Theoretical Aspects of the Mode of Transmission in Cultural Inheritance

Theoretical Population Biology 55, 208225 (1999)

Article ID tpbi.1998.1400, available online at http:www.idealibrary.com on

Theoretical Aspects of the Mode of Transmission in Cultural Inheritance Kiyosi Takahasi Department of Biological Sciences, Graduate School of Science, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Received May 30, 1997

This study investigates how evolutionary factors interact to determine the relative importance of vertical versus nonvertical mode of transmission in cultural inheritance. Simple mathematical models are provided to study the joint evolution of two cultural characters, one determining the viability and the fertility of individuals, and the other determining the vertical transmission rate of the first trait. Ordinary local stability analyses indicate that intrademic processes should lead to a greater reliance on vertical cultural transmission. On the other hand, when newly arisen variants are adaptive and favored in biased cultural transmission, interdemic processes may lead to a decrease in vertical transmission. This is because biased nonvertical transmission may effectively propagate the adaptive variants, further increasing the average growth rate of the population. These results are verified under several distinct sets of assumptions. It is also inferred that the degree and intensity of transmission bias may be the important determinants of cultural processes. ] 1999 Academic Press

INTRODUCTION

(1989a), and Feldman and Zhivotovsky (1992) for general treatments of vertical transmission. Under the prevalence of vertical transmission, cultural changes should show similar evolutionary dynamics to genetic evolution, as long as natural selection overcomes counteracting effects of cultural selection (Cavalli-Sforza and Feldman 1981) or biased cultural transmission (Boyd and Richerson, 1985); biased transmission, another non-Mendelian property of cultural inheritance, occurs when certain cultural elements are accepted in preference to others (see below). Cooccurrence of various modes of transmission may, on the other hand, complicate the evolutionary dynamics of cultural changes. This can be illustrated by a simple generalized model of cultural transmission (Cavalli-Sforza and Feldman, 1981). Consider a dichotomous trait taking character states H and h with Darwinian fitness of H being 1+s relative to 1 for h. Assuming random mating, the conditions for the protection of H from loss are

The essence of culture in the context of evolutionary biology is a system of inheritance via social learning. Viewed in this way, cultural dynamics can be investigated in a similar way as population geneticists study genetic transmission and evolution (Cavalli-Sforza and Feldman, 1981; Boyd and Richerson, 1985). While genetic and cultural inheritance systems share many features in common, some differences are found in nonMendelian properties inherent in cultural inheritance. Cultural elements, the cultural analog of genes in genetic inheritance, are transmitted through various pathways (Cavalli-Sforza and Feldman, 1981). Oblique and horizontal transmission are said to have occurred when the elements have been transmitted from (nonparental) elders to the young (oblique transmission) or passed within a generation (horizontal transmission). Vertical transmission occurs from parents to their (biological or genetic) offspring as is the case with ordinary genetic transmission; see Findlay (1991, 1992a), Findlay et al.

0040-580999 K30.00 Copyright ] 1999 by Academic Press All rights of reproduction in any form reserved.

s>0 with haploid genetic transmission,

208

(1)

209

Modes of Cultural Transmission

(1+s)(g+2b)>1 with vertical and oblique cultural transmission, (2a) and (1+s)(1+g) } 2b>1 with vertical and horizontal cultural transmission, (2b) where b is the expected proportion after vertical transmission of H offspring from the heterogeneous mating H_h, and g is the rate of oblique or horizontal transmission (see Eqs. (3.4.4) and (3.4.5) in Cavalli-Sforza and Feldman, 1981). Conditions (2) can be rewritten as 1 &1 g+2b

(3a)

1 &1, (1+g) } 2b

(3b)

s> and s>

respectively. Note that in (3) negative s is possible if g and b are sufficiently large. Note also that the condition (1) for genetic transmission is a special case of conditions (3) with g=0 and b=12 (but also see Crow, 1979). Above conditions clearly indicate that while genetically transmitted traits can be maintained in a haploid population only if they are selectively favored, deleterious cultural traits may be protected from loss if nonvertical transmission occurs with sufficient effectiveness. Other studies have also confirmed that with nonvertical cultural transmission, cultural dynamics may greatly differ from those expected from ordinary theory of genetic evolution (e.g., Boyd and Richerson, 1985; Findlay et al., 1989b; Richerson and Boyd, 1992). Acceptance of the small family ideal associated with recent demographic transitions in European society may be an example of nonvertically transmitted maladaptive cultural practices (see Cavalli-Sforza and Feldman, 1981, Sect. 3.9; Richerson and Boyd, 1984). Since these studies assume constant obliquehorizontal transmission rates, however, they are unable to explain, e.g., why nonvertical transmission to favor maladaptive cultural practices might have evolved. Therefore, from the theoretical view point, it would be interesting to know how the evolution determines the relative importance of vertical versus nonvertical transmission in cultural systems. In this paper I provide a number of mathematical models to describe evolutionary changes in the degree of relative importance of the

respective modes of cultural transmission. Models follow the joint evolution of two culturally transmitted traits, one determining the viability and fertility of individuals, and the other determining the vertical transmission rate, or the proportion of individuals who acquire the first trait via vertical cultural transmission. Although it is assumed throughout that the rate of vertical transmission is determined culturally, I expect that some special cases are also appropriate for genetic determination of the trait because of the formal resemblance between genetic and unbiased vertical cultural transmission. Similar arguments are found in the epidemiological studies of the host-parasite coevolution. A growing literature suggests that it is generic in parasitic transmission that the greater transmissibility results in the evolution of greater virulence (e.g., May and Anderson, 1983; Ewald, 1983; Toft and Karter, 1990; Bull, 1994). Several theoretical and conceptual studies have addressed the problem in terms of the modes of parasitic transmission (Fine, 1975; Ewald, 1987; Bull et al., 1991; Yamamura, 1993; see also Axelrod and Hamilton, 1981; Nowak, 1991; Lipsitch et al., 1995). Yamamura (1993) has investi gated a game theoretical model for the joint evolution of parasite exploitation of the host and rate of vertical parasitic transmission. He found that there is a critical value of the transmission rate above which the evolutionarily stable degree of exploitation is negative and a mutualistic relation between hosts and parasites eventually evolves. As discussed elsewhere (e.g., CavalliSforza and Feldman, 1981; Richerson and Boyd, 1989), a comparison of cultural and parasitic transmission may be fruitful in the evolutionary studies of cultural inheritance. I first investigate the evolution of dichotomous traits assuming that the vertical transmission is uniparental. These assumptions are introduced solely to avoid mathematical complexity and to obtain a clearer view of the system. Later in the analysis the more generalized models with multiple states or biparental vertical transmission are investigated based on the results of the simplified analysis.

