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We present a mathematical model, based on the compilation and statistical processing of radiocarbon dates, of the transition from the Mesolithic to the Neolithic, from about 7000 to 4000 BC in Europe. With the arrival of the Neolithic, hunting and food gathering gave way to agriculture and stock breeding in many parts of Europe; pottery-making spread into even broader areas. Archaeological evidence, radiocarbon dates and genetic markers are consistent with the spread of farming from a source in the Near East. However, farming was less important in the East; the Eastern and Western Neolithic have distinct signatures. We use a population dynamics model to suggest that this distinction can be attributed to the presence of two waves of advance, one from the Near East, and another through Eastern Europe. Thus, we provide a quantitative framework in which a unified interpretation of the Western and Eastern Neolithic can be developed.