The Role of Advection in Modeling Population Migrations

May 24, 2014 by

(Thanks to Vitaliy Rayz for his help with this post.)

As I argued elsewhere, advection as well as diffusion should be taken into account in mathematical models of language expansion. Here, we examine one study of population movement that does just that. This study, co-authored by Kate Davison, Pavel Dolukhanov, Graeme R. Sarson, and Anvar Shukurov of University of Newcastle upon Tyne and published in Journal of Archaeological Science in 2006, aims to model the spread of Neolithic farmers in Europe from a localized area in the Near East. This work is particularly relevant to the linguistic issue at hand, as Bouckaert et al. argue that Indo-European languages were carried from Anatolia into Europe and elsewhere by those very Neolithic farmers. Yet, Bouckaert et al. make no reference to this significant article, even though it was published six years prior to the their own piece in Science.

The Davison et al. article provides a more reasonable model of population dispersion throughout Europe than what Bouckaert et al. propose. First, Davison et al. take geographical space to be highly anisotropic, which means that its properties vary in different directions. They are therefore able to take into account spatial variation in both human mobility and the carrying capacity of the land. Their calculations also encompass variations in altitude and latitude, which do not seem to play any role in Bouckaert et al.’s model. Second, Davison et al. focus on the role played in the advance of farming by coastlines and major river paths, such as the Danube and the Rhine, which must have been especially attractive to the farmers due to the presence there of fertile alluvial soil. Such waterways are accounted for by introducing an advection term confined to the proximity of such major rivers and coastlines. Bouckaert et al., in contrast, distinguish only “land” and “water”, the latter category including both open seas and rivers, without noting that one form of water tended to impede movement while the other more often facilitated it. Third, Davison et al.’s model is validated against radiocarbon dates of archeological finds, especially those involving the Linear Pottery (LBK) and the Impressed Ware traditions along the Danube-Rhine corridor and the Mediterranean coastline, respectively. Bouckaert et al.’s model, in contrary, runs afoul of such archeological evidence as wheeled vehicles, as discussed in a GeoCurrents post.

Bouckaert et al. refer to their model as “a feature-rich geographical model where different locations can have different migration patterns” (Supplementary Materials, p. 11); however, it is not nearly as “landscape based” as the authors make it sound, as the only landscape feature that they take into account is the distinction between “land” and “water. Bouckaert et al. claim that “the approach [they] have developed could in principle be extended to incorporate other geographic features such as mountains or deserts” (Supplementary Materials, pp. 12-13); yet they do not incorporate anything of the sort in their calculations. As mentioned in a GeoCurrents post, Bouckaert et al. have some Indo-European migration fronts advance along ridges well in excess of 20,000 feet in elevation. Nor do they consider the differences between lower latitudes (warmer climate, higher carrying capacity, easier to move through) and higher latitudes. It is not clear whether their model can adequately incorporate such factors, or how it could be done.

Let’s consider Davison et al.’s proposal more closely. They model population dispersion from the Near East as a Boundary Value Problem. Jericho, one of the earliest large settlement sites with complicated masonry structures and fortifications, is used as the point of origin of the population spread. The Ural Mountains (about 60° E) form a natural eastern boundary, the western boundary is set at 15° W (to include the most westerly reaches of Ireland and Spain), while the northern and southern boundaries are taken to be 75° N and 25° N, respectively. The authors “adopt zero flux conditions at each of the four boundaries” (p. 646), meaning that population movement does not cross those limits. In other words, migration does not lead into the Atlantic or the Arctic Oceans, across the Ural Mountains, or into the Sahara desert. Thus, in effect Davison et al. already do what Bouckaert et al. say that they model “could in principle” do, that is incorporate landscape features such as mountains and deserts, as well as the sea/land distinction.

basic_equasion

In order to calculate the population density (N) at any given point in space and time, Davison et al. use the basic equation reproduced on the left. According to this equation, the change in population density is produced by three factors (from right to left in the equation): the diffusion term, the basic logistic growth term, and the advection term. As we shall see below, geographical factors such as altitude, latitude, opportunity for sea travel, major rivers, and coastlines, play an important role in defining all of these terms. Thus, compared to Bouckaert et al., Davison et al. take the anisotropic nature of geographical space seriously: factors such as latitude and altitude play an important role throughout their calculations, while water is seen as sometimes impeding human migration and sometimes facilitating it.

