Bret, that's a far more likely scenario. People won't travel far to steal if there's plenty to steal where they're at.
During the so-called Rodney King riots in LA, I lived in an upper-middle class neighborhood in Long Beach that was surrounded by lower class neighborhoods. There was rioting within a couple of miles to north, south and west of me, but I never saw a single looter. It wasn't because they feared the Jewish doctors and lawyers who were my Bixby Knolls neighbors. It was simply because they saw no reason to leave their own neighborhoods when they could loot the stores close to home.