Can an “Invasive Species” Earn the Right to Stay?

You mean, invasive as in humans? Invasive species are by nature Alpha, to the indigenous Beta. They don’t earn the right to stay, they take it, and it’s the indigenous Betas that must earn the right to stay once invaded if they aren’t killed in the fight, and they tend to end up subservient to the invaders.

Non-territorial species are by definition nomadic and Beta as they get pushed out by Alpha invaders.

This is different from the macaques as they are an occupying species and usually non-aggressive. Mammalian occupation due to population growth is not the same as being invasive.