O R I G I NA L A RT I C L E doi:10.1111/evo.12403

ASSORTATIVE SOCIAL LEARNING AND ITS IMPLICATIONS FOR HUMAN (AND ANIMAL?) SOCIETIES Edith Katsnelson,1,2 Arnon Lotem,3 and Marcus W. Feldman1 1

Department of Biology, Stanford University, Stanford, California 94305 2

3

E-mail: [email protected]

Department of Zoology, Faculty of Life Sciences, Tel Aviv University, Tel-Aviv 69978, Israel

Received February 1, 2013 Accepted February 28, 2014 Choosing from whom to learn is an important element of social learning. It affects learner success and the profile of behaviors in the population. Because individuals often differ in their traits and capabilities, their benefits from different behaviors may also vary. Homophily, or assortment, the tendency of individuals to interact with other individuals with similar traits, is known to affect the spread of behaviors in humans. We introduce models to study the evolution of assortative social learning (ASL), where assorting on a trait acts as an individual-specific mechanism for filtering relevant models from which to learn when that trait varies. We show that when the trait is polymorphic, ASL may maintain a stable behavioral polymorphism within a population (independently of coexistence with individual learning in a population). We explore the evolution of ASL when assortment is based on a nonheritable or partially heritable trait, and when ASL competes with different non-ASL strategies: oblique (learning from the parental generation) and vertical (learning from the parent). We suggest that the tendency to assort may be advantageous in the context of social learning, and that ASL might be an important concept for the evolutionary theory of social learning. KEY WORDS:

Assortment, homophily, model choice strategies, phenotypic asymmetry, social learning strategies.

The tendency of people with similar traits to preferentially interact, or homophily, is a known principle in social science (Lazarsfeld and Merton 1954; McPherson et al. 2001; Rogers 2003). Homophily affects the spread of behaviors between people who share traits (Centola 2011). It is recognized as an important factor in social networks and cultural transmission, and simulation studies suggest that it may contribute to cultural diversity (Axelrod 1997; Centola et al. 2007; Flache and Macy 2011). However, a synthesis of homophily with the evolutionary research on social learning is lacking. This is surprising because social learning is a central element in shaping cultures, and is recognized as the means by which behaviors are transmitted in humans and other animals (Cavalli-Sforza and Feldman 1981; Boyd and Richerson 1985; Heyes and Galef 1996; Laland and Janik 2006; Whiten and Mesoudi 2008; Aoki and Feldman 2014). Because a learner’s behavior is often directly influenced by that of its model, the process of model choice is critical for the  C

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social learner’s success (Boyd and Richerson 1985; Laland 2004; Rendell et al. 2011). Moreover, this process determines dynamics which lead to the profile of traditions within and among societies (Boyd and Richerson 1985; Henrich and Boyd 1998; Henrich 2004; Ihara and Feldman 2004; Kobayashi and Aoki 2012). Indeed, the growing literature on social learning explores the success and prevalence of various strategies of model choice both theoretically (e.g., Henrich 2004; Eriksson et al. 2007; Wakano and Aoki 2007; Ihara 2008; Kendal et al. 2009a; Fogarty et al. 2011; Kobayashi and Aoki 2012) and empirically (e.g., Dugatkin and Godin 1993; Whiten et al. 2005; Duffy et al. 2009; Horner et al. 2010; Chudek et al. 2012). These strategies are often based on variants of “success bias” (a bias for learning from successful individuals) or “conformity bias” (a bias for adopting the behavior of the majority); if behaviors that are successful for some are likely to be successful for others or to be common, such social learning strategies are expected to be advantageous (although the

C 2014 The Society for the Study of Evolution. 2014 The Author(s). Evolution  Evolution 68-7: 1894–1906

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success of conformism is not guaranteed, e.g., Eriksson et al. 2007; Wakano and Aoki 2007; Kandler and Laland 2009). Under success or conformity bias, some individuals have a greater influence than others over all potential social learners, which results in reduced diversity within populations (Boyd and Richerson 1985; Henrich and Boyd 1998; Henrich and Gil-White 2001; Ihara and Feldman 2004; Kandler and Laland 2009; Kendal et al. 2009b, c; Whitehead and Richerson 2009; Morgan and Laland 2012). There are cases however, in which “the most successful behavior” cannot be clearly identified or defined absolutely; for example, people can be similarly successful in different professions or differ in their success within a profession; different hobbies may provide similar or different psychological benefits to different individuals; females and males might behave differently although their ultimate success could be similar. Accordingly, in many cases behavioral polymorphism exists in the population while there is no “correct” or “wrong” behavior for all individuals. Model choice based on success bias in this case might not be appropriate; it might even be counterproductive if the target behavior requires attributes possessed by only some individuals in the population. For example, the success of athletes depends on special physical and mental traits; seeing athletes as role models and being inspired to pursue an athletic career will result in failure for most people, but may have an important motivating role for potential athletes who share these traits. Hence, a behavior may provide similar benefits for those that share traits, and different benefits for those with different traits (Rogers 2003; Creanza et al. 2012). Because social learning is subject to the cost of learning inappropriate or irrelevant behavior (Rogers 1988; Feldman et al. 1996; Giraldeau et al. 2002), a mechanism for filtering out unsuitable models is expected to be advantageous (Enquist and Ghirlanda 2007; Rendell et al. 2010, 2011). We present a framework for exploring the evolution of assortative meeting, which is quantitatively equivalent to homophily (Eshel and Cavali-Sforza 1982), in the context of social learning: we suggest that assortative meeting may increase the chance that an individual adopts the most appropriate behavior given its traits, thus, promoting the evolution of assortment. Assortative social learning (hereafter, ASL) is a model choice strategy, where assorting on a trait that varies in the population acts as an individual-specific mechanism for filtering relevant models for social learning. An interesting analogy may be drawn between assortative meeting and assortative mating (Jiang et al. 2013): mating assortatively with respect to a trait may evolve when hybrids or intermediate morphs of this or associated traits are disadvantaged (Huber et al. 2007). Moreover, assortative mating is important for understanding cultural transmission and can lead to polymorphism (Creanza et al. 2012).

