Introduction

Social learning, defined as ‘learning that is influenced by observation of, or interaction with [...] a conspecific or its products’ [1], is widespread in the animal kingdom [2] and a seemingly extraordinary capacity for complex forms of social learning [3] is often credited for human demographic success. The evolution of social learning and cultural transmission, therefore, has been a target of research for decades (e.g., [2,4–12]). Much of this work has focused on understanding the spread of behaviours, which are passed to naïve individuals either innately (or genetically) or socially, or which are newly innovated.

The transmission of behaviours directly from parents to offspring, a type of social learning known as vertical transmission [6,11,13] or vertical learning [14], appears to be particularly important in the context of human learning and human cultural evolution [16]. Hewlett & Cavalli-Sforza [16] investigated the source of a large number of technologies and behaviours in the Aka, a foraging population living in the Central African Republic, and concluded that over 80% of the cultural traits or behaviours possessed by individuals in these populations were transmitted vertically, or in other words, learned from biological parents. A similar study of the sources of beliefs and habits of Stanford undergraduates came to similar conclusions [17], although such estimates are the subject of some controversy (see e.g., [14,15]).

The circumstances under which vertical learning is adaptive are, nevertheless, likely to be important to our understanding of the evolution of human social learning in particular. Theoretical models have shown that in an unchanging constant environment, vertical learning increased the speed at which beneficial phenotypes could fix in a population compared to spread via oblique social learning, i.e. the transmission of behaviour from an older (unrelated) individual to a younger individual [18]. When the environment instead changed rapidly, they showed that vertical learning was disfavoured compared to oblique social learning. Similar results were obtained by McElreath & Strimling [14] who suggested that environmental stability was among the important forces driving the evolution of vertical learning.

These results echo the findings from work that contrasts the adaptive value of innate transmission of behaviour, social learning, and innovation. This work showed that innate transmission was favoured by natural selection when the rate of environmental change was very slow, social learning was favoured at intermediate rates of environmental change, and that high rates of innovation were favoured when the environment changed rapidly [6].

One simplifying assumption common in work on this topic is that an individual can engage in vertical learning, oblique social learning, or innovation in order to adapt to their environments, only once in their lifetimes. Further, many such models assume that individuals remember just one behaviour at a time. Here, the adoption of a new behaviour implies forgetting the one used before. Ethnographic evidence, however, suggests that enculturation, or culture learning, in many human populations is more complex. For example, Aunger [15] studied several populations of horticulturalists and foragers in the Central African Republic and suggested that their learning was phased with a strong emphasis on vertical learning early in life, and a strong emphasis on oblique social learning later in life. Similarly, Hewlett et al. [19] showed that individuals amassed cultural information in a way that was phased across life stages in Congo Basin hunter gatherers, and that the timing of the phases of different types of cultural learning showed some cross-cultural variation.

In a recent study [20] we considered a situation where an individual first obtains a repertoire of cultural variants from their parent, and then has the capacity to add to, or subtract from, this cultural repertoire throughout its lifetime. In particular, we focused on the evolution of social learning in temporally changing, recurring environments by allowing for the interplay between social learning, in form of payoff-biased social learning, and innovation, alongside a capacity for ‘memory’, i.e., the storage, retrieval, and forgetting of information. Individuals acquire cultural variants by innovation or social learning, maintain a bounded set in their repertoires, and sample and express traits from that repertoire. Memory decouples the immediate value of a cultural variant from its “long-term” value. That buffering favours social learning in broad range of environmental conditions: higher propensities for social learning can produce faster population-level adaptation since variants adapted to alternative environmental states can be preserved without affecting individual fitness and be expressed once states recur. At the same time, the strength of forgetting evolves to prevent the unbounded accumulation of especially low-benefit variants, producing repertoires whose sizes and compositions are tuned to the environmental stability experienced by the population. Thus, the composition of the cultural repertoires, i.e. the number of cultural variants and their adaptation values, is an outcome of the evolutionary dynamics.

