Flight Instructor providing instruction at a non towered airport reported a NMAC with a cropduster aircraft that was not communicating on CTAF.

Narrative

My student and I were working on patterns in ZZZ using Runway XX with multiple cropdusters and a Warrior in the area. On the roll of around our 5th pattern; I noticed a cropduster on the departure end of the runway; about 2 miles; moving northbound around 1;400 feet MSL. We lost sight of the aircraft on the climb when he was underneath us; but I assumed he passed us by the time we started crosswind; around 1;500 feet MSL. We did start the turn a little later than we normally would to create more room between us. During our turn; the Warrior gave us a heads up that the cropduster was underneath us about 100 feet low. We continued the climb and regained sight of it when we turned into the downwind. I took radio calls on the takeoff roll; talking constantly to make sure the cropduster knew where we were; but of course they were not listening. We continued patterns and came back to ZZZ1 without any issues.Cause: Cropduster not having radios and us taking off towards them; assuming they were going to stay away from us.Suggestion: Strong radio communication and not assuming the crop dusters will do the right thing.