Using simple models we demonstrate that in a phenotypically polymorphic population, ASL leads to the maintenance of stable behavioral polymorphism, and that it is expected to be a stable strategy for a wide range of parameters. In contrast to the prediction from several other social learning mechanisms, ASL itself accounts for stable behavioral polymorphism, which is maintained independently of the concurrent existence of individual learning. We suggest that the tendency to assort, for example, in human societies, may be advantageous in the context of social learning. ASL might be prevalent in humans (Centola 2011) and possibly in other social animals, especially in those that have complex social structures, such as primates. The Model

We investigate the evolution of ASL in an asexual population in which each member is specified by alternative forms T and t of a dichotomous phenotype T. T could represent the combined expression of potentially many traits, which might be determined by different factors (genetic or nongenetic), and through the individual’s development and maturation. We explore two models that differ in whether the form of T has a substantial heritable component (other differences are discussed below). By heritability we mean the level of accuracy with which the form of T is transmitted from parent to offspring via any vertical transmission mechanism (e.g., genetic, epigenetic, cultural). In model 1, the nonheritable trait model, we suppose that the heritability of T is low enough to be considered zero. This simple framework is useful because it applies to cases in which genetic or vertical inheritance is not expected to generate an adaptive association between T and some focal behavior. Examples may be any polymorphic trait with negligible heritability such as sex, traits influenced by maternal effects, or those affected by environment and/or experience. We investigate the evolution of assortative meeting when the trait of an offspring cannot be predicted from that of the parent. As T is not heritable, we suppose that the probability of an individual having form T is determined by a parameter r, 0 < r < 1, that is fixed, for example, by the environment. The probability that an individual is t is 1 − r. The environment is assumed to be stable with respect to this measure so that r does not vary over time. For example, in the case of sex, the chance of being a female is close to that of being a male so that r ∼ = 0.5. Hence, T and t may have the same = 1−r ∼ (when r = 0.5) or different (when r = 0.5) frequencies in the population. In model 2, the heritable trait model, we suppose that the probability of an individual having one of the two alternative forms of T is the sum of the probability of having a parent with this phenotype weighted by the heritability level (h), and the external probability (r and 1 − r for T and t, respectively) weighted by 1 − h. Although heritability can be high, we assume it cannot be

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perfect (h < 1), thus, T/t cannot go to fixation and the population is phenotypically polymorphic. In addition to phenotype T, each individual has one of the alternative forms B and b of a dichotomous behavior B. The combination of phenotype T/t and behavior B/b determines fitness. Behaving appropriately with respect to the phenotype will result in the highest fitness for that phenotype. For individuals with trait T, behavior B is advantageous relative to b (and is thus appropriate), whereas for individuals with the trait t, behavior b is advantageous relative to B. Although the fitness of Tb is smaller than that of TB, and the fitness of tB is smaller than that of tb, the fitness relations between the traits may vary. We define fitness of TB as 1, whereas the fitness of Tb, tB, and tb are 1 + s1 , 1 + s2 , and 1 + s3 , respectively (see Table 1a). For the fitness relations described above, we assume: −1 < s1 < 0; −1 < s2 < s3 < 1. Accordingly, if s3 = 0, there is no superior combination of phenotype and behavior because all individuals can potentially achieve maximal fitness if they choose the most appropriate behavior for themselves (TB and tb are equally fit). However, if s3 < 0, individuals with trait t can never achieve a fitness as high as that of individuals with trait T that use the appropriate behavior, and vice versa when s3 > 0. Individuals survive to adulthood and reproduce according to these fitness. Those that survive reproduce once and serve as potential models for social learning of the behavior by the offspring generation as described below. Behaviors B/b are transmitted in the population by social learning from the parental generation (i.e., the survivors of selection in the previous generation). There are two social learning strategies that an individual may use to choose a model from which to learn. We refer to the social learning strategy as M, with possible alternatives M and m, where M individuals use a nonassortative social learning strategy (NASL) whereas m individuals use an ASL strategy. We explore the evolution of ASL by assuming that the form of M is vertically transmitted but the mechanism of transmission is not specified (e.g., it could be genetically transmitted or learned from the parent). In model 1 (nonheritable trait model), NASL individuals copy the behavior of a random individual from the parental generation (Henrich and Boyd 1998; Eriksson et al. 2007; Kendal et al. 2009a). Hence, they use oblique social learning (Cavalli-Sforza and Feldman 1981) and we refer to them as NASLO . Behaviors B or b are adopted by NASLO individuals according to the respective frequencies of these behaviors in the parental generation. In model 2 (heritable trait model), NASL individuals use a vertical social learning strategy, that is, they copy the behavior of their parent, and we refer to them as NASLV . Such a strategy is expected to be beneficial due to the heritable component of the trait T. ASL strategy: we assume that ASL individuals are able to recognize individuals in their parents’ generation that have the same trait as themselves (T or t) and copy the behavior (B or b)

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of the former (i.e., they are capable of phenotypic matching, see discussion). We include a self-perception error (e < 0.5), which is the probability that an individual perceives itself differently from what it actually is, and accordingly copies the behavior of another individual with a different trait from its own (if e ≥ 0.5 assortment does not exist because there is the same or higher chance to learn from a model whose trait differs from its own). For example, the probability that an assortative social learner (m) with trait T adopts behavior B (resulting in phenogenotype mTB) is

