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 (F1) and migratory one (F2). The conversion between F1 and F2 is specified by such a way that if the total farmers F1 + F2 are overcrowded, F1 actively changes to F2, 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 F0) depending on the development of farming technology in a monotonically increasing way. It determines whether the total farmers are either over crowded (F1 + F2 >F0) or less crowded (F1 + F2 <F0) ( [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 cm(F0)) of travelling wave solutions, and that cm(F0) is monotonically decreasing as F0 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 F0 = 0 and F0 → ∞, and to prove cm(0)>cm(∞).

中文翻译：

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反应扩散系统中的行波模拟欧洲新石器时代过渡中农民和狩猎采集者的相互作用

最近我们提出了一个单稳态反应-扩散系统来解释新石器时代欧洲从狩猎采集生活到农民生活的转变。该系统由狩猎采集人群的三组分系统描述（H), 久坐的农民 (F1) 和迁移的 (F2）。之间的转换F1和F2是通过这样一种方式指定的，如果总农民F1+F2人满为患，F1积极改变F2，而如果它不那么拥挤，情况是反之亦然. 为了在系统中包含此属性，系统包含一个关键参数（例如F0) 以单调递增的方式依赖于耕作技术的发展。它确定农民总数是否过度拥挤（F1+F2>F0) 或不那么拥挤的 (F1+F2<F0) ( [9, 20])。先前的数值研究表明，系统行波解的结构在性质上类似于 Fisher-KPP 方程的一个，即农民的渐近扩展速度等于最小速度（例如C米(F0)) 的行波解决方案，以及C米(F0) 单调递减F0增加。后者的结果表明，农业技术的发展抑制了农民的扩张速度。作为对该属性的部分分析结果，本文的目的是考虑两个极限情况，其中F0= 0 和F0→ ∞，并证明C米(0)>C米(∞)。