Collective Behavior of Self-Propelled Particles with Heterogeneity in Both Dynamics and Delays

Heterogeneity is a common feature in natural swarms, such as variations in mobility, perception ability, etc., among particles. In this study, we investigate the collective behavior considering both dynamic heterogeneity and time delay heterogeneity, based on a model with symmetric attractive forces. Three patterns emerge from our model: translating, ring and rotating states. Through mean-field approximation and bifurcation analysis, we theoretically derive the properties of these three patterns and identify the conditions for switching between these patterns in the parameter space. Based on this analysis, we find that separation occurs in all three patterns, and we analyze the reasons for the separation of different states. Furthermore, we validate the theoretical analysis through particle simulation experiments.