  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.
