Worker allocation between defence and non-defence tasks in social insect colonies: Thresholds and stable...

We study how a social insect colony reallocates workers between local entrance defence and other colony tasks during a short disturbance. To make this trade-off explicit, we extend a patrol-recruit framework by separating workers in a waiting pool from workers engaged in non-defence tasks such as foraging and nest work. On a behavioural time scale short relative to colony demography, we formulate a four-class model for patrollers, alarmed recruiters, waiting workers, and other-task workers, and reduce it, via workforce conservation, to a three-dimensional system. We show that all biologically feasible solutions remain bounded and that the colony cannot lose its waiting pool or entrance patrols through internal dynamics alone. The reduced system always admits a non-recruiting equilibrium and can admit at most two recruiting equilibria. Their existence depends on colony size and a scalar balance relation for the waiting pool, while recruiter invasion occurs only above a colony-size threshold. We further show that at most one recruiting equilibrium can be stable, and that the candidate stable branch is selected by a simple comparison between recruitment efficiency and turnover between waiting and other-task workers. These results identify parameter ranges with local bistability between the non-recruiting equilibrium and a recruiting equilibrium and distinguish lean from buffered defence configurations. Numerical simulations agree with the analysis and indicate convergence to steady worker-allocation patterns rather than sustained oscillations.