Everybody Lied!

When a psychiatrist was found murdered in his apartment, 4 of his patients were questioned about his death.

I.  The police knew from the testimony of witnesses that each of the 4 patients had been alone with the psychiatrist in his apartment just once on the day of his death.

II.  Before the 4 patients were questioned they met and agreed that every statement each patient made to the police would be a lie.

Each of the patients made 2 statements, as follows:

Avery - 1. None of us 4 killed the psychiatrist.  2.  The psychiatrist was alive when I left.

Blake - 3.  I was the second to arrive.  4.  The psychiatrist was dead when I arrived.

Larry - 5.  I was the 3rd. to arrive.  6.  The psychiatrist was alive when I left.

David - 7.  The killer did not arrive after I did.  8.  The psychiatrist was dead when I arrived.

Which one of the 4 patients killed the psychiatrist?  Explain your reasoning.