When a species enters a new environment, it may find there a number of other organisms that keep it in check. There may be sufficient food and water. While there may be competition for those resources, there are enough for the new species to get by within bounds. In that happy case, the new species reaches what is called equalibrium in its new home.
If the existing predators are too successful, competition too fierce, or climate conditions are unbearable, the species will quickly decline and fail in its new environment.
Some species arrive in a new environment; find its climate to their liking with few predators and weak competitors. Assuming plenty of food and water, the sky is the limit. The new species will thrive, pushing out competitors and making itself one of the dominant species in its new home. Those are plants, animal and micro organisms we call invasive.
Invasive species may be either native or alien. Sometimes a native species can become invasive if conditions favor the native plant, such as in soil disturbed by recent fire or construction activity. In those cases, the “pioneer” species will be dominant and appear invasive until conditions return to pre-disturbance and its neighbors regain their place in the environment.
Alien invasive species are much more of a problem because the local environment lacks the predators and competitors that kept it in bounds in its old home. It can take many generations for a new equilibrium to occur. In the mean time many native species can become extinct in their local environment never to return.