How long does a Pokémon stay in one location?

If a friend encounters a Pokémon, it doesn't disappear directly. So I should be able to catch it, as well.

My question is, how long does it stay there? Can I still expect to find it there a few hours later, or even a day later?

Pokemon do not "move" in GO. Once they spawn in a location, they tend to stay there until despawn.

Pokemon have an attribute which controls spawn behavior (likely TimeTillHiddenMs and ExpirationTimeMs in the protobuf files), which determines their spawn length. For the most part, Pokemon will despawn either 15, 30, or 45 minutes after initially spawning, although some outliers do exist as this is a server-side variable.

Once a Pokemon spawns in an area, they do not move in that area until they despawn. So generally they will be in the same spot for 15 minutes.

What's not clear is when they despawn, there does seem to be a better than random chance they will respawn in an adjacent area. I'm not sure how big a game area is, but it is at least several city blocks.

I'm not convinced of the answer above. All the 'good' Pokemon I come across seem to move around frequently, and quickly until they show up on the screen. One moment it'll appear, and it'll be gone just a couple minutes into the search, or the foot-prints for a single Pokemon will increase or decrease without my even moving.