GA pilot reported multiple conflicts with three Stearman aircraft while on the pattern for landing at a non-towered airport requiring evasive action to gain separation. The Stearman aircraft were talking with each other while performing high performance maneuvers; but they were not responding to the reporter's radio communications on CTAF.

Narrative

I made the first call to enter the pattern at ZZZ in the left downwind at 12 miles; again at 6; 3; and 2 miles. Knowing there were 3 Stearman aircraft in the pattern already; I communicated my position frequently. The three were communicating with each other while not responding to my communications. Shortly before entering the left downwind; I located one of the biplanes below pattern altitude and directly to my 9 o'clock and climbing. I asked if the aircraft had me in sight. Upon not receiving a response I announced I will be executing a right 360 degree turn to gain separation from the traffic. Once I had reentered the downwind after completing the 360; I heard one aircraft announce they were departing Runway XX. I located them and saw them complete an immediate high performance climb while turning crosswind at the runway threshold. I requested if the aircraft had me in sight and received no response. I continued downwind while monitoring the traffic turning in behind me. Upon realizing I was not far ahead of him; the aircraft trailing announced his exiting of the pattern by overflying the runway and orbiting to the east of the airport in the right downwind. The aircraft I was following is on short final at this point. My passenger pointed out another stearman above traffic pattern altitude and to my 3 o'clock about 1/8 of a mile away flying parallel to me southbound. I made another announcement of my position at this point while also; requesting the intent of this aircraft. The aircraft then began a spiraling descent so as to follow me in. The aircraft I was following departed the runway and completed another steep climbout. The aircraft circling in the right pattern had turned his smoke on and was doing several abnormal maneuvers at and below pattern altitude. Upon landing I continued to the exiting point of the runway; once I exited; I noticed the stearman behind me was halfway down the runway from me and executing a low approach at around 10 ft. AGL; and executed another high-performance climbout and turn to downwind. These maneuvers continued for another 10 minutes after my landing. Throughout my single pattern there was an increased risk due to the extremely abnormal and hazardous traffic pattern operations demonstrated by the three stearman.