Pet Battles - Searching for pets question

I finally got into the pet battling, and I've been battling trainers and getting some pets to max level. I recently wanted to get a rare quality of the Thundertail Flapper out on the Isle of Thunder (worked amazingly well against Major Payne's mechanical pet), but couldn't find many of them on the map, and have never seen one mixed in with another critter - though I finally did find one this morning.

So my question is, is there a method to making the pets you want appear? More specifically:

- Will pets eventually despawn and repopulate on their own, or will they always stay on the map until they're defeated/killed?
- If I forfeit and kill it outright with melee, does that help the respawn process at all?
- Should I battle/kill all of the surrounding pets that aren't what I need just to get them off the map as well, assuming they're all potential placeholders?
- I've heard some pets only spawn at a certain time, like the Restless Shadeling, but are all pets this way?

Source: http://www.mmo-champion.com/threads/1282229-Pet-Battles-Searching-for-pets-question

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