Stability of choice in the honey bee nest-site selection process

We introduce a pair of compartment models for the honey bee nest-site selection process that lend themselves to analytic methods. The first model represents a swarm of bees deciding whether a site is viable, and the second characterizes its ability to select between two viable sites. We find that the one-site assessment process has two equilibrium states: a disinterested equilibrium (DE) in which the bees show no interest in the site and an interested equilibrium (IE) in which bees show interest. In analogy with epidemic models, we define basic and absolute recruitment numbers (R(0) and B(0)) as measures of the swarm's sensitivity to dancing by a single bee. If R(0) is less than one then the DE is locally stable, and if B(0) is less than one then it is globally stable. If R(0) is greater than one then the DE is unstable and the IE is stable under realistic conditions. In addition, there exists a critical site quality threshold Q(*) above which the site can attract some interest (at equilibrium) and below which it cannot. We also find the existence of a second critical site quality threshold Q(**) above which the site can attract a quorum (at equilibrium) and below which it cannot. The two-site discrimination process, in which we examine a swarm's ability to simultaneously consider two sites differing in both site quality and discovery time, has a stable DE if and only if both sites' individual basic recruitment numbers are less than one. Numerical experiments are performed to study the influences of site quality on quorum time and the outcome of competition between a lower quality site discovered first and a higher quality site discovered second.