p (TB) p (tB) (1 − e) + e, p (TB) + p (Tb) p (tB) + p (tb) where p (·) are frequencies in the parental generation. Here, p(T B) is the frequency of behavior B among parents with p(T B)+ p(T b) phenotype T (i.e., the probability of learning behavior B as a p(t B) result of correct assortment), and p(t B)+ is the frequency of p(tb) behavior B among parents with phenotype t (i.e., the probability of learning behavior B as a result of erroneous assortment). In practice, self-perception error results in an assortment error: it is the probability that the individual perceives itself as having a different trait from the one it actually has, and accordingly assorts with individuals whose trait differs from its own. Selfperception error is potentially an intrinsic cost of ASL. It represents the cognitive problem of having a correct representation of oneself. We envisage that such an error is likely to capture the main evolutionary cost of the ASL strategy (see discussion). Because ASL individuals assort based on trait T, correct assortment may result in learning inappropriate behavior (e.g., with probability p(T b) correctly assorting ASL individuals with trait T learn p(T B)+ p(T b) behavior b). A parent transmits its social learning strategy (M or m) to its offspring (asexual, haploid population), which determines the strategy by which the model for social learning of behavior B/b is chosen (NASLO or ASL in model 1, and NASLV or ASL in model 2). The phenotype (T/t) is determined by r in model 1 or by r, h and the parent’s phenotype in model 2. Table 1 summarizes the recursion relations for both models that follow from these assumptions in a very large population (no sampling drift). Methodological note Analytic solutions of some special cases that demonstrate the core elements of our model have been obtained, while deterministic numerical simulations are used for exploration beyond these cases. Simulation details are given in Supporting Information 1. Analytical solutions are given in Supporting Information 2, and are specifically referred to throughout the text. Population measurements at equilibrium are based on offspring frequencies f (·).

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Table 1. f(i) and p(i) represent offspring and parent frequencies, respectively. M/m specify the social learning strategy (NASL/ASL); T/t the form of phenotype T; B/b the type of behavior B.

(a) Phenogenotype

Offspring frequency

Fitness

Parent frequencies

MTB

f (MTB)

1

mTB

f (mTB)

1

MTb

f (MTb)

1 + s1

mTb

f (mTb)

1 + s1

MtB

f (MtB)

1 + s2

mtB

f (mtB)

1 + s2

Mtb

f (Mtb)

1 + s3

Mtb

f (mtb)

1 + s3

f (MTB) NF p (mTB) = f (mTB) NF (1 + s1 ) p (MTb) = f (MTb) NF f (mTb) p (mTb) = NF (1 + s1 ) (1 + s2 ) p (MtB) = f (MtB) NF (1 p (mtB) = f (mtB) + s2 ) NF f (Mtb) p (Mtb) = NF (1 + s3 ) (1 + s3 ) p (mtb) = f (mtb) NF

(b) ∗∗ Offspring frequency in the next generation ( f (i) ) Model 1 f (MTB) = p (M) r p (B)  (1 − e) + f (mTB) = p (m) r p(TB) p(T )

p(tB) e p(t)

f (MTb) = p (M) r p (b)  (1 − e) + f (mTb) = p (m) r p(Tb) p(T )

p(tb) e p(t)

f (MtB) = p (M) (1 − r ) p (B)  (1 − e) + f (mtB) = p (m) (1 − r ) p(tB) p(t) f (Mtb) = p (M) (1 − r ) p (b)  (1 − e) + f (mtb) = p (m) (1 − r ) p(tb) p(t) ∗

NF(normalizing factor, which      f Mtb + f mtb (1 + s3 ).

∗∗

is

the



p(Tb) e p(T )

p (MTB) =

Model 2   p(MTB) = (1 − h) r p (MB) + hp (MTB) p (M) (1 − h) r p(MB) + h p(MT) p(M) p(M) p(MT)    p(mT) p(TB)(1−e) p (m) r (1 − h) + h p(m) + p(tB)e p(T ) p(t) (1 − h) r p (Mb) + hp (MTb)   p(Tb)(1−e) p (m) r (1 − h) + h p(mT) + p(m) p(T )



p(tb)e p(t)



(1 − h) (1 − r ) p (MB) + hp (MtB)   p(t B)(1−e) p (m) (1 − r ) (1 − h) + h p(mt) + p(m) p(t)



(1 − h) (1 − r ) p (Mb) + hp (Mtb)   p(tb)(1−e) p (m) (1 − r ) (1 − h) + h p(mt) + p(m) p(t)

p(TB) e p(T )

population

fitness)

∗

=



p(TB)e p(T )

p(Tb)e p(T )





     f (MTB) + f (mTB) + f MTb + f mTb (1 + s1 ) + ( f (MtB) + f (mtB)) (1 + s2 ) +

r and 1 − r are the probabilities that an individual has the phenotype T or t, respectively. e is the chance that an assorting social learner makes an

assortment error. When one or more of the phenogenotypes are not mentioned, the frequency is the sum of the two possible forms (e.g., p(M) =           p(MTB) + p MTb + p(MtB) + p Mtb ; p Tb = p MTb + p mTb , and so on).

Results BEHAVIOR FREQUENCIES FOR EACH SOCIAL LEARNING STRATEGY

For each social learning strategy (NASL or ASL) in each model (model 1—nonheritable trait or model 2—heritable trait), we first examined the behaviors (B and b) at equilibrium where only one social learning strategy exists in which case, when h = 0 model 2 reduces to model 1. Thus, for h = 0, the analysis in this section applies to both models (the notation NASL here refers to both NASLO and NASLV ).

Nonassortative social learning (NASL) Here, we investigate the frequencies of behaviors B and b when the population contains only nonassortative social learners (NASL),

that is, when M is fixed. We use f (·) to denote offspring frequencies; p(·) are those in the parental generation. 1. NASL with h = 0. Following the equations in Table 1, the frequencies when p (M) = 1 satisfy 1 + (1 − r ) s2 p (B) p  (B) × = . p  (b) p (b) 1 + r s1 + (1 − r ) s3

(1)

2 −s3 , we must have fixation of Accordingly, for any r = s1 s+s 2 −s3 s2 −s3 2 −s3 ). Hence, for these either B (if r > s1 +s2 −s3 ), or b (if r < s1 s+s 2 −s3 cases, NASL will result in fixation of only one behavior, regardless of the starting proportions of the behaviors. This means that depending on r, s1 , s2 , s3 , the population may become fixed on the correct or wrong behavior for the majority trait (because trait T is