Here, we build on this model to investigate the role that vertical learning plays in adaptation to changing environments, allowing individuals to engage in vertical learning, oblique or horizontal social learning and innovation throughout their lifetimes. This flexible model allows for the accumulation of cultural information from different sources, provides a ‘role for individuals in their own education’, as suggested by [15], and brings a focus on processes involved in remembering and forgetting cultural information.

The precise mechanism of innovation might also affect the evolution of social and vertical learning. Models exploring the evolution of social learning have formulated innovation in a number of ways. For example, it is often assumed that individuals can interact with the environment and, through ‘trial-and-error’ learning, immediately discover a behaviour that matches the environment well. Often, although such individuals will be well adapted to the environment, they will also incur a cost for trial-and-error learning (e.g., [10,21]). Such costs reflect the dangers of such experimentation or, alternatively, the time involved. The implicit assumption that good information is readily available through innovation means that, in very changeable environments, relying on innovation rather than social learning or even vertical learning is reasonable. However, if good information is, instead, very difficult to find and innovation is relatively ineffective, reliance on innovation would only be reasonable across a narrow range of environmental stability parameters. Here, we investigate the interaction between innovation effectiveness and the evolution of social and, in particular, vertical learning.

In agreement with previous work, our model shows that vertical learning is advantageous when the environment is relatively stable and disadvantageous when the environment changes extremely rapidly. However, our model also suggests that vertical learning is most advantageous where environmental change happens at intermediate rates. The relative benefits of vertical learning at intermediate rates of environmental change depend on the process of innovation and how cultural variants are maintained. Where innovation is easy, any information that is lost can be replaced when needed. In this case, vertical learning, which can maintain this information, is useful but not essential. However, when innovation is ineffective and replacing lost information takes many timesteps, vertical learning provides a larger advantage.

1 Methods

The model used in the following is built on [20] with the main difference that (i) vertical learning can be present or absent (see Sect 1.4) and (ii) different models of innovating new cultural variants have been implemented (see Sect 1.2.2). For convenience of the reader we redescribe the modelling details in this section.

1.2 Expression dynamic

Each time step, all individuals must express a variant whose adaptation value determines their individual fitness levels. For this purpose, individuals make an informed decision based on their cultural repertoire. First, they evaluate their cultural repertoire and choose a variant from it. Second, depending on the adaptation value of this chosen variant they either express it or, if the chosen variant is deemed not beneficial enough, engage in social learning or innovation. The decision to learn or innovate is controlled by the genetically transmitted propensity for social learning, . The learnt or innovated variant is then expressed, regardless of the adaptation value and, if not present already, added to the cultural repertoire. In the following we provide a detailed description of these decisions.

1.2.1 Choice of cultural variant from individual repertoire.

An individual, j, chooses variant i from its repertoire, M^j, with a probability proportional to the variant’s adaptation value, i.e. the conveyed benefit, in the current environment, a_i(k), relative to the rest of the repertoire

(1)

The rationale behind decision rule (1) stems from the tendency of human decision-making to be error-prone, which means that individuals might not consistently select the option with the highest adaptation value from the available choices for various reasons. Additionally, we investigate the impact of an alternative decision rule frequently examined in optimal decision-making contexts, specifically the softmax rule ([22], p. 5). The results of this exploration are detailed in S4 Sect of the supplementary material.

Based on the adaptation value of the chosen variant i, individual j decides whether to express variant i or not: with probability a_i(k) individual j expresses i, and with probability j decides to engage in social learning or innovation. Put differently, a high adaptation value of the chosen variant will prevent learning or innovation in this time step.

1.2.2 Learning.

If individual j has decided to not express the variant chosen from its repertoire it engages in social learning with probability and in innovation with probability .

