Gene-Culture Coevolution: Models for the Evolution of Altruism with Cultural Transmission (Pheno/Genotypes/Hamilton's Rule) MARCUS W

Home , Kin selection

Proc. Nati. Acad. Sci. USA Vol. 82, pp. 5814-5818, September 1985 Evolution Gene-culture coevolution: Models for the evolution of altruism with cultural transmission (pheno/genotypes/Hamilton's rule) MARCUS W. FELDMANt, L. L. CAVALLI-SFORZAt, AND JOEL R. PECKt tDepartment of Biological Sciences, Stanford University, and tDepartment of Genetics, Stanford Medical School, Stanford, CA 94305 Contributed by L. L. Cavalli-Sforza, May 21, 1985

ABSTRACT Models ofsexual haploids under kin selection the rate of cultural transmission of an altruistic trait can ini- are constructed. The trait of altruism is transmitted vertically tially succeed are not as simple as in the genetic case, but from parent to child, but not in a strictly genetic manner. Two that under special conditions, natural interpretations are systems of altruism are considered: parent-to-offspring and available. sib-to-sib. In the former case it is shown that even when Ham- ilton's conditions for the success of genetically determined al- Transmission Model with Sexual Haploids truism are met, genes that increase the transmission of altruism may not invade the population. With sib-to-sib altruism, There are two genotypes, A and a and two phenotypes indi- such genes will always increase initially. cated by the subscripts 1 and 2. Thus, there are four pheno/genotypes, Al, A2, a,, and a2, whose frequencies are ul, Hamilton (1) proposed a model for the evolution of altrusim u2, U3, and u4 in the parental generation at mating (7). Trans- in which altruists increase the survival rate of their relatives mission of the phenotype is unipareptal, and Table 1 gives while decreasing their own. In this model, altruism was un- the probabilities that a transmitting parent of the respective der simple genetic control and this, together with the way in phenotype produces offspring of the respective pheno/geno- which the fitness effects were combined, led to a simple and types. Mating is random and the mating and transmission are intuitively plausible rule for the initial success of altruism. summarized in Table 2. The way in which the fitness effects are combined is known IfU, u2, U3, uTare the frequencies of the pheno/genotypes to influence the conditions for evolution of altruism (2, 3). in the offspring after vertical (genetic and cultural) transmis- Departures from simple genetic transmission of altruism sion, then should also play a role in determining the conditions for its increase. l bA1W + bA2X [la] In this paper, we shall study the evolution of altruistic traits that are transmitted both by genetic means and by ver- TU = (1- bA,)W + (1- bA2)X [lb] tical (i.e., parent to child) cultural transmission. There is no ba1Y + ba2Z [ic] general agreement as to the best way to construct such a UT= model. Indeed, there has been considerable controversy in u4T = (1 - bal)Y + (1 - ba2)Z' [1d] recent years concerning the appropriate theoretical method- ology whereby the evolution oftraits under both genetic and where cultural control may be studied. Lumsden and Wilson (ref. 4; see also Wilson, ref. 5) have taken an approach which (they W = Ul + U1U2 + U1U3 + (U1U4 + U2U3)/2 [2a] leads to the of differences claim) production major between X = + U2 + U2U4 + (U1U4 + u2u3)/2 populations as a result of small between-population genetic ulu2 [2b] differences. Lumsden and Wilson also claim that their ap- Y = u3 + u3u4 + U1U3 + (U1U4 + u2U3)/2 [2c] proach demonstrates that cultural transmission should accel- erate the rate of genetic evolution. The arguments that lead Z = u4 + u3u4 + U2u4 + (ulu4 + U2U3)/2. [2d] to these conclusions have been questioned by Maynard Smith and Warren (6). Furthermore, Maynard Smith and Warren (6) show that assumptions that appear to be just as Parent-to-Offspring Altruism: Model I plausible as those used by Lumsden and Wilson lead to very different results. Models for parent-to-offspring altruism may be specified in a Here we shall take a different approach and introduce a number of different ways. Here we shall use the additive class of evolutionary genetic models that take into account combination of gain and loss in fitness specified by Cavalli- both vertical cultural transmission of a trait and kin selection Sforza and Feldman (2) that is most commonly used in kin on the trait. Thus, our analysis is an attempt to generalize selection studies. Individuals of phenotype 1 suffer a fitness some of the evolutionary theory of altruism to the situation loss y. Broods with a randomly chosen parent of phenotype where the trait under kin selection is transmitted from parent 1 enjoy an average fitness increase proportional to /. One to child, but not necessarily in a strictly genetic manner, We way to compute the resulting fitnesses uses the conditional show (i) that nongenetic transmission entails significant de- kinship probabilities developed in ref. 2 to produce the ex- partures from Hamilton's rules used in strictly genetic models to specify conditions on the fitness gain to receivers of Table 1. Pheno/genotypic transmission rates altruism that allow the increase of the altruistic phenotype Phenotype of Probability of offspring pheno/genotype and (ii) that the conditions under which a gene that increases transmitting parent Al A2 a, a2 The publication costs of this article were defrayed in part by page charge be marked "advertisement" 1 bAl 1-bAl bI 1-bak payment. This article must therefore hereby 2 in accordance with 18 U.S.C. §1734 solely to indicate this fact. bA2 1-bA2 b,2 1-ba2

