We propose expansion simulations (ExpSim), which can also be subject to directional trends; however, not through an inherent directional bias, but through geographical constraints, forcing an expansion in one direction. Geographical constraints could be barriers, such as mountains or oceans, which are difficult to traverse, or simply disfavoured regions, where for example an important crop does not grow or land is ascribed negative cultural values. We capture this constrained expansion scenario in a simulation based on the following ideas (figure 3): languages are represented by areas consisting of cells of a grid. These areas randomly expand over time into free neighbouring cells, i.e. cells that are not occupied by another language or blocked by geographical constraints. In a sensitivity analysis (see electronic supplementary material, S6), we relax this assumption and allow up to three languages to occupy the same cell. With increasing area a language becomes more likely to split into two new languages, which in turn continue to expand separately. This simulation carries some resemblance to the model presented by Gavin et al. [26], but with crucial differences: while their model simulates language areas sequentially, the ExpSim model simulates splits, inducing a tree structure in the simulated languages.

Two illustrations of the expansion simulation: (a) every language is represented by a coloured area, which over time grows into free surrounding areas and splits up into new languages. The result is a phylogenetic and geographical expansion, as we can further see in (b): the expansion started at the black star and grows into new free space (visualized at three time points). Free space in this case is only available in a 72° sector. The phylogenetic tree is visualized by the black edges, leading from the root (black star) to the different areas, representing the extant languages.

As illustrated in figure 3, the ExpSim simulation is defined by two parallel processes: growing and splitting of areas. Growing is controlled by a parameter pgrow. At every step of the simulation, an area adds a random cell from its neighbourhood with probability pgrow. We implement heterogeneity in the growth rate by drawing pgrow uniformly from the interval [0, 1] for each language. At a certain randomly chosen size a language splits up into two. In our experiments, we choose the split-size uniformly at random and for every language independently between 70 and 100 cells. Again, the exact values do not significantly influence the results. We chose the distribution of the split-size to roughly match the expected tree size in the MigSim simulations. Again, we run the simulation for 5000 steps. In order to make space and time comparable, we define a step to correspond to 1 year and a cell to correspond to 100 × 100 km.

To introduce a directional trend into the simulation, we model geographical constraints. In reality, these constraints (e.g. deserts, mountains or seas) may have arbitrary shapes, but to systematically investigate the properties of constrained expansions we confine the simulations to geometric examples: the first language starts to expand in the corner of a circular sector of angle α and is only allowed to grow within this sector. Leaving the sector completely open (α = 2π) allows a concentric expansion in all directions. Choosing the sector very tightly (α → 0), forces the expansion to move in one direction out along this narrow sector. In this way, we can control α to achieve different levels of directional trend in the expansion. To compare this trend to the inherent bias parameter in the MigSim scenario, we will introduce a measure for the observed trend in §3.3.

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