Despite concerns about the richness of plant invaders on islands, and their likely effects on local systems, impacts of these species seem to be small. However, this may be due to an absence of information on impacts, including changing species occupancy and forecast occupancy, rather than lack of impact per se. Here we use the plant invaders on the sub-Antarctic Prince Edward Islands (PEIs) and spatially explicit modeling of presence–absence survey data to demonstrate that the geographic extent of many invasives is increasing and is forecast to lead to occupancy of >60% of the islands’ surface area by 2060, with ongoing climate change. In keeping with theory, proximity to human activity, neighboring populations (i.e. propagule pressure) and residence time, along with more minor contributors such as elevation, explain >50% of the variation in the occupancy of each of the six main invasive species on the islands. Human disturbance and changing climates seem to have led to recent increases in the rate of range expansion. Our results suggest that impacts of island plant invaders may be more significant than previously estimated, largely owing to prior data deficiency. More specifically they also suggest that control plans for the PEI (and other Southern Ocean Islands, SOIS) should first target less widely distributed species, which are invasive elsewhere. They also indicate that for the other SOIS, and for Antarctica, surveillance and anticipatory control plans should be in place.