摘要
Recently we have proposed a monostable reaction-diffusion system to explain the Neolithic transition from hunter-gatherer life to farmer life in Europe. The system is described by a three-component system for the populations of hunter-gatherer (H), sedentary farmer (F <sub>1</sub> ) and migratory one (F <sub>2</sub> ). The conversion between F <sub>1</sub> and F <sub>2</sub> is specified by such a way that if the total farmers F <sub>1</sub> + F <sub>2</sub> are overcrowded, F <sub>1</sub> actively changes to F <sub>2</sub> , while if it is less crowded, the situation is vice versa. In order to include this property in the system, the system incorporates a critical parameter (say F <sub>0</sub> ) depending on the development of farming technology in a monotonically increasing way. It determines whether the total farmers are either over crowded (F <sub>1</sub> + F <sub>2</sub> >F <sub>0</sub> ) or less crowded (F <sub>1</sub> + F <sub>2</sub> <F <sub>0</sub> ) ([9, 20]). Previous numerical studies indicate that the structure of travelling wave solutions of the system is qualitatively similar to the one of the Fisher-KPP equation, that the asymptotically expanding velocity of farmers is equal to the minimal velocity (say c <sub>m</sub> (F <sub>0</sub> )) of travelling wave solutions, and that c <sub>m</sub> (F <sub>0</sub> ) is monotonically decreasing as F <sub>0</sub> increases. The latter result suggests that the development of farming technology suppresses the expanding velocity of farmers. As a partial analytical result to this property, the purpose of this paper is to consider the two limiting cases where F <sub>0</sub> = 0 and F <sub>0</sub> → ∞, and to prove c <sub>m</sub> (0)>c <sub>m</sub> (∞).