农民和狩猎采集者三分量反应扩散模型的扩散性质
Spreading properties of a three-component reaction-diffusion model for the population of farmers and hunter-gatherers
摘要：
本文研究了一类三分量反应扩散系统的农民-狩猎-采集者模型解的传播性质。从理论上讲，该模型描述了最初本地化的农民人口向狩猎采集者占据的地区的地理扩散。该模型是由Aoki，Shida和Shigesada于1996年提出的，通过数值模拟，他们根据参数的不同将传播行为分为四种不同的类型。尽管有如此有趣的观察结果，但还没有进行数学上严格的研究来证明这些数值观察结果的合理性。主要的困难来自于这样一个事实，即比较原则不适用于整个系统。本文对这四种传播行为进行了理论论证。此外，我们表明，对数相位漂移发生标量KPP方程。
英文摘要：
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, and by numerical simulations, they classified the spreading behaviors into four different types depending on the parameter values. Despite such intriguing observations, no mathematically rigorous studies have been made to justify these numerical observations. The main difficulty comes from the fact that the comparison principle does not hold for the whole system. In this paper, we give theoretical justification to all the four types of spreading behaviors. Furthermore, we show that a logarithmic phase drift occurs as in the scalar KPP equation.
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