Most alien (plant) species are well behaved. European settlers brought oats, wheat, barley, many varieties of turf grass; your name it they brought it to America. Think of boxwood, lilacs, tulips and apples. These are all alien species and stay put pretty well where they are planted.
Certain native (plant) species can become invasive under the right circumstances. Sumac is a good example. It is native to North America but when it moves into disturbed ground its rhizomes spread quickly and it forms dense thickets. Many of the brambles, like black cap raspberries, push out native prairie grasses and wildflowers if they are not controlled by fire or herbicide.
Soil type, moisture and temperature can influence whether a species (alien or native) is invasive. Japanese knotweed is incredibly invasive in the UK and the Pacific Northwest but be less aggressive in places where cold winters and drier soils limits its rate of spread.
Finally, microbes can limit the reproduction and spread of plants. Native plants evolved with their surroundings and so are eaten by and have some level of defense against predation by the microbes in their neighborhood. Alien plants may be ignored by microbes in their new surroundings and so gain a competitive advantage over native plants.
Now let’s talk animals. All animals need food, water and shelter. Like plants, animals can be alien or native; invasive or well behaved. The success of an animal is limited either by environmental constraints (food, water & shelter) or predation by microbes and other animals. For every animal native to its environment, there are other animals that use it as food. Even top predators have predators.
Alien animals may find that they have few if any animals around them that see them as food. The New Zealand Mud Snail has been found in fresh water streams in southern Wisconsin. It thrives on the nutrient loaded surface water runoff from farm fields. None of the fish in those streams know how to eat the snail or they find it unpalatable. Regardless, with no natural predators, this new arrival is causing havoc in its new home.
The real problem with invasive species is that they upset the natural balance in an ecosystem resulting in lower lower diversity of species, lowering of native populations, and disruption of multiple food chains. It is often these disturbances more than their relative success that cause the most damage to the environment. It can take generations or centuries for their new neighbors to evolve predators and disease organisms that bring the new invader into ecological check.