"Nobody is condoning any bad behaviour but it's just something you have to deal with.
"He is what he is and he's the character that he is, and you just have to try and harness that."
The Uruguay international was also banned for eight games in 2011 for racially abusing United's Patrice Evra.
Ayre, who was about to board a flight to Australia when "got the call to come back" after Suarez bit Ivanovic in April, said the club have been working with Suarez over the summer to try and reform the Uruguayan's volatile character.
"We've worked a lot with Luis since the last incident with Ivanovic and he's responded well to that," Ayre said.
"He's been prepared to commit to that sort of work on his character, and you saw him back last night and it looks like the good bits of the old Luis which is terrorising defenders, and we're pleased that he's back and pleased that he's contributing."
Ayre added that Liverpool has a "process" to deal with any "crisis" like the Suarez incident, but said that "it's not a Luis Suarez process, it's a process for any crisis, whether it's on the pitch or off the pitch."
Suarez pushed to leave Liverpool this summer but the club rejected two bids from Arsenal.
"As we said all along through the summer, we expected him to remain a Liverpool player and he has remained a Liverpool player," Ayre said.