5814 Downloaded by guest on October 1, 2021 Evolution: Feldman et aL Proc. NatL Acad. Sci. USA 82 (1985) 5815

Table 2. Mating and transmission Transmitting Other Mating Probability of offspring pheno/genotype parent parent* frequency A1 A2 a, a2 Al Aj,A2 Ul(Ul + U2) bAl 1 -bAl 0 0 A1 al,a2 U1(U3 + U4) bAl/2 (1 - bA1)/2 ba1/2 (1 ba)/2 A2 AlA2 U2(U1 + U2) bA2 1 - bA2 0 0 A2 al,a2 U2(U3 + U4) bA2/2 (1 - bA2)/2 ba2/2 (1 - ba2)/2 a, A1,A2 U3(U1 + U2) bAl/2 (1 - bA1)/2 ba1/2 (1 - bai)/2 a, al,a2 U3(U3 + U4) 0 0 baI 1- bal a2 U4(U1 + U2) bA2/2 (1 - bA2)/2 ba,2/2 (1 - ba2)/2 0 0 1- a?-Z U4(U3 + U4) ba,2-Z ba2-. *Offspring probabilities are the same for both phenotypes of this parent.

pressions in the square brackets below. The frequencies u;, where w is the sum of the right sides of the equations. U2, U3, and u4 in offspring after transmission and selection are as follows. Initial Increase of the Altruistic Phenotype is 1 and child Consider the situation where phenotype 2 is fixed. What are U1 arL0 - + A P(parent phenotype A1)] U1UlL ~~~~~P(child is Al) the conditions that allow phenotype 1 to increase in frequency when it occurs first at low frequency near where pheno- would be where all A U2 0C U2T[ + ;P(parent is phenotype 1 and child A2) 1 type 2 is fixed? The simplest situation + PprnisP(child is AD) genotypes are A1 and all a genotypes are a2; i.e., where bA1 = bA2 = 1 and ba, = ba2 = 0. This situation is identical to that [3] of strict genetic transmission of altruism. Here there is initial fixation of a2 (i.e., U4 = 1) which is unstable to the introduc- + is phenotype 1 and child a,)1 U3U3 0C U3T[13[ _ . P(parent tion ofA1 when P/2 > y, the usual Hamilton condition (1, 2). P(child is a,) More generally, for phenotype 2 to be fixed with A and a coexisting, it is required that = = 0. Then there is a is 1 and child 1 bA2 ba2 p P(parent phenotype a2) neutral points with + = 1. U4 °C UT{1 + line of equilibrium a2,a4, a2 a4 P(child is a2) J From any such point, the local increase of A1 and a1 is de- scribed by the linear system in u1 and U3: In other words, Uj = bAj- + 2) (a2 +2 Wu' = ul(l - l) +bAl ,1 y( [5al U2U3 1 y -3u3;+ + + UU+ + -u2b P[bAl(ui + UlU3 2 4 2 U3 2 (1U+22)4u12 + b + + U2-3 [4a] p EA2 2 4 4/]' =..( +-U+ + )( + [Sb] = ba~l 2 2 U4)U3. WU2+U2 + p a1- bAj)(1 +l UIU3 + 122 + UI4 + 4~3 The characteristic polynomial of Eqs. 5 is -2_(A(- y + 2) LbAl(12 + + ba, `4 + + (1 - bA2)( 2U+ + 4 [4b] + + W3= u3T(l - y)' 22ba(-13)2 [6]