EVOLUTION UNDER UNIPARENTAL TRANSMISSION Consider two culturally transmitted characters, A and B, with possible character states (or phenotypes) A i and B j , respectively. Each individual is assumed to express only one of the states for each trait. The trait A determines the vertical transmission rate r of trait B, or the

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Kiyosi Takahasi

proportion of individuals who acquire the trait B via vertical transmission. The trait B determines the fertility and viability of individuals; an individual with state B j has expected fecundity f j and viability v j . It is assumed that while the trait B may be acquired either through vertical or nonvertical transmission, the trait A is exclusively transmitted vertically. The life cycle starts with a parental population that consists of phenotypes A i B j with their frequencies P ij ( ij P ij =1). Each of these adults reproduces asexually with the expected fecundity f j , and soon after (or simultaneously with) reproduction, the vertical transmission of trait A occurs. This yields an offspring population that consists of phenotypes A i , where the symbol indicates that the individuals have not yet acquired the trait B at this stage of the life cycle. Vertical transmission of the trait B follows, and this is further followed by obliquehorizontal transmission of the trait. An offspring with state A i has a probability r i ( >0) of acquiring the trait B via vertical transmission, while the remaining proportion 1&r i acquires the trait through nonvertical pathway; i.e., via obliquehorizontal transmission. This completes the cultural transmission of traits A and B. To make the argument clear, we shall deal with the cases of oblique and horizontal transmission separately; vertical transmission of the trait always occurs anyway. Viability selection then follows, yielding the adult population of the next generation with phenotypic frequencies P $ij . I first formalize the model by assuming that vertical transmission is uniparental. Note that this assumption automatically leads to unbiased vertical transmission. Under this assumption the master recursion for the trait frequency is derived as W } P$ij =[r iP ij f j +(1&r i )( k P ik f k ) } g j ] v j , (4a) where k is a dummy variable, g j is the obliquehorizontal transmission parameter (see below), and W is the average growth rate of the population. The corresponding recursion for numbers of individuals is n$ij =[r i n ij f j +(1&r i )( k n ik f k ) } g j ] v j ,

(4b)

where n ij is the number of individuals with phenotype A iB j . Within the brackets of Eqs. (4), the first terms indicate the contribution to the next generation through vertical transmission of trait B. For example, in Eq. (4b), among the n ij f j offspring produced by parents of phenotype A iB j , a proportion r i acquires the state B j via vertical transmission. This leaves (1&r i )( k n ik f k ) offspring with state A i that are undetermined regarding the trait B; their phenotype at this stage of the life cycle is

A i . Among these uncultured offspring, a proportion g j acquires the state B j via oblique or horizontal transmission, whichever available (see the second term in the bracket). Since every individual is assumed to acquire one and only one of the states for the trait B,  j g j =1 holds. The sum of the first and second terms multiplied by the viability gives the expected number of adults in the next generation. Dividing both sides of (4b) by n= ij n ij yields Eq. (4a) with W=n$n. While I here assume that the viability of an individual is determined by its own character state, there may be cases where the parental character state governs the offspring viability. These situations are analyzed briefly in Appendix A. I now proceed to define the obliquehorizontal transmission parameter g j . Figures 1 and 2 describe the case where the trait B is dichotomous (j=0, 1). Consider a group of transmitters and another group of receivers; biased (or unbiased) transmission occurs from the transmitter group to the receiver group. Then g j is the expected frequency of individuals with state B j in the receiver group after the transmission (see Fig. 1). Given the frequency x j of B j individuals in the transmitter group, it is assumed that the relation g j =(1+G j ) x j [ k (1+G k ) x k ]

(5a)

holds for all j, where the parameters G j (&1
(5b)

Under horizontal transmission, the corresponding transmitter group consists of offspring who have already acquired the trait B, as well as the trait A, via vertical transmission. This requires that, for some i, r i >0. Then x j =( i r iP ij f j )( ik r i P ik f k ) and hence g j =(1+G j )( i r i P ij f j )[ k(1+G k )( i r i P ik f k )]. (5c)

Equations (4) and (5) together determine the evolutionary dynamics of the system.

211

Modes of Cultural Transmission

assuming that the other trait is fixed on one of the two states. Evolution of Trait A Under Fixation of Trait B

FIG. 1. Biased transmission that occurs from the group of transmitters to the group of receivers in the case of dichotomous trait ( j=0, 1; G 1 =G and G 0 = &G with &1
In the following, I assume that both traits A and B are dichotomous; i.e., i, j # [0, 1], and G 1 =G and G 0 = &G. The obliquehorizontal transmission parameter can then be simplified as g 0 =(1&G) q 0[(1&G) q 0 +(1+G) q 1 ]

(5d)

with oblique transmission, where q j =P 0j +P 1j , g0 =

(1&G)(r 0P 00 +r 1P 10 ) f 0 (1&G)(r 0P 00 +r 1P 10 ) f 0 +(1+G)(r 0P 01 +r 1P 11 ) f 1 (5e)

with horizontal transmission, and g 1 =1&g 0. We see from nonvertical transmission parameters (5) that this type of biased transmission is also frequency dependent, and is formally identical to the model of selective mating presented by Kirkpatrick (1982). A model of cultural selection with this type of biased transmission is discussed in detail by Takahasi (1998). In the following, I first investigate the evolution of one of the two traits

FIG. 2. The degree of transmision bias for a dichotomous trait B ( j=0, 1; G 1 =G and G 0 =&G with &1
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Let p i = j P ij denote the frequency of A i individuals in the population. Then assuming that the population is fixed on B j , we have p$i =p i for all i. This is intuitively obvious because the population is monomorphic with respect to both fecundity and viability, and cultural transmission of the trait A is unbiased. Evolution of Trait B Under Fixation of Trait A Let q 0 =P 00 +P 10 and q 1 =P 01 +P 11 denote the frequencies of B 0 and B 1 individuals, respectively. I then investigate the evolution of the trait B assuming that the population is fixed on A 0 . Substituting appropriate values in Eq. (4a), the recursion equation for q 1 is derived as q$1 = r 0 q 1 w 1 +(1&r 0 )(q 0 f 0 +q 1 f 1 ) g 1 v 1 , r 0(q 0 w 0 +q 1 w 1 )+(1&r 0 )(q 0 f 0 +q 1 f 1 )(g 0 v 0 +g 1 v 1 ) (6a) where w j =f j v j is the fitness of B j individuals. I further assume in the following analysis that the natural selection operates through differential viability only; fertility of individuals is assumed to be a constant (f j =f for all j). With this assumption, the recursion (6a) is simplified as q$1 =

r 0q 1v 1 +(1&r 0 ) g 1v 1 . r 0(q 0v 0 +q 1v 1 )+(1&r 0 )(g 0v 0 +g 1v 1 )

(6b)

The assumption of constant fertility is made merely for mathematical convenience. Differential fertility is introduced in Appendix B, where the possible role of nonvertical transmission in demographic transitions and the diffusion of birth control is discussed. Note that in the present case where all individuals in the population maintain the same vertical transmission rate and fertility, obliquehorizontal transmission parameters (5d) and (5e) become identical. Hence there is no need to distinguish between these two modes of nonvertical transmission, and either oblique or horizontal transmission results in identical equilibrium properties. Setting 2q 1 = q$1 &q 1 =0 in Eq. (6b), we find three equilibrium states in terms of trait frequencies.

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Kiyosi Takahasi

Equilibrium E 0. Fixation of B 0 with equilibrium frequencies q^ 0 =1 and q^ 1 =0. The equilibrium always exists and is stable if r0 }

v1 1+G v 1 +(1&r 0 ) } } <1. v0 1&G v 0

(7a)

The condition shows that if the vertical transmission rate r 0 is low and the state B 0 is favored by biased transmission (G<0), then fixation of a maladaptive variant (v 0 v 1 ). Such dependence of stability on the vertical transmission rate may be better illustrated by rewriting the condition (7a) as v 0 1+G r0 < & v 1 1&G

\

+<\

1+G 1& for G<0 1&G

+

(7b)

or r0 >

v0

1+G

1+G

\ v & 1&G+<\1& 1&G+ for G>0.

