The membership year runs from 1 October to 30 September of the following year. If you pay the initiation fee after 1 May, (either you are a new member or a member that is returning from over a year or more's absence) your first year's membership is good until 30 September of the following calendar year.
Existing members pay full renewal fees regardless of what date they send in their membership fees. In other words, if you send in your renewal on 1 September, your membership expires 30 days later.
While AMA endeavors to send out invoices for memberships in August- September of each year, the responsibility to renew lies on the shoulders of the member to renew their dues by the end of September.