assume liness is measured in Malthusian parameters, such that if b At and d At are the probabilities of an individual giving birth and dying (respectively) during the infinitesimal interval At, we have m At = (b-d) At. To simplify matters, I suppose that while reproduction is biparental, cultural transmission is uniparental (see, e.g., Cavalli-Sforza and Feldman, 1981), with the probability of an offspring developing fitness phenotype m depending on the transmitting parent’s genotype A,A, and phenotype m’. Thus we write tjo(m Im’) as the probability of an offspring developing fitness m, given that the transmitting parent has phenogenotype AiAj(m’). In a diploid population of N individuals, there are 2N genes at the A locus. Denoting the fitness density of heterozygous individuals by 2d,(m’) dm’, we have ni(m’) = 2Nx q5Jm’)

(1)

as the number of copies of allele Ai in individuals with phenotypic value lying in [m’, m’+ dm’]. Assume for the moment that over the time interval At, d= 0 for all individuals, in which case m At = b At and the increase in phenoallelotype A,(m) over the interval At is 2N 1 j m’cjq(m’) t,hu(m1m’) dm’ At -n,(m). i

(2)

Then for At sufficiently small, we obtain the continuous approximation

dni(m) -~A,(m)=2N~~m’~,(m’)~,(m~m’)dm’ dt i describing the instantaneous rate of change in n,(m). Let p,(m) = ni(m)/2N be the phenoallelotpe density, Pi=/ pi(m) dm

(4)

the frequency of allele Ai, and ni=

5

n,(m) dm

(5)

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the number of copies of allele Ai. Then the expected contribution individuals carrying allele Ai to fitness class [m, m + dm] is b,(m) =i, c j m’q5ij(m’) $,(m 1m’) dm’. 1J

371 of

(6)

Substitution in Eq. (3) then gives lii(Wl) = b,(m)n,.

(7)

Integrating in Eq. (1) over all possible values of m’ and summing over i yields

T j ni(m’) dm’= 2NF 7 j 4iiW) dm’ =2N

(8)

as expected. The rate of change in the total population size N is then

=N~~~m’~o(m’)~,(m~m’)dm’dm zj = Nfi’,

(9)

where 6’ is the mean biological fitness of the adult population. Combining Eqs. (7) and (9) yields @,(m)=l NZ CNfii(m)- ni(m)fil = pibi(m) - p,(m)&

(10)

as the instantaneous rate of change in the phenoallelotype density pi(m). If mi is the average fitness, in Malthusian parameters, of allele Ai, then mi=~,~J~~(m’)m’dm’=jbi(m)dm. 'I

Integrating in Eq. (10) over all possible values of m and substituting from Eq. (11) gives the well-known result P,=~@i(m)dm=pi(mi-ti’)

(12)

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as the rate of change in the frequency of allele Ai. Although this was derived assuming that over the interval dr, d = 0 for all individuals, it is also true when all individuals have the same death rate. Despite the apparent similarity between Eq. (8) and the classical equations of gene frequency change, there are important differences. In the purely biological case it is often appropriate to treat genotypic values as fixed parameters (e.g., Crow and Kimura, 1970, pp. 190-191; see also Nagylaki and Crow, 1974), so that the average excess for fitness m,--ti’ depends only on the distribution of genotypes in the population at any given time. By contrast, in the biocultural case the probability of an individual developing a particular trait depends both on its genotype and on its exposure to the trait during socialization. As the population evolves, the phenotypic distribution changes. This in turn may influence the exposure of individuals to a particular trait during socialization, and hence its likelihood of adoption. Consequently, the average fitness of a particular genotype will generally not be constant, and the average excessfor fitness of a given allele will depend on the joint distribution of phenotypes and genotypes.

THE FUNDAMENTAL THEOREM UNDER GENERALIZED BIOCULTURAL TRANSMIWON

Fisher’s original statement of the fundamental theorem applies to large populations in which fitness is determined by a single locus. Although his formulation is not restricted to the case of random mating (see Ewens, 1990), I shall here assume that random mating holds, namely that 40~s c&(m) dm = s p,(m) dm .j pj(m) dm,

(13)

so that

giving h(m) = 2~~ p,(m), 2&(m) = 2(pl(m)pz + p,(m)pl), 2p2p,(m). Differentiation with respect to time yields

and Mm)

=

ill(m) = WI PI(m) + PI b,(m)), &&Am) = 2(dlh)p2

+ PI(

d22h) = W2 h(m) + h Mm)),

+ A(m

+ pAm)i4),

(15)

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where dV(m) is the instantaneous rate of change in the density of phenogenotype AiAi(m). The mean biological fitness of the population is rii=x

ij f

mq5,i(m)dm.

