Theoretical Population Biology 100 (2015) 6–12
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Modeling abrupt cultural regime shifts during the Palaeolithic and Stone Age Kenichi Aoki Organization for the Strategic Coordination of Research and Intellectual Properties, Meiji University, Nakano 4-21-1, Nakano-ku, Tokyo 164-8525, Japan
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Article history: Received 23 September 2014 Available online 7 December 2014 Keywords: Catastrophic bifurcation Innovation Carrying capacity Creative explosion Toolkit diversity Neural hypothesis
abstract The coupled dynamics of the size and the mean cultural/technological level of a population, with positive feedback between these two variables, is modeled in the Malthusian–Boserupian framework. Bifurcation diagrams, with innovativeness or the cultureless carrying capacity as the parameter, show that abrupt transitions in the mean cultural level are possible. For example, a gradual evolutionary change toward greater innate innovativeness would produce an associated gradual increase in mean cultural level, until a threshold is crossed that triggers an abrupt cultural regime shift. Hence, the model may help explain the apparently sudden and dramatic efflorescences of Palaeolithic/Stone Age culture during the Late Pleistocene, without having to invoke major contemporaneous genetic changes in cognition. The results of statistical studies on the association between population size and toolkit diversity among ethnographic societies are also discussed. © 2014 Elsevier Inc. All rights reserved.
1. Introduction There are two contrasting views on how population size and culture/technology are causally related. For Malthus, ‘‘population [size] equilibrates with resources at some level mediated by technology’’, whereas for Boserup, ‘‘technological change is itself spurred by increases in population [size]’’ (Lee, 1986, p. 96). In other words, the Malthusian position is that population size is limited by the available technology, and the Boserupian one is that technological change is dependent on population size. In fact, ‘‘the two theories are not contradictory, but rather complementary’’ (Lee, 1986, p. 96), and a theory of cultural/technological change should incorporate the reciprocal effects of population size on culture/technology, and vice versa. Many Palaeolithic archaeologists and anthropologists currently emphasize the Boserupian perspective in interpreting ‘‘sudden’’ and ‘‘dramatic’’ changes in stone tools or other cultural artifacts during the Late Pleistocene, in particular the ‘‘creative explosions’’ (Kuhn, 2012) of the African late Middle Stone Age and the European Upper Palaeolithic (e.g. Shennan, 2001; Henrich, 2004; Kline and Boyd, 2010; Zilhão et al., 2010; Mesoudi, 2011; Clark, 2011 and Kuhn, 2013). I take the liberty here and below – consistent with Lee (1986, p. 97) – of using the rubric ‘‘Boserupian’’ to indicate the directionality of the arrow of causation noted above,
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.tpb.2014.11.006 0040-5809/© 2014 Elsevier Inc. All rights reserved.
without distinguishing among the various possible mechanisms or processes. In fact, theoretical studies have repeatedly shown that population size can have a large effect on cultural evolutionary rate and cultural diversity (e.g. Shennan, 2001; Henrich, 2004; Strimling et al., 2009; Mesoudi, 2011; Lehmann et al., 2011; Aoki et al., 2011; Kobayashi and Aoki, 2012; Aoki, 2013 and Fogarty et al., 2015), as can interconnectedness of subpopulations (e.g. Powell et al., 2009; Perreault and Brantingham, 2011 and Premo, 2015). Transmission chain experiments conducted in the laboratory also provide some support for a link between population size and cultural complexity (e.g. Derex et al., 2013; Muthukrishna et al., 2014 and Kempe and Mesoudi, 2014; but see Caldwell and Millen, 2010). However, archaeological evidence on the role of demographic factors in cultural evolution is inconclusive or even contradictory. Two recent studies of Late Pleistocene South Africa are particularly relevant. Clark (2011) looked for signatures of population growth/demographic stress in an increase of diet breadth (e.g. the use of non-preferred prey animals), obtaining some support for an association with the heightened creativity of Howieson’s Poort. But, as Clark (2011) is careful to note, this association is open to an alternative interpretation, namely that rapid cultural change produced new tools, which were used to exploit novel resources. Klein and Steele (2013) (see also Klein, 2008, Box 1) observed that edible shellfish remains from Middle Stone Age middens are significantly larger than those from Later Stone Age middens. If shellfish size reflects human collection intensity, then this finding suggests that the precocious appearance of modern behaviors in the Still Bay and
K. Aoki / Theoretical Population Biology 100 (2015) 6–12
Howieson’s Poort may not have been associated with population growth. In addition, statistical analyses of ethnographic hunter–gatherers have failed to show an association between population size and the number of food-getting tool types (e.g. Collard et al., 2005 and Read, 2006). However, ethnographic food-producing societies (e.g. small-scale farmers and herders) do conform to the theoretical prediction that population size and toolkit diversity should be positively correlated (Kline and Boyd, 2010; Collard et al., 2013). Details are given later. Possible explanations for these contrasting results have been suggested, including higher degrees of specialization in the latter societies. A fundamental problem in human evolution is how to account for an apparently abrupt cultural change, without invoking a major genetic change in cognition (e.g. innovativeness), for which there is at present no strong evidence (Klein, 2008). In spite of the negative results of some empirical studies, it is clearly worthwhile to investigate theoretically the joint dynamics of culture and population size. However, it is difficult to discern what their explicit mathematical form might be. In a paper that presages the recent archaeological/anthropological discussions, Lee (1986) presents a semi-quantitative graphic model for population size and technology that synthesizes the contrasting viewpoints of Malthus and Boserup. He demonstrates the existence of alternative stable equilibria and/or regimes – a small population with a low technology and a large population with a high technology – and the possibility of transitions between them. Recently, Richerson and Boyd (2013) discussed the importance of adopting such an interactive approach in understanding progressive and regressive cultural changes during the Palaeolithic. ‘‘Perhaps toolkit complexity waxed and waned with the demographic fortunes of populations subject to highly variable conditions. . . . Perhaps . . . human populations were bistable. A high population density equilibrium would generate a fancy technology and . . . it could maintain high population density. A small population . . . would have a simple toolkit and a slow response to variation and hence would remain small’’. (pp. 290–291). Richerson and Boyd (2013) also note the possibility of hysteresis, whereby forward and backward transitions between alternative stable regimes occur under different exogenous conditions. See also Richerson et al. (2009). In a paper that predates Richerson and Boyd (2013), Ghirlanda and Enquist (2007) posit a highly specific functional form for the endogenous effect of population size on the ‘‘amount of culture’’, and vice versa. In particular, innovations are assumed to be produced in proportion to population size, and the carrying capacity is proportional to the amount of culture. Their model predicts either a stable equilibrium in the two variables, population size and amount of culture, or an explosive increase of both. The outcome is determined by three parameters, one of which can be regarded as a measure of innovativeness. However, their model does not yield bistability, for the reason explained later. The goal of the present paper is to describe and analyze a minimal dynamical model – in the spirit of Ghirlanda and Enquist (2007) – for population size and cultural level (or cultural complexity) that instantiates the verbal model of Richerson and Boyd (2013). In the interest of simplicity, the model makes arbitrary assumptions without empirical foundation, except perhaps in a qualitative sense. But even so it is not amenable to the kind of thorough treatment that is possible, say, for the spruce budworm model (e.g. Murray, 1989, pp. 4–8). Hence, we resort to numerical examples to demonstrate the possibility of bistability, ‘‘catastrophic bifurcations’’, and hysteresis (Scheffer and Carpenter, 2003). These numerical examples are selective, since my purpose is to demonstrate the possibility, not the likelihood, of their occurrence. Nevertheless, the analyses suggest that bistability is observed only