(7c)

1

\

v 0 1+G & v 1 1&G

+<\

1&

1+G , 1&G

+

(8)

R 0 is the threshold rate of vertical transmission above (below) which the equilibrium is stable if G>0 (G<0). If, for example, the state B 0 is less favored in the biased transmission (G>0), then the stability of the fixation equilibrium demands higher vertical transmission rate (r 0 >R 0 ). The parameters and variables are listed with their explanations in Table I. Equilibrium E 1. Fixation of B 1 (q^ 0 =0, q^ 1 =1). The equilibrium always exists and is stable if r0 }

v0 1&G v 0 +(1&r 0 ) } } <1, v1 1+G v 1

(9a)

or r 0 0 and r 0 >R 1 for G<0,

(9b)

where R1 =

v1

1&G

1&G

\v &1+G+<\1&1+G+ 0

The List of Parameters and Variables Symbol

Explanation

fj

fertility of B j parents

gj

obliquehorizontal transmission parameter

Gj

degree of transmission bias to favor the state B j

Hi

degree of transmission bias to favor the state A i

pi

frequency of A i individuals in the population

P ij

frequency of A iB j individuals in the population

qj

frequency of B j individuals in the population

ri

rate of vertical transmission of A i individuals

Rj

threshold rate of vertical transmission above (below) which the equilibrium E j is stable if G>0 (G<0)

vj

viability of B j individuals

wj

fitness of B j individuals, defined as w j =f j v j

gives the threshold rate. Due to the symmetry between the character states B 0 and B 1, replacing the parameters for the state B 1 to those of B 0 (e.g., v 1 Ä v 0 , G Ä &G, and so on) in the conditions (9) yields the stability conditions (7) for the equilibrium E 0 . Equilibrium E p. Polymorphism of B 0 and B 1, with equilibrium frequencies

Then defining R0 =

TABLE I

(10)

q^ 1 =

r 0(1&G)(v 0 &v 1 )+(1&r 0 )[(1&G) v 0 &(1+G) v 1 ] &r 0 } 2G(v 0 &v 1 ) (11)

and q^ 0 =1&q^ 1. The equilibrium exists if 0
(12)

where R 0 and R 1 are the threshold rates defined by (8) and (10), respectively. The equilibrium is stable whenever it exists. Note that in condition (12), subscripts 0 and 1 of the threshold rate R designate which of the two states B 0 or B 1 is fixed, while the subscript 0 of the vertical transmission rate r indicates that the state A 0 is fixed (see Table I). Conditions (7), (9), and (12) indicate that, for any set of parameter values, one and only one of the three equilibria is necessarily stable with the other two being unstable or nonexistent. Numerical iterations of the recursion (4a) never produced limit cycles, and it is suggested that the equilibrium is globally stable and the system converges to this stable equilibrium regardless of initial conditions. Figure 3 illustrates the stability regions of the equilibria along the vertical transmission rate r.

Modes of Cultural Transmission

FIG. 3. Stability region of the three equilibria along vertical transmission rate r for cases (a) v 0 v 1 >1>(1+G)(1&G) or v 0 v 1 > (1+G)(1&G)>1; (b) v 0 v 1 <1<(1+G)(1&G) or v 0 v 1 <(1+G) (1&G)<1; (c) (1+G)(1&G) v 0 v 1 >1. The threshold rates R 0 and R 1 are defined by (11) and (13) in the text. It is indicated that when natural selection and biased transmission favor the same character state (v 0 v 1 >1>(1+G)(1&G), or vice versa) or when these two evolutionary forces operate in a mutually antagonistic manner with relatively weak transmission bias compared to the intensity of natural selection (v 0 v 1 >(1+G)(1&G)>1, or vice versa), the system converges to a monomorphic equilibrium where the adaptive (i.e., more viable) state is fixed, irrespective of the rate of vertical transmission (cases (a) and (b)). On the other hand, if the degree of transmission bias is so strong that it should overcome the antagonistic effect of viability selection ((1+G)(1&G)>v 0 v 1 >1, or vice versa), then different equilibria may be reached depending on the vertical transmission rate (cases (c) and (d)).

The figure shows that when natural selection and biased transmission favor the same character state (v 0 v 1 >1> (1+G)(1&G), or vice versa) or when these two evolutionary effects operate in a mutually antagonistic manner with relatively weak transmission bias compared to the intensity of natural selection (v 0 v 1 >(1+G) (1&G)>1, or vice versa), the system converges to a monomorphic equilibrium where the adaptive (i.e., more viable) state is fixed, irrespective of the rate of vertical transmission (cases (a) and (b) in Fig. 3). On the other hand, if the intensity of transmission bias is so strong that it should overcome the antagonistic effect of viability selection ((1+G)(1&G)>v 0 v 1 >1, or vice versa), then different equilibria may be reached depending on the vertical transmission rate (cases (c) and (d) in Fig. 3). Such dependence of stability on the vertical transmission rate occurs because, in the present model, while vertical transmission is assumed to be unbiased and always favors the evolution of adaptive variants (as in ordinary haploid genetics), obliquehorizontal transmission may establish the maladaptive, less viable state if the associated transmission bias is strong enough to overcome the counteracting effect of natural selection. Consequently, if the maladaptive variant is preferred in biased

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213 obliquehorizontal transmission, low vertical transmission rate may produce stable fixation of the maladaptive variant (r 0 R 1 in case (c) and r 0 >R 0 in case (d)); indeed, conditions (7) and (9) show that the extreme case r 0 =1 always produces the fixation of adaptive state. Above results are obtained under the assumption that the population is monomorphic for the vertical transmission rate (p 0 =1). Considering that there are situations where the stability of the equilibrium depends on the fixed rate of vertical transmission r 0 (cases (c) and (d) above), one may suspect that joint introduction of an innovative (or, mutant) state A 1 with r 1 ( {r 0 ) may drive the initial population away from the otherwise stable equilibrium (i.e., when condition (7) holds). Then assuming that the population is initially fixed on phenotype A 0B 0 with condition (7) so that this equilibrium state is stable with respect to perturbations of the trait B, it is interesting to know whether (i) joint introduction of states A 1 with r 1 >R 0 >r 0 and B 1 with v 1 >v 0 to the population results in a new equilibrium (see case (c) in Fig. 3), and whether (ii) introduction of states A 1 with r 1
214

Kiyosi Takahasi

Set P 00 =1+= 00 , P 01 == 01 , P 10 == 10 , and P 11 == 11 , where | = ij | < <1 and  ij = ij =0; = ij indicate the small frequency deviations from the equilibrium. Writing the recursion (4a) in a matrix form, we have =$=M=+ O( |=| 2 ), where ==(= 01 , = 10 , = 11 ) T and the coefficient matrix M has the form

m 11 0 m 13 M= 0 1 m 23 .