(16)

Hence the instantaneous rate of change in ti is

Recalling that xi fii = 0 and xi pi = 1, and substituting Eqs. (15) into Eq. (17), eventually yields dt?i

;i;=21mCPl(m)+P2(m)ldm,

(18)

which on further substitution from Eq. (10) gives drii

,=2SmCp,b,(m)+p,b,(m)-m’)(m)ldm,

(19)

where d(m) is the phenotype density. Let b(m) = p,bl(m) + d4m) =~jm'O,(m')$B(mIm')dm'. ij

(20)

In Eq. (20), b(m) is the rate of change in the fitness density b(m) due to the combined influences of differential biological fitness and genetic and cultural transmission of the fitness phenotype. Since we can write the conditional probability eV(rn 1m') as tiq(rn 1m') = ev(rn, m')/q4v(m') with J dm dm' t,hij(m,m') = 1 and #V(m, m') denoting the joint density of parental and offspring phenotypic values, Eq. (20) becomes (21)

Substitution in Eq. [19] then gives jmb(m)dm--fiti' mm'dmdm'Il/,i(m,m')-mm' = 2(E(mm')-tiCi'),

> (22)

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FINDLAY

with E the expectation operator. We therefore obtain the fundamental theorem of natural selection in biocultural populations drii z = 2 Cov(m, m’). In words, the instantaneous rate of change in the mean biological fitness of a population under biocultural transmission is equal to twice the covariance between parental and offspring fitnesses. Fisher’s (1930) fundamental theorem is a special case of Eq. (23) that applies when transmission of the phenotype is ‘purely genetic. Under purely genetic transmission tiv(rn 1m’) = tiii(m), since the probability of a parental genotype A,A, producing an offspring with fitness m depends only on genes passed to the offspring, and is independent of the parental phenotype m’ per se. Substitution in Eq. (6) then gives b,(m) =-!-F tjq(rn) 1 mr#ii(m’) dm’.

(24)

’ J

Defining 1 my = K

s

m’dij(mf) dm’

(25)

as the mean fitness (genotypic value) of genotype A,A, in the parental population, Eq. (24) can be rewritten as

hi(m) =i c IClJm) tip%.

(26)

’ J

Summing over both alleles then yields b(m) = C IC/@) 4pti. (i

(27)

In the one locus case, there are three possible genotypic values: m,,, ml2 and m,, . Let cj(rn1mu) be the probability of an offspring developing phenotype m, given a genotypic value mu. Then under purely genetic transmission with random mating, the probability of an A 1A, parent producing an offspring with phenotype m is

on recalling that pi = xi dii. Generally one assumesthat

4(m Imu) = pexp[-(m-m,j)2/2V,], J&

(29)

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that is, that phenotypic values are normally distributed with mean mu and an environmental variance V, assumed independent of the genotype (e.g., Cavalli-Sforza and Feldman, 1976). Using Eq. (28) and its analogs for J/&m) and $&rn) and Eq. (27) we obtain

b(m) = 411wlC~14(m I ml) + p24(m Iwdl

+~12m,,C~I~(niIm,,)+p,~(mIm,,)+~(mIm,,)l + d22m22CP1WIm12)+ h&m Im2dl.

(30)

Substituting Eq. (29) into Eq. (30) and integrating over all possible fitness values gives 2jmb(m)dm=2C411

ml13 +h2m12hmll

+ p2m2, + m12)+ tiz2mz2m21. (31)

The mean biological fitness of the population is

fi = hlmll + 2h2m12+ 422m22.

(32)

Assuming random mating and substituting Eqs. (3 1) and (32) into Eq. (22) eventually yields dti ~=2Cpi(mi-fi)2, I

(33)

which in the single locus case with random mating is equal to the additive genetic variance in fitness (Crow and Kimura, 1970, pp. 206-207).