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within a limited range of parameter values. Within such a limited parametric range, we have two regimes – a small population at a low cultural level and a large population at a high cultural level – and the forward and backward shifts between these locally stable regimes follow different paths. Importantly, transitions between alternative stable equilibria may be sudden on an archaeological timescale. Theoretical results are obtained that may help in interpreting the creative explosions of the Palaeolithic/Stone Age. Of particular interest is the prediction that a gradual evolutionary increase of innovativeness can eventually trigger a saltational increase in cultural level. The threshold for innovativeness would depend on other conditions such as environmental productivity. This scenario is not inconsistent with the ‘‘neural hypothesis’’, a recent version of which invokes ‘‘a neural change that promoted the extraordinary modern human ability to innovate’’ (Klein, 2008, p. 271). However, the neural change would not, we argue, be attributable to just one ‘‘fortuitous mutation’’ in a major gene 50,000 years ago. 2. Model Assume the Henrich (2004) model of directly-biased cultural transmission, with the cultural level of an individual given by z – for example, a quantitative measure of skill or cultural trait diversity – and the mean cultural level of the population given by z¯ . In this discrete-generations model, each of the N newborns simultaneously and independently tries to copy the individual of the parental generation with the maximal value of z, which we write as zmax . The actual z value acquired by each newborn follows the Gumbel distribution. We modify the Henrich model, as done by Mesoudi (2011), so that the negative deviation of the mode of this Gumbel distribution from zmax is a function of z¯ , specifically −α¯z , where α > 0. In other words, we are assuming that it becomes increasingly more difficult for a newborn to improve on the cultural level of its exemplar as the mean cultural level of the population increases. This requires us to set z¯ > 0. Then, approximating the original difference equation (Mesoudi, 2011, Eq. (2)) by a differential equation, we have dz¯
= −α¯z + β (ε + log N ) , (1) dt where β > 0 is a measure of the dispersion of the Gumbel distribution and ε ≈ 0.577 is Euler’s constant. The Henrich (2004) model and its extensions (e.g. Powell et al., 2009; Mesoudi, 2011 and Kobayashi and Aoki, 2012) are widely used as representations of cultural evolution among hunter–gatherers, and that is my reason for adopting it. Eq. (1) entails that ddtz¯ is more likely to be positive when N is large. A larger population facilitates an increase in mean cultural level, because the maximal value of z in the offspring generation is then probabilistically more likely to exceed that of the parental generation, zmax . Hence, the mechanism by which population size drives cultural change may not be what Boserup had in mind (e.g. Lee, 1986 and Shennan, 2002), but the arrow of causation points in the same direction. Next, assume the logistic model of population growth, where the carrying capacity, M (¯z ), is a sigmoid function of z¯ with an inflection point at z ∗ > 0. Specifically, dN dt
= rN 1 −
N M (¯z )
,
(2a)
where M (¯z ) = K + D
∗ ec (¯z −z )
(2b) ∗ . 1 + ec (¯z −z ) We can regard K as representing the carrying capacity of the ‘‘cultureless’’ state (z¯ → 0), provided cz ∗ is sufficiently large. On
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K. Aoki / Theoretical Population Biology 100 (2015) 6–12
the other hand, when culture explodes (z¯ → ∞), the carrying capacity saturates at K + D. Parameter c measures the sharpness of the transition between the low and high carrying capacities. In writing Eqs. (1) and (2) we have assumed that the cultural level of an individual, z, does not contribute to the biological fitness of that individual, perhaps because resources are shared, but that the mean cultural level of a population, z¯ , determines – has a positive effect on – the carrying capacity of the population. We should also note that, rigorously speaking, N in Eq. (1) refers to the population size of the offspring generation after reproduction (Kobayashi and Aoki, 2012), whereas N in Eq. (2) gives the population size in the parental generation before reproduction. However, the approximation is acceptable if we assume, as is done here, that the intrinsic growth rate, r > 0, is small. 3. Existence of alternative stable equilibria Make the transformation of variable x = c (¯z − z ∗ ), which yields dx dt
= −α x − α cz + β c (ε + log N ), ∗
(3)
Fig. 1. Phase plane for the coupled dynamics with the transformed cultural level, x = c (¯z − z ∗ ), on the horizontal axis and the population size, N, on the vertical axis. Parameter values in this numerical example are λ = α/β = 0.2, c = 6, z ∗ = 18, K = 10, and D = 30, which satisfy the necessary condition, inequality (A.4), for the existence of three valid equilibria. The exponential curve is the ‘‘Boserup line’’ along which dx = 0, and the sigmoid curve is the ‘‘Malthus line’’ along which dt dN = 0 (Lee, 1986). For these parameter values, the two null clines do in fact dt intersect three times to yield three valid equilibria. The two outside equilibria are locally stable, whereas the middle equilibrium is unstable, which entails bistability.
and dN dt
= rN 1 −
N x
e K + D 1+ ex
.
(4)
ˆ which must The equilibria of the coupled system, xˆ and N, dN simultaneously satisfy dx = 0 and = 0, are given by the dt dt intersection(s) of the exponential curve N (x) = meλx/c ,
(5)
where λ = α/β and m = e N ( x) = K + D
ex 1 + ex
λz ∗ −ε
, with the sigmoid curve
.
(6)
(Incidentally, Eq. (5) is similar to Case 4(d) of Lee, 1986.) Hence, xˆ solves f (x) ≡ K + D
ex 1 + ex
λ/c − m ex = 0.
(7)
Appendix A shows that Eq. (7) yields one and only one equilibrium if λc ≥ 1. Moreover, λc < 1 is a necessary condition for the existence of three equilibria. Fig. 1 depicts the phase plane for our two-variable model, with the transformed cultural level, x = c (¯z − z ∗ ), on the horizontal axis and population size, N, on the vertical axis. In this numerical example, the exponential curve which is a plot of Eq. (5) and the sigmoid curve which is a plot of Eq. (6) intersect three times to yield three valid equilibria – zˆ¯ > 0, which entails xˆ > −cz ∗ – and to partition the phase plane into six regions. The directions of change of x and N in the six regions of the phase plane are given by Eqs. (3) and (4) and indicated approximately by the short arrows. They show that the two outside equilibria are both locally stable, whereas the middle equilibrium is unstable. Hence, we have bistability in this case. The locally stable equilibria on the right and left represent a large population at a high cultural level and a small population at a low cultural level, respectively. Transitions between the alternative stable equilibria can occur endogenously, following a large perturbation of population size. For example, suppose the large population at the high cultural level is decimated by a natural disaster. This event is represented in Fig. 2 by a vertical downward displacement from the stable equilibrium on the right. Numerical solution of Eqs. (3) and (4) from this initial state yields a trajectory, as shown in Fig. 2, that converges to the stable equilibrium on the left. The reason why the
Fig. 2. Numerical solutions of the coupled dynamics are plotted in the phase plane, showing convergence to the alternative equilibrium in a bistable situation after a large perturbation of population size. Parameter values in this numerical example are α = 1, β = 5 (i.e. λ = 0.2), c = 6, z ∗ = 18, K = 10, and D = 30, yielding three valid equilibria as in Fig. 1. The two locally stable equilibria are indicated by filled circles, and the unstable equilibrium by an open circle. Small dots trace out two trajectories. The lower trajectory shows what may happen after a sudden large drop in population size, and the upper trajectory illustrates the possible consequence of a sudden large rise in population size.
basin of attraction of this alternative equilibrium is large is that we are assuming a small value of the intrinsic growth rate, r, relative to the parameters, α and β , governing cultural change. See Richerson et al. (2009) for a thoughtful review of timescales. Similarly, the merger of two small populations at the low cultural level can result in a transition to the high cultural level if the combined population is large enough. I have already mentioned that there is no empirical foundation for the specific functional forms assumed in Eqs. (1) and (2). It is necessary to evaluate how these choices affect the possibility of bistability and catastrophic bifurcation. This is done in Appendix B. 4. Innovativeness as the exogenous condition Innovativeness can be defined in the Henrich (2004) model as the probability, φ , that the value of z acquired by a newborn exceeds zmax (Kobayashi and Aoki, 2012). For our modified model, we have
φ = 1 − exp (− exp (−α¯z /β)) .
(8)
Hence, smaller values of λ = α/β correspond to greater innovativeness.
