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We are concerned with the persistence of both predator and prey in adiffusive predator-prey system with a climate change effect, which is modeledby a spatial-temporal heterogeneity depending on a moving variable. Moreover,we consider both the cases of nonlocal and local dispersal. In both thesesituations, we first prove the existence of forced waves, which are positivestationary solutions in the moving frames of the climate change, of eitherfront or pulse type. Then we address the persistence or extinction of the preyand the predator separately in various moving frames, and achieve a completepicture in the local diffusion case. We show that the survival of the speciesdepends crucially on how the climate change speed compares with the minimalspeed of some pulse type forced waves.

Persistence of species in a predator-prey system with climate change and either nonlocal or local dispersal