This article seems to take more of an anthropologic perspective. It's true that if we wait long enough, the nature will settle down the issue on it's own. However, are we willing to make the short-term sacrifices? The main issue of the so-called invasive species is that they don't have a natural competitor in a new environment, therefore they will become dominant if left alone. Their overgrowth not only disrupt the variety of the local vegetation world, but also the balance of the animal world, sometimes including us.

For instance, the potato famine in ireland: The potato plants across Europe were originally transplanted from Peru, they became the main source of food in Ireland. Since there was no diversity of the plants, when potato blights happened, it destroyed all the potato plants, and caused a catastrophic famine with 1 million death toll. So, in a way, the nature did take care of itself, but with a great cost of human lives. This might not be the best example of the invasive species, since the active human agriculture was involved.

In the end, humans always lose the war to the nature no matter what. The control of the invasive species may help a little, but unless human migrations are completely restricted, the invasive species will have their places.