"Population-dynamics science tells us two things broadly about invasive animal population trends. Typically, the population of an invasive species that moves into a new area and is freed from predation will boom — it expands beyond what is called the "carrying capacity", and diminishes the food and resources it needs to sustain itself in the process.
In the case of humans, modern medicine and technology have facilitated that boom. Things that used to kill us don't, and we can exploit resources much further afield and faster than during all of our history.
But, in animals at least, then comes the crash — where the ratio of animals to resources becomes so skewed that a period of fierce competition for limited resources breaks out, and only the fittest survive."