The Neolithic transition began the spread of early agriculture throughout Europe through interactions between farmers and hunter-gatherers about 10,000 years ago. Archeological evidences indicate that the expanding velocity of farming into a region occupied by hunter-gatherers is roughly constant all over Europe. In the late twentieth century, from the contribution of the radiocarbon dating, it could be found that there are two types of farmers: one is the original farmer and the other is the converted farmer which is genetically hunter-gatherers but learned agriculture from neighbouring farmers. Then this raises the following questions: Which farming populations play a key role in the expansion of farmer populations in Europe? and what is the fate of hunter-gatherers (e.g., become extinct, or live in lower density, or live in agricultural life-style)? We consider a three-component reaction–diffusion system proposed by Aoki, Shida and Shigesada, which describes the interactions among the original farmers, the converted farmers, and the hunter-gatherers. In order to resolve these two questions, we discuss traveling wave solutions which give the information of the expanding velocity of farmer populations. The main result is that two types of traveling wave solutions exist, depending on the growth rate of the original farmer population and the conversion rate of the hunter-gatherer population to the converted farmer population. The profiles of traveling wave solutions indicate that the expansion of farmer populations is determined by the growth rate of the original farmer and the (maximal) carrying capacity of the converted farmer, and the fate of hunter-gatherers is determined by the growth rate of the hunter-gatherer and the conversion rate of the hunter-gatherer to the converted farmer. Thus, our results provide a partial answer to the above two questions.
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