EVOLUTION OF THE SOCIAL-LEARNER-EXPLORER STRATEGY IN AN ENVIRONMENTALLY HETEROGENEOUS TWO-ISLAND MODEL

Authors


Abstract

Social-learner-explorer (SE) is a learning strategy that combines accurate social learning with exploratory individual learning in that order. Arguably, it is one of the few plausible learning strategies that can support cumulative culture. We investigate numerically the factors that affect the evolution of SE in an environmentally heterogeneous two-island model. Conditions favorable to the evolution of SE include a small exogenous cost of social learning, the occurrence of migration after social learning but before individual learning, the ability to adaptively modify the behavioral phenotype in the postmigration environment (asymmetrical individual learning), and a relatively high migration rate. The implications of our model for the evolution of SE in humans are discussed. Of particular interest is the prediction that behaviors affecting fitness would have to be socially learned in the natal environment and then subsequently modified by individual learning in the postmigration environment, suggesting a life-cycle stage dependent reliance on the two types of learning.

Cumulative culture is a distinguishing feature of the human (Homo sapiens) condition (Tomasello 1999; Alvard 2003; Henrich and McElreath 2003; Laland and Hoppitt 2003), which requires organisms that are equipped with the ability to accurately absorb the extant culture and then to build creatively on it. Social learning is the generic term for the psychological processes such as local enhancement and imitation that support cultural inheritance. Individual learning comprises the psychological processes such as trial-and-error and insight that generate innovations and novel behaviors (Cavalli-Sforza and Feldman 1981; Galef 1988; Whiten and Ham 1992; Heyes 1993). Borenstein et al. (2008) argue that “social-learner-explorer” (abbreviated SE), a learning strategy that combines accurate social learning with exploratory individual learning in that order, is supportive of cumulative culture. A similar strategy was previously postulated by Boyd and Richerson (1985) in their “guided variation” model.

There is wide theoretical agreement that social learning and individual learning (as opposed to the innate determination) of behavior are adaptations to temporally and/or spatially variable environments (Boyd and Richerson 1985, 1988; Rogers 1988; Boyd and Richerson 1995; Feldman et al. 1996; Henrich and Boyd 1998; Laland et al. 2000; Richerson and Boyd 2000; Alvard 2003; Henrich and McElreath 2003; Wakano et al. 2004; Aoki et al. 2005; Wakano and Aoki 2006, 2007; Enquist et al. 2007; Nakahashi 2007; Aoki and Nakahashi 2008; Borenstein et al. 2008; McElreath and Strimling 2008). Most models of the evolution of learning—the “standard models” (Aoki and Nakahashi 2008)—assume that social learning and individual learning are mutually exclusive strategies, or have taken the mixed strategy approach. This theoretical stance is consistent with the view that social learning and individual learning require distinct psychological abilities, between which an evolutionary trade-off exists (Tomasello and Call 1997). Clearly, such models cannot fruitfully be applied to the evolution of the SE strategy.

The issue is not just a theoretical one. In a comparative survey of the primate literature, Reader and Laland (2002) found that the recorded instances of innovation (individual learning) and of social learning tend to covary positively across species. Moreover, an experimental study on pigeons suggested a positive correlation at the organismal level, that is, pigeons that are innovative also appear to be adept at social learning (Bouchard et al. 2007). Boyd and Richerson (1988, 1995), Enquist et al. (2007), and Lehmann and Feldman (2009) describe various strategies that are capable of both social learning and individual learning. Borenstein et al. (2008) have introduced a model that allows for social learning and individual learning to evolve independently of each other. Whether the empirical findings noted above are regarded as definitive, these new theoretical approaches are clearly preferable to one that assumes a priori that social learning and individual learning are mutually exclusive strategies.

Borenstein et al. (2008) investigated the effects of temporal environmental fluctuations on the evolution of the SE strategy. In the current article, we adapt their model to the case of spatial environmental heterogeneity. We consider the simplest such case, namely a model of two subpopulations that occupy environmentally dissimilar sites. Recent advances in the genetic study of modern human origins suggest that the ancestral African population may have been structured (reviewed by Harding and McVean 2004; Garrigan and Hammer 2006). It is possible that environmental variability across the range of this subdivided population may have driven the evolution of “the cognitive mechanisms …[that] fostered the input, analysis, and mental representation of highly varied external information and the output of versatile, novel response” (Potts 1998, p. 130).

On the basis of extensive numerical work, five factors emerge as having significant effects on the evolution of the SE strategy: the exogenous costs of social learning and individual learning, symmetrical versus asymmetrical individual learning, the timing of migration relative to social learning and individual learning (i.e., the life cycle), the migration rate, and the degree of environmental overlap between the sites occupied by the two subpopulations. Conditions for the evolution of the SE strategy are apparently quite stringent, and in the situations in which the SE strategy is disadvantaged, simpler learning strategies involving only social learning or individual learning may evolve. Moreover, thresholds and bistability are observed at some values of the migration rate.

