The invasive has to outcompete. In the case of some invasive species. Like the cane toad, theyre unsafe.
Plenty of invasive species become food to the right predators. The problem is often humans have eradicated all predators in the local area.
Isolated islands are an an exception where humans ended up the outright dominant predator and therefor all subsequent invasive species have no real problems other than human management, such as in Australia and New Zealand