Spatial dynamics for a non-monotone reaction-diffusion system with spatio-temporal delay

In this paper, we study the spreading speed and traveling wave solutions of a non-monotone reaction-diffusion system with spatio-temporal delay. By constructing a pair of auxiliary systems and using the Schauder 's fixed point theorem, the existence of the spreading speed is proved, which is consistent with the minimum wave speed of the traveling wave solution. The results show that the spreading speed and the traveling wave solution are convergent upward.