Spatial dynamics for a non-quasi-monotone reaction-diffusion system with delay and quiescent stage

In this paper, we study the spreading speed and traveling wave solutions of a non-quasi-monotone delayed reaction-diffusion model for a single species population with separate mobile and stationary states. By using comparison arguments, Schauder's fixed-point theorem and a limiting process, we establish the existence of the spreading speed and characterize it as the minimal wave speed for traveling wave solutions. The upward convergence of the spreading speed and traveling wave solutions are also established by applying a fluctuation method. In particular, the effects of the delay and transfer rates on the spreading speed are investigated.