This talk is concerned with propagation phenomena in a nonlocal delayed reaction-diffusion equation in time almost periodic media.

More specifically, we investigate the existence and nonexistence of transition waves and the upper and lower bound estimates of spreading speeds. By studying properties of exponential decaying solutions of the corresponding linearized problem, constructing appropriate upper and lower solutions, and using comparison arguments, we first prove that no matter the birth rate function is monotone or not, there is a critical wave speed such that a transition wave exists as soon as the mean value of the wave speed is greater than this critical speed. Moreover, in the monotone case we confirm the asymptotic behavior of the transition wave. Under some further assumptions, we establish the upper and lower bound estimates of spreading speeds for solutions of the Cauchy problem with slow and fast exponential decay initial values and also obtain the nonexistence of transition waves. Finally, a brief discussion is given to demonstrate that the critical wave speed obtained in the present paper coincides with the minimum speed and spreading speeds observed by others for some special cases. This is a joint work with Dr. Ning Wang and Professor Shigui Ruan.