Penguins normally immigrate when the weather in the north becomes too severe and blizzards are jeopardizing the eggs and cause extreme climate drop that limits their resources. They move to other penguin communities and share living space until the storms die off.
Penguins tend to emigrate if they are kicked out of their community and/or aren't providing enough for their community so they leave and live elsewhere and have to provide for themselves.
Birth rates increase when the moms leave the fathers with the eggs and go off to find resources (fish) to eat and then come back to find their mates and their babies hatched and ready to feed. If resources weren't available to the mothers, birth rates would drop due to the lack of resources the mother was able to supply.