Model Description

Posit two sites (islands) each occupied by a subpopulation comprising an infinite number of asexual organisms. Migration occurs at rate m (0 ≤m≤ 1/2) when the sizes of the two subpopulations are equal. Site i inhabited by subpopulation i is in environmental state yi (where i= 1 or 2 and y1 < y2) with tolerance w≥ 1 (which is the same for both sites). That is, organisms with integer phenotype z at site i survive viability selection (see below) if and only if yiwzyi+w (the fitness function is rectangular). Deaths that occur when the phenotype lies outside this range constitute the endogenous cost suffered by a strategy.

A strategy is defined by the triplet of numbers g, k, and b, where g is the genotypic value, k is the probability of social learning, and b is the breadth of (exploratory) individual learning. We write the frequency of strategy g, k, b in subpopulation i among the newborns of generation t as φt(g, k, b; i). (We can regard g, k, b as a haplotype where g, k, and b are alleles at three completely linked genetic loci.) The distribution of phenotype z among the surviving adults in subpopulation i in generation t will be denoted by ηt(z; i). The surviving adults serve as exemplars, which are copied by the naïve social learners of the next generation.

In addition to these basic variables, it is useful to define for each subpopulation the frequency of organisms with strategy g, k, b and initial phenotype x (phenotype after genetic determination or social learning, but before individual learning, see below), which we write as αt(x, g, k, b; i). Similarly, the frequency of organisms with strategy g, k, b and mature phenotype z (phenotype after individual learning, see below) will be denoted by χt(z, g, k, b; i). The two variables—αt(x, g, k, b; i) and χt(z, g, k, b; i)—are joint distributions in g, k, b, and x or z, but we will refer to them as distributions in x and z, respectively. The frequency of strategy g, k, b is in fact the marginal distribution

image((1a))

and

image((1b))

Hence, we distinguish three classes of phenotype distributions: the initial and mature phenotype distributions associated with each strategy (αt(x, g, k, b; i), χt(z, g, k, b; i)), and the postselection phenotype distribution that includes contributions from all extant strategies (ηt(z; i)).

The initial phenotype distribution is related to the frequency of a strategy as

image(2)

where δxg is Kronecker's delta (δxg= 1 if x=g and 0 otherwise). Equation (2) entails that with probability 1 −k the genotypic value g is innately expressed as the initial phenotype, and that with probability k the initial phenotype is acquired by social learning from the previous generation (oblique transmission). The mature phenotype distribution is

image(3)

where λ (z; x, yi, b) is the individual learning function. The individual learning function is a probability distribution on the mature phenotype, z, which depends on the initial phenotype, x, the environmental state of site i, yi, and parameter b. Specific forms, which may be symmetrical or asymmetrical, will be defined below.

The life-cycle events are genetic determination or social learning, individual learning, migration, viability selection, and asexual reproduction. We consider three different life cycles by varying the timing of migration. However, in all scenarios we assume that genetic determination or social learning occurs before individual learning, and that asexual reproduction immediately preceded by viability selection occurs last. We assume that the sizes of the two subpopulations are equal at birth, and because migration always precedes viability selection in the various versions of the model, equality still holds when migration occurs.

When migration precedes genetic determination or social learning (early migration), the effect of migration can be summarized by the recursions

image((4a))
image((4b))

The asterisk indicates a variable after migration. When migration occurs after innate determination or social learning but before individual learning (middle migration),

image((5a))
image((5b))

Finally, when migration occurs after individual learning (late migration),

image((6a))
image((6b))

Now define the viability function for mature phenotypes at site i

image(7)

where d and c are the unit exogenous costs associated with social learning and individual learning, respectively. Note that these exogenous costs, which derive from the need to develop and maintain the neural substrates for learning—increased brain size (Reader and Laland 2002)—and from “unintended” errors made during learning, are superposed on the endogenous costs. Then,

image((8a))

where the double asterisk indicates variables after selection, and

image((8b))

is the mean viability. The viability of strategy g, k, b (as opposed to phenotype z) at site i can be expressed as

image(9)

In the case of late migration, we substitute the postmigration mature phenotype distribution given by equation (6) into equation (8) and equation (9). For middle migration, the value of χ*t (z, g, k, b; i) to be plugged into these equations is

image(10)

And for early migration, we substitute

image(11)

Finally, from equation (1b) and the assumption of asexual reproduction, the frequency of strategy g, k, b among the newborns of the next generation becomes

image(12)

Similarly, the frequency of phenotype z among the surviving adults of the current generation becomes

image(13)

Special assumptions

Subsequently, we limit our attention to strategies for which k= 0 or 1 and b= 0 or 1. Because parameter k is the probability of social learning, our first restriction entails that we consider only strategies where social learning is absent (k= 0) or obligate (k= 1). (Note that the genotypic value is irrelevant to the determination of the initial phenotype when k= 1.) Parameter b is to be interpreted as follows. The setting b= 0 corresponds to the case in which individual learning does not occur, so that the initial and mature phenotype distributions are identical. When b= 1, however, individual learning follows genetic determination or social learning so that the two distributions differ. This yields four generic strategies: (1) g, 0, 0, which we call “genetic” and abbreviate as G; (2) g, 0, 1, called “genetic-explorer” and abbreviated GE; (3) g, 1, 0, called “social learner” and abbreviated S; and of course (4) g, 1, 1 is the social-learner-explorer (SE) strategy.

