Spreading speeds and uniqueness of traveling waves for a reaction diffusion equation with spatio-temporal delays

A class of reaction diffusion equation with spatio-temporal delays is systematically investigated. When the reaction function of this equation is nonlinear without monotonicity, it is shown that there exists a spreading speed c⁎>0c⁎>0 for this equation such that c⁎c⁎ is linearly determinate and coincides with the minimal wave speed of traveling waves, and that this equation admits a unique traveling wave (up to translation) with speed c>c⁎c>c⁎ and no traveling wave with c