  In this paper, we investigate the expanding patterns and spreading speed of
solutions of farmer and hunter-gatherer model which is a three-component
degenerate reaction-diffusion system. Ecologically speaking, since the
lifestyle of agriculture and settlement allows for a larger population, after
an initially localized population of farmers migrated into a region occupied by
hunter-gatherers, the Neolithic transition from hunter-gatherers to farmers
happened. This model was proposed by J. Elias, K. Humayun and M. Mimura in
2018. By numerical simulations, a travelling wave solution with a certain
speed, depending on the parameter values, is observed. Despite such observation
and studying on the well-posedness of the system, no mathematically rigorous
studies on the spreading properties have been made. The main difficulty comes
from the fact that the comparison principle does not hold for the whole system.
In this paper, we give theoretical proof of the existence of two different
expanding patterns. Furthermore, we show a lower estimate for the spreading
speed.
