There's not just one species of cockroach, there are many. Also, whether or not a species is called invasive is relative; in the area it originally evolved, it is a natural part of the ecosystem and is not called invasive, but if it has migrated to a new area (typically as a result of human activity) and managed to flourish, to the detriment of native species, then it is called invasive. In North America, there is a common species of cockroach called the German Cockroach, which is actually originally from Asia. It's an invasive species in North America, and it really flourishes in human structures. Other cockroaches, for example, wood roaches, are native to North America, and are not considered invasive in this area.