Greater than the sum of its parts? Modelling population contact and interaction of cultural repertoires

Evidence for interactions between populations plays a prominent role in the reconstruction of historical and prehistoric human dynamics; these interactions are usually interpreted to reflect cultural practices or demographic processes. The sharp increase in long-distance transportation of lithic material between the Middle and Upper Palaeolithic, for example, is seen as a manifestation of the cultural revolution that defined the transition between these epochs. Here, we propose that population interaction is not only a reflection of cultural change but also a potential driver of it. We explore the possible effects of inter-population migration on cultural evolution when migrating individuals possess core technological knowledge from their original population. Using a computational framework of cultural evolution that incorporates realistic aspects of human innovation processes, we show that migration can lead to a range of outcomes, including punctuated but transient increases in cultural complexity, an increase of cultural complexity to an elevated steady state and the emergence of a positive feedback loop that drives ongoing acceleration in cultural accumulation. Our findings suggest that population contact may have played a crucial role in the evolution of hominin cultures and propose explanations for observations of Palaeolithic cultural change whose interpretations have been hotly debated.

In a population of N individuals that at time t=0 has a cultural repertoire of size zero, the expected number of lucky leap innovations at time t that have been produced in this process, n lucky , is thus: In our simulations, tools can be lost as a result of stochastic drift. This stochastic loss occurs at a rate that is dependent on the number of existing tools and inversely proportional to the population size. If we consider only lucky leaps, the rate of spontaneous loss from the tool repertoire is: Subtracting this loss term from the rate of accumulation of lucky leaps given in equation 1a, we get: We assume that the population has no tools at time t=0 (n 0 =0), and we treat this difference equation as a continuous time differential equation with solution at time t: At large t, the curve described by this function approaches an asymptote: This is Equation 1 in the main text. Here we have made the simplifying assumption that loss of a tool is final, i.e. that reinvention of the same tool does not occur. In the stochastic simulations, however, this applies to lucky leaps but not to toolkit tools or combination innovations. If a toolkit or combination tool is lost but the lucky leap associated with it remains in the cultural repertoire, the population is capable of reinventing the lost toolkit or combination tool.
We assume that there are L toolkit innovations associated with each lucky leap, where L is sampled from a uniform distribution U (1,11). Then, if L is the expected value of L (namely 5.5), the expected rate of change of toolkit innovations is Δn toolkit Δt = P lucky ⋅N ⋅ L , provided the L toolkit innovations associated with a lucky leap tool are invented immediately following the lucky leap innovation. This assumption is a reasonable approximation of the stochastic dynamics when the rate of toolkit innovation is orders of magnitude higher than the rate of lucky leap innovations. Combining the expected rates of change of lucky leaps and toolkit innovations, a population of N individuals thus has an expected number of lucky leaps (n lucky ) and toolkit innovations (n toolkit ) at time t given by: When lucky leap innovations are combined to produce innovative combinations, these combinations are useful to the population with probability P CombUseful , leading to an expected rate of change of Δn comb Δt = P lucky ⋅N ⋅n lucky ⋅P CombUseful , where n comb is the expected number of innovative combinations and n lucky is defined in equation (1). As with toolkit innovations, we make the simplifying assumption that all combinations with a given lucky leap are tested immediately following that lucky leap's innovation. When innovations can combine in this way to form new tools, the expected size of the tool repertoire of a population of size N at time t is n total = n lucky + n toolkit + n comb : This combination scheme represents the notion that groundbreaking ideas are often widely applicable to other existing technologies, and it is relatively conservative since only lucky leaps can be combined to produce new innovations. For simplicity we assume that all potentially useful combinations and toolkit tools are innovated immediately upon the lucky leap's invention, that is, the population can produce all useful combinations of tools whenever a new tool becomes available, so that an individual may test more than one combination per time step.
Because the rate of cultural loss of tools is likely to decrease as population size increases (e.g. [2]), we assume that tools can be lost at each time step with probability P SpontLoss /N. Toolkit innovations and combination tools can be individually lost, but if a lucky leap tool is lost, the toolkit and combination tools associated with it are also lost.
With probability n lucky n total , the tool that is lost is a lucky leap, and thus its associated toolkit and combinations are lost with it, so when a lucky leap is lost, the total number of tools lost is L t + C t + 1, where L t and C t are, respectively, the mean number of toolkit innovations and combination innovations associated with a lucky leap innovation at the time of its loss, t, and the 1 accounts for the lucky leap itself. In our model, each combination tool is formally associated with only one of the lucky leaps from which it is composed, and thus the loss of that lucky leap leads to the loss of the combination. The Electronic Supplementary Material from [1] also examines the alternate assumption in which a combination tool is lost following the loss of either of its component lucky leaps, which does not qualitatively alter the results. With probability 1− n lucky n total , the tool lost is a toolkit or combination innovation, and the number of tools lost is 1. This leads to the difference equation: where C inv is the average number of new useful combination tools that can be made possible by a new lucky leap upon its invention (which is dependent on the existing number of lucky leaps, n lucky ), and where ! C inv is the expected value of C inv .
The last term in (8) is the expected loss term for a population of size N to be subtracted from the expected number of innovations generated at time t. This loss term can be simplified, and replacing n total by n lucky + n toolkit + n comb , equals to: Since ! C inv and C t change with time because of their dependence on (the timedependent) n lucky , the solution of equation (8) with ! ! C t = C inv 2 .
Since combination tools are made up of a new lucky leap and the lucky leaps already in the cultural repertoire, when a lucky leap occurs in a population at steady state, the average number of combination tools that turn out to be useful is ! ! C inv = P CombUseful ⋅n lucky * .
Using the expression for this term, we find