Dynamical behavior of a colony migration system: Do colony size and quorum threshold affect collective-decision?

Social insects are ecologically and evolutionarily most successful organisms on earth, which can achieve robust collective behaviors through local interactions among group members. Colony migration has been considered as a leading example of collective decision-making in social insects. In this paper, a piecewise colony migration system with recruitment switching is proposed to explore underlying mechanisms and synergistic effects of colony size and quorum on the outcomes of collective decision. The completed dynamical analysis for the non-smooth system (including the dynamics on subsystems, switching surface, and full system) is performed, and the sufficient conditions significantly related to colony size for the stability of equilibria are provided. The theoretical results suggest that large colonies are more likely to emigrate to a new site. More interesting findings include but not limit to: (a) the system may exhibit oscillation when the colony size is below a critical level; (b) the system may also exhibit a bistable state, and colonies migrate to a new site or the old nest depending on their initial sizes of recruiters. Bifurcation analysis shows that the variations of colony size and quorum threshold greatly impact the dynamics. The results suggest that it is important to distinguish two populations of recruiters in modeling. This work may provide important insights on how simple and local interactions achieve the collective migrating activity in social insects.