both answers are actually correct. Alligators by nature will/should stop feeding during the winter months. If the temps are low then the alligator will/should stop feeding same as it would in the winter months.

They dont/shouldnt stop feeding just because its the winter but because its the temps, there bodies sence the cold season/temps coming on and this trips the stop feeding response. The stop feeding response to the cool temps is to keep the alligators alive, as feeding in the cold temps wont allow the alligators to digest there food which would lead to death.
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Great Lakes Zoological