In addition, we assume dichotomous variation in the genotypic values, that is, g can be either g1 or g2, yielding a total of eight specific strategies. Moreover, we assume for the numerical work that the two specific G strategies, g1, 0, 0 and g2, 0, 0, are each adapted to (at least) one of the two environmental states, in particular that

image(14)

for i= 1 and 2. We note from Appendix equation (A2) and Appendix 3 that we can generalize equation (14) to inequality yiwgiyi+w without affecting the viabilities of the specific G strategies. However, the viabilities of the two specific GE strategies, g1, 0, 1 and g2, 0, 1, may change.

Let us now formally define the individual learning function. When b= 0

image(15)

In this case, individual learning does not occur, and the mature phenotype z is identical to the initial phenotype x. When b= 1 and individual learning is symmetrical, we set

image(16)

That is, the mature phenotype z may take any one of the three integer values around and including the initial phenotype (x− 1, x, and x+ 1) with equal probability.

For asymmetrical individual learning, we posit

image((17a))

if x < yi (the initial phenotype is smaller than the environmental state)

image((17b))

if x=yi (the initial phenotype is equal to the environmental state), and

image((17c))

if x > yi (if the initial phenotype is greater than the environmental state). When the initial phenotype coincides with the environmental state (x=yi), the effect of asymmetrical individual learning is the same as symmetrical individual learning (eq. 17b). On the other hand, when the initial phenotype deviates from the environmental state, asymmetrical individual learning shifts the center of the mature phenotype distribution toward the environmental state yi by one phenotypic unit (eq. 17a and 17c). For example, when x < yi the mature phenotypes are x, x+ 1, and x+ 2 (see Fig. 1).

Figure 1.

Top figure shows the initial phenotype distribution, concentrated in this example on z=x, and the phenotypes that can survive in environmental state y, which are ywzy+w. Bottom figure shows the mature phenotype distribution after asymmetrical individual learning. Because asymmetrical individual learning shifts the center of the distribution toward the environmental state by one phenotypic unit, one-third of the organisms adopting this strategy in this example survive.

Results

In Appendix 1, we give formal derivations of the viabilities of the four generic strategies when there is no migration (m= 0). Due to our assumption in equation (14), the specific G strategy gi, 0, 0 has viability 1 at site i (where i= 1 or 2) and is not exceeded in viability by any of the other strategies. Let us first consider the case in which the distance between environmental states, Δyy2y1, is greater than the tolerance, w, (i.e., Δy > w) so that strategy g1, 0, 0 cannot survive at site 2 and vice versa. If in addition the exogenous costs of learning, d and/or c, are taken into account, strategy gi, 0, 0 has the highest viability at site i (where i= 1 or 2), and our evolutionary model entails that this strategy will be fixed at that site. In other words, the globally stable equilibrium point in this case is

image(18)

where the caret denotes equilibrium.

For small values of the migration rate, m, we can invoke the theorems of Karlin and McGregor (1972a,b). Thus, there exists a locally stable equilibrium that either coincides with equation (18) or is located in its neighborhood.

If Δyw, on the other hand, the two specific strategies g1, 0, 0 and g2, 0, 0 have equal viability 1 at both sites 1 and 2 regardless of the migration rate (see Appendices 2 and 3). Hence, the neutrally stable equilibrium for all migration rates will comprise both strategies in proportion to their initial frequencies.

Numerical results were obtained according to the following protocol. Initial frequencies of the eight specific strategies among newborns in each subpopulation were set equal (uniform configuration), or alternatively were assigned random values (random configuration). Initial postselection phenotype distribution of the parental generation in each subpopulation was assumed to be uniform across all viable phenotypes. The recursion equations were then iterated until equilibrium was reached. Specifically, simulations were terminated when two quantities—the sum of the absolute differences in strategy frequencies between two generations for both islands, and the sum of the absolute differences in postselection phenotype frequencies between two generations for both islands—both fell below 10−6.

