From watching interviews from TellTaleGames called "Playing Dead" Mark shows up as a sign that time is passing.
Right away the game tells you in the intro of Episode 2 that it has been three months since Episode 1. The designers/writers had basically wanted to let you know that the game/story/characters have progressed since last time and situations change.
The only reason Lily decides to let Mark join along is because he had food that he could share with the rest of the group. Yes, you don't see when they get Mark but they do talk about it if you pay attention enough in dialogue options.