\

0

0

m 33

+

(13)

Eigenvectors corresponding to the three eigenvalues, * 1 =m 11 , * 2 =1, and * 3 =m 33 , are 1 e1= 0 ,

0 e2= 1 ,

\+ \+ 0

m 13(* 3 &* 1 ) and e 3 = m 23(* 3 &* 2 ) ,

\

0

1

+

(14)

respectively. The matrix (13) shows that the maximal eigenvalue is either unity or greater than unity. It is further found that when the maximal eigenvalue is unity, the equilibrium is neutrally stable to the second order (analysis not shown; see Nagylaki, 1992, pp. 140141, for a method of second-order analysis). Two other eigenvalues are * 1 =m 11 =r 0 }

v1 1+G v 1 +(1&r 0 ) } } v0 1&G v 0

(15)

v1 ; v0

(16)

and * 3 =m 33 =r 1 }

the same eigenvalues are obtained with either oblique or horizontal transmission. Suppose that the equilibrium is stable against perturbations of the trait B, so that the condition * 1 <1 is always satisfied (see condition (7a) above). Then if * 3 >1, the equilibrium becomes unstable and innovative states A 1 and B 1 may be established. This corresponds to the situation (i) described above (see also case (c) in Fig. 3); joint introduction of state A 1 with r 1 >v 0 v 1 and state B 1 with v 1 >v 0 into the population violates the stability of fixation of A 0 B 0 . Note that the condition * 3 >1 (or r 1 >v 0 v 1 ) and the assumption * 1 <1 (or 1> v 0 v 1 >(1+G)1&G)) automatically satisfy the inequality r 1 >R 0 (see definition (8)). However, there are values

of r 1 which satisfies r 1 >R 0 but still r 1 v 1 and hence * 3 <1 is automatically satisfied (as well as * 1 <1). The stability of the equilibrium cannot be revealed by the ordinary linear analysis because the maximal eigenvalue is unity and the system is neutrally stable with respect to perturbations of the trait A (but not B; see Eqs. (13) and (14)). As we have seen above that p$i =p i under fixation of B, emergence of the neutral stability is reasonable. We find that such neutral stability is also the case in situation (iii), where * 3 <1 as well as * 1 <1 is assumed and hence the maximal eigenvalue again becomes unity. In these situations, less significant evolutionary effects such as random drift or interdemic competition may ultimately determine the evolutionary outcome of the cultural processes. These results of the stability analysis are mostly confirmed also for the multistate model given and analyzed in Appendix C. From the above analytical results, one may suspect that once the fixation of a more viable, adaptive character state (e.g., B 1 with v 1 >v 0 ) associated with a high vertical transmission rate (e.g., r 1 >v 0v 1 ) is attained, no maladaptive state can be established in the population. Hence the population is expected to maintain a greater reliance on vertical transmission. However since the above analyses assume * 1 <1 so that the equilibrium is stable against perturbations of the trait B, the possibility is not excluded where the reliance on vertical transmission may be reduced when * 1 >1. To investigate this possibility, the uniparental transmission model (4a) was studied numerically assuming that the natural selection acts through differential viability. On doing this, I posited multiple states for the trait A by assigning r i =iN A for each of the states A i (i=0, 1, 2, ..., N A ), where N A +1 denotes the number of possible states for the trait (see also Appendix C). Specifically, N A =100 was used. Hence the vertical transmission rate was assumed quasi-quantitative. The viability trait B was kept dichotomous; i.e., N B =1. The evolutionary dynamics were studied numerically by iterating the oblique transmission version of the uniparental transmission model (4a), and following deterministic changes in the average vertical transmission rate (r = N i=0 r i p i ) and the average viability (v =q 0 v 0 +q 1 v 1 ). When the system converged to a stable monomorphic equilibrium with respect to the trait B, an innovative state B j ( j=0, 1) was introduced at a low frequency, with v j and G j each determined A

215

Modes of Cultural Transmission

FIG 4. An example of dynamical changes over 1,000 generations in the average rate of vertical transmission rate, r = N i=0 r i p i , and the average viability, v =q 0 v 0 +q 1v 1 . The uniparental transmission model was numerically studied for multiple states for the trait A with r i =iN A (i=0, 1, 2, ..., N A ). Specifically, N A =100 was used. The viability trait B was kept dichotomous, N B =1. The evolutionary dynamics were studied by iterating the oblique transmission version of the biparental transmission model (4a), and following deterministic changes in the average vertical transmission rate and the average viability. The initial value of r was set around 0.1. See text for a detailed description of the method. A

by generating pseudorandom numbers. The iteration was then continued until another monomorphic equilibrium (with respect to the trait B) was reached. No stable polymorphisms of the viability trait were observed. Since we are especially interested to see if the increase in the rate of vertical transmission should be observed, the initial value of r was set as low as 0.1. Under these assumptions, each set of the iterations was repeated for 1,000 generations. An example of evolutionary dynamics of the average viability and the average rate of vertical transmission is illustrated in Fig. 4. The figure shows typical evolutionary changes of r over time. That is, starting from the initial value around 0.1, the average rate of vertical transmission increases steadily over time, and although the population may experience temporal decrease in the average rate, a relatively greater reliance on vertical transmission is maintained once it is attained.

RANDOM MATING AND BIASED BIPARENTAL TRANSMISSION So far I have investigated the coevolution of the rate of vertical cultural transmission and the viability trait assuming that the vertical transmission is uniparental. The assumption necessarily entails that the vertical transmission is unbiased, and hence may give fundamentally

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different dynamics from models that allow biased vertical transmission. In this section, I extend the analysis to incorporate biased vertical transmission by assuming panmixia and biparental vertical transmission. It is assumed that the traits are dichotomous and viability of an individual is determined by its own phenotype; there is no difference in fertility among individuals. Further the analysis is confined to the local stability of the fixation of A 0 B 0 . Consider a monoecious panmictic population, with biased vertical transmission scheme as summarized in Table II. To describe biased vertical transmission, I assume that a proportion r i (1+G)2 of the offspring produced from a partially heterogeneous mating A i B 0_ A i B 1 acquires the state A i B 1 via vertical transmission. This is a special case of biased transmission (5), where a group of transmitters consists of an A i B 0 parent and an A i B 1 parent (see Fig. 1). Note also that the biased transmission parameters G are assumed to be the same for both vertical and nonvertical transmission. The corresponding parameter for the biased vertical transmission of the trait A is H ( =H 1 , &1
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Kiyosi Takahasi

TABLE II Mating Table for the Panmictic Population with Biased Vertical Transmission Offspring distribution after the vertical transmission of traits A and B Mating

Frequency

A0 B 0

A0 B1

A1 B0

A1 B 1

A0 &

A1 &

A 0 B 0_A 0 B 0

P 200

r0

0

0

0

1&r 0

0

A0 B1

2P 00 P 01

r 0(1&G) 2

r 0(1+G) 2

0

0

1&r 0

0

A1 B0

2P 00 P 10

r 0(1&H) 2

0

r 1(1+H) 2

0

(1&r 0 )(1&H) 2

(1&r 1 )(1+H) 2

A1 B1

2P 00 P 11

r0 (1&G)(1&H) 4

r 0 (1+G)(1&H) 4

r 1 (1&G)(1+H) 4

r 1 (1+G)(1+H) 4

(1&r 0 )(1&H) 2

(1&r 1 )(1+H) 2

P 201

0

r0

0

0

1&r 0

0

A1 B0

2P 01 P 10

r0 (1&G)(1&H) 4

r 0 (1+G)(1&H) 4

r 1 (1&G)(1+H) 4

r 1 (1+G)(1+H) 4

(1&r 0 )(1&H) 2

(1&r 1 )(1+H) 2

A1 B1

2P 01 P 11

0

r 0(1&H) 2

0

r 1(1+H) 2

(1&r 0 )(1&H) 2

(1&r 1 )(1+H) 2

P 210

0

0

r1

0

0

1&r 1

2P 10 P 11

0

0

r 1(1&G) 2

r 1(1+G) 2

0

1&r 1

P 211

0

0

0

r1

0

1&r 1

A 0 B 1_A 0 B 1

A 1 B 0_A 1 B 0 A1 B1 A 1 B 1_A 1 B 1

transmission as described in the previous models follows, yielding the recursions for the phenotypic frequencies as listed in Appendix D. Setting P 00 =1+= 00 , P 01 == 01 , P 10 == 10 , and P 11 = = 11 in (D1) in Appendix D as in the previous section, I again find that the coefficient matrix M has a form given by Eq. (13). Two of the eigenvalues of the matrix are * 1 =r 0 (1+G) }

