GA flight instructor reported while in pattern; traffic made nonstandard pattern entry; disregarded position calls; which resulted in an NMAC.

Narrative

I was performing touch and goes with a student. I was established in the downwind when I noticed Aircraft Y; behind us on my traffic display. He had entered the pattern in the upwind rather than crossing midfield as suggested in the AIM. From the upwind; he turned crosswind then downwind behind me. The separation on my traffic display was decreasing. Both airplanes were making the recommended traffic calls. I suspect that Aircraft Y was not listening on the radio and therefore not aware that I was ahead of him. As the separation continued to decrease; I turned base to avoid a mid-air collision when Aircraft Y turned base behind me. My traffic display indicated that Aircraft Y was 200 ft. above me. I took evasive action by decreasing power and diving. Aircraft Y crossed above us when I said on the radio 'we're both on base; you're right above us; you turn final; I'll extend base'. He said 'oh I see you now.'The closest I could get the ADS-B playback too was 300 ft. vertical separation; but my traffic display indicated 200 ft. vertical separation. I was a high wing while Aircraft Y was flying above me in a low wing. Aircraft Y did not enter the traffic pattern in the recommended manner; was not listening to the radio; and lost all situational awareness as a result. Aircraft Y's poor airmanship led to a near mid-air collision and required me to take evasive action.