Monostable pulsating traveling waves in discrete periodic media with delay

The purpose of this article is to investigate various qualitative properties of pulsating traveling waves for a delayed lattice dynamical system with global interaction in periodic habitat. Under some appropriate assumptions, we first establish a Harnack type of inequality for its wave profile system. Then, we show that all wave profiles remain strictly increasing in the propagation process. Based on the Harnack's inequality and monotonicity result, the other effort of the article is devoted to the exponential decay of the non-critical pulsating traveling waves towards the unstable steady state. Finally, we obtain the uniqueness of all non-critical pulsating traveling waves.