First of all to be correct, which I feel is important, the name "invasive species" is incorrect. "Abductees" is better for the first generation, as they were taken from a place and put into another. So they do not invade anything, they just try to live in another environment where they have been brought into.
Second point is: where do they grow. You'll see that they become abundant, most of if not almost always, in places that are not close to nature in the sense most ecologist use it. They do well in so called "disturbed habitats", in essence: unnatural, artifical habitats you see all over europe. Like heath, weathfields, pastures and all other manmade habitats. These do or do not belong as much or as little as the newcoming species as both have humans as the common introducer of them. If left alone, in most cases, all these artificial habitats will turn into forests mostly. But the same people who are combatting immigrant species are protecting those manmade habitats. And so they have to constantly do this. The reason how these habitats came to be (cultural use) have dissapeared, so you have to spend a lot of money into in countrysides which become dependant on economy. If money fails, nature will take over with all species that are available....
Another strange thing to me is the broad consensus that you can only be native based on history. If you were here untill 1400, you will always belong. I have seen ecologist almost crying when a long lost species was reintroduced after centuries. Yet, if animalright activists combat the massmurder of grey squirrels in England or Lomardy they are accused by the very same people of being "over sentimental".
I feel most of the reasoning behind conserving nature is based on emotions and not based on reason at all.
Now: If you came here by humans a century later (1500 AD) and have lived here for 40 generations (trees for example) you will rarely if ever be considered at home. What is so good about history, the past, that it dictates this? Why not the most logical, biological factor? Fitness. Survival of the fittest, remember. Not that you have shown you could NOT survive anymore, but the very fact that you can exist and persist inspite of humans right now!
Many times, if I say this, yet another human standard is used in arguments. Somehow it is "unfair" that newcomers, for some time, is more fit than some species that happen to come to a region earlier. Since when is nature "fair"? If I want to kill a lion because it kills a helpless, defenseless calf than all of a sudden fairness is thrown out of the window and it is "nature""...Doublestandards are the rule and not the exception which is a telltale sign of not to strong argumentation for a case. Nature knows no fairness.
In my view, the argument about invasive species is unscientific, is based on personal preferences of a few very conservative souls. It predates Darwin and has its origin in Puritan England. Which highly valued all things British/native.
I have wondered what can explain this best and it is simple nationalism. Everything we consider "belonging to us or our country" is accepted. Everything else rarely so. Therefor it involves the very same things we see in cultural nationalism: people have a big interest in what happens in their country(side), and cut it into provinces with borders (habitats) and manage it that way so they feel some sort of control over (unwanted) changes, are unwelcoming to newcomers and particularly develop a singlesided view on those newcomers and only scrutinize them for what they do wrong. A very biased view develops, use double standards (same traits for indigenous species are called different or not mentioned at all, or simply accepted), people resist change, people admire the past and idealise the past.
And, like extreme nationalism or xenophobia, they do not look at the individual and feelings of sentient beings. They have declared it unwelcome, have rendered it worthless and (in my view ruthless) extermination is often chosen as a rigtheous way of conduct. In my view, it is not too different from ethnic cleansing among humans. It operates via the same mechanism: "This is MY country....You people do not belong here because it was my country till 1500 and you are different/not one of us....so I reclaim what is ours for us alone and if you don't go, we'll kill you." Something like that. The only difference is the species.
In another discussion I mentioned this already: the only constant in the Universe is change. Currently, we are more aware of this than in other times I think. Climate changes and so habitats, if they exist. THey move north. So what is native in any given place in europe in 2100? when the temps have risen by 2-5 C? Accepting newcomers and change and quit trying to control nature is better, more flexible solution. But it is difficult, because instead of changing someone or something else, you have to change yourself...