Pulsating waves of a partially degenerate reaction–diffusion system in a periodic habitat

We investigate a partially degenerate reaction–diffusion system in a periodic habitat and prove the existence and stability of pulsating waves. More specifically, we show that if the wave speed is greater than the spreading speed, then there exists a pulsating wave connecting the stable positive periodic steady state to the unstable trivial one. Further, this pulsating wave attracts exponentially in time all solutions with initial functions in its bounded neighborhood with respect to a weighted maximum norm.