v1 1+G v 1 +(1&r 0 ) } } v0 1&G v 0

(17a)

with oblique transmission, or * 1 =r 0 (1+G) }

v1 (1+G) 2 v 1 +(1&r 0 ) } (17b) } v0 1&G v 0

with horizontal transmission, and

differ depending on which of the two modes of nonvertical transmission is assumed. This is because in the present model, biased vertical transmission alone may produce frequency changes in the trait B. Hence the frequencies in the group of transmitters in horizontal transmission, which consists of offspring who have acquired the trait B via (biased) vertical transmission, may differ from those of adults, who constitute the group of transmitters in oblique transmission (see second terms in (17a) and (17b); see also Fig. 1). Also, for the same reason, a high vertical transmission rate may nonetheless lead to stable fixation of a maladaptive variant (v 0
(1+G)(1+H) v 1 } . * 3 =r 1 } 2 v0 Note that while same eigenvalues are obtained either oblique or horizontal transmission in uniparental transmission model (see eigenvalues and (16)), the eigenvalue * 1 for the present model

for G<0

and

r 0 >R 0 for

(19)

(18) with the (15) may

G>0,

where the threshold R 0 is defined in the present biparental model as R0 =

v 0 1+G 1 1 & } 1& 1+G v 1 1&G 1&G

\

+\

+

&1

(20a)

217

Modes of Cultural Transmission TABLE III

with oblique transmission, and R0 =

\

v0 1 1+G 1+G } 1& & v 1 1+G 1&G 1&G

+\

+

&1

(20b)

Changes in Average Vertical Transmission Rate r and Average Viability v over 1,000 Generations Generation

with horizontal transmission. Note that it is possible to obtain an inequality 0v 0 v 1 >(1+G)(1&G)

for

G<0

for

G>0

r 0.100\0.004 v 0.507\0.217

100

200

500

1,000

0.431\0.246 0.704\0.174

0.626\0.230 0.729\0.177

0.820\0.162 0.771\0.150

0.869\0.120 0.778\0.160

Note. Values are the mean \ standard deviation calculated from 100 runs of numerical iterations of a multistate version of the biparental transmission model. See text for detail.

(21a)

or 1+G
0

(21b)

with oblique transmission. The corresponding condition for horizontal transmission model becomes 1+G>v 0 v 1 >(1+G) 2(1&G)

for

G<0

(22a)

1+G
for

G>0.

(22b)

or

As expected, these conditions indicate that for the inequality 00 so that the biased transmission favors the state B 1 over B 0 , the condition v 0 >v 1 is required (see (21b) and (22b)). Suppose first that the condition (21b) holds with oblique transmission (or (22b) with horizontal transmission) so that the fixation of B 0 with v 0 >v 1 (1+G) ( >v 1 ) is stable if r 0 >R 0 (see condition (19)). Substituting the condition (21b) (or (22b)) into (18), we have * 3 R 0 ) and the state B 0 with condition (21b) or (22b) avoids invasion of the new state B 1 (see situation (ii) above and case (d) in Fig. 3). On the other hand, if the condition (21a) or (22a) holds instead and if * 1 <1, coevolutionary invasion of A 1 with r 1 >R 0 and B 1 with v 1 >v 0 (1+G) may occur, because we may then have * 3 >1 (see case (c) in Fig. 3).

This is the situation where the population initially fixed with relatively low rate of vertical transmission and the state B 0 with (21a) or (22a) should eventually be invaded by the new state B 1 associated with a state A 1, leading to an introduction of a relatively high vertical transmission rate r 1 ( >R 0 >r 0; see (19)). These evolutionary scenarios are just like those obtained under the assumption of uniparental transmission (see situation (i) above). Since the conditions depend on the transmission bias parameter G (and on H, also) in more complex ways, it is not as straightforward to interpret the above results as in the uniparental transmission model. Obviously, the complications are due to the assumption of biased biparental transmission. Indeed, vertical transmission alone may lead to stable fixation of a maladaptive state if the transmission bias is so strong as to overcome the counteracting effect of natural selection. To clarify the analytical results of the biparental transmission model, a multistate version of the dichotomous model (D1) in Appendix D (see also Table II) was studied numerically. To parallel the results with genetic determination of the rate of vertical transmission (see Discussion below), vertical transmission of the trait A was assumed unbiased (H i =0 for all i). With other things being equal to the uniparental case (see above), coevolutionary changes in the average rate of vertical transmission and the average viability were recorded over 1,000 generations for 100 runs. The average values in generations 0, 100, 200, 500, and 1,000 are listed with the standard deviations in Table III. From the table, we see that the vertical transmission rate increases steadily over time, and that relatively high viability is maintained during the course of evolution. In summary, as inferred from the analytical results, numerical studies suggest that the intrademic processes should lead to the evolution of higher vertical transmission rates in both uni- and biparental systems (see also Fig. 4). However, since the

218

Kiyosi Takahasi

neutral stability is attained when the viability trait is monomorphic, the possibility is not excluded where relatively weak effects may govern the evolution of vertical transmission rate.

INTERDEMIC COMPETITION The above analysis indicates that due to the eigenvalue of unity (see (13)), the evolution within a population may be dominated by nondeterministic effects such as random drift. In addition, interdemic competition may play a significant role. In the ordinary genetic evolution, natural selection that operates through differential proliferation andor extinction between demes seldom dominates the countervailing individual selection (e.g., Aoki, 1982). On the other hand, it is argued that in cultural processes, group selection should often be rendered effective through the action of cultural selection that cancels out the effects of individual selection acting on maladaptive cultural variants (Boyd and Richerson, 1985, Chaps. 7 and 8, 1990). To examine the possible role of interdemic competition in the evolution of vertical transmission rate, I here investigate the evolutionary rate of the character state B 1 under vertical transmission rate r 0. Assuming that the trait A is fixed on the state A 0 and substituting P 0j =q j and P 1j =0 in the master recursion (4a) for the uniparental transmission model or (D1) for the model with biparental transmission, I calculate the derivative d 2q 1dr 0 =d(q$1 &q 1 )  dr 0 and investigate the dependence of 2q 1 , the frequency change of the state B 1 in a generation, on the vertical transmission rate r 0 ; I obtain

sgn

d

\dr 2q + =sgn(&G) 1

(23)