THE FUNDAMENTAL THEOREM UNDER SPECIFIC MODES OF CULTURAL TRANSMISSION

Equation (23) is a general description of the evolution of mean biological fitness under biocultural transmission of the phenotype. It differs from Fisher’s original formulation for purely biological populations in at least two important respects. First, so long as the covariance between parental and offspring titnesses is nonzero, the mean fitness of the population will change. Note that this is a phenotypic rather than a genotypic covariance. Consequently, mean fitness can evolve even if the additive genetic variance in fitness is zero, in direct contrast to Fisher’s conclusion that additive genetic variance in fitness is necessary for evolution to occur. Second, whereas variances are always positive, covariances may be either 653138/3-9

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negative or positive. Equation (23) therefore implies that mean biological fitness can decline over evolutionary time. The actual dynamics are likely to depend on the form of the conditional densities tiij(m 1m’), which in turn will depend on the particulars of the socialization process. Previous results (e.g., Cavalli-Sforza and Feldman, 1983; Feldman and Cavalli-Sforza, 1988) indicate that the dynamics of biocultural systems may be quite sensitive to the mode of cultural transmission. In this section, I examine the implication of specific modes of cultural transmission to the fundamental theorem. Vertical Cultural Transmission Vertical transmission refers to the transmission of a trait from parents to offspring (Cavalli-Sforza and Feldman, 1981). Empirical studies suggest that this is an important mode of cultural transmission in both human (Cavalli-Sforza et al., 1982; Hewlett and Cavalli-Sforza, 1986) and nonhuman (Galef, 1976; Mundinger, 1980) populations. A simple form of uniparental vertical transmission is given by the linear equation E(m 1m’)v = aii + Piirn’,

(34)

where E(m 1m’)ii is the expected fitness of an offspring produced by parental phenogenotype AiAj(m) and aij, /Iii are the standard regression parameters which depend only on the parental genotype. This is similar to the model of vertical transmission derived by Cavalli-Sforza and Feldman (1981, p. 275), except that transmission is assumed to be uniparental. Empirical studies suggest that vertical transmission often results in offspring adopting a phenotype similar to that of their parents (e.g., Hewlett and CavalliSforza, 1986), in which case the a’s and /I’s will be positive. Since E(mIm’),=~m~Jm~m’)dm,

(35)

we have E(m)=fi=x{d,i(m’) E(mIm’),jdm’ ij = Cr+ E(/?m’),

(36)

where

W-W = C Bii f m’4v(m’) dm’ = C POm,qS,, ij ij

(37)

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with E the expectation operator and rno the genotypic value of genotype A,A,. Similarly,

=E(am')+E(bm'*)

(38)

with E(am’)=~a,~ 17

m'q4,(m')dm'=Ca,im,4,, ij

E(~m'2)=~/?~~m~2)u(m')dm'=~/3~E(m'2),jq5~, ij ij

(39)

where E(m'2)v is the expected value of the square of the fitness of genotype A,A,. Substituting Eqs. (36)-(39) into Eq. (23) yields $=2[E(ctm')-6i'oi+E(flmr2)-m'E(Bm')l

In Eq. (40), Qii, mu, and I’, denote the frequency, genotypic value, and environmental variance of genotype AiAj, respectively, fi’ is the mean fitness of the population, and clii and /I# are the genotype-dependent regression parameters. In the case where vertical transmission is independent of genotype so that all genotypes are equally effective at transmitting and equally likely to adopt a particular phenotype, txii = tl and fiii = j) for all i, j and Eq. (40) reduces to (41) where Vo = E(mi) - fi’* is the genotypic variance, VE the environmental variance, and VP the total phenotypic variance in fitness. Hence, mean fitness can evolve in the absence of genotypic variation for the effects of cultural transmission. Comparing Eq. (41) with Fisher’s original equation drii -$=

v A

(42)

and recalling that I’, > VA, we see that vertical cultural transmission may either increase or decrease the rate of change in mean biological fitness

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relative to the purely genetic case depending on the magnitudes of the regression coefficient #I and the nonadditive contributions to the total phenotypic variance. In particular, when V, is small relative to V,, vertical transmission need not be particularly strong (as measured by the magnitude of /I) to acceleratethe rate at which the population adapts to its environment relative to the purely genetic case. Inspection of Eq. (40) shows that, as suggestedabove, evolution of mean fitness can occur even if the genotypic variance in fitness is zero. Zero genotypic variance implies rnv - fi’ = 0 for all i, j, in which case Eq. (40) reduces to (43) If each genotype has the same environmental variance, I’, = V, for all i, j, and Eq. (43) further reduces to

with p the regression of offspring on parental fitness averaged over all genotypes. In both cases,the rate of change in mean fitness is always positive and proportional to the environmental variances Vii and to the regression coefficients pii. If all the VU’sare zero, mean fitness does not change. These results indicate that under linear vertical transmission of the type given by Eq. (34), mean fitness can evolve in the absence of genotypic variance (and hence, additive genetic variance) for both fitness and the effects of cultural transmission. Moreover, even for the LX’Sand /I’s all positive, the rate of change in mean fitness may be negative. Hence, in the general case where both fitness and cultural transmission are genotypedependent, mean fitness can decline over evolutionary time. Of course, vertical transmission need not be linear. A generalized nonlinear model is given by E(m 1m’)u = au + fiijm’ + yiim12