+ + 3[bal(U3 UlU3 2 4 4 The condition for the initial increase of phenotype 1 is that the larger root of Eq. 6 be greater than unity in absolute value. If we set G = bA1(l + u2) + ba1(l + U4), this condition is + b2U3U4 + UU4+ U2U3 [4c] 4 4/]' G - (G2 - 8bAlbai)½2 p . - [7] and > 1 + 2bAlbal

Wu';=ua When bA, = bal = b, for instance, Eq. 6 reduces to the simple form + U3U4 + + U2U3 P[(1-ba)(U2 + + UiU4 P - 2 > y + (b-' 1). [8] + (1-ba)QU3!4 + UlU3 + 23 [44d] Since b s 1, condition 8 illustrates that there is a transmission load which makes the increase of altruism more difficult Downloaded by guest on October 1, 2021 5816 Evolution: Feldman et aL Proc. Natl. Acad. Sci. USA 82 (1985)

as its transmission deteriorates-i.e., b decreases. With per- and the local stability ofthe root of Eq. 11 to the introduction fect transmission (b = 1) condition 8 takes the form expected of allele A is governed by the roots ofthe characteristic poly- by simple application of Hamilton's rule. In general, inequal- nomial Q(X): ity 7 reveals that it is more difficult for altruism to increase under the present form of cultural transmission than under simple genetic assumptions. This agrees with conclusions Q(X) = (kw) drawn earlier (6, 8). P03 - - kW 1 - yb* + + +2yb*(2 [b. 122 Parent-to-Offspring Altruism: Model II Awilinreas whe raei 2hagrroto ()irae In Model I, the choice of the parent to interact with the off- A will increase when rare if the larger root of Q(A\) is greater spring is made at random from the parental pair. In Model II, than unity. This occurs if the offspring gains only when the transmitting parent is altruistic. In this case Eqs. 4 are altered to become Q(1) = (b* - b)[(d3 + 1)( -2) wui = u~(1 - v) + 2- b*) - y2b < 0. [131 + 3bAl(u2 + U1u2 + U1u3 + 2 + U2U3)3 [9a] A cursory examination ofcondition 13 suggests that a simple relationship between (y - /3/2), the Hamilton expression, = T+ 3(1 Wu' 2( UU 24 and (b* - b), the transmission difference, is not likely to bAj) validate the inequality. The following four numerical exam- X U2 + UlU2 + UlU3 + - + 2 [9b] ples indicate the complications that arise here. Example 1: P = 1.0, y = 0.6, b = 0.8, b* = 0.9. Here ,//2 < y and b* > b, yet A increases when rare. Wul = Uj(1T)- Example 2: /3 = 1.0, y = 0.4, b = 0.8, b* = 0.9. Here /3/2 > y and b* > b, and A increases when rare. + pbai(3U + UlU3 + U3U4 + y + 2U3' [9C Example 3: /3 = 0.3, y = 0.2, b = 0.2, b* = 0.1. Here /3/2 < y and b > b*, yet A increases when rare. Example 4: /3 = 1.1, y = 0.5, b = 0.1, b* = 0.2. Here /3/2 and > 'y and b* > b, yet A does not increase when rare. Remark. In the special case bal = ba2 = b, with bA, = bA2= + - Wu-= u4T /3(1 ba2) b*, Model I and Model II produce the same characteristic polynomial 12. In more general cases, there will be differ- X (U2 + Ulu3 + U3U4 + + 2 [9d] ences between the models, which are not pursued here. Clearly the nongenetic transmission of the "altruistic" phenotypes interferes with the validity of any straightfor- where w = 1 - y(uT + U3) + /(W + Y) (see Eq. 2). In this ward application of Hamilton's rules. Nevertheless, there case ,B/2 is replaced everywhere in Eqs. 5 by 13 and