For example, Dichrostachys cinerea is highly invasive in Cuba (introduced range) and also very abundant (aggressive) in South Africa (center of origin).
Yes, even in their centre of origin, plants have their own niche and can be considered invasive if they enter unfamiliar landscapes. Rapid population expansion of native species can threaten ecosystems and economics just as much as exotic species when habitats are anthropogenically disturbed. This is particularly disastrous for island ecosystems :-)
The indigenous Pragmites europea (common reed) invades oligothrophic wetlands after eutrophication in Europe and is an alien invasive in North America.
Pteridium aquilinum is invasive (weedy) in anthropogenic vegetation both as indigenous and exotic species.
Ulex europeus is another example.
As Andrew seems already to imply, invasion by alien and/or indigenous organisms is rarely exclusively driven by the presence of specific species, but rather by rapid expansion of extensive new environments with an open ground and/or an open canopy cover. For example in the distribution area of Chromolaena in South Africa (attached) you find a multitude of other invasive plants species over large areas and great numbers. Transformation of land use and land cover was particular fast by the development of large areas for sugar cane fields and pulpwood plantations in parallel with the new connectivity provided by infrastructure serving dispersal. Another example, is the Kafue flats were indigenous woody species (including the Dichrostachys mentioned by Narcisco) are spreading to such an extent that the process is labelled bush encroachment, while in the wetter floodplain parts an alien shrub is invading over the same period.
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In literaure on weeds, no strict definition is applies for invasive.
In general, an invasive species is not native to a specific location (an introduced species), has a tendency to spread and cause damage to the environment, human economy or human health. However a native species is also considered as invasive if it colonize natural area beyond a threshold.
I agree with the last comment: I do not know a definition of the term "invasive" alone. What is defined is the term "invasive alien species", unfortunately with different meanings. In the context of the CBD invasive alien species are those that threaten biodiversity, Richardson et al. (2000) term those alien species invasive that spread with a given rate. In your question you seem to apply the first definition. But how can a native species have a negative impact on biodiversity? Even if it forms large monospecific stands they should be considered part of the native biodiversity. If, however, native species colonize habitat where they were previously not occurring I would assume habitat change as the cause and the plant's behaviour as an effect. Some authors have suggested to use the term colonization instead of invasion to describe native species' spreading into new habitat (e.g., Richardson et al. 2000 in Diversity and Distributions).
Some researchers have advanced the idea that an invasive species can be native too (e.g. Mack, 1985; Gouyon, 1990; Joenje, 1987; Mooney et Drake, 1989; Le Floch et al., 1990), some considering the capacity to invade new habitats or the explosive growth of the population.
Center of Origin is the native place of an organism; for instance, plants here in this discussion. The population of an organism in its native range may be small (not dominant) or large (dominant). Considering large populations of an organism in its native range 'invasive' is fallacious as it threatens the definition of 'abundant' or 'dominant'. An organism rare in its native range may be invasive in alien landscapes. Similarly, an organism abundant in its native range may also be invasive in alien range. I do not know of any rule which suggests that an invasive organism is more abundant in its alien range in comparison to its native range, or vice-versa. An organism can have both positive or negative interactions with other organisms in its native range. That does not mean that the organism with negative interaction is invasive. For example, epiphytic Ficus spp grow on other trees and mostly kill the host, but they are not invasive. I think 'invasive in native range' is word play. I agree with Uwe that colonization is a better word.
As is evident from previous comments, the metaphor "invasive" is applied more or less loosely depending on the context of the comment. Early on (1980s-90s) invasion biologists/ecologists used the term interchangeably with 'introduced.' Since then they have attempted keep 'invasive' subsidiary to 'introduced.' Invasive species were given the primary distinction of being harmful to some interest. Governments subsequently defined invasive roughly the same way, although details varied. If we accept that strict definition, a 'native' species cannot be 'invasive.' More interestingly, given almost any current definition of 'species' used by systematists and evolutionary biologists, the adjective 'invasive' is a nonsensical metaphor. Species can't carry out invasions. In philosophical terms, it's a category mistake. Whether we even need a special word for species that reproduce or spread rapidly within their putative native range seems questionable. If they are spreading unwanted into cultivated areas 'weed' or 'pest' still make the point. If harm to something valued is not the issue, comparative quantification of population dynamics over time seems more informative than name calling.
In some way, we have several examples of this in the Canary Islands, e.g. with Aeonium haworthii (Crassulaceae), which is endemic to the western part of Tenerife and it's invading other areas of the same island (mainly in the Northeast) after being cultivated in gardens. I attach the pdf file of a short article on such species and this little invasion.
In our islands there are other examples of native and even endemic plants than can be, at least partially or locally, invasive in the archipelago. One good example is Rumex lunaria, which is endemic to the Canaries but it has been introduced in one of the islands, Lanzarote, where it's spreading in some areas. On the other hand, this plant easily colonizes recent lava flows and disturbed habitats, and many times shows an agressive ecology. I can imagine that if this species is introduced in any part of the world with similar environments, it would be for sure an important alien plant...
Ruben's answers exemplify the problems here. In the second, "native to the Canaries" and "native to Lanzarote" don't mean the same thing the same way. The Canaries are not the same kind of geographical unit as one island. Furthermore, if we consider the Canaries to be an administrative unit we get one result; if they are a biogeographical unit we get another. That is compounded by Ruben's first answer, where something can be called native to part of one island, but not to another part of the same island. The blade used to cut this Gordian knot is usually evidence of "human agency" – but that blade has two edges. If a history of human agency renders a plant nonnative, then nothing humans plant, even within some documented historical home range, can be considered native in the same way as a plant growing from a line untouched by human hands or technology. Once the continuity of plant-propagule-plant is interrupted by a human action, intentional or accidental, nativeness (as generally conceived) is "lost". The whole matter of natives, aliens and invasions has more to do with trying to separate "natural" from "cultural" than it does with explaining anything fundamental about biota. Given the present contention over cultural decolonization, turning to another social metaphor doesn't solve the problem - and by the way, the same person (H.C. Watson) who first applied "native" and "alien" to plants as we use them today also used the term "colonists" to denote a particular level of population establishment. All that in the 1840s! His system survived the Darwinian revolution unscathed. Amplified, actually, because he wasn't concerned with suppressing or eradicating alien populations; merely identifying them as such. See attached.