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From the equations in Table 1b, an explicit solution for the equilibrium of model 2 is hard to derive. We used simulations to explore the evolution of behavioral strategies for different parameter combinations of s1 , s2 , s3 , r , and h. When h = 0, the simulations results matched the analytical prediction for NASL from the previous section. For 0  h < 1, most tested cases (99.9%) resulted in fixation of one behavior. As with h = 0, the behavior that fixed did not depend on the starting conditions of the different behaviors but only on the parameters. Moreover, as with h = 0, special parameter combinations that did not result in fixation of one behavior, produced neutral polymorphic equilibria, at which the proportions of B and b depended on the starting frequencies of these behaviors (see Sections 1.2.1 and 2.1 in Supporting Information). Figure 1A shows an example of the B/b frequencies. Altogether, our results suggest that a population of NASL (NASLO or NASLV ) is expected to fix on one behavior. Assortative social learning Here, we investigate behaviors B and b in a population with only assortative social learners (ASL), that is, with m fixed. We can obtain analytic solutions for the equilibrium frequencies in two important special cases: 1. There is no self-perception error (e = 0). In this case, for both models (for any heritability level h < 1 in model 2), from Table 1, we have 

p (mTB) f (mTB) =  = p (mTb) f (mTb) =

f (mTB) f (mTB)+ f (mTb)(1+s1 ) f (mTb)(1+s1 ) f (mTB)+ f (mTb)(1+s1 )

f (mTB) , f (mTb) (1 + s1 )

f (mtb) (1 + s3 ) f (mtb) .  = f (mtB) (1 + s2 ) f (mtB)

(2)

Because −1 < s1 < 0 and s2 < s3 , the frequencies of mTB and mtb increase over time, whereas f (mTb) and f (mtB) approach zero. Hence, whether the phenotypic trait is heritable or not, both B and b are maintained at equilibrium (see Section 2.2.1

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−0.9

−0.7

−0.7

−0.5

−0.5 h=0

−0.3

−0.1

0.1

0.1

0.3

0.3

−0.1

−0.5

−0.9

−0.1

ASL −0.9

−0.9

−0.7

−0.7

−0.5

−0.5

−0.3 −0.1

e=0.05 −0.3 −0.1

0.1

0.1

0.3

0.3

−0.1

−0.5

−0.9

h=0.2

−0.3

−0.1

−0.5

−0.9

e=0.2

−0.1

−0.5

−0.9

s1 (Tb selection coefficient)

B

The highest e of ASL with behavioral polymorphism at equilibriums −0.9 −0.7 −0.5 −0.3 −0.1 0.1 0.3

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

−0.1 −0.5 −0.9 s1 (Tb selection coefficient) Figure 1.

In each plot, s3 = 0.5 (tb selection coefficient). Selection

coefficients s1 and s2 are on the x- and y-axis, respectively. (A) Upper row: population behavioral profile for a population of nonassortative social learners only (NASL) for different levels of heritability (h) when r = 0.3. Bottom row: population behavioral profile of assortative social learners only (ASL) for different levels of self-perception error (e). Black—fixation of behavior b; white— fixation of behavior B; gray—polymorphism with both behaviors. (B) The highest value of self-perception error, e, (for the different selection coefficients combinations) for which ASL resulted



and similarly

−0.9

s2 (Tb selection coefficient)

2. Model 2-heritable trait (NASLV ) with 0  h < 1.

NASL

A

s2 (Tb selection coefficient)

controlled by r and r ≤ 0.5, t is the trait of the majority). When the population is fixed on B, the majority uses the wrong behav2 −s3 ior. In the special case r = r ∗ = s1 s+s , there is a continuum of 2 −s3 polymorphic equilibria of all four phenogenotypes MTB, MTb, MtB, Mtb, with frequencies that are determined by the starting frequencies of the different behaviors. Note that these polymorphic equilibria are neutral, and that in a real (finite) population they are not expected to be stable to random drift. Thus, even though NASL is an unbiased strategy, it is expected to result in fixation of one behavior in the population.

in behavioral polymorphism at equilibrium. The dashed line is 1+s1 1+s2 1 = 1+s3 (with s3 = 0.5). Values of e are indicated by the graystyle colorbar. (Behavior fixation patterns in ASL did not change with r or with h, see main text. For figures 1a-ASL and 1b, we used r = 0.3 and h = 0.6.)

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in Supporting Information for explicit solution of the frequencies of f (mTB) and f (mtb) at equilibrium). It is convenient to use this simple example to demonstrate how ASL may stabilize behavioral polymorphism in the population: because individuals with different traits have different behaviors that are more appropriate for them, after selection, the fraction of individuals using the appropriate behavior among individuals with a certain trait (either T or t) increases. Although this is true for both ASL and NASL, ASL individuals use the trait (T or t) as a filter to choose from whom to learn. Following equation (2), when e = 0, all individuals with trait T use behavior B, whereas all individuals with trait t use behavior b at equilibrium. Because when e = 0, ASL individuals learn only from parents with the same trait as themselves, individuals with traits T learn behavior B whereas individuals with traits t learn behavior b. In other words, if e = 0, the frequency of adoption of behavior B among those with trait T is one, whereas the same frequency among those with trait t is zero and vice versa for behavior b. As we demonstrate numerically below, e > 0 leads to complex dynamics, and whether behavioral polymorphism is maintained at equilibrium depends on the selection coefficients of the different trait–behavior combinations (s1 , s2 , s3 ). 2. r = 0.5, h = 0, and s1 = s2 , s3 = 0. For both models (model 2 reduces to model 1 when h = 0 and f (m) = 1), for e < 0.5, the equilibrium can be shown to be of the form of f (mT B) = f (mtb); f (mT b) = f (mt B) (see Section 2.2.2 in Supporting Information). That is, ASL results in a stable behaviorally polymorphic equilibrium even for high values of self-perception error. This special case demonstrates that depending on the parameters, a stable behavioral polymorphism may be maintained at equilibrium even for high values of selfperception error and it may be relevant to the case of sex-biased social learning (see discussion). To explore the parameter space beyond these special cases, we used numerical simulations. Equilibrium frequencies when f (m) = 1 did not depend on the starting conditions (see Section 1.2.2 in Supporting Information). Although at equilibrium the population could be behaviorally monomorphic, for every combination of selection values (s1 , s2 , s3 )there is a sufficiently small self-perception error, e, such that B and b coexisted at equilibrium (see an example in Fig. 1B). The highest value of e, for which polymorphism existed at equilibrium was greater than zero in more than 95% of the parameter combinations, and in more than 44% it was the highest value tested, that is, 0.45 (Fig. 1C). This value of e did not depend on r or h (except for very rare exceptions that resulted from the computational conditions that define polymorphism, Supporting Information 1). Thus, behavioral polymorphism can be stable when T and t have different frequencies, that is, when r = 0.5 and/or