K. Aoki / Theoretical Population Biology 100 (2015) 6–12
Fig. 3. Bifurcation diagram showing the dependence of xˆ = c (zˆ¯ − z ∗ ) on parameter λ. Smaller values of λ correspond to greater innovativeness (see text for details). Other parameter values are the same as in Fig. A.1. The solid and broken lines connect the locally stable equilibria and the unstable equilibria, respectively. There are two alternative locally stable equilibria for each value of λ in the approximate range between 0.1677 and 0.2037. Outside this interval, there is one globally stable ˆ obtained by equilibrium. To each value of xˆ , there exists an associated value of N, substitution of the former into Eq. (6).
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following scenario for changes in the mean cultural level when a gradual evolutionary increase in innovativeness occurs—i.e. when λ, assumed to be a quantitative genetic trait, gradually decreases. Initially, a small decrease in the value of λ will be associated with a small increase in the equilibrium mean cultural level, along the lower solid line representing equilibria at the lower level. The endogenous dynamics described by Eqs. (3) and (4) will then drive the system to the new equilibrium. However, when λ crosses a critical threshold – ≈0.1677 in the case illustrated here – known as a ‘‘catastrophic bifurcation’’, there will be a relatively rapid cultural regime shift, again driven by the endogenous dynamics, to an equilibrium at the higher cultural level represented by the upper solid line. The important point is that an incremental genetic change in innovativeness can trigger an abrupt behavioral transformation when a threshold is crossed. Put another way, an abrupt behavioral transformation need not be associated with a major genetic change. Empirically, this relatively sudden shift in cultural level may be observable in the archaeological record as a creative explosion. If for some reason innovativeness subsequently declines, the mean cultural level of the population will gradually decrease along the upper solid line of equilibria. Then, when λ crosses the other critical threshold – ≈0.2037 in the case illustrated here – a regime shift occurs to the lower stable line. The forward and backward switches follow different paths, i.e. we have hysteresis. 5. Cultureless carrying capacity as the exogenous condition
Fig. 4. Bifurcation diagram showing the dependence of xˆ = c (zˆ¯ − z ∗ ) on parameter K , which represents the carrying capacity of a population without culture (see text for details). Other parameter values are the same as in Fig. A.2. The solid and broken lines connect the locally stable equilibria and the unstable equilibria, respectively. There are two alternative locally stable equilibria for each value of K in the approximate range between 5.86 and 19.37. Outside this interval, there is one ˆ globally stable equilibrium. To each value of xˆ , there exists an associated value of N, obtained by substitution of the former into Eq. (6).
Fig. A.1 shows five exponential curves which are plots of Eq. (5) for five different values of parameter λ, with the two other parameters in that equation fixed at c = 6, z ∗ = 18. The sigmoid curve is the plot of Eq. (6) drawn for the fixed parameter values K = 10, D = 15. The intersection(s) of each exponential curve with the sigmoid curve define the possible equilibria of x = c (¯z − z ∗ ) and N for that particular value of λ. From top to bottom, λ = 0.21, 0.2037, 0.19, 0.1677, and 0.155. We observe that the number of equilibria changes from one to two to three, then back to two and one again as λ decreases, suggesting hysteresis. This is made explicit in Fig. 3, which shows how xˆ = c (zˆ¯ − z ∗ ) depends on λ. Eq. (7) was solved numerically to obtain the equilibrium values of the (transformed) mean cultural level. The two solid lines denote alternative locally stable equilibria – high and low cultural level regimes – whereas the broken line gives the unstable equilibria. Here, as in Fig. 4, the width of the parametric region of bistability will depend on the values of other parameters. My motivation for focusing on innovativeness is that it is likely to be an important aspect of hominid learning strategies that has been under positive natural selection (e.g. Feldman et al., 1996; Aoki and Feldman, 2014; see also Klein, 2008). Fig. 3 suggests the
Whether or not the diachronic or synchronic differences in cultural level that are seen in Late Pleistocene hominids can be attributed to differences in innovativeness, or more generally cognition, is currently a controversial issue. Less contentious is the possible consequence of an environmental change affecting the availability of food. We can use the cultureless carrying capacity, K , to represent environmental productivity. Fig. A.2 illustrates the effect on the number of equilibria of varying the parameter K . From top to bottom, K = 30, 19.37, 10, 5.86, and 2, with the other parameters fixed at λ = 0.205, c = 6, z ∗ = 18, D = 20. Fig. 4 gives the corresponding bifurcation diagram with K on the horizontal axis and xˆ = c (zˆ¯ − z ∗ ) on the vertical axis. Here again we observe sudden dramatic changes in the equilibrium mean cultural level at the catastrophic bifurcations. 6. Discussion We have described and analyzed a minimal model for the coupled dynamics of mean cultural level, z¯ , and population size, N. For the dynamics of the mean cultural level, we have adopted the Henrich (2004) model as modified by Mesoudi (2011). These dynamics entail a positive feedback such that the mean cultural level increases when it is low relative to population size and decreases when it is high relative to population size. Similarly, for the dynamics of the population size, we have assumed logistic growth in which the carrying capacity is an increasing function of mean cultural level. In addition, our model contains several parameters, which are of interest from an evolutionary or an ecological standpoint and which can produce catastrophic bifurcations when they are varied. The parameters we have specifically focused on are a measure of innovativeness, λ, and the carrying capacity of the cultureless state, K . As the analyses show, two locally stable equilibria of the mean cultural level and the population size can exist within a given intermediate range of either λ or K , whereas there is one globally stable equilibrium outside this range (Figs. 3 and 4, respectively). Within or outside of a parametric region of bistability, a small change in parameter value will result in a small
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K. Aoki / Theoretical Population Biology 100 (2015) 6–12
change in the values of the mean cultural level and population size at equilibrium. However, if a similarly small change in parameter value entails a threshold being crossed, there will be a relatively large shift in these equilibrium values followed by convergence to the new equilibrium. For example, innovativeness may have been under positive natural selection in hominids living in a temporally fluctuating environment (Boyd and Richerson, 1985; Feldman et al., 1996; Aoki and Feldman, 2014). Most evolutionary changes are gradual, and an evolutionary advance in innovativeness is not likely to have been the exception. Then, as innovativeness improved, there would have been an associated gradual increase in cultural level, until the threshold was crossed and an abrupt regime shift was induced. This theoretical argument provides an alternative explanation to the neural hypothesis (e.g. Klein, 2008), in terms of evolutionary changes in innovativeness, for the apparently sudden and dramatic changes in Palaeolithic cultures. Namely, an abrupt behavioral transformation need not be associated with a major genetic change. Hence, anthropologists and archaeologists may not be rewarded in their search for ‘‘adaptive shifts’’ underlying key advances in Palaeolithic/Stone Age cultures (e.g. Wynn, 2002). Alternatively, natural selection for greater innovativeness may have been transient, as during range expansion, particularly into a spatially variable environment (Aoki and Nakahashi, 2008; Wakano et al., 2011). In this case, an upward cultural regime shift may have been followed by a downward cultural regime shift after range expansion ceased. Still another possibility is that innovativeness in modern humans is facultative and waxes or wanes in response to necessity. As