Pulsating traveling fronts and entire solutions in a discrete periodic system with a quiescent stage

In this paper, we study pulsating traveling fronts and entire solutions for a spatially discrete periodic reaction–diffusion system with a quiescent stage. Here, the entire solutions are defined in the whole space and for all time t ∈ R . Under the monostable assumption, the existence and asymptotic behavior of pulsating traveling fronts and spatially periodic solutions connecting two periodic steady states are first established. Combining the leftward and rightward pulsating traveling fronts with different speeds and a spatially periodic solution, the existence and qualitative properties of entire solutions are then proved. Finally, some numerical simulations are carried out to confirm the theoretical results.