We are concerned with the development and analysis of a predator–prey system designed for heterogeneous insular environments; populations are native preys exposed to introduced and invading predators and competitor preys. We first look at the unstructured model; this yields a singular system of ordinary differential equations having interesting dynamical features, such as finite time extinction or persistence of populations. Next we build a spatially heterogeneous structured model upon developing a reaction–diffusion system; then, using numerical experiments we analyze some typical effects of spatial heterogeneities on the persistence or extinction of native or introduced and invading species. The case of Kerguelen sub-Antartic heterogeneous islands where both domestic cats and alien preys have been introduced is taken as an example.