In this paper, we investigate the spreading properties of solutions of farmer and hunter-gatherer model which is a three-component reaction-diffusion system. Ecologically, the model describes the geographical spreading of an initially localized population of farmers into a region occupied by hunter-gatherers. This model was proposed by Aoki, Shida and Shigesada in 1996. By numerical simulations and some formal linearization arguments, they concluded that there are four different types of spreading behaviors depending on the parameter values. Despite such intriguing observations, no mathematically rigorous studies have been made to justify their claims. The main difficulty comes from the fact that the comparison principle does not hold for the entire system. In this paper, we give theoretical justification to all of the four types of spreading behaviors observed by Aoki et al. Furthermore, we show that a logarithmic phase drift of the front position occurs as in the scalar KPP equation. We also investigate the case where the motility of the hunter-gatherers is larger than that of the farmers, which is not discussed in the paper of Aoki et al.

在本文中，我们研究了农民和狩猎采集模型解的传播特性，这是一个三组分反应扩散系统。在生态学上，该模型描述了最初本地化的农民人口向狩猎采集者占领的地区的地理分布。该模型由 Aoki、Shida 和 Shigesada 于 1996 年提出。通过数值模拟和一些形式线性化论证，他们得出结论，根据参数值，存在四种不同类型的传播行为。尽管有这些有趣的观察，但没有进行数学上严格的研究来证明他们的主张。主要的困难来自于比较原则并不适用于整个系统的事实。在本文中，我们对 Aoki 等人观察到的所有四种传播行为给出了理论依据。此外，我们表明前部位置的对数相位漂移发生在标量 KPP 方程中。我们还调查了狩猎采集者的运动量大于农民的运动量的情况，这在 Aoki 等人的论文中没有讨论。