Scenario I: The greedy species eat out a critical resource for themselves, the individuals not able to replace it, and must die out. A poll of other species make a feast of the remains.
Scenario II: A critical resource is gone, but a small portion of the greedy species in some locations adopts and survives. They are among the feasters, but their aggregate impact is now modest.
Scenario III: The greedy species start to cannibilize or competatively kill each other, just before critical resources are gone, so that a small portion of them survives without much change in "the way of life" (except that they become increasingly aggressive towards each other).
Scenario IV: For some reason, the greedy habits or genes are supressed by some cooperative mechanisms, at least for a considerably long time. But then a "libertarian political economy" takes over anyway, the species rapidly expand and "prosper" for a few generations, and then they found themselves in one of the above scenarios.
Scenario V: The greedy habits or genes are supressed by some cooperative mechanisms, but when the "culture of greed" makes a progress, cooperation and other restricting mechanisms evolve as well. This scenario might get good traction after a number of "boom and bust" cycles has happened to the same species.
Scenario VI: The greedy species does not have restrictive mechanisms in the genes or in the social structure, but the environment has the functionality of detecting its own stress, and is able (for example) to provoke a sharp "boom and bust" cycle on the greedy species, forcing them either to diminish their impact, or join the controlling environmental system. On the global scale, this is called Gaia.
It is an interesting exercise to ponder relative actuality of these scenarios ;-]
