| Abstract: |
| The Neolithic migration of farmers in regions previously inhabited by hunter-gatherers is a long time studied ecological problem. We propose a new reaction-diffusion model that consists of equations governing the spatio-temporal evolution of sedentary and migrating farmers and hunter-gatherers in the Neolithic transition. Ecologically, the model stems from the fact that a lifestyle of agriculture and settlement, as it allows for a larger population, is evolutionary advantageous than hunting and gathering. Therefore, in our modelling framework, we assume that farmers do not migrate unless the population density pressure forces them. The population density pressure is, in a sense, linked with a certain level of development of farming and food-producing technology and associated sedentary lifestyle traits such as pottery making, domestication of various plants and animals, and related social and cultural changes, security, trading. We prove the global well-posedness of the system and show numerically that for a suitable value of a stay-or-migrate-threshold the model reproduces the spread of farming that corresponds to the archeological findings in Europe. Moreover, due to different time-scales between the intra-population dynamics (slow) and the stay-or-migrate decision making (fast) we consider a singular limit of the proposed problem when the conversion stay-or-migrate-rate grows above all the limits. We show that in the limit we obtain a nonlinear diffusion problem of interesting spatio-temporal properties such as the breakdown of radial symmetry of the solution. |
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