  In this paper, we investigate the spreading properties of solutions of the
Aoki-Shida-Shigesada model. This model is a three-component reaction-diffusion
system that delineates the geographical expansion of an initially localized
population of farmers into a region occupied by hunter-gatherers. By
considering the scenario where farmers and hunter-gatherers possess identical
motility, Aoki et al. previously concluded, through numerical simulations and
some formal linearization arguments, that there are four different types of
spreading behaviors depending on the parameter values. In this paper, we
concentrate on the general case for which farmers and hunter-gatherers possess
different motility. By providing more sophisticated estimates, we not only
theoretically justify the spreading speed of the Aoki-Shida-Shigesada model,
but also establish sharp estimates for the long-time behaviors of solutions.
These estimates enable us to validate all four types of spreading behaviors
observed by Aoki et al..