The outcomes of a large set of simulations are reported in six tables (Tables 1–6). As noted above, the two genotypic values were assumed to coincide with the two environmental states, that is, gi=yi for i= 1 and 2. We used the uniform configuration for the initial frequencies of the eight specific strategies in these simulations. Each table displays the generic strategy or strategies (G, GE, S, or SE) present at equilibrium when the migration rate, m, and the distance between environmental states, Δyy2y1, are varied. Because the tolerance, w, was set to 1 in all simulations, Δy= 1, 2, 3, and 4 can be termed wide (environmental) overlap, narrow overlap, contiguous, and one-removed, respectively. The first group of three tables (Tables 1–3) shows our results with symmetrical individual learning. The second group (Tables 4–6) assumes asymmetrical individual learning. The “(A)” and “(B)” tables compare the effects of the exogenous costs of social learning and individual learning. In the “(A)” tables d= 0.01 < c= 0.1 (social learning is less costly than individual learning), whereas in the “(B)” tables d= 0.1 > c= 0.01 (social learning is more costly than individual learning).

Table 1.  Strategies present at stable equilibrium with symmetric individual learning and early migration. Shown is the dependence on migration rate, m, and distance between environmental states, Δy=y2y1. Genotypic values coincide with environmental states (gi=yi). Initial configuration of specific strategy (haplotype) frequencies is uniform. Initial postselection phenotype distribution is uniform. Tolerance set to w=1. Threshold between two in same column. See text for details. G, genetic; GE, genetic-explorer; S, social learner; SE, social-learner-explorer.
mΔy=1Δy=2Δy=3Δy=4
(A) d=0.01, c=0.1
 0GGGG
 0.009GGGG
 0.010GG, SG, SG, S
 0.011GSSS
 0.1GSSS
 0.2GSSS
 0.3GSSS
 0.4GSSS
 0.5GSSS
(B) d=0.1, c=0.01
 0GGGG
 0.09GGGG
 0.1GG, SG, SG, S
 0.11GSSS
 0.2GSSS
 0.3GSSS
 0.4GSSS
 0.5GSSS
Table 2.  Strategies present at stable equilibrium with symmetric individual learning and middle migration. Shown is the dependence on migration rate, m, and distance between environmental states, Δy=y2y1. Genotypic values coincide with environmental states (gi=yi). Initial configuration of specific strategy (haplotype) frequencies is uniform. Initial postselection phenotype distribution is uniform. Tolerance set to w=1. Threshold between two s, bistability in vicinity of two * in same column. See text for details. G, genetic; GE, genetic-explorer; S, social learner; SE, social-learner-explorer.
MΔy=1Δy=2Δy=3Δy=4
(A) d=0.01, c=0.1
 0GGGG
 0.1GGGG
 0.16GG*GG
 0.17GS*GG
 0.2GSGG
 0.3GSGG
 0.4GSGG
 0.5GSGG
(B) d=0.1, c=0.01
 0GGGG
 0.1GGGG
 0.12GGGG
 0.13GGEGG
 0.2GGEGG
 0.27GGE*GG
 0.28GS*GG
 0.3GSGG
 0.4GSGG
 0.5GSGG
Table 3.  Strategies present at stable equilibrium with symmetric individual learning and late migration. Shown is the dependence on migration rate, m, and distance between environmental states, Δy=y2y1. Genotypic values coincide with environmental states (gi=yi). Initial configuration of specific strategy (haplotype) frequencies is uniform. Initial postselection phenotype distribution is uniform. Tolerance set to w=1. Threshold between two ’, bistability in vicinity of two *’ in same column. See text for details. G, genetic; GE, genetic-explorer; S, social learner; SE, social-learner-explorer.
mΔy=1Δy=2Δy=3Δy=4
(A) d=0.01, c=0.1
 0GGGG
 0.1GGGG
 0.16GG*GG
 0.17GS*GG
 0.2GSGG
 0.3GSGG
 0.4GSGG
 0.5GSGG
(B) d=0.1, c=0.01
 0GGGG
 0.1GGGG
 0.12GGGG
 0.13GGEGG
 0.2GGEGG
 0.27GGE*GG
 0.28GS*GG
 0.3GSGG
 0.4GSGG
 0.5GSGG
Table 4.  Strategies present at stable equilibrium with asymmetric individual learning and early migration. Shown is the dependence on migration rate, m, and distance between environmental states, Δy=y2y1. Genotypic values coincide with environmental states (gi=yi). Initial configuration of specific strategy (haplotype) frequencies is uniform. Initial postselection phenotype distribution is uniform. Tolerance set to w=1. Threshold between two in same column. See text for details. G, genetic; GE, genetic-explorer; S, social learner; SE, social-learner-explorer.
mΔy=1Δy=2Δy=3Δy=4
(A) d=0.01, c=0.1
 0GGGG
 0.009GGGG
 0.010GG, SG, SG, S
 0.011GSSS
 0.1GSSS
 0.2GSSS
 0.3GSSS
 0.4GSSS
 0.5GSSS
(B) d=0.1, c=0.01
 0GGGG
 0.06GGGG
 0.07GGEGG
 0.09GGEGG
 0.1GGEG, SG, S
 0.11GGESS
 0.12GGESS
 0.13GSSS
 0.2GSSS
 0.3GSSS
 0.4GSSS
 0.5GSSS
Table 5.  Strategies present at stable equilibrium with asymmetric individual learning and middle migration. Shown is the dependence on migration rate, m, and distance between environmental states, Δy=y2y1. Genotypic values coincide with environmental states (gi=yi). Initial configuration of specific strategy (haplotype) frequencies is uniform. Initial postselection phenotype distribution is uniform. Tolerance set to w=1. Threshold between two , bistability in vicinity of two * in same column. See text for details. G, genetic; GE, genetic-explorer; S, social learner; SE, social-learner-explorer.
mΔy=1Δy=2Δy=3Δy=4
(A) d=0.01, c=0.1
 0GGGG
 0.1GGGG
 0.11GG*GG
 0.12GS*GG
 0.2GSGG
 0.25GSG*G
 0.26GSSE*G
 0.3GSSEG
 0.31GS*SEG
 0.32GSE*SEG
 0.4GSESEG
 0.5GSESEG
(B) d=0.1, c=0.01
 0GGGG
 0.06GGGG
 0.07GGEGG
 0.1GGEGG
 0.12GGEGG
 0.13GGEGEG
 0.2GGEGEG
 0.3GGEGEG
 0.4GGEGEG
 0.5GGEGEG
Table 6.  Strategies present at stable equilibrium with asymmetric individual learning and late migration. Shown is the dependence on migration rate, m, and distance between environmental states, Δy=y2y1. Genotypic values coincide with environmental states (gi=yi). Initial configuration of specific strategy (haplotype) frequencies is uniform. Initial postselection phenotype distribution is uniform. Tolerance set to w=1. Threshold between two , bistability in vicinity of two * in same column. See text for details. G: genetic, GE: genetic-explorer, S: social learner, SE: social-learner-explorer.
mΔy=1Δy=2Δy=3Δy=4
(A) d=0.01, c=0.1
 0GGGG
 0.1GGGG
 0.13GG*GG
 0.14GS*GG
 0.2GSGG
 0.3GSGG
 0.4GSGG
 0.5GSGG
(B) d=0.1, c=0.01
 0GGGG
 0.1GGGG
 0.11GGGG
 0.12GGEGG
 0.2GGEGG
 0.3GGE*GG
 0.31GS*GG
 0.4GSGG
 0.5GSGG

