Even if not on an island extinction is on the menu depending on how large the population was to begin with and the velocity of the crash. This is because the males in the population can't reproduce by themselves and the weakest are often the young. So when food runs out, even if it is temporarily, you may see one last generation of very fit males and possibly some cannibalism and after that it is game over. Calhoun did some pretty gruesome (and borderline unethical) experiments that are the closest thing we have to lab controlled overpopulation experiments. The results presented are pretty sobering.