If you evolve an Eevee while it is your buddy pokemon and you have walked at least 10km with it, it will turn to Umbreon during the night and to Espeon during the day. But what are the exact timeframes here, i.e. what times are considered day and night by Pokemon Go? I haven't been able to find an exact answer to this.
Pokemon-go – What are the exact timeframes to get Espeon/Umbreon via the buddy trick
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Related Solutions
The Silph Road has done quite a bit of research into spawning mechanisms and their findings have been corroborated by much of the community. It isn't necessarily guaranteed to be 100% accurate but the evidence supporting it is overwhelming. In either case, all their work is reproducible and peer-reviewed so it's largely considered to be the most accurate non-official information available.
From what I understand of their findings, the world map is littered with "spawn points" (marked by the rustling grass animations), which spawn one pokemon at regular intervals. Most spawn points trigger once per hour but I've heard of some spawning as often as every half hour (although it is possible that the location simply has two spawn points very close together). A pokemon spawned in this way lasts for 15 minutes before despawning and will be available to any player who is in range during that time. Every spawn location has different timings so one might trigger at :02 past the hour and another might trigger at :29.
I have not yet seen any evidence to suggest that these points may cease spawning at any time for any reason. We do know that they are tied to locations that historically have increased mobile data usage, which is why there are tons in the downtowns of larger cities but very few, if any, out in the country.
Assuming this is all correct, it is theoretically possible to momentarily deplete an area of pokemon by catching everything that is currently spawned but that will only affect you and will only last until the next spawn point triggers.
If I were to speculate, I would say that the "dry area" effect is a description of the 45 minutes per hour that there are no pokemon at a given spawn point. If your area only has one or two spawn points within radar range (especially if they all trigger at roughly the same time), the effect would be highly visible, which is likely what that other user was seeing. I don't know whether the map turning grey is related to this but I would suspect it isn't. Instead, I would chalk that up to a graphical glitch but I don't know that for certain.
The reason you see these, is most likely because of server trouble. They appear around Pokémon that are standing there. Normally you would see a Pokémon standing there in the middle, but because of server issues, you aren't actually seeing the Pokémon. See the following picture.
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Best Answer
This likely uses the same mechanism the game uses to determine whether to show "day" or "night" maps. I believe it gets the local sunset/sunrise time from the phone. In this case the evolution will be Espeon from sunrise to sunset, and Umbreon from sunset to sunrise.
Update: Now that the evolve button shows what you're going to get (if it's not a random draw among the first three), as the other answer here describes, that's a much better way to tell when to evolve.