  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.
