Spreading properties of a three-component reaction-diffusion model for the population of farmers and hunter-gatherers

Abstract

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.