That's only true for the rats. For the pigs and chickens (and rice), there will be fewer of them because they do not themselves have the capacity to exist at current population levels. The reason they're doing so well right now is purely that we want them to.

We also actively cull them, for the rats. Recently watched an episode of survivorman where he was on this tropical island infested with rats. There's absolutely nothing on that island yet there's rats all over the place. Having no humans they may explode in population.