Rats are absolutely everywhere, you just don't normally see them.
Mice also burrow and make mounds. I had a similar situation where I'd removed half a concrete path, they were burrowing under what's left from the new lower part. They seem to be attracted to any bank that they can dig horizontally into. In my case they kicked out a 1920 penny, which I cleaned and polished. So it was nice.
You need a solid edging. Personally I'd stuff some bricks or concrete under. Technically it may be your neighbour's land, but they won't know and are unlikely to care.