Let us summarize the numerical results, supplementing the information presented in these six tables with the added insight gained by varying the initial conditions, the exogenous costs of learning, and the degree of environmental overlap.

Result 0: The six tables confirm the mathematical prediction that the G strategy will be fixed, regardless of the timing of migration, when the migration rate is low (upper region of tables) or when there is wide environmental overlap (Δy= 1).

Result 1: The main issue addressed in this article is the evolution of the SE strategy. Five conditions must be satisfied simultaneously for this to occur: (1) the exogenous cost of social learning must be small (d≪ 1); (2) individual learning must be asymmetrical; (3) migration must occur after social learning but before individual learning takes place (middle migration); (4) the migration rate must be relatively high; (5) the environments of the two sites must be narrowly overlapping, contiguous, or one-removed (2 ≤Δy≤ 4) (although not shown in Table 5A, the SE strategy can be fixed when Δy= 4 provided the migration rate is sufficiently high and the exogenous costs of learning are sufficiently low).

Strong support for conditions (2) and (3) is provided by simulations in which the exogenous costs of learning were ignored (d=c= 0). We find in this case that the SE strategy can be fixed when conditions (2) and (3) are both satisfied. Otherwise, it appears that the SE strategy can exist at a polymorphic equilibrium but cannot reach fixation. Such a polymorphic equilibrium is neutrally stable, with the implication that the SE strategy will be eliminated when the exogenous costs of learning are introduced.

Result 2: If at least one of the three conditions (1), (2), or (3) of Result 1 does not hold, but conditions (4) and (5) are satisfied, either the GE strategy or the S strategy may be fixed.

Result 3: Nevertheless, with middle migration and symmetrical individual learning (Table 2) or with late migration (Tables 3 and 6), the G strategy is fixed when Δy≥ 3 regardless of the migration rate.

Result 4: When the migration rate, m, is varied whereas the other parameters (d, c, Δy) are held constant, the six tables suggest a threshold below and above which two different strategies are fixed (polymorphism is also possible, see Result 5 below). However, simulations from alternative initial conditions reveal that there may sometimes be a range of transitional migration rates that supports bistability.

In the tables, a pair of vertically placed daggers () indicates the existence of a threshold, whereas a pair of vertically placed asterisks (**) indicates the presence of bistability for a range of migration rates. For example, when Δy= 2 in Table 2B, a threshold exists between m= 0.12 and m= 0.13 such that strategy G is fixed below and strategy GE is fixed above this threshold. On the other hand, when Δy= 2 in Table 2A, bistability occurs for migration rates between and extending beyond m= 0.16 and m= 0.17.

