The DUP influence in each district of the game starts off at 100% and will gradually lower as you do the side missions, as well as grab the other collectibles. Whenever it drops to 30% or below, the District Showdown will be available for that district. These show up as a phone icon on your map and entail you fighting with some DUP soldiers. This will be the last major fight with the DUP in that district, so they will be sending everything they have.
Completing these District Showdowns will limit the DUP driving around in that district, as well as earning a new vest for Delsin to wear.