0

for both the uniparental and biparental transmission models. In deriving the equation, it is further assumed that the traits are dichotomous and individual viability is determined by its own character state. From Eq. (23), we see that the derivative is negative if G>0, indicating that if the state B 1 is preferred in biased transmission, 2q 1 becomes a monotone decreasing function of r 0. Further if the state B 1 is more viable relative to B 0 and hence 2q 1 >0 (see, e.g., Eq. (6b) above), the population with lower vertical transmission rate can propagate the adaptive state B 1 more rapidly throughout the population. The result is also confirmed by a model with continuous time approximation, as given and analyzed in

Appendix E. If, then, higher frequency of the state B 1 in a population increases the relative growth rate of the population, natural selection that operates upon the differential growth rate between populations should favor those populations that have a greater reliance on nonvertical transmission (see Boyd and Richerson, 1990; see also Boyd and Richerson, 1985, Chaps. 7 and 8). Note that the decrease in vertical transmission due to the interdemic competition is possible only when the cultural innovation, the cultural analog of genetic mutation, produces more adaptive variants that may raise the relative growth rate of the population; otherwise, the population with a greater reliance on nonvertical transmission may propagate the maladaptive innovations and natural selection should favor the populations with a higher vertical transmission rate. Thus it may be concluded that the direction and the intensity of transmission bias should have profound effects also on interdemic processes in cultural inheritance.

DISCUSSION General Remarks This paper provides several mathematical models that follow the joint evolution of two cultural characters, one determining the viability and the fertility of individuals, and the other determining the vertical transmission rate of the first trait. Ordinary local stability analyses suggest that intrademic processes should promote the evolution of a greater reliance on vertical cultural transmission. Similar results are obtained under various assumptions, e.g., uni- or biparental transmission, oblique or horizontal transmission, viability or fertility selection, and so on. The results are further exemplified by numerical studies (see Fig. 4). On the other hand, it is inferred that when newly arisen variants are adaptive and favored by biased cultural transmission, interdemic processes may lead to decrease in vertical transmission. This is because biased nonvertical transmission may effectively propagate the adaptive variants, further increasing the average growth rate of the population. It is widely accepted that in ordinary genetic evolution, group selection should only provide less significant evolutionary effects that rarely dominate the countervailing individual selection (see, e.g., Aoki, 1982). On the other hand, in cultural processes, cultural selection may often cancel out the effects of individual selection acting on maladaptive cultural variants, further rendering cultural group selection more effective (Boyd and Richerson, 1985, Chaps. 7 and 8; see also Boyd and Richerson, 1990).

Modes of Cultural Transmission

The present study may provide support for a long-standing view that human social attributes might have been formed through the action of cultural group selection that have had significant effects during the course of human evolution (see Richerson and Boyd, 1989; Dugatkin, 1992; Findlay, 1992b; Wilson and Sober, 1994; Soltis et al., 1995). This also implies that the degree and intensity of transmission bias may be an important determinant of cultural changes (Takahasi, 1998; see also below). While the above argument emphasizes the importance of interdemic processes, this is not to deny the possible role of other factors, for example, of environmental fluctuation, in promoting the evolution of nonvertical cultural transmission. It seems possible that in a temporally varying environment, individuals are given a greater chance of acquiring adaptive cultural variants with a greater reliance on nonvertical transmission, thereby increasing the individual fitness. To see if environmental fluctuation should indeed lead to a reduction in the relative importance of vertical transmission, we need to formulate the effects of temporally varying environment, as done by Feldman et al. (1996) in their analysis of the evolution of social learning in a fluctuating environment. Vertical versus Nonvertical Transmission : An Appropriate Contrast ? Models presented above assume that while the vertical transmission rate of the trait B could vary, the trait A is exclusively transmitted vertically. In contrast, when the transfer of the trait A occurs exclusively via oblique transmission, coevolutionary invasion of states A 1 and B 1 into a population initially fixed on A 0 B 0 , as depicted above, does not occur. Mathematically, this is because =$11 =O(= 2 ) and corresponding eigenvalue * 3 =m 33 of the matrix M becomes zero (see (10) and (13) or (18)). This seems to imply that the evolutionary origins of vertical cultural transmission might have been possible only under genetic (and hence vertical) transmission of the trait A. Otherwise we have * 3 =0 and increase in the rate of vertical transmission should not be observed. It should also be noted that the incorporation of the type of oblique (and horizontal) transmission presented by Eshel and Cavalli-Sforza (1982) may produce qualitatively different results. It has been repeatedly argued that the initial evolution of sociality (altruism, cooperativeness, etc.) is more likely if the social interaction occurs between individuals who meet assortatively (e.g., Axelrod and Hamilton, 1981; Eshel and CavalliSforza, 1982). An example of such assortment occurs

219 between relatives. In the present context, this type of nonrandom interaction should result in some nonzero (positive) value for the eigenvalue * 3 , and hence should promote the coevolutionary invasion of cultural variants (see also Cavalli-Sforza and Feldman, 1983; Aoki and Feldman, 1994; Takahasi, 1997). Likewise, nonvertical transmission of trait B may occur between individuals with assortative encounters, further rendering its evolution more likely. Considering that the transfer of adaptive information to unrelated individuals is an altruistic act, and since it is usual in cultural transmission studies to neglect such nonrandom effects in nonvertical transmission (see Cavalli-Sforza and Feldman, 1983; Aoki and Feldman, 1994, for exceptions), a more appropriate comparison may be made between transmission with and without assortment, rather than between vertical and nonvertical transmission as discussed in the present study. Such a view point may also give advantages in comparative studies of social learning and transmission in animal species. While vertical transmission has probably been the major transmission mode in most situations of cultural inheritance throughout the human evolutionary history (Cavalli-Sforza and Feldman, 1981; Boyd and Richerson, 1985; Hewlett and Cavalli-Sforza, 1986; Mace and Pagel, 1994; Guglielmino et al., 1995), social information transfer in nonhuman animals mainly occur horizontally (but also see Tanaka, 1995). It has been further inferred that the ancestral state of cultural transfer was horizontal (Laland, 1993; Laland et al., 1993, 1996). While the capacity for active information transfer (King, 1991) may be limited to the human species (Tomasello et al., 1993), the few reported instances, if any, of teaching behaviors in nonhuman animals seem to occur between relatives (Caro and Hauser, 1992). These observations may lead us to speculate that active information transfer has indeed played a decisive role in human evolution, having promoted the formation of social groups composed of genetically related individuals in the human lineage (Cavalli-Sforza and Feldman, 1983; Takahasi and Aoki, 1995). This might have further induced an evolutionary transition in the relative importance of the respective modes of cultural transmission. While the present study assumes that the rate of vertical transmission, as well as the viability trait, is determined culturally, such evolutionary changes may be investigated by postulating genetic inheritance of the trait instead. Since the models discussed above have been restricted to the case with exclusive vertical cultural transmission of the trait A, the analysis includes the ordinary genetic inheritance of the trait as a special case (e.g., H=0; see Table II), and may give some insights

