Anon-autonomouscompetingsystemisinvestigatedinthispaper,wherethespeciesxcandiffusebetweentwopatchesofaheterogeneousenvironmentwithbarriersbetweenpatches,butforspeciesy,thediffusiondoesnotinvolveabarrierbetweenpatches,furtheritisassumedthatalltheparametersaretimedependent.Itisshownthatthesystemcanbemadepersistentundersomeappropriateconditions.Moreover,sufficientconditionsthatguaranteetheexistenceofauniquepositiveperiodicorbitwhichisgloballyasymptoticstablearederived.