Small, herbivorous mammals have a hard time of it when it comes to being attacked by larger carnivorous ones. Life expectancies are generally short in the wild, counter-balanced by the ability of most animals to breed like... well, rabbits. Many of their behavioural traits are also shaped by the need to avoid being eaten, including, not just general watchfulness, but trying to do things like foraging where (or when) predators are least likely to be a problem.
On the other hand, predators naturally have an interest in getting round this problem, and the result is a never-ending battle between the habits and skills of predators and prey alike. Over the course of thousands of years, this settles into a sort of equilibrium, changing perhaps only as new species evolve and as the geography or climate change on even longer time scales. Unless, of course, we humans get involved.
Humans can mess up the balance of life among wild animals in quite a number of ways, but the one we're concerned with today is that of invasive species. We purposefully carry species from one part of the world to others, especially when it comes to domesticated animals. Pre-Columbian America had no cows, no domestic sheep (although, of course,
two native wild species), no pigs, and no horses or cats. There were dogs, but only because humans had brought them across much earlier. The introduction of all of these animals, and the changes in land use required to support them, have had significant effects on the local wildlife in at least some parts of America, and that's before we include animals deliberately introduced for other reasons, such as fur-farming or hunting, never mind those that we didn't introduce purposefully at all, such as rats.