Both use definition #4, a common meaning of
swamp:
3. (of a boat) to (cause to) sink or be filled with water:
The little boat was in danger of swamping.
The next huge wave swamped the boat.
4.

to overwhelm:
I was swamped with work.
Meaning #4 was derived from meaning #3, and generally means something unpleasant.
In your first example, there are too many off-spring for the predators to eat. The predator's appetite is overwhelmed.
In your second, they are assuring you that you will be able to eat all the courses without feeling overly full; your appetite won't be overwhelmed. I haven't heard this particular expression before, but my appetite has been 'swamped' at holiday meals.