condi- are two other ways to express condition 13 that are informa- tion 8 becomes ,B> 'y + (b'1 - 1), which, of course, does not tive. Write Wa = 1 - yb + /3123, the average fitness of a, and reduce to the Hamilton condition for b = 1. a2 prior to the introduction of A. Then condition 13 is Initial Increase of a New Allele That Affects Transmission (b* - b)[w(wy- ) + (1 bb*)/3y < 0 [14a] Here we suppose that a, and a2 are initially in equilibrium (under cultural transmission as in ref. 9). We seek the conditions under which the allele A will increase in frequency Thus, only if b + b* = 1, a very special case, does this pro- when introduced near the al,a2 equilibrium. This equilibrium duce a condition where Hamilton's rules apply. is given by U13 and u4 = 1 - 13, where 23 is the unique admis- Alternatively, if we denote by fA1 the limiting fraction of sible root of the quadratic equation phenotype 1 among allele A as the equilibrium (123,124) is approached, condition 13 reduces to U3 [(3/2 + y)(bA2 - ba2)-/] + U3[/(bA2 + ba2)/2 + (1 - y)(bA2 - ba2) - 1 + yba2] 2 (QA - U3) > y(b* - b). [14b] + ba2(1 - y) = 0. [10] Sib-to-Sib Altruism Convergence to the root of Eq. 10 has not been proven in general. It should be noted that under transmission alone, in Here the fitness gain accrues to individuals with a sib ofphe- the absence of selection, stable cycles may occur (10), al- notype 1. In this case the conditional kinship approach pro- though under viability selection they do not. In the present duces the following recursions after transmission and selec- kin selection case, some 200 randomly chosen sets of bA2, tion: ba2, /, and y did not produce a single case of cycling or other anomalous behavior upon numerical iteration. We have suc- the initial increase under the ceeded in analyzing problem U1UI C UluT[11 - special condition ba1 = ba2 = b, say, and bAl = bA2 = b*. Then Eq. 10 reduces to + P P(offspring is Al and sib is phenotype 1) /3u + u3(1 - yb - Pb) - b(l - y) = 0, [ill P(offspring is A1) Downloaded by guest on October 1, 2021 Evolution: Feldman et aL Proc. NatL Acad Sci. USA 82 (1985) 5817

U2 X 2 [ t2l=ba(1+Yy) 4 lU4j,

+ P(offspring is A2 and sib is phenotype 1)1 and P(offspring is A2) = bal(1 2 + 14) + P(baa4 + 4 b 22) U3 ° U3 [ When bAl = ba = 1, the eigenvalues of T are 1 - y + / and + P(offspring is a1 and sib is phenotype 1)1 (1 - y + P)/2. hence, the condition for initial increase ofAl, P(offspring is a,) a, is 3> y. When bA,= bal = b, the eigenvalues are b(l - y + (3b) and b(l - y + 3b)/2. The condition for initial increase U4 °C UT 1 is then p8b > y + (b- - 1). [16] + pP(offspring is a2 and sib is phenotype 1)1 P(offspring is a2) J This condition differs from that in the parent-to-offspring case. It is especially interesting that when b = 1, the initial- In other words, increase condition does not reduce to Hamilton's rule. Note, however, that when b1 = b2 = 1, condition 16 is identical to Wu' = uT(1 ') + f3{bA,(Ul + U2)[bAjUl + (bAl + bal)U3/4] the parent-offspring result when only the parent transmitting phenotype 1 (altruism) is the fitness donor (that is, Model + bA2(Ul + U2)[bA2U2 + (bA2 + ba2)U4/4] II).