(45)

In this case, Eq. (40) becomes

(46) where E(m’3),i is the third moment of the fitness distribution of genotype A,A,. The conclusions reached above for the linear case also apply, with the difference that here the higher order characteristics of the fitness

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distributions may exert significant effects depending on the magnitude of the nonlinear coefftcients yii in the regression equation. Oblique Cultural Transmission In addition to transmission from parents to offspring, non-parental members of the parental generation may play an important role in social learning and enculturation (e.g., Diamond, 1986), particularly in species with highly developed social structures including extended family units (Hinde et al., 1985; Goodall, 1986). This mode of transmission has been termed “oblique” by Cavalli-Sforza and Feldman (1981). For human populations, there is some evidence that oblique transmission may take the form of “trend-watching” such that the greater the exposure to a trait during socialization, the greater the likelihood of its being adopted (see Lumsden and, Wilson, 1981). Unlike the case of vertical transmission, the social environment now includes parents, aunts, uncles, and other members of the extended family unit, all of whom may participate in the socialization process. Cavalli-Sforza and Feldman (1981, p. 131) propose a model of oblique transmission based on random contacts between unenculturated juveniles and potential teachers in the parental generation such that the probability of a juvenile adopting a trait is proportional to the frequency with which the individual encounters teachers who use the trait. We can model this situation by E(m 1m’)ii = aU+ pgii’. This is similar to the vertical transmission equation (34) except that here transmission depends on the distribution of fitness phenotypes in the population as a whole rather than the parental phenotype: the greater the abundance of high fitness phenotypes in the population, the greater the mean fitness and the more likely an unenculturated individual is to adopt a high fitness phenotype. Proceeding as in Eqs. (35b( 39) above eventually yields ~=2~40[(mr-ti’)(a,+j?v6z’)]. Iy There are several important differences between Eq. (48) and Eq. (40). First, note that in the former case, if there is no genotypic variance in fitness, mu- rii’ = 0 for all i, j and mean fitness does not change. This is somewhat closer in spirit to Fisher’s original formulation except that as in the case of vertical transmission it is the genotypic rather than the additive genetic variance that counts. In principle, mean fitness can evolve in the absence of additive genetic variation for fitness provided that there are

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other sourcesof genotypic variation, including dominance or (in the multiple locus case) epistatic effects. Note further that mean fitness may either increase or decrease, depending on the genotype distribution and the regression parameters. Second, even if there is genotypic variation for fitness, mean fitness remains unchanged unless there is also genotypic variation for the influence of cultural transmission. If the likelihood of an individual adopting or transmitting a particular phenotype through oblique transmission is independent of its genotype, then aii = c1and /Iij= /I for all i, j, in which case Eq. (48) becomes ~=(a+~iii)~Po(m,-rFi)=O. zy

(49)

+Consequently, for evolution of mean fitness to occur under oblique transmission, there must be genotypic variation for both fitness and the effects of cultural transmission. As with the case of vertical transmission, we can formulate a generalized nonlinear model of oblique transmission given by E(m 1m’)g = aii + /?@i’ + ygS2,

(50)

which eventually yields d% ~=2~~,[(rn,-6z’)(~l~+~~ti’+ygii’~)]. ij

(51)

Once again, genotypic variation for both fitness and the effects of cultural transmission is required if mean fitness is to evolve.

