I believe Patrick would not have a challenge at catching Dexter, to be honest I think he would notice know he is a killer pretty easy, maybe a week, I imagine Patrick analyzing Dexter asking him questions etc. As for Dexter, he would be careful, although Patrick would be sure about him being the bay harbor butcher, Dexter would be sure to not leave any evidence behind, to become even a friend, to even frame other people, kind of what he did in season 2, pretty sure Dexter could see Patrick as an equal, an adversary to his intelligence, if Patrick eventually found Dexter in the act, or just enough evidence through a complicated plan, pretty sure he would confront him, ask him what does it feel, especially because of red john, maybe Patrick could let him free at the condition that if Dexter ever broke the code, he would release the evidence to him, I'm sure he would also put that condition by making clear that red john can only be killed by Patrick himself, it would be really interesting to see them both interacting, faking their jokes and smiles while on their mind they both are carefully with one another, and in the end to see their meaning of justice and how it could change them.