## Abstract

The effect of environmental change on the rate of innovation and level of cultural complexity in a population is a theoretically understudied piece of an important puzzle at the heart of cultural evolution. Many mathematical models of cultural complexity have focused on the role of demographic factors such as population size or density. However, statistical studies often point to environmental variability as an important factor determining complexity in many cases. The aim of this study is to explore the interaction between environmental fluctuations and the rate of cultural innovation within a population and to examine the relationship between rates of innovation and the probability of maintaining a complex cultural repertoire in a changing environment. Two models are presented that draw on previous models used to examine rates of genetic mutation. The models show that, as in a genetic system, the stable rate of cultural innovation in a population decreases with environmental stability and increases in unstable environments. This effect is similar but quantitatively different for different modes of cultural transmission (success bias, conformity bias and random oblique learning). The model shows that innovation can increase diversity but that this relationship depends critically on learning mode and learning parameters.

This article is part of the theme issue ‘Bridging cultural gaps: interdisciplinary studies in human cultural evolution’.

## 1. Introduction

The theoretical underpinnings of cultural evolution have been the focus of considerable attention in recent decades. Much of this theoretical work has adapted and extended well-understood models from theoretical population genetics to study the dynamics of cultural evolution (e.g. [1–7]). One pressing question at the heart of cultural evolution is: what are the forces driving cultural complexity and cultural loss? This question has been the centre of heated controversy for over a decade (e.g. [8–13]). Mathematical models and laboratory experiments have consistently suggested that demographic factors such as population size [8,9,14–16] or connectedness [17] should drive cultural complexity. These models suggest that larger populations can be expected to have more complex toolkits and smaller populations should, on average, have less complex toolkits [8,9,17–20]. Regional and worldwide statistical analyses have shown little support for this idea in food-gathering societies where proxy measures for environmental risk have been shown to correlate with toolkit complexity [10,21–23], but see [24] for a discussion of population size measures used), although this does not appear to be the case for food-producing societies [25–27]. In one notable case, diversity in pottery design was shown to correlate inversely with population density, which may have led to an increase in the benefit of conformity and collective action [28]. It is, therefore, crucial that we understand the interaction between the environment, innovation and cultural complexity. However, despite the important role that environmental fluctuations appear to play in maintaining cultural complexity in hunter–gatherer societies in particular, as well as their putative role in determining the rate of innovation [29], mathematical models of cultural complexity rarely model environmental factors explicitly [27,30,31]. Here I present two models that describe the effects of environmental fluctuations on the rate of innovation and go on to examine the effect of innovation, environmental fluctuation and social learning on a measure of cultural diversity. The models use well-understood theory describing the interaction between the environment and mutation in genetic systems and help to build on a body of literature on innovation and environmental factors on which the debates surrounding the driving forces of cultural complexity can draw.

## 2. The models

The first model presented here is a neutral modifier model with the modifier acting on the rate of innovation in a cultural system. This class of models has been used to understand evolution of mutation rates, recombination rates, and rates of migration in genetic systems [32–34], epigenetic systems [35] and cultural niche construction systems [36]. Here they are used to examine the evolution of rates of innovation and the maintenance of cultural diversity in fluctuating environments.

We first consider an infinite population of individuals living in a temporally varying environment that can switch between environment 1 (E1) and environment 2 (E2). E1 and E2 favour different cultural toolkits (or cultural repertoires), which consist of a different set of subsistants [37], designed to extract energy from each distinct environment. These repertoires are labelled **R**: *R* and *r* and the fitness associated with them in each of two environments is given by table 1.

A ‘cultural modifier’ trait (which could be either genetic or cultural itself) controls the rate of innovation such that an individual with *M* has an innovation rate given by , and an individual with *m* has an innovation rate given by Each generation, the population undergoes mating, learning, innovation and selection, after which the environmental state may change. Mating is random but learning may be either random or governed by learning rules. Here learning may be random oblique (as a baseline expectation), conformist or success biased as described by Aoki *et al*. [19]. Note that these modes of transmission are modelled separately but in real human populations they may, of course, be used in concert. Success bias entails taking a random sample of *n* individuals and choosing the most successful (or most fit in the current environment) among that sample as a cultural role model. Success bias is found in small-scale human societies in a variety of domains [38]. This formulation also reflects the likelihood of restricted neighbour–neighbour interactions, for example, between individuals in large populations. Conformist learning is defined as a disproportionate tendency to copy the most common variant in a subsample of *n* tutors.