+ (U3 + + ba1)Ul U4)[bA,(bA, Initial Increase of a New Allele + bA2(bA2 + ba2)U2]/4}, [iSa11 Take a, and a2 in equilibrium to start. Initially, therefore, ul Wu= u + (3{(1 - bAj)(ul + U2)[bAUl + (bA, + ba1)U3/41 = U2 = 0 so that in Eqs. 1 and 2 W = X = 0. Thus, as before, UT = ba1143 + baU4-2ba2)U4.and uT = (1 - ba,)u3 + (1 As in + (1 - bA2)(Ul + U2)[bA2U2 + (bA2 + ba2)U4/41 the analysis of the parent-to-offspring case, we may simplify + (U3 + U4)[(1- bA,)(bA, + bal)Ul matters by taking bA, = bA2 = b* and ba1 = ba2 = b. Thus, b is the transmission rate associated with the resident a gene and + (1 -bA2)(bA2 + ba2)U2]/4}, [15b b* that associated with the rare A gene. In this case, the equilibrium of a1 and a2 to begin the process is specified by WU= U(1 - y) + P{bal(U3 + U4)[balU3 + (bAl + bal)Ui/4] U3 = b[l + (3b - y]/[l + (P - y)b]. The local analysis of the u1, U2 transformation near this equilibrium produces the con- + ba2(U3 + U4)[ba2U4 + (bA2 + ba2)U2/4] clusion that A increases when rare if + (Ul + U2)[bal(bA, + ba1)U3

+ ba2(bA2 + [15c - - b) > 0. [171 ba2)U4]/4}, ] (22 y(b* and Now P/2 > y is the condition derived by Hamilton for the initial increase of an altruistic allele under purely genetic transmission. We see that if this condition holds, then a new Wi4 = u47 + P{(1 - ba,)(03 + U4)[balU3 + (bAI + ba,)Ui/4] gene A that imparts greater uniparental transmission of the + (1 - ba2)(,3 + U4)[ba2U4 + (bA, + ba,)U2/4] kin-selected trait (i.e. b* > b) will succeed. Thus, in addition to Hamilton's condition it must also be true in this case that + (Ui + U2)[(1- ba)(bA, + ba,)U3 altruism be more easily learned by individuals carrying A + (1 - ba2)(bA2 + ba2)U4]/4}, [15d] than by individuals carrying a. where W is the sum of the right-hand sides. Conclusions Initial Increase of the Altruistic Phenotype There are obviously several ways to construct population models for the learning ofaltruism. We have considered only As in the case of parent-to-offspring altruism, if bA = 1, ba, vertical uniparental transmission and additive combination = O bA2= 1, and ba2 = 0 (the strictly genetic situation), then of fitness loss and gain. The effects of departures from these the condition for initial increase of Al from fixation of a2 is assumptions are under investigation, but they seem a natural P3/2 > By. More generally, if bA2 = ba2 = 0 and initially the place to start, and from our results it is clear that nongenetic population is fixed for phenotype 2 with A and a segregating transmission of the altruistic phenotype causes departures at frequencies u2 and a4, respectively, the initial increase of from Hamilton's rule. These departures appear to be most phenotype 1 is governed by the eigenvalues of matrix T = pronounced when the transmitter of the altruism and the al- with truist are the same, as in the parent-to-offspring case. It is Ijtfjjj also seen that whether altruism is performed by a randomly chosen parent or by just the transmitting parent may also be tl = b [(1 - )( + + P(bAia2 + + ba ) important. The effects ofother types ofcultural transmission 2±) bA4 (9) remain to be considered. We have taken care to differentiate two classes of prob- A2 (bAl + bal) ] tl2 = bA,[( "2j9 lems. The first involves the increase of a rare altruistic phe- 24 notype starting from a situation in which alleles with differ- Downloaded by guest on October 1, 2021 5818 Evolution: Feldman et aL Proc. NatL Acad. Sci. USA 82 (1985)

