I realized that there is no such modifier affecting pet battles. If you want to capture a specific breed or a rare pet, you will have to farm until you find it, with only the pure statistical odds helping (or hurting) you. Some players claim the game was "better" or "more difficult" back when this was the case for quest items. I don't know if I agree, but I know that the harder you have to work at something, generally, the more rewarding achieving your goal feels. Hence, I like the fact that you never know when you'll find the pet that you're searching for - it adds a flare of hardcore-ness to pet collecting