in the model of Lee (1986), it is in principle straightforward to extend our model to accommodate three or more locally stable equilibria. For example, we could replace the sigmoid function for the carrying capacity in Eq. (2) with a double sigmoid function. More simply, we could assume a multiple step function that equals K for z¯ < z1∗ , K + D1 for z1∗ ≤ z¯ < z2∗ , K + D2 for z2∗ ≤ z¯ < z3∗ , etc. A double sigmoid function would intersect the exponential curve Eq. (1) at most five times, in which case there would be three locally stable equilibria for population size and cultural level. Let us now briefly consider what can be said about statistical associations between population size and toolkit diversity, assuming that our model is relevant to the situation. Empirical studies (e.g. Collard et al., 2005; Read, 2006; Kline and Boyd, 2010 and Collard et al., 2013) of ethnographic societies report univariable (simple linear) regression coefficients and partial regression coefficients of toolkit diversity on population size. Note first that real-world societies will often deviate randomly from equilibrium. Hence, within the range of parameter values where only one equilibrium is predicted by our model, the partial regression coefficient (which controls for parameter values) is expected to be non-significant, although the univariable regression coefficient may be significant if the societies differ in parameter values. On the other hand, when two or more locally stable equilibria are predicted, and societies are to be found at or near these equilibria, a positive univariable regression coefficient and a positive partial regression coefficient are both expected, as when heterogeneous samples are pooled (see Fig. 1). Multiple regression analyses conducted by Collard et al. (2005) on their sample of hunter–gatherer populations, with toolkit diversity evaluated in three different ways as the dependent variable, yielded non-significant partial regression coefficients on population size. In other words, there was no observable association between population size and toolkit diversity. Read (2006) reached the same conclusion. The study by Collard et al. (2005) incorporated various control variables, including effective temperature as a measure of ‘‘risk’’. If ‘‘necessity is the mother of invention’’, then effective temperature may perhaps also be regarded as a measure
of facultative innovativeness in our model. Hence, with reference to Fig. 3 where the horizontal axis measures innovativeness, we might plausibly argue that these hunter–gatherer societies fall outside the parametric region of bistability and belong to the same cultural regime. For the food-producers, on the other hand, the univariable and partial regression coefficients of toolkit diversity on population size were both significantly positive. Moreover, none of the other independent variables in the multiple regression analyses yielded significant partial regression coefficients (Kline and Boyd, 2010; Collard et al., 2013). This may suggest that these small-scale farming and herding societies have been sampled from more than one cultural regime. The model presented here neglects the ‘‘organizational aspects of a society’’ (e.g. craft specialization, social stratification, political leadership), which are known to correlate with population size among ethnographic societies (Carneiro, 1967). It is easy to see how craft specialization might result in a diversification of tool types. Hence, when modeling societies that have ‘‘progressed beyond’’ egalitarian hunter–gatherers (Hewlett et al., 2011) characterized by a minimal division of labor by sex and age (Kaplan et al., 2000), it may be useful to introduce an intervening variable or variables, such as specialization, between population size and mean cultural level (complexity). Acknowledgments This work was begun during a visit to the Centre for Advanced Study in Oslo, Norway. I wish to express my sincere thanks to Bjørg Egelandsdal, Gro Amdam, Marije Oostindjer, and other members of the Centre for their hospitality during my stay. I benefited greatly from discussions with Ron Lee, with whom I shared an office, and with Shripad Tuljapurkar, whose office was just down the hall. I am grateful to Mayuko Nakamaru and Jeremy Kendal for directing me to the relevant literature, and to Alex Mesoudi, Laurent Lehmann and Stefano Ghirlanda for helpful comments on the manuscript. This work was supported in part by Monbukagakusho grant 22101004. Appendix A Set y = ex > 0. Then Eq. (7) is equivalent to
λ
λ
g (y) = K + (K + D)y − m y c +1 + y c
.
(A.1)
The first and second derivatives of Eq. (A.1) are g (y) = K + D − m ′
λ c
λ
+1 y + c
λ c
y
λ −1 c
,
(A.2)
and
λ
λ
g ′′ (y) = −m y c −2 c
λ c
λ +1 y+ −1 .
(A.3)
c
First assume λc ≥ 1. Then, g ′ (y) decreases monotonically from g (0) = K + D to −∞, and there exists one y∗ > 0 such that g ′ (y∗ ) = 0. Hence, g (y) increases from g (0) = K , attains a local maximum g (y∗ ) > 0, then decreases monotonically to −∞. Thus, there is one and only one equilibrium. ′
Second assume λc < 1. Then, limy→0 g ′ (y) = −∞ and g ′ (y) has
−λ one local maximum at y∗∗ = cc +λ > 0 i.e. where g ′ (y∗∗ ) = 0. If ′ g (y) < 0 for all y > 0, then there is one and only one equilibrium. However, if g ′ (y∗∗ ) > 0, then there may be as many as three ′
K. Aoki / Theoretical Population Biology 100 (2015) 6–12
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where γ , δ, r , a, b are positive constants. Eqs. (B.1) and (B.2) correspond to Eqs. (4a) and (4b), respectively, of Ghirlanda and Enquist (2007), but we have partly changed the notation to avoid confusion. The null clines of this system are N (X ) =
γ X, δ
(B.3)
and N (X ) = aX + b.
Fig. A.1. The effect of varying parameter λ on the number of equilibria. The five exponential curves (Boserup lines) are, from top to bottom, drawn for λ = 0.21, 0.2037, 0.19, 0.1677, and 0.155, respectively. We observe that the number of equilibria changes from one to two to three, then back to two and one again as λ decreases. Other parameters are fixed at c = 6, z ∗ = 18, K = 10, D = 15.
Fig. A.2. The effect of varying parameter K on the number of equilibria. The five sigmoid curves (Malthus lines) are, from top to bottom, drawn for K = 30, 19.37, 10, 5.86, and 2, respectively. We observe that the number of equilibria changes from one to two to three, then back to two and one again as K decreases. Other parameters are fixed at λ = 0.205, c = 6, z ∗ = 18, D = 20.
equilibria. Thus, a necessary condition for three equilibria is K +D m
>
λ
1+ c 1 − λc
Both are straight lines, so this model yields at most one valid equilibrium, ruling out bistability. If we replace the linear function for the carrying capacity in Eq. (B.2) by a sigmoid function of X analogous to Eq. (2), we can readily visualize the situation where the two null clines, given by Eq. (B.3) and the modified Eq. (B.4), will intersect three times. It is also clear that the two outside equilibria in this case will both be stable. Hence, the specific functional form assumed in Eq. (1) – specifically the concave log N term of the Henrich (2004) model – is not a necessary condition for bistability and catastrophic bifurcation. The dependence on N can be linear as in the Ghirlanda and Enquist (2007) model, or even convex as can be shown by graphic argument. On the other hand, several constraints apply to the functional form of the carrying capacity. To keep things simple, assume that the null cline defined by the dynamics of the cultural level or the amount of culture is given by a monotone function such as Eq. (5) or Eq. (B.3), respectively. Clearly, a step function or a differentiable function that approximates it is a suitable choice for the carrying capacity, as it can intersect the first null cline more than once. If we assume the carrying capacity to be a twice-differentiable function of the cultural level (or the amount of culture), it must possess an inflection point. Moreover, the second derivative must be positive to the left, and negative to the right, of this inflection point. The sigmoid function in Eq. (2) satisfies these requirements. However, the carrying capacity need not be bounded, nor does it have to be monotone increasing. The above considerations suggest the following minimal model for the coupled dynamics of the cultural level, z¯ , and the population size, N, that yields explicit conditions for bistability. We write dz¯ dt
1− λ c
.
