  Bees undergo a self-organised process known as shimmering, where they form
emergent patterns when they interact with each other on the nest surface as a
defence mechanism in response to predator attacks. Many experimental studies
have empirically investigated how the transfer of information to neighbouring
bees propagates in various shimmering processes by measuring shimmering wave
strength. However, there is no analytical modelling of the collective defence
mechanism in nature. Here we introduce the first analytical tri-state
Inactive-Active-Relapse (IAR) model to formulate the intrinsic process of bee
shimmering. The major shimmering behaviour is shown to emerge under theoretical
conditions which is demonstrated numerically and visually by simulating
1,000,000 bee agents, while the number of agents is scalable. Furthermore, we
elaborate on these mathematical results to construct a wave strength function
to demonstrate the accuracy of shimmering dynamics. The constructed wave
strength function can be adapted to peak between 50-150ms which supports the
experimental studies. Our results provide a foundation for further theoretical
understanding of bee shimmering wave dynamics and could serve as inspiration
for modelling other self-organised phenomena across scientific applications.
