A healthy ecosystem has checks and balances to keep each species in line. Visualize a very simple food cycle - let's say trees, white-tailed deer, black bear. Trees sprout and grow but the stand can't get too thick because of deer browsing the seedlings. The deer population is held in check by the bears. And the bears eat the berries on the trees and poop out the seeds, allowing new trees to grow.
Now, this is highly oversimplified, but you can see what I mean about checks and balances. You throw something non-native in the mix and the whole cycle suffers.
But it occurred to me yesterday that native species can become invasive. In reality, the simple food web that I described above doesn't work all that well where I live. Black bears (and other top predators) are rare and the white-tailed deer population has exploded. The understory of the forest here has almost no seedlings because the deer browse everything, leaving no baby trees to replace the ones that are sick or dying.
In my opinion, white-tailed deer in the mid-Atlantic are an invasive species. Now, the very definition of an invasive species, at this point, implies that a species is non-native. But I would argue that when a native species population explodes out of control with nothing to check it, it's invasive too.