220 also into the evolution in dual inheritance systems (Boyd and Richerson, 1985; Durham, 1991; see also Feldman and Cavalli-Sforza, 1976, 1984). Evolution of Transmission Bias in Cultural Inheritance Preceding arguments show that biased transmission plays a crucial role in determining the evolutionary dynamics of cultural changes. Earlier in this paper, I mentioned, as the two critical non-Mendelian features inherent in cultural inheritance, nonvertical mode of information transfer and biased transmission. While evolutionary changes in the vertical transmission rate are studied extensively in the present article, the intensity of transmission bias is assumed to be constant and its evolution is largely ignored. Transmission bias in cultural inheritance may be interpreted as a propensity or predisposition that prefers to accept a certain set of cultural elements over others (see Hinde, 1987; Richerson and Boyd, 1989; Takahasi, 1998). While it has been repeatedly argued that the bias may be produced by psychological mechanisms that are (or have been) shaped by natural selection to maximize individual (or inclusive) fitness (see Flinn, 1997), evolution of transmission bias has been one of the most problematic issues among the researchers on human behavioral evolution, for some propensities do not seem to make any adaptive sense (see Hinde, 1987; Barkow, 1989; Richerson and Boyd, 1989; Logan and Qirko, 1996). Indeed, the results of the present study do not exclude the possibility that a maladaptive state can be established or even fixed within a population if biased transmission dominates the effects of counteracting natural selection. By the same token, Durham (1991) has incorporated cultural selection into his theory of dual inheritance systems (see also Boyd and Richerson, 1985) and investigated how culture interacts with genetics to drive the phenotypic evolution. However, these studies do not tell us why and how some biases have evolved to prefer maladaptive variations (see Hinde, 1982, Chap. 15; Takahasi, 1998). By investigating a model of indirect bias, Boyd and Richerson (1985, Chap. 8) have indicated that there are situations where a runaway of transmission bias occurs (see also Richerson and Boyd, 1989; Takahasi, 1998), just like a runaway of a female preference in the models of sexual selection (see Lande, 1981; Kirkpatrick, 1982). Their results may provide a possible explanation for the evolved predispositions that favor maladaptive variations, but as Takahasi (1998) argues, runaway dynamics should be observed only when the predispositions are

Kiyosi Takahasi

also transmitted culturally. The problem still requires further investigation, especially on proximate mechanisms that give rise to biased cultural transmission. As the present study clearly illustrates, transmission bias in cultural inheritance determines the evolutionary fate of cultural processes in crucial ways, and a thorough understanding of the cultural aspects of the human species may never be possible until we understand the nature of biased cultural transmission.

APPENDIX A When the Viability of an Individual is Determined by Its Parental Phenotype While the models described in the text assume that the viability of an individual is determined by its own character state, there may be situations where the parental phenotypes govern the offspring viability. That is, offspring produced by A i B j individuals have a viability v j . Then A i B j parents contribute r i n ij f j v j =r i n ijw j individuals to the next generation via vertical transmission, because among n ij f j offspring produced by A i B j individuals, a proportion r i acquires the state B j via vertical transmission and further these individuals have a viability v j . Then after the vertical transmission of trait B, (1&r i ) n ij f j offspring whose parents are A i B j are left undetermined with respect to the trait B. While a proportion g k of these offspring acquire the state B k via obliquehorizontal transmission, their viability is v j because their parental phenotype is v j . Hence the number of A i B j individuals in the next generation contributed by the nonvertical transmission becomes  k[(1&r i ) n ik f k g j v k ]=(1&r i )( k n ik w k ) } g j , where w k =f k v k . Finally summing up the contributions through vertical and obliquehorizontal transmission, we obtain the recursion equation in the number of individuals as n$ij =r i n ij w j +(1&r i )( k n ik w k ) } g j .

(A1a)

Dividing both sides of (A1a) by the population size n yields the master equation for the present model as W } P$ij =r i P ij w j +(1&r i )( k P ik w k ) } g j . (A1b) Following the same analytical procedure, we see that after redefining some parameters and variables, same mathematical results derived under the previous assumption should apply also in the present case. One of the

221

Modes of Cultural Transmission

alterations is made on the threshold rate of vertical transmission R 0 (and R 1 also); it is here redefined as

\

R 0 = 1&

1+G 1&G

+<\

v 1 1+G & v 0 1&G

+

(A2)

instead of the definition (8) in the text. Equilibrium frequencies q^ 0 and q^ 1 for the polymorphic equilibrium E p are also altered; they are &r 0(1&G)(v 0 &v 1 )+(1&r 0 ) } 2Gv 0 q^ 1 = 2G(v 0 &v 1 )

(A3)

instead of (11) in the previous model, and q^ 0 =1&q^ 1 . With these alterations, the whole argument based on the previous model, where the individual viability is determined by its own phenotype, is applicable to the present situation.

r}

Fertility Selection and Demographic Transition In this section I present and analyze a model of fertility selection by assuming that v j =v for all j. As an example of maladaptive cultural practices in human societies, Cavalli-Sforza and Feldman (1981, Sect. 3.9) discuss the diffusion of the small family ideal associated with recent demographic transitions in Europe (see also Richerson and Boyd, 1984). By briefly reviewing available demographic data, they conclude that the decrease in birth rate in 19th century Europe would have required oblique or horizontal transmission of the small family ideal. An equivalent conclusion is drawn from the present model. Assume that the practice of birth control is transmitted vertically from mothers to their daughters, and that women of previous generations other than mothers are responsible for oblique transmission of the practice. Further assuming that the viability of offspring is not affected by whether their mothers accept the small family ideal or not (i.e., v j =v for all j), I modify the uniparental transmission model (6a) to obtain the recursion for the spread of the small family ideal as q1 f1 +(1&r) g 1 . q 0 f 0 +q 1 f 1

(B1)

Here, q 1 is the frequency of women who have accepted the small family ideal, r is the vertical transmission rate

f1 1+G +(1&r) } >1, f0 1&G

(B2)

where G is the degree of transmission bias to favor the practice. Following Cavalli-Sforza and Feldman (1981), I posit f 1  f 0 =0.4. As expected, the condition (B2) obviously requires G>0, and the condition can be rewritten as r<2G(1.4 G+0.6). Hence as argued by Cavalli-Sforza and Feldman (1981), the invasion of the small family ideal requires effective biased nonvertical transmission. The corresponding condition with the horizontal transmission parameter (5e) becomes

r}

APPENDIX B

q$1 =r }

of the practice, f 1 and f 0 are the fertility of mothers who have or have not accepted the practice, respectively, and g 1 is the oblique transmission parameter (5d). From Eq. (B1), the condition for invasion of the small family ideal is obtained as

f1 1+G f 1 +(1&r) } } >1, f0 1&G f 0

(B3)

which is more stringent than the condition (B2) for oblique transmission, given that f 1 >f 0 . The condition can be rewritten as r<(3.5G&1.5)(2G).

APPENDIX C Traits with Multiple States The analysis for dichotomous traits can be extended in part to multistate traits under certain assumptions. We here present partial results of the local stability analysis of fixation of A 0 B 0 , assuming that the individual viability is determined by its own phenotype and that the natural selection operates through differential viability only. Further assuming that vertical transmission is uniparental, the assumption of obliquehorizontal mode of nonvertical transmission does not produce any difference in the results presented below. Similar conclusions may be derived also in other situations. The stability of the equilibrium can be investigated by setting P 00 =1+= 00 and P ij == ij for other sets of i and j, where = ij denote the small deviations from the equi<1, in the master equation (4a). librium and |= ij | =O(=)< We further set i=0, 1, 2, ..., N A and j=0, 1, 2, ..., N B; N A +1 and N B +1 are the numbers of possible states for traits A and B, respectively.