0 < h < 1. A qualitative examination of the results suggests that when ratios between the fitness effects of the two behaviors within 1 ∼ 1+s2 each trait are similar (i.e., when 1+s ), behavioral poly1 = 1+s3 morphism is stable for high values of e. As can be seen from Figure 1B, stable polymorphism with high values of e exists for a wide range of parameters around the dashed line (which represents this ratio). Thus, there is a wide range of parameters for which ASL results in stable behavioral polymorphism when the value of self-perception error is high. Behavioral polymorphism with a high value of e can be stable even in cases in which the fitness that accrues from using the appropriate behavior for one trait is lower (or higher) than that from using the appropriate behavior for the other trait (i.e., when s3 = 0). Although the existence of polymorphism does not depend on r or h, the equilibrium frequencies of the different phenogenotypes do depend on these values (although in some special cases, e.g., when e = 0 and s3 = 0, the phenogenotype frequencies do not change with h). Behavioral polymorphism could be stable when most individuals use the wrong behavior for their phenotype ( f (mT b) + f (mt B) > 0.5) and when one of these “wrong” types has the highest frequency in the population. Compared with a population of NASL, stable behavioral polymorphism in a population of ASL did not necessarily result in higher population fitness. In addition, values of e that led to fixation of one behavior in a population of ASL could lead to fixation of a different behavior in a population of NASL; in these cases, the population fitness of NASL offspring was higher than, or in rare cases, the same as the fitness in the ASL. Thus, although the self-perception error in our model is smaller than 0.5 (the proportion of correct assortment is always higher than that of incorrect assortment), at the population level, ASL does not necessarily lead to an “optimal” or better solution than NASL. When the equilibrium entailed behavioral polymorphism, the frequency of learning behavior B among offspring with trait T was larger than the frequency of learning this behavior among those B) B) B) B) (1 − e) + p(t (1 − e) + p(T e > p(t e, with trait t (i.e., p(T p(T ) p(t) p(t) p(T ) p(T B) p(t B) see Table 1b), which for e < 0.5, can be written as p(T ) > p(t) ; and the frequency of learning behavior b among those with trait t was larger than the frequency of learning this behavb) > p(T . Note that ior among those with trait T, that is, p(tb) p(t) p(T ) p(T B) p(T b) p(t B) p(tb) while p(T ) , p(T ) , p(t) , p(t) might take different values, we B) B) b) p(T b) − p(t = p(tb) − p(T (because p(T B)+ = must have p(T p(T ) p(t) p(t) p(T ) p(T ) p(t B)+ p(tb) = 1). That is, when ASL results in stable behavp(t) ioral polymorphism, there is a higher frequency of learning one behavior among offspring for which this behavior is more appropriate given their trait, and a higher frequency of learning of the other behavior among individuals with the other trait. To summarize, ASL results in maintenance of behavioral polymorphism in the population for a large range of parameters, and this does not occur in the NASL models.

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NASL

s2 (tB selection coefficient)

−0.9

−0.9

−0.7

−0.7

−0.5

−0.5

−0.3

−0.3

−0.1

−0.1

0.1

0.1

−0.1

−0.5

−0.9

−0.1

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The trait behavioral profile in model 2, demonstrated by the difference between the frequency of T from r (indicated by the

Figure 2.

colorbar) for different parameter combinations. Upper row: NASL; bottom row: ASL. In all plots, s3 = 0.2 (tb selection coefficient), r = 0.4, and selection coefficients s1 and s2 are on the x- and yaxis, respectively.

PHENOTYPIC FREQUENCIES FOR EACH SOCIAL LEARNING STRATEGY

In both models 1 and 2, an individual is born with a certain form of T, while the behavior B is learned according to the social learning strategy. Although in model 1 (nonheritable trait), the frequencies of T and t are fixed and determined by r alone, in model 2 (heritable trait) these are also affected by h. It is interesting to explore whether the different social learning strategies affect the proportions of T and t, even though only B is learned. Because when h = 0, r (the proportion of T due to the nonheritable component) is the only determinant of T, it is convenient to use it as a baseline which compares the phenotypic frequencies that arise in populations of NASLV and ASL when h > 0. For each parameter combination, we calculated the difference between the frequency of T at equilibrium and r. In both NASLV and ASL populations, the absolute difference from r increased or did not change with increasing h (excluding rare cases in NASLV of polymorphism which depended on the starting conditions). However, depending on parameter combinations, the direction and strength of the differences from r were different for the two social learning strategies. Figure 2 shows an example of the different patterns in NASLV and ASL.

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Although the two different social learning strategies affect only the behavior of the individuals (an individual is born with a certain trait and learns only the behavior), the population frequencies of T and t are affected by these strategies. This demonstrates a niche-constricting effect through which populations may differ in the frequencies of an innate trait as a result of a mechanism that affects the adoption of another trait (in our case, the behavior). In other words, the social learning strategy, which directly affects only the behavior of the individual, feeds back to affect the trait frequency in the population. STABILITY OF ASL TO INVASION OF NASL STRATEGIES