• Social learning: An individual j observes the levels of benefit gained by all individuals during the last time step and chooses to copy a variant proportional to its observed benefit. In other words, it engages in payoff-biased social learning (e.g., [2]). Then individual j expresses this learned variant, x, receives the benefit a_x(k), and adds x to its repertoire M^j (if it is not already contained in it).
• Innovation: In general, innovation events introduce new variants into the system. Here we focus on the impact of three different innovation processes that differ in the probability with which highly adaptive variants can be introduced. Two further innovation processes where we alter the assumption about the benefit provided in the alternative environment, are described in the supplementary material (see S5 Sect), which are referenced where informative.
  1. – Type I innovations: The adaptation value of the new variant is uniformly chosen from the interval [0,1].
  2. – Type II innovations: With a probability of 0.95 the adaptation value of the new variant is uniformly chosen from the interval [0,0.8] and with probability 0.05 from the interval [0,1]. This implies that innovations of highly adaptive cultural variants are rare and the vast majority of the innovations provide a low/medium benefit.
  3. – Type III innovations: With a probability of 0.95 the adaptation value of the new variant is close to zero, i.e. the innovation effectively failed, and with probability 0.05 it is chosen from the interval [0,1]. As for type II innovation, introducing highly adaptive cultural variants are rare but now the vast majority of innovations provide no benefit.
  
  Individual j expresses the innovated variant, x, receives the benefit a_x(k), and adds x to its repertoire M^j.

Last, we track how often a variant i has been expressed by individual j and denote this number of expressions by .

1.3 Forgetting

In each time step, individuals can lose knowledge. defines the probability that individual j removes one cultural variant from its repertoire M^j with the restriction that the variant expressed in the current time step cannot be forgotten, i.e. the minimum size of the repertoire is 1. Rarely expressed variants are more likely to be forgotten compared to frequently used variants. More precisely, if individual j “decides” to forget, it “chooses” variant z to be removed from its repertoire with probability

(2)

where and i denotes the variant expressed in this time step that can never be forgotten. We note that Eq (2) is not the only plausible choice for a forgetting rule (see [20], for a discussion of alternative choices).

1.4 Birth-death process and vertical learning

In each time step, one individual is chosen for reproduction and one for death. The reproducing individual is chosen proportionally to the benefit generated in this time step, i.e. individual j is chosen with probability

(3)

The process of death is age-dependent, i.e. the probability that an individual dies is proportional to its relative age such that older individuals are more likely to die than younger individuals. This way, we ensure that individuals do not become unreasonably old to isolate the effects of vertical learning from the preservation of cultural variants through the long life spans of some individuals. In the supplementary material, we demonstrate that the age distribution of populations experiencing age-dependent death is narrower than for a conventional Moran model with random death (see S1 Sect). If individual j is chosen for reproduction, it passes its - and -value to its offspring. This happens faithfully with a probability and , respectively. Put differently, with probability , the offspring will possess

and with probability

Note that the normally distributed errors are truncated to avoid values larger than 1 or smaller than 0. To investigate the importance of vertical learning, in particular for adaptation, we consider two scenarios. In the first scenario, we do not allow for vertical learning, i.e. offspring are born without knowledge about any cultural variant and individuals have to acquire their first variants through social learning or innovation. In the second scenario, we assume that offspring inherit the entire cultural repertoire of their parent but it does not inherit information about the parent’s experiences of the variants in the repertoire—the -values describing the number of times variant i was expressed by parent j. Thus, at first, all vertically learned variants are equally susceptible to forgetting as their numbers of expressions are set to 1. Contrasting these extreme scenarios has the advantage of revealing the strongest effects of vertical learning. Our definition of vertical learning is in line with others (e.g., [6,13,23]), with repertoires consisting of more than a single variant due to the capacity for memory.

2 Results

By definition, the presence or absence of vertical learning determines the way in which naive individuals acquire their first cultural variants. In populations that do not engage in vertical learning, naive individuals obtain their first cultural variant either through social learning or innovation. If the environment remains unchanged, the adaptation value of this first variant determines the likelihood of further learning or innovation events. For example, a high adaptation value implies a low likelihood of future learning or innovation events and the opposite is true for a low adaptation value (see Sect 1.2.2).

Without vertical learning, the first environmental change an individual experiences will always result in a learning or innovation event as no cultural variant adapted to the changed environmental conditions can be contained in its repertoire. For all subsequent environmental changes an individual can either rely on its repertoire or engage in learning or innovation. Thus, the quality of the available social information and the nature of the innovation process determine, in part, the size of the individual repertoires. As we show below, the absence of vertical learning leads to relatively low average repertoire sizes. In contrast, vertical learning results, on average, in large individual repertoires when the environment is unstable and small repertoires when it is stable [20].

