# Spreading Waves in a Farmers and Hunter-Gatherers Model of the Neolithic Transition in Europe

## Abstract

The Neolithic transition began the spread of early agriculture throughout Europe through interactions between farmers and hunter-gatherers about 10,000 years ago. Archeological evidence produced by radiocarbon dating indicates that the expanding velocity of farming is roughly constant all over Europe. Theoretical understanding of such evidence has been performed from mathematical modeling viewpoint. However, the expanding velocity determined by existing modeling approaches is faster than the observed velocity. For understanding this difference, we propose a three-component reaction–diffusion system which consists of two different types of farmers (sedentary and migratory) and hunter-gatherers from the viewpoint of the influence of farming technology. Our purpose is to study the relation between the expanding velocity of farmers and the farming technology parameter (say, \(\gamma \)). In this paper, we mainly focus on the one-dimensional traveling wave solution with minimal velocity and show that the minimal velocity decreases, as \(\gamma \) increases. This can be compatible with the observed velocity when farming technology is developed. Our results suggest that the reason for the slowdown of the Neolithic transition might be related to the increase in the development of farming technology.

## Keywords

Neolithic transition Reaction–diffusion model of farmers and hunter-gatherers interaction Traveling wave solutions## Mathematics Subject Classification

01A10 35K57 35C07## Notes

### Acknowledgements

The authors are grateful to the anonymous referees for their useful suggestions and comments which helped improve the exposition of the paper. MHK acknowledges the support of GCOE program of MIMS, Meiji University, Japan, during doctoral study. MM is partially supported by JSPS KAKENHI Grants Nos. 15K13462 and 16H01728. JCT is supported by MOST and NCTS of Taiwan.

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