Powell et al. (2009) Late Pleistocene demography

Powell, A., Shennan, S., & Thomas, M. G. (2009). Late Pleistocene demography and the appearance of modern human behavior. Science, 324, 1298-1301. 

The model is rooted heavily in that of Henrich (2004).

Unrealistic elements in Henrich (2004) to address:

• All individuals belong to a single, large population
• late Pleistocene, and indeed, modern hunter-gatherer metapopulations consist of small, highly dispersed subpopulations connected by migratory activity
• Naive individuals can accurately identify the most skilled member of the preceding generation as an oblique model
• in large populations this would be impossible and, we suggest, even in small populations, identification of the most skilled adult will be inaccurate
• Naive individuals always copy oblique models
• cultural skills and behaviours are often first, and sometimes only, learned from parents 

The model 

• G subpopulations, each with N adults. 
• N = 25, assuming total group size of about 100 individuals. 
• Henrich's [\alpha / \beta] condensed into single term \alpha = "skill complexity"

Each generation, the following steps occur. 

1. Within each subpopulation, a new generation of offspring (size N) is created.
2. Parents are chosen for each offspring by random sampling of current population (with replacement). 
3. Vertical transmission: Naive offspring receive a z value from their parent according to Gumbel distribution (defined in Henrich 2004). 
4. Oblique transmission: All offspring randomly select an individual from among the adults in their subpopulation with z-values greater than they received from their parent, with probability proportional to the magnitude of the z-value difference. If no such model exists this step is skipped. The individual only retains the z-value from the oblique model if it exceeds the one it received from its parent. 
5. The now "enculturated" offspring replace their parents. 
6. Migration: See below. 

Density-dependent Migration: 

• Subgroups were placed at random on a 2-D grid at density D. 
• Each individual does a Gaussian random walk, and if they "hit" another subgroup they migrate to that group, otherwise they stay in their original subpopulation. 
• The standard deviation of the Gaussian distribution is the "migratory range" M, which is defined as a proportion of the average nearest neighbor distance. For M > 0.33, the mean global migration rate approximate the global subpopulation density. 
• Unless otherwise stated, M is set to 1.0 (i.e., equal to the average nearest neighbor distance).

Results

• All adults in all subpopulations were initialized with a z-value of 10.0, and run forward for 100 generations  If the mean z-value in the final generation was greater than 10, the result was deemed "cumulatively adaptive." 
• These results indicate that the accumulation, or maintenance, of culturally inherited skill is not dependent on the absolute meta-population size, but rather on the degree of interaction of the constituent subpopulations, given population substructure and that G > ~50. 
• However, when G < ~50, skill accumulation will, to an extent, be depen- dent on G, and thus the size of the metapopulation. This result may have some bearing on debate concern- ing the erosion of cultural complexity in Holocene Tasmania (30, 32, 33). As a conservative measure, we fixed G at 100 in all subsequent simulations.

Heterogeneous contact with other subpopulations: 
Partitioned the simulated world into two regions differing in density by an order of magnitude.
--Skill accumulation was consistently higher in the high-density region even though the two regions were contiguous.
They also partitioned the world and kept the density constant but varied the migratory activity (M).
--Similarly, skill accumulation was consistently higher in the well-connected region.

Fig. S3. An illustration, from a single iteration and shown at 25-generation intervals, of the spatial structuring of skill accumulation in a heterogeneous migratory range world. Individuals in the left-hand side of the simulation world in each subplot have migratory range M_high (1.0) and those on the right- hand side have M_low (0.1). Each subpopulation is marked by a circle, centred on the spatial location of the group and with diameter proportional to its mean z-value. Regional mean z-values are also given at the top of each subplot.

A quantification of the effect of increasing migration activity in terms of the effective number of adult individuals available as transmission models within each subpopulation.
--To achieve this, they inverted the simulation process: for given values of D and \alpha, they simulated widely over N to find the minimum number of adults needed in each subpopulation for adaptive cumulative evolution to occur.