Particularly, in unstable environments when individuals are likely to experience multiple environmental changes in their lifetimes, the low average repertoire sizes are in stark contrast to those expected in populations that do engage in vertical learning. Such populations have been shown to produce relatively large and highly structured repertoires [20]. By “structure” we mean the shape of the distribution of adaptation values within individual repertoires. Highly structured repertoires contain a large proportion of highly adaptive variants, i.e., the distribution of adaptation values is strongly skewed toward 1. Such structures arise partly from vertical learning of whole repertoires combined with low forgetting rates, which buffer against the stochastic loss of highly adaptive variants in either environmental state and thereby facilitate adaptation from standing, unexpressed variation. Additionally, the specifics of the way individuals choose a variant from their repertoires influence the development of the structure of the individual repertoires. For example, rule (1) imposes pressure on the repertoires to become structured because only such structures yield the choice of a highly adaptive variant on average. Alternative decision rules that are less sensitive to repertoire structure could alter the evolutionary dynamics by reducing the dependence on highly structured repertoires (see S4 Sect).

In more stable environments, however, populations with and without vertical social learning evolve higher rates of forgetting (see Figs A, B, and C in S3), and as a result are characterized by similar average individual repertoire sizes. Nevertheless, we observe differences in the composition of these repertoires. Without vertical learning, cultural variants adapted to the alternative environment are lost at the very latest after the death of the last individual that experienced this alternative environment. The intergenerational transmission facilitated by vertical learning means these variants can be found in the repertoires of individuals that have not experienced the alternative environment, effectively prolonging the lifetime of information about the alternative environment.

In summary, besides influencing the way in which naive individuals acquire their first cultural variant, the presence or absence of vertical learning affects (i) the size and structure of the individual repertoires, and (ii) the persistence of information about the alternative environment differently in different environmental scenarios. These differences dictate whether vertical learning is advantageous for the population. To quantify the advantage or disadvantage of vertical learning we show in Fig 1 the difference in the average fitness level of populations engaging and not engaging in vertical learning for all innovation types. Negative values indicate that vertical learning leads, on average, to a lower fitness level. The reverse is true for positive values.

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Fig 1. Advantage of vertical learning.

Relationship between environmental variability and average fitness difference between populations with and without vertical learning. Values are calculated as the average expression value of variants by all individuals in the last generation (i.e. 200 time steps) of a simulation. Fitness values of populations without vertical learning are subtracted from populations exhibiting vertical learning. Population averages are further averaged over all 3000 simulations. Dark grey boxes represent areas, where vertical learning is disadvantageous, white boxes represent areas with a benefit of vertical learning, light grey boxes, where present, indicate intermediate dynamics. The three panels represent the three innovation regimes.

https://doi.org/10.1371/journal.pcbi.1013785.g001

In general, the absence of vertical learning proves advantageous in unstable environments because the smaller cultural repertoires contain cultural variants well-adapted to both environmental states which allows for consistently better variant choices (see Fig 2). At the same time information about the alternative environmental state is kept in the population with certainty.

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Fig 2. Repertoire size and features of sub-optimal variant choice.

Results are shown for different values (x-axis) and three innovation regimes. Filled dots correspond to = 10, open dots correspond to = 200. A) Average count of variants: Values are calculated as the average variant count per individual in the last time step of a simulation. B) Average proportion of individual’s sub-optimal variant: Values are calculated as the age-normalized proportion of sub-optimal variant choices of individuals living in the last generation of a simulation. C) Average magnitude of sub-optimal choice: Values are calculated as the age-normalized average distance between individual’s sub-optimal variant choice and the best variant in their repertoire during the last generation of a simulation. D) Average adaptive value of expressed variants (dot) versus the best variant (star) in an individual’s repertoire: Values are calculated for the last time step of a simulation.

https://doi.org/10.1371/journal.pcbi.1013785.g002

In more stable environments the presence of vertical learning proves advantageous. In this case, the average size of the individual repertoires is similar to the case of no vertical learning but information about the alternative environment persists for longer in the population facilitating adaptation through standing, unexpressed variation (see Fig 4). Intuitively, the probability of preserving information about the alternative environment until the next environmental change decreases with increasing environmental stability, even when forgetting is low.

