A grocery store is different than an airline as most of the food does not spoil immediately as do unsold airline seats.

An airline sets it's fares, and the quantity of each fare that is made available, based on filling up the airplanes. If they aren't selling enough seats, they lower the fare/make more of the lower fares available. If they are filling up the flights too quickly with the lower-priced fares they raise the fares/reduce the number of the lower fares. None of that changes when a new competitor enters the market.

What does change with a new competitor is the number of available seats that need to be filled. With a significant increase in the number of seats, fares will have to drop in order to fill those additional seats.

It's not about who the competitors are, it's about the number of seats being added.