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It happened so many times in geologic history like when S.America and N. America collided or when rodents rafted from Africa to S.America
All them resulted into the invasive species being the dominant not because they are, but because the ones that failed to live in the new environment were quickly forgotten.
Anyway, to answer your question, assuming we just allowed invasive species to come, the result would be 2 things
1. The local populace would accept them in a sense and become the new dominant predator (Sabertooth tigers made the giant dogs extinct)
2. The invaders would destroy the local fauna and flora so badly that they too would be driven extinct as soon as their easy food run out. Then the environment would rise again.
The fire ants in N.America, snakes and rats and spiders and lionfishes would simply replace the natural predators in a given niche
But the goats in galapagos, the rabbits in Australia, and kudzu vines would most definitely drive everything including themselves to extinction if a predator didn't rise up to control their population
Like seriously, the Kudzu vines causes a monoculture whereever it go. With monoculture means lack to diversity and one single disease shall result into an epidemic. Not to mention the soil erotion due to lack of nitrogen recycling plants like legumes and stuff
