General aviation instructor pilot reported a near miss with another aircraft in the traffic pattern of non-towered airport DYB. The instructor took control and maneuvered to avoid the other aircraft based on the ADS position on the flight instruments; then re-entered the pattern and landed safely.

Narrative

Prior to departure checked for traffic on final visually and ADS-B traffic on the MFD and no traffic was observed (distance ring on the MFD was set to 2.5 miles). While turning crosswind and climbing from ~800 ft to pattern altitude we noticed a traffic conflict appear on the G1000 and received a traffic audio traffic alert. The traffic was nearly the same altitude flying westbound through the DYB traffic pattern on the south side of the airport. I immediately took control of the plane and executed an evasive maneuver to the right to slow convergence of the aircraft and determined the best altitude was to climb. Once we no longer had a risk to flight safety we obtained the call sign from the G1000 and I re-entered the pattern and executed a landing. Prior to starting turn to crosswind there was no evidence of the traffic on the G1000 and with the plane behind us were unable to acquire visually. We did make calls before departure indicating runway and remaining in the pattern and made a call turning crosswind. Aircraft X started near Ladson and ended shortly after the NMAC. I didn't pick up again until the plane was SE of Atlanta. Why the plane was flying at pattern altitude directly through the DYB traffic pattern with no radio calls is unknown. They also did not appear to make any evasive maneuvers.