The population thus consists of individuals with phenotypes *RM*, *Rm*, *rM* and *rm*, the frequencies of which are represented as *x*_{1}, *x*_{2}, *x*_{3} and *x*_{4}, respectively. We can describe the fitness of each phenotype and assess the optimal rate of innovation in certain environmental conditions. We can extend this model to examine the probability of a polymorphism in **R**, which would indicate the presence of a large repertoire suited to all environments.

### (a) Mating

Mating is random and so the probability of parents meeting is proportional to their frequency in the population (table 2). Vertical learning happens directly after mating and production of a new generation of individuals.

### (b) Learning

After mating, newborn individuals learn a repertoire through vertical transmission first, followed by oblique transmission. This reflects the mode of learning in a number of hunter–gatherer groups including, for example, the Penan and Bedamuni who learn first from their fathers and then from uncles or unrelated adults [39]. Hewlett & Cavalli-Sforza [40] suggested that up to 80% of learning in the Aka in some domains is vertical although the value may be much lower in other groups. We define the proportion of learning that is vertical to be *P*_{V} and the proportion that is oblique to be *P*_{o} with and examine values of *P*_{o} that are anthropologically relevant. Using the mating probabilities in table 2, we can describe the proportion of offspring with each phenotype after vertical learning. We will denote these proportions as with *i* = {1,2,3,4}. These are

After vertical transmission, oblique learning can take place. If this learning is random, then a role model is chosen in proportion to the frequency of that type in the adult population. For convenience, we assume that the modifier trait **M** is either genetic or vertically transmitted and so oblique transmission describes transmission of the **R** trait only, although in reality the mode of transmission of a trait modifying the rate of innovation is an empirical question. Thus, after random oblique transmission we have

If transmission is not random but, rather, success biased then we have
if *R* is favoured, or

if *r* is favoured. Here,
is the probability that at least one of an individual's sample of *n* role models has the favoured variant whose frequency is given by and denotes the probability that all of the sample have the non-favoured variant. Finally, if transmission is conformist, we have

where or the probability that over half of an individual's sample of *n* role models has a certain **R** trait whose frequency is given by *x _{i}* +

*x*. By varying the parameter

_{j}*n*or the size of the pool of role models available to an individual, it is possible to assess the effect of population connectedness on the optimal rate of innovation and the frequency of the

*R*and

*r*traits.

### (c) Innovation

After learning, population-wide innovation can occur. The rate of innovation depends on the frequency of the modifier traits in the population and the repertoire traits with which they are associated. We assume that individuals can innovate either repertoire *R* or *r* with equal probability. This probability is denoted by for the *M* trait and for the *m* trait. Thus the final proportions of each phenotype in the population after vertical learning, oblique learning and innovation are given by

### (d) Selection

Selection acts on offspring after learning and innovation and is based on the fitness associated with the **R** trait only (in other words, the modifier trait **M** is neutral). The proportion of surviving offspring of each type that forms the next generation is given by
in environment E1, where and

in environment E2, where .

### (e) Simulations

Following the work of Carja *et al.* [35] on the evolution of mutation, recombination and migration in fluctuating environments, simulations began with a population fixed on the *M* modifier trait. The rate of innovation associated with this trait was sampled randomly from a uniform distribution between 0 and 1. This population was allowed to evolve until a stable equilibrium was reached. Trait *m* was then introduced at low frequency with an associated innovation rate that was the product of the resident rate and a random number generated from an exponential distribution with mean 1. After 1000 generations the frequency of *m* in the population was assessed and rates higher than the initial rate were used to indicate invasion and lower, eventual extinction. If the *m* trait did invade, the value associated with *m* became the resident value and a new simulation began with that value represented by trait *M.* After 500 trials in which the resident innovation rate cannot be invaded, the resident is considered to be the stable rate in that simulation [35].