(A.4)
= −γ z¯ + δ N ,
dN
= rN 1 −
Appendix B
where
The justification for Eq. (1) is that the Henrich (2004) model, on which it is based, has entered the literature as a convenient model of cultural evolution with directly biased social learning and a dependence on population size. Eq. (2) seems qualitatively reasonable, given that the carrying capacity of a population may increase with cultural level, but is also limited by non-cultural environmental factors. It is instructive to compare our model with the extensive growth model of Ghirlanda and Enquist (2007). In this latter model, the coupled dynamics for the amount of culture, X , and the population size, N, are given by
M (¯z ) =
dt and dN dt
= −γ X + δ N , = rN 1 −
(B.1)
N aX + b
,
(B.2)
(B.5)
and
dt
dX
(B.4)
K K +D
N
,
(B.6a)
for z¯ < z ∗ for z¯ ≥ z ∗ ,
(B.6b)
M (¯z )
and γ , δ, r , K , D, z ∗ are positive constants. Two equilibria exist and are both locally stable if and only if Kδ
γ
< z∗
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Theoretical Population Biology journal homepage: www.elsevier.com/locate/tpb
Modeling abrupt cultural regime shifts during the Palaeolithic and Stone Age Kenichi Aoki Organization for the Strategic Coordination of Research and Intellectual Properties, Meiji University, Nakano 4-21-1, Nakano-ku, Tokyo 164-8525, Japan
article
info
Article history: Received 23 September 2014 Available online 7 December 2014 Keywords: Catastrophic bifurcation Innovation Carrying capacity Creative explosion Toolkit diversity Neural hypothesis
abstract The coupled dynamics of the size and the mean cultural/technological level of a population, with positive feedback between these two variables, is modeled in the Malthusian–Boserupian framework. Bifurcation diagrams, with innovativeness or the cultureless carrying capacity as the parameter, show that abrupt transitions in the mean cultural level are possible. For example, a gradual evolutionary change toward greater innate innovativeness would produce an associated gradual increase in mean cultural level, until a threshold is crossed that triggers an abrupt cultural regime shift. Hence, the model may help explain the apparently sudden and dramatic efflorescences of Palaeolithic/Stone Age culture during the Late Pleistocene, without having to invoke major contemporaneous genetic changes in cognition. The results of statistical studies on the association between population size and toolkit diversity among ethnographic societies are also discussed. © 2014 Elsevier Inc. All rights reserved.
1. Introduction There are two contrasting views on how population size and culture/technology are causally related. For Malthus, ‘‘population [size] equilibrates with resources at some level mediated by technology’’, whereas for Boserup, ‘‘technological change is itself spurred by increases in population [size]’’ (Lee, 1986, p. 96). In other words, the Malthusian position is that population size is limited by the available technology, and the Boserupian one is that technological change is dependent on population size. In fact, ‘‘the two theories are not contradictory, but rather complementary’’ (Lee, 1986, p. 96), and a theory of cultural/technological change should incorporate the reciprocal effects of population size on culture/technology, and vice versa. Many Palaeolithic archaeologists and anthropologists currently emphasize the Boserupian perspective in interpreting ‘‘sudden’’ and ‘‘dramatic’’ changes in stone tools or other cultural artifacts during the Late Pleistocene, in particular the ‘‘creative explosions’’ (Kuhn, 2012) of the African late Middle Stone Age and the European Upper Palaeolithic (e.g. Shennan, 2001; Henrich, 2004; Kline and Boyd, 2010; Zilhão et al., 2010; Mesoudi, 2011; Clark, 2011 and Kuhn, 2013). I take the liberty here and below – consistent with Lee (1986, p. 97) – of using the rubric ‘‘Boserupian’’ to indicate the directionality of the arrow of causation noted above,
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.tpb.2014.11.006 0040-5809/© 2014 Elsevier Inc. All rights reserved.
without distinguishing among the various possible mechanisms or processes. In fact, theoretical studies have repeatedly shown that population size can have a large effect on cultural evolutionary rate and cultural diversity (e.g. Shennan, 2001; Henrich, 2004; Strimling et al., 2009; Mesoudi, 2011; Lehmann et al., 2011; Aoki et al., 2011; Kobayashi and Aoki, 2012; Aoki, 2013 and Fogarty et al., 2015), as can interconnectedness of subpopulations (e.g. Powell et al., 2009; Perreault and Brantingham, 2011 and Premo, 2015). Transmission chain experiments conducted in the laboratory also provide some support for a link between population size and cultural complexity (e.g. Derex et al., 2013; Muthukrishna et al., 2014 and Kempe and Mesoudi, 2014; but see Caldwell and Millen, 2010). However, archaeological evidence on the role of demographic factors in cultural evolution is inconclusive or even contradictory. Two recent studies of Late Pleistocene South Africa are particularly relevant. Clark (2011) looked for signatures of population growth/demographic stress in an increase of diet breadth (e.g. the use of non-preferred prey animals), obtaining some support for an association with the heightened creativity of Howieson’s Poort. But, as Clark (2011) is careful to note, this association is open to an alternative interpretation, namely that rapid cultural change produced new tools, which were used to exploit novel resources. Klein and Steele (2013) (see also Klein, 2008, Box 1) observed that edible shellfish remains from Middle Stone Age middens are significantly larger than those from Later Stone Age middens. If shellfish size reflects human collection intensity, then this finding suggests that the precocious appearance of modern behaviors in the Still Bay and
K. Aoki / Theoretical Population Biology 100 (2015) 6–12
Howieson’s Poort may not have been associated with population growth. In addition, statistical analyses of ethnographic hunter–gatherers have failed to show an association between population size and the number of food-getting tool types (e.g. Collard et al., 2005 and Read, 2006). However, ethnographic food-producing societies (e.g. small-scale farmers and herders) do conform to the theoretical prediction that population size and toolkit diversity should be positively correlated (Kline and Boyd, 2010; Collard et al., 2013). Details are given later. Possible explanations for these contrasting results have been suggested, including higher degrees of specialization in the latter societies. A fundamental problem in human evolution is how to account for an apparently abrupt cultural change, without invoking a major genetic change in cognition (e.g. innovativeness), for which there is at present no strong evidence (Klein, 2008). In spite of the negative results of some empirical studies, it is clearly worthwhile to investigate theoretically the joint dynamics of culture and population size. However, it is difficult to discern what their explicit mathematical form might be. In a paper that presages the recent archaeological/anthropological discussions, Lee (1986) presents a semi-quantitative graphic model for population size and technology that synthesizes the contrasting viewpoints of Malthus and Boserup. He demonstrates the existence of alternative stable equilibria and/or regimes – a small population with a low technology and a large population with a high technology – and the possibility of transitions between them. Recently, Richerson and Boyd (2013) discussed the importance of adopting such an interactive approach in understanding progressive and regressive cultural changes during the Palaeolithic. ‘‘Perhaps toolkit complexity waxed and waned with the demographic fortunes of populations subject to highly variable conditions. . . . Perhaps . . . human populations were bistable. A high population density equilibrium would generate a fancy technology and . . . it could maintain high population density. A small population . . . would have a simple toolkit and a slow response to variation and hence would remain small’’. (pp. 290–291). Richerson and Boyd (2013) also note the possibility of hysteresis, whereby forward and backward transitions between alternative stable regimes occur under different exogenous conditions. See also Richerson et al. (2009). In a paper that predates Richerson and Boyd (2013), Ghirlanda and Enquist (2007) posit a highly specific functional form for the endogenous effect of population size on the ‘‘amount of culture’’, and vice versa. In particular, innovations are assumed to be produced in proportion to population size, and the carrying capacity is proportional to the amount of culture. Their model predicts either a stable equilibrium in the two variables, population size and amount of culture, or an explosive increase of both. The outcome is determined by three parameters, one of which can be regarded as a measure of innovativeness. However, their model does not yield bistability, for the reason explained later. The goal of the present paper is to describe and analyze a minimal dynamical model – in the spirit of Ghirlanda and Enquist (2007) – for population size and cultural level (or cultural complexity) that instantiates the verbal model of Richerson and Boyd (2013). In the interest of simplicity, the model makes arbitrary assumptions without empirical foundation, except perhaps in a qualitative sense. But even so it is not amenable to the kind of thorough treatment that is possible, say, for the spruce budworm model (e.g. Murray, 1989, pp. 4–8). Hence, we resort to numerical examples to demonstrate the possibility of bistability, ‘‘catastrophic bifurcations’’, and hysteresis (Scheffer and Carpenter, 2003). These numerical examples are selective, since my purpose is to demonstrate the possibility, not the likelihood, of their occurrence. Nevertheless, the analyses suggest that bistability is observed only