Here, we explore the stability of ASL to invasion by NASL. When self-perception error, e, is zero it can be shown that ASL is always stable against invasion by NASLO or NASLV in both models 1 and 2, as the leading eigenvalues of the local stability matrices arising from the recursions for both models are always less than 1 (see Section 2.3 in Supporting Information). Note that when the population is not fixed on ASL or NASL, the two models are different even when h = 0 in model 2. Because in both models a population of ASL only is always behaviorally polymorphic when e = 0, for every parameter combination, there is always a sufficiently small e for which ASL is stable to invasion of NASL, and both B and b are present in the population. For model 1, in the special case r = 0.5; s1 = s2 , s3 = 0, ASL is stable for any e < 0.5 (see Section 2.3.1 in Supporting Information). For this case as well we showed above that a population of ASL is always behaviorally polymorphic. Our simulations indicate that this is also the case in model 2 when h = 0. Thus, when heritability is low enough, parameter combinations exist for which ASL is stable against invasion of NASL even for high values of self-perception error, e. We used numerical simulations to further explore invasion dynamics beyond these special cases (see Section 1.3 in Supporting Information). As predicted by the local stability analysis, in both models, ASL was always stable to invasion by NASL when e = 0. In both models, ASL may be stable to invasion even when f (mT b) + f (mt B) > 0.5 (the majority of ASL individuals are using the wrong behavior). We searched for the largest value of self-perception error (e) for which ASL remained stable against invasion by NASL for different values of s1 , s2 , s3 , r, and h in model 2 (see example in Fig. 3A). In model 2 with h = 0, the maximum e for which ASL was stable against invasion of NASLV was lower than in model 1 in 80.1% of the parameter combinations, the same in 19.7%, and higher in less than 1%. Thus, in a pure population of either ASL or NASL, h = 0 reduces model 2 to model 1, but when both social learning strategies exist, the two models differ with

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Figure 3. Invasion by NASL of a population of ASL. In all plots, r = 0.3, s3 = 0 (tb selection coefficient); first row: model 1 (nonheritable trait); two bottom rows: model 2 with heritability values of 0 and 0.3. (A) The highest value of self-perception error, e, for which ASL is stable to invasion by NASL (e values are indicated by the colorbar).

Selection coefficients s1 and s2 are on the x- and y-axis, respectively. (B) An example of equilibrium frequencies of the different phenogenotypes. s1 = −0.71, s2 = −0.57, s3 = 0. Arrows indicate the highest self-perception error (e) for which ASL was stable against invasion of NASL.

ASL being more resistant to invasion by NASLO than by NASLV (compare first and second rows of Fig. 3A, B). We discuss this result in the General Discussion. In model 2, the maximum e for which ASL was stable against invasion of NASLV tended to decrease with h: although for consecutive values of h in a given parameter combination this value of e often did not change, when it did change, it decreased in >97% of cases. It is interesting to note that in model 1 when NASLO invaded a polymorphic population fixed on ASL, the result was a population polymorphic for M and m (social learning strategy), but monomorphic with respect to B. For example, in the top panel of Figure 3B, with e = 0.45 b is fixed. In model 2, however, invasion by NASLV of a population of ASL could result in fixation of NASLV or in polymorphism of both behaviors and social learning strategies (see example in Fig. 3B second panel e = 0.3, third panel e = 0.2). In these cases, there was a

behavioral polymorphism among ASL individuals, but only one behavior for NASLV individuals (excluding rare exceptions close to our numerical definitions of polymorphism). Thus, in model 2, a population of assortative and vertical social learners may maintain both social learning strategies and both behaviors (although the behavioral polymorphism is maintained only among ASL individuals). THE EVOLUTION OF ASL

Here, we address the evolution of the ASL strategy. As shown above, in many cases, ASL allows stable polymorphism of B and b whereas a population fixed on NASL is expected to retain only B or b, but not both. In the special polymorphic cases of NASL, behavioral polymorphism is not expected to be maintained in finite populations due to random drift (see above). In these cases, before one behavior is eventually fixed, we would expect a transient phase in which B and b drift. After fixation, the lost

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behavior may be introduced again as a rare innovation or by migration from a group in which a different behavior was fixed, resulting, again, in a behaviorally polymorphic transient phase. This situation is not analogous to the classical cases of protected polymorphism of ASL and NASL where fixation of both would be strictly locally unstable. In model 1, we identified the neutral case 2 −s3 when r = r ∗ = s1 s+s ; in this case, if s3 = 0, it can be shown 2 −s3 analytically that for all nonzero frequencies of B and b, ASL will invade an NASL population for any parameter combination with e < 0.5 (see Section 2.4 in Supporting Information). This includes the case r∗ = 0.5, (s1 = s2 ) for which we show that ASL is stable to invasion by NASLO and leads to behavioral polymorphism for any e < 0.5 (see above and Sections 2.2.2 and 2.3.1 in Supporting Information). Thus, this is an example where even for high values of e, ASL invades a behaviorally polymorphic population of NASLO at any stage before fixation of one behavior, is stable to invasion by NASLO once it has fixed, and maintains behavioral polymorphism. In most cases, however, a population of NASL at equilibrium is expected to be fixed on one behavior. We thus explore the evolution of ASL when the population is fixed on one behavior, in which case assorting is meaningless, and the two social learning strategies are neutral in both models and for any parameter combination. As there is no advantage to either of the social learning strategies in this case, in a finite population their frequencies would be expected to drift until one of them is eventually fixed. To simulate different transient stages we used different frequencies of the social learning strategies (ASL and NASL) in a population in which only one behavior exists (either B or b) and introduced the other behavioral strategy at a low frequency to represent the case of a rare behavioral innovation (see Section 1.4 in Supporting Information). Depending on the proportion of ASL and on the parameter values, when the error is low enough and the proportion of ASL in the population is large enough, introducing a new behavior could create an opportunity for ASL to increase and maintain a stable behavioral polymorphism. However, if the frequency of ASL is not high enough, one behavior may fix before the frequency of ASL has increased sufficiently, even when self-perception error is potentially low enough to allow the maintenance of behavioral polymorphic population of ASL. Figure 4 shows an example of these dynamics. For different parameter values, the result may be fixation of one behavior, of one learning strategy, or any combination of both. Still, for any parameter combination, there was a low-enough error and a high-enough starting proportion of ASL to allow ASL to be maintained and result in a behavioral polymorphism. In line with the results in the previous section, in model 2 with h = 0, it is harder in most cases for ASL to reach fixation in model 2 than in model 1.