Let’s consider the terms of their basic equation one at a time, starting with the logistic growth term. This component describes changes in population density if the population remains static over time; that is, if nobody moves anywhere. The two variables in this term are the intrinsic growth rate of the population (γ) and the carrying capacity of the land (K), both of which “may vary in space (and time), to model the variation in the habitats’ ability to support the population” (p. 644). In particular, K is modulated with a linear function of latitude, as the harsh climate of more northern areas means that fewer people per unit of area can be supported.

diffusivity_map

The last term on the right-hand side of Davison et al.’s basic equation “describes the diffusion resulting from random migration events, quantified by the diffusivity ν” (p. 645). The diffusivity coefficient depends on the distance an average member of the population can move over a given span of time and can be thought of as an expression of human mobility.* Because it is harder to travel in the harsher climates of higher latitudes, the diffusivity coefficient, like the carrying capacity, is modulated with a linear function of latitude; both K and ν decrease by a factor of 2 between Greece and Denmark. Altitude too affects the diffusivity, as it is harder to travel across mountainous areas. The shades of grey in the map reproduced on the left show the distribution of diffusivity near the Danube estuary due to altitude variations. As can be seen from this map, diffusivity is highest along a river path, followed by lowland areas. Highlands, such as the Carpathian Mountains north of the Danube, have a much lower diffusivity coefficient, and some mountainous areas are effectively impenetrable, as diffusivity vanishes at altitudes exceeding 1,000 meters. Sea areas too are modeled as having lower diffusivity than flat lowland areas; however, contrary to Bouckaert et al.’s assumption, seas are not isotropic. Davison et al. allow for limited sea travel; in their model “the diffusivity decreases exponentially with distance from land, over a length scale of 10 km” (p. 647).** As a result, population can diffuse—albeit at a slower pace—across relatively narrow straits, such as the English Channel or the Danish straits, which “allows the British Isles to be populated (albeit with some delay), and Scandinavia to be populated via Northern Germany and Denmark” (p. 647). In contrast, under Bouckaert et al.’s basic assumptions (which they do abandon in their alternative “sailor model”), it is not clear how the British Isles and other islands come to be populated at all, and their maps do indeed show many Mediterranean islands as never having been reached by Indo-European speakers. Moreover, a significant distinction is drawn by Davison et al. between the Aegean Sea and the Black Sea. Because of numerous small islands, most of the Aegean constitutes areas of lower diffusivity yet not impenetrable to migration. In contrast, the Black Sea cannot be crossed; the only way for population spread around it must take the coastal sailing route or proceed by land where possible.

diffusion_only_dispersal_map

Crucially, Davison et al. argue that the two terms on the right-hand side of the equation—the basic logistic growth term and the diffusion term—are insufficient to describe the dispersal of agricultural populations in Neolithic Europe. Assuming values for their model parameters—the intrinsic growth rate of the population γ, the carrying capacity K, and background diffusivity ν—based on estimates from previous literature, Davison et al.’s simulation arrives at the isochron map reproduced on the left. This map shows population dispersing from the point of origin in the Near East, like waves from an object dropped into water. These waves are deflected by mountains such as the Alps and the Caucasus, as well as the impenetrable seas. For example, one migration route is modeled as having passed through a narrow passage on the south coast of the Black Sea and then via sea travel along the coastline. As a result the population spread through the Caucasus region must have been slow. Notably, the major river paths, those of the Danube and the Rhine, marked on the map by a blue line, do not deflect the diffusional pattern, neither facilitating nor impeding migration, according to this diffusion-only model. Also crucial are the late dates at which the population front reaches Scandinavia and the British Isles, 5,000 and 5,500 years, respectively. Still, by allowing sea travel in waters close enough to the shore, this model correctly predicts that farming first entered Sweden from southern Scandinavia by the sea route, rather than through Russia and Finland, which agrees with archaeological evidence.