7
within a limited range of parameter values. Within such a limited parametric range, we have two regimes – a small population at a low cultural level and a large population at a high cultural level – and the forward and backward shifts between these locally stable regimes follow different paths. Importantly, transitions between alternative stable equilibria may be sudden on an archaeological timescale. Theoretical results are obtained that may help in interpreting the creative explosions of the Palaeolithic/Stone Age. Of particular interest is the prediction that a gradual evolutionary increase of innovativeness can eventually trigger a saltational increase in cultural level. The threshold for innovativeness would depend on other conditions such as environmental productivity. This scenario is not inconsistent with the ‘‘neural hypothesis’’, a recent version of which invokes ‘‘a neural change that promoted the extraordinary modern human ability to innovate’’ (Klein, 2008, p. 271). However, the neural change would not, we argue, be attributable to just one ‘‘fortuitous mutation’’ in a major gene 50,000 years ago. 2. Model Assume the Henrich (2004) model of directly-biased cultural transmission, with the cultural level of an individual given by z – for example, a quantitative measure of skill or cultural trait diversity – and the mean cultural level of the population given by z¯ . In this discrete-generations model, each of the N newborns simultaneously and independently tries to copy the individual of the parental generation with the maximal value of z, which we write as zmax . The actual z value acquired by each newborn follows the Gumbel distribution. We modify the Henrich model, as done by Mesoudi (2011), so that the negative deviation of the mode of this Gumbel distribution from zmax is a function of z¯ , specifically −α¯z , where α > 0. In other words, we are assuming that it becomes increasingly more difficult for a newborn to improve on the cultural level of its exemplar as the mean cultural level of the population increases. This requires us to set z¯ > 0. Then, approximating the original difference equation (Mesoudi, 2011, Eq. (2)) by a differential equation, we have dz¯
= −α¯z + β (ε + log N ) , (1) dt where β > 0 is a measure of the dispersion of the Gumbel distribution and ε ≈ 0.577 is Euler’s constant. The Henrich (2004) model and its extensions (e.g. Powell et al., 2009; Mesoudi, 2011 and Kobayashi and Aoki, 2012) are widely used as representations of cultural evolution among hunter–gatherers, and that is my reason for adopting it. Eq. (1) entails that ddtz¯ is more likely to be positive when N is large. A larger population facilitates an increase in mean cultural level, because the maximal value of z in the offspring generation is then probabilistically more likely to exceed that of the parental generation, zmax . Hence, the mechanism by which population size drives cultural change may not be what Boserup had in mind (e.g. Lee, 1986 and Shennan, 2002), but the arrow of causation points in the same direction. Next, assume the logistic model of population growth, where the carrying capacity, M (¯z ), is a sigmoid function of z¯ with an inflection point at z ∗ > 0. Specifically, dN dt
= rN 1 −
N M (¯z )
,
(2a)
where M (¯z ) = K + D
∗ ec (¯z −z )
(2b) ∗ . 1 + ec (¯z −z ) We can regard K as representing the carrying capacity of the ‘‘cultureless’’ state (z¯ → 0), provided cz ∗ is sufficiently large. On
8
K. Aoki / Theoretical Population Biology 100 (2015) 6–12
the other hand, when culture explodes (z¯ → ∞), the carrying capacity saturates at K + D. Parameter c measures the sharpness of the transition between the low and high carrying capacities. In writing Eqs. (1) and (2) we have assumed that the cultural level of an individual, z, does not contribute to the biological fitness of that individual, perhaps because resources are shared, but that the mean cultural level of a population, z¯ , determines – has a positive effect on – the carrying capacity of the population. We should also note that, rigorously speaking, N in Eq. (1) refers to the population size of the offspring generation after reproduction (Kobayashi and Aoki, 2012), whereas N in Eq. (2) gives the population size in the parental generation before reproduction. However, the approximation is acceptable if we assume, as is done here, that the intrinsic growth rate, r > 0, is small. 3. Existence of alternative stable equilibria Make the transformation of variable x = c (¯z − z ∗ ), which yields dx dt
= −α x − α cz + β c (ε + log N ), ∗
(3)
Fig. 1. Phase plane for the coupled dynamics with the transformed cultural level, x = c (¯z − z ∗ ), on the horizontal axis and the population size, N, on the vertical axis. Parameter values in this numerical example are λ = α/β = 0.2, c = 6, z ∗ = 18, K = 10, and D = 30, which satisfy the necessary condition, inequality (A.4), for the existence of three valid equilibria. The exponential curve is the ‘‘Boserup line’’ along which dx = 0, and the sigmoid curve is the ‘‘Malthus line’’ along which dt dN = 0 (Lee, 1986). For these parameter values, the two null clines do in fact dt intersect three times to yield three valid equilibria. The two outside equilibria are locally stable, whereas the middle equilibrium is unstable, which entails bistability.
and dN dt
= rN 1 −
N x
e K + D 1+ ex
.
(4)
ˆ which must The equilibria of the coupled system, xˆ and N, dN simultaneously satisfy dx = 0 and = 0, are given by the dt dt intersection(s) of the exponential curve N (x) = meλx/c ,
(5)
where λ = α/β and m = e N ( x) = K + D
ex 1 + ex
λz ∗ −ε
, with the sigmoid curve
.
(6)
(Incidentally, Eq. (5) is similar to Case 4(d) of Lee, 1986.) Hence, xˆ solves f (x) ≡ K + D
ex 1 + ex
λ/c − m ex = 0.
(7)
Appendix A shows that Eq. (7) yields one and only one equilibrium if λc ≥ 1. Moreover, λc < 1 is a necessary condition for the existence of three equilibria. Fig. 1 depicts the phase plane for our two-variable model, with the transformed cultural level, x = c (¯z − z ∗ ), on the horizontal axis and population size, N, on the vertical axis. In this numerical example, the exponential curve which is a plot of Eq. (5) and the sigmoid curve which is a plot of Eq. (6) intersect three times to yield three valid equilibria – zˆ¯ > 0, which entails xˆ > −cz ∗ – and to partition the phase plane into six regions. The directions of change of x and N in the six regions of the phase plane are given by Eqs. (3) and (4) and indicated approximately by the short arrows. They show that the two outside equilibria are both locally stable, whereas the middle equilibrium is unstable. Hence, we have bistability in this case. The locally stable equilibria on the right and left represent a large population at a high cultural level and a small population at a low cultural level, respectively. Transitions between the alternative stable equilibria can occur endogenously, following a large perturbation of population size. For example, suppose the large population at the high cultural level is decimated by a natural disaster. This event is represented in Fig. 2 by a vertical downward displacement from the stable equilibrium on the right. Numerical solution of Eqs. (3) and (4) from this initial state yields a trajectory, as shown in Fig. 2, that converges to the stable equilibrium on the left. The reason why the
Fig. 2. Numerical solutions of the coupled dynamics are plotted in the phase plane, showing convergence to the alternative equilibrium in a bistable situation after a large perturbation of population size. Parameter values in this numerical example are α = 1, β = 5 (i.e. λ = 0.2), c = 6, z ∗ = 18, K = 10, and D = 30, yielding three valid equilibria as in Fig. 1. The two locally stable equilibria are indicated by filled circles, and the unstable equilibrium by an open circle. Small dots trace out two trajectories. The lower trajectory shows what may happen after a sudden large drop in population size, and the upper trajectory illustrates the possible consequence of a sudden large rise in population size.
basin of attraction of this alternative equilibrium is large is that we are assuming a small value of the intrinsic growth rate, r, relative to the parameters, α and β , governing cultural change. See Richerson et al. (2009) for a thoughtful review of timescales. Similarly, the merger of two small populations at the low cultural level can result in a transition to the high cultural level if the combined population is large enough. I have already mentioned that there is no empirical foundation for the specific functional forms assumed in Eqs. (1) and (2). It is necessary to evaluate how these choices affect the possibility of bistability and catastrophic bifurcation. This is done in Appendix B. 4. Innovativeness as the exogenous condition Innovativeness can be defined in the Henrich (2004) model as the probability, φ , that the value of z acquired by a newborn exceeds zmax (Kobayashi and Aoki, 2012). For our modified model, we have
φ = 1 − exp (− exp (−α¯z /β)) .
(8)
Hence, smaller values of λ = α/β correspond to greater innovativeness.