Bistability is readily identified when simulations from alternative initial conditions, but at the same migration rate, result in the fixation of two different strategies. On the other hand, thresholds are difficult to demonstrate numerically. Near a threshold migration rate, we expect that selection will be relatively weak and hence that convergence to equilibrium will be comparatively slow. This criterion was used to deduce thresholds.

Result 5: With early migration (Tables 1 and and 4), a threshold value of the migration rate apparently exists, at which a neutrally stable polymorphism of the G strategy and the S strategy is observed. For migration rates below and above this threshold, the G strategy and the S strategy, respectively, are fixed. This situation is indicated by three vertically placed daggers () as in Table 1A at m= 0.009, m= 0.01, and m= 0.011, where m= 0.01 is the threshold migration rate. Convergence is extremely slow at migration rates near the threshold value, e.g., m= 0.01 ± 0.00001.

Result 6: With early migration, dependence on the migration rate is identical for 2 ≤Δy≤ 4 in each of Tables 1 and 4A and for 3 ≤Δy≤ 4 in Table 4B. This includes the rate of convergence and the frequencies of the G strategy and the S strategy at polymorphic equilibrium. These results generalize to the case of Δy > 4, so that for each table the pattern is maintained for all distances between environmental states.

Result 7: Tables 2 and 3 with symmetrical individual learning and with middle or late migration are identical, because the mature phenotype distributions are the same whether migration occurs before or after symmetrical individual learning.

Discussion

SE is a learning strategy that combines accurate social learning with exploratory individual learning in that order (Borenstein et al. 2008; see also Boyd and Richerson 1985). Arguably, it is one of the few plausible learning strategies that can support cumulative culture. In this article, we investigated the conditions that permit the evolution of the SE strategy in an environmentally heterogeneous two-island model. Our findings based on extensive numerical work are summarized above as Result 1. Let us provide an intuitive justification for these results.

First, we note Result 0 that the G strategy will be fixed when the migration rate is low or when there is wide environmental overlap (Δy= 1). (This result has also been justified mathematically.) Condition (4) and part of condition (5) immediately follow.

Second, individual learning must be asymmetrical, which is condition (2). When individual learning is symmetrical, the SE strategy yields the nonsurviving mature phenotypes y1w− 1 and y2+w+ 1 in environmental states y1 and y2, respectively (Borenstein et al. 2008). By contrast, when individual learning is asymmetrical, the SE strategy can bias exploration toward the appropriate environmental state (Fig. 1) and thus (partly) avoid this endogenous mortality cost.

Third, we can rule out late migration. This is because, when individual learning precedes migration, immigrants lose the advantage associated with asymmetrical individual learning noted directly above, which is the ability to adjust their mature phenotypes toward the new environmental state. Early migration is excluded for a different reason. When social learning occurs after migration, the S strategy suffers only the mortality associated with the exogenous cost of social learning, because both residents and immigrants alike acquire only those phenotypes that are viable (in the environmental state in which selection takes place). The SE strategy is also burdened with the exogenous cost of individual learning. Together, these considerations imply condition (3), that is, middle migration.

Fourth, even with middle migration, if the distance between the environmental states of the two sites is too large (specifically, Δy > 4, under our assumptions b= 1 and w= 1), none of the immigrant phenotypes are viable. Hence, the remaining part of condition (5) follows.

Finally, condition (1) reflects the utility of the GE strategy, which is more flexible than the G strategy and enjoys a selective advantage over the S and SE strategies when the exogenous cost of social learning is relatively large (e.g., compare Table 5).

Let us now compare our model and/or its predictions with alternative but related models of the evolution of learning. It is natural to begin with the temporal environmental fluctuation model of Borenstein et al. (2008), of which the current model is the spatial analog. Immediately, we notice a major difference. In the model of Borenstein et al. (2008), the SE strategy with symmetrical individual learning can be fixed when the environmental states are contiguous (Δy= 3). This is true, as shown in their discussion, when the exogenous costs of learning are taken into account. In the current model, by contrast, the SE strategy is selected for only when individual learning is asymmetrical (Table 5A).

We believe the underlying reason for this discrepancy is that in the temporal version only one of two environmental states exists at any one time, whereas in the spatial version both environmental states coexist. Hence, in the former case a strategy can persist only if it has positive viability, no matter how small, in both environmental states. Under certain conditions, the SE strategy is the only strategy that satisfies this requirement (because it produces the mature phenotypes y1+w+ 1 and y2w− 1, which can survive an environmental change from state y1 to y2 and from y2 to y1, respectively). In the latter case, such a small advantage is not sufficient to compensate for the endogenous costs noted above.

“Critical social learner” (Enquist et al. 2007) is another strategy that combines social learning and individual learning. The two types of learning occur in the same order as for the SE strategy (and the “cultural transmission” strategy of the guided variation model, Boyd and Richerson 1985). However, the behavior that is acquired by social learning is either unmodified by individual learning (if judged to be adaptive) or subsequently ignored (if judged to be maladaptive), so that there is no opportunity for cumulative behavioral modification.

