An invasive species is any organism that becomes overpopulated and negatively impacts the environment as according to Wikipedia, sometimes by predation and sometimes by spreading diseases. When it comes to animals, it would be parsimonious and provocative to say that dogs aren’t just the first domesticated animal but also the first invasive animal at that. If you have dogs hunting on animals such as antelope in Mongolia, blue sheep in China and wild ass in India then that’s not only predation but also negatively impacting the environment especially if they’re endangered that dogs would be the first invasive carnivore.
Another reason why they’d be the first invasive carnivore and animal’s that cats have yet to reach outside of Africa and Anatolia in those days, if cats weren’t mentioned in Ancient Greek, Chinese and Indian texts then it shows they wouldn’t arrive there yet and cats would constitute as the second wave of invasive animals after dogs, pigs and mice. If dogs aren’t native to Africa and probably descend from Middle Eastern dogs, in addition to being invasive predators to Barbary macaques in Morocco and vervet monkeys in Uganda then they’d qualify as an invasive species there too.
Cats wouldn’t be domesticated until around 10,000 BC, centuries after dogs were domesticated (if they’re first domesticated in 14,200 BC) so they’d be part of the second wave of invasive animals after dogs, mice and pigs. Pigs, mice and dogs constitute the first wave since they were around 15,000 to 12,000 years ago with mice and dogs being the earliest invasive animals wrecking havoc on native wildlife whilst leading commensal lives with humans though it could also be said of pigs to some extent as well.
Cats, goats, sheep, rats and cattle would constitute the second wave of domestic animals with cats and goats being the second wave of invasive animals, especially with regards to their ecological impact on native wildlife. Cats preying on animals, goats eating endangered plants and rats doing both in addition to spreading diseases. I would say second wave as their domestication and spread occurred centuries after the domestication and spread of dogs, mice and pigs. There were even fancy rats in Japan sometime in the 17th century, which shows how far and early they’ve spread.
Rabbits and ferrets would constitute the third wave of invasive species and domestic animals, centuries after cats were domesticated and in the case with these two they’ve hurt and decimated native wildlife in New Zealand. In fact, ferret ownership’s restricted in New Zealand to keep them from preying on other animals. Rabbits are also a pest in Australia where there are attempts at curbing their populations through viruses. Rabbits may not be the last animal to be domesticated but they’re part of the third wave of invasive species along with the ferret.
Mongooses, foxes and cane toads would constitute the fourth and fifth waves of invasive species since mongooses did have a long history of being used as pest control and foxes were used for fur as early as the Middle Ages but only became invasive species in select countries fairly recently. Same with the cane toad which’s invasive in the Philippines and Australia. I might have to write a book on invasive species if I wanted to and return to this subject matter, maybe revising it with a bit more detail but that’s all I know about invasive species.