The origins of early farming and its spread to Europe have been the subject of major interest for some time. The main controversy today is over the nature of the Neolithic transition in Europe: the extent to which the spread was, for the most part, indigenous and animated by imitation (cultural diffusion) or else was driven by an influx of dispersing populations (demic diffusion). We analyze the spatiotemporal dynamics of the transition using radiocarbon dates from 735 early Neolithic sites in Europe, the Near East, and Anatolia. We compute great-circle and shortest-path distances from each site to 35 possible agricultural centers of origin—ten are based on early sites in the Middle East and 25 are hypothetical locations set at 5° latitude/longitude intervals. We perform a linear fit of distance versus age (and vice versa) for each center. For certain centers, high correlation coefficients (R > 0.8) are obtained. This implies that a steady rate or speed is a good overall approximation for this historical development. The average rate of the Neolithic spread over Europe is 0.6–1.3 km/y (95% confidence interval). This is consistent with the prediction of demic diffusion (0.6–1.1 km/y). An interpolative map of correlation coefficients, obtained by using shortest-path distances, shows that the origins of agriculture were most likely to have occurred in the northern Levantine/Mesopotamian area.

The classical wave-of-advance model of the neolithic transition (i.e., the shift from hunter-gatherer to agricultural economies) is based on Fisher's reaction-diffusion equation. Here we present an extension of Einstein's approach to Fickian diffusion, incorporating reaction terms. On this basis we show that second-order terms in the reaction-diffusion equation, which have been neglected up to now, are not in fact negligible but can lead to important corrections. The resulting time-delayed model agrees quite well with observations. [ S0031-9007(98) [11,12]. Thus HRD equations, instead of the usual PRD equations, should be regarded as the first choice from a conceptual perspective. Hyperbolic reaction-diffusion equations have been very recently applied to the spread of epidemics [13], forest fire models [14], and chemical systems [15].An interesting application of PRD equations arose after archaeological data led to the conclusion that European farming originated in the Near East, from where it spread across Europe. The rate of this spread was measured [16], and a mathematical model was proposed according to which early farming expanded in the form of a PRD wave of advance [17]. Such a model provides a consistent explanation for the origin of Indo-European languages [18], and also finds remarkable support from the observed gene frequencies [19]. However, this PRD model predicts a velocity for the spread of agriculture that is higher than that inferred from archaeological evidence, provided that one accepts those values for the parameters in the model that have been measured in independent observations [17]. Here we will analyze this problem by means of a HRD model.Let p͑x, y, t͒ stand for the population density (measured in number of families per square kilometer), where x and y are Cartesian coordinates and t is the time. We assume that a well-defined time scale t between two successive migrations exists. We begin, as usual [20], noting that, between the values of time t and t 1 t, both migrations and population growth will cause a change in the number of families in an area differential ds dx dy, i.e., ͓p͑x, y, t 1 t͒ 2 p͑x, y, t͔͒ds ͓p͑x, y, t 1 t͒ 2 p͑x, y, t͔͒ m ds 1 ͓p͑x, y, t 1 t͒ 2 p͑x, y, t͔͒ g ds ,where the subindices m and g stand for migrations and population growth, respectively. We denote the coordinate variations of a given family during t by Dx and Dy. The effect of migrations on the evolution of p͑x, y, t͒ will be derived here by means of a simple extension of Einstein's model of Fickian diffusion [21]. The migration term in Eq. (1) can be written as ͓p͑x, y, t 1 t͒ 2 p͑x, y, t͔͒ m ds ds Z 12`Z 12`p ͑x 1 Dx, y 1 Dy, t͒f͑Dx, Dy͒dDx dDy 2 dsp͑x, y, t͒ ,where f͑Dx, Dy͒ is the fraction of those families lying at time t in an area ds, centered at ͑x 1 Dx, y 1 Dy͒, such that they are at time t 1 t in an area ds, centered at ͑x, y͒. Therefore, Z 12`Z 12`f ͑Dx, Dy͒dDx dDy 1 ,

There is a long-standing controversy between two models of the Neolithic transition. The demic model assumes that the Neolithic range expansion was mainly due to the spread of populations, and the cultural model considers that it was essentially due to the spread of ideas. Here we integrate the demic and cultural models in a unified framework. We show that cultural diffusion explains ∼40% of the spread rate of the Neolithic transition in Europe, as implied by archaeological data. Thus, cultural diffusion cannot be neglected, but demic diffusion was the most important mechanism in this major historical process at the continental scale. This quantitative approach can be useful also in regional analysis, the description of Neolithic transitions in other continents, and models of many human spread phenomena.he Neolithic transition, a major episode in human history, is defined as the shift from a hunter-gatherer economy (Paleolithic) into another one based on agricultural activities (Neolithic) (1). In the Near East, this transition took place ∼12,000 y ago, and from there it spread across Europe until ∼5,000 y ago (2-4). Archaeologists have provided many data that make it possible to measure the speed of the spread of the Neolithic transition, but they disagree on which of the following possibilities is correct: (i) it was mainly a demic process (range spread of farmers) (5); (ii) it was mainly a cultural one (transmission of the plants, animals and knowledge of farmers to hunter-gatherers (6)); or (iii) it was mainly demic in some regions and mainly cultural in others (7). It is important to note that many authors have clearly argued for the importance of both demic and cultural diffusion. For example, Ammerman and Cavalli-Sforza (8), when introducing their demic diffusion model in 1973, wrote that demic and cultural diffusion are not mutually exclusive, and discussed the interactions between the Neolithic and Paleolithic populations that would have led to cultural diffusion and genetic clines. These authors also made some crucial statements: "The real question may well be to evaluate the relative importance of demic and cultural diffusion in different regions of Europe" because "in some areas both are likely to have contributed to the spread of farming," but "what is necessary before such an attempt can be made is the introduction of much more specific models" (ref. 4, pp. 6, 135, and 62, respectively). This is precisely the problem to which the present paper aims to contribute: we will here present a model, and apply it to determine the importance of demic and cultural diffusion on the spread rate at the continental scale. We will also outline how our model could be applied to solve the same problem at regional scales in future work.Several aspects of transitions in human prehistory have been analyzed during the past decade using increasingly refined mathematical models (9-16). On the other hand, genetic studies have led to an increasing consensus that demic dispersal was important in the Neolithic transition in ...