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General Discussion Our models suggest that assortment may be advantageous in the context of social learning, where the tendency to assort acts as a mechanism for individual-specific filtering of relevant models. Unlike the nonassortative social learning strategies (NASL) which are expected to result in behavioral homogenization, ASL results in behavioral polymorphism and is stable to invasion of NASL for a wide range of parameters. In both models, nonheritable and heritable, for any parameter combination, there is a sufficiently low self-perception error that allows a stable population of ASL in which B and b coexist, that is, there is a behavioral polymorphism. For intuition regarding the maintenance of behavioral polymorphism, consider the simple case of no self-perception error (see Results section 1, ASL when e = 0). In this case, ASL individuals with a certain trait learn only from parents with the same trait, whereas at equilibrium all parents use the appropriate behavior (see eq. (2). Because the appropriate behavior differs between traits, both behaviors are maintained. As e increases, ASL individuals also learn from others with a different trait. Yet, it seems that as long as the frequency of learning a certain behavior is higher in those for which this behavior is appropriate, behavioral polymorphism is maintained (as this happened when polymorphism was maintained at equilibrium). Although the evolution of social learning is often modeled in the context of changing environment (Boyd and Richerson 1985; Rogers 1988; Feldman et al. 1996; Aoki et al. 2005; Enquist and Ghirlanda 2007; Rendell et al. 2009; Aoki and Feldman 2014), learning is obviously important in other contexts as well (e.g., Lotem et al. 1992; Hammer 1995). In our model, the advantage of ASL results from variation in phenotype among individuals, which leads to variation in fitness of the different behaviors (Katsnelson et al. 2012). Our model does not assume or depend on environmental change. On the contrary, we assume that the environment is stable for long enough to allow the patterns predicted by our model to evolve. Indeed, despite the conservative equilibrium criterion (see Section 1.1 in Supporting Information), equilibrium was often achieved after a small number of generations (see Sections 1.3 and 1.4 in Supporting Information). It is most likely that throughout their history, both human and animal societies have experienced long periods of a physically and/or culturally stable environment, at least with respect to some trait– behavior combinations (e.g., d’Errico et al. 2012). Moreover, our models do not depend on the simultaneous existence of individual learners that can repeatedly innovate/produce a behavior: although individual learning (or innovation) may be the means by which behaviors initially arise, once behavioral polymorphism has appeared, and regardless of the starting frequencies, ASL itself is sufficient to maintain it. Indeed, once evolved, social learning may lead to the spread of a new behavior whereas individual

proportion of b

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1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1 h=0

model 1

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model 2

proportion of m Figure 4.

Introducing one behavior to a population fixed on the other behavior with both social learning strategies present. Arrows

go from the starting frequencies to the equilibrium frequencies of ASL (x-axis) and behavior b (y-axis). Upper row: b is introduced into a population fixed on B; lower row: B is introduced into a population fixed on b. Solid arrows: e = 0; dashed arrows: e = 0.25. In all panels, s1 = −0.5, s2 = −0.7, s3 = −0.2, r = 0.3.

learning (or “innovation”) remains rare and the environment does not change too often (Cavali-Sforza and Feldman 1981; Boyd and Richarson 1985; Rogers 2003; Reader and Laland 2003; Reader 2004). The emerging behavioral polymorphism also does not require a negative frequency-dependent relation between the behaviors. It only requires that individuals with different traits have different behaviors that are more appropriate to them given their respective trait. Compared with NASL, we may assume ASL is cognitively challenging as it requires phenotypic matching: each individual knows its own traits and is able to recognize others that share those traits (see Methods). If this ability is cognitively complex and costly, there may be error in self-perception. Unlike NASL, which may result in learning from individuals with different or the same trait, self-perception error specifically leads to erroneous assortment, and thus results in learning from an individual with a different trait. Once ASL individuals identify themselves (correctly or not), detecting a model with the same perceived trait is not assumed to be a major obstacle: in highly social societies for which our model is relevant, individuals are expected to know, and maintain, close ties with tens of individuals (Dunbar 2012) and simultaneously maintain a much larger number of weak social ties that are also potentially influential (Granovetter 1973). Accordingly, as long as the frequencies of traits are not negligible (r  0.1 in our simulations), a potential model already exists among the individuals one is familiar with or knows about. (Note that we have ignored any costs to the transmitter, which could be

considerable, see Fogarty et al. 2011 and Creanza et al. 2012.) Model choice is most likely a result of a complex combination of various mechanisms (e.g., conformism, success bias, a tendency to learn from kin, from proximate individuals, etc.), which are artificially separated and categorized in models for practical reasons. In reality, however, several strategies for model choice, such as success bias, kin bias, and ASL, may interact and be influenced by the specific social structure of the community/species (Rogers 2003; Centola 2011), thus affecting the level of ASL. Our model assumes a population which is phenotypically polymorphic for trait T (neither trait can go to fixation) and the parent–offspring correlation of this trait cannot be perfect (both hold because h < 1). Phenotypic polymorphism with imperfect parent–offspring association is expected to be common, and may be a result of various mechanisms of which no/partial heritability is just one. We expect that ASL may evolve in such cases. For example, trait polymorphism that is maintained by some frequency-dependent mechanism may be the background on which ASL operates. Another example is sexual reproduction under which the association in traits between parents and offspring is not straightforward. Although the population in our model is asexual (a common simplification when studying social learning evolution, e.g., Rendell et al. 2009), the core principles of ASL are expected to apply to sexual populations. ASL in this case may even be stable to invasion of NASLV for higher heritability levels of T: if different parents have different traits, learning from