DISCUSSION

Fisher’s (1930) fundamental theorem occupies a central place in population genetics.Since its original derivation there has been considerable interest in refining it and assessingits validity in cases where some or all of the original assumptions do not hold (Wright, 1955; Kimura, 1958; Moran, 1964; Li, 1967; Price, 1970; Nagylaki, 1974;Nagylaki and Crow, 1974;Crow and Nagylaki, 1976; Ewens, 1979; Nagylaki, 1989). These investigations have concentrated on situations in which the phenotype is biologically transmitted from parents to offspring. The situation where the phenotype is both biologically and culturally transmitted seemsto have escapedmuch attention despite the prevalence of cultural transmission in both human and nonhuman species (Galef, 1976; Mundinger, 1980; Lumsden and

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Wilson, 1981) and despite the fact that many sociobiological arguments for the evolution of behavior either implicitly or explicitly assume that the fundamental theorem-or some extension thereof-holds. In this paper, I derive a fundamental theorem of natural selection for biocultural populations. This theorem differs from Fisher’s in at least two important respects. First, under biocultural transmission of the phenotype, the instantaneous rate of change in the mean fitness of a population may be negative, implying that populations can become more poorly adapted biologically over time despite the action of natural selection. Second, under at least one common form of cultural transmission (transmission from parents to offspring), additive genetic variance for fitness is not necessary for the evolution of mean finess to occur. These results have important implications. In the one-locus, random mating case, mean fitness is a Lyapunov function for the dynamics of purely biological systems (Hofbauer and Sigmund, 1988). When two or more loci are involved, mean fitness is not generally a Lyapunov function (e.g., Moran, 1964) although for many important casesit appears possible to construct a generalized fitness function (not necessarily mean fitness) that is a Lyapunov function (Ewens, 1969; Akin, 1979). But for both one-locus and multiple-loci cases, the implication is that if one has knowledge of the fitness of different genotypes and such purely biological features as recombination and epistasis, one can predict the direction in which the population will evolve. Not so in the biocultural case. My results indicate that in order to generate accurate predictions about population evolution, one must have information about both the strength and mode of cultural transmission and the extent to which each depends on the genotype. In general one cannot construct a Lyapunov function for biocultural dynamics that includes only considerations of biological fitness. Likewise, one cannot in general assume that the fundamental theorem-or some direct extension thereof-holds when constructing sociobiological arguments for the evolution of social behavior. Consequently, predictions based on such arguments may be inaccurate (see also Lumsden and Wilson, 1981; Boyd and Richerson, 1985; Findlay et al., 1989a,b). On the other hand, these results imply that the observation that populations, particularly human populations, are not particularly well-adapted biologically (e.g., Vining, 1986 and ensuing commentary) is not a legitimate test of the validity of the sociobiological enterprise, if we take the latter to mean an attempt to explain human behavior in terms of its effects on biological fitness. Without specification of the strength and mode of cultural transmission and the degree to which these properties depend on an individual’s genetic makeup, there is no expectation that mean fitness will increase or that populations will become progressively better adapted

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to their environment even in the presence of natural selection. This does not mean that biological fitness and the genetic basis (if any) of population variation are irrelevant to an understanding of the dynamics of culturally transmitted traits, only that in themselves they do not tell the full story. Previous results (e.g., Maynard Smith and Warren, 1982; Cavalli-Sforza and Feldman, 1983; Feldman et al., 1985; Aoki, 1986) indicate that cultural transmission tends to decrease rates of genetic evolution. My results suggest that this reduction need not imply a decreased rate of biological adaption. For example, when the additive genetic variance in fitness is small, the rate of increase in mean fitness under purely genetic transmission is also small. With vertical cultural transmission however, mean fitness can increase rapidly under the same conditions if the regression of offspring on parental fitness is large and positive and there is environmental variance in fitness. SUMMARY

I derive a general equation for the evolution of mean biological fitness in large, randomly mating populations in which the phenotype is subject to both biological and cultural transmission and in which fitness and the effects of cultural transmission are genotype-dependent. The equation is dSi x = 2 Cov(m, m’), where drii/dt is the instantaneous rate of change in the mean biological fitness of the population and Cov(m, m’) is the covariance between offspring and parental Iitnesses. This formulation differs from Fisher’s fundamental theorem in at least two important respects. First, because covariances can be either positive or negative, mean fitness can decline over evolutionary time, in contrast to Fisher’s result that fitness always increases.Second, since Cov(m, m’) is a phenotypic rather than a genotypic covariance, natural selection can bring about changes in mean fitness even in the absenceof additive genetic variance for fitness. Comparison of simple models of vertical (from parents to offspring) and oblique (from members of the parental generation to offspring) cultural transmission indicates that in the former case, mean fitness can evolve in the absence of genotypic variation for both fitness and the effects of cultural transmission, while in the latter case, both are required for evolution to occur. ACKNOWLEDGMENTS I thank Ken Aoki, Charles Lumsden, and David Sloan Wilson for rewarding discussions on biocultural evolution and Warren Ewens for comments on an earlier version of the manuscript. This work was supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada.

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