Fu Sheng-Chen, Mimura Masayasu, Tsai Je-Chiang
Department of Mathematical Sciences, National Chengchi University, 64, S-2 Zhi-nan Road, Taipei, 116, Taiwan.
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, Tokyo, Japan.
J Math Biol. 2021 Mar 2;82(4):26. doi: 10.1007/s00285-021-01585-3.
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.
新石器时代的转变始于约10000年前农民与狩猎采集者之间的互动，使得早期农业在欧洲得以传播。考古证据表明，农业向狩猎采集者占据地区的扩张速度在欧洲各地大致恒定。在20世纪后期，根据放射性碳年代测定法的研究发现，存在两种类型的农民：一种是原生农民，另一种是转变型农民，他们在基因上是狩猎采集者，但从邻近农民那里学会了农业。于是引发了以下问题：哪些农业群体在欧洲农民群体的扩张中起关键作用？狩猎采集者的命运如何（例如，灭绝、低密度生存或采用农业生活方式）？我们考虑了青木、志田和重定提出的一个三组分反应扩散系统，该系统描述了原生农民、转变型农民和狩猎采集者之间的相互作用。为了解决这两个问题，我们讨论了行波解，它给出了农民群体扩张速度的信息。主要结果是，根据原生农民群体的增长率和狩猎采集者群体向转变型农民群体的转化率，存在两种类型的行波解。行波解的轮廓表明，农民群体的扩张由原生农民的增长率和转变型农民的（最大）承载能力决定，而狩猎采集者的命运由狩猎采集者的增长率和狩猎采集者向转变型农民的转化率决定。因此，我们的结果为上述两个问题提供了部分答案。