In the model of Boyd and Richerson (1988), individual learning precedes social learning. Organisms resort to social leaning only when the information obtained by “experimental” individual learning is indecisive. Such a learning strategy is likely appropriate in many situations. However, as in the case of critical social learner, it provides no opportunity for cumulative behavioral modification. Moreover, it is formally equivalent to a mixed strategy, which entails a trade-off between social learning and individual learning (Wakano and Aoki 2007).

Lehmann and Feldman (2009) are interested in the increase in the quantity and variety of culturally transmitted elements that characterize human technological advance. With this in mind, they investigate the evolution of a learning strategy called “producer,” which is able to generate innovations in proportion to the number of cultural elements acquired from the parental generation. However, the innovations attributable to each producer are not necessarily related, in intention or function, to the extant cultural elements that it has absorbed. The SE strategy, on the other hand, uses individual learning to effect incremental changes in the socially learned behavioral phenotype, which seems to be more consonant with the description of cumulative culture due to Tomasello (1999, p. 512): “modifications to an artifact or a social practice made by one individual or group of individuals often spread within the group, and then stay in place until some future individual or individuals make further modifications.” The two approaches clearly complement each other.

As noted in the Introduction, these four models (Boyd and Richerson 1988; Enquist et al. 2007; Borenstein et al. 2008; Lehmann and Feldman 2009), which combine social learning and individual learning in biologically relevant ways, are preferable to the standard models in which the two types of learning are viewed as mutually exclusive competing strategies. Nevertheless, the standard models possess a simpler structure and have the advantage of being analytically more tractable. Thorough analyses of the standard model are given by Aoki et al. (2005) (see also Feldman et al. 1996; Wakano et al. 2004) for the case of temporal environmental variability and by Aoki and Nakahashi (2008) for the case of spatial environmental variability. It is of interest to relate the predictions of the current model to those obtained from these standard models to see whether the latter remain valid and useful.

Aoki et al. (2005) posit three pure strategies: “innate” (behavior determined genetically), social learner, and individual learner. Innate and social learner may behave either adaptively or maladaptively, but individual learner always acquires the adaptive behavior. The environment changes periodically or probabilistically between generations, never reverting to an earlier state (infinite environmental states model). The (additive) costs of maladaptive behavior, social learning, and individual learning are s, d, and c, respectively. Then, assuming d < c < s, it is shown that the strategies innate, social learner, and individual learner are favored by natural selection when environmental changes occur at long, intermediate, and short intervals, respectively.

In the current model, the environment is assumed to vary spatially rather than temporally. As pointed out by Boyd and Richerson (1985, 1988, 1995; see also Aoki and Nakahashi 2008), a higher rate of migration between the environmentally heterogeneous sites of a subdivided population is analogous to a greater instability of the temporally changing environment. In the columns labeled Δy= 2 and Δy= 3 of Table 5A, we notice a pattern similar to that observed by Aoki et al. (2005). Namely, the G strategy, the S strategy, and the SE strategy are fixed under natural selection at low, intermediate, and high migration rates, respectively. Clearly, the G strategy is equivalent to innate, and the S strategy is synonymous with social learner. Hence, if the SE strategy can be matched with individual learner, the predictions of the two models correspond, at least in this instance.

Aoki and Nakahashi (2008) posit two pure strategies: social learner and individual learner. An arbitrary number of sites, each with a different environment, are connected by migration (island model, circular and linear stepping stone models). Migration occurs after social learning but before individual learning, which coincides with the middle migration case of the current model. Social learner may behave adaptively or maladaptively, but individual learner always behaves adaptively. It is shown that individual learner is more likely to persist (fixation or polymorphism) at higher migration rates, which is again consistent with Table 5A of the current model.

We conclude that the standard models can generate valid and useful results. However, the generality of these results must be viewed with caution, because as Tables 1–4, 5B, and 6 show, the dependence on the migration rate can take many different forms. For example, in an apparently recurrent pattern, the G strategy, the GE strategy, and the S strategy are fixed at low, intermediate, and high migration rates, respectively (Tables 2B, 3B, 4B, and 6B). Thus, the GE strategy, which includes an individual learning component, may be advantageous in relatively stable environments, whereas the S strategy is advantageous in relatively unstable environments, contrary to the consensus view (Laland et al. 2000; Richerson and Boyd 2000; Alvard 2003; Henrich and McElreath 2003).

Finally, we briefly discuss the implications of the current model for the evolution of SE in humans (Homo sapiens). Recall that our motivation for considering SE is that it is one of the few plausible learning strategies supportive of cumulative culture. Although culture apparently exists among nonhuman animals, cumulative culture may be limited to humans. Interestingly, cumulativeness is perhaps lacking from the lithic tradition of even our nearest hominid relatives, the extinct Neandertals (Homo neanderthalensis) (Akazawa et al. 1998; Klein 2009). It is beyond the scope of this discussion to argue why the capacity for cumulative culture evolved in humans but not in nonhuman animals including the Neandertals. Rather, our modest goal here is to suggest the kinds of information on the (ancestral) human condition that are relevant to testing some aspects of the current model.

