I don't remember whether I noticed it in Tokyo or not, but soon after I got to Osaka I started seeing a lot of plastic bottles, full of water, outside houses along the side of the road.  I think I assumed the first one I saw was meant to weigh down a sign or something in case of strong wind, but they're all over the place. Apparently, people put them out to keep cats away.  The cats see their reflections on the side or something, and are freaked out, is what I was told.  I think I remember reading or hearing about this somewhere else - pointing more towards cats' intense dislike of water, which makes more sense, though how full, sealed bottles are going to do the job is such a mystery that this must be crossing the line into superstition.  Most of the bottles are a little too dirty to be reflecting much of anything, anyway.
So why would cats be such an issue?  Don't Japanese love cats? They star in almost as many cute photos and Youtube videos as rabbits, definitely.  If not more.  With pancakes and things on their heads.
 The internet proves that there are plenty of pampered house pets in Japan, but alongside that - I definitely noticed in the park in Tokyo where I would go running that there were what seemed like a lot of outdoor cats hanging around. Usually I saw them because they were being fed by people who were sitting at a little bit of a distance, as if the cats would be easily scared away.  I was only in Tokyo for a week, though, so I didn't see all that much.
The alley/street that my guest house in Osaka opens into has several hanging around all day. Normal, right? But then, down south a couple of miles in Nagai Kouen (Nagai Park), where I am going to try to run while I'm here... well, you know how in Boston we have squirrels?  In Osaka, apparently, they have cats. I'm being completely serious.  I saw several people feeding them while I was doing laps, but there were others, all over the place. Just hangin' out.
I also noticed several tufts of sparrow feathers on the sidewalk that looked like they were not naturally shed.  I bet there isn't much of a rat problem, though.