The environment changes every *c* generations where *c* is fixed and allows the environment to alternate between two environmental states. In other words, a periodically varying environment was implemented.

### (f) The effect of population connectivity

We can assess the relative importance of the regime of environmental change and the population connectedness in determining rates of innovation by examining the parameter *n*, which describes the connectedness of the population with regard to learning. Where individuals subsample the population in order to choose a role model, the size of that subsample can be varied to assess the effect of this parameter on the eventual rate of innovation.

### (g) A model with ‘heterozygosity’

The model described above allows the population to evolve and maintain distinct repertoires in different environmental conditions. The implication of this formulation is that the populations are in some way specialized: some individuals can use repertoire *R* when appropriate and others can use repertoire *r*. In some cases, a population may not be specialized in such a way and an individual may be capable of using a repertoire favoured in one environment or a larger repertoire favoured in all (or many) environments. To examine this possibility, the model can allow a kind of repertoire ‘heterozygosity’ to exist where the possible variants of the **R** trait in the population are *R*, *Rr* or *r.* In other words, an individual can use the *R* repertoire only, the *r* repertoire only or both. Note that in contrast to genetic models, neither the *R* repertoire nor the *r* repertoire is dominant here—*Rr* suggests that both *R* and *r* can be used, in other words, that the individual has a more complex repertoire suited to both environments. The frequency of these phenotypes can be represented by *x*_{1}, *x*_{2} and *x*_{3}, respectively, as is consistent with the previous formulation.

It is necessary to introduce a cost to learning and maintaining both the *R* and *r* repertoires relative to learning and maintaining just one repertoire. This is represented by *δ*_{c}—the cost of learning two repertoires, relative to one. We can normalize the cost of learning one repertoire to 1 and assume that the cost of learning *R* is equal to the cost of learning *r*. Again, we consider the effects of random oblique learning and success-biased learning as defined above and examine the effects of these modes on the probability of maintaining a polymorphism in **R** under different regimes of innovation and environmental change. This formulation implies that an individual, not just a population, may have a smaller or larger repertoire and examines the population level effects of this individual variation. It is assumed that there is little overlap between *R* and *r.* The mating frequencies in this ‘heterozygosity’ model are given in table 3 and equations covering mating, learning, innovation and selection are presented below.

After vertical learning, the frequencies of the three types in the population are denoted by where *i* = {1,2,3}. These values are

Following vertical learning, oblique learning occurs with a probability *P*_{o}. Here we investigate the effects of random oblique and success-biased learning. These are described by the equations:
for random oblique learning. Note that the equations make the further assumption that it is prohibitively difficult to switch between specialisms, in other words, the rate of switching directly from *R* to *r* through oblique learning after vertical leaning is 0.

For success-biased learning, following the scheme set out above for random oblique learning we get
when *R* is favoured, and

if *r* is favoured. We can then allow the population to innovate at a rate *µ*. Here, again, individuals can change their type from *R* to *Rr*, *r* to *Rr* or from *Rr* to any of the available types, assuming that an individual ‘forgets’ or neglects part of the larger repertoire. The frequencies of the types after innovation are given by where *j* = {1,2,3}.

Finally, selection acts on the population with different selection affecting the types in different environments. Environment 1 favours the *r* trait and so selection proceeds as follows: and
where *δ*_{c} is the cost of maintaining two repertoires compared with maintaining one and *s*_{1} is the fitness benefit of having trait *r* in environment 1. Similar logic can be applied to environment 2, which favours *R:*

The simulations begin with the environment in state 0 and allow the environment to change every *c* time steps. In this case, the innovation rate is varied in order to assess the effect of innovation rates on cultural complexity under different regimes of environmental change and under the influence of different social learning mechanisms. It would be an interesting and worthwhile extension of this model to combine the approaches used in the first model presented with those of the second in order to assess the optimal rate of innovation alongside the frequency of large repertoires.

