Which outcome is most likely if a nonnative species is introduced to an ecosystem

Alright! In most cases it will induce nutrition or sth. like that to the ecosystem because it's simply going to be eaten or die or whatever. Just a large amount of animals will be able to survive. They need to be lucky enough to stand the environment (temperature & stuff) and they need to be able to find food and water sources.
In this rare case (I heard it's less than 10%) it might have the luck to have no natural foes so it can spread. The fact that they might not have any natural foes might cause a strong reduction of their food source. So if you have a big ape that keeps on eating dodos and its population keeps on growing and there's nothing that stops it... then one day there will be no dodos left. You see, an alien species usually will not be specialized on one prey, (or food source) because if they were so specialized, they'd be specialized on a food source that you can find at their homes and not at their new environment. (so they would have died out in advance already). So they're likely to survive a little longer.

You have the cats and camels in Australia for example... there are huge masses of them and there are no real natural enemies to "stop" them.

You see, in the usual case there will always be a balance.
Imagine a fox only eats rabits... once there are no rabbits left, the population of foxes will shrink because there's almost no food. Population of rabbits will grow again which makes it possible for foxes to find food easily which will increase their population and as a consequence decrease the population of rabbits ;)
So in the end it is still possible that a balance will be established... but in the worst case it might just create an irreversable unbalance. And... it has usually never been a good idea to invent another species as a natural foe of the other one.