Predicting cirrus cloud properties with confidence requires a sound understanding of the relative roles of homogeneous and heterogeneous ice formation. This study explores the effect of mineral dust and contrail-processed aviation soot particles as ice-nucleating particles (INPs) competing with liquid solution droplets in cirrus formation. We study aerosol-cirrus interactions by accounting for atmospheric variability in updraft speeds and INP number concentrations. Our results confirm the dominant role of mineral dust in ice nucleation events in cirrus clouds. In addition, we show that pre-existing thin cirrus may suppress ice formation when updraft speeds are low. We find that homogeneous freezing of liquid solution droplets dominates clear-sky aerosol-cirrus interaction above a threshold updraft speed determined by total number concentrations and ice nucleation abilities of INPs. When mineral dust particles exceed number concentrations of 10 L^-1, they reduce homogeneously nucleated ice crystal numbers significantly and even prevent homogeneous freezing for frequently observed local updraft speeds between 10 and 20 cm s^-1. When both mineral dust and aviation soot particles coexist with solution droplets, dust typically prevents ice nucleation by aviation soot. Aviation soot exerts a notable impact on cirrus ice numbers only if updrafts are weak, large soot particles are present in number concentrations that are considerably higher than typically observed in emission measurements, and/or number concentrations of mineral dust and other INPs are low. Overall, our results elucidate the role of aviation soot-cirrus interactions in the presence of other INP types.