## 3. Results

### (a) The optimal rate of innovation

The optimal rate of innovation decreases as environmental stability increases, and is lower when oblique learning keeps useful information in the population. In general, the more social learning is relied upon, in other words, for higher values of *P*_{o}, the lower the optimal rate of innovation becomes (figure 1). This optimal value decreases further when oblique learning is more effective and for both success-biased learning and conformist learning. This happens because less frequent innovations are needed to maintain useful information when success bias or conformist learning is in operation than when random oblique learning or vertical learning alone is used. The interaction between effective social learning and innovation is important. Social learning allows the spread of new traits and keeps traits in the population that might otherwise be lost at times when they are not under direct natural selection. Therefore, under different circumstances, effective social learning may be a conservative or an innovative force.

An increase in the connectedness of the population, in other words, an increase in the number of role models from whom an individual can learn, makes social learning more effective and, in turn, decreases the optimal rate of innovation (figure 1). Therefore, the effect of population connectedness in this model is to decrease the optimal rate of innovation but to maintain previously existing information in the population. Depending on the regime of environmental change, an increase in population connectedness may increase or decrease cultural diversity either through its effect on conservation of traits or innovation. This points to an important nuance regarding the effects of innovation. It is not the case that high rates of innovation in response to environmental changes or risks will necessarily lead to high cultural complexity, nor will population connectedness (or population size) drive cultural accumulation in isolation. The cultural repertoire of a population is a complex product of the population's environment, innovativeness, learning biases and connectivity.

### (b) Maintaining cultural diversity

The cultural diversity model examines a case where individuals may be specialists or generalists with the ‘general’ cultural repertoire being larger and representing a higher level of cultural diversity. Individuals with large cultural repertoires are favoured in all environmental conditions. On the other hand, the ‘specialists’ maintain just one cultural repertoire and are culturally adapted to one environment only. Note that there is an implication in economics that ‘specialists’ benefit not just themselves, but also other individuals in the population at times where their skills are needed—this is not the case in this model. The results of the model depend qualitatively on two things. First, the cultural diversity observed in the system depends on the rate of innovation in most cases. Second, the shape of this relationship depends on the rate of environmental fluctuation. For example, figure 2*a* shows that for vertical learning (dashed lines) when the environment is very unstable (the environmental period is 2), the frequency of the generalist repertoire in the population does not change significantly with the innovation rate. However, when the environment is more stable (environmental period is 100), the frequency of the generalist repertoire increases in the population with the innovation rate. To see why this might be, it is instructive to examine the case where there is little innovation and the system relies heavily on learning. This is the case on the far left of figure 2*a* where the value of *µ* = 0.01. For a stable environment with low innovation, the population tends towards high frequencies of a single repertoire, spread by learning, and higher rates of innovation (towards the right of the *x*-axis) favour diversity.

Oblique learning changes this relationship. Where the population relies on some oblique learning, even where the environment is very unstable, novelty is spread rapidly increasing diversity as rates of innovation increase. For a heavy reliance on social learning, this effect is striking (figure 2*b*). It is clear here that in increase in reliance on social learning leads to a decrease in diversity compared with vertical learning alone even as innovation rate increases.

## 4. Discussion

The aim of this study was to explore the interaction between environmental fluctuations and the rate of cultural innovation within a population and to examine the relationship between rates of innovation and the probability of maintaining a complex cultural repertoire in a changing environment. The models showed that, as in a genetic system, the rate of cultural innovation in a population decreases with environmental stability and increases in unstable environments. This effect was similar for different modes of cultural transmission (success bias, conformity bias and random oblique learning). However, there were clear quantitative differences between these modes of learning. The efficiency of information spread, or the effectiveness of the social learning strategy in use by a population, seems to alter the amount of innovation that is necessary to maintain an effective cultural repertoire in the face of rapid environmental fluctuations. Previous work has shown that cultural accumulation occurs more rapidly in certain populations when the rate of environmental fluctuation is high [27]. The model presented here suggests that this may be partially the result of increased rates of innovation in unstable environments.

