Second, those that do end up successful (from their point of view anyway) do so in a several ways. For example, they might be released from an herbivore or from competition from another plant species that's present on their home turf. Or perhaps there's a genotype that happens to be super duper adapted to the habitat in which it has landed. It can also happen that people think the plant makes a great garden plant, and it escapes and becomes a problem, like purple loosestrife, Lythrum salicaria.
And, as unlikely as it may seem (I love how truth can be stranger than fiction sometimes) it could happen that an introduced species crosses with another introduced species of the same genera that's from someplace else and this produces hybrids that are REALLY invasive, which is the case with Salt Cedar (Tamarix sp.) I'll do a whole post on that cool system sometime.
Of course, the effects on the invaded plant communities are varied. A paper recently came out that showed two species of invasive plants with showy, insect-pollinated flowers, Opuntia stricata (a prickly pear cactus) and Carpobrotus affine acinaciformis (a succulent) attracted more pollinators to the area, and thus increased the overall rate of pollination for native species.
So the native plants received some benefit, during this slice of time in the process of the invasion anyway. The study did not show whether this increased attention by pollinators translated into the tangible benefit of higher seed set, or bigger seeds or something like that. However, there were differences in the types of pollinators the invasives attracted. The succulent attracted more generalist pollinators that everyone benefited from, while the cactus had a higher percentage of insects that came in only to pollinate it.
The authors concluded that invasion by the cactus was more detrimental because it could monopolize pollinators, while invasion by the succulent could actually benefit native plants by bringing in generalist pollinators. I think this study is interesting because it shows some of the more subtle effects of non-native plants.
Reference: Bartomeus I., Vila M., Santamaría L. Contrasting effects of invasive plants in plant-pollinator networks. Oecologia 155(4): 761-770
image: Botanic Gardens Trust