Entrance saturation as a control point for recruitment robustness in social insect colonies

Nest entrance crowding and inhibitory feedback limit recruiter-worker encounters in social insects, yet explicit capacity constraints are rarely modeled. We introduce a minimal capacity-limited recruitment mechanism with worker-dependent saturation that represents entrance congestion and inhibitory signaling, and embed it in a four-compartment model together with an aggregate model. Analytically, we derive boundary replacement thresholds that separate maintenance from loss of alarm and recruiters, and a colony-level replacement ratio $R_0$ for the aggregate model. The aggregate dynamics are globally convergent: every trajectory approaches an equilibrium, and periodic orbits are absent. Coexistence equilibria are determined by a cubic in available workers, so at most three occur. In the full model they are organized by a quartic, and up to four can occur under restricted regimes. The model yields testable predictions. Increasing saturation lowers $R_0$ , renders it concave in colony size, attenuates the marginal effect of direct recruitment at high availability, and can reverse sensitivities to nest departure and return. Therefore, the stable interior equilibrium may occur at either low or high worker availability. In the full model, crowding reduces recruiter replacement thresholds but leaves the alarm threshold unchanged, which yields a permanence condition for coexistence. Numerical analyses map multistability windows and show that stronger saturation narrows multistability, suppresses or bounds periodic orbits, expands the basin of attraction of the worker-only state, and recovers bilinear behavior as saturation vanishes. These results link entrance design and flow control to colony-level robustness against over-recruitment and suggest interventions that ease entrance crowding, strengthen direct recruitment, or increase recruiter retention.