In very stable environments vertical learning provides a small advantage that is not ascribed to the preservation of information. During the long periods of stasis, parents transmit their highly adaptive cultural variants and thus, their offspring do not engage in other forms of learning to acquire their initial cultural variants avoiding the introduction of low-benefit variants through, for example, innovation.

To aid interpretation of the results we define two parameter ranges (see dark grey and white areas in Fig 1) that indicate whether the differences between the cultural dynamic in populations with and without vertical learning are primarily caused by (i) or (ii) defined above.

However, we note that the distinction between the two ranges is not clear-cut (see e.g., light grey areas in Fig 1). There exist some intermediate rates of environmental change where the dynamics underlying adaptation balance, and both factors—the size of the repertoire and the persistence of information—play an important role. In the following sections we detail how a balance between dynamics dominated by (i) or (ii) affects the role of vertical learning in the adaptation process when innovation operates in different ways.

We quantify the size and structure of the individual repertoires by recording the average number of variants adapted to each environmental state along with the number of sub-optimal variants that individuals choose to express from their repertoires, i.e. the fraction of times individuals do not choose the variant with the highest adaptation value available to them from their repertoire, the magnitude of sub-optimal choices, i.e. the average distance of a sub-optimal variant choice from the best variant in an individual’s repertoire (see Fig 2B and 2C), and the average adaptive values of variants that are actually expressed by individuals alongside their best variants (Fig 2D). Additionally, we illustrate the overall structure of the individual repertoires by plotting the distributions of the adaptation values of all cultural variants contained in the population (see Fig 3 and for more details A, B, and C in S2).

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Fig 3. Cultural repertoires.

Repertoire composition at different values ( = 10 in 1^st and 3^rd row; = 200 in 2^nd and 4^th row) with vertical learning (top panels, in yellow) and no vertical learning (lower panels, in red). Each bar shows the fraction of cultural variants that fall within a given adaptation interval, ranging from 0 (lowest adaptation value, left) to 1 (highest adaptation value, right). Fractions refer to variants adapted to the current environment. Fractions are calculated for single repertoires and further averaged over all simulations. The three columns represent the three innovation regimes.

https://doi.org/10.1371/journal.pcbi.1013785.g003

To quantify the persistence of information about the alternative environment, and so the potential of adaptation through unexpressed standing variation, we calculate the probability that at least one individual in the population possesses a variant adapted to the alternative environment at the point of an environmental change (see Fig 4).

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Fig 4. Preservation of unexpressed variants.

Relationship between environmental variability and the probability of preserving at least one variant adapted to the ‘alternative’ environment until the next change occurs. Simulations are run until the burn-in phase and stopped once the first change occurs (before learning takes place). For each simulation it is recorded whether the population consists of at least one individual that knows about a variant adapted to the ‘alternative’ environment. Values are averaged over all 3000 simulations. Yellow lines correspond to simulations with vertical learning, red lines correspond to simulations without vertical learning. The three panels represent the three innovation regimes.

https://doi.org/10.1371/journal.pcbi.1013785.g004

3 Discussion

The model presented here allows for the acquisition of cultural variants through learning or innovation, the generation of individual cultural repertoires through memory and forgetting, and their persistence across generations through vertical learning. The model explores the benefit of vertical learning as the environment changes and when innovation is easier or more difficult. Previous analyses of the evolution of vertical learning have shown that vertical learning is favoured when the environment is very stable and information gained and used by this generation is likely to be useful to the next (e.g., [6,18,23]). Here we have shown that with the capacity for memory, and assuming that learning can occur multiple times in an individual’s lifetime, the benefit of vertical learning critically depends on the interplay between the size of individual repertoires and the duration of environmental stasis. Overall, we find that vertical learning is most advantageous when environments vary at intermediate rates and innovation is particularly difficult.

