Original Sin Dexter vs Season 1 Joe

We all know Dexter usually mauls Joe alive in these debates but i feel like this one is alot closer due to Joe at this point already having a couple kills under his belt same with dexter at this point,

Scenario:Joe kills beck and starts his cycle in miami and like usual he’s leading the police in all sorts of directions, would you g dexter even figure out it’s someone like joe?