Overall, however, this diffusion-only model does not correlate well with dates of Neolithic farmers’ arrival to Western and Northern Europe established by radiocarbon dating of archeological finds. These discrepancies prompt Davison et al. to include an additional term into their equation for advection, which accounts for the enhanced ability, and motivation, of the population to move in particular directions. It is important to note that “such an advection arises naturally if the random walk that underlies diffusion is anisotropic; e.g., if the length of step depends on the direction in which it is taken” (p. 645). In the absence of the advection term, a certain amount of variation in the rate of spread can be attributed to variations in either the intrinsic growth rate or the diffusivity, or both. But can the variations in these two parameters explain the full range of variation in the rate of spread? The answer, according to Davison et al., is negative. The intrinsic growth rate cannot vary significantly: its upper bound is determined by the facts of human biology (a woman can give birth to only a certain number of children over a period of time). Moreover, it has been observed that the intrinsic growth rate varies very little in very diverse environments. If so, the burden of explaining variation in the speeds of the propagation of the population front must be carried by the diffusivity coefficient, which must vary by a factor of 16–100. Davison et al. take that magnitude of variation to be implausible. In other words, diffusion that is so non-uniform over space effectively means advection: too much bias of migration in a particular direction means that movement is not random, thus making advection a more appropriate mechanism than diffusion. Compare this to a coin toss: a genuine, evenly-weighted coin should land heads or tails randomly. Over a large number of tosses, the number of times that the coin would land heads-up would approach 50 percent. Over a short sequence of tosses, however, the coin may well land heads-up many times in a row, thus seeming to exhibit a certain bias. But if a coin lands heads-up over an implausibly large number of times, it can no longer be considered a random phenomenon—and the coin is probably a fake. The advection term, introduced by Davison et al., means that the propagation speed along a major river is much higher than across land, approximately by a factor of 5; the propagation speed along the Mediterranean coast is higher by an order of magnitude.

diffusion_advection_dispersal_map

With the advection term as part of the equation, Davison et al. produce a second isochron map, reproduced on the left. The crucial difference between this map and the previous one is that in this diffusion-cum-advection model, the lines of population advance no longer cross the major river path (shown again in blue); instead, they run alongside it as population diffuses away from the major rivers. Even though the advection is restricted to the narrow vicinity of waterways in this model, it significantly affects the global propagation speed of agricultural populations. For example, an examination of the 500-year isochrones in the two maps shows that the population has moved approximately 500 km into Africa in the diffusion-only model, but this distance is significantly increased in the diffusion-cum-advection model. The advance of the population over the loess plains of Central Europe along the Danube and Rhine valleys causes the population to reach France, Belgium, and Denmark sooner than in the diffusion-only model. Although “still having suffered a 500-year delay due again to the inhibiting effects of the water barrier” (p. 649), the population front reaches the British Isles and Norway after only 3,500 years, that is 1,500–2,000 years earlier than in the diffusion-only model. These patterns and dates are much more in accord with the radiocarbon dates of archeological finds than those of the diffusion only model. While the authors point out a number of problems with their model, the relative success of the computation that takes into account the advection term, even as it is confined to the close proximity of rivers and coastlines, highlights not only the role of major river paths and coastlines in the advance of farming, but also the importance of advection in modeling human migrations.

IE_tree

While Davison et al.’s calculations correlate well with the archeological record, the dates proposed by Bouckaert et al. for the splits in their phylogenetic tree of Indo-European languages are a poor fit for Davison et al.’s diffusion-cum-advection model. For example, according to this model, it takes only about 2,000 years for the migration from Anatolia to cross the English Channel and to reach the southern shores of the British Isles. Yet according to Bouckaert et al., it takes over 3,500 years from the highest-order split of Indo-European into Anatolian and non-Anatolian branches to the split between Celtic and Italic languages (though presumably the latter split took place on the continent, with a subsequent migration of Insular Celts into Britain taking place even later).

_____________

* The only difference between this part of Davison et al.’s equation and the standard diffusion formula is due to the fact that they take diffusivity to be a function of geographical coordinates.

** According to Davison et al., “both the intrinsic growth rate γ and the carrying capacity K vanish in the sea” (p. 647).


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