In the absence of vertical learning, cultural adaptation after an environmental change requires either rapid innovation or social learning from other individuals who have retained or discovered adaptive cultural variants. Additionally, naive individuals have to engage in either social learning or innovation when vertical learning is absent. Our results show that vertical learning provides a slight benefit to the offspring in highly stable environments, when their parent’s knowledge is up-to-date and can be copied with low risk. This effect is most obvious for scenarios in which a single cultural variant can be highly adapted to both environmental states (see S5 Sect) and the generation of individual repertoires is obsolete. The major benefit of vertical learning in our model comes from its ability to allow unexpressed cultural variation suitable for alternative environmental states to persist and be carried across generations. This means that at intermediate rates of environmental change, individuals need not rely on innovation in order to adjust to their new environments, rather, they can rely on the cultural variants contained within their repertoires. The benefit of this is especially visible when innovation is most difficult (here, we referred to this case as ‘type III innovation’). Reliance on innovation in this case would mean that individuals must wait a considerable time before they discover a suitable cultural variant. This ‘waiting time’ can be eradicated or greatly shortened by vertical learning, which maintains larger repertoires. When the number of individuals who have repertoires containing useful cultural variation decreases, social learning plays a crucial role—even where vertical learning is possible. It is sufficient that one or two individuals in the population maintain useful variation, this can be spread rapidly through the population via social learning. We show that the preservation of variants depends critically on the evolution of forgetting, which is not solely dominated by the process of innovation: An almost deterministic variant choice rule can relax selection pressure on forgetting, thereby increasing the probability of preserving variants adapted to the alternative environmental state. As a consequence, the advantage of vertical learning is increased.

The preservation of a large number of cultural variants over long periods of time is a double-edged sword. In intermediate and stable environments, this is beneficial. In rapidly changing environments, however, these preserved repertoires become increasingly large and less structured compared to smaller repertoires beginning to interfere with an individual’s ability to choose cultural variants effectively (see also, [20]). Therefore when the environment changes rapidly and cultural repertoires become large and less structured, vertical learning is detrimental. However, we show that this disadvantage can be mitigated by a less error-prone variant choice rule (see S4 Sect).

In our model, different innovation types lead to nuanced effects. We contrasted three types of innovation: one that produces many good innovations, one that produces many innovations of moderate value and occasional innovations of very high value, and one that produces many poor innovations and occasional innovations of very high value. In the latter cases, the probability of producing an extremely good variant was the same. We found that producing many moderate innovations was more detrimental than producing many poor innovations. Poor innovations were easily recognised by individuals and rarely, if ever, chosen from the repertoire. When many moderate innovations were produced, individuals chose these at a much higher rate. This led to the counter-intuitive result that if innovation is to fail, it is best that it fails completely. We note that this result interacts critically with our assumptions about how individuals choose variants from their repertoires.

These findings are largely in concordance with the results of previous models which also suggest a role for vertical learning in relatively stable environmental conditions. A noteworthy outcome of our model is that, unlike in some previous work (e.g., [18]), the benefit of vertical learning does not continue to increase, nor does it stabilise at a high value, as the environment becomes increasingly stable and unexpressed variants are erased from the repertoire before the next change occurs due to high forgetting rates. Based on our model, for each case we have considered, there appears to be some value of environmental stability for which the fitness levels of populations that engage in vertical learning and those that do not, equalise, or become very similar. The point at which this happens is different for different types of innovation.

We note that our model, as with the model of Ammar et al. [20], has made several simplifying assumptions about the ways in which memory, forgetting, and decision making proceed. We have relaxed several assumptions of previous models on similar topics, for example by allowing several learning events per lifetime, by allowing individuals to engage in vertical learning, social learning, and innovation, and by allowing individuals to possess sophisticated and dynamic memories. Although this brings our model in closer alignment with previous work on, for example, enculturation, it would certainly be possible to refine and improve our representation of cognitive processes involved in learning and memory, perhaps with implications for our results.

Supporting information