The spread of viruses in growing plaques predicted by classical models is greater than that measured experimentally. There is a widespread belief that this discrepancy is due to biological factors. Here we show that the observed speeds can be satisfactorily predicted by a purely physical model that takes into account the delay time due to virus reproduction inside infected cells. No free or adjustable parameters are used.

We review the recent theoretical progress in the formulation and solution of the front speed problem for time-delayed reaction-diffusion systems. Most of the review is focused on hyperbolic equations. They have been widely used in recent years, because they allow for analytical solutions and yield a realistic description of some relevant phenomena. The theoretical methods are applied to a range of applications, including population dynamics, forest fire models, bistable systems and combustion wavefronts. We also present a detailed account of successful predictions of the models, as assessed by comparison to experimental data for some biophysical systems, without making use of any free parameters. Areas where the work reviewed may contribute to future progress are discussed.

The Neolithic transition is the shift from hunting–gathering into farming. About 9000 years ago, the Neolithic transition began to spread from the Near East into Europe, until it reached Northern Europe about 5500 years ago. There are two main models of this spread. The demic model assumes that it was mainly due to the reproduction and dispersal of farmers. The cultural model assumes that European hunter–gatherers become farmers by acquiring domestic plants and animals, as well as knowledge, from neighbouring farmers. Here we use the dates of about 900 archaeological sites to compute a speed map of the spread of the Neolithic transition in Europe. We compare the speed map to the speed ranges predicted by purely demic, demic–cultural and purely cultural models. The comparison indicates that the transition was cultural in Northern Europe, the Alpine region and west of the Black Sea. But demic diffusion was at work in other regions such as the Balkans and Central Europe. Our models can be applied to many other cultural traits. We also propose that genetic data could be gathered and used to measure the demic kernels of Early Neolithic populations. This would lead to an enormous advance in Neolithic spread modelling.

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The earliest dates for the West Mediterranean Neolithic indicate that it expanded across 2,500 km in about 300 y. Such a fast spread is held to be mainly due to a demic process driven by dispersal along coastal routes. Here, we model the Neolithic spread in the region by focusing on the role of voyaging to understand better the core elements that produced the observed pattern of dates. We also explore the effect of cultural interaction with Mesolithic populations living along the coast. The simulation study shows that (i) sea travel is required to obtain reasonable predictions, with a minimum sea-travel range of 300 km per generation; (ii) leapfrog coastal dispersals yield the best results (quantitatively and qualitatively); and (iii) interaction with Mesolithic people can assist the spread, but long-range voyaging is still needed to explain the archaeological pattern.T he Neolithic transition in Europe spread at an average rate of about 1 km·y −1 (1, 2). This process can be modeled by the socalled wave-of-advance model, which describes a progressive land-based expansion due to population growth and short-range migratory activity (3). Ancient DNA studies provide support for a mainly demic expansion in many parts of Europe (4), involving two main pathways: one up the Danube, connected with the spread of the Linearbandkeramic (LBK) culture (5, 6), and the other along the Mediterranean shores (7).Current radiocarbon dates indicate a coastal spread in West Mediterranean Europe taking place at a much faster rate (above 5 km·y −1 ) than one would expect on the basis of the classical wave-of-advance model. An alternative approach is needed to explain this process. The maritime pioneer colonization model (8, also refs. 9, 10) postulates a sea-based expansion that involves voyaging along the coast in the form of cabotage (with the possibility of making a short stop here and there along the way). This model drew upon new and more reliable carbon-14 dates [including accelerator mass spectrometry (AMS) determinations on short-lived samples to avoid the old wood effect], and is consistent with a demic expansion and the observed pattern (alternative approaches to the West Mediterranean spread are discussed in ref. 11). During the past 15 y, quality dates for the Early Neolithic in the West Mediterranean have continued to come in. As a result, the overall pattern is now more refined but remains consistent with the maritime pioneer model.Voyaging during the Early Neolithic is well documented in the Eastern Mediterranean (e.g.,. From the distribution of obsidian artifacts in the Cyprus, Aegean, and Tyrrhenian basins, we know that its quantity tends to fall off with distance from a given source and that long-distance crossing of the open sea between these three basins is extremely rare (15). In short, it is fair to say that early voyaging in the eastern and central parts of the Mediterranean was kept on a comparatively short leash. In the West Mediterranean, obsidian is far less common at Early Neolithic sites; it occurs in small...