The problem is twofold:
“Enough time” is potentially thousands of years, depending on the species comp already there and how quickly they’re able to adapt
Invasive, sort of by definition, are aggressive and highly competitive. To the point where they will tend to outcompete more laid back natives trying to fill the same niche. It may seem like this wouldn’t matter (the niche is still filled, after all. What’s the practical difference?) But this can cause two major problems: 1. Trophic cascades. Every other species that interacts with that niche is adapted to the old species. The switch may well have negative impacts on other species, and this can lead to large chunks of local ecology completely collapsing and the entire system failling. 2. It leads to a drop in biodiversity. In general, a more biodiverse system is more robust and better able to respond to changing conditions.
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