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parents (NASLV ) may become more similar to oblique learning (NASLO ). Whether M and T are acquired from the same or different parents, the effect of M (m or M) is expected to be similar to that in the asexual model. However, sexual reproduction may lead to unexpected dynamics that are hard to predict and worth exploring in future research. For example, in a study of assortative mating and niche construction, Creanza et al. (2012) showed that for some parameter combinations, polymorphism of two forms of two traits may exist even when there are no mutations in either trait. Although models 1 and 2 generally give similar results, they provide some different insights. In model 1, the phenotype represents a complex of traits with negligible heritability. Because there is no association between the trait of a parent and that of its offspring, the a priori competing nonassortative social learning strategy in this case is oblique (learning from any individual in the parent generation). In model 2, however, depending on the level of heritability, an individual is more likely to have the trait of its parent. The a priori competing nonassortative social learning strategy in this case is vertical (learning from the parent). We can view this kin-based social learning (Schwab et al. 2008) in light of ASL: with the increased chance of similarity among kin, learning from kin is a filtering mechanism that does not require phenotypic matching. The prevalence of ASL versus NASLV will be determined by the level of self-perception error, e, (which leads to erroneous assortment in ASL) versus the level of heritability, h. Yet, like NASLO and unlike ASL, NASLV alone cannot result in stable behavioral polymorphism. The case h = 0 has an interesting outcome: although in a population of NASL only, model 2 reduces to model 1 (there is no difference between NASLO and NASLV ), NASLV is generally more successful competing against ASL than NASLO (see Fig. 3). It is possible that this is because NASLO individuals learn from all individuals in the parent generation (including ASL parents), whereas NASLV learn only from their NASLV parents. ASL parents may maintain the behavior that would gradually disappear in an NASL population. Because NASLO individuals lack the trait-based filtering mechanism, learning from all parents possibly results in a higher chance of mismatch between phenotype and behavior relative to NASLV individuals. In other words, in this case (h = 0) NASLO individuals are exposed to a behavioral profile shaped by ASL without having the filtering mechanism of ASL. Humans who are similar on some dimension may assort by a variety of traits that range from physical characteristics to subtle hobby preferences (McPherson et al. 2001; Centola et al. 2007); and transmit other traits based on assortment (Centola 2011). Although human societies are the main motivation for our models, these models are not limited in application to humans. Animals are known to assort in sexual and social contexts (see below), and to have the ability of self-referent phenotypic matching (Hauber

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and Sherman 2001). However, even if it exists, detecting ASL in animals might be difficult, because regardless of their own success/status, animals are likely to show success bias when given a choice between demonstrators that differ in these attributes (e.g., stronger vs. weaker; Duffy et al. 2009; Horner et al. 2010). Such a bias is expected, especially if the learner can potentially “improve” its state by greater resemblance to the successful model (e.g., growing, getting stronger). To test ASL, demonstrators should be different but not superior in traits that are relevant for ASL in that species. Moreover, we may not know or recognize traits by which animals assort. For example, chimpanzee friendships were recently found to be based on homophily in personality traits (Massen and Koski 2014). If this homophily is also associated with increased rates of transmission of behaviors, this would be an indication that ASL is present. Thus, although the existence of ASL in animals remains an open question, social assortment is known to exist. Other examples include sex- and age-based assortment that has been demonstrated in dolphins (Lusseau and Newman 2004) and primates (Langergraber et al. 2009; RamosFernandez et al. 2009; Brent et al. 2011). Sex is an intuitive example of a nonheritable phenotype T (h = 0, r = 0.5) according to which individuals assort because sex-dependent payoffs from different behaviors are likely to exist in humans and other species. Our model suggests an evolutionary basis for sex-based ASL of behaviors that are likely to result in different levels of success depending in one’s sex (phenotypic matching in this case is expected to be simple). Once sex-based ASL exists, sex-based social learning may extend to behaviors whose payoffs are not necessarily sex-dependent, leading to sex-linked differences in a range of behaviors. In our model, we did not define any prior advantage from the mere act of assorting (such as synergism, Fu et al. 2012). In other words, we do not assume that homophily per se is advantageous; rather we demonstrate that the advantage that emerges from assortment is due to the opportunity it creates for social learning from relevant models. An interesting possibility is that assortment creates opportunities for both ASL and synergetic effects that drive the evolution of homophily even further. Here, we suggest that the filtering mechanism assortment provides within social learning may give assortment an important advantage. Indeed, people are known to assort, and to transmit a behavior based on assortment (Centola 2011). As people are rarely identical (e.g., there are sex, size, and skill differences), and as human variability in success as a result these differences is expected to be common, such a filtering mechanism is likely to have an advantage. Once it exists, ASL might be a good “default” mechanism when it is not clear a priori whether the success deriving from a behavior is linked to one’s traits or not. This may contribute even further to the importance of homophily as a principal behavioral pattern.

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Assortment may play an important role in various social dynamics and their evolution, such as cooperation and altruism (Eshel and Cavalli-Sforza 1982; Durrett and Levin 2005; Rankin and Taborsky 2009) or in the spread of adaptive and/or maladaptive behaviors (Smith and Christakis 2008; Centola 2011) as shown for other social learning strategies (Giraldeau et al. 2002; Ihara 2008; Lehmann and Feldman 2009). It may also constitute a barrier against diffusion of beneficial behaviors (Rogers 2003). Importantly, once evolved, assortment may explain the emergence of different cultures in complex societies (Axelrod 1997; Centola et al. 2007; Flache and Macy 2011). Simulations suggest that cultural influence among homophilus individuals can lead to the formation of separate groups, which once they become dissimilar, are less likely to influence each other, resulting in overall cultural diversity. Diversity in these models results from contagious dynamics among homophilus individuals as partially similar individuals increasingly become more similar (they can change their attributes in all their traits), which may lead to distinct groups in each of which the cultural characteristics and participating individuals are a result of a random starting point and driven by homophily. Diversity in these models is dependent on model specifications and might be fragile, yet rare innovations within social groups are unlikely to spread. Moreover, whether this polymorphism represents a stable equilibrium is unclear (Flache and Macy 2011). Virtual networks and instantaneous exchange of ideas and information add new dimensions to the importance of ASL in understanding the spread of behaviors in modern human society (Centola et al. 2007; Centola 2011). Yet, the advantage conferred by assortment in social learning might be rather general, and might have an important role in driving the evolution of assortment itself.

ACKNOWLEDGMENTS We wish to thank E. Louidor, N. Creanza, M.R. McLaren, M. Arbilly, and the Feldman laboratory members. We are grateful for the valuable comments of two anonymous reviewers and O. Ronce on previous versions of this article. This research was supported in part by the Morrison Institute for Population and Resource Studies at Stanford University. AL was supported in part by the Israel Science Foundation grant no. 1312/11. DATA ARCHIVING The doi for our data is 10.5061/dryad.p63jh.

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Supporting Information Additional Supporting Information may be found in the online version of this article at the publisher’s website: Numerical Simulations details and analysis.

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Associate Editor: O. Ronce