As with initial work in the genetic sphere, these models focus on a non-random, periodic environmental fluctuation. There is considerable scope for extensions of this model to account for temporally random fluctuations as well as fluctuations with more than two possible environmental states. Given the results of the model, we might expect to see that a high rate of environmental change, represented in the model by the parameter *c*, should allow the evolution of high levels of innovation and thus generate or maintain cultural complexity. This can be gleaned from data by comparing the cultural complexity in populations with the environmental variability to which they are subject. This has been addressed to some extent by Fogarty & Creanza [27]. Analysing a number of papers that collated data on the interaction between the environment and measures of cultural complexity, this study showed that the extent to which environmental factors are correlated with cultural complexity relies heavily on subsistence strategy. However, many studies focus on hunter–gatherer populations alone (e.g. [21–23,41]).

To assess the nature of a relationship between environmental fluctuation, innovation rate and diversity, it may be possible to use longstanding data on cultural complexity and a variety of environmental parameters from Torrence [42]. This dataset has been used to examine the ‘risk hypothesis’, which suggests that the cultural complexity of a population should be proportional to the environmental risk experienced by that population. Torrence used a population's latitude as a proxy for a direct measure of this ‘riskiness’. Subsequently, Collard *et al*. (e.g. [21]) suggested that above ground productivity among other measures may be better approximations of risk, and aspects of the dataset have been critiqued by Henrich [43]. However, latitude is closely related not only to the average temperature experienced by a population but to the annual temperature *range* experienced by a population [44], which may be a useful proxy for environmental variability. The dataset used by Torrence and subsequently by others contains information about latitude and an approximation of the annual temperature range can be gained from that information. Intuitively, a similar correlation exists between the measures of ‘cultural complexity’ and a measure of environmental risk thus defined as have been found between cultural complexity and latitude. However, a complete analysis would require information on cultural complexity, latitude, altitude, as well as downwind distance from the sea for all relevant populations—all of which play a role in determining the temperature range [44].

One of the most interesting findings of this study is that innovation may increase diversity under certain circumstances or innovation may have little or no effect on cultural diversity. A similar relationship between innovation and cultural complexity has been shown in previous studies notably by Kandler & Laland [45]. This work showed that independent innovations, that is, innovation that does not build on previous knowledge, does in general lead to an increase in cultural diversity. However, conformist learning tended to weaken this relationship. The model presented here shows that effective forms of learning such as conformist transmission may, in fact, lead to lower rates of innovation in a population. The results of the ‘heterozygosity’ model presented above are consistent with this previous work showing that effective learning decreases the need for a pool of cultural diversity to buffer against a changing environment and that this effect changes the relationship between cultural diversity and innovation.

Necessarily, there are a number of important aspects of innovation that have not been accounted for in the models presented above. For example, it has been shown that the ‘magnitude’ of innovation is linked to the evolution of innovative behaviour [46], that recombination can be an important aspect of innovation [29,31] and that the increasing costs of innovation as information becomes more complex is a crucial aspect of the relationship between innovation and cultural accumulation [47]. It is also important to note that learning processes are also likely to be under selection and that this possibility is not accounted for here. Finally, the results seen in this infinite population model may change in a more realistic finite population where more innovation may be needed to maintain important traits in a slowly changing environment. However, these models represent a step towards further understanding important but sometimes unintuitive links between the environment, innovation and cultural complexity, and may begin to offer a theoretical insight into one of the most contentious questions in cultural evolution at present: What primarily determines cultural complexity—the environment in which a population lives and learns, or the strength and number of the social ties between individuals within that population? A thorough understanding of the effects of the environment on innovation and learning will be crucial in answering that question.

## Data accessibility

This article has no additional data.

## Competing interests

I declare I have no competing interests.

## Funding

I received no funding for this study.

## Acknowledgements

The author thank the editors and two anonymous reviewers for their careful reading of the paper, and Gilbert Smith for comments on an earlier draft.

## Footnotes

One contribution of 16 to a theme issue ‘Bridging cultural gaps: interdisciplinary studies in human cultural evolution’.

- Accepted December 12, 2017.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.