ent learning capacities are both present. The second exam- In the sib case, a regression ofrecipient sib on donor sib may ines the increase, when rare, of an allele that affects the be computed. The values for the sibs are b* for Al, b for a, teaching or learning of altruism. The former of these prob- and 0 for A2 and a2. A straightforward calculation reveals lems is the usual one addressed in kin-selection theory, but that the sib-to-sib regression is 1/2, the usual value for the strictly in terms of increase of an allele from a genetically initial increase condition under sib-to-sib kin selection. (and phenotypically) monomorphic state. As to the second problem, when only four transmission coefficients (reduced This research was supported by National Institutes of Health to two in most of the analysis) are used, it is not possible to Grants GM10452, GM20467, and GM28016. separate teaching and learning abilities. It should be stressed that the case bA1 = bA2, bal = ba2 analyzed in detail here is 1. Hamilton, W. D. (1964) J. Theor. Biol. 7, 1-52. special and that the four-parameter transmission problem is 2. Cavalli-Sforza, L. L. & Feldman, M. W. (1978) Theor. Popul. unlikely to yield such elegant solutions. Biol. 14, 268-280. A final remark that may suggest a connection between the 3. Uyenoyama, M. K. & Feldman, M. W. (1982) Am. Nat. 120, above results and those in the standard kin-selection litera- 614-627. ture is in order. Uyenoyama and Feldman (11) and Uye- 4. Lumsden, C. J. & Wilson, E. 0. (1981) Genes, Mind, and Cul- noyama et al. (12) suggested that initial increase of the rare ture (Harvard University Press, Cambridge, MA). > 5. Wilson, E. 0. (1975) Sociobiology: The New Synthesis (Belk- allele should occur if PbD-R y, where bD--R is the regres- nap Press of Harvard University Press, Cambridge, MA). sion of the recipient's additive genetic value on that of the 6. Maynard Smith, J. & Warren, N. (1982) Evolution 36, Suppl. 3, donor of the altruism. The regression must be computed in 620-627. the limit as the original equilibrium is approached. In the par- 7. Feldman, M. W. & Cavalli-Sforza, L. L. (1984) Proc. Natl. ent-to-offspring case, a regression is obtained by considering Acad. Sci. USA 81, 1604-1607. the parent as the recipient and the offspring as the donor. 8. Cavalli-Sforza, L. L. & Feldman, M. W. (1983) Proc. Natl. Offspring are phenotype 1 and value b* with probability (u, Acad. Sci. USA 80, 4993-4996. + u2) and phenotype 1 of value b with probability (U3 + U4). 9. Cavalli-Sforza, L. L. & Feldman, M. W. (1981) Cultural Phenotype 2 has value 0. Only transmitting parents are Transmission and Evolution (Princeton University Press, 1 if 1 and if Princeton, NJ). counted and given value ofphenotype 0 ofpheno- 10. Feldman, M. W. & Cavalli-Sforza, L. L. (1976) Theor. Popul. type 2. The limiting value, as (23, 24) is approached, of the Biol. 9, 238-259. regression of parent on offspring in this sense is (QA - 11. Uyenoyama, M. K. & Feldman, M. W. (1981) Theor. Popul. u3)/2(b* - b), which is the quantity relevant to the initial Biol. 19, 87-123. increase condition 14b. We have not devised a natural re- 12. Uyenoyama, M. K., Feldman, M. W. & Mueller, L. D. (1981) gression in which these donor/recipient roles are reversed. Proc. Natd. Acad. Sci. USA 78, 5036-5040. Downloaded by guest on October 1, 2021