Time-Delayed Theory of the Neolithic Transition in Europe

Abstract

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 ,

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