222

Kiyosi Takahasi

For i=0, the recursions for the deviations = ij are NA

=$0j == 0j } * 0j + : = kj } m j, k (NB +1)+j k=1

+O(= 2 ), j=1, 2, ..., N B ,

(C1a)

are unity. Given that * 0j =m jj <1 for all j, the equilibrium is unstable if * ij =r i } v jv 0 >1 for (at least) a set of i and j with i, j{0. Since 1>* 0j r 0 } v jv 0 (see (C1b)), this requires both r i >r 0 and v j >v 0. The multistate version of the uniparental transmission model (4a) is also investigated numerically in text (see also Fig. 4).

where * 0j =m jj =r 0 }

vj 1+G j v j +(1&r 0 ) } } . v0 1+G 0 v 0

APPENDIX D (C1b)

The transmission bias parameters G j for multiple states have the property &1
=$i0 == i0 } * i0 + : = ik } m i(NB +1), i(NB +1)+k +O(= 2 ), (C2a) k=1

Recursions for the Biparental Transmission Model Assuming that the traits A and B are transmitted independently and that the natural selection operates through differential viability only (see Table II), the recursion equations for dichotomous traits read W } P$00 =[r 0[P 200 +P 00 P 01 (1&G) +P 00 P 10 (1&H) +P 00 P 11 (1&G)(1&H)2

where

+P 01 P 10 (1&G)(1&H)2] * i0 =m i(NB +1), i(NB +1) =1.

(C2b)

Equations (C2) show that the eigenvalue that governs the invasion of state A i is unity. For i, j{0, the recursions become =$ij == ij } * ij +O(= 2 ),

(C3a)

+(1&r 0 )[P 200 +2P 00 P 01 +P 00 P 10 (1&H) +P 00 P 11 (1&H) +P 201 +P 01 P 10 (1&H) +P 01 P 11 (1&H)] g 0 ] fv 0 ,

(D1a)

W } P$01 =[r 0[P 00 P 01 (1+G)+P 00 P 11 (1+G)(1&H)2 +P 201 +P 01 P 10 (1+G)(1&H)2

where

+P 01 P 11 (1&H)] vj * ij =m i(NB +1)+j, i(NB +1)+j =r i } . v0

(C3b)

The dichotomous model can be recovered by setting N A =N B =1. For instance, compare the eigenvalues given by (C1b) and (C3b) with corresponding eigenvalues * 1 and * 3 in the previous model (see (15) and (16) in text); indeed, Eqs. (C1), (C2), and (C3) show that the eigenvalues of the coefficient matrix are given by its diagonal elements as in the previous model. We see from (C1) that for the equilibrium to be stable against the perturbations of the trait B, the inequalities * 0j =m j j <1 must hold for all j. Further if * ij =r i } v jv 0 >1 for any set of i and j with i, j{0, the equilibrium becomes unstable. On the other hand, if * ij <1 for all sets of i and j with i, j{0, the system then has a set of eigenvalues * i0 that

+(1&r 0 )[P 200 +2P 00 P 01 +P 00 P 10 (1&H) +P 00 P 11 (1&H) +P 201 +P 01 P 10 (1&H) +P 01 P 11 (1&H)]g 1 ] fv 1 ,

(D1b)

W } P$10 =[r 1[P 00 P 10 (1+H)+P 00 P 11 (1&G)(1+H)2 +P 01 P 10 (1&G)(1+H)2+P 210 +P 10 P 11 (1&G)] +(1&r 1 )[P 00 P 10 (1+H)+P 00 P 11 (1+H) +P 01 P 10 (1+H) +P 01 P 11 (1+H)+P 210 +2P 10 P 11 +P 211 ] g 0 ] fv 0 ,

(D1c)

223

Modes of Cultural Transmission

Further setting G=G{, s f =s~ f {, and s v =s~ v {, and letting { Ä 0, I derive the time derivative of the state frequency as

and W } P$11 =[r 1[P 00 P 11 (1+G)(1+H)2 +P 01 P 10 (1+G)(1+H)2+P 01 P 11 (1+H)

dq 1 =q 1(1&q 1 ) } [r 0(s~ f +s~ v )+(1&r 0 )(2G +s~ v )]. dt

(E1)

+P 10 P 11 (1+G)+P 211 ] +(1&r 1 )[P 00 P 10 (1+H)+P 00 P 11 (1+H) +P 01 P 10 (1+H)+P 01 P 11 (1+H) +P 210 +2P 10 P 11 +P 211 ] g 1 ] fv 1 ,

From (E1), we can calculate the time T required for a certain frequency change, say, from q 1(0)=Q 0 to q 1(T)=Q T , is derived as

(D1d) T=

where parameters and variables are defined as in the text.

|

T

_

|

QT Q0

dq 1 q 1(1&q 1 )

_

=[r 0 (s~ f +s~ v )+(1&r 0 )(2G +s~ v )] &1 } ln

APPENDIX E Continuous Approximation for the Time Required for a Certain Frequency Change In ordinary population genetics, the evolutionary rate of a character is often measured by investigating the time required for a certain gene frequency change. This can be done by employing continuous time approximations. The approximation usually assumes that the natural selection is weak. In our models of biased cultural transmission, an additional assumption of weak transmission bias is also required. I commence with the uniparental transmission model (4a), assuming that the trait A is fixed on the state A 0 . In the analysis below, it is assumed that nonvertical transmission of the trait B occurs via oblique pathway and the viability of an individual is determined by its own phenotype. Let t=n{ be a new time scale, where n is the unit in generation. Assuming weak selection that acts on differential fertility and viability, we set f 1  f 0 =1+s f and v 1 v 0 =1+s v where s f , s v =O({); then w 1 w 0 =1+s f + s v +O({ 2 ). We further assume weak transmission bias; G=O({). Substituting (5d) and these relations into the recursion (4a) and noting that

q$1 =q 1(t+{)=q 1(t)+

dt=[r 0(s~ f +s~ v )+(1&r 0 )(2G +s~ v )] &1

0

dq 1 } {+O({ 2 ), dt

I obtain

q 1 QT . 1&q 1 Q0 (E2a)

&

With biparental transmission, this becomes s~ f G + +s~ v +(1&r 0 )(2G +s~ v ) 2

_\ + q _ ln _ 1&q & .

T= r 0

&1

&

QT

1

1

Q0

(E2b) We see from Eqs. (E2) that for any parameter values, the relative evolutionary rate of the system is governed by r 0(s~ f +s~ v )+(1&r 0 )(2G +s~ v ) in the uniparental transmission model or r 0 (G +s~ f 2+s~ v )+(1&r 0 )(2G +s~ v ) in the biparental transmission model. A little algebra show that in either uni- or biparental transmission model, evolution proceeds faster with a greater reliance on nonvertical (or, in the present case, oblique) transmission if s~ f &2G <0. Recalling that here it is set G=G{, s f =s~ f {, and s v =s~ v{ with { Ä 0, and that no differential fertility is assumed in the text (hence s f =0), the condition is identical to the corresponding one in the text (see Eq. (23)). While both methods produce equivalent results, the superiority of the method in the text is that it does not require the additional assumption of weak natural selection nor transmission bias.

ACKNOWLEDGMENTS dq 1 } {=q 1(1&q 1 ) } [r 0 (s f +s v ) dt +(1&r 0 )(2G+s v )]+O({ 2 ).

This research was supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists, a Grantin-Aid for Encouragement of Young Scientists from the Ministry of Education, Science, Sports and Culture, Japan, and NIH Grant GM28016 to M. W. Feldman.

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