You can speed up increase of any species by following these advices:
At the beginning, fill the requirements just for two first representatives of a given species. For the following, use the Romance Sweet.
It's good to buy some more Romance Sweets. You wouldn't need to go to the shop frequently.
As soon as the first egg will be delivered, wait until you regain control over the parents and give them Romance Sweets. Then engage them to romance again. This way you'll make easier going through the maze, because every new animal of given species means that you can make less mistakes while playing the minigame. It's especially important when dealing with more complicated mazes. Remember also, that unhatched pinatas don't count as residents.
Minigames aren't very difficult but even then there's no need to rush.
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