Propagation dynamics of the lattice Leslie-Gower predator-prey system in shifting habitats

In this paper, we are concerned with the propagation dynamics of a discrete diffusive Leslie-Gower predator-prey system in shifting habitats. First, we discuss the spreading properties of the corresponding Cauchy problem depending on the range of the shifting speed which is identified respectively by (i) extinction of two species; (ii) only one species surviving; (iii) persistence of two species. Then, we give the existence of two types of forced waves, that is, I type forced waves invading the state where only one species exists in supercritical case and critical case, and II type forced waves invading coexistence state for any speed.