The first prediction concerns the exogenous cost of social learning, which must be small. If language had emerged prior to the separation of the human and Neandertal lineages (Krause et al. 2007), it is possible that unintended errors in social learning may have been minimized. Hence, the exogenous cost of social learning may have been reduced to a level compatible with the evolution of SE.

The second prediction is that individual learning must be asymmetrical, resulting in behavioral adjustments toward those that “work better” in the given environment. This point agrees with the conventional definition of individual learning and so will not be pursued further.

The third prediction to evaluate is that migration must occur after social learning but before individual learning of behaviors that affect fitness. Humans often emigrate as young adults after absorbing the culture of their native society. Hence, some social learning usually precedes migration. The question is whether they adjust their behavior after migration primarily by individual learning. If social learning is used instead, the S strategy but not the SE strategy may evolve. Put more generally, an SE strategy that evolves in a spatially heterogeneous environment is predicted to show a life-cycle stage (or age) dependent reliance on the two types of learning. Foraging theory is another area in which the value of learning is being investigated and an effect of life history has been noted (Eliassen et al. 2007).

The fourth prediction is that the migration rate must be relatively high. We have already noted that a model of a structured population inhabiting environmentally heterogeneous sites may apply to our African ancestors (Potts 1998; Harding and McVean 2004; Garrigan and Hammer 2006). However, Garrigan and Hammer (2006) suggest a low migration rate between demes, based on the distribution of coalescence times for various genes. Fortunately, this is not a fatal discrepancy. Recall that in Table 5A, the exogenous costs of social learning and individual learning were set to d= 0.01 and c= 0.1, respectively. If the latter is also small, say c= 0.01, then SE can evolve at low migration rates.

The fifth prediction places a restriction on the degree of environmental overlap—narrow overlap, contiguous, and one-removed (2 ≤Δy≤ 4) are permissible—between the two sites. Hence, we require data on the extent of spatial environmental heterogeneity. However, this result cannot be interpreted in isolation from our assumptions regarding tolerance (w= 1), breadth of individual learning (b= 0, 1), and the rectangular shape of the viability function. In addition, with more than two sites each in a different environmental state, we expect to observe a dependence on the pattern of migration, for example, island, stepping stone, as well.

Borenstein et al. (2008) have shown that SE can evolve in a temporally fluctuating environment. The current article shows that conditions for the evolution of this learning strategy in a spatially heterogeneous environment may be more stringent. In actuality, ancestral human and other hominid populations likely experienced a combination of temporal and spatial environmental variability. It remains to be seen how the two forms of environmental variability might interact.


Associate Editor: M. Doebeli

ACKNOWLEDGMENTS

I thank Y. Taguchi for discussion. Research was supported in part by Monbukagakusho grant 17102002.

Appendices

Appendix 1

VIABILITIES OF THE FOUR GENERIC STRATEGIES AT SITE I WHEN THERE IS NO MIGRATION

For simplicity, we suppress the exogenous costs of learning in the following derivations. Then, from equations (2), (3), and (9) when m= 0

image((A1))

The viability of the generic G strategy is

image((A2))

and 0 otherwise.

The viability of the generic S strategy is

image((A3))

The viability of the generic GE strategy with symmetrical individual learning is

image((A4))

and 0 otherwise.

The viability of the generic GE strategy with asymmetrical individual learning is

image

Hence

image

if gyi− 1,

image

if g=yi, and

image

if gyi− 1. Thus,

image((A5))

and 0 otherwise.

The viability of the generic SE strategy with symmetrical individual learning is

image((A6))

Finally, the viability of the generic SE strategy with asymmetrical individual learning is

image((A7))

Appendix 2

VIABILITIES OF THE FOUR GENERIC STRATEGIES AT SITE I DO NOT DEPEND ON THE MIGRATION RATE IN THE CASE OF EARLY MIGRATION

Substituting equation (11) into equation (9), canceling the factor φ*t (g, k, b; i) in the numerator and the denominator, and suppressing the exogenous costs of learning yields equation (A1). That is, in the case of early migration, equation (A1) holds for all migration rates.

Appendix 3

VIABILITY OF GENERIC STRATEGY G AT SITE I DOES NOT DEPEND ON THE MIGRATION RATE

In Appendix 2, we have already given the proof for the case of early migration.

For the case of middle migration, substituting equation (10) into equation (9) and suppressing the exogenous costs of learning gives

image((A8))

From equation (5)

image

where ji, which on substitution of equation (2) reduces to

image((A9))

Finally, substituting equation (A9) into equation (A8) and noting equation (15) yields equation (A2).

For the case of late migration, from equation (6), (3), (2), and (15)

image((A10))

Hence, substituting equation (A10) into equation (9) and suppressing the exogenous costs of learning yields

image

which is identical to equation (A2).

Ancillary