I whiffed on my question about zebra mussels. Here’s another version. Some species seem to be so hardy that they elbow out other species and easily win the competition for resources. I think that would be my understanding of an invasive species, particularly if it is introduced into a relatively established ecosystem. Do these species have similar characteristics elsewhere? Where they’re native, for example? Are they only invasive in the context? Or do they tend to take over niches everywhere they occur? If not, what keeps them in check?
Piggy-backing the OP: Cane Toads in Australia would be a good example of an invasive species I suppose. What is their situation in their native environment?
The number of checks on species in their native environment are many. An invasive species must have multiple and massive predation on it in its native ecosystem. In general, species will produce enough offspring to ensure survival for another generation – if they produce too few offspring for conditions, the species goes extinct, if they produce too many it is a waste of resources.
I do not know specifically what eats zebra mussel larvae, or adult zebra mussels, or if introducing a zebra mussel predator into compromised ecosystems would cause more damage (since the natural checks on the predator are also missing). Remember what happened on Guam when they brough in snakes to “control” the rat population.
If an invasive species were an invasive species everywhere, then it would already be everywhere and would have nowhere to be invasive to. Species must inevitably reach some equilibrium state with their habitat, at which point they are no longer invasive.
In the realm of plants, “exotic invasive” species are not invasive everywhere, largely because of climate. If it’s too cold the plant is killed off or dramatically weakened in winter. Too much heat can keep it in check or kill it as well.
In the case of kudzu, what keeps it in check where it’s native?
Weather as well as insects.
And grazing animals–it’s edible.
And zebra mussels? And garlic mustard? Or purple loosestrife? Where do they grow - in balance with their surroundings?
Right. They grow in balance in their surroundings, where they’re native.
That’s really my question. What type of ecosystem is their natural habitat? They appear to be invasive just about everywhere they occur.
Well, as my grandma would say, “If you look it up yourself, you’ll remember it better.” 
I’m pretty sure there are comprehensive Wiki articles on zebra mussels, kudzu, and any other invasive species you’d care to look up. That would give you a starting point for your research into their growth habits and native ecosystems.
…she said kindly, with no malice or snark intended. 
Evidently fire ants aren’t even noticed in their native environments in South America (according to a recent show I saw on Discovery Channel), and part of that is because there are flying insects that attack them. Some guy in Alabama introduced the insects and it’s proving to kill off the fire ant mounds around his lab.
They are native to Central and South America. They are common enough, but not a pest. There are a few animals that have figured out how to eat them here, like coati mundis, that flip them on their backs to avoid the toxic skin glands. They are a bit of a pest in Florida, where they are introduced.
It’s late, so can’t post too lucidly now, but here’s a start toward understanding. Will try to post more tomorrow, this encompasses my day to day thoughts with botanical concerns.