K. Aoki / Theoretical Population Biology 100 (2015) 6–12
Fig. 3. Bifurcation diagram showing the dependence of xˆ = c (zˆ¯ − z ∗ ) on parameter λ. Smaller values of λ correspond to greater innovativeness (see text for details). Other parameter values are the same as in Fig. A.1. The solid and broken lines connect the locally stable equilibria and the unstable equilibria, respectively. There are two alternative locally stable equilibria for each value of λ in the approximate range between 0.1677 and 0.2037. Outside this interval, there is one globally stable ˆ obtained by equilibrium. To each value of xˆ , there exists an associated value of N, substitution of the former into Eq. (6).
9
following scenario for changes in the mean cultural level when a gradual evolutionary increase in innovativeness occurs—i.e. when λ, assumed to be a quantitative genetic trait, gradually decreases. Initially, a small decrease in the value of λ will be associated with a small increase in the equilibrium mean cultural level, along the lower solid line representing equilibria at the lower level. The endogenous dynamics described by Eqs. (3) and (4) will then drive the system to the new equilibrium. However, when λ crosses a critical threshold – ≈0.1677 in the case illustrated here – known as a ‘‘catastrophic bifurcation’’, there will be a relatively rapid cultural regime shift, again driven by the endogenous dynamics, to an equilibrium at the higher cultural level represented by the upper solid line. The important point is that an incremental genetic change in innovativeness can trigger an abrupt behavioral transformation when a threshold is crossed. Put another way, an abrupt behavioral transformation need not be associated with a major genetic change. Empirically, this relatively sudden shift in cultural level may be observable in the archaeological record as a creative explosion. If for some reason innovativeness subsequently declines, the mean cultural level of the population will gradually decrease along the upper solid line of equilibria. Then, when λ crosses the other critical threshold – ≈0.2037 in the case illustrated here – a regime shift occurs to the lower stable line. The forward and backward switches follow different paths, i.e. we have hysteresis. 5. Cultureless carrying capacity as the exogenous condition
Fig. 4. Bifurcation diagram showing the dependence of xˆ = c (zˆ¯ − z ∗ ) on parameter K , which represents the carrying capacity of a population without culture (see text for details). Other parameter values are the same as in Fig. A.2. The solid and broken lines connect the locally stable equilibria and the unstable equilibria, respectively. There are two alternative locally stable equilibria for each value of K in the approximate range between 5.86 and 19.37. Outside this interval, there is one ˆ globally stable equilibrium. To each value of xˆ , there exists an associated value of N, obtained by substitution of the former into Eq. (6).
Fig. A.1 shows five exponential curves which are plots of Eq. (5) for five different values of parameter λ, with the two other parameters in that equation fixed at c = 6, z ∗ = 18. The sigmoid curve is the plot of Eq. (6) drawn for the fixed parameter values K = 10, D = 15. The intersection(s) of each exponential curve with the sigmoid curve define the possible equilibria of x = c (¯z − z ∗ ) and N for that particular value of λ. From top to bottom, λ = 0.21, 0.2037, 0.19, 0.1677, and 0.155. We observe that the number of equilibria changes from one to two to three, then back to two and one again as λ decreases, suggesting hysteresis. This is made explicit in Fig. 3, which shows how xˆ = c (zˆ¯ − z ∗ ) depends on λ. Eq. (7) was solved numerically to obtain the equilibrium values of the (transformed) mean cultural level. The two solid lines denote alternative locally stable equilibria – high and low cultural level regimes – whereas the broken line gives the unstable equilibria. Here, as in Fig. 4, the width of the parametric region of bistability will depend on the values of other parameters. My motivation for focusing on innovativeness is that it is likely to be an important aspect of hominid learning strategies that has been under positive natural selection (e.g. Feldman et al., 1996; Aoki and Feldman, 2014; see also Klein, 2008). Fig. 3 suggests the
Whether or not the diachronic or synchronic differences in cultural level that are seen in Late Pleistocene hominids can be attributed to differences in innovativeness, or more generally cognition, is currently a controversial issue. Less contentious is the possible consequence of an environmental change affecting the availability of food. We can use the cultureless carrying capacity, K , to represent environmental productivity. Fig. A.2 illustrates the effect on the number of equilibria of varying the parameter K . From top to bottom, K = 30, 19.37, 10, 5.86, and 2, with the other parameters fixed at λ = 0.205, c = 6, z ∗ = 18, D = 20. Fig. 4 gives the corresponding bifurcation diagram with K on the horizontal axis and xˆ = c (zˆ¯ − z ∗ ) on the vertical axis. Here again we observe sudden dramatic changes in the equilibrium mean cultural level at the catastrophic bifurcations. 6. Discussion We have described and analyzed a minimal model for the coupled dynamics of mean cultural level, z¯ , and population size, N. For the dynamics of the mean cultural level, we have adopted the Henrich (2004) model as modified by Mesoudi (2011). These dynamics entail a positive feedback such that the mean cultural level increases when it is low relative to population size and decreases when it is high relative to population size. Similarly, for the dynamics of the population size, we have assumed logistic growth in which the carrying capacity is an increasing function of mean cultural level. In addition, our model contains several parameters, which are of interest from an evolutionary or an ecological standpoint and which can produce catastrophic bifurcations when they are varied. The parameters we have specifically focused on are a measure of innovativeness, λ, and the carrying capacity of the cultureless state, K . As the analyses show, two locally stable equilibria of the mean cultural level and the population size can exist within a given intermediate range of either λ or K , whereas there is one globally stable equilibrium outside this range (Figs. 3 and 4, respectively). Within or outside of a parametric region of bistability, a small change in parameter value will result in a small
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K. Aoki / Theoretical Population Biology 100 (2015) 6–12
change in the values of the mean cultural level and population size at equilibrium. However, if a similarly small change in parameter value entails a threshold being crossed, there will be a relatively large shift in these equilibrium values followed by convergence to the new equilibrium. For example, innovativeness may have been under positive natural selection in hominids living in a temporally fluctuating environment (Boyd and Richerson, 1985; Feldman et al., 1996; Aoki and Feldman, 2014). Most evolutionary changes are gradual, and an evolutionary advance in innovativeness is not likely to have been the exception. Then, as innovativeness improved, there would have been an associated gradual increase in cultural level, until the threshold was crossed and an abrupt regime shift was induced. This theoretical argument provides an alternative explanation to the neural hypothesis (e.g. Klein, 2008), in terms of evolutionary changes in innovativeness, for the apparently sudden and dramatic changes in Palaeolithic cultures. Namely, an abrupt behavioral transformation need not be associated with a major genetic change. Hence, anthropologists and archaeologists may not be rewarded in their search for ‘‘adaptive shifts’’ underlying key advances in Palaeolithic/Stone Age cultures (e.g. Wynn, 2002). Alternatively, natural selection for greater innovativeness may have been transient, as during range expansion, particularly into a spatially variable environment (Aoki and Nakahashi, 2008; Wakano et al., 2011). In this case, an upward cultural regime shift may have been followed by a downward cultural regime shift after range expansion ceased. Still another possibility is that innovativeness in modern humans is facultative and waxes or wanes in response to necessity. As in the model of Lee (1986), it is in principle straightforward to extend our model to accommodate three or more locally stable equilibria. For example, we could replace the sigmoid function for the carrying capacity in Eq. (2) with a double sigmoid function. More simply, we could assume a multiple step function that equals K for z¯ < z1∗ , K + D1 for z1∗ ≤ z¯ < z2∗ , K + D2 for z2∗ ≤ z¯ < z3∗ , etc. A double sigmoid function would intersect the exponential curve Eq. (1) at most five times, in which case there would be three locally stable equilibria for population size and cultural level. Let us now briefly consider what can be said about statistical associations between population size and toolkit diversity, assuming that our model is relevant to the situation. Empirical studies (e.g. Collard et al., 2005; Read, 2006; Kline and Boyd, 2010 and Collard et al., 2013) of ethnographic societies report univariable (simple linear) regression coefficients and partial regression coefficients of toolkit diversity on population size. Note first that real-world societies will often deviate randomly from equilibrium. Hence, within the range of parameter values where only one equilibrium is predicted by our model, the partial regression coefficient (which controls for parameter values) is expected to be non-significant, although the univariable regression coefficient may be significant if the societies differ in parameter values. On the other hand, when two or more locally stable equilibria are predicted, and societies are to be found at or near these equilibria, a positive univariable regression coefficient and a positive partial regression coefficient are both expected, as when heterogeneous samples are pooled (see Fig. 1). Multiple regression analyses conducted by Collard et al. (2005) on their sample of hunter–gatherer populations, with toolkit diversity evaluated in three different ways as the dependent variable, yielded non-significant partial regression coefficients on population size. In other words, there was no observable association between population size and toolkit diversity. Read (2006) reached the same conclusion. The study by Collard et al. (2005) incorporated various control variables, including effective temperature as a measure of ‘‘risk’’. If ‘‘necessity is the mother of invention’’, then effective temperature may perhaps also be regarded as a measure
of facultative innovativeness in our model. Hence, with reference to Fig. 3 where the horizontal axis measures innovativeness, we might plausibly argue that these hunter–gatherer societies fall outside the parametric region of bistability and belong to the same cultural regime. For the food-producers, on the other hand, the univariable and partial regression coefficients of toolkit diversity on population size were both significantly positive. Moreover, none of the other independent variables in the multiple regression analyses yielded significant partial regression coefficients (Kline and Boyd, 2010; Collard et al., 2013). This may suggest that these small-scale farming and herding societies have been sampled from more than one cultural regime. The model presented here neglects the ‘‘organizational aspects of a society’’ (e.g. craft specialization, social stratification, political leadership), which are known to correlate with population size among ethnographic societies (Carneiro, 1967). It is easy to see how craft specialization might result in a diversification of tool types. Hence, when modeling societies that have ‘‘progressed beyond’’ egalitarian hunter–gatherers (Hewlett et al., 2011) characterized by a minimal division of labor by sex and age (Kaplan et al., 2000), it may be useful to introduce an intervening variable or variables, such as specialization, between population size and mean cultural level (complexity). Acknowledgments This work was begun during a visit to the Centre for Advanced Study in Oslo, Norway. I wish to express my sincere thanks to Bjørg Egelandsdal, Gro Amdam, Marije Oostindjer, and other members of the Centre for their hospitality during my stay. I benefited greatly from discussions with Ron Lee, with whom I shared an office, and with Shripad Tuljapurkar, whose office was just down the hall. I am grateful to Mayuko Nakamaru and Jeremy Kendal for directing me to the relevant literature, and to Alex Mesoudi, Laurent Lehmann and Stefano Ghirlanda for helpful comments on the manuscript. This work was supported in part by Monbukagakusho grant 22101004. Appendix A Set y = ex > 0. Then Eq. (7) is equivalent to
λ
λ
g (y) = K + (K + D)y − m y c +1 + y c
.
(A.1)
The first and second derivatives of Eq. (A.1) are g (y) = K + D − m ′
λ c
λ
+1 y + c
λ c
y
λ −1 c
,
(A.2)
and
λ
λ
g ′′ (y) = −m y c −2 c
λ c
λ +1 y+ −1 .
(A.3)
c
First assume λc ≥ 1. Then, g ′ (y) decreases monotonically from g (0) = K + D to −∞, and there exists one y∗ > 0 such that g ′ (y∗ ) = 0. Hence, g (y) increases from g (0) = K , attains a local maximum g (y∗ ) > 0, then decreases monotonically to −∞. Thus, there is one and only one equilibrium. ′
Second assume λc < 1. Then, limy→0 g ′ (y) = −∞ and g ′ (y) has
−λ one local maximum at y∗∗ = cc +λ > 0 i.e. where g ′ (y∗∗ ) = 0. If ′ g (y) < 0 for all y > 0, then there is one and only one equilibrium. However, if g ′ (y∗∗ ) > 0, then there may be as many as three ′
K. Aoki / Theoretical Population Biology 100 (2015) 6–12
11
where γ , δ, r , a, b are positive constants. Eqs. (B.1) and (B.2) correspond to Eqs. (4a) and (4b), respectively, of Ghirlanda and Enquist (2007), but we have partly changed the notation to avoid confusion. The null clines of this system are N (X ) =
γ X, δ
(B.3)
and N (X ) = aX + b.
Fig. A.1. The effect of varying parameter λ on the number of equilibria. The five exponential curves (Boserup lines) are, from top to bottom, drawn for λ = 0.21, 0.2037, 0.19, 0.1677, and 0.155, respectively. We observe that the number of equilibria changes from one to two to three, then back to two and one again as λ decreases. Other parameters are fixed at c = 6, z ∗ = 18, K = 10, D = 15.
Fig. A.2. The effect of varying parameter K on the number of equilibria. The five sigmoid curves (Malthus lines) are, from top to bottom, drawn for K = 30, 19.37, 10, 5.86, and 2, respectively. We observe that the number of equilibria changes from one to two to three, then back to two and one again as K decreases. Other parameters are fixed at λ = 0.205, c = 6, z ∗ = 18, D = 20.
equilibria. Thus, a necessary condition for three equilibria is K +D m
>
λ
1+ c 1 − λc
Both are straight lines, so this model yields at most one valid equilibrium, ruling out bistability. If we replace the linear function for the carrying capacity in Eq. (B.2) by a sigmoid function of X analogous to Eq. (2), we can readily visualize the situation where the two null clines, given by Eq. (B.3) and the modified Eq. (B.4), will intersect three times. It is also clear that the two outside equilibria in this case will both be stable. Hence, the specific functional form assumed in Eq. (1) – specifically the concave log N term of the Henrich (2004) model – is not a necessary condition for bistability and catastrophic bifurcation. The dependence on N can be linear as in the Ghirlanda and Enquist (2007) model, or even convex as can be shown by graphic argument. On the other hand, several constraints apply to the functional form of the carrying capacity. To keep things simple, assume that the null cline defined by the dynamics of the cultural level or the amount of culture is given by a monotone function such as Eq. (5) or Eq. (B.3), respectively. Clearly, a step function or a differentiable function that approximates it is a suitable choice for the carrying capacity, as it can intersect the first null cline more than once. If we assume the carrying capacity to be a twice-differentiable function of the cultural level (or the amount of culture), it must possess an inflection point. Moreover, the second derivative must be positive to the left, and negative to the right, of this inflection point. The sigmoid function in Eq. (2) satisfies these requirements. However, the carrying capacity need not be bounded, nor does it have to be monotone increasing. The above considerations suggest the following minimal model for the coupled dynamics of the cultural level, z¯ , and the population size, N, that yields explicit conditions for bistability. We write dz¯ dt
1− λ c
.
(A.4)
= −γ z¯ + δ N ,
dN
= rN 1 −
Appendix B
where
The justification for Eq. (1) is that the Henrich (2004) model, on which it is based, has entered the literature as a convenient model of cultural evolution with directly biased social learning and a dependence on population size. Eq. (2) seems qualitatively reasonable, given that the carrying capacity of a population may increase with cultural level, but is also limited by non-cultural environmental factors. It is instructive to compare our model with the extensive growth model of Ghirlanda and Enquist (2007). In this latter model, the coupled dynamics for the amount of culture, X , and the population size, N, are given by
M (¯z ) =
dt and dN dt
= −γ X + δ N , = rN 1 −
(B.1)
N aX + b
,
(B.2)
(B.5)
and
dt
dX
(B.4)
K K +D
N
,
(B.6a)
for z¯ < z ∗ for z¯ ≥ z ∗ ,
(B.6b)
M (¯z )
and γ , δ, r , K , D, z ∗ are positive constants. Two equilibria